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Transcript
Magnetic Resonance Imaging
via Radio Frequency Gradient with
Examples from NMR and Pure NQR
by
Guowang John Zhang
M.S., Electrical and Systems Engineering
University of Connecticut, 1989
M.S., Electrical and Computer Engineering
Graduate School of Academia Sinica, 1987
B.S., Electronic Technology
Tsinghua University at Huanghua Men, 1983
Submitted to the Department of Nuclear Engineering
in partial fulfillment of the requirements for the
degree of
Doctor of Philosophy in the Field of Radiological Science
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
February 1998
@ Massachusetts Institute of Technology 1998. All rights reserved.
Signature of Author: /
Department of Nuclear Engineering
December 9, 1997
Certified by:
Dai-. Cory, Associate Professor
--Department of Nuclear Engineering
Thesis Advisor
P
Certified by:
/
Sow-Hsin Chen, Professor
)partment of Nuclear Engineering
Thesis Reader
/
.
Certified by:
Kevin W. Wenzel, Assistant Professor
SDepartment of Nuclear Engineering
Thesis Reader
Accepted by:
//L awrence M. Lidsky, Professor
Chairman, Departmeal Committee on Graduate Students
CC
~
r61.sA
Magnetic Resonance Imaging
via
Radio Frequency Gradient
with
Examples from NMR and Pure NQR
by
Guowang John Zhang
Submitted to the Department of Nuclear Engineering
on December 9, 1997, in partial fulfillment of the requirements for the degree of
Doctor of Philosophy in the Field of Radiological Science
Abstract
In this thesis, we explore the use of RF gradients in NMR imaging. RF gradients
have advantages over gradients of the static magnetic field in that they are non-secular and
offer more experimental freedom (for example they permit amplitude, phase and frequency
variations). In the case of pure NQR imaging, RF gradients preserve an undistorted
lineshape.
RF gradients also present significant new challenges in both the probe design and
the spin dynamics. All of these issues are addressed in this thesis.
Potential applications of RF gradients include NMR imaging and RF gradient
spectroscopy (where the RF gradients are used to average internal Hamiltonians and to
select a unique coherence pathway). Pure NQR imaging is also a potentially exciting
application due to the large spectral changes that are observed with physical modifications,
such as radiation dose, pressure and temperature.
Additional complications arise in pure NQR as compared to high field NMR since
the principle axis system is defined by the crystal orientation rather than an external field.
The RF field breaks this symmetry and introduces another level of complexity to the spin
dynamics.
Thesis Supervisor: David G. Cory
Title: Associate Professor of Nuclear Engineering
Acknowledgments
I would like to sincerely thank Professor David G. Cory for his valuable guidance
throughout this research and great help on my personal career and life. I am grateful to Dr.
Werner Mass for the assistance on this project.
The valuable and amusing discussions with my laboratory fellows and my friends,
Dr. Jianyu Lian and Dr. Xinghu Gan, will be remembered.
I sincerely thank my friends, Dr. Howard Cohen, who helped me correct the first
several chapters of my thesis, and Dr. Zhongxue Gan, who fully encourage me to finish
my study at MIT.
My very special thanks go to my father, mother, aunt and younger sisters for their
full support on my everything: career, life and etc.
Also this thesis is used to deeply cherish the memory of my grandparents and my
lovely daughter, Bianca!!!
Contents
Abstract
Acknowledgments
1. Fundamental of NMR Imaging
1.1 Nuclear Magnetic Resonance Phenomenon
1.1.1 Nuclear Magnetic Resonance
1.1.2 Interactions of Spin Systems
1.2
Principles of Nuclear Magnetic Resonance Imaging
1.2.1 Magnetic Field Gradient
1.2.2 Basic Imaging Equation and k-Space
1.2.3 NMR Fourier Imaging
1.2.4 NMR Projection Imaging
1.3 Introduction of Radio Frequency Imaging
1.3.1 High Field NMR Radio Frequency Imaging
1.3.2 Pure NQR Radio Frequency Imaging
2. RF Imaging in High Field NMR
2.1 RF Coils
2.1.1 NMR Probes with Homogeneous RF Coils
2.1.2 RF Gradient Coils
2.2
2.1.3 Nutation Experiments
RF Gradients
2.2.1
2.2.2
2.2.3
2.2.4
2.2.5
Characteristics of Bo and B, Gradients
Converting Mixed Radial B, Gradients to Pure Linear Gradients
New Multiple-Pulse Cycles
Spatial Encoding and 1-D Imaging
Converting B, Gradient to Bo Gradient
2.2.6 Example of RF Gradient: Coherence Pathway Selection
2.3
RF Imaging
2.3.1 Principle of RF Imaging
2.3.2 RF Fourier Imaging
2.3.3 RF Back Projection Imaging
3. Introduction and Principle of NQR
3.1 Introduction of Nuclear Quadrupole Resonance Imaging
3.1.1 Nuclear Quadrupole Resonance Phenomenon
3.1.1.1 Background
3.1.1.2 Study of Impurity or Defect
3.1.2 Nuclear Quadrupole Resonance Imaging
3.1.2.1 Difficulties of Pure NQR Imaging
3.1.2.2 Zeeman Perturbation NQR Imaging
3.1.2.3 Rotating Frame NQR Imaging
3.2
3.1.2.4 Pure NQR Imaging
Principle of Nuclear Quadrupole Resonance
3.2.1 Definition of Nuclear Quadrupole Resonance (NQR)
3.2.2 Electric Quadrupole Moment (eQ)
3.2.3 Electric Field Gradient (eq)
3.2.4 Asymmetry Parameter (r1)
3.2.5 Energy States (pm)
3.2.6 Energy Levels (Em)
3.2.7 Energy Transitions (0om)
4. Spin Dynamics of NQR
4.1 Definition and Transformation of Different Reference Systems
4.1.1 Laboratory Axis System (X,Y,Z): LAS
4.1.2 Principle Axis System (x,y,z): PAS
4.1.3 Rotation Axis System or Rotation Frame (x',y',z'): RAS
4.1.4 Transformation Between the LAS and the PAS
4.1.5 Transformation Between the PAS and the RAS
4.2
RF Field Representation in Various Axis Systems
4.2.1 Definition of the RF Field in the LAS
4.2.2 Transformation of RF from the LAS to the PAS
4.2.3 Transformation of RF from the PAS to the RAS
4.2.4 Total Hamiltonian of the NQR and the RF Field
4.3
Fictitious Spin Analysis
4.3.1
Energy Transition Mechanism
4.3.2 Fictitious Spin
4.3.3 Full Coupling between RF Field and Energy Transitions in NQR
4.3.4 Oscillation of NQR Signal
4.4
Classical Analysis of Spin Nutation
4.4.1
Spin Dynamics
4.4.2
Spin Nutation
4.4.3
Signal Detection - FID
4.5 Density Matrix Analysis of Spin Dynamics
4.5.1 Theory of Density Matrix
4.5.2 Calculation by the Density Matrix
4.5.3 Significant Points from Density Matrix Analysis
4.6 Wave Function Analysis of Spin Dynamics
4.6.1 Theory of Wave Function
4.6.2 Expectation Value of Spin System
4.6.3 Significant Points from Wave Function Analysis
5. Spin Selection of NQR
5.1 Orientation Selection of Spins in NQR
5.1.1 Orientation Dependence
5.2
5.3
5.1.2 Available Spins after Orientation Selection
5.1.3 Mechanism of Spin Orientation Selection
Spin Selection by Spin-Locking
Spin Selection by DANTE-Based Sequence
5.3.1 Principle of DANTE Sequence
5.3.2 The X Spin Selection
5.3.3 The YZ Spin Selection
5.3.4 Discussion of Spin Selection by DANTE Sequence
5.3.5 Experimental Result of Spin Orientation Selection
6. Experimental Setup of Pure NQR RF Imaging
6.1 Pure NQR System
6.2 RF Transmitter
6.3
RF Receiver
6.4
RF Probe
7. Experiment and Simulation Results of Pure NQR RF Imaging
7.1
Pure NQR Nutation without Spin Orientation Selection
7.2
7.3
7.4
Pure NQR Nutation with Spin Orientation Selection
One Dimensional Pure NQR Imaging
Conclusion of Pure NQR Imaging
8. RF Probe Design and Diagnosis
8.1 RF Resonance Circuits
8.1.1 Inductance of Coil
8.1.2 Basic RF Resonance Circuits
8.1.3 Decoupled RF Resonance Circuits by Geometry
8.1.4 Decoupled RF Resonance Circuits by Active Switching of Modes
8.2
RF Probe Design for Pure NQR Experiments
8.2.1 Transformer Probe with One RF Field
8.3
8.2.2 Transformer Probe with Two Perpendicular RF Fields
8.2.3 Transformer Probe with Two Parallel RF Fields
8.2.4 One RF Field Probe with Four Straight Wires
8.2.5 Two RF Field Probe with Eight Straight Wires
Diagnosis of Probe Characteristics with Two Decoupled RF Coils
8.3.1 Bench Test: Quality Factor and Decoupling Attenuation
8.3.2 Shape of Two RF Magnetic Fields
8.3.3 Spatial Orthogonality of Two Fields
8.3.4 Coupling Efficiency between Fields and Spins
8.3.5 Isolation: Electrical Coupling between two Coils
8.3.6 Switching Time between Two Coils
References
Appendix
A. Basic Equations for Calculation
Coordinate Transformation
Angle Transformation Equations
Angle Rotation Transformation
B. Homogeneous and Quadrupole Field: Lab -> Rot Frame
Homogeneous Field: Bl,Lab(t,qp) -> B,Rot(()
Quadrupole Field:
Bl,Lab(t,O) -> Bl,Rot()
Quadrupole Field:
Bl,Lab(t,6,P() -> Bl,Rot(,)
C. Spin Nutation by a Homogeneous RF Pulse
Representations of Spin and RF Pulse
Spin Nutation by Bloch Equation Method
Spin Nutation by Exponential-Operator Method
D. Spin Motion Transformation from Rotating to Lab Frame
E. RF Transformation from LAS to PAS
F. RF Transformation from PAS to RAS
G. Wave Function Derivation
Time-Dependent Schrodinger Equation
Calculation of Matrix Elements
Calculation of Expectation Values of Spin System
H. Spin Nutation by a Quadrupole RF Pulse
Spin Nutation by Bloch Equation Method
Spin Nutation by Exponential-Operator Method
Chapter 1
Fundamental of NMR Imaging
1.1
Nuclear Magnetic Resonance Phenomenon
1.1.1.1 Nuclear Magneitc Resonance
In the most general case, NMR experiments require a quantum mechanical
description (usually in terms of the density matrix (p) approach) since the measured bulk
magnetization is from a collection of nuclear spins. A spin system can be represented by a
density matrix (po) at equilibrium and for a high temperature (T)[1] has the simple form,
po = (
,
1+
(1.1)
where k isthe Boltzmann constant. The dynamics of the spin system may be explored
based on Liouville's theorem, the time rate of change of density matrix at a fixed point in
phase space is,
dp(t)
dt
dt
I [p(t), Hl.
(1.2)
l
In the event that a Hamiltonian (H) isindependent of time, the solution of the above
equation is,
p(t)= e
po -e
(1.3)
The signal at the given time (t) is the bulk single spin, transverse, dipolar
magnetization,
(M(t)) = yhTr p(t) - i.
(1.4)
The total Hamiltonian (iHT)
of the system may be divided into a Zeeman
Hamiltonian ( Ho), an RF Hamiltonian ( H) and the internal Hamiltonians (
HT = Ho + lI +
H;e'
H',teal .
1
)
(1.5)
The Zeeman term is normally the largest and defines the axis of quantization of spin = 1/2
system, the internal Hamiltonians define the spectral features and the RF Hamiltonian
provides a means of experimentally manipulating the system.
1.1.2 Interactions of Spin Systems
For an ensemble of single spins with spin I, acted on by a static external field (Bo),
the Zeeman Hamiltonian is:
(1.6)
Ho = -TBo0 z.
The presence of the strong magnetic field (Bo) provides a cylindrical symmetry about the zaxis in the NMR experiments. The Zeeman Hamiltonian introduces a processional motion,
e
e-
"I
(1.7a)
= Ix cos(yB0 t) + IVsin(yB0 t)
I e
eh
= -Ix
sin(yBot) + I, cos(yB0t).
(1.7b)
A rotating RF field (B,) acts on an isolated spin I via the Hamiltonian,
iLab=
IB(,f
cos ot -
.sin cot) = -Be'*ixIe i
.
(1.8a)
It is convenient to transform this rotating field to the rotating frame, where the RF
Hamiltonian is time independent,
HjioR = -e-'t~abe
e.i'
= -1B
In this rotating frame the RF transforms spin terms via a simple rotation dynamics,
(1.8b)
.Ix e
e
e '
e
Ie
= I(1.9a)
= I, cos(yBt) + Iz sin(yBt)
= Iz cos(yBt) -
-h -I -
JV sin(yBt).
(1.9b)
(1.9c)
In considering only the spectral features of resonance's it is sufficient to consider
the secular parts, which commute with the Zeeman Hamiltonian and have no effects on the
transverse spins during the free evolution, of the internal Hamiltonians.
The magnetic dipole-dipole interaction describes the interaction between the
magnetic fields of one spin by an adjacent spin,
Hd 1Y
2 r3 h (1 - 3cos 2 )(31lz 2z
-I 1I2 ).
(1.10)
Since molecular motions average this to zero, it will not be considered further here.
The surrounding electrons partially shield the nuclear spin from the external field,
so that the effective field is dependent on chemistry and orientation,
BNuces
B
-
Bo
B(1-
),
(1.11)
where ( is the shielding factor. The internal Hamiltonian for this chemical shift is,
Hc = -yhoBoI z .
(1.12)
The shielding tensor, a, is second-rank and the magnitude of the shielding depends on the
orientation of the molecule relative to the applied field. For rapidly tumbling molecules, the
directional (anisotropic) part of the a averages out, so that in a liquid the chemical shift may
be treated as a simple number.
The scalar interaction is through bond, electron mediated coupling of two spins
(I,S) with a Hamiltonian of the form,
Hs= hJisIz Sz.
(1.13)
The quadrupole interaction representing the interaction between the nuclear spin
(1>1/2) and the electric field gradient at the nucleus will be discussed in chapter 3 in great
details.
1.2
Principles of Nuclear Magnetic Resonance Imaging
1.2.1 Magnetic Field Gradient
NMR imaging is based on the simple concept that the Larmor frequency is directly
proportional to the local magnetic field strength which may be made spatially varying by
carrying out the measurement in an external static field (Bo) and a linear gradient field
(BG=Bzz). The frequency ((o) at a particular position (r) is then spatially dependent,
(1.14)
oo(F) = y -(B,+G. T) ,
where G is the grad of the gradient field component parallel to the B0,
dB
dB
Bz+
G= VB = dx + dy
dB,
z.
--z.
(1.14a)
The spectrum is encoded by a magnetic field gradient therefore provides a direct
measurement of the distribution of spin density.
1.2.2 Basic Imaging Equation and k-Space
The basic NMR imaging equation can be described by the Bloch equation for
transverse magnetization in the presence of a Bo linear magnetic field gradient[2][3]:
dM
dt
=- -1 +i
r M.
(1.15)
T2
Ignoring the effects of diffusion, flow, chemical shift, T2 relaxation, the solution of the
above equation at a location r=(x,y,z) and at time t is,
-iyf G dt
M_(T,t) = p(F)e 0
(1.16)
where the initial value of the magnetization is proportional to the spin density p(r) of the
sample at the position r. The NMR signal (S(t)) is the integral of the transverse
magnetization M_ within the excited volume:
-ty
S(t)= M(F,t)d = p(F)e
G ,dt
o
(1.17)
dF.
Since the signal appears as the Fourier Transform of the spin density if a suitable change of
variables is performed, the spin density p(r) can be calculated by taking an inverse Fourier
transform of the NMR signal (S(t)).
From the above equations, a reciprocal space vector (k)[4] can be defined as,
k= y. J
(1.18)
(t)dt = y -6. T.
0
This wave number, k, may be related to the pitch of a spatial spin magnetization wave since
spins in a magnetic field gradient evolve with different Larmor frequencies and therefore
develop a spatially dependent phase. This results in a sinusoidal magnetization grating
across the sample. The k vector is the wave number of this grating with a period (A),
k =
1.2.3
27r
(1.19)
NMR Fourier Imaging
A two dimensional Fourier encoding of the spin density may be described from
independent spin evolution in two orthogonal magnetic field gradient,
s(T,,
=
Jp(F)exp -iyJG-Fdt
0
d1.
.
(1.20)
=f p(x,y)e-(xkx+yk dxdy = S(kx,k,)
In such experiments it is necessary to define wave numbers, kx and ky, for both encoding
directions,
kx = y Gx(t)dt = GxTx
0
(1.21a)
(1.21b)
k, = yJG,(t)dt = G,T,
0
where gradients, Gx and Gy, are on Tx and Ty constantly and respectively. The gradient
Gx,y or the on-time Tx,y of the gradient can be changed separately resulting in the same
coverage in the k-space. A 2D Nyquist condition determines the sampling limited
resolution in both directions.
As shown in equation (1.20), a 2D Fourier transformation of the spin density,
p(x,y), is the detected signal while the inverse Fourier transformation of the signal,
S(kx,ky), gives out the spin density in equation (1.22b),
1.2.4
S(kx, k) = F[p(x,y)]
(1.22a)
p(x, y) = F'[S(kx,k)].
(1.22b)
NMR Projection Imaging
The basic principle of 2-D projection imaging is to project the spin density of the
Projection
sample along an axis that is rotated through a set of measurements.
reconstruction in NMR imaging may be understood by defining a new coordinate (x',y')
rotated by an angle 0, as shown in figure 1.1:
Y
% Line Integral Here
%
X
Figure 1.1 Two Cartesian Coordinates, (x,y) and (x',y'), with an angle 0.
The line integral is parallel to the y' axis.
x = x' cos - y' sin0
(1.23a)
y = x' sin + y' cosO.
(1.23b)
The Radon transformation of a two dimensional slice of an object in the (x,y) plane
is the complete set of projections within the plane. Each element of the projection
corresponds to a line integral of the object function at an angle 0 and at an offset x'.
In NMR such a line integral may be obtained by a suitable rotation of the magnetic
field gradient. The gradient along the x' axis is generated as a vector sum of the x and y
gradients,
dB _ dBzd
dx'
dx dx'
dBz dy = coso dB,
Bz +sin0 dB z
dx
dy dx'
dy
(1.24)
and has a corresponding reciprocal space vector, kx,
kx'. = 7f G.(t')dt' =y dB t.
dx'
(1.25)
0
Therefore, the dot product G,,*r' in the signal equation can be replaced by the scalar Gx'x',
and the vector k,. by kx, so that the signal generated from the spin density and gradient in
equation (1.17) is,
S(t, ) = I p(x, y) exp(-ik, F)dxdy
=
Jj p(x, y) exp(-ikx. (x cos
(1.26a)
+ y sin 9))dxdy
By two changes of the variables, first from (x,y) to (x',y') and then from t to the k-space
vector, kx,, as is defined in equation (1.25), the signal may be rewritten as,
S(kx,,O) = f p(x' , y' )exp(-ikx. x')dx' dy'
= (p(x' ,y')dy')exp(-ikx, x')dx' =J P(x')exp(-ikx,x')dx'
(1.26b)
Note, that the final makes use of the projection operator, P, the integral of the spin density,
P(x') = p(x', y')dy' = Ip(x,y)6(x' -xsinO - ycosO)dxdy
(1.26c)
=P(x' , )
The signal is the Fourier transformation of the projection along the axis defined by the 0,
S(kx ,O) = IP(x',O)e-kx x'dx ' = F[P(x',e)]x.,
(1.26)
where F[] represents the Fourier transformation. The set of projections, {(P(x',8))}, is the
Radon transform of the spin density, p(x,y), or object function.
The spin density, p(x,y), can be reconstructed to an image by using the filtered
back projection, B[], or inverse Radon transform,
p(x,y) = B[F-[S(kx.,O) kx. ]].
1.3
(1.27)
Introduction of Radio Frequency Imaging
Another way of introducing spatial heterogeneity into the spin dynamics is through
the use of an RF (B ) gradient. In such a case, the spatial dependency appears in the
nutation frequency [u=yB(r)]. In this thesis, RF gradient imaging techniques will be
explored for both high field NMR imaging and pure NQR imaging.
1.3.1 High Field NMR Radio Frequency Imaging
It is appealing to treat a B, gradient as a B0 gradient in a different reference frame,
and then to argue that a B, gradient is perhaps useful from a technical point of view; that is
the hardware may be considerably simplified, gradient switching time is generally not an
issue, pre-emphasis and zo compensation are not needed, and the lock channel is not
effected by the gradient pulse. However, there are important differences between a DC and
an RF gradient, and these differences prevent the direct analogy to B o gradients from being
generally useful, and at the same time the differences lead to new experiments that are not
directly analogous to familiar Bo gradient methods. The major physical differences
between Bo and B,fields are the way they couple to the spin system and the fact that a DC
field is secular while an RF field is non-secular.
The spin dynamics are dependent on the RF field as well as the Bo field and so it is
possible to build imaging schemes based on spatially varying RF fields [5][6][7][8]. This
method is much simpler and is nearly insensitive to susceptibility inhomogeneity[7][8].
The principle of RF imaging was proposed by Hoult[5], and based simply on a set
of RF coils with various gradients. Since nutation angles can be any angles (<180')
resulting in that the transverse magnetization is less than the maximum magnetization from
900 nutation, the signal-to-noise ratio becomes smaller than the Bo gradient imaging. Also
the speed and safety are of great important for RF imaging because there are a series of
nutations and long RF pulses (a large number of periods). But RF gradient methods permit
greater flexibility in experiment design and the RF imaging method enables us to perform
fast imaging without rapidly switching gradients and with good sensitivity.
As we have seen the internal interactions of spin system, most of the interactions
occur between the external static field Bo and the component of spins in the Z direction, Iz.
In turn, the RF field is coupled with the transverse components of spins, Ix and/or I,.
RF gradient methods have been developed and demonstrated in relation to spatial
localization and imaging [5][6][7][8]. The relatively poor time efficiency and high RF
power in RF imaging have been improved a lot[6]. Also, some problems of standard
NMR imaging methods based on static field gradients, such as susceptibility distortions
from internal gradients [7][8], are solved by applying the RF imaging.
Other applications of B, gradients have been discussed widely, for example
heteronuclear couplings [9][10], molecular diffusion measurements [11][12][13], solvent
suppression [14] [15], and coherence pathway selection [16] [17] [18].
1.3.2 Pure NQR Radio Frequency Imaging
Pure Nuclear quadrupole resonance (NQR) Imaging can only be performed via RF
imaging if coherent methods are to be used. The NQR resonance frequency and its
associated lineshape are a function of the electric quadrupole moment (eQ) and the electric
field gradient (EFG) and are very sensitive to defects including those introduced by
radiation[19]. In addition, NQR is sensitive to pressure and temperature changes[20].
Pure NQR Imaging has been proposed as a means of mapping out the concentrations of
impurities or defects in solids[21].
NQR imaging could be carried out by either Zeeman perturbation [22][23] or the
combination of the "rotating-frame zeugmatography" and projection/reconstruction methods
[24] [25] [26][27][28].
Rotating frame Pure NQR imaging (pNQRI) is particularly
advantageous since it is experimentally simple, avoids the use of external static magnetic
fields and magnetic field gradients, and measures an undistorted NQR resonance. Thus the
full spectroscopic information remains unconcealed and can be used to characterize the
materials. The spin dynamics of Pure NQR are complicated, however, when compared to
more familiar NMR dynamics, since the spin quantization axis is tied to the molecular
structure and not to any external field direction.
The lack of a global quantization axis results in a spread of interaction angles
between the spin and an applied RF field, and an inhomogeneity is thus introduced as the
angle (0) between the RF field (B 1) and maximum Electric Field Gradient (EFG). This
results in a spread of nutation frequencies ( ) from yB 1 to zero. Since spatial information
is encoded in the nutation frequency, this spread corresponds to a decreased spatial
resolution. The point spread function (PSF) of pure NQR in powder samples compared to
that for NMR depends not only on the gradient strength but also on the powder
distribution,
and spreads asymmetrically,
introducing
significant low
frequency
contributions. Therefore, pure NQR images typically have a low spatial resolution and
rather large distortions. Those could be improved by deconvolution based on post-data
processing to restore the original spatial information of the powder sample[24]. However,
as with any deconvolution scheme one trades signal-to-noise ratio for resolution. Here, we
wish to explore the potential of experimentally returning the image PSF to a sharp well
defined impulse. The new approaches that we have developed are based on pure NQR
Imaging of selected spin packets. By using a pair of RF fields, we introduce a spin
selection procedure which selects those spins that are perpendicular to the RF field gradient
for imaging. Thus, a sharp PSF for RF imaging is created by the second RF gradient.
Details about the high field NMR and pure NQR RF imaging will discussed in the
following chapters. Chapter 2 will describe RF gradient NMR Imaging, including imaging
principles and techniques, RF homogeneous and gradient coils, RF field gradients and two
dimensional RF imaging. Especially, the usage of composite RF pulses is provided in
imaging and other applications. Chapter 3 will describe the principles of NQR. Chapter 4
will use classical and quantum approaches to discuss spin dynamics in pure NQR in three
coordinates. Some significant differences between NQR and NMR and the complexity of
pure NQR are pointed out. Chapter 5 describes methods for spin orientation selection from
spin dynamics. Chapter 6 will discuss the experimental setup for pure NQR RF imaging,
where a specified pure NQR system is constructed, and special probe and RF resonance
circuits are used. Based on the experimental and simulation results, the significance of the
pure NQR imaging is concluded. Chapter 7 provides several designs of RF probes and
methods to characterize two decoupled RF coils. Two new types of RF probes are design
and developed. One is an RF transformer probe to generate RF homogeneous and gradient
fields in multi-dimensions. The other is a combination of two decoupled RF homogeneous
and quadrupole coils to create a desired planar field gradient by composite RF pulses for
two dimensional imaging. A series of tests to diagnose the characteristics of the decoupled
coils are made.
Chapter 2
RF Imaging in High Field NMR
RF imaging encodes spatial information in the rotating frame so that the flip angle
due to a pulse length is dependent upon position. Our contribution will be to show how a
quadrupole RF field gradient can be used to encode a 2-D image.
2.1
RF Coils
2.1.1 NMR Probes with Homogeneous RF Coils
It is most profitable to start by reviewing what is commonly known concerning an
NMR probe containing a single homogenous RF coil, and for convenience we will follow
the major NMR signals throughout a simplified block diagram of a spectrometer (see Fig.
2.1). The synthesizer generates a constant, pure frequency which will be defined as having
a 0Ophase. This is split into two separate paths going to the transmitter and the receiver.
Following the transmitter path, the signal is phase shifted by an amount ,t, amplified by
+G, dB, and sent to the probe through a switch. The probe is shown schematically as
having two switchable pathways, one for transmitting the excitation pulse, and a second for
receiving the free induction decay (FID) (which is shown as a voltage source in series with
the coil). The current through the RF coil can be approximately calculated by assuming that
the LC series combination is exactly on resonance so that the RF power is dissipated
through the coil's series resistance.
synthesizer
PreAm
mixer
power
splitter
L
phase
GdB
receive
transmit
Figure 2.1. Simplified Block Diagram of Spectrometer
00
90
Knowing the current through the RF coil, the B, field can be determined. One
important issue is to realize that for high sensitivity coils a DC calculation of the magnetic
field profile is also appropriate at RF frequencies. For highest sensitivity, the coil should
have a high Q (which is a measurement of the coil's efficiency at producing a magnetic
field), the Q should be limited solely by the coil's resistance, and the capacitance between
turns of the coil should be negligible. The RF field is given by Ampere's law:
(2.1)
B, = !ouNI,
where t o is the permeability of free space (41t x 10-7 henry/meter), and N is the number of
turns on the coil (modeling the coil as a solenoid and neglecting end effects). Overall the
RF magnetic field due to unit current is,
B, =
__
r
1020 e-' COS(oot).
(2.2)
The rotating frame Hamiltonian associated with this B, field is,
Hr = yh oN 1020 (Ix cos t - I,.sin 0t),
(2.3)
2r
where the factor of 2 arises from the conversion of an oscillating magnetic field to a sum of
rotating and counter-rotating fields. The spin system only interacts with the rotating field
component.
To follow through the remainder of the spectrometer, consider the experiment
where a n/2 pulse has been applied to the sample. The probe is switched to the receive
mode and the induced emf, r, in the coil is given by the principle of recipicallity [29] and
by treating M(r) as a collection of local rotating magnetic dipoles,
7
--
(T)"
dt
]
(
)
d
=
-i
(rdF,
dt
(2.4)
where B, is the spatially dependent RF field normalized to unit current through the RF coil
disregarding the time and phase dependence. In high field NMR the detection bandwidth is
very narrow compared to the resonance frequency and so we are only looking at the DC
component of the induced emf riding and the audio frequency signal is equivalent to the
spatial integral of the rotating frame magnetization scaled by the dot product of the B, field.
The spin magnetization is a rotating field 900 from the transmitter field,
M(t) = Moe
2
(2.5)
e-"t.
For an on-resonance signal the B, field and the spin magnetization are both uniformly
rotating 90' out of phase, and so the signal going to the preamp is,
(2.6)
77 = Mo cos(wot + 0,),
and after being mixed with the receiver phase-shifted signal from the synthesizer, the two
audio outs are:
Audio(real)
=
Audio(imaginary) =
M o cos40
(2.7)
-M o sin 0
Two of key points are that (1) the spin dependence of the B Hamiltonian is directly
traceable back to the phase of the transmitter RF, and (2) the instantaneous phase of the RF
rotating field in the coil is spatially constant.
Now allow the picture to become slightly more complicated by introducing some
capacitance across the RF coil. This acts as a spatially dependent phase shift 0( F) of the
RF field, so that the B, field is now,
G
GN
B, (
)
= N
'
10 2
e '- e- iO( ;) cos(wOot).
(2.8)
This spatially varying phase is carried over into the spin magnetization (following a it/2
pulse),
M(t, ) = Moe
2
e-l')eiwot,
(2.9)
and finally the induced emf in the same spatially varied coil is again given by Eq. (2.10):
(2.10)
77 = -M"Oo cos(coot + 0).
So here we see that having an RF field with a spatially varying phase does not introduce
any complications into the NMR experiment and indeed in single coil probes the presence
of this will only be felt as a decrease in the signal amplitude (associated with a decrease in
the coil's Q). The spin evolution throughout the sample will have an extra spatially
dependent phase relationship, but the voltages that are induce into the coil add
constructively. That is, the direction of the applied B, field varies across the sample, but
the coupling of the spin magnetic moments and the coil have exactly the same spatial
dependence and so the NMR signal adds coherently. It is this fact that makes surface coil
studies so robust and valuable. Here, however we will be concerned with detection via a
homogeneous RF coil, the gradient coil being used to introduce spatially varying dynamics
into the spin system but generally not for detection.
Finally, let us consider the situation where the sample is excited with a spatially
varying RF field but detected with a coil that is spatially uniform. The magnetization will
therefore have the form of Eq (2.9) above, and the induced emf in this coil will be,
7 = -Moao
0
(2.11)
cos(coot + 0,)f e-i(')d.
Notice that the phase dependence of the signal/receiver coil combination no longer vanishes
and when integrated over the sample, the FID can actually vanish for an ideally symmetric
uniform sample.
Throughout this discussion an RF Hamiltonian will be composed of two parts, a
vector describing the spatially dependent RF field, and a separate portion detailing the
phase of the transmitter pulse, for example,
iRF
-
Bj,(-)
I,cos ,
-
sin,
O(2.12)
2.1.2 RF Gradient Coils
In exploring gradient coils, one needs to think simultaneously in a few reference
frames. The most important ones being the laboratory frame (u,v,z) where we define these
directions as being aligned with the physical gradients directions (G,) of the RF gradient
coil (that is along the direction of increasing RF field strength in the laboratory frame), and
the usual rotating frame (x,y,z) where x is defined as the direction (B,) of the RF field for
an on-resonance pulse. The z direction is taken as being along the static magnetic field
direction and is identical in the two reference frames.
As opposed to the RF field from a homogeneous coil, which can be correctly
described locally as a uniform field in a given direction (see Fig. 2.2), a coil can not create
a linear field gradient in only one direction. The RF field must obey Maxwell's equations
of magnetostatics,
V-B
VX
= 0
= 0*
VxB = 0
(2.13a)
For our purposes, it is sufficient to remember that magnetic fields are sourceless, and that
their flux lines must close on themselves. In general, then a magnetic gradient should obey
the two simple relations,
dB + dBb+ dB = 0
da
db
dc
dB,
db
_
dBb
(2.13b)
(2.13c)
da
Although these relations are widely discussed in the case of a Bo gradient, for B,
coils the arguments must be changed since the two fields couple into the spin system in
very different ways. A Bo field (a DC magnetic field) can only couple to the spins along
the z axis, that is only the I1 component of the Bo gradient Hamiltonian can influence the
dynamics of the spin system (aB/au, dBz/av, dBzaz). So regardless of how complex a Bo
field is created, from an NMR point of view the effective Bo field always has a simple spin
dependence (it is always directly proportional to Iz), and also generally a simple spatial
dependence. B, fields (RF magnetic fields), however, couple into the spin system from
any orientation within the transverse plane; it is only the Iz spin state component of a B,
field Hamiltonian that can not couple. So the six components, dBx/Du, dBx/jv, DBx/z,
B/ u, dB/ v, and B/d z, all contribute. The result is that in general a B, field will have
both a complicated spin dependence and a complex spatial dependence.
The characteristics of the RF gradient field that we will be primarily concerned with
are the symmetry of the gradient field (for the examples discussed here, the field is either
planar or radial), and the spin state dependence of the gradient Hamiltonian (it is spatially
varying, "mixed," or spatially uniform, "pure").
m,
---
"
---
U
HOMOGENEOUS FIELD
Mif-'man-"---
I
ma
UU
QUADRUPOLAR GRADIENT FIELD
PLANEAR GRADIENT FIELD
Figure 2.2 Radio Frequency Magnetic Field
Table 2.1. Characteristics of B Gradient Coils
B Gradient
Z Component
Comments
HRF
of B
RF
Planar Gradient
dBx
dBz
du
dz
dB
Spatially
(x coso,-
Spin
Dependence
x
-Y--r
"u)x
Radial Gradient
XB ,
-y
dBBx = d,
du
sin,)
Uniform
(rt)x
(ixcost,
0
sin)
+yaB -(r v)x
Spatially
pin
Varyg
Spin
Dependence
(Ix
cos + I sin 0,
Since the RF coupling of a B, gradient probe is axially symmetric, there are three
geometrical choices of particular interest in the design of the gradient coil, the
characteristics of which are listed in Table 2.1. The details of constructing these coils are
24
not of interest here, but it should be noted that the current distributions for generating these
fields are known. In classifying these coils there is no need to distinguish between the
laboratory frame directions u and v, or between the rotating frame directions x and y.
Figure 2.2 shows the shape and spatial variations of these options in a schematic fashion,
along with that of a homogeneous field for comparison. There are lower symmetry
configurations (some of which have been used), but they do not introduce anything new
into the discussion that follows. The z-components of the coils may be specified from Eq.
(2.3) and are included in the Table 2.1, but they do not influence the spin dynamics.
It might at first appear that adopting a planar B, gradient configuration is preferable
to a radial B, gradient since the spin dynamics are then analogous to those with a Bo
gradient. However, note that a Bo gradient retains the cylindrical symmetry of the NMR
experiment, while a planar B, gradient breaks this. A radial symmetric gradient coil retains
the cylindrical symmetry of the NMR experiment. We are interested in an experimental
configuration where the normal spatially homogeneous RF coil is surrounded by an RF
In practice the
gradient coil and both coils are tuned to the same frequency.
transmitter/receiver is switched between the two and so only one is active at a time. With
this setup the RF path length between the transmitter and the two coils are not the same and
the apparent transmitter phase is different between the two fields. If a planar gradient
geometry is employed then this phase difference must be taken into account during the
experiment (a method for setting this phase is discussed latter), where as for a radial
gradient geometry the spin dynamics are independent of this phase (for a spatially uniform
sample). In other words, the radial B, gradient does not break the cylindrical symmetry of
the experiment, and if the sample is also cylindrically symmetric then only the
homogeneous B, field imparts a directionality to the experiment.
A homogeneous coil presents a linear B, gradient with a spin dependence that we
will call "pure" to indicate that every spin packet in the sample sees a gradient effective field
with the same spin dependence. It is only the amplitude of the gradient field that varies
across the sample.
For the remainder of the discussion, we will be concerned with a probe of the type
shown in Fig. 2.3, which consists of two RF coils, an inner coil that generates a spatially
uniform RF field and a RF gradient coil that is correctly modeled as four upright wires
located at the edges of a square. The magnetic field B(F) generated by a single wire along
the z-axis with current +I and at a distance R from the origin and an angle 6 from the x axis
can be derived from the vector potential A(f):
R
Homogeneous Coil
(Inner 4 Wires)
Quadrupole Coil
(Outer 4 Wires)
0
X
Figure 2.3 A Probe with Two Coils: Homogeneous and Quadrupole Coils
A()
'/Io__/ln,/
=A()2 in
(u - RcosP) 2 1+(v
-
Rsin)
2
/
(2.14)
(2.14)
'
(-v + Rsin tO
B()
= V x A=
Rsind)2
27 (u- Rcos) 2 +(v-
u - Rcos
(2.15)
The magnetic field in Eq. (2.16)'s for four such wires arranged symmetrically with 6=45o
is given by their superposition which at the origin corresponds to a radial field gradient,
and the exact shape and phase dependence of this field is shown in figures 2.2
[30][31][32][33],
B(F) =
2R2
0
-1
0 -F.
0
0/
(2.16a)
The total magnetic field can be rewritten in another form in order to analyze the coupling
properties with spin system:
B(F) = Bx
+
B, = Bxi
+
B, = gi + gv9
,
(2.16b)
where gu,=Bx/au=o1/2R 2 and gv=-aB/cv=gl/2R2 .
Key features to realize are that each RF pulse generates two orthogonal RF
2 and aB/av=-1oI/2R 2), that the spin system responds to both of
gradients (aBx/Bu=g1/2R
0
these, and that the RF of these two fields are exactly in phase.
This built-in phase
coherence allows each oscillating gradient field to be decomposed individually into rotating
and counter-rotating fields. If, for example, the two RF fields were 900 out of phase from
each other then the sum would correspond to a rotating field and only if the field was
rotating in the correct direction could it couple into the spin system.
Not only the strength of the RF field, but also the phase of the RF field is spatially
dependent, and we will describe this as a "mixed" spin dependence to indicate that the spin
dependence of the gradient Hamiltonian varies across the sample. It is profitable to think of
these two fields ( Bx and B,) as originating from two separate coils, one that has a field
aBx/u with phase 0O(aligned with the x homogeneous RF field), and a second coil with a
field aB/ v with phase 90 ° (orthogonal to the x homogeneous RF field). When the RF
going to the coil is phase shifted by
t, the phases of both gradient components vary
resulting in the Hamiltonian given as the following equation (2.17) or in Table 2.1,
=- yhgu( cos , - i, sin 0,= -A(g,u cos 0, + gvsin O,)x -
g
cos0, +
sin
.
(2.17)
(gvvcos , - gu sin 0,),
2.1.3 Nutation Experiments
The Hamiltonians governing RF pulses are given for each of the coils in Table 2.1.
It is, of course, the spatial encoding of spin magnetization that will be of interest in
exploring B, gradients and nutation sequences are a direct measure of this. A general
nutation experiment is outlined in Fig. 2.4, and consists of a string of RF pulses with data
acquisition occurring stroboscopically between pulses. The resonance of interest should
normally be on resonance, and the overall length of the acquisition should be much less
than T2. The nutation response as a function of the number of nutation pulses is given by
the normal density matrix calculation,
p(t) = U(t)poU-'(t),
(2.18)
where po is the starting magnetization, and the propagator describes the nutation due to the
pulsed RF field,
)
U(t,) = exp-it B
cos
-sin,(2.19)
Equation (2.19) describes the propagator for a single RF pulse; with the spins onresonance, the only interaction is the RF field and this naturally commutes with itself, so
the density equation may be rewritten as,
(2.20)
p(t = [U(t )] PU-(t)] ,
where tp is the pulse width. The total nutation is n times of the small flip angle by the t,.
tp
0
0
Figure 2.4 A General Nutation Pulse Sequence.
Table 2.2. Nutation Experiments by Using Different Coils
Detection
Detection
Coil
PO
Homogeneous Coil
Gradient Coil
Homogeneous
Iz
Mo sin('Bntp)
0
Iy
Mo cos(yBnt)
0
Nutation
Iz
0
41
4wrsin(ygrntp)
(ygrntp)3
(ygrntp)2
+ 2[
-
Gradient
(ygrnt)2 cos(ygrnt)
(ygntp)
7r 2
2
(ygrntp
ircos(ygrnt)
2rcos ygrrnt)
3
((ygntp)2
-
2
r sin(yg2rnt,
7[ -2+(Ygrrntp)] sin(ygrntp)
(2
(yg,nt,)
.r3
ygrnt
+
3
The observable signal is dependent on the spatial properties of the B, field of the
observation coil as described in Eq.(2.4) with the spin magnetization given by the normal
projections of the density operator. With a RF homogeneous/RF gradient probe there are
four possible nutation experiments since the RF pulses may be applied to either the gradient
or homogeneous channel and the detection may be with either the gradient or homogeneous
channel. There is an additional degree of freedom in the initial spin magnetization (po = Iz,
or po = ly), and the results of these eight experiments when applied to a cylindrical sample
are summarized in Table 2.2 with 0,=0o.
Notice that a Fourier Transformation of the nutation signal yields a measure of the
RF field strength, and that to scale this correctly in frequency units the effective dwell of
the acquired data should be set to the RF pulse length. The interval between RF pulses is
only present for sampling convenience and as the experiment is normally run this delay
should be short enough so as not to influence the spin dynamics.
08
0
5
A
II
5
-0
A\
"
<01
1
0
L s
i
•
150
i
F
00oo
\
0
25
V
/
[;
I
O 8S
-0
600
800
1000
1200
2
(b) Experiment from Quadrupole Coil
/
i
i
50
-o
2
2 ov 400
(a) Experiment from Homogeneous Coil
0
6
04
'-IS
10
'
150
so
04
250
oo
5
(d)
Calculation
from
Quadrupole Coil
-1
(d) Calculation from Quadrupole Coil
(c) Calculation from Homogeneous Coil
S'V V
08
S06
04
\
-0
5
150
-1-
-
(e) Experiment & Calculation Comparison
for Homogeneous Coil
'Ls
0 2
00
0
-0
400
600
800
1000
1200
2
(f) Experiment & Calculation Comparison
for Quadrupole Coil
Figure 2.5 Nutation Data of Homogeneous & Quadrupole RF Coils from Experiment and
Calculation with T2p. The vertical and horizontal axes are relative amplitude and time in gs.
Some experimental results are shown in Fig. 2.5 and 2.6 of nutation studies on the
homogeneous and gradient coils from which it is clear that the probe is well described as a
radial gradient probe with a separate homogeneous coil. The pulse sequence is shown in
figure 2.4. Figure 2.5 (a), (c) and (e) display the nutation data from the homogeneous coil,
and the (b), (d) and (f) are from the quadrupole coil. Figure 2.6 presents the nutation
spectra of the nutation data respectively. In figure 2.5 and 2.6, the (a)'s and (b)'s indicate
the experimental results by using the homogeneous and quadrupole coil respectively shown
in figure 2.3, and the (c)'s and (d)'s represent the calculation results with a T 2p decay based
on the table 2.2. The T2p decays, which have contribution from T, and T 2 relaxation's and
the gradient, are 1.55 ms and 0.2 ms for the homogeneous and quadrupole coils.
1
1
0.8
08
0.6
0 6
0.4
04
0.2
0.2
10
20
30
40
50
2.5
7.5
5
10
12.5
15
17.5
(b) Experiment from Quadrupole Coil
(a) Experiment from Homogeneous Coil
1
1
0.8
08
0.6
0.6
0.4
0.4
0.2
0.2
10
20
30
50
40
5
2.5
7.5
10
12.5
15
17.5
(d) Calculation from Quadrupole Coil
(c) Calculation from Homogeneous Coil
1
1
0.8
08
0.6
06
0.4
04
0.2
0.2
10
20
30
40
50
(e) Experiment & Calculation Comparison
for Homogeneous Coil
2.5
5
7.5
10
12.5
15
17.5
(f) Experiment & Calculation Comparison
for Quadrupole Coil
Figure 2.6 Nutation Spectra of Homogeneous & Quadrupole RF Coils from Experiment
and Calculation with T2 p. The vertical and horizontal axes are relative amplitude and kHz.
All spectra were acquired on a Bruker AMX-400 spectrometer with a prototype B,
gradient probe of the design shown in figure 2.3. A single transmitter capable of delivering
20 W into 50 Q was actively switched between the two RF channels. The switching was
accomplished by actively detuning the channel that was not being used. Switching times
are of the order of 15 lps and the isolation between channels is better than 50 dB, as shown
in figure 8.22 and 8.32 respectively. With 20 watt of power the homogeneous coil delivers
5.87 G (tn/2 pulse length of 10 gs) at about 25 kHz, and the gradient coil delivers 6.98
G/cm (it/2 pulse length at the edge of the sample of 50 gs for a 5mm sample tube) at about
5 kHz. The particular geometry that we employ creates a gradient that passes through zero
at the middle of the sample. The nutation pulse widths (tp) in figure 2.4 are 2 gs and 10 gs
with respect to the homogeneous and quadrupole coils separately.
2.2
RF Gradients
2.2.1 Characteristics of Bo and B, Gradients
Conventionally, Bo gradients are used in many experiments. Bo gradients comprise
variations in the static Bo field. Typical gradient coils produce linear field gradients along
each lab frame Cartesian axis gx=dBz/dx, gy=dBz/dy and gz=dBz/dz. The strength of these
fields is small compared with the uniform Bo field (generally less than 1 part in 104). On
the other hand, Bo gradients have some practical disadvantages. Since the fields are static,
the skin depth is large and gradient pulses produce substantial eddy currents in the magnet
dewar and coils. This eddy current requires long settling times before applying RF pulses
or beginning signal acquisition and perturbs the lock system. Because of large Bo field
physically strong gradient coils are required resulting in long rise times for their large
inductance and high power for necessary currents [34].
Table 2.3. Differences between Bo and B, Gradients
Bo Gradient
B1 Gradient
m..............................................................................................................................
...... Symmetry of Coupling
Along I only
Within the I, Iyplane
Secular
Yes
No
Switching Time
> 100 Is
200 ns
Pre-emphasis Needed
Yes
No
Zo Compensation Needed
Yes
No
Affects Lock Signal
Yes
No
Radio frequency gradients instead exploit spatial variation in the B, field strength
and perform dephasing about the RF axis since the nutation frequency varies as a function
of position in the sample. For a given RF coil, these gradients may be described in
equation (2.13) and Table 2.1. RF gradients do have a number of intrinsic advantages over
B0 gradients. They are frequency selective and can be applied independently to different
nuclear species. They use standard RF hardware and are easily fitted into most pulse
sequences. The major physical differences between Bo and B, fields are the way they
couple to the spin system and the fact that a DC field is secular while an RF field is nonsecular. These features are summarized in table 2.3.
2.2.2 Converting Mixed Radial B, Gradients to Pure Linear Gradients
For many experiments there is an advantage to having a linear rather than radial
spatial dependence on the gradient, and it is almost always the case that a spatially uniform
spin dependence is desired. In most NMR experiments the spin evolution is restricted to
being in a plane and so at the point that detection occurs, the summation of all of the spin
magnetization is measurable. With a spatially heterogeneous spin dependence, the spin
magnetization will generally be spread over the entire sphere (Ix, Iy,Iz) and only a subset of
the total spin magnetization is observable at any time. Certain experiments do employ the
entire sphere for dephasing unwanted spin magnetization, but almost universally the
desired spin magnetization is restricted to a plane. Having the spin magnetization spread
over a sphere is also a complication when one desires to apply homogeneous RF pulses
latter to the sample, since the angle between the RF field and the spin magnetization will
vary throughout the sample and the action of the RF pulse will be spatially modulated.
Although it is profitable to develop B, gradient methods for use with coil
geometries that produce pure, linear gradients, finding approaches to exploit the radial
geometry are appealing. The radial gradient extends symmetrically in all directions and so
experiments where more than one gradient are required can be performed with the same
hardware, and with a radial gradient there is no unique phase angle between the
homogeneous and gradient RF fields. Notice that in both spin space and in real space there
are no unique directions of the radial field. The task then is to selectively retain one
component of the gradient Hamiltonian while averaging out the perpendicular component.
The T-pulse refocusing experiments in Table 2.4 converts a mixed radial gradient into a
pure linear gradient by refocusing one component of the spin dynamics.
The averaged Hamiltonian theory approach (AHT) is used to describe the effective
spin dynamics from a homogeneous/gradient RF cycle. Since it is convenient to perform
the calculation such that the toggling frame transformation is uniform throughout the
sample and to treat the interaction of interest as a perturbation, the coherent averaging cycle
will be composed of the homogeneous RF pulses, and the gradient pulses will be treated as
a perturbation[35][21]. In addition, the influence of the multiple-pulse cycle on various
internal Hamiltonians is of interest, particularly inhomogeneous offsets or scalar couplings.
A very appealing experiment is to use a multiple-pulse cycle to suppress the
evolution due to all internal Hamiltonians, and to accomplish this uniformly throughout the
sample. At the same time an additional goal is to use a combination of gradient and
homogeneous RF pulses to generate effective Hamiltonians that behave as pure, linear
gradients. This is quite different than the Bo case where due to the additive properties of B o
and internal Hamiltonians, it is perhaps preferable to keep the internal Hamiltonians.
There are, at least, two general approaches to this multiple-pulse cycle, the first
employs the selective refocusing ability of 7t pulses to choose the gradient direction, and the
second relies on a train of RF pulses and second averaging. The it pulse experiments are
similar to Carr-Purcell Cycles in that they average interactions that are perpendicular to the
direction of the 7t pulse, shown in following.
B ,Q
BI,Q
BI,0
B
tp
tp
RFHomo
tp
B
BI,Q
B
tp
RFHomo
BI,Q
SGrad
2
RFQua
2
RFQud
BI,Q
]
I1
B
BQ
,
L
__
tp
tp
Ix
ly
Ix
- Iy
HRn
HRFI
(a)
RFGrad
BQ
1241_
tp
Ix
y
wl
L
tp
-Ix
Iy
HRn
(b) ir,
Figure 2.7 Homogeneous t Pulse and Gradient Pulses Composition.
x
(a) The nt Pulse in the x direction resulting in the change of Iy.
(b) The it Pulse in the y direction resulting in the change of Ix.
By letting the phases of the two gradient pulses be same (4~= 02) and adding the two
Hamiltonians (H,,+H2) in the equation (2.17) for a quadrupole gradient field, either I,
or Ix component of the averaging Hamiltonian ( HRF) shown in equation (2.21) is canceled
with respect to the homogeneous 7c pulse in the x or y direction so that a radial B, gradient
is converted to a pure linear gradient field,
H
HRF
RF1
+HRF
2
RF2 =
(guCOS
/2
= -Ay(gvcos
, + gvsin ,)I
, - g,usin #)Iy
for
for
x
((2.21)
,y
2.2.3 New Multiple-Pulse Cycles
In setting out to design a multiple-pulse cycle that converts a mixed, radial B,
gradient to a pure linear B, gradient, it is useful to recall that the homogeneous B, field that
is also available in the probe will define the symmetry of the experiment. The multiplepulse sequence will therefore be a composite pulse made up of pulses on each of the two
RF coils.
According to the principle in figure 2.7 and equation (2.21), table 2.4 lists a
number of composite pulses and the averaged gradient interaction for both a mixed, radial
and a pure, linear gradient field. The first thing to notice is that all of the cycles create
linear gradients with pure spin dependencies; that is the spin dependence of the effective
gradient Hamiltonian is constant throughout the sample. Notice also that a linear, pure
gradient is created regardless of the phase of the gradient RF to the homogeneous RF; this
greatly simplifies the spin dynamics of the experiments. In the case of a linear gradient
field, the phase difference between the gradient and homogeneous RF fields leads to an
amplitude modulation of the effective gradient field which can be varied from 0 to 1. This
property leads to many approaches for setting the phase difference. If the two gradient
pulses differ in phase, then the direction of the effective field is modulated by this
difference which does not depend on the phase difference between the gradient and
homogeneous fields.
The most robust and generally useful sequence is the fifth listed and its averaged
gradient Hamiltonian is shown in the following for the radial RF gradient field,
go HRF
-
x
-
g, - g,
U COs
2
(2.22)
- r - go
+ gvsin 2
2
Icos
2
1, sin
2
.
(2.23)
The averaged Hamiltonian demonstrates that not only the effective field direction,
but also the gradient direction in the laboratory frame can be rotated simply by changing the
phase angle between the gradient pulses. This is extremely useful since now the gradient is
always of equal strength, the direction is easily varied and the absolute direction is rarely of
interest, so there is no setup step. Likewise with setting the relative phase difference
between the gradient and homogeneous RF fields, since by setting the two gradients equal
the effective field is automatically aligned with the 7t pulses shown in equation (2.21).
The sequences in Table 2.4 that contains an equal number of x and -x it pulses are
also compensated for the RF inhomogeneity of the homogeneous RF coil.
Table 2.4 Averaging Hamiltonians of Composite Multiple Pulses
Pulse
Sequence
go - - go
>
H
of Linear RF Gradient
0
-guIx cosO
-gUucos
g- -
-
gP
- , -
0
o
g9
g(
g( -- rx - g90
xix
cos
-{g,ucos 0 + gv sin O}Ix
[ g ucos
x
2
sin
-
}
-gug os - -
x {Icos
ucos
2
2
-gucos
HRF of Radial RF Gradient
2
x
cs 2
- gucos
2
+ gvsin
-
2
2
sin
+ gvsin
2
sin
+ gvsin
..........................................
.................
........................................................................
.....................................................................
go - 0)TX - g,,-2
0
-guCos 0+
&U Cos
-
cos
Xsin
gUCos+P2+gvsin2+
gucos
+g
sin
v
Cos 2
*For simplification, the coefficient y* is neglected.
An alternative approach is to again take advantage of the uniformity of the
homogeneous B, field, but to employ this as a second averaging interaction where the
strength of the field is used to dominate the spin dynamics. As opposed to other second
averaging schemes, here the "strength" of the two interactions is the field strength times
time since they influence the spin system sequentially.
The quaternion formulism is a very useful way of discussing composite pulses.
Here the composite pulse is replaced by an effective pulse about an effective field. The
details these effective pulse lengths and directions have been calculated and the results are
identical to an averaged Hamiltonian result where the eigenvalues of the overall propagator
are calculated and the average Hamiltonian is defined as the logarithm of the eigenvalues.
The equivalance is a consequence of both methods limiting the spin system dynamics to
successive rotations of a single two level system. The quaternion for the composite pulse
is described by Eq.(2.22), which is the expected pure, planar gradient. That the composite
pulse has the described properties will be born out in the imaging experiments to be
described below.
The quaternion allows a simple approach to calculate errors in composite pulse
cycles and the influence of pulse errors in setting the nt pulse lengths is explored. This is
an important consideration since coupling between the gradient and homogeneous RF coils
will lead to a systematic deviation in the strength of the homogeneous RF field.
2.2.4 Spatial Encoding and 1-D Imaging
A good way of insuring that we understand the composite pulses listed in Table 2.4
is to employ these in imaging experiments with a sample of known geometry. The
experiment is again a nutation sequence where we take advantage of the possibility to
reorient the gradient (see Eq. (2.23)). A set of 1-D images suitable for back-projection
reconstruction can be generated by setting the phases of the two gradient pulses in Eq.
(2.22) equal (qp = 0) and varying this phase (0) incrementally over 1800.
Setting the
phases equal forces the spin evolution from the RF gradient to be about the Ix axis,
regardless of the orientation of the gradient in the laboratory frame (u,v). The gradient
direction in the laboratory frame is varied by changing the phases of both gradient pulses in
step, Eq. (2.23) then takes the simplified form,
HRF
= -[gu
cos 0 + g,vsin O]Ix,
(2.24)
where 0 is the phase of both of the RF gradient pulses in the experiment,
go - Tx - go - go -
-r.- go.
(2.25)
IU
I.
II
I.
-_U
--.
___.-.O=y
"II
GRF r tp
Figure 2.8 Composite Pulse Sequence of ID Imaging/Nutation Experiment
Upper: A Series of RF Gradient and Homogeneous n Composite Pulses.
Lower: Two/Four RF Gradient and One/Two Homogeneous 7 Composite Pulses.
In NMR detection, quadrature phase sensitive detection (QPD) of the transverse
spin magnetization is used to record absorption made data. In ID RF imaging, the
magnetization is rotated in the YZ plane in the laboratory frame by the nutation pulses along
the X direction, therefore, the Y and Z components of the magnetization are needed. In
order to acquire a quadrature detected nutation signal two ID data set are needed. For one
of these a 900 prepulse is necessary to align the spins with the Y axis.
The results shown in figure 2.9 (a) and (b) are two 1-D images of phantoms, which
consist of one 5 mm tube and two capillary tubes filled with water. The homogeneous 7/2
and 7c pulse lengths for pre-excitation and composite pulses are 10gs and 19.95gs
respectively, and the encoding pulse length for the quadruple is 10gs.
The spectra/image recorded directly from the experiment has a large DC
contribution from pulse feedthrough. The DC peak is removed in a post-processing step.
Two-dimensional images can be acquired via the back-projection reconstruction
methodology discussed above, or as Fourier Imaging with a two-dimensional acquisition in
which the laboratory frame gradient direction is changed by 900 from the evolution to the
detection period. These will be described latter.
1
0.8
0.6
0.5
04
0.2
-2.5
2.5
-2
(a) One 5 millimeter Tube with Water
-1
1
2
(b) Two Capillary Tubes with Water
Figure 2.9 One Dimension RF Imaging Experiments
2.2.5 Converting B1 Gradient to Bo Gradient
There have been many suggestions recently for using Bo gradients to speed up
NMR experiments and to aid in suppressing artifacts. In the following we explore a few of
these experiments from a B1 gradient point of view. In many cases the new experiments
are directly analogous to the Bo experiment and the B, gradient is imbedded in a composite
z-rotation. Some of the these experiments have been investigated previously by taking
advantage of residual field inhomogenieties in conventional high resolution NMR probes.
A z-rotation can be generated by the composite pulse [36],
-
,
-<i
(2.26)
where the it/2 pulses in the x direction may be thought of as tilting the plane of spin
evolution. If the 6 pulse shown in figure 2.8 is a composite RF gradient in the y direction,
then the overall composite pulse in equation (2.27) is identical to a B o gradient pulse. This
pulse sequence can be calculated by propagator based on equation (2.19):
U(t,) = e 1 ethe
() =
u(~).-I
(2.27)
2
U-(t) = Rx -
R,(0) -Rx
(2.28)
where R x,Ry and Rz are rotation matrices about the x, y and z axis.
The gradient pulse in the above scheme should have the form of a pure planar
gradient field, as given (for example by Eq.(2.22)), so with the probe geometry under
discussion here, a composite z-gradient would consist of,
-2g- - -r -g-
-g)
2
,
(2.29)
by using a radial gradient field with the zero RF phase resulting in a phase encoding Bo
imaging technique shown in figure 2.10 and equation (2.29).
A reasonable question to pursue is the range of validity in replacing a Bo gradient
with a comparable composite B1 gradient sequence. For single spin= 1/2 systems they are
exactly equivalent and may be used interchangeably (neglecting pulse error artifacts), for
coupled spin systems additional care must be taken since an RF gradient can introduce
coherence transfers that would not be observed with a Bo gradient pulse. This is again
simply a consequence of the non-secular nature of a B, gradient.
In the next section, the applications of the above composite pulse will be explored
in regards gradient enhanced spectroscopy, as well as approaches that use the RF gradient
field directly without trying to twist it into a Bo gradient.
G
Bo
Gradient
0
tp
__ _
_
- - - - -
- - -
--_ _
- - - - -
One of the most widely touted applications of B0 gradients in high resolution
spectroscopy is a means of eliminating (or reducing) phase cycling[37][38][39] for
coherence pathway selection.
By combining gradients and coherence
transformations, a
wide range of robust methods have been introduced to select only that portion of the overall
spin magnetization that follows a particular coherence transformation pathway. Gradients
characterized and reproducible dephasing mechanism into the experiment where the
strength of the gradients is sufficiently strong that a coherence can be completely dephased
(be made unobservable) in a time short compared to any relaxation time.
Radio frequency (RF or B) [16[43][15][44]o[45]gradients can also be used for
these applications and one obvious approach is to combine Bogradients to form an effective
Bspingradient pulse. The RF gradients have a special attraction for coherence pathway
selection because the effective coherence number changes with the phase of the RF field,
because the RF fields can themselves introduce coherence transformations and because RF
gradients can be used to suppress the evolution of internal Hamiltonians so that the spin
dynamics become simplified.
As an example, the COSY experiment is based on a coherence transfer from one
antiphase term, I,x 12, , to a term that is antiphase on a second spin actively coupled to the
first, I1z I2y. In the simplest case this is accomplished by an evolution period in which the
scalar coupling between the spins is active and a t/2 pulse. The overall experiment being,
-- t 1 -
-t
2
,
(2.30)
with acquisition taking place during t2. For a homonuclear two spin system (Iz + 12z)[46],
at the end of the evolution period t2 , the observable magnetization includes both diagonal
and transverse resonance's and there is quadrature detection in the acquisition domain, but
not in the evolution domain. The result of a 2D Fourier transformation is a spectrum that is
folded about the carrier frequency in m . To acquire quadrature detected signal in 0o1, a
second acquisition can be performed with the mixing pulse phase shifted by 900 to retain
the cosine modulated signals in t1 [37][38][39].
Now consider the desirable experiment of acquiring a quadrature detected signal in
0,
as well as in (02[40][41][42][43]. Based on the product operators at the end of the
evolution period t2 , it is clear that the quadrature information exists at this point. The
reason that the original experiment fails to acquire quadrature data in one scan is that the
mixing pulse selectively transfers only one of the two quadrature spin states and the second
is transferred to non-detectable coherences. An ideal experiment would be to transfer both,
but this is not possible. However it is possible to break the sample into spatially
heterogeneous regions and to vary the coherence pathway across the sample. Let's
consider the following experiment,
- t -
- t2 ,
(2.31)
where the phase of the mixing pulse is spatially modulated. Some portions of the sample
will yield a cosine modulated signal and other portions a sine modulation, and the overall
result will be a quadrature/quadrature detected signal. If 0(r) can be made to vary over the
range of 0 to 2r with equal weight to all angles then the detected magnetization will have
the form calculated by Bax[40], which only contains the rephased p-quantum pathway.
The trick remains of how to generate a it/2 pulse that is spread out in phase. The
solution up to now has been to apply a composite pulse of the form ( )z-(E/2)x-(+l)z
where the z-pulses have taken the form of a B0 gradient, shown in following figure.
RF
Homogeneous
t
2
1
G1
G2
Bo or B1
Gradient Pulse
Figure 2.11 Pulse Sequence for Obtaining Quadrature Selection in t, in a
Single Scan in COSY Experiments by Using Gradient Field.
The same approach can be taken with B, gradients[16][43][15][44][45] where a
composite z pulse is made up from a B, gradient in the above figure 2.10 and 2.11, the
experiment result is shown by Maas and Cory[16].
There is an alternative approach where we recognize that the symmetry of the B,
gradient field immediately has the desired form and actually the gradient nature of the field
is not a benefit. So the mixing pulse can be replaced by a "radial" RF pulse which is
approximated by a radial gradient pulse. The closer to 7t/2 the pulse can be made the more
complete the coherence transfer will be; this is accomplished by using composite 2t/2
pulses. The spins that experience other than a ct/2 pulse contribute more heavily to the
diagonal but do not introduce distortions into the experiment.
In the phase cycling version and the z-rotation experiment it is immediately clear
how to select the anti-echo pathway, simply change the direction of the phase cycling or zrotation during the echo period (following the mixing pulse). For the radial experiment, the
pathway selection is governed by the polarization direction of the two RF coils which is
necessarily identical always yielding an echo (not the anti-echo). The n-type pathway can
only be selected by inverting the sense of polarization of the gradient coil, and this is most
simply accomplished by placing a T) x pulse immediately preceding the gradient pulse.
The two experiments use very different aspects of the B, gradient coil for pathway
selection and also have very different strengths and weaknesses. In particular the radial
pulse method only requires a very short gradient pulse and is good at suppressing n-type
coherences. It does suffer from introducing additional diagonal intensity.
Notice that the radial gradient should be particularly robust in regards to introducing
axial peaks. If there is relaxation so that a homogeneous RF pulse would introduce
undesired axial resonance's the relative symmetries of the gradient and homogeneous RF
fields insure that any Iz magnetization that is excited by the gradient pulse will phase
average to zero when detected by the homogeneous RF coil.
Another very powerful feature of this method is that one may spin the sample! Up
to now all gradient methods in spectroscopy suffered from the requirement that since the
gradient was employed to burn a magnetization grating into the transverse spin
magnetization, and the sample had to remain static until the grating was refocused. Here
we are not employing a grating and therefore there are no limitations of the sample motion.
2.3
RF Imaging
2.3.1 Principle of RF Imaging
As shown earlier a complex nutation spectrum may be acquired in a 2 step process,
O(r)= y .G,,RF
(2.32)
r
M',(, F) = p(T). sin[O(r)]
for po = p(7F)
(2.33a)
M2 (, r) = p(F) cos[ (r)]
for po = p(F)-.
(2.33b)
GRF(,PP)
RF
Planner Composite
Gradient Pulse
Fl
RIF
Homog eneous
Pul se
RF
Quadru pole
Puls
F, ,1
e
F"
-
~-
Figure 2.12 A Gradient Pulse, GRF(4,tp), Made by Composite Pulses
The two results may be combined to form a Fourier pair,
M
(T, F) = M2 (,
F) - iMl,(T, F) = p(F) - e-i(r)
(2.34)
Thus, the integration form, S(t) over the sample gives out the Fourier
Transformation of the spin density p(r) in variable 'C,and this permits one to define the
normal, kr wave-number,
S(z) = Jp(Fr) e-'r)dr = p(F) "e-k'rdr
=
(2.35)
S(k,)
kr = GRFr.
(2.36)
The RF gradient pulse, GRF(,)p), is created by a sequence of composite RF pulses.
When the phases, 0 and (p, of the quadrupole field are the same, O=p, a pure planar
gradient pulse, GRF(qp,(p), is formed, as shown in figure 2.12.
2.3.2 RF Fourier Imaging
is set to both 00 and 900
In two dimensional RF imaging, the RF gradient phase, (p,
to encode the spin density in the X and Y directions as HRF(x) and HRF(y) respectively,
=
HRF
-h[Gxx cosp + Gy sin ]l,
GxxIx
HRF(x) =
=
HR,(y)
-GyyIx
p =0 0 .
for
p = 900
(2.37)
7r),
/
RFHome!
for
00
.
900 X
(==o9o
I
"I
to
t
T
Sampling
IN
t2
t3
M
0
'I
Time
Figure 2.13 Fourier RF Imaging Pulse Sequence with Composite RF Gradient GRF(4,q).
The pulse sequence for RF imaging is shown in figure 2.13. The prepulse permits
a complex data set to be acquired. The on-time of the RF gradient pulse, GRF((P,cP), is
-z-
incremented to encode the spin density. For (p=0, a Hamiltonian, HRF(x), with the x
position as a variable is created, and the spin density is encoded along the x direction.
Meantime, the information in the y direction is encoded for p=900.
By letting the nutation angles a, = )hGxxnT and a 2 = FhGyymT the spin dynamics
followed the above pulse sequence at different times is :
to:
I =I -Iz
I
=
I
= -- i
RFHomo
for
0
RFomo = 900
I[cosaIz - sin aI,I
=
-I cosai
-
+ sina ]'
RFHomo = 0
for
RFHomo = 90
+-[COSa cos a 2 - sin a1 sin a2i2z
I-[cosa 1 sin a 2 + sin a 1 COS a2
RFHomo
0°
for
(+[cos a,cos a 2 - sin a sin a 2 ]i ,
+\[cosa,
sina 2 + sina, cos a 2
RFomo = 900
)
In RF Imaging, only the projection of the magnetization on the transverse plane
contributes to the detectable signal. So the complex signal must be reconstructed from two
measurements:
S=
, =-I[cosa sina
=2= -- I[cos
'Tota
I
=
12 -
ii =
-le
-
2
+ sin a Cos a 2]
for
, cos 2 - sin a sin 2 ]I
0
RFomo
(2.38)
RFomo = 900
(2.39)
i(a+2).
The nutation signal, integrated over the sample, is:
FID, = -p
-[cos a, sina 2 + sin a, Cosa 2 ]d
FID2 = - p [COS al COS a 2 - sin a, sin a 2 ]dr
;
for
for RFom
00
90
FIDrToal = FID2 - iFID1 = -jj p(x, y)- e-i(a+a2)dxdy,
o
(2.40)
(2.41)
which may be rewritten in terms of 2D wave-number as,
S(n,m) = -I
kx = yiGxnT
k = yGmT
p(x, y)e-i4(GxxnT+GymT)dxdy =
-J
p(x, y)e-i(kxx+k Idxdy
(2.42)
(2.43)
Reconstruction of the spatially varying spin density is simply accomplished by an
Inverse Fourier Transformation:
p(x, y) = FT-' S(n,m)}.
(2.44)
2.3.3 RF Back Projection Imaging
In much the same way, back projection imaging method may be applied by
incrementing the phase, p, of the RF gradient, GRF((p,p), over 1800. The Hamiltonian in
equation (2.24) becomes,
HRF = - jGxx cosq + Gy sin
p}Ix = -hGRFx'I
(2.45)
x = xcos p + ysin Vp.
(2.46)
\\
i
RFHome
I
Sampling
I
RFQuad
T
to
M
tl
I
/
A '=/
t2
/'/
-
Time
Figure 2.14 Projection RF Imaging Pulse Sequence with Composite RF Gradient GRF.
The pulse sequence is indicated in figure 2.14 and spin dynamics is shown as
below by letting the nutation angle a = hGRFmT. The encoding in the radial direction is
realized by varying the pulse length, nT, and the angular information is by the phase, p,
to:
tl
I =I-iZ
; for
=
-I- I
I =
= -
00
=900
RFomo =
RFo
lcos(a)i - sin(a)i^]
cos(a)
+ sin(a)1'
RFH
for
omo
= 00
RFHomo =900
The detected spins in the transverse for 0o and r/2 prepulses and the combination are,
I
=
I = -Isin(a)I,
=
I2 = -Icos(a)I,
iTotal =
2 - ii1
-Ie
- ia
for
; for
RF
=0
(2.47)
(2.47)
RFomo = 900
(2.48)
.
In the same as the Fourier RF imaging, the detected signal is,
S(n,m) = -
p(x, y)e-i4GRFx nT"dx dy = -
P(m,x')e-ik'xx dx ,
(2.49)
where P(p,x') is projection function along the y' axial direction, and kx,= YGRFxnT.
The spin density, p(x,y), may be reconstructed by Filtered Back Projection from
the signals, S(n,m), where the n represents the duration of the RF gradient and m
represents the phase change of the RF gradient,
p(x, y) = B[FT' {S(n,m) . I}].
(2.50)
As an example of RF imaging for high field NMR, a series of ID Fourier images
(or projection profiles) are displayed in figure 2.15 acquired using composite RF pulses
made up of homogeneous RF magnetic field and quadrupole RF gradient magnetic field
with the sequence in figure 2.14.
An imaging phantom has two tiny holes containing water located at 1.25 and 1.875
millimeter away from the center. The homogeneous t/2 and 7t pulse lengths for preexcitation and composite pulses were 10gs and 19.75gs, and the encoding pulse length on
the quadrupole coil is 10gs on a Bruker AMX400 spectrometer.
By increasing the encoding pulse length with different RF phase, (p, a series of 1D
Fourier images were obtained and the 2D Fourier image was reconstructed.
A set of projection profiles obtained by increasing the phase by 30' are shown in
figure 2.15. The DC peak is an artifact of RF feedthrouth and was removed by subtracting
it from all the profiles. The 2D image of the phantom reconstructed from the 12 projection
profiles is shown in figure 2.16.
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
-1.82
0
1
23
-2
W4,L
-1
1
2
(a) 150 Projection Profile: Original and Processed
1
0.8
0.6
0.4
0
2
1.
-1.3
-1
-2
85
(b) 450 Projection Profile: Original and Processed
1
1
0.8
0.8
0.6
0.6
0.4
0.2
~Ak I-YYL
-0
4
. 3
~
0.2
~
'1~Ahi1.±Jh
- 'y._INy
-:
U
-0.2
I-
"
'
"
.
-1
1
,,
_; ,
2
.
(c) 750 Projection Profile: Original and Processed
-0
'
-2
.S
7
-1
1
I
2
(d) 1050 Projection Profile: Original and Processed
1
0.8
0
0.6
0.6
0.4
0.4
0.2
0
8
2
a...
.-- -1
I.(e) 1350 Projection Profile: Original and Processed
-x
I
-0
85
6
1.
3
-2
1
1
2
7
0.8
0.6
0.4
0.2
-1
22
0
1.82
-2
-1
(f) 1650 Projection Profile: Original and Processed
2
,
1
0.8
0.6
0.4
0.2
.
1.2'~
1.
21
. 81
-2
-
1-2
-
1
4,
1
2
(g) 1950 Projection Profile: Original and Processed
Figure 2.15 Projection Profiles of Two Capillaries at 150, 450, 750, 1050, 1350, 1650 &
1950. The Is' column is original profile from experiment, the
2 nd
is processed profile.
The Reconstructed Image from 12 Filtered Back Projection
-2,5
-2
-1,5
-1
-0,5
0.5
1
1.5
2
2.5
-2
-1
0
x mm
1
2
Figure 2.16 The image is reconstructed from 12 experimental projection profiles, some of
which are shown in the second column of figure 2.15 after data processing with 30 degree
increment. The two capillaries can be seen more clearly with more projection.
Chapter 3
Introduction and Principle of NQR
3.1
Introduction of Nuclear Quadrupole Resonance Imaging
3.1.1 Nuclear Quadrupole Resonance Phenomenon
3.1.1.1
Background
NQR is based upon the interaction of the electric quadrupole moment (eQ) produced
by nonspherical charge distribution in the nucleus[47] and the electric field Gradient (eq) of
valence electrons[48]. In NQR the transitions between energy levels are produced by
applying an oscillating RF field with a frequency near 0o to satisfy the Bohr condition. The
RF field interacts with the magnetic dipole moment of the nucleus resulting in a timedependent perturbation since the nucleus has no electric dipole moment[20].
By observing the energy transition frequencies, the electric quadrupole moment
(eQ) of a nucleus can be studied to measure the deviation of the electric charge distribution
of the nucleus from spherical symmetry[49]. In NQR the values of both the electric field
gradient (eq) and asymmetry parameter (fl) can be used to obtain information about the
electronic structure/distribution within a molecule when it is under the influence of electrical
forces from neighboring molecules in some solid[50][51][52]. Based on a broadening of
the quadrupole resonance the charge and size effects of impurities or defects within
Thus charge effect may be used in dosimetry of
materials can be obtained.
radiotherapy [19]. The electric charge distribution of nuclear compounds of the first,
second, third and fifth group elements of the periodic table have been interpreted[49].
The values of the electric field gradient (eq) and the asymmetry parameter
(T1)[50][51][52] can provide the constitution of the crystalline unit cell49][53][54], which
pertains to the number and orientation of the molecules in the unit cell, the chemical
equivalence or inequivalence of different sites for the same nucleus in the unit cell, and the
change in the orientation of molecules that take place when phase transition occurs.
3.1.1.2
Study of Impurity or Defect
The charge effect arises if the impurity core in the materials or the impurity ion in
the ionic crystal has a different charge from the cores and ions. The additional field
gradient at a nucleus is caused by the extra charge on the impurity, and results in a different
distribution of the field gradient and a broadening of the quadrupole resonance.
Charged impurities may be introduced by x-ray or y-ray irradiation where charged
free radicals, such as electrons and charged molecular ions, are produced. These will give
rise to the "charge effect" by producing distributions in the field gradients at the nuclei. In
addition, the electrons and free radicals can lead to a broadening of linewidth due to their
magnetic interaction with the nuclear magnetic dipole moments. The broadening due to
both the magnetic and charge effects would be comparable for nuclei with smaller
For example, after
quadrupole moments such as "B, 14 N or 35C1 nuclei.
paradichlorobenzene is irradiated with 60Co, which has 5.25 year half life and
1.173(99.8%) and 1.332(100%) MeV y-rays, at 20'C, there is a thirty-seven percent
decrease in intensity of the 35C1 resonance for irradiation's of 2.7x108 roentgens[48].
When x(y)-rays interact with matters, they knock out some electrons by
photoelectric absorption (PE), Compton scatting (CS) and pair production (PP) with
respect to different energy ranges. In the case that the energy of gamma rays is less than 10
MeV, the PE and CS interactions are dominant and result in molecular ions (free radicals),
such as y-ray interactions with water molecules in tissues:
Excitation
H20*
Dexcitation
H
+OH
H. +OH
+
H 20
lonization
H20
e-
H20
H+
> H
+OH-
(3.1)
e aq
H*
where the OHs are free radicals. The free radicals are going to react with the base or sugar
(R) in DNA resulting in an indirect action, sublethal or lethal damage of tissues,
OH e +R -=>
ROH(
R
(3.2)
This kind of radiation response can be modified chemically, physically and/or biologically
in the application of radiation damage and radiotherapy.
Dosimetry by NQR may be applied to biological systems exposed to ionizing
radiation because of the introduced changes in the electric field gradient. Since the line
width of free induction decay is physically related to the broadening of NQR and in
practical terms approximated to the inverse of spin-spin relaxation time (1/T2), radiation
dosimetry can be carried out by exhibiting a consistent change of the T2 as a function of the
delivered dose in certain materials. As shown in figure 1.1, Hintenlang and Higgins[19]
demonstrated the variation in the 14N spin-spin relaxation time, T 2 , as a function of the
60Co
radiation dose delivered by
gamma rays in hydrated urea.
T2
800IL6
o
S680
.Ei
560
440
440
>C
0Dry
A
Urea
o
o
Hydrated Urea
0
MLI
.*
X
320
200
O
120
60
radiation Dose (Gy)
o
300
D
Figure 3.1 Variation of the 14N2T as a Function of Radiation Dose by 60 Co y-rays[19].
Vargas, Pelzl and Dimitropoulos[55] studied the irradiation defects produced by
gamma in polycrystalline samples. The linewidth and the intensity of the 35 C1 NQR signal
were investigated as a function of the radiation dose. They concluded that (1) an irradiation
defect in an ionic compound modifies the electric field gradient (q) in the ways of an excess
charge, an excess magnetic moment and a mass defect and (2) the linewidth and the
intensity of the NQR spectrum are affected by an elastic and chemical misfit.
The size effect is significant only if the foreign ion cores in metals or foreign ions in
ionic crystals have sizes appreciably different from those of the original ion cores or ions in
the sample. A distortion of the electron shells of the ions containing the resonant nuclei
then occurs leading to changes in the field gradients at the nuclei[56].
Size impurities occurs in molecular crystals if a neutral impurity molecule is
introduced. An impurity molecule produces a gradient change at neighboring nuclei
directly and indirectly on the intermolecular contributions [57] [58] [59].
3.1.2 Nuclear Quadrupole Resonance Imaging
3.1.2.1
Difficulties of Pure NQR Imaging
In order to analyze the spin dynamics of pure NQR, three coordinate frames are
required: the Laboratory Axis System (LAS, the Principle Axis System (PAS) and the
Rotating Axis System (RAS). In powdered samples, since each atom has its own PAS, the
directions of nuclear spin Iz's are distributed equally in all directions of the LAS.
The
interaction of the spin in a magnetic field is simply a torque, thus the spin dynamics are
dependent on the angle between the internal electric field gradient, EFG, and the external
RF magnetic field, B
,LAS'
The nutation frequency, , and nutation angle, In, of spins depend upon the angle,
0, between the EFG and the B ,LAS.
to a maximum,
The nutation frequency varies continuously from zero
-=yB1 , for the spins in the direction perpendicular to the RF field,
Y
=
/31sin 8
=
,1sin
=
yB, sin(O)t,
=
co1sin(O)t w
(33)
The intensity I(() of a nutation spectrum in figure 3.2 shows significant DC and
low frequency contributions. These components come from the distribution of the nutation
frequency as a function of the orientation distribution of the EFG.
Without Spin Selection
With Spin Selection
0.8
0.8
0.6
0.6
0.4-
0.4
0.2
0.2
500
1000
1500
2000
500
1000
1500
2000
(a) Normalized Nutation Spectrum (b) Original Nutation Spectrum
Figure 3.2 Spectrum Intensity
Linewidth & Intensity are smaller with spin selection than without spin selection
In our proposal, the spin selection selects those spins that are perpendicular to the
RF magnetic field gradient and parallel to the RF homogeneous field, as an example. This
removes the DC and low frequency terms of the point spread function and a sharp PSF for
RF imaging by the RF gradient is generated, as shown in figure 3.3(a). Figure 3.3 (b)
indicates that the trade-off of the spin selection is low detection efficiency due to the small
fraction of spins selected for imaging [21]. At this point we prefer well ordered spin
dynamics and are willing to accept the loss in sensitivity. Obviously, a future goal is to
select all spins that are perpendicular to the RF field and hence maintain a good signal-tonoise ratio and simple spin dynamics.
From the above analysis it is clear that in a powdered sample the nutation frequency
is represented by a distribution reflecting the RF inhomogeneity resulting from the
orientations of the EFG in the LAS. Therefore, one of the difficulties in pure NQR
imaging is to remove the angle dependence between spins and the RF field.
The first NQR imaging experiment in a powered solid was reported by Matsui et.
al.[22]. The study took advantage of the fact that the spectral width of a Zeeman perturbed
NQR powder pattern is proportional to the local Zeeman field. By combining the "rotatingframe zeugmatography" method with pure NQR and projection images Rommel et. Al.
[25] [27] [26][24][28] achieved the spatial information, which is amplitude-encoded in the
free-induction decay by the aid of the gradient, G1 , of the local RF amplitude, B 1. The
rotating frame NQR imaging is particularly advantageous because one makes use of really
pure NQR without any magnetic fields or field gradients. Thus, the full spectroscopic
information remains unconcealed and can be used for the characterization of materials.
However, the problem without an external DC field is the RF field inhomogeneity
within a powder sample. That inhomogeneity is introduced by the angles, 0, between the
B, and the EFG resulting in that the nutation frequency is the function of the angle and that
the line width is broadened as discussed earlier. Rommel et. al. developed a deconvolution
algorithm based on post-data processing to restore the original spatial information. The
trade off is the decreasing of signal to noise ratio in great degrees. Therefore, we have
been developing new approaches based on the pure Nuclear Quadrupole Resonance
Imaging by first using a homogenous RF field to make the spin selection then using an
inhomogeneous RF field to make the spatial encoding. Those two RF fields are controlled
by using pin-diode switching of modes. After correcting this broadening effect by using
spin selection methods only those spins perpendicular to the RF field gradient for RF
imaging remain and the nutation function becomes a purely sinusoidal function. Based on
the RF imaging described in chapter 2, the spatial information of a powdered sample is
encoded by varying the pulse length of the RF gradient for the pure NQR imaging.
3.1.2.2
Zeeman Perturbation NQR Imaging
As shown in figure 3.6, with transitions between two adjacent energy levels, the
spectral line width (Awo) of a Zeeman-perturbed ( Bo
O) powder pattern is proportional to
the applied Zeeman field ( Bo). If a Zeeman field gradient (Go) is applied in x direction, the
Zeeman field and the spectral line width become functions of spatial location (x):
B= x'G,
(3.4)
Ao = w(Bo) = w(x)
As the line width increases, the spectral amplitude (A) decreases as a function of
Zeeman field (Bo). If we define a nuclear spin density, p(xi), which is independent of
Zeeman field, Boi, the amplitude, Ai, of the spectrum at the position xi is proportional to
the spin density p(xi) and a field dependent parameter, W(Boi), and can be written as:
(3.5)
Ai = W(Bo,) p(x,).
The observed spectral amplitude, At, is the total of all amplitudes, Ai, for i= 1,2,,,N
from N points in the x direction, that is:
N
N
A =
A, =
1=1
(3.6)
W(Boi) -p(xi).
t=l
In order to image the spin density p(xi), the field dependent parameter W(Boi)
should be pre-calculated by applying different Zeeman field Boi related to different xi and
could be converted into the position variable xi:
(3.7)
W, = W(Bo,) = f(x,).
, = f(wi)
Then, each spectrum, Ati, measured under the uniform Zeeman fields Boi
corresponds to a powder pattern[22]. The N sets of linear equations, Ati, for i=1,2,,,N are
obtained and the spin density, p(xi), at position xi for all i's is determined by solving the
linear equation system:
Bo=B
N
N
A, = W, .p(xj) = W, .p(xj)
J=1
J=1
At
LLJ.N
N
N LP(XN)
for
i= 1,2,,,N
(3.8)
(3.9)
p(x, )
W,
W,
... W,
At,
p(xN)
WN
WN
... WN
A
(3.10)
By applying the same principles to carry out NQR experiments in two dimensions,
the NQR imaging technique can be realized. For the first time, this method provides a tool
for obtaining spatial distributions of NQR sensitive quantities such as temperature and
pressure in disordered solids. The trade off is that an external field is required, which
could influence the electric field gradient, resulting in the distortion of the spectroscopic
and/or structural information from the sample.
3.1.2.3
Rotating Frame NQR Imaging
The rotating frame nuclear quadrupole imaging (pNQRI) technique combines the
rotating-frame zeugmatography with pure NQR so that there are no Zeeman splitting effects
[25][27][26][24][28]. Based on the principle of NQR for the pure NQR experiments, an
RF field is applied to perturb the NQR interaction resulting in the energy transitions of
nuclear spin systems. The signal with the transition frequencies is collected in the form of
either free induction decay or spin echo[49].
Rotating-frame zeugmatography[5] is a flip-angle encoding technique. Nonuniform
RF fields are applied so that the flip angle depends on its position with respect to the RF
field gradients. The spatial information is encoded either in the amplitudes or the phase of
the transverse magnetization, which produces a free induction signal. However, only
amplitude-encoding is used in the NQR experiments.
The FID signal, sj(t,tp), at Oj resonance lines can be written in the following
form[27][60] after an RF pulse along the z axis excites the magnetization:
t,
=
sin(t)esin
0 (t)e
-
= y-B 1o+y-G,
0 1(z)=y-Bl(z)=y- Blo+fG(z).dz
k
0
(3.11)
in(tp sin Oco1(z))Odz
0
) G1 (z )=Const
,
(3.12)
where simply the zero asymmetry parameter (1r=0) is assumed, B1 is one half the
amplitude of the RF pulse, pj is the nuclear density, tp is the RF pulse duration, and t is the
sampling time.
Amplitude encoding by flipping angle variation modulates the magnetization
reached after the RF pulse duration (tp). The spectra Sj(co,tp) of the Fourier transform of
sampling time (t) with respect to the signal sj(t,tp) are,
S,(w, -
FT,
(tt)
JpW(
1
(3.13)
in)2
p (z)sin2 Osin( (z)sinO)dOdz
0
2 6
(3.13a)
5(z) = - tp 0),(z).
The inner integral can be represented by a series of Bessel functions and can be
approximated with the "k-space", k,= V3y -G,-t,, component[27]:
S
S(,k
-e
-[J)2
2
(3.14)
f p,(z)
sin kzz -
dz
For a constant gradient, G1, a second Fourier transform with respect to the variable
kz yields the projections of the object in the z direction for each resonance line o,
p,(z) FT [S,(, ,k)].
(3.15)
The second dimension for solid NQR is obtained simply by using the projection
procedure. A set of projections in the desired directions varied by small angles is produced
by rotating the object relative to the direction of the RF gradient step by step. At each step
the rotating-frame zeugmatography procedure is performed. The NQR images can be
reconstructed by using a filtered back projection technique[61].
In order to apply the pNQRI in powder geometries, it is necessary to account for
the orientational distribution of the EFG tensor with respect to the RF coil axis. Since the
orientational dependence distorts the spatial profiles to a certain degree, the point spread
function (PSF) of the imaging system has a broadening effect. Therefore, a numerical
deconvolution procedure was developed to produce the true spatial profiles[24]. A nutation
FID with a pulse length, tp, results in absorption after the Fourier transformation. A set of
absorption's is generated by varying the nutation pulse length tp, which is called pseudo
FID and may be written simply as:
S(t) = pI(x)dxl sin 2 sin[ (x) sin O]dO
=
Jp(x)
J ((x)) -
J 2n+
(x))
1
,=, (2n - 1)(2n + 1)(2n + 3)
3
ldx(3.16)
(3.16)
=I p(x)f(x t,)dx
(3.16a)
)Gxtp
=
4(x)
{
f(x -t) = Isin2 0sin[ (x) sin O]dO
S8 j
3
(3.16b)
J2n+l((X))
(x))-
,=, (2n - 1)(2n + 1)(2n + 3)
By assuming that the spatial resolution is beyond the orientational correlation length of the
powder, the nutation pseudo FID S(tp) could be rewritten by using a substitution:
t,
x
=
tdeu
(3.17)
x0 e
fp(xe-V)f(xOevtde"')d(xoe-
S(tdeu)
=
fa(V)
=
xe-Vp(xoe-V)
=
f(Xotdeu-V
f(u
-
S(u) =
v)
Jfa(v)fb(u
- v)dv.
)
(3.17a)
(3.18)
The result is a convolution expression with respect to a transformation from linear to
logarithmic scales. Based on the convolution theorem and Fourier transformation, the spin
density, p(x), could be calculated by deconvoluting:
p(X)=
x
Fu
Fu[S(u)]
Fuv[fb(U- V)]
(3.19)
From the above analysis of pNQRI and post-data processing, the spatial profiles of
a powder sample may be restored. However, the numerical deconvolution used to increase
spatial resolution dramatically reduces the signal-to-noise ratio.
3.1.2.4
Pure NQR Imaging
We have developed a new approach for pure NQR Imaging on RF Imaging, which
first uses a homogenous RF field to make the spin selection then applies an inhomogeneous
RF field to make the spatial encoding by using pin-diode switching of modes, as shown in
figure 3.3. Therefore, the full spectroscopic information, which remains unconcealed, can
be obtained without external DC field. The spin selection procedure eliminates the RF field
inhomogeneity within a powder sample from the orientation influence between the RF field
and the EFG and increases the spatial resolution and signal-to-noise ratio.
RF
Spin
Imaging
Selection
Data
Image
Collection
Reconstruction
Figure 3.3 Diagram of Pure NQR RF Imaging
The spin selection could be realized by using Spin Locking, DANTE sequence,
Jump and Return, Shaped Pulse or etc. By comparing the detection efficiency and signalto-noise ratio, an approach of pre-selection pulses by a homogenous RF field in the X
direction may be applied. Then, for example, a nutation pulse for imaging is followed after
the spin selection in the Y direction. As shown in figure 3.4, a series of FIDs by varying
the pulse length of the RF field gradient can be collected resulting in a set of pseudo FIDs
with respect to a one dimensional spatial profile after the image reconstruction.
Homo
RF
Gradient
x
2
N
Nx
M
Figure 3.4 One Dimensional Pure NQR Imaging Pulse Sequence
3.2
Principle of Nuclear Quadrupole Resonance
3.2.1 Definition of Nuclear Quadrupole Resonance (NQR)
Nuclear Quadrupole Resonance measures the interaction between the electric
quadrupole moment (eQ) and the electric field gradient (eq) [20][49][62]. The Hamiltonion
of NQR is:
H-Q
e2
Q(VE)
Q, (VE),
d 2V
-eQ
21(21-1)
2
d 2V
12
dy "2
X
72 + dyy
!
21
e Qq
41(21- 1)
(3.20)
-2
2V
i2
dzdz
+ i 2)
71
z
(3.21)
where Q and VE are the electric quadrupole moments and the electric field gradient
tensors, I is the nuclear spin operator, and I+ = Ix + i,
& I_ = Ix - iI, are step up and
down nuclear spin operators. Semi-classically speaking, the interaction energy (EQ) of the
nuclear charge distribution (p,(F)) with an atomic/molecular electron charge distribution
(p,(F,)) or potential (V) is calculated in equation (3.22), where the V(r) is an expanded
Taylor series about the center of the nucleus. Thus, its integration returns the quadrupole
contribution to the interaction energy. The angle between the Z axis of V and the Z axis of
nuclear symmetry is defined as the angle between Iz and I of the nuclear spin[47].
P(rn)Pe(T) dnde
EQ
Vn
1
Ve
-eQ
4
-
I
3m
2
p (F)V(F )d3r
I
-1(I+
I(2I-1)
1)
(3.22)
d2V
Kdzzz=o
3.2.2 Electric Quadrupole Moment (eQ)
The Electric Quadrupole Moment is produced by an unsymmetrical/nonspherical
distribution (or a prolate/oblate ellipsoid distribution) of electric charge in the nucleus
resulting from the second order vibration and/or nuclear rotations[47], as shown in figure
3.5. The nuclear electric potential Vn(r) is calculated classically from its charge density
pn(r) as below [47][20][1]:
1
p,(r')d3
(T-Fl
4re
eZ+ 47"Eo
)72
21(3cos2 0- 1)d3
(3.23)
where the I - rl is expanded based on Taylor series by assuming the r>>r', angle 0
between r and r' is determined by the relationship between the nuclear spin I and the
component Iz with the direction of the electric gradient field, given by cos0=Iz/I[47]. The
integral in the first term gives the total charge eZ, and the second term, the electric dipole
moment, vanishes for nuclei under ordinary circumstances. The interesting term in the
multipole expansion is the quadrupole term, which is defined classically and in quantum
terms as the nuclear quadrupole moment (eQ):
eQ- Jp,()r2(3cos2 0 -1)d 3r
ej
Classical
(3z 2 - r 2 )dr
(3.24)
Quantum
where yfis the wave/distribution function of the nucleus, r =x
+y2
+Z2
is the
position, and the integration is performed over the entire volume of the nucleus. The eQ
equals zero if a nuclear spin I is either equal to 0 because the nucleus has no orientations or
equal to 1/2 since the two possible spin orientations of the nucleus differ only by reversal
of the spin direction and thus correspond to the same effective nuclear charge distribution
resulting in no orientation-dependent electrostatic interaction[20]. For spin 1/2 nuclei, also
the electrostatic energy does not split the m, degeneracy[1].
Y
X
Y
X
Oblate
Prolate
Figure 3.5 Nuclear Charge Distribution
3.2.3 Electric Field Gradient (eq)
The Electric Field Gradient (eq) is produced by the valence electrons or electron
charge distribution in the molecule [62],
(3.25)
,
eq - eqzz = e.
(e dzdz),
where V is the electrostatic potential (electron charge distribution) at the nucleus due to the
surrounding charge, Ez = -dV/z is the electric field at the nuclear position in the z
2
direction, and qzz =
V/dzdz is the gradient of the electric field. The charge distribution
V[49] is determined by the electrical forces in a molecular crystal that (1) bind the atoms
within the molecules, and (2) bind neighboring molecules to form the crystal. The field
gradient (q) at atom A in a molecule is related to the electronic ground state wave function
I,the nuclear charge ZB of any atom in the molecule other than A, the angle 0 AB between
the bond axis or highest-fold rotation axis for A and the radius vector RAB from A to B, and
the angle eAn between the bond or principle axis and the radius vector
eq = eqA
e
[
1
OAB
3
B#A
,-
AB
r3
n
rAn
to the nth electron.
1 yd3r
(3.26)
An
3.2.4 Asymmetry Parameter (Ti)
Asymmetry Parameter (Ti) is defined to specify the EFG and to measure the
departure of the field gradient from cylindrical symmetry[20][49][62]:
d 2 V/dxdx - 2 V/ydy
d2 V/azZ
(3.27)
0 axially symmetric
77=(0, 1) axially asymmetric
1
2
d V
dx
2
_2V
-
d2 V
dxdx
=0
(3.27a)
axially asymmetric
-
ox
y
dyd
V
field gradient.
1
eq
d2 V
= eq for 77= 0
& oz
&
2
&
d2 V
dydy
-
d2V
dzz
for 77= 1,
(3.27b)
(3.27c)
where r=0 corresponds to axial symmetry around the z axis in the principle axis system
(PAS), which means that there is no difference for the x and y axes in the xy plane.
3.2.5 Energy States (Pm)
of the Hamiltonian, H., in equation (3.21) under the
The eigenfunctions (pm)
conditions of the nonaxial field gradient without a Zeeman field are formed by combining
the pure NQR energy states. For nuclear spin I=1 [49], the energy states, 1-1>, 10> and
1+1>, compose the eigenfunctions under the conditions of axial field gradient ( 7 = 0)
without the Zeeman field ( Bo = 0):
S((+11-(-11)2
for
9P-1
TP+1
=
(3.28)
I=1
((+11 + (-1|)/2
For nuclear spin 1=3/2, letting p=(1+2/3) 1/2, the energy states of the H. under
77
0 and Bo # 0 are formed by mixing the pure NQR energy states 1-3/2>, 1-1/2>, 1+1/2>
and 1+3/2>, which are the eigenfunctions under 7= 0 and B, = 0 [63][64]:
p+l
3
F20
3
2
p-i
1
2p
1',,
2/
T- 3
2
T- I
TPM
40m
2
2
-/
+1
p+l
p-l 3
p
2p -2-+
9+3
2
1
2/p
2p
1\,
for
=-3
2"
(3.29)
7-/
3.2.6 Energy levels (Em)
The total eigenvalues (Em)
of the NQR Hamiltonian (HQ) and the Zeeman
Hamiltonian (Hz) is calculated as[62][65],
Emor
I=the
spin energy levels .are calculated and shown in figure 3.6 [62] (3.30)
For spin I=1, the energy levels are calculated and shown in figure 3.6 [62]:
2
e2 q + (hB° cos 0) 2+ e q4
4
E+
cos 0) +
Em-
e2Qq
4
for I=1. (3.31)
-2e 2Qq
4
Splitting
Splitting
m=+l
E+I
.-
;Qq/4
co+
1O
11
U
-_-(__
m=O
(a)
I
i
_0_. _
(b)
E-1
2
Eo= -2e Qq/4
(c)
Figure 3.6 Energy Levels for I=1 under Different Conditions.
(a) Axial Field Gradient without Zeeman Field (b) Nonaxial Field Gradient without
Zeeman Field (c) Nonaxial Field Gradient with Small Zeeman Field
The energy levels (Eo) for m=0 are the same under all conditions. However, the
energy levels (E+, and E_) for m=+l may be split under the different conditions. The Tr
splits the E,, and E_,.
A small Zeeman field Bo, which has an angle 6 with Iz and
/lz = -yhBo * I, will perturb the pure NQR system resulting in further splitting of the
energy levels. This is important to indicate the reason that the pure NQR can provide
undistorted spectroscopy. Under the condition of axial field ( 77 = 0) gradient without the
Zeeman field ( Bo = 0) the E,, and E , are the same, that is,
E, = 41(21-1)
e2 Qq [3m 2 - I(I+ 1)=
E_,
e2Q
(3.32)
(.
4
1
For nuclear spin I=3/2, there are four energy levels: E3/2,E1/2, E3/2 and E. ,2. They
are calculated in equation (3.33) and shown in Figure 3.7[62][49]. In axial field gradient,
there are only two different levels since the energy levels, E1/2and E_1/2 , are the same, and
the E3/2, E3/2 are the same. In nonaxial field gradients ( 7
0), the four energy levels are
mixed up and two new energy levels are formed[63][64],
e 2Qq 1 + 72 3
=
Em =4
F
e 2Qq
(3.33)
1+2/3
9+3/2
3/2
MJixing
e Qq/4
ixing
m=_t3/2
.
1f3
)
e 2 Qq/4
..-
-
WQ
-
p1/
- OQq/4
m 1_ 1/2
7n/)CQ4
(b) Nonaxial Field Gradient
(a) Axial Field Gradient
Figure 3.7 Energy Levels for 1=3/2
3.2.7 Energy Transitions (com)
Transitions between different energy levels may be induced by an RF (B1 )
perturbation. However, the observed transitions between different energy levels depend
upon both the selection rule and the energy difference[62][49] because of
(mlI
or
ylm')
0 only for
Iml - ImI= 1 and
om
0 for E.m
Em.. The frequency of the
transitions between energy levels, Em and Em., is,
S
Em - E.
h
(3.34)
As shown in the figure 3.6 and 3.8, for nuclear spin I=1, the transitions with
respect to E , <: E0 , and E_,I=> Eo result in two frequencies o), & o4,, and the frequency
difference (Aco) between them. When 7 = 0, there is no energy transition between the E ,
and E_,, as shown in figure 3.6(a). The transition frequency between the E, and Eo is o,
under 1 = 0 & B = 0 and named the NQR frequency[66],
{
M
=
E, - E°
4h
=
=
e 2 Qq
Afo=.+
- o_ =
13)
)
CO~A
for
(3.35)
f
(3.36)
{
for
3e 2 q 2
[
)
3e2Qq
E( - Eo/h
.4
3e2Qq .
(3.37)
O
Figure 3.8 Spectrum of Energy Transitions for I = 1.
The spectral amplitudes, Am, are related to the RF Hamiltonian (HRF) and the wave
functions ( T) of the energy states of NQR and are given in equation (3.38). The relative
values of the amplitudes depend on the power and the direction of the RF field. For
example, for nuclear spin I=1, A ,= 0 & A-,=max if the RF is in the X direction, A.,=max
& A_--O if the RF is in the Y direction in the LAS, and A,=A, for powder sample due to
isotopic distribution in orientation of spins[49],
Am
HRF,.
)2
(3.38)
2
AA = A_,
m=J A:
oc
(Pl IHRF
for
(3.39)
I=1.
o)
For a nuclear spin 3/2, there are two different transitions between the 9 3/2 and p1 /2
and between the _-3/2 and , 2. Since the energy levels of the E3/2 and E_3/2 and of E1/2 and
are identical[49][62], as given,
E_1 2 are the same, the observed energy transitions
W 3
-)Q
+
(E+3/ 2 -E+ 112 )/h
2
3
2
CP-3/2
S(E-3/ 2
- E-1 /2 )h
E+-3/2 =
e 2Qq
72
2h
3
3
2
(W-+
E+-1/2 = -(1+
(3.40)
2
e Qq/4
,(1+rI/3)
1/-3/2
CP-1/2
for I=-.
3)
3
/2
e2Qq/4
(b)
Figure 3.9 Energy Transitions for I = 3/2
(a) Energy transition between E-3/ 2 and E-1/2 (b) Energy transition between E3/2 and E-/2
No energy transition between E-3/2 and E+/2 and between E+3/2 and E-1/2
No observed transition between E3/2
+1/2
-1/2 and E+1/2
+3/2 and between E1/2
-2 and E+3/2
Chapter 4
Spin Dynamics of NQR
4.1
Definition and Transformation of Reference Systems
In order to explore spin dynamics on pure NQR where the RF field breaks the
symmetry of a sample, transitions between energy levels may be introduced by application
of an RF field with a frequency, co, to satisfy the Bohr condition.
The interaction
Hamiltonian[20], HRF, and the allowable transitions are,
B1 .i=-
HRF ~-
h
(B
x +B y.I .+Bz .iz)
*4
(4.1)
s Imm
(mIm) = m'=
(mi
x
+im)=
I(I+1)-m(m
(4.2)
l)m(m l)
Because spin dynamics in pure NQR is complicated due to the powder distribution,
one needs to relate the applied RF field with arbitrary distributed nuclear spins to three
different coordinate frames. They are the Laboratory Axis System, the Principle Axis
System and the Rotation Axis System, refereed to as LAS, PAS and RAS respectively.
z.
)z
(a)
LAS
(b)
PAS
(c)
RAS
Figure 4.1 Three Axis Systems in Pure NQR
4.1.1 Laboratory Axis System (X,Y,Z): LAS
The laboratory axis system, LAS, is defined based on the applied RF field and
gravity, that is, the X axis in the LAS is the direction of an applied RF field and the Z axis
is the direction of the gravity. If another RF field is applied in direction perpendicular to
the X axis, the direction of the Y axis can be used in this direction.
The RF field, B1,Lab(t), is defined in the laboratory frame and oscillates at a
frequency, o), with a phase, (p, along the X axis. Following the normal method the RF
field can be decomposed into two rotating fields: B1,LabL(t) rotates counter clockwise and
BI,LabR(t)
rotates clockwise about the z axis:
B,Lab(t) = 2B1 cos(ct +9p)i = Bl,bL(t) + Bl,LbR(t)
(4.3)
BLabL(t) = B,[cos(wt + (p)x + sin(ct +
(4.3a)
B1,LabR(t) = B,[cos(ot + (p)x - sin(wt + 9p)
^]
4.1.2 Principle Axis System (x,y,z): PAS
To simply describe the EFG, five independent gradient tensors in the LAS may be
reduced to three orthogonal dependent EFG tensors, Vxx, Vyy and Vzz[20],
V 2 V = V, + Vyy + VZ = 0,
Vxx-
Vyy <-VI
l.
(4.4)
(4.4a)
A convenient way to choose any two dependent EFG tensors is to select the z axis
of the PAS along the direction of maximum field gradient, Vzz, and the x axis of the PAS
along the direction of minimum field gradient, Vxx. The nuclear spin, Iz, is defined along
the z direction of the PAS. Since each atom in a powder sample has its own PAS, the
direction of each nuclear spin, Iz, is distributed equally in all directions. In other words,
the spin system is quantized in the EFG. When the EFG is symmetric about the z axis,
r=0 (Vxx=Vyy), only the z axis is defined. So in this particular case, the x may be defined
as forming a plane with the z axis, which contains the X axis of the LAS.
4.1.3 Rotation Axis System or Rotation Frame (x',y',z'): RAS
As the standard NMR, it is very convenient to refer the motion of spins to a
coordinate system rotating at its natural frequency. In this rotation frame, the z' axis is
defined in the same manner as the z axis in the PAS and the x' axis is in the direction of the
DC term of the scaled RF field transformed from the LAS to the PAS. For simplification
by assuming that the LAS and PAS are identical, the RF field, B ,LAS , can be transformed
to a rotating frame in equation (4.5) as described in Appendix A and Appendix B. The RF
field, B1,RAS, in the RAS can be represented by two components that Bi,RASL is constant in
time domain and B 1,RASR is time dependent. In NMR, the latter can be neglected because
the Larmor frequency of spins is at +ol and the effect of the 2o component is averaged
out. However, it becomes time independent in NQR as discussed by the fictitious spin.
(B
B,s = R I(ot)Bl,s =
,RASLAS
BI,RASL
B,RASR
= B, sin p
i 0
Bx, = B,[cos(p) + cos(2ot + )]
B,,, = B,[sin() - sin(2ct + 9)]
(4.5))
(4.5a)
B [COS(P)x' + sin(p)' ](4.5b)
B,RASL
R - B,LASL
=
Bl,RASR
R-'Bl,LASR
= Bl [cos(2
ot
+ 9)^' - sin(2t + 9)'
4.1.4 Transformation Between the LAS and the PAS
Spin dynamics in NQR are related to the EFG and an applied RF field. In general,
the RF field, B],LAS , has two components, B ,LASX and BI,LASY, described by an angle 3 in
the LAS. The angle between the B1,LAS and the EFG is represented by 0, as shown in
figure 4.2(a). An angle, cx, is introduced for r77~0 in figure 4.2(b).
Oz
IzEFC
x
B
,
B 1 LAS
1,LASX
X--(b)
(a)
Figure 4.2 Definition of the LAS and PAS and the Relationship
(a) Two physical vectors: Electric Field Gradient (EFG) and RF Field (Bl,LAS)
(b) Relation between the PAS (x,y,z) and the LAS (X,Y,Z).
In order to obtain the mathematical relationship that reflects the definitions and
relationships in figure 4.2, a series of transformations between the LAS and the PAS is
written as,
PAS = RTot, -LAS
LAS
=
R~t,,.PAS
(4.6)
Roa = R ()R (0) R
RTotal=
R(
()
)
cos
-sin
Rz(T
)
R-
sin 0
1
0
(4.7)
(4.7a)
0 cos 0)
cosp -sing
= sin T cosT
0
cos
2
3)
)R(-
,
0
0
R(0) =
)(--)
O
0
0
(4.7b)
1
SBILASX
=
Bsx + B
1,LAS
sin
(4.8)
,LSY
-
4.1.5 Transformation Between the PAS and the RAS
In the RAS, an RF field becomes static in the x'y' plane for the applications of the
Bloch Equation. Since it has the same z axis as the PAS, only the x'y' plane is rotating at w
frequency about the z axis in the counter-clockwise direction. Thus, we have,
RAS
PAS
4.2
= R7''(ot). PAS
= Rz(ct) RAS
(49)
Scos
-sin 0'
ct
Ct
R, (ot) = sin ot
0
coscot
0.
0
1)
(4.10)
RF Field Representation in Various Axis Systems
4.2.1 Definition of the RF Field in the LAS
An applied RF field, B ,LAS, in the LAS is represented by the Bl,LASX and Bl,LASY'
B,LAS(t)= B,LASXX +B,LAsy Y= B1 xy(cos#
BASX
BI,LASy
= B1 cos(t+)
= B,, cos(Ct + )
for co = c
sin
0)
(4.11)
(4.11 a)
Bjxy = /
+
±sx
B ,s.
(4.11 b)
1
4.2.2 Transformation of RF from the LAS to the PAS
The RF field, BiPAS, in the PAS is transformed from the BILAS by a series of
transformation as described in Appendix E,
(cos a sin0)
B1
BlPAS =R
=R (a)R(O)R(,~
(a)R(6)R(
IRz(-)B,as = B xvI sin a sin 0
(4.12)
)
cos
sin 0
,PAS
BI,PAS
=
R, (a)z
(4.12a)
0
Bxy
y
(0)
R, 2 (2- Rz(- )BlAS=
cos &J
Those results clearly show that the RF field, B ,PAS, in the PAS does not depend on
the 3 but only depends upon the 0. If ir=0, the y term in equation (4.12a) becomes zero.
4.2.3 Transformation of RF from the PAS to the RAS
In NMR, the RF field, B,,PAS(t), in the RAS is decomposed into a counterclockwise field, B,,PASL(t), and a clockwise field, B1,PASR(t), in the transverse plane, as
given (by letting B,Asx = Bl,ASY = ,2B, cos(ot + 9)),
'cos a si
B,PAS
=2B1 cos(t±+9) sin asi
cos
2
Bi,PASL
+
B1,PASR
(cos(t +
BI,PASL -
B1,PASxL
BI,PASyL
=
BI,PASxR
B,PASyR =
+ a)
B,sin
a)
-sin(wt +9 - a)
0
B1,PASz
(4.13)
B, sin 0 sin(t + q+ a)
cos(cot +p-
BI,PASR =
BI,PASz
2B, cos0
0
cos(wct + 9)
(4.13a)
Bl,PASxL
=
B1,PASxR
=
B1,PASyL
=
B1,PASyR
=
B, sin Ocosa(cos(ot + 9)
sin(cot + (9)
B1 sine cosa(cos(cot + 9) -sin(t + (9)
B sin sina(-sin (c t + q) cos(ct + 9)
B, sin 8 sin a(sin(cot + 9)
0)
0)
(4.13b)
cos(Wt + 9)
The two RF fields, Bl,PASL(t) and BI,PASR(t), are transformed into the B1,RASL and
and -oQ based on the Fictitious Spin.
B1,RASR in the RAS with respect to +0Qo
They
become time independent as derived in Appendix F,
where the B
B1,RASL
SRz1 (WQ)Bl,PASL
B1, RASR
SRzI(-Q)Bl,PASR
B1,RASz
Rzl( ±Q)Bl,PASz
,PASz
(4.14)
with an angular frequency,
Q,, is neglected since its effect is averaged
out as CoQ>>yB . Since the two components, Bl,RASL and Bl,RASR, in equation (4.14) are
identical, they have the same effects on a spin system as indicated by the Fictitious Spin.
Therefore, the total RF power except for the z component is used in NQR. The RF field,
BI,RAS , in the RAS is given as,
+ B,RASR
Bi,RA
B,RAS =2B,sin O(cos9
sin 9
B,RA S =B,RASL
9)
9)>
= 2B,
0)
for
7/=0.
(4.15)
(4.16)
From the above analysis, a time dependent RF field, B ,LAS, applied in the LAS is
transformed into a time independent field, B ,RAS, in the RAS, which could be easily used
to calculate the spin motions by the Bloch equation.
4.2.4 Total Hamiltonian of the NQR and the RF Field
To describe the physics of spin dynamics by Quantum mechanics, in the PAS for
ri=0, the total Hamiltonian of the pure NQR and an RF field is defined as,
HPAS = HQ
(4.17)
+ H1,PAS
3
/=
( -e
A(3II_ 2)
-,
HII,As
4.3
2
Qq
for 7"=0 & A= 4i(2
1-
)
2e
2
Qq18)
128)
Bl,PAS = -2fyhB cos(ot + q)(sin Oi + cos OZ).
(4.19)
Fictitious Spin Analysis
In NQR, since the spin is greater than 1/2, the spin dynamics become complicated.
To understand the difference between the NMR and NQR, a simple classical method,
fictitious spin, is introduced. Two significant results, the oscillation of NQR signal and the
full coupling between an RF field and NQR energy transitions, are obtained.
4.3.1 Energy Transition Mechanism
Two energy levels with IAm = 1 for Iz=m are coupled with an RF field, thus, the
various transitions could be induced[66]. For example, the transitions between the 10> and
I-1> and between the 1+1> and 10> for I=1, and the transitions between 1-1/2> and 1-3/2>
and between 1+3/2> and 1+1/2> for 1=3/2 can be observed, as shown in figure 3.9.
The energy levels with an axially symmetric quadrupole interaction are given by the
eigenvalues of the NQR Hamiltonion[67][66]:
HQ=Em
Em
e2 q ((312
41(21 - 1)
^2
e2
((Pm.
m)=
eq
41(2I - 1)
E+ =
Eo
E m =_
k+3/2
=
(4.20)
_I)
[3m2 -I(I + 1)]
for
Iz = m
(4.21)
e2Qq/4
2e 2 Qq/4
for I= 1
(4.22a)
+e2Qq/4
for I= -
(4.22b)
+e2Qq/4
2
The observed transition frequency between adjacent energy levels, m #>(m - 1) for
Iz=m, such as between the 1+3/2> and 1+1/2> and between the 1-1/2> and 1-3/2> for 1=3/2,
are given as[66][67]:
Em -
m
m
a
-
h
-
3e 2 q
(4.23)
3eq (2m - 1)
4I(21 - 1)h
W+3/2 =(E+
3 / 2 - E+
11 2)
)1h
1)-1/2 = (E-1/2 - E-3/2)/
for
2
=
2
- e 2Qq =
I
for I=
2
-C
(4.23a)
2
(4.23a)
where oo_- e 2Qq/2h for 1=3/2, and the transition frequencies are +oQ and -oQ with
respect to the transitions between the 1+3/2> and 1+1/2> and between the 1-1/2> and 1-3/2>.
This means that the spin in NQR is rotating in the transverse plane in both clockwise and
counter-clockwise directions.
4.3.2 Fictitious Spin
The transition between any adjacent energy levels, m and m-1, can be described by
+(oQ between the +Iml and +ImI-1 for m20 and by -co between the -Iml-1 and -Iml for m0O
because of m = -I, -I+1, , ,I-1, I for spin I.
Only two pairs for ±m are treated since selective pulses are used in NQR. Any
adjacent degenerate pairs can be written as +m= C +1/2 and ±(m-1)= ± C T 1/2 for the
same C and m_0[67]. Figure 4.3 presents an coupled adjacent degenerate pair, ±m and
±(m-1). Then those pairs can be treated as two pairs of 1/2 states in figure 4.4. So a spin
I is treated as 1/2 spin system, namely fictitious spin.
3
±
1
2 2. Sm=-3
1'
2
- +- +1+!
+m=:C
(m -1)
=
1
1
2
+CT--1
2
2
=
2
++
2 2.
2
2
m=l
(4.24)
m
-m
E- m
Em
+CQ
Coupling
Em-1
Q
-
V
m-1
E_(m-l)
-(m-l)
(b) Coupling between -m & -(m-l)
(a) Coupling between m & m-1
Figure 4.3 Coupling between two adjacent degenerate pairs of energy levels:
Em <---> Em,1 and Em <---> E-(m-_l)
-m
m
+rm
E+ m
(a)
E-m
+
Em
+CO Q
OQ
+ (m-1)
(m-l)
4
E+ --(m-l)
Em-1
m(m-l)
(C)
(b)
t 1/2
SE+
C-o Q !
m
-1/2
E+ (m-)
(d)
+1/2
A
Em
E-(m-(1)
4
-1/2
E-m
-Co
+-
+COQ
E-(m-1)
-COQ
Q
+1/2
-1/2
Figure 4.4 Fictitious Spin Representation and Transition Frequencies.
(a) and (d) represent two types of coupling between two adjacent energy levels.
(b) and (c) represent the fictitious spin between two adjacent energy levels.
4.2.3 Full Coupling between RF Field and Energy Transitions in NQR
An RF field, B ,Lab(t), oscillated at a frequency, 0 Q in the X direction may be
decomposed into two rotating fields, Bl,LabL(t) and Bl,Lab(t), rotating counter clockwise and
clockwise about the Z axis:
B,,ab(t) =
2B, cos(coQt + p()X =
BlbbL(t) = B [cos(OQt +
l,LabR(t) =
BLbL(t)-
B,LabR (t)
q)X + sin(ot + 9)]
(4.25)
(4.25a)
Bl[cos(Qt + P)X - sin(cQt + p])Y]
The Bl,LabL(t) and Bl,Lab(t) are transformed into the RAS by fictitious spin. For the
transition between the +1/2 and -1/2 at +03Q, Bl,abL(t) becomes time constant Bl,RotL,
Bl,RotL
B(+01,LabL
1
-cos OQt
sin O)t
0r cos(Ot+ (PL)
=B -sinpQt
coswQt
0
0
0
COS
(4.26)
sin(wt+qL) =B, sinTL
1).
0
1 0
where the RF phase, (p, is defined as p,,Lwhich in the counter clockwise direction is
positive in the LAS.
For the transition between the -1/2 and +1/2 at -oWQ, B1,LabR(t)
becomes B ,RotR, which is also time independent,
BIRotR = R
- wQ)BlLabR
Qt
(cos
= B sin wt
-sin owt
coscoet
O
0
0' cos(coQt - PR)
COs PR)
00 -sin(,t - gpR) = B, sin pR
1
O
(4.26a)
0
where the RF phase, p, is defined as (PR' which in the clockwise direction is negative in the
LAS. The results in the RAS indicates that there is no directional difference. Therefore,
the RF phases, qPL and (pR, are the same and can be represented by (p in equation (4.26b),
and the
B1,RotL
and Bl,R,tR interact with spins in the same way,
B,Rot =- B,RotL
+
B,RotR = 2B(cosT
sin p
0).
(4.26b)
In NQR, the analysis by the fictitious spin indicate that both RF components,
B1,LabL and B1,LabR, rotating in opposite directions at (,ocause the transitions of the nuclear
spin between the 1+1/2> and IT 1/2> states at the frequencies ±oQ. This means that the
full RF power is completely used for spin transitions. However, in NMR, the power of
clockwise RF energy is wasted[66].
4.3.4 Oscillation of NQR Signal
The spin dynamics can be described by the Bloch equation (4.27) or by the
Exponential Operator (4.28) in the RAS (Appendix C),
It
(4.28)
dt
H--RAL
1,RASL
RASL'FB
=
0BI4
[cos (P + sin (P,
-
jSB[csPI
(4.28a)
+in rPi.
-Isin p sin(yBt )
IX
Icos sin(yBt,)
lPM
(4.29)
,
Icos(Atj)
where tw is the RF pulse length and (pis the phase. By varying the t, to make the nutation
angle
the Iz is equal to zero and the spin is in the transverse plane.
-a=yBztw=it/2,
A spin I with r=0 can be decomposed into s' and s" in equation (4.30)[66] and the
transitions between two adjacent levels can be decomposed into two different transitions as
in figure 4.5: 1+1/2)
and 1-1/2) < 1+1/2) . Thus, the transition frequencies
> -1/2)
(co) are the same but the direction (sign) is opposite.
13/2>
1-3/2>
Co'
s coupling
P&.no
_V
11/2>
coup
ing
co
ing
cO"
1-1/2>
(b)
(a)
1-1/2>
1+1/2>
' coupling
coupling
''
CO
2I
cou ling
1-1/2>
(c)
s
Co"
'
coupling
1+1/2>
(d)
Figure 4.5 Transition for nuclear spin 1=3/2.
(a) and (b) represent two types of coupling between two adjacent energy levels.
(c) and (d) represent the fictitious spin between two adjacent energy levels.
There is no coupling between (a) and (b) and between (c) and (d).
(4.30)
i =s +s
s =sx+sy+sz
' = S'x ++
S(E+3/2>
.30a)
Sz(4
s = S'x+S v+S
z
-E+1/2)
h
=
2e2 q/h
= -2e 2Qq/h
(4.31)
SO)Q
The two decomposed RF components, B1,RASL , and B1,RASR , interact with the s' and
s" separately without any loss of RF power since the two spin components, s' and s",
with respect to the two transitions have different natural frequencies, corresponding to +OQ
and -oQ. The interactions between the RF fields and spins in the RAS result from equation
(4.29), as shown in figures 4.6(a) and 4.7(a). Since the right-hand rule is used in equation
(4.32), the signs are different in the LAS. The solutions are,
ds/olt = +'x
d~ /la = -"
(4.32)
,,AS
x
1,RASR
-s'sin ()sin(Bt,)
Sx
(4.33a)
, =Y s'cos(p)sin(yBt,)
sRA=
. s'
cos(yt.)
s"sin( p)sin(yB t,)
,s= S, = -s" cos()sin(Bt,)
S
x
z)
.
(4.33b)
t scos(y t)
The spin motion in the LAS is given in equations (4.36) by reference to
transformation (Appendix D). The two fictitious spins are nutated into ±Y' directions in
the RAS by a n/2 RF pulse in the X direction with pq=0o in figure 4.6(a). The two spins in
the transverse plane are rotating in opposite direction at the frequency oQ and form an
oscillated spin in the X direction in the LAS at the frequency oQ starting from zero to minus
maximum, as shown in figure 4.6(b). When (p= 90', since the B1,RASL , and B1,RASR are
along the Y' axis in the RAS, the s' and s" are nutated into the opposite direction and are in
the different X' directions for a r/2 pulse in figure 4.7(a). During the free precession of
those spins, a spin is formed and is oscillating along the Y direction in the LAS from zero
to minus maximum, as shown in figure 4.7(b).
The excited spin in the LAS is oscillating at a frequency GQ in the X or Y direction
for an RF field along the X axis with respect to the 00 or 900 of its phase. This is one of
the differences with NMR.
(4.34)
Is = RIRAS
SAs
Rz (
LAS=
R(_)")s
)'RAS
RAS
(4.35)
S(-I cos(p) sin(,yBt.)sin(wtot)
s
,LA =
= -Isin((p)sin(yBltw)sin(Qt)
+
Icos(yB tw)
St=O
s/"
9
s-.
s'
S
"t=t w
-s
+mWt
'-t
/
t=tw
(4.36)
s'
s"
t=O Y
t=O
Y
B1,RASR
X
B1,RASL
(a) 900 Nutation with 0ORF Phase
(b) Free Precessions in the LAS
Figure 4.6 Spin s' and s" Dynamics for a RF Phase (p= 00
Spin s' and s" are nutated into the + &- Y' axis in the RAS by an RF field in the X
direction of the LAS with a zero degree of RF Phase, and rotating in counter clockwise and
clockwise directions in the transverse plane of the LAS at angular frequency 0Q resulting in
an oscillating sinusoidal signal along the X direction due to a cancellation in the Y direction.
Z' t=0
AStw
/'/s"I
s'
+m t
t 0
B1,RASL
B I,RASR
t=tw
Y
s
(a) 900 Nutation with 90o RF Phase
',
s"--- tt--0
(b) Free Precessions in the LAS
Figure 4.7 Spin s' and s" Dynamics for a RF Phase (p= 90'
Spin s' & s" are nutated into the + &- X' axis in the RAS by an RF field in the X direction
of the LAS with a ninety degree of RF Phase, and rotating in counter clockwise and
clockwise directions in the transverse plane of the LAS at angular frequency (OQ resulting in
an oscillating sinusoidal signal along the Y direction due to a cancellation in the X direction.
4.4
Classical Analysis of Spin Nutation
4.4.1 Spin Dynamics
The simplest method for spin dynamics is a classical approach based on the Bloch
Equation. The calculation is made only from the counter clockwise part in equation (4.37),
where 0 is the angle between the EFG and an RF field, a is an angle between the x axis
and B 1 projection on the xy plane, (pis the RF phase and the magnitude of the RF field is
reduced to half based on equation (4.15),
di/dt =
i=
(4.37)
x B,RAS
++
I
z
(4.37a)
& I(0) = I
B1,RA = B, sin O(cos(a + T)
sin(a + p)
(4.37b)
0).
The nutated spins in the RAS calculated in Appendix C are,
TIsin(a + ) sin(yBt,w sin 8)'
±Icos(a+ T)sin(yB 1t, sin e)
I cos(Bt, sin e)
I,
ls=
(4.38)
where Ix and I, are either - or + with respect to the +0 Qor -oQ and tw is the pulse length.
The spin motion in the PAS is given in equation (4.39) based on the fictitious spin
as described in Appendix D. If qp=O and 1=0, the result in equation (4.39a) is the same as
A. Abragam's equation (62) and T.P. Das & E.L. Hahn's equation (4.15).
-Icos(a + p)sin(yBl sin(O)tw)sin(wat)'
=
PAS
(4.39)
-I sin(a + 'p)sin(yB sin(O)tw) sin(wot)
I cos(yB, sin(O)t,)
sin(O)tw)sin( ot)
0
IPAS
I
for
0
(4.39a)
0
For the signal detection, the spin motion is transformed from the PAS to the LAS,
LAS
)R ' 121(R
lR
R', ITal
"PAS
= R-(-f)R;
(x
IYi
Isin(yB1
(4.40)
O)
' sin psin p - cos Pcos sin00
sin Otw) sin(ot) -cos /sinq -sin cosy sin O ,
(4.41)
)
cos Tcos
L
Iz
where the
.
'()R (i)x
P is determined by the RF transmitting coil(s) in equation (4.8).
As can be seen in equation (4.41), the a vanishes since the spin depends only upon
the EFG and B 1. The spin components, I x , Iy and Iz , in the LAS will contribute to the
signal, which depends upon the way of detection, as given,
LAs
for
= sin(yB1 sin Ot)sin(ct) -sin /siI
(9=00
cos
(4.42)
Ssin/3
= Isin(yB, sinOtw)sin(coQt) -cos P
for
9 = 900
0 )
When the transmitting coil in the LAS is only in the X direction, P=0o, this result
depends upon the (p. Thus, when qp=O0 , the signal is in the X and Z directions, but when
(p=n/2 the signal is only in the Y direction,
ILAS
= Isin(B1 sin t,w) sin(o t) -sin'
0
cos )
for
0
(4.42a)
= Isin( B1 sin Ot)sin(o)t) -1l
for
=
[9 = 90°
\0
When the transmitting coil in the LAS is only in the Y direction, P=90', the Y and Z
directions have signal for qp=0 but only the X direction has signal for qp=t/2,
for
I4s = Isin(AB sin Ot)sin(wQt) -sin 0
P = 900
cos0 )
(4.42b)
= Isin(yB, sin Otw)sin(ow,t) 0
90
= 90
for
(p = 900
4.4.2 Spin Nutation
From the previous calculation, both the nutation frequency, on, and nutation angle,
Wn, of the spins depend upon the angle, 0, defined by the EFG and an RF field,
yB1 sin 8
C0,sin 0
yB, sin(O)t,,
0, sin(0)t,
(4.43)
The intensity, I(o) in equation (4.45), of the spectrum in figure 4.8 could be
normalized becoming a Lorentzian function of the (On, and results in a broadening of the
point spread function. By integrating the I(O) within a small angle about 0=ir/2, a sharper
PSF is obtained (as discussed in chapter 5), as shown in figure 4.9[21].
I
= 1
f I(o,)d,
-(Onw,
1
0
ir
2r
(4.44)
dp sin 8d
(4.45)
/)')
I(an)
1
I(n
)
0.8
8
0.6
0 6
0.4
0.4
0.2
0 2
04
02
06
08
Figure 4.8 Spectral Intensity
of Powder Sample
with rich low frequency.
02
0.4
06
08
Figure 4.9 Spin Selection
By integrating for
a sharper PSF.
4.4.3 Signal Detection - FID
For a general situation, the signals in the X, Y and Z directions may be calculated
separately by integrating the whole sample in a spherical coordinate in the LAS. If defining
three functions, Iso, IcO and Ie, and calculating the integral over the solid angle,
dM = sin OdadO, in the spherical coordinate, we obtain Ms0 , Mco and M,, nutation
functions by neglecting Mo=2rtI in equation (4.46)'s,
Is(,(, t.)
=
Isin 0 sin(yB1 sin Ot)
Ice(0, tw)
=
Io(0,tw)
=
Icos0sin(yB1 sin Ot )
I sin(yB sin Otw)
Mso(t.) =
Iso (0, tw)Sin OdOda =
Jsin2
(4.46)
sin(y 1 sin Ot)dO = 3.15 J(t,)
(4.46a)
0
Mc (t.) =
Ic (O,t)sin dda = sincossin
(4.46b)
sin(B 1 sinOtw)dO = 0
0
M,(t.) = f I,(, tw)sinOdOda = jsin sin(yB1 sinOtw)dO = 3.15J(t),
(4.46c)
0
where J1(tw) is a first order and first kind Bessel function of the pulse length, t,.
3.15J (t )
Ms o(t w)
f
(a)
M400)
1ts
VLs
is clos100e to00
(t00
200
600
800
1000
1200
1400
(b) Mce(tw) is equal to zero
(a) Mso(t ) is close to JI(t,,)
3.15 Ji(t )
-M o(tw)
400
15
15
e(tw)
M s e(t w)
_
/
/Mc(t
)
5
-101
(c) Mie(tw) is same as JI(tw)
(d) Comparison of the Functions
Figure 4.10 Nutation Functions: M,,(tw), McO(tw) and M10(tw) vs. RF Pulse Length, tw.
The nutation functions, Mso, Mco and Mi0, have different characteristics, that is, the
Mso and Mi0 are approximate to a first order and first kind Bessel functions of the tw, but
the Mce is zero all the time for any tw, as shown in figure 4.10.
Therefore, the magnetization of the powder sample in pure NQR is represented in
the LAS by equation (4.47),
M,,sin / sin
Mx
S=
M
=in
-Mse cos cos
t-M,,cos sinTp
M
-
Msosinf cosT .
(4.47)
0
Since the magnetization, either Mx or My, oscillates in one direction, the detectable signal
depends upon the direction and the phase of the applied RF field.
4.5
Density Matrix Analysis of Spin Dynamics
The classical analysis provides a simple way to calculate the spin dynamics based
on the Bloch equation to see the spin behavior. But to understand the physics and
mechanics of the interactions between an RF field and spins, we may go through the
procedures of quantum computation, that is, density matrix or/and wave function.
4.5.1 Theory of Density Matrix
The spin dynamics could be described by using a density matrix p based on the
time-dependent Schrodinger equations for 3/2 spin system in both the presence and absence
of the RF pulses[49]. If the NQR Hamiltonian is HQ and the additional RF perturbation is
HI,PAs in the PAS described in 4.2.3 and 4.2.4 sections, then if a transition matrix R and a
free-precession matrix D represent the transformation matrices in the presence and absence
of RF pulses respectively, the Schrodinger equations under those two conditions will be,
ih dR/dt
=
(I
ih dD/dt =
S
HI,PAS
+I,PAS)R
(4.48)
HQD
=
A(3
=
-AI - B,PASL
2)
B,PASL = B, sin O(cos(wt + T + a)
(4.48a)
sin(cot + p + a)
0),
(4.48b)
where A=e 2Qq/12 for spin 1=3/2 and the RF field B1,PASL is the part of equation (4.13a).
Therefore, the RF Hamiltonian could be rearranged to simplify the latter calculation,
H, PAS=
BPASL
,P
= -h[
+
cos(at + p + a) lI
,
on
+e-(wt+p+a)
+
kw2 1^"
sin(ot + p + a)].
(4.48c)
jei(wt+q+a)]
In order to solve the transition matrix R, the equation (4.48) becomes,
ihdR*/dt = H
ihdD/dt =
(4.49)
*PASR
HD
R= Qt
h R
R=e
(4.49a)
HtQt
tQt
h H1,PASe
HI,PAS =
(4.49b)
e
-
2
where eif Qt/ h I++e - rQt/h = eeI i( l z T1/2)owQt/h
=e
-
J
,(
1
e
+ et
+1
+o+T )e -icovt(Iz 2
. By solving equation (4.49), we have,
,PASdt
0
(4.49c)
-1-h
o
If n RF pulses are applied, the net transformation matrix S will be given as, where
the suffixes refer to the successive pulses,
S = D,RnDn_-1 _I ... D 2 R 2 D1R1 .
(4.50)
Following the passage of the pulses the density matrix, p(t), will be related to the
initial density matrix, p(0),
p(t) =S-p(o) -S-'
k T
T[e /k']
(4.51)
k
21+1
where prior to applying the RF field the spin system was in thermal equilibrium, however,
in reality the high temperature approximation is used since the absolute temperature T is
greater than 1K in NQR, and k is the Boltzmann constant.
In general, where the RF pulse and the detection use the same coil(s), by using
Tr[x,y,z] = 0, the expectation values of Ix , I and Iz are calculated as,
(Ix,,Z)
T,[(t)Ix,,z=
T, S "p(O) -S-1x,V,z]
S
TS - I Q S
kT(2I + 1)
1
xz]
.
(4.52)
3/2
=-
(mS-H- S 'x,,z
kT(21 + 1)m=- 3/2
m)
4.5.2 Calculation by the Density Matrix
Since calculation of the expectation value is very complicated, some procedures are
described here only for one pulse experiment. The key part of the expectation value is to
compute the (mS -HIQ S- Ix,y,zm) term,
Iq
1 3/2
= kT(mISkT(21 + 1) m=-3/2
HIQ S-Iq m),
(4.53)
where q is x, y or z respectively. By inserting Y Im)(m = I, the calculation becomes
doable, where since Et/h we assume
etiEnt/h =
1 in equation (4.53c) by neglecting the HQ
effect during the RF pulse. The calculation is,
(mlS -H -S-' Iql m) = (m DR -HQ R+D+ qIm)
1 1X (mIDIml)(ml R m2 )
=
m 1 m2
x(m
2
m3
"4
HQ m3)(m3
R+ m4)(M4 D+Iq| m )
(4.53a)
E,nl t
(mIDim,)
=
HQIm 3 )
(m2
e
6 mm1
(4.53b)
EmGm2
=
Ern4 t
=
(mlD
4 D+Im)
I
e
( i m)
(m
4
Iqm)
ot
Et
im)=
o
k
t
k
oo
ASdt
(niR*lm)= (nle
(4.53c)
h (n|R*lm) h= (nlR*lm)
h R*|m)= e
(IR+n)= (nle
(mR'|n)
<<
(n
m).
H PAsdt
0
k=0
(4.53d)
For the calculation of equation (4.53d), it is expanded by exponential expansion, then
calculated by different order k, for example k= 1,
(
ti H
(n
A
PASdt
(0
k
k=1
(nlI+m)f0 e-i(t++a)e
ho)n
)
21dt
2dt
(4.53e)
+
t ,ioQttn+l)
2
0
Since different n gives different (nII1+m), the RF frequency, o, is equal to -oQ and +CoQ
with respect to the couplings between the 1-3/2) and 1-1/2) and between the 1+3/2) and
+ 1/2), and the observed transitions only occur between the two adjacent energy levels
with Vim| = 1 and energy difference,
H/2,PASdt m) =
-1/2 o
hO /-3/2
2
-1/2
==
K{+3/2
+1/2
2
2
ti
h(on
Hi"ASdtlm) = -A
f+P/2
0
_
hct twe
+3/2
2
dt
t(+a)
-1/2,m
I
{ -3/
2 \+1/2
S
t)e,(+ae
2,m
i et(a+9+aeiOQetd
dt
)e
m) e:F
I+I/2
twe
0
+1/ 2,m
2+3 / 2,m
(4.53f)
Recursively, after the calculation for different k and by reusing the exponential series, the
result for equation (4.53d) and (4.53c) is given as:
+ RI m) = cos-l
3,m
+isin -
ne
(+a)
2
2*m
co
2
2
(4.53g)
2
The final expectation values of the spin system for one RF pulse are the ensemble
signal of the spin system in the PAS,
I
=
Mo cos(a + 9p)sin( -yB,sin(O)t,)sin(coQt)
M sin(a + T)sin yB, sin(O)t,) sin(ot)j
(4.54)
This result is the same as the classical calculations as in equation (4.39) except for
some constant differences.
4.5.3 Significant Points from Density Matrix Analysis
According the above calculation of the density matrix, some significant facts can be
presented:
1 The transition frequency is positive or negative oQ with respect to energy states
between the 1+3/2) and 1+1/2) and between the 1-3/2) and 1-1/2), which results
in that the full RF power is used and the signal oscillates in the transverse plane.
2 Transitions can be observed only between the two adjacent energy levels with
Vlml = 1 and energy difference.
3 The effect from the transition frequency during the RF pulse can be neglected by
assuming that the RF pulse length is short enough.
The above conclusions from the density matrix analysis are the same as the classical
fictitious spin analysis and the Quantum wave function analysis in next section.
4.6
Wave Function Analysis of Spin Dynamics
The classical and density matrix analysis give almost the same results except some
constants. However, the density matrix provides more inside information about the
interaction between an RF field and the spin system in pure NQR. The wave function is
another quantum approach to describe the physics and mechanics between the interaction
and provide some significant results.
4.6.1 Theory of Wave Function
Based on the Schrodinger equation in the interaction picture[1][68], the total
Hamiltonian in equation (4.17) can be transformed into the RAS by defining,
HQt
ih
(4.55)
h lRAs
VPAS -e
dt
ih
d
Hkt
e
= HQe
Ht
fRAs + ihe
iQt dRA
(4.55a)
fiQt
HPAS
e=
ih
HPAs
ih A
HQt
S
h
dHs
ih dAS
=-HI,PASe
=e h Hl,PAse
=
HRS
=e
h
RAS
(4.56a)
HQt
Qt
h VRAS
(4.56b)
(4.57)
RAs
HQIt
H,RAs
RAS
(4.56)
PAS
HQt
ihe
+HlPAS)e
PAs=
HQt
HI,PASe
h
(4.57a)
By applying equation (4.17) and (4.55) to the Schrodinger equation (4.56), we could
obtain the pure NQR Hamiltonian (4.57a) in the RAS, which is "time-independent".
Spin dynamics has been calculated by wave function xy for spin 1=3/2 in terms of
the eigenfunctions of the NQR based on the time-dependent Schrodinger equation
[60][63][64]. The eigenfunctions can be expressed in equation (4.58), and is used to
calculate the expectation value of Ix,y,z at any given time t 2t w,
Ernt
E
+
h
cm(t))me
=
i
.E,n
+32
cm(t) me
t
0 < t<t
'
m=-32
(4.58)
+32
E,n (t-t,
cm(tm)Ome
-
-
)
t>t
h
m=-32
To determine <Ix,y,z>, the coefficients, ci(tw), are calculated by solving the timedependent Schrodinger equation,
dyf
0o<t<t,
H + HIPAS)V
: A
V
dt
HQ
=
A(3I2I
H1,PAS
=
-
0
(4.59)
! t t W
2)
(4.59a)
IB,PASL
(B, sin Ocos(ot + qp + a)
Bl,PASLz = B1,PASL + Bl,PASz
=L Bsin Osin(ot + c + a)
(4.59b)
2B cos 0 cos(ot + 0)
in the interval 0 < t < t, . The RF field, B,PASLz , is the part of equation (4.13a). Using the
orthogonal properties of 4i (i=-3/2,-1/2,1/2,3/2), a set of equations of the coefficients can
be achieved, as derived in Appendix G,
+3/2
(t)
ih dc,
dt
where wnm = (E
in Appendix G.
n
X
Cm (t)e''t Vnm
0O
t < t,
(4.60)
m=-3/2
0
t>t w
- E,)/h = +O)Q & 0 ,and V
= On-,PAS
Hm
m is
matrix element calculated
The derivative equations of the coefficients are indicated in equation
(4.60a) where since the monm is +(o or -m, the BI,PASL is used for the transition between
the 1+3/2> and 1+1/2> with respect to the coefficient c+3/ 2 and c+1/2 and the B1,PASR is for the
transition between the 1-3/2> and 1-1/2> with respect to the coefficient c-3/ 2 and c-1/2,
-
3
,(+a)
2
sinne
2
i -- OI sin e
2
C+I 2
c±3 /2
for
0
tt,
(4.60a)
sin2
sin
-
=
C± / 2
c)
02
for O< t < t .
(4.60b)
2
The solution for the above derivative equations can be solved by assuming the
initial values c+3/ 2 = 0, cl/2 =1//2 and
V2
1
(3
SCos
(-
V2
2
=
C3/2 (tw)
1 /2 (tw
C±1l/2
2
= 1, as shown in Appendix G,
e (+)sin -c, sin(O)t)
2
=
C3/2 (t)
ICm(t)
W.
for
0o t
tw
(4.61)
sin(O)t
Se
T(a)sin -3-i sin(O)t,)
Ssin(
1
-02 sin(6)tw
t 2 tw .
(4.61a)
)=
4.6.2 Expectation Value of Spin System
Because the nuclear magnetization is proportional to the expectation value of a spin
system, the expectation value of spin operators, Ix, Iy and I z, can be calculated by applying
the wave function of the spin system,
3/2
(
3/2
A
Ix
, Y'Z)
=
W
V
)c m (tw)e""'(nlxy,zlm),
'Y'Z
(4.62)
n=-3/2 m=-3/2
where Om = i m ) and 0* = (ml. By neglecting the terms with
,nm=0,
the calculation given
in Appendix G is,
i)I,
(2 / cos(a + P) sin( -3yB,sin(O)t,
2 -- sin(a + )sin(,yB, sin(O)t,
(4.63)
The results are the same as the classical and density matrix results in equation
(4.39) and (4.54) respectively in the PAS, except for some constants. The above result
could be transformed into the LAS in the same manner as in the classical calculation.
4.6.3 Significant Points from Wave Function Analysis
Some significant facts could be obtained by deriving equations from (4.60) to
(4.60a) based on calculating the matrix elements Vnm by equation (G.5). The results in
Table G. 1 in Appendix G show:
1. An RF field applied in the transverse plane in the LAS will perturb all the energy
levels, such as 1-3/2>, 1-1/2>, 1+1/2> and 1+3/2> for 1=3/2, and cause energy
transitions between adjacent energy levels for Vm = I1.
2. A transition could be observed if the adjacent energy levels have different energy
values, that is,
On,,
= (E, - E
)
/ h # 0.
3. The full RF power, both counter-clockwise and clockwise power, is used to
perturb the spin system without power loss since the clockwise RF field causes the
transition between the 1-3/2> and 1-1/2> and the counter-clockwise RF field causes
the transition between the 1+1/2> and 1+3/2> with respect to -coQ and +OQ.
The calculation of expectation values of the spin system also shows that the energy
transition only occurs between adjacent energy levels with Vm = ±1 and the transition
could be observed only if the adjacent energy levels have different energy values. Also the
transition frequency is -oQ or +oQ with respect to the state couplings between the 1+1/2>
and 1+3/2> or between the 1-3/2> and 1-1/2>.
The final result shows that the signal generated by NQR is oscillating in the
transverse plane in the PAS.
The above conclusions from the wave function analysis are almost the same as the
fictitious spin analysis in classical mechanics and exactly the same as the density matrix
analysis.
Chapter 5
Spin Selection of NQR
5.1
Orientation Selection of Spins in NQR
The lack of a global quantization axis in pure NQR results in a spread of interaction
angles (0) between the spins and the applied RF field. Therefore, the effective field which
describes the nutation axis for a given RF field takes on a range of orientations and lengths
in a powdered sample, and this range does not change with the variations in the RF field
strength. In the following sections, the potential of experimentally removing this
heterogeneity will be discussed. In RF imaging the implication is that the image PSF may
be restored to a sharp, well-defined impulse by selecting only those spins with specific
orientations to the RF field.
5.1.1 Orientation Dependence
As described in chapter 4, by defining co = yB,, the nutation frequency (cOn) and
nutation angle (Vn) depend upon the angle (0) between the EFG and the RF field,
0,
=
yB, sin 0
=
o, sin
Y,
=
B, sin(O)t,
=
ol sin(O)t,
(5.1)
The nutation signal from a powder sample in a homogeneous RF field was
calculated in chapter 4, and is well approximate to a first order Bessel function of the first
kind, as shown in figure 5.1.
This orientation dependence of the nutation frequency results in a broadening of the
nutation spectrum, which in an image corresponds to a broadening of the point spread
function (PSF) or a blurring of the image. In pure NQR imaging, where typically each
voxel contains a full powder distribution of the EFG, this powder orientation effect can be
the most important limitation to the spatial resolution.
M,(t,) =
Jsin2
0sin(yB
sin(O)tw)dO = 3.15J,(tw)
(5.2)
0
3.15 Jl(tw Bessel Function Approximation
----
Mxy
1 5
-
--------
M
t
Exact Calculation
1
0.5
t
0
1
-0.
0
40
(us)
0
5
-1
Figure 5.1 Nutation function (Transverse Magnetization vs. RF Pulse Length, tw) indicates
the orientation dependence between the spins and the applied RF magnetic Field.
5.1.2 Available Spins after Orientation Selection
A powder sample consists of a multitude of randomly oriented single crystals. In
the case of axial symmetry (1=0), the spin distribution of the crystals is a constant over a
sphere. Each point on the sphere corresponds to a definite orientation of the applied RF
field (B,) with respect to the PAS. The nutation frequency described in equation (5. la) is a
constant at a given angle (0) and is mapped over the sphere, as shown in figure 5.2.
Since any orientation selection of spins removes those spins which are not in the
desired directions from the detected pool, the signal must decrease. The effect on the signal
to noise ratio can be seen by looking at the intensity, I(0), of a spectrum as a function of
the nutation frequency, o. This was calculated from equation (5.3)[21],
f I(m)d
o
d
= 7
p sin 0d0
o
(5.3a)
o
As the nutation frequency is defined in equation (5.1 a), the relationship between the angle
0 and the frequency will be derived,
sin 6
coso
=
=
coso =
m/c1
0/l-( 1/) 2
1- (0-/0)1
(5.3b)
do)
dO =
1 - (cO/cOI)
(5.3c)
2
1/2 ol
1/2 ol
1/2 ol
Figure 5.2 Stereographic projection of curves o=constant with respect to e=constant
when ri = 0. The nutation frequency varies from 0 to o, while 0 is from 0 to t/2.
By substituting equation (5.3b) and (5.3c) into (5.3a),
(01
01
J
SI(o)d) =
2 do.
(5.3d)
By comparing the both insides of the integral, the spectrum of the entire powder sample is,
I(co) =
o
(5.3)
1(- (O/I0)2
To sharpen the PSF while maintaining as much signal to noise ratio (SNR) as
possible we must potentially select those spins that have their PAS pointed 900 (or close to
90') from the direction of the RF field. The number of selected spins may be calculated by
the integral from 00 to 7t/2 of the spectrum,
fI(o)dw= f
Selected Spins(O) =
co1 sin
), sin 0 0
1
I(cm)
-
(C/
(5.4a)
do.
)
)2
Number of Selected Spins
0.8
0.6
0.4
0.2
o()1
0.2
0.4
0.6
O)1 sino
1
Figure 5.3 Number of Selected Spins vs. the Angle 0o by Spectral Integral.
I(co)
1
0.8
FWHM
0.6
0.4
0.2
n=O
C(o(1o
0.2
0.4
0.6
0.8
1
Figure 5.4 (a) Spectra with respect to all spins and different selections
After spin selection about 0=7t/2 by increasing the n in the function f(O) = sin"(0), both the
linewidth and SNR are decreased.
By selecting the spins around the direction perpendicular to the applied RF field, the
spectral width decreases as the angle, 0 o, increases. The trade off between the resolution
and sensitivity is obvious, but unfortunately we can not implement such a sharp cut off in
spin packets.
The simplest spin selection to implement is to weight the spectrum by a power of
sin(0),
ISelect(cO) =
sin'"0
(5.4b)
/
Equation (5.3) and (5.4b) present the intensity of the spectrum, which are
Lorentzian functions of the nutation frequency, with respect to the entire spins and selected
spins. The spectrum of entire spins has a very broad linewidth and the maximum SNR
while the spectra of selected spins have a narrower linewidth but a lower SNR. Figure
5.4(b) shows how the PSF changes as a function of n.
n=O--,
m 0.8
FW at One Eighth Maximum
00
FW at A Quarter Maximun
FWHM,
0.6
0.4
0.4
0
n=32
, n=1024
SNR
0.6
0.7
0.8
0.9
1
Figure 5.4 (b) Three curves of relative frequency vs. Relative signal to noise ration. The
three vertical lines represent the three selections shown in figure 5.4 (a).
The SNR axis is normalized to the maximum SNR. The vertical axis is normalized to the
maximum frequency of the full width at one eighth maximum.
5.1.3 Mechanism of Spin Orientation Selection
In order to study the spin distribution before and after the orientation selection of
spins in a powder sample, a simple relationship between an RF field and the spins will be
developed by defining them in the LAS then transferring to the RAS.
The orientation dependence of the nutation frequency can be described by an
effective RF field.
In the LAS, the spin distribution in figure 5.5(a) for an axial
symmetrical powder sample, rl=0, may be simply represented by the angle, 0, between the
spins and the applied RF field along the X axis in figure 5.5(b).
(a) Uniform Distribution
(b) RF Field Effect
Figure 5.5 Simplification of Spin Distribution within Powder Sample in the LAS for ql=0.
A
LAS
Z
A
z
EF ,EFG
PAS
RFA
X
RF
(a) RF Field in the LAS
(b) RF Field in the PAS
PAS
(c) RF Decomposition in the PAS
(d) Effective RF Field in the RAS
Figure 5.6 RF Magnetic Field Effect in Different Axis Systems
To study the orientation dependence of the interaction between the spins and the RF
field, the RF field may be transferred from the LAS into the PAS. In the LAS there is one
RF field which is acting on different spins with the different angle (0) in figure 5.6(a). On
the other hand, when it is viewed in the PAS, each spin experiences a different RF field
with the different angle (0) in figure 5.6(b). The RF field can be decomposed into sinO
and cos9 components in figure 5.6(c). But only the sin0 component along the x direction
influences the spin motion in the RAS since the effect of the cosO component is averaged
out, as shown in figure 5.6(d).
It is clear that the orientation dependence of nutation frequency comes from the
angle 0 dependence of the effective RF field for each spin. In reality, the spin selection can
preserve the spins in the Y direction, in the Z direction or in the YZ plane for an imaging
RF field along the X direction by using Spin-Locking, DANTE, Jump/Return, Shaped
pulse methods and etc.
5.2
Spin Selection by Spin-Locking
The simplest way to select the spin orientation is the spin-locking. The basic spin
locking technique involves using a (90) o-(Spin-Lock)
90 pulse sequence shown in figure
0
5.7. The first 900 pulse along the x direction flips spins along the z axis to the y direction.
The long RF pulse along the y direction preserves the flipped spins in the y direction. The
last 90 RF pulse in the -x direction rotates the preserved spins back to the longitudinal
direction for nutation or imaging experiments.
FR
T2P
2 x
tsp
-1x
>> T2p-x
Figure 5.7 Spin Locking Pulse Sequence
In pure NQR, the first pulse differentiates spins in the PAS in figure 5.8.
The
second pulse with duration time >> T2 locks the spin components in the y direction and
meantime destroys the spin components in the x and z directions by the RF inhomogeneity
from the orientation dependence resulting in the coherent spins along the y direction.
A
A
a
-
m
m
m
-
'I
(b) After a 900 Pulse along X Axis
(a) Spin Uniform Distribution
Figure 5.8 Spin Differentiation by an RF Pulse
To examine the spin dynamics of the spin locking pulse, a simple model could be
used after the first excitation pulse. Because of the axial symmetry about the RF pulse
along the Y axis the sphere of spin distribution in figure 5.8(b) is represented in figure
5.9(a). Figure 5.9(b) shows the sine component in the ZX plane and cosO component
along the Y direction after decomposition.
A
AA
A
A
(a) 0 varies from 00 to 900.
(b) Spin Decomposition
Figure 5.9 Simplification of Differentiated Spins for a Locking Pulse
To calculate the spin motion in the spin locking, the spin has its initial state in the
LAS as given in equation 5.5. The flip angle, xiV, for the Y and Z component is (rC/2)sin0
with respect to the first 7r/2 RF pulse. The spin distribution immediately after this pulse is,
iLs(O)=(cosO
sinasinO
cosasinO)
100
(5.5)
K
)=
ILs(-
x
cos 0
cosasin0cosiV
(5.6)
.
sinasin0cosy 1 -cosasin0sinyt
+ sinasin0sin,
By applying the second RF pulse in the Y direction the spin Y component remains the
same, if the T,
is neglected, and the X and Z components are destroyed by RF
,
inhomogeneity from the 0 and vanish by T2p decay. The final states after the locking pulse
are given in equation (5.7) and figure 5.10(b),
IL4s(- 2
+t
))=(0
sinasin0cosy, -cosasin8siny
e
1
0).
(5.7)
Dependence
"Y
A
0
Independence
(b) Spins After Locking Pulse
(a) Spins Before Locking Pulse
1
0.75
0.5
0.25
-0.5
-0.75
(c) Nutation after Spin Locking: Normalized Amplitude vs. Pulse Length (gs).
Figure 5.10 Spin Locking Leaves Spin Components in the Y Direction.
But in reality, the RF pulses applied only in the X direction in the LAS perform the
x or y pulses in the RAS by varying the RF phase with respect to 0' or 900 respectively.
The exact spin state after the spin locking followed by a nutation pulse in the LAS is,
sin 0(sin 2 a - cos 2 a) cos i1, sin i
As = cos
1, cos y 2 sin f
3
+ sin
f1 COS V 3
2
cos8(cos a - sin
101
2 a)
-
2
2 sin a cos a cos V, sin V2 , (5.8)
cosy 1 sin V/2
where
142
is the spin locking pulse and W3=yBt,sin6 is the nutation angle with the
orientation dependence. By using phase cycling, which contains two spin-locking
sequences as shown in figure 5.7, but the second of which has inverse locking phase and
results in a negative nutation pulse phase, that is,
VIIx V2 IV3 X l X 21V3
(5.9)
the spin-locking-cycling result shown in figure 5.10 (b) and (c) is given as,
IS(t) = (0
siny, cosy
y1 =
r/2sinO
=
B sin Ot,
Y13
3
0)
(5.10)
By examining the result of spin-locking in figure 5.10 (b) and (c), the spins
selected in the Y direction contain two kinds of spins: one is the spin component with
orientation dependence and the other is the spins with the orientation independence. When
the third nutation or imaging pulse is applied, since the spin systems experience the same
RF distribution due to orientation dependence with different initial spin values, the nutation
angle,
13, is
still the 0 dependence.
Although the spins had been selected and their
components do align in the Y direction, a pure orientation of spin, such as a perpendicular
plane or axis(es) to the nutation RF field, is not obtained since some spin components with
the 0 dependence have been selected at same time. As an interesting example, the spinlocking method points out the complication of the interaction between the applied RF field
and the spin system of powder in pure NQR.
5.3
Spin Selection by DANTE-Based Sequence
5.3.1 Principle of DANTE Sequence
A DANTE sequence is appropriate for selectively exciting only those spins which
experience a specific RF field strength[69]. The most straight forward implementation of a
string of no pulses generates an RF profile with the form (cos4) n. For the most purposes,
the DANTE sequence can be described by linear-response theory since ideally it is
constructed from infinitesimal rotations. Alternatively, the technique can be viewed as an
102
example of second averaging. The method consists of two alternating rotations about
orthogonal axes. The spin magnetization is aligned with the first rotation axis for a Ir/2
pulse, and this rotation is arranged to be large and to vary with the Hamiltonian upon which
the selectivity is to be based. The second rotation is normally small and uniform
throughout the sample. As an example, an Iz rotates due to chemical shifts and a short RF
pulse is applied to yield an I, rotation. The essence of the DANTE method is revealed by
treating the dependent rotation (the Izrotation in the above example) as a phase shift of the
uniform rotation (the short RF pulses in the above example).
To use a DANTE sequence to selectively excite spins based on the RF field strength
that they experience, the dependent rotation must be based on the RF field strength[69].
Since the effective RF field of pure NQR in a powder sample depends upon the orientation
of the spins, which is represented by an angle, 0, between an RF field and spins, the RF
field strength that the spins experience is directly related to the orientation of them. By
controlling the flip angle, A, from an RF field with the variation of either the field strength,
B1, or the pulse length, tw,
(5.12)
Vf = yBzt, sinO,
and by using the DANTE technique, the spins in some orientations could be selected.
5.3.2 The X Spin Selection
By letting yBtw=7t/2 and the interval, T, between pulses much larger than the T 2
decay, as shown in figure 5.11,
T>> T2
x
twI Y
Figure 5.11 Spin Selection Sequence along the X Direction
the spin motions is calculated based on the procedure in section 4.4.1 in the RAS. The
spin after the first n/2 pulse is,
103
RAS-
IR4
=
2
0
cos( sin ,
sin -sinO
2
(5.13)
after the T2 relaxation decay, only the z component remains,
(5.14)
0 cos( -sin O} .
RASX +T= 0
In order to make an effective spin orientation selection, a series of t/2 pulses with T
interval are applied, and the state is the nth order of the above z component,
IRASI
X+ T)
= 0 0 cosn( sin 8}.
(5.15)
With increasing order, n, the efficiency of the selection becomes higher, as shown
in figure 5.12 for n=0, 4, 16 and 32. Following the spin selection, a nutation RF pulse is
applied in the Y direction in the LAS (see figure 5.11),
B,LAS = (0
B
(5.16)
0).
By a series of transformation, Rx(ao)Ry(tn/2-0), this RF field has its new PAS
representation,
BPAS =
Rx(a)Ry
sin a cos0
)-B. == B,. cos a
- sin a cos )
0
-06
(1
Rx (a)Ry(
A0O
0
0
0
0
cosa
sina
)
sin 0
-sina
0
cosa ).-coso
0 cos'0
1
0
0 sinO)
(5.17)
(5.17a)
Since the z component of the RF field in the PAS has no effect for the z spin, the
nutation angle, Nn, depends upon the RF field strength in the transverse plane,
104
Vy,
2
= yBtw sin 2 aCOS2 0 +6COS a .
(5.18)
The nutation magnetization in the detection direction along the Y axis in the LAS
with the initial state given in equation (5.15) is calculated by using the Bloch equation in the
RAS, by transforming from the LAS and by integrating the Y magnetization over the
sample,
MAs =
cos"n
cos0 f(a,O)sin(iBtwf(a,O))sinOdacd
(5.19)
f(a,O) = sin 2 acos2 0+cos2 a.
(5.19a)
Figure 5.12 shows that when the number of pre-selection pulses (7t/2) is increased,
the orientation selection of spin in figure 5.12 (a), (c), (e) and (g) becomes more efficient
and the nutation functions in figure 5.12 (b), (d), (f) and (h) are more approximate to a
sinusoidal function. Therefore, the lineshape of the nutation spectrum by taking a Fourier
Transformation of the nutation functions should be narrower resulting in a higher imaging
resolution. The resolution is increased twice but the signal is reduced almost 13 times as
trade off by comparing zero pre-selection pulse with 32 pre-pulses.
*.
wW1
T
Aak
1
Fig. 5.12 (a) Spin Distribution in the LAS without Pre-Selection Pulse
Fig. 5.12 (b) Nutation without Pre-Selection Pulse
105
,
.
1
1
+
1jN
1
Fig. 5.12 (c) Spin Distribution in the LAS with 4 7t/2 Pre-Selection Pulses
1
-0.5
-- 0
.
I\
A
V V
i
\
I
+1
i
+1
Fig. 5.12 (d) Nutation with 4 ir/2 Pre-Selection Pulses
.71
63
Fig. 5.12 (e) Spin Distribution in the LAS with 16't/2 Pre-Selection Pulses
An
_
o
M
---
o.
o
.9
\v7
!
. 75
.
A5 /A A
.
.
.
.
V v
±1
.
IvI v
Fig. 5.12 (f) Nutation with 16 t/2 Pre-Selection Pulses
c~
1
+4
+
1
1
Fig. 5.12 (g) Spin Distribution in the LAS with 32 7t/2 Pre-Selection Pulses
0.7
A
0-
2
o
-
-0, 0. -7!>
-.
-- O .
-- 0 .75
Fig. 5.12 (h) Nutation with 32 r/2 Pre-Selection Pulses
Figure 5.12 Spin Distribution and Nutation with n=0, 4, 16 & 32 Pre-Selection Pulses
106
5.3.3 The YZ Spin Selection
From the last section, the spin selection in one direction reduces the signal to noise
ratio a lot. If the pre-selection pulse, yB tw, is a it or 21r pulse and the nutation pulse has
the same direction,
Figure 5.13 Spin Selection Pulse Sequence on the YZ Plane
the spin after the n it and n T relaxation decay for the transverse components in the PAS
only has the z component,
nas(n .(rIx +T)) = {0
(5.20)
0 cosn(isin9)}.
This spin becomes more orientation independent when the number of t pulses is
increased, n=4, 32 and 128, by examining figure 5.14, which shows that the thickness of
the spins around the YZ plane decreases.
Fig. 5.14 (a) Spin Distribution in the LAS with 4 it Pre-Selection Pulses
Fig. 5.14 (b) Spin Distribution in the LAS with 32
107
it
Pre-Selection Pulses
Fig. 5.14 (c) Spin Distribution in the LAS with 128
Pre-Selecion Pulses
Fig. 5.14 (c) Spin Distribution in the LAS with 128 7 Pre-Selection Pulses
Pulses
Figure 5.14 Spin Distribution in the LAS with n=4, 32 & 128 Pre-Selection TC
When the nutation RF pulse remains in the same direction as the pre-selection
pulses, the nutation angle, xV, has a simple orientation dependence in equation (5.21), and
the magnetization in the PAS after the nutation pulse is given in equation (5.22),
V/,
(5.21)
= yB,,t, sin 0
(5.22)
M,PAS = cos" (n sin ) sin(yB tw sin0).
The detected magnetization in the X direction in the LAS is converted from the PAS
by using equation (5.17a) then is integrated over the powder sample. The final result is
presented in equation (5.23) and shown in figure 5.15, which describes the different
nutation functions for different number of pre-it pulses,
.
Mx, s = cos(rsin)sin()sinB,t sin 0)sin2 OdO
15
n=O
n=0
n=32
(5.23)
n=128
n=1024
n= 128
n= 1024
05
05
0 1 A0
00
00
V 000
0A F00
R00
12
-0.5
-0 5
-1
-1
(a) Original Nutation
00
12
(b) Normalized Nutation
Figure 5.15 Nutation Functions with n=O, 32, 128, 1024 Pre-Selection 7t Pulses
Transverse Magnetization vs. Pulse Length (gs)
108
The nutation function with 1024 pre-selection t pulses, as shown in figure 5.15, is
much closer to a sinusoidal wave with a T2p decay by comparing with the nutation function
without pre-orientation selection of spins. In other words, the linewidths of the nutation
spectra with the pre-selection t pulses are much narrower than without the pre-selection 7t
pulses as indicated in figure 5.16(b), which are normalized for the better comparison of the
line shape. The line widths are 562 Hz and 263 Hz with respect to 0 and 1024 pre-pulses.
Since the line width of T2p=609 gs decay is 261 Hz, the line width with 1024 pre-pulse is
almost the same as the T2pdecay and twice narrower than zero pre-pulses. The trade off is
the intensity decrease of the detected signal, as shown in figure 5.16(a). The signal
intensities are 0.2443, 0.1948, 0.156 and 0.099 for n=0, 32, 128 and 1024 respectively.
The intensity with 1024 pre-pulses is about 40% of the zero pre-pulses. By comparing
with other methods in the linewidth and signal to noise ratio, the spin selection in the YZ
plane is the better choice. Meantime, the low frequency and DC effects of the spectrum are
removed by this selection procedure.
1
0.25
n= 0
n=32
n=128
n = 1024
0.2
0.8
0.15
06
0.1
0.4
0.05
0.2
1000
2000
3000
4000
5000
6000
n=O
n=32
n=128
n = 1024
1000
(a) Original Spectrums
2000
3000
4000
5000
6000
(b) Normalized Spectrums
Figure 5.16 Nutation Spectrums with n=0, 32, 128, 1024 Pre-Selection
it
Pulses
and with T2p=609gs Relaxation Decay. The signal intensity decreases 60% (a)
and the linewidth reduces 50% (b) for n=0 & 1024.
5.3.4 Discussion of Spin Selection by DANTE Sequence
From the above two sections, when a series of rt/2 pre-selection pulses is applied in
the X direction of the LAS, the spins along the X direction are preserved as shown in
figure 5.12. While a number of 7t pre-selection pulses is used, the spins on the YZ plane
and along the X direction are selected as shown in figure 5.14.
109
0.8
0.6
0.4
.4
.2
0.2
j
25
75
50
100
125
150
175
25
(a) it/2 Pre-Selection for 0 from 0 to 1800
50
75
100
150
125
175
(b) t Pre-Selection for 0 from 00 to 1800
1
0
0
-75
-50
-25
25
75
50
-75
(c) in/2 Pre-Selection for 0 from -90 to 900
-50
-25
25
75
50
(d) ir Pre-Selection for 0 from -900 to 900
Figure 5.17 Distribution of Preserved Magnetization by 4, 32, 128 & 1024 rt/2 & Rt
Pre-Selection Pulses vs. the angle 0 between a RF Field and the EFG.
1
1
.8
.6
-4
.2
0.8
0.6
0.4
2 5 S 07
5_0 C_2 l5
.7 5
I
n
0.2
2550
7
10
CL2
2550
7 50L 0.2
S7
575
1
0.
0.
0.4
0.
N
8
6
2
2550
751L0 1.2
5s a7
s
5 a7
5
(a) Comparison of ir/2 and 7t Pre-Selections for 0 from 0Oto 1800 and n=4, 32, 128, 1024
0.
0.
0-
0.
-7559-25
-755-25
-75S925
~IL
0.8
0.5
0.
0.8
0.6
0.4
0.2
255075
255075
(b) Comparison of r/2 and
it
255075
-755525
255075
Pre-Selections for 0 from -90' to 900 and n=4, 32, 128, 1024
Fig. 5.18 Comparison of Preserved Magnetization of x/2 and t Pre-Selections
for n = 4, 32, 128 & 1024 Pre-Selections by Varying the Angle 0.
110
In figure 5.17, the distributions of the preserved magnetization vs. the angle, 0, are
displayed under curves with respect to 4, 32, 128 & 1024 t/2 and nc pre-selection pulses
separately. The center peaks in the figure (b) and (c) are used for imaging. The figure (c)
shows the figure (a) in different form but the amount of preserved magnetization clearly.
The center peaks in the figure (d) point out the amount of selected magnetization, which is
not used for imaging and is going to influence the imaging in certain ways if the encoding
RF field gradient is not only in the X direction.
Figure 5.18 show the difference of the two types of pre-selection for different
numbers of RF pulses, that is, n = 4, 32, 128 and 1024, in the two different 0 ranges.
Figure 5.19, as examples of figure 5.18, provides the comparison of the selected imaging
magnetization of their center peaks. The figure (a) tells us that the amount of the
magnetization selected by t pre-selection is much larger than by r/2 pre-selection for the
same number of pulses. On the other hand, by looking at the figure (b), almost the same
amount of the preserved magnetization for imaging could be obtained either by 32 r/2 preselections or 1024 nc pre-selections. However, the shape of the curves in figure (b) points
out that the orientation selection of 7t pulses is much better than t/2 pulses.
1
n/2
r
Pre-Selection
P r e -S e l e c t i
on
0.5
(a) 1024 ir/2 and 7t Pre-Selections and rt Pre-Selection Pulses
8
o.
.
n/2
Pre-Selection--
-
32 Pulses
t Pre-Selection
-____
1024 Pulses
(b) 32 ic/2 and 1024 cnPre-Selection Pulses
Figure 5.19 Comparison of Preserved Magnetization of ir/2 and 7t Pre-Selections for n = 4,
32, 128 & 1024 Pre-Selections by Varying the Angle 0 between a RF Field and the EFG.
5.3.5 Experimental Result of Spin Orientation Selection
The experiment in figure 5.20(b) presents a set of nutation data after eight 7t preselection pulses of spin orientations are applied. It is made on a Necolet NMR system by
111
using a solenoidal homogeneous RF coil. The pulse sequence described in figure 5.13 is
5s-(40gsli-5ms),-(5gslix-Sampling)n applied in this experiment, where the ir/2 pulse length
is 20gs, and the ir pulse is 40gs. Since the T 2* is about 0.85ms, the repeating time of the
it pulse is 5ms.
For Barium Chlorate Monohydrate (Ba(CI0 3) 2 H20) powder, the T,
relaxation time is about 1.6 second. Thus, the delay time for each scan is 5 second.
After the eight it selection pulses are applied in the X direction and T 2* relaxation
occurs, only the spins in and close to the YZ plane remain except some spins along the X
axis. Thus, a series of nutation pulses, 5 s, in the X direction are applied, meantime the
transverse magnetization is collected, as shown in figure 5.20(b). By comparing this data
with the nutation without pre-selection in figure 5.20(a), the experimental results indicate
that the orientation dependence of spins in pure NQR is improved. Figure 5.20 (c) and (d)
compare those in different ways.
1000
750
A
500
250
80
20
-200
-250
-500
-750
S
-400
20
4
60
80
100
-600
(b) Nutation with Pre-Selection
(a) Nutation without Pre-Selection
1000
750
without Selection Pulse
with Selection Pulse
- - -without Selection Pulse
_with Selection Pulse
500
250
20
-250
60
80
-0.5
-500
-750
-1
(d) Normalized Nutation w/o Pre-Selection
(c) Original Nutation w/o Pre-Selection
Figure 5.20 Nutation Experiments w/o 8 nt Pre-Selection Pulses (Horizontal Axis=gs).
112
Chapter 6
Experimental Setup of Pure NQR RF Imaging
The proposed Pure NQR Imaging method uses a homogeneous RF field to select
spins in a desired direction(s) then applies composite RF gradient pulses to make the spatial
encoding. The procedure of this new pure NQR imaging is described as,
Spin Selection
RF Imaging
DANTE
Image
-I Multiple-Composite
Reconstruction
RF Gradient
RF Homogeneous
Figure 6.1 Diagram of Pure NQR Imaging.
For spin selection, the DANTE-based sequence with multiple 7r pre-selection pulses
by a homogenous RF field may be applied, as shown in figure 6.2. After a series of 1C
pulses and transverse magnetization decay, T2 , the spins, which are perpendicular to the
RF field, are preserved for imaging. By increasing the number of the selection pulses, the
orientation dependence of the nutation frequency can be reduced to a desired degree. But in
reality, there is a T2P relaxation that limits the linewidth of the nutation spectrum and signal
to noise ratio. As a compensation, the linewidth of the nutation spectrum after 1024
i7
pre-
selection pulses could reach one half of the linewidth without pre-selection. But as trade
off, the signal to noise ratio reduces fifty percent (50%) of zero pre-selection.
."
Field
T >> T 2
n
RF Composite
tw
Gradient Field
Figure 6.2 Pure NQR Imaging Pulse Sequence.
RF homogeneous pulses preserve the spins, which are perpendicular to the RF field.
RF composite gradient pulses encode the spatial information.
113
The nutation pulses for imaging, after the spin selection, encode the position to the
nutation angles by varying the pulse length, tw, of the RF composite gradient. Therefore,
the amplitudes of the detected signal form a Fourier relationship with the spin density of the
selected spin. A 1D profile of the spin density is obtained by an inverse Fourier
Transformation. For projection imaging, a series of FIDs by varying the pulse length, tw,
and the phase, 4, of the RF gradient are collected by the RF homogeneous coil resulting in
a set of spatial profiles. By processing the spatial profiles, a 2D image of the powder
sample can be reconstructed.
Because the pure NQR experiment is different with NMR, the experiment setup of
pure NQR will be described. The biggest difference between pure NQR and NMR is the
probe, which will be discussed in chapter 8 in great details. For the pure NQR probe, a
decoupled RF resonance circuit with active switching of modes is presented. Some
simulation and experiment results will provide the evaluation of the theory, the design and
the method, which are involved in this pure NQR imaging approach.
6.1
Pure NQR System
The experiment system only consists of a simplified pulse spectrometer and a probe
since the pure NQR experiment does not need an external DC (Bo) field, as shown in figure
6.3[70][2]. The pulse spectrometer outputs a pulse with about 310 watts (350 peak to peak
voltage) power on 50U load at 29.33MHz for the probe, and collects a FID signal with an
amplitude about micro-volt voltage from the probe. It is composed of control and data
processors, a programmable pulse generator, a transmitter in figure 6.4 and a receiver in
figure 6.5. The probe acts as an antenna, which converts both the received high RF power
to excitation energy absorbed by nuclear spins and the emitted energy by returning the
equilibrium energy states of nuclear spins to very low RF voltage based on the EMF. It
consists of a transmitter/receiver (T/R) switch, a temperature controller, a pin diode
switching and an RF resonance (LC) circuits, as shown in figure 6.6.
293C Pulse
Programmer
Transmitter
Probe
Nicolet
11 80E
Data Processor
Receiver
Figure 6.3 Pure NQR System: Spectrometer and Probe
114
6. 2 RF Transmitter
The transmitter consists of a frequency source, a modulator and amplifiers. To
ensure coherent operation of the instruments, the RF source for the transmitter, the
reference RF for the phase-sensitive detectors, and the timing for the control pulses should
all ultimately be synchronized and delivered from the same master oscillator. Modulation is
achieved by using double-balanced mixers (DBM).
In this system, the modulator outputs four frequencies, 6.83, 11.25, 18.08 and
29.33 MHz, by mixing an intermediate frequency (IF), 11.25MHz, and a reference
frequency, 18.08MHz. After a high pass filter, an RF pulse with a carrier frequency,
which is nuclear spin dependent, at a 0.23 Volt level is generated by a gating switch. Since
the frequency of the RF pulse is not clean enough, a band pass filter is connected just in
front of an RF power amplifier. The band pass filter is made up by a quarter wave length
coaxial cable at the carrier frequency (29.33 MHz) with short at one end. The wave with
this frequency, which has 900 phase shift at the end of the cable, will be fully reflected
without energy loss[71]. The pulse coming out from this band pass filter has the "pure"
carrier frequency with an amplitude at 0.206 volts. The noise of its zero output is cleaned
by a pair of diodes for a second stage narrow band power amplifier. The final output of the
transmitter is about 360 volt peak to peak value, as shown in figure 6.4.
L.-O-
H.P.
4V0.6V
11.25
_0
ArFMxrR
16
--
z/
R29.33MHz
1
SYN
18
18.08
IV
0.27V
29.33MHz
18.08MHz
11 25MHz
V
RFIN
Power
-I'
Spliter
RF1
Switch
/=
.
Spl-erReceiver
=
--
TTL
RF2
0.23 V
Du
29.33MHz
0.206 V
136V
360V
B..d
EIN
Figure 6.4 A Block Diagram of Pure NQR Transmitter
6.3
RF Receiver
NQR receivers consist of a pre-amplifier(s) to make the RF signals large enough, a
double-balanced mixers to heterodyne the carrier frequency, 29.33 MHz, with offset
115
frequencies of the collected signal to the intermediate frequency, IF, 11.25MHz, several
RF amplifier stages at the IF, a quadrature phase sensitive detector to distinguish signals
having positive and negative offsets from the carrier frequency, two double-balanced
mixers to demodulate the intermediate frequencies with signal offsets to audio frequencies
of the signals, and a couple of audio amplifier stages to input enough intensity of the signal
for analog to digital converter after passing through low pass filters.
In figure 6.5, a low noise figure and high gain pre-amplifier is used to obtain a
reasonable signal to noise ratio at a necessary voltage level since signals from pure NQR
experiments are very weak at about micro volts. Also the IF band pass filter and the AF
low pass filter should have good performance for the clean spectrum of the signal.
Duplexer
29.33zfJ,
7
Prem
7
F
Mixte
Me
4
A
CA
< 30 MV
IF
11.25
+10 dB
.: f PSC
33MHz
i29
Power
_ .. Spliter
18.08Transmitter
S
- 1
dB
11.25-=h
MHz
Audio A
CHAL.
B.P.
47 41MHz
PT
MH
MHz
f
MAR
4
RFQuadraure
17 dB
RF probe
includes
transmplifi circuits
e
r (Tplifi
Phaswitch
deliercircuits to convert
switching to detune
two KHz2
RFa6.resonance
and
RFDetector
resonanceto (LC)
L.P.
.CHB
Hz-2energy
electrical
between
power
Audio B
to
magnetic
QP
field,
as
shown
in
figure
6.6.
Figure 6.5 A Block Diagram of Pure NQR Receiver
6. 4 RF Probe
RF probe includes a transmitter/receiver (T/R) switch to deliver the transmitter
power to the RF resonance circuit and to couple the received signal to the receiver, a
temperature regulator to control the temperature of the sample at about 300 K0 , a pin diode
switching to detune two RF resonance circuits and RF resonance (LC) circuits to convert
energy between electrical power to magnetic field, as shown in figure 6.6.
The T/R switch is made up of two pairs of diodes and a quarter wavelength (X14)
coaxial cable. The diode pair connected between the RF power amplifier and RF resonance
circuit transfers high RF power to the LC circuit and block the FID from it, and the pair
connected to the ground protects the pre-amplifier from the RF power.
The quarter
wavelength cable also protect the pre-amplifier to minimize the transmitted RF power at the
input of the pre-amplifier. In other words, when the transmitter outputs high power, the
116
diode pair after the RF amplifier is short and the pair before pre-amplifier is also short.
Therefore, the RF power passes through LC circuit and the shorted cable reflects RF power
back from the pre-amplifier end to protect the pre-amplifier. During the data acquisition,
since the collected signal is at microvolts level, both pairs of diodes are open. Thus, the
signal goes from the LC circuit through the cable to the pre-amplifier[2].
During the experiment, a high RF power is inputted to the resonance circuit with a
limited resistance. Some RF power becomes heat and raises the sample temperature. Since
the frequency of pure NQR is very sensitive to the change of sample temperature (one
degree temperature change vs. one kHz frequency change), the temperature of sample is
controlled at about 300 Ko by blowing air into a temperature/RF isolation box.
An RF resonance circuit is composted of capacitors and a inductor. The RF coil
should have the highest possible quality factor (High Q). The high Q coil will convert the
maximum electrical power to the magnetic field strength and give out the best SNR. To
improve the sensitivity, the coil should be as small or as close to the sample as possible; in
other words, it should have the highest filling factor.
In pure NQR imaging, normally two detuned RF resonance circuits are needed.
There are two methods used here. One is geometrical isolation by placing two coils
perpendicular to each other. The other uses pin diode switching. In the latter, to decouple
a quadrupolar RF gradient coil from a RF homogeneous coil, a set of pin diode switching
modes is connected to its RF resonance circuit. These two sets of pin diodes are controlled
by an RF channel selector. The selector has two outputs to the sets of pin diode switching.
One channel is -200 volts to put the pin diodes off to tune this circuit on the resonance with
a good matching, meantime the other is +0.7 volts to turn the pin diodes on to detune this
circuit off the resonance. It is the same in the other way round.
Figure 6.6 A Block Diagram of Pure NQR Probe
117
Chapter 7
Experimental and Simulation Results of Pure
NQR RF Imaging
Several experimental and simulation results were achieved based on the setup in
chapter 6 for pure NQR imaging. Those results indicate that this potential pure NQR
imaging technique is applicable to certain materials to obtain the full unconcealed
spectroscopic and spatial information without external DC field by eliminating the RF field
inhomogeneity within a powder sample and with a reasonable signal to noise ratio.
The following results from simulation and experiments will give us the evaluation
of the theory, the design and the method, which are involved in this pure NQR imaging
approach.
7.1
Pure NQR Nutation without Spin Orientation Selection
The calculation of NQR nutation in a homogeneous coil without the B o field is
shown in figure 7.1(b), where the distribution of the EFGs within a powder sample is
indicated. An experiment of pure NQR for Barium Chlorate Monohydrate (Ba(C10 3) 2 H20)
powder, which has NQR frequency at about 29.33MHz, is performed on a Necolet NMR
system by using a solenoidal homogeneous coil with one centimeter diameter. The RF
field strength is 117 Gauss. The nutation data are presented in figure 7.1(a) with a 2.5 gs
separation time.
The averaged t/2 pulse length is about 5gs because the magnetogyric ratio, ycl, is
417.2Hz/G,.
The exponential decay rate including the T 2p decay and the RF
inhomogeneity of powder orientational effects is about 100OOs.
There is a great
contribution from the RF inhomogeneity generated by spin orientations.
By examining the nutation data of the experiment and calculation, the two data sets
are matched well, especially in a short time period. Figure 7.2 is the spectra of the nutation
data in different forms. The absorption (imaginary) parts are basically the same.
Therefore, the experiments prove the theory of pure NQR, in turn, the simulation results
can predict some experiments.
118
1
1
0 75
0.5
A'
0 25
' ' ' '
-0
-0
5
0
A
2 4
-0
75
0
yaiC
0
'''9S
25
25
-0
5
25
(N\
%
-
-
kkLS
P
-0.5
75
V"
(a) Nutation Experiment
1
(b) Nutation Calculation
Experiment
Calculation
0.75
0.
5
0.25
-0.25
-0.5
-0.75
(c) Comparison of Nutation Experiment and Nutation Calculation
Figure 7.1 Powder Sample Nutation without External B0 Field.
Spin orientations contribute signal decay effect.
KHz
10 20 30
'
(0 70
KHz
[
10
ZO
40
50 60
70
KHz
10 20
30
40
50
60
70
(c) Spectra Amplitude
(a)Spectra Real Part
(b) Spectra Imaginary Part
Figure 7.2 Nutation Spectra of Power Sample from Experiment and Simulation.
Spin orientations contribute DC and low frequency terms and line broadening effect.
7.2
Pure NQR Nutation with Spin Orientation Selection
An experiment result in figure 7.3 (c) and (d) presents the nutation and its spectrum
with eight xt pre-selection pulses according to the previous analysis of orientation selection.
The probe used for this experiment is a new type of design based on the principle of
transformer. The pulse sequence as described in figure 5.13 is 5s-(40gslx-5ms) 8-(5gslx Sampling)n, where rt/2 pulse length is 20gs, and rt pulse length is 40gs. The nutation
119
frequency is about 12.5 kHz. Since the T2* is about 0.85ms, which represents 2.35 kHz
linewidth in figure 7.3(b), the repeating time of the
it
pulse is 5ms. For Barium Chlorate
Monohydrate, Ba(C10 3) 2 H20, powder, the T, relaxation time is about 1.6 second.
Therefore, the delay time used in the sequence is 5 second.
1000
50
500
Z
-250
20
4
60
80
2.35 KHz
\
100
-500
--.....
/
12.5 KHz
-750
Nutation without Pre-Selection (gs)
(a)
..
(b)Spectrum without Pre-Selection
600
400
200
1.86 KHz
20
60
so
100
-200
-400
12.5 KHz
(d) Spectrum with Pre-Selection
(c) Nutation with Pre-Selection(gls)
without Selection Pulse
with Selection Pulse
0.5
" I86/2 35=0 791
12.5 KHz
-1
(f) Spectral Comparison w/o Pre-Selection
(e) Nutation Comparison w/o Pre-Selection
Figure 7.3 Nutation Experiments w/o 8 n Pre-Selection Pulses.
The Lineshape is reduced 21% by 8 1 Pre-Selection Pulses. When the number of preselection pulses is increased, the lineshape will become narrower as desired.
After the eight n selection pulses are applied inthe X direction with T2* relaxation, a
series of nutation pulses, 5gs in width, in the X direction are applied, meantime the
transverse magnetization is collected, as shown in figure 7.3(c). By comparing with the
data in figure 7.3(a) without spin selection, the experimental results indicate that the
orientation dependence of pure NQR is improved as shown in figure 7.3(e). Because of
only few pre-selection pulses, this improvement only occurs in a short period of the time.
The linewidths of the spectra are reduced from 2.35kHz in figure 7.3(b) to 1.86kHz in
120
figure 7.3(d), about 21% as shown in figure 7.3(f). If the number of the pre-selection
pulse is increased, this orientation dependence will be removed. This experiment shows
that the DANTE based sequence can be used to select the orientation of spins.
One Dimensional Pure NQR Imaging
7.3
We can use the following simulation of pure NQR imaging in one dimension to
predict some experimental results based on the experimental and calculation results from the
previous sections. A one dimensional imaging experiment is setup in such way so that the
Ba(C10 3)2 H20 sample is placed at both ends of a transformer probe, which has a circular
cone shape as described in figure 8.6(c) and its image is represented in figure 7.4.
Figure 7.4 An Image Related with Half of Sample Geometry in One Dimension
In figure 7.5(c), a 1D pure NQR imaging experiment without the spin selection is
carried out on the pure NQR system. By determining the ninety degree pulse lengths at
both ends from the experiments, 20gs and 60gs, the RF field strengths are 30G and 10G.
Thus, the RF gradient is approximate to 3.3 Gauss per centimeter for a 6cm length coil.
Those parameters are used to simulate the ID pure NQR imaging in figure 7.5 (a) and (b)
with 5[gs encoding pulse length.
Figure 7.5 (a), (b) and (c) show the imaging data in time domain. By examining
the figure (d) and (e), they present the difference in different ways with respect to 0 and
1024 it pulse pre-selection. The figure (f) indicates the difference between the simulation
and the experiment for a relative short encoding time.
0
04
2
S
0
400
SOO 600
4
-02
-0
-0 4
-0 4
(a) Without Pre-Selection
(b) With Pre-Selection (c) Experiment without Pre-Selection
121
-Without Pre-Selection
With Pre-Selection
o
400oo
50oo
-
600
(d) Comparison of Original Nutation Simulation w/o Pre-Selection
IWith
Pre-Selection
0.5
/Without Pre-Selection
O
10
O
4oo0000
7
ooO
V
6oo
-0.5
--1
(e) Comparison of Normalized Nutation Simulation w/o Pre-Selection
3
0
/," Calculation without Pre-Selection
. 75
'
Experiment
0.5
0.
25
25
-0.
15
0
25
-0.5
(f) Nutation Comparison of Simulation and Experiment without Pre-Selection
Figure 7.5 ID Nutation of Simulation and Experiment w/o Pre-Selection.
Figure 7.6 gives their spectra by a Fourier Transformation since the data in time
domain is hard to display the image features. It is clear to see that there is a blurring effect
on the reconstructed images in figure 7.6 (a) and (c), which have no spin selection.
However, the reconstructed image with the spin selection in figure 7.6(b) has almost the
same shape as the original image in figure 7.4.
By comparing the non-selection image with the selection image in figure 7.6 (d) or
(e), it is clear that the high frequency parts at the edges are smoothed out and that there are
some extra low frequency remaining within the image. In other words, by applying a
series of spin selection pulses, the blurring effect and additional low frequency are
removed. This result indicates that the pure NQR imaging technique we proposed here will
provide a new way to examine the characteristics of materials in a wide range directly.
122
1
o
. 8
0.6
0~
-2
cm
(a)
Reconstructed
D
Image
without
Pre-Selection10
(a) Reconstructed 1D Image without Pre-Selection
cm
-2
6
8
10
(b) Reconstructed ID Image with Pre-Selection
0.
8
0.
0.
O a
cm
8
10
(c) Reconstructed ID Image from 2 Layer Samples without Pre-Selection
,Without Pre-Selection
With Pre-Selection
cm
-2
2
4
6
10
8
(d) Comparison of Original Reconstructed Images w/o Pre-Selection
,With Pre-Selection
SWithout
Pre-Selection
.N
-- 2
cm
8
10
(e) Comparison of Normalized Reconstructed Images w/o Pre-Selection
123
Calculation Without Pre-Selection
- Experiment
cm
S
4
1O
(f) Reconstructed Images of Calculation & Experiment without Pre-Selection
Figure 7.6 Reconstructed iD Images from Calculation and Experiment w/o Pre-Selection.
7.4
Conclusion of Pure NQR Imaging
Pure NQR imaging can be used to measure the spatial distributions of the selected
nuclear spins with spin (I) > 1/2 since the NQR frequency and its associated lineshape are a
function of the electric quadrupole moment (eQ) and the electric field gradient (EFG) and
are very sensitive to impurities and defects including those introduced by radiation,
pressure and temperature changes.
Traditional NQR imaging is carried out either by external magnetic field or by data
deconvolution to remove the spin orientation dependence with respect to an applied RF
field. In order to remain an unconcealed and undistorted NQR spectroscopy and increase
the signal-to-noise ratio for resolution, we have proposed the pure NQR imaging method
and have developed the technology, which first uses a homogeneous RF field to select
spins then applies an inhomogeneous RF field to make the spatial encoding by using pindiode switching of modes.
Since the spin quantization axis is tied to the molecular structure other than any
external field direction, the spin dynamics of pure NQR are complicated. The lack of a
global quantization axis results in a spread of interaction angles, 0, between the spin and
the RF field, B 1, and an inhomogeneity is thus introduced as the angle. This results in a
spread of nutation frequencies from yB 1 to zero. Since spatial information is encoded in
the nutation frequency, this spread corresponds to a decreased spatial resolution. The point
spread function, PSF, of pure NQR in powder samples depends not only on the gradient
strength but also on the powder distribution, and spreads asymmetrically, introducing
significant low frequency contributions.
The spin selection eliminates the RF field
inhomogeneity from the orientation dependence and increases the spatial resolution by
remaining a reasonable signal-to-noise ratio. The spin selection could be accomplished by
Spin Locking, DANTE sequence, Jump and Return and Shaped Pulse. By the DANTE
method, different selection pulse sequence could be designed to select the different spins in
124
a desired orientation. In practices, in the LAS the spins along the axis or a plane, which is
perpendicular to the applied RF field, are selected for nutation or imaging experiments.
By comparing the detection efficiency, line width (imaging resolution) and signalto-noise ratio from the experimental and simulation results, an approach of n pre-selection
pulses by a homogenous RF field may be applied. According to the analysis in chapter 5,
the nutation function with 1024 7t pre-selection pulses is much more closer to a sinusoidal
wave. In other words, the linewidth with the pre-selection is much narrower than without
pre-selection. The line widths are 562 Hz and 263 Hz with respect to 0 and 1024 prepulses. Since the line width of T2p=609 gs decay is 261 Hz, the line width of this spin
selection is almost the same as the T2p decay and twice narrower than non spin selection.
The trade off is the intensity decrease of the detected signal. The intensity reduces about
60%. Meantime, the low frequency and DC effects of the spectrum are removed by the
selection procedure.
After the spin selection, a series of FIDs by varying the pulse length of the RF field
gradient can be collected resulting in a set of pseudo FIDs with respect to a one dimensional
spatial profile. Based on the principle of RF imaging, a two dimensional image can be
obtained by varying either a 2 nd RF gradient pulse length with 900 phase shift or the phase
of this RF gradient pulse. According to the experiment and simulation of the RF imaging,
both results indicate that the RF imaging by composite RF pulses can be used in pure NQR
imaging.
For pure NQR imaging, a specified pure NQR system is constructed, and special
probe system and RF resonance circuits are used. Two new types of RF probes are
designed and developed, which are the major part of the pure NQR imaging. One is an RF
transformer probe to generate RF homogeneous and gradient fields in multi-dimensions.
The other is a combination of two decoupled RF homogeneous and quadrupole coils to
create a desired planar field gradient by composite RF pulses for two dimensional imaging.
The characteristics of a RF imaging probe switched between a cylindrical
quadrupole B gradient coil and a homogeneous B, coil will be described in chapter 8.
Such a probe can normally generate gradient fields simultaneously in two orthogonal
directions with a spatially varying spin state dependence. The complications arise from this
superposition as discussed in chapter 2 and new sets of composite pulses for creating a B,
gradient with a spatially uniform spin dependence are introduced.
Overall, the experiment and simulation have demonstrated that the pure NQR RF
imaging technique is applicable to characterize the types, location and concentrations of
impurities and defects in a wide range of materials.
125
Chapter 8
RF Probe Design and Diagnosis
For RF imaging in high field NMR and pure NQR, a planar RF gradient pulse,
composed of multiple homogeneous and quadrupole (radial gradient) RF pulses, is applied
to make the spatial encoding by varying its length, t,. The following section discusses the
novel specialized probe.
8.1
RF Resonance Circuits
For RF imaging, two decoupled RF resonance circuits with capacitors and coil(s),
are required. There are tow general approaches to decouple resonance circuits, that is
geometrical isolation and is pin diode switching.
8.1.1 Inductance of Coil
In most nuclear resonance experiments, the inductor, L, is a single layer solenoid
air core with inductance of[72],
L=
0.394 - r 2 N 2
9-r+10-1
CD
CD
CD
,
CD
(8.1)
D
D
CD
C
D
C
CD
r
Figure 8.1 Single Layer Air Core Inductor
A solenoid coil with N turns, r radius and I length using d diameter wire.
where N is the number of coil turns, r is the coil radius in centimeter and I is the coil length.
The coil length must be greater than 0.67r for this equation to be accurate. For an optimum
quality factor, Q, the 1should be approximately equal to the coil diameter.
126
8.1.2 Basic RF Resonance Circuits
There are two common RF resonance circuits[70], a parallel resonance circuit and a
series resonance circuit, as shown in figure 8.2.
L
CQ
'
Cm
(b) A Series Resonant Circuit
(a) A Parallel Resonant Circuit
Figure 8.2 Two Common RF Resonance Circuits
In pure NQR experiments, the basic resonance circuit has two tuning capacitors, Ct,
and Ct2, connected the both ends of a coil, L, to the ground and a series matching capacitor,
Cm,, as shown in figure 8.3. Those two tuning capacitors balance the coil so that any noise
to the coil will pass one or both of them directly to the ground. The total reactance, Zto,, of
the circuit in equation (8.2) can be obtained,
Cm
L
Ct2
Cti
Figure 8.3 The RF Resonance Circuit for Pure NQR Experiments
1
Z = jcOL +
jct2
. o 2 LC 2 -1
-j
m2LCt2 -1
Z, .1/joC,
=C
Z = Z
z +1/ oc,,
Z=-
1
m
+ Z =-
(8.2a)
CoCt2
(8.2b)
1
o(m2LCC,Ct2 - C, - Ct2)
W 2LCt2 (Ct
w
+ Cm)-(Ct,
+ C2
+ Cm)
(8.2)
2LC,,Ct2 - C,, - Ct2 )
To obtain both the resonance condition and the matching condition at resonance
both the denominator and the numerator are equal to zero,
127
w 2 LC
-(Ct + Ct2)
2 LCt2(Ct
+ Cm))-(Ctl,, +
+ Cm)
=0
=0
SRe sonance
for [Matching
1
4L
(8.3)
Re sonance
- Ctl Ct2 (Ctl +Ct2
(8.4)
=
Matching
Mat =
(Ctl +Ct2 +Cm)
L-Ct2 tl
By examining the resonance condition in the equation (8.4), it is decided by the L and an
equivalent capacitor, Cs=(Ct, Ct2)/(C,I+Ct2), which is the series of the two tuning
capacitors, Ct, and Ct2, in the resonant circuit. The matching condition is determined by the
series of the second tuning capacitor, Ct2, and an equivalent capacitor, Cp=(Ct,+Cm), which
is the parallel of the first tuning capacitor, C,,, and the matching capacitor, Cm.
8.1.3 Decoupled RF Resonance Circuits by Geometry
For pure NQR RF imaging, two RF coils are required. Each of the coils has its
own resonance circuit. But the two RF coils must not be coupled. If the two RF coils are
isolated geometrically, such as the two RF fields created by the coils are perpendicular to
each other, as shown in figure 8.4, those two modes are geometrically isolated from one
another. So even though the two coils are tuned to the same resonance frequency they do
not couple and no active switching of modes is needed.
LCHomo
LC Gradient
Figure 8.4 Two Isolated RF Coils are Perpendicular to Each Other.
8.1.4 Decoupled RF Resonance Circuits by Active Switching of Modes
However, in most cases, a set of active pin diode switching is required to
decouple/detune the two RF coils, as shown in figure 8.5(a).
128
The mutual inductance
between the two RF coils will mix the two field modes and interface with the simple
imaging picture. To completely detune those two resonance circuits, two pin diodes, D,,
are connected at the ends of each RF coil. The series capacitor, CDC, has a large enough
capacitance to permit the RF to flow through while still acting as a DC block. When these
diodes are on, one end of the coil is grounded and the other is connected to tuning,
matching and CDc capacitors. Thus, an off-resonance circuit in figure 8.5(b) is formed
with a very low resonance frequency. While the diodes are off, the two diodes act as two
capacitors with about 0.8pF in figure 8.5(c). The two DC levels ,+0.7V and -200V, are
generated by the RF channel selector.
CtC t2Dp
D
RF1L1
Cm
___6pF
C
L2
or
LRF1
L1
RF2
Ctt
R1L-200V
DC
JCf"33U6p7
-200V
36pF
LRF2
or7
2
t
0.7V
Dp
(a) Two Decoupled RF Coils by a Set of Pin Diode Switching
Cm
LRF
Cm
164pF
Cti
C
1-
164pF
CDC
Ct
LRF
Ct2CD
(c) On-Resonance Circuit with -200V
(b) Off-Resonance Circuit with +0.7V
Figure 8.5 Two Decoupled RF Resonance Circuit for RF Imaging
8.2
RF Probe Design for Pure NQR Experiments
8.2.1 Transformer Probe with One RF Field
An RF transformer can be used to shape field lines since a conducting surface
displaces field lines. For example, a flux concentrator is shown in figure 8.6(a), where the
field in the sample area, C or H, can be arranged as either a homogeneous or gradient field
by the choice of geometry in figure 8.6 (b) or (c).
129
C Hole
-
-
_-
----
H Hole
(b) Homogeneous Field (c) Gradient Field
(a) Aluminum Block
Figure 8.6 A Transformer Probe for One RF Magnetic Field
Since the aluminum block can be treated as an ideal conductor, the RF current
generates a time varying magnetic field inside the C hole. Therefore, eddy currents are
setup in the conducting loop around the two holes (the C and either the H or G). This eddy
current creates a magnetic field in the transformer coupled sample area.
Because the fluxes at the both ends of the H or G hole should be the same, the
magnetic field inside the H hole must be homogeneous because the cross sections of the H
are the same through the entire hole. However, the radius of the cross section of the G
hole varies linearly so that the magnetic field inside the G hole changes from the one end to
the other by a gradient, which is a function of the coil length, as shown in figure 8.7.
0.4
0. 2
-- - -
Ladder-Shaped
Circular Cone
1
2
3
4
5
Figure 8.7 Normalized Magnetic Field vs. Distance from Small End to the Large
Solid = Circular Cone in Figure 8.6(c) & Dashed = Ladder-Shaped in Figure 8.8
As an example, if letting the radii be 0.3cm and lcm at each end respectively and a
6cm length of the G hole, the cross sections are r0.32 and it for the small and big ends of
the G in figure 8.6(c). Since the flux, B sr 0.32, at the small end is equal to the flux,
BiLtr, at the large, the ratio, B, /Bls, of those two magnetic fields is the square ratio, 0.32,
of the radii, about 0.1. The field along the G hole varies as a square function of the
distance from the small to large end, as shown in figure 8.7 (solid curve).
In some experiments, for some practical reasons, a ladder-shaped hole is necessary
to create an inhomogeneous magnetic field from the shape of the sample holder. Figure
130
8.8(a) shows the architecture, the (b) is the right view, and the (c) is the back view. The
magnetic field along the slot is represented by the dashed curve in figure 8.7 for 0.3 and 1
centimeter heights at the ends receptively and 6 centimeter length.
(a) Architecture
(b) Right View
(c) Back View
Figure 8.8 A Ladder-Shaped Transformer Probe for an RF Gradient Field
Besides the RF gradient field formed by a geometrical shape, another way is to
control the direction of the eddy current around the H hole in figure 8.9 to create an RF
gradient field. In figure 8.9, as the currents in the two coils, which forms a Maxwell pair,
have opposite directions, the magnetic fields inside the H hole are also in opposite
directions. By superimposing them together, a RF gradient field is generated.
_- C Hole
--------- --------:r
--- -- -- - ----------- -- -H Hole
Figure 8.9 A RF Gradient Transformer Probe by the Maxwell Pair Coils
8.2.2 Transformer Probe with Two Perpendicular RF Fields
For some pure NQR imaging experiments, an encoding RF gradient field in the Y
direction in the LAS is required while a homogeneous selection RF field is in the X. The
homogeneous RF field is generated in the same way as figure 8.6 (a) and (b) except the
shape of the hole shown in figure 8.10(a).
The gradient RF field in the direction
perpendicular to the homogeneous RF can be created by a ladder-shaped hole geometrically
or by Maxwell pair eddy currents around the hole, as shown in figure 8.10 (a) and (b).
131
0 X
z
1
Right View
(a) A RF Gradient from a Ladder-Shaped Hole
---------- ----------
Right View
(b) An RF Gradient from a Maxwell Pair Eddy Current
Figure 8.10 A Homogenous Field in the X direction and an RF Gradient in the Y direction
Formed by a Ladder-Shaped Hole and Maxwell Pair Coils
132
The sample located at the intersection between the two sample holes experiences a
homogeneous RF field from the X direction and an RF gradient field in the Y direction.
According to the principle of electromagnetic field, since the cross section at the
intersection between the two sample holes is different with the rest part of the holes, the
magnetic field, B, at the intersection must not be the same as the rest of it. To find out the
distribution of the B along the sample holes, we start from the Maxwell equations
[71][73][74] by assuming an infinite length of the holes in all directions and the boundary
conditions shown in figure 8.11,
1 dB
VxE+-
= 0
(a)
Faraday's Induction Law
Vx B--
= 0
(b)
Ampere's Circuit Law
VB
=0
(c)
Magnetic Field Gauss' Law
V E
=0
(d)
Electic Field Gauss' Law
c dt
c dt
(8.5)
where magnetic field, B, and electric field, E, are real vector functions of the position and
time, c is the speed of the light, and it is assumed in vacuum. By taking the curl of the (a)
in equation (8.5), using the identity equation (8.6) for any vector, A,
Vx(VxA) = V(V A) -V
2
(8.6)
substituting the (b) and the (d) of equation (8.5) into equation (8.6), an electric field
Laplacian equation can be found. In turn, a magnetic field Laplacian equation could be
derived in the similar way. Thus, the E and the B are separated in equation (8.7) to find a
solution of the magnetic field in figure 8.11.
Y
B = (0, 0)
B = (1,0)
-00
B = (1, O)
(0, 0 )
+00
B = (0, 0)
0m
Figure 8.11 Horizontal Cross Section of Sample Holes with Boundary Conditions.
133
In pure NQR, since the magnetic field is a sinusoidal wave at a radio frequency, 0),
the wave function of the magnetic field can be rewritten in equation (8.8) by defining a
wave number k= co -e,
where
V2
.t is
the permeability and E is the permittivity,
=0
I2
(8.7)
(8.8)
(V2 +k 2 )f = 0.
For a simplification, the magnetic field, B, is decomposed in Bx and By as the
functions of their position x and y along the X and Y directions respectively. Therefore,
the equation (8.8) becomes two completely separated equations and can be replaced by,
(8.9)
(V 2 + k 2 )B(x,y) = 0,
where B(x,y) is either the Bx field or the By field.
To solve this differential equation, the B(x,y) may be separated into equation
(8.10). By substituting it in equation (8.9) then dividing by equation (8.10), we have,
B(x,y) = bx(x)b,.(y)
1 d 2bx
1 d 2b
bx d 2
by 32
x+
Since (1/bx)d 2 bx/ x
2
=
.
(8.10)
k2
(8.11)
.
is a function only of x and y independent, it is defined as a constant
in equation (8.12),
1 d2bx
bx dx2
o2b
d
b
Oy2
2
1
-
(8.12)
2
2
k2 + k2 = k 2 .
(8.13)
134
By solving the equations, a general solution of equation (8.9) is the combination of
all possible solutions with constant parameters,
e+k,x .e-k2y
B(x, y) = (
+ k2 ,
e+'k'x .e
e
±ik x
(8.14)
*e ±+k2 Y
where kl=kl', k'" & k,"' with respect to k2=k 2', k2" & k2 "' to satisfy equation (8.13).
The special solution for the B x and By can be obtained by determining the constant
parameters based on the boundary conditions in figure 8.11. The distribution of the
magnetic fields at the intersection in figure 8.11 are displayed in figure 8.12 with the field
strength and orientation. Continuous curves show the same level of the field strength.
Because of the geometry change at the intersection, the field varies in both magnitude and
orientation. But around the center area inside the circle, the field can be approximated the
same as sy its both ends. Thus, the intersection effects of the sample holes may be
neglected in figure 8.12 (a) and (b). Figure 8.12(c) shows the two fields perpendicular to
each other around the center area. Those results are the expectation of this design.
Two experimental results to test the design with the structure in the figure 8.10(b)
by using two solenoidal coils generating two orthogonal RF magnetic fields are shown in
figures 8.13, which displays the projection and distribution of those fields by a 2D Fourier
Transformation of nutation data. The experiments are carried out on a Bruker AMX122
Spectrometer with three channels. Two series of nutation pulses are inputted into each coil
while the sampling is performed in one of them. The figure (a) and (c) show the data
collection in one coil and the (b) and (d) are in the other. The nutation pulse length is 2gs
for both channels, the dewell time is 20ps and the sampling points are 128x128.
Hanzontal
Magentc
Fields
B-(BBy)
-15
-1
-05
0
05
1
15
-1.5
-1
.5
-1,5
-1
-05
0
0.5
1
5
- /rtd
-1.5
-1
1
05
Magb Rl's
-0,5
fB-.
0.5
1
5
1t5
XCM
XCn
(b) Coil in the Y Direction
(a) Coil in the X Direction
135
Harizontal &Vertical Magentic Fields
1.5
-
1
-
1
c
rrr
-05
41
5
rr
r-r-
Er
0
- rrrrr
r r -
1
r
v,
-1.5
-1.,5 -1
-0.5
0
x cm
0.5
1
1.5
(c) Distribution of Magnetic Fields from Both Coils Separately
Figure 8.12 Magnetic Field Distribution along the Sample Holes for either Coil on.
(a) Lx is on and (b) Ly is on.
(c) Two Fields are Perpendicular to Each Other around the Central Area.
The averaged B, strengths are 7.34 Gauss and 5.87 Gauss respectively since the
t/2 pulse lengths are 8ts and 10gs. The peaks in figure 8.13 are located at 7.34 Gauss
along the horizontal axis and 5.87 Gauss along the vertical axis. Since there are many
factors, which influence the field distribution, in experiments, the distributions of the two
channel are not the same (symmetry). However, the results are consistent.
These experimental results indicate that the design of two orthogonal RF magnetic
fields is applicable for NMR and pure NQR Imaging if it is necessary to develop a
commercial system. Those schemes have the advantage of a geometrical simplicity and
very robust performance. The two resonance circuits in figure the 8.4 are well isolated
geometrically from one another. The only consideration is to ensure that the external tuning
elements do not couple. The disadvantage is the rather low filling factor which makes the
136
entire experiment of low relative signal to noise ratio when coupled with the geometric EFG
direction selection.
120
120
100
100
80
80
60
60
40
40
20
S-20.
0
20
40
60
80
100
0
120
(a) Projection and Sampling X Channel
1.5 108
5. 10
..
1. 10 7
'
40
20
60
80
100
120
(b) Projection of Sampling in Y Channel
1
10
100 5. 10
100
50
100
100
(c)Distribution of Sampling in X Channel (d)Distribution of Sampling in Y Channel
Figure 8.13 Projection and Distribution of Two Orthogonal RF Fields
8.2.3 Transformer Probe with Two Parallel RF Fields
The pure NQR imaging needs one RF magnetic field to make the spin selection and
another to encode the position of the selected spin. Those two RF fields are quite different.
The first RF field is a homogeneous field in the X direction while the second is a gradient
field in either the X or Y direction. The gradient field could be formed by the shape of the
sample holder, such as figure 8.6(c), 8.8 and 8.10(a), or by the architecture of the coil(s),
such as figure 8.9 and 8.10(b). Since those two RF fields are not turned on at the same
time, a simple transformer probe to produce two fields in the same direction is shown in
figure 8.14, including one coil hole and one sample hole with a thin chink in between.
When a homogeneous RF field is needed for the spin selection, the same current as
in coil L1 is inputted into coil L2 from upper port. Thus, those two coils act as one
solenoidal coil and generate a homogeneous RF field inside the sample. During the
137
encoding period, the current direction of the L1 remains the same while the current in L2
passes from the lower port to the upper. Therefore, those two coils form a Maxwell pair
coil and produce an RF gradient field inside the sample. One advantage is that the isolation
for the two RF coils is not as an issue any more since the two coils are combined into one
resulting in only one resonance circuit, which only needs an extra switch[42].
Upper Port
Out
Lower Port
I
In
L2
L1
Sample
Figure 8.14 Homogeneous and Inhomogeneous RF field in Same Direction.
Solenoidal coil creates homogeneous field and Maxwell Pair creates gradient field.
The transformer probe can transfer an RF magnetic field from one place to another
and change the distribution of the field by its shape. This scheme has the advantage of a
geometrical simplicity and very robust performance. The disadvantage is the rather low
filling factor.
8.2.4 One RF Field Probe with Four Straight Wires
A simplest way to generate either an RF homogeneous or gradient field is a
combination of four straight wires. A magnetic field, B(F), at a position, F = xx + y5,
generated by a single straight wire along the z-axis with +I current and at a distance, R,
from the origin and an angle, 5, from the +x axis can be derived from the curl of a vector
potential, A(T),
J(x - Rcos)
2A(T)
I
2
(ol
(T)= Vx
=
(8.15)
+(y - Rsin ) 2
(-y + Rsin Oa
1
21r (x - Rcos
02i)
2
+)
+(y
2
-Rsin)
x - RcosO
0
. (8.16)
When the wire is placed at the origin along the z axis, here R is equal to zero, the magnetic
field is simplified as,
138
=
2
y2 (-Y
X
2B(i)
2~ 7 x
x
(8.17)
0).
2
Figure (8.15) shows the distribution of this field with respect to an electric current coming
in and going out the xy plane. The directions and magnitudes of those fields are
represented by arrows and the amplitudes are the same on a contour curve.
Magnet
for ngaingcurrent
1-
Magnet
for outgoing
acurrnt
4
,
2
r I
t t t
r \\
\ \
I ,
-2
-4
-6
-6
-4
-2
0
2
4t
6
aight Wire
-6
-4
-2
2
0
4
6
Figure 8.15 Magnetic Field of A Str aight Wire with respect to In/Out Going Current.
When some straight wires with electric currents are structured together, a desired
RF field could be created. In this experiment, four straight wires are symmetrically placed
at a distance R from the origin and at four different angles, that is, O=45o , 1350, 225' and
3150, from the +x axis with an arranged current directions. Therefore, the magnetic field is
given by their superposition. When the currents passing through the wires are -I (going in
the xy plane) at O=450 & 135' and +1 (coming out the xy plane) at 1=2250 & 3150, a
homogeneous coil in figure 8.16(a) is constructed so that a spatially uniform RF field is
generated and is shown in figure 8.17(a), 8.18(a) and 8.19(a) in different ways.
R
/
(b) Quadrupole Coil
(a) Homogeneous Coil
Figure 8.16 Four Straight Wire Coils with Differently Arranged Current Directions
139
If the four upright wires located at the edges of a square are correctly modeled by
applying the current of two adjacent wires in an opposite directions, a RF gradient coil in
figure 8.16(b) is manufactured and called quadrupole coil. The magnetic field in Eq.
(8.18) for such four wires is given by their superposition which from the origin
corresponds to a radial field gradient, and the exact shape and phase dependence of this
field are displayed in figures 8.17(b), 8.18(b) and 8.19(b)[30][31][32][33],
'1
B(T) =
O
0
0R2
2R 2
(8.18)
0 -o .
0,
10
The field is rewritten in another form to analyze the coupling properties with spins,
B(F) = Bx + B,, = Bx~ + B,, = gxx - g,jy
dBx
X =
(8.19)
2R 2
dx
(8.20)
2R 2
dy
where gx and gy are the x and y gradient components and have the same value (gI1/2R 2).
Figure 8.17 shows the superimposed field strengths and orientations by arrows in
the xy plane. Contour curves in figure 8.18 describe the same amplitudes of the fields
along the curves. Figure 8.19 demonstrates a homogeneous field along the x direction and
a radial linear gradient field around the center area, where a sample is placed, in the xy
plane.
. .
.
S)
-- -
-
-I
. -. ".- -
-' . -
-.
i.
-
+
.
.
.
.
-
-
.
.
.
.
.
.
---
-
. ... .,
.
.
,
.
,-
t
't
. , . .
(b)t ua
tl,
.
,
'-4
-I-'
._
-
-"
-
. . ,
.
.
..
I
'
P
.
,I
.
i l tFiel
t
'
.
(b) Quladnmpole Field
(a) Homogeneous Field
Figure 8.17 Directions and Amplitudes of Fields at Horizontal x & y Vertical y
140
Fieldoffourstraight
wires: Contour
Homogeneous
Magnet
MagnetFieldoffourstraightwires:Contour
Quadrature
xcm
xcm
(b) Quadrupole Field
(a) Homogeneous Field
Figure 8.18 Contour Curves of Magnetic Fields
-
. .. - -
0
-I
--
-
I
I
I
'
\
I
t
I
a
A
(b) Quadrupole Field
(a) Homogeneous Field
Figure 8.19 Vectors and Contour Curves of Magnetic Fields around the Origin
at the x (Horizontal axis) & the y (Vertical axis)
The gradient of the quadrupole field in equation (8.18) can be decomposed into
aBx/Dx and B/ay along the x and y direction respectively, which have the relationship,
dB,
91_3dx
BB-
=
II
(8.21)
2R 2
and obey the magnetic field Gauss' law of the Maxwell equation. Key features to realize
are that each RF pulse from the quadrupole coil generates two orthogonal RF gradients,
aBx/ax and DBy/ay, that the spin system responds to both of these, and that the two RF
fields are exactly in phase. This built in phase coherence allows each oscillating gradient
field to be decomposed individually into rotating and counter-rotating fields. If, for
example, the two RF fields were 90' out of phase from each other then the sum would
correspond to a rotating field and only if the field was rotating in the correct direction could
it couple into the spin system.
141
This quadrupole (or radial) RF field in equations (8.18) is a oscillating field vectors
with a radio frequency, o, and a phase, 9, and could be rewritten from the Cartesian
coordinate, (x,y), to a polar coordinate, (r,15), form,
=Grr(cosoi-sini5)cos(ot+qp)=(Bx B,
B
B,
(8.22)
0)
= G,rcosOcos(+)
= -GrsinOcos(t + p)
B,
dB,
x _oI
g'
Gr= dr=
2
(8.22a)
,R
(8.22b)
where F = xi + y5 and 6 = tg-'(y/x). After it is decomposed in a clockwise and a counter
clockwise rotating RF fields, we can transfer the rotating fields to a rotation frame by a
rotation matrix, Rz(ot) (See Appendix B). By neglecting the terms with rotation frequency
2o, the quadrupole field in the rotation frame is given as,
BRo,
Rz(aot) B ,
cos(O - p)
gxxcos p + gysinq('
= GrI sin(O -q ) =1gyycos - gxxsing . (8.23)
O
0
By changing the field phase, (p,from 0Oto +/-90 o then combining them together, a rotating
magnetic field in the rotation frame is created as,
(g xx )
9)=o
BRot
jgyy
=
B
- g yy )
=
gx
P=y
g yy
-x x=
(8.24)
-90oB0
Figure 8.20 A Counter Clockwise or Clockwise Rotation Field in Rotation Frame
by Varying the field Phase from 00 to -90 oor +900.
142
Figure 8.20 shows a counter clockwise or clockwise rotation with respect to the
second field phase (p=+9 0 o or -90o by letting gxx # g.y.
Not only the strength of the RF field, but also the phase of the RF field is spatially
dependent, and we will describe this as a "mixed" spin dependence to indicate that the spin
dependence of the gradient Hamiltonian varies across the sample. It is profitable to think of
these two fields (,Bx and B3,) as originating from two separate coils one that has a field
aBx/Ix with phase 0O(aligned with the x homogeneous RF field), and a second coil with a
field aB/By with phase 90o (orthogonal to the x homogeneous RF field). When the RF
going to the coil is phase shifted by (p, the phases of both gradient components vary
resulting in the Hamiltonian given as,
=
-gxx(x cos
-
sin)
-
gy( cos +
sin
.
(8.25)
= -(gxx cos + gy sin O)I - (gycos 0 - gxx sin O),
8.2.5 Two RF Field Probe with Eight Straight Wires
As described before, the pure NQR imaging requires two RF fields, one of which
is a homogeneous field and the other is a gradient field. Also, in some experiments, a
composite RF pulse is used to generate a planner field gradient. Therefore, a homogeneous
RF coil and a quadrupole RF coil could be constructed together, as shown in figure 8.21.
The inner four wires consist of the homogeneous coil and the four outers form the
quadrupole coil. When either of them is turned on, a homogeneous RF field along the X
direction or an RF field gradient along the radial direction is generated respectively.
Using the configuration in figure 8.21, the spin orientation selection could be made
by the homogeneous RF coil, and the radial position would be encoded by the quadrupole
coil so that a Fourier relationship is formed between the detected signal by the quadrupole
coil and the spin density with respect to different radius (r). An angular encoding is carried
out by varying the phase of the quadrupole phase as discussed in figure 8.20. Therefore, a
radial projection relationship between the detected signal by the homogeneous coil and the
spin densities at different angle i is established. Theoretically speaking, by combining
those two radial and angular profiles, a 2D image may be obtained.
143
Quadrupole Coil
Homogeneous Coil
Inner-
-Outer
Figure 8.21 A Probe with Two Coils: Homogeneous and Quadrupole Coils
The most popular application of this probe is a composite pulse, which is made up
of a combination of one or two homogeneous RF pulse(s) and multiple quadrupole RF
pulses. Based on the averaged Hamiltonian theory and the properties of the interaction
between the quadrupole field and the spin system in equation (8.25), a refocusing
homogeneous 7r pulse is employed to choose the quadrupole field direction from equation
(8.23), thus, the effective Hamiltonian only remains one interaction term in equation
(8.25). As discussed in chapter 2.2.2, the simplest composite RF pulse consists of two
radial RF pulses with a homogeneous n pulse in between so that a linear RF gradient pulse
is generated. When the phase of the tcpulse is either 00 or 900, a x or y linear RF gradient
pulse is created, which is shown in figure 2.7 and described in equation (2.21). Other
versions of the composite RF pulse are discussed in the Table 2.4.
In practice, the probe configured in figure 8.21 has a coupling effect because of a
mutual inductance between the two coils. In order to remove the coupling effect, a set of
pin diodes is used in both resonance circuits indicated in figure 8.5. The diagnosis of the
coupling effect can be made by sample spinning technique.
8.3
Diagnosis
of Probe Characteristics with Two Decoupled
RF Coils
For NMR and NQR RF imaging experiments, we have developed a probe with two
decoupled RF coils, which are associated with two detuned RF resonance circuits as
shown in figure 8.5. However, in practices, we have to know how those two coils behave
144
during experiments. Therefore, a series of tests on this probe will be accomplished to
characterize it. Since the probe provides a homogeneous and a quadrupole fields, the
shapes of the two fields can be described by their nutation spectrum. The spatial
orthogonality will be indicated by sample spinning. The length of a n/2 pulse provides the
efficiency of the coupling between the coils (magnetic fields) and sample (spins). Also the
parameters of isolation (electrical coupling of the two coils), switching time between the
two coils and quality factor are evaluated. Most of the tests are carried out on a Bruker
AMX 400 spectrometer with water sample.
8.3.1 Bench Test: Quality Factor and Decoupling Attenuation
The probe for RF imaging with those coils are tuned at 400.135MHz resonant
frequency, f, with 50 Q matching resistance. The quality factors and decoupling
attenuation's are obtained by using a HP 8505A Network Analyzer. The quality factor, Q,
is a measurement of the efficiency of a resonant circuit and is defined as:
Q=
r
- =
Af 3dB'
(8.26)
,
where o=2nf, L and r are the inductance and resistance of a coil. In practices, it is easily
to calculate the Q by using the fc dividing the spectral width, Af3dB, at 3 dB attenuation.
3dBL
fc
Figure 8.22 Reflection Curve of Resonance and Matching
As the Af 3dB are 3.98MHz are 4.24Mhz, while the biggest attenuation's are 60dB
and 55dB at f, = 400.135MHz, the quality factors are 100.54 and 94.37 without loading
with respect to the homogeneous coil and quadrupole coil by adjusting matching and tuning
capacitors. Those Q values are good enough for RF imaging.
In RF imaging, one of the two RF coils is On (resonance) while the other should be
Off (resonance) based on the design of the probe. The reflection power for either coil is
145
close to OdB when it is Off, and is greater than 50dB when it is On. This results represent
a good decoupling efficiency of the inductance between those two coils.
8.3.2 Shape of Two RF Magnetic Fields
The shape of the RF field tells us the strength distribution of it inside sample. It can
be described by a nutation spectrum. The linewidth of the spectrum represents the
inhomogeneity of the RF field since the frequency depends upon the RF field strength
linearly by the gyromagnetic ratio, y,.
The nutation of a homogeneous RF field is an
absorption sinusoidal signal with a T2p decay if assuming the RF on resonance. Thus, the
spectrum of it is a Lorentzian line. The experimental and calculation results of the nutation
spectrum from the homogeneous coil in figure 8.23 (a) and (b) show the filed strength
about 4 Gauss and the T2pabout 1.25ms. They are the same as shown in figure 8.23(c).
4
35
4
3
3
3
25
2
2
15
2
11
1
05
5
10
15
20
25
30
5
10
15
20
25
30
5
10
15
20
25
30
(c)Comparison
(b) Calculation
(a) Experiment
Figure 8.23 Nutation Spectra of the Homogeneous RF Coil from Calculation and
Experiment. The vertical axis is relative amplitude and the horizontal is frequency in kHz.
The nutation of the quadrupole coil is calculated based on the term of gradient
excitation and gradient detection for initial magnetization in the z direction in Table 2.2.
Since the field of the quadrupole coil is a radial field, which varies from zero at the center to
its maximum at the edge of a symmetric sample, the spectrum of the nutation calculation is
presented in figure 8.24(a). In reality, the transient nutation suffers a relaxation, T2 p,
contributed from both T, and T2 relaxation's. Therefore, the spectrum calculated with a T2p
decay is shown in figure 8.24(b). The experimental nutation spectrum is in figure 8.24(c)
and is compared with calculation in figure 8.24 (e) and (f). Since the shimming of magnet
is not perfect, there is a difference near the peak between the experiment and calculation.
However, the shapes of the spectra are almost the same.
The spectra in figure 8.25 represent the different nutations of the homogeneous coil
by varying the pulse length and the RF power. Figure (a), (c), (e) and (g) show the
different nutation spectra with different nutation pulse lengths, 2gts, 5gs, 10gs and 20gs
146
under an RF power 20 W. The peak frequencies of them are the same at about 23.5 kHz.
Thus, the homogeneous RF field strength is about 5.52 Gauss at 20W RF power. The
field strength for the 2W RF power in figure (b), (d), (f) and (h) is about 1.76 Gauss at a
7.5MHz peak frequency. The RF field strength is reduced by 3.13 times while the RF
power is decreased by 10dB. The horizontal axes from the (a) to (h) are frequency in kHz.
Figure 8.25 (i) and (j) show the spectra at different nutation pulse lengths in gs.
6
4
2
2 5
5r
7 5
10
12 5
15
17 5
(a) Calculation without T2p
2 5
5
7 5
10
12 5
15
2 5
17 5
5
7.5
10
12 5
15
17 5
(c) Experiment Spectrum
(b) Calculation with T2p
25
10 12 5 15 17.5
' 25
5 7 5
0 12.5 1
75 1
25
10 12.5 15 17 5
5 r 7 .5
5 r 7.5
Comparing (a), (b) & (c)
(d) With and without T2p (e) Calculation & Experiment (f)
Figure 8.24 Nutation Spectra of the Quadrupole RF Coil from Calculation & Experiment.
The vertical and horizontal axes are relative amplitude and frequency in kHz respectively.
1
1
0
0.8
8
0.6
06
0.4
0.4
0.2
0.2
50
100
150
200
r
250
(a) 2 Is Length with 20 W Power
20
40
60
80
80
100
100
120
120
(b) 2 gs Length with 2 W Power
1
1
0
60
40
20
0.8
8
0.6
0.6
0.4
0.4
0.2
0.2
i50
20
40
60
80
100
10
20
30
40
(d) 5 gs Length with 2 W Power
(c) 5 gs Length with 20 W Power
147
50
0.8
0.6
0.4
0.2
10
20
30
5
40
10
20
15
25
(f) 10 gs Length with 2 W Power
(e) 10 pgs Length with 20 W Power
1
0.8
0.6
04
0.2
20
15
10
5
2
4
6
8
10
(h) 20 pgs Length with 2 W Power
(g) 20 ts Length with 20 W Power
0.6
0.4
0.2
2
u's
5
2
10 us
us
6
us
20 us
10 us
is5 us
(j) Different Lengths with 2 W Power
(i) Different Lengths with 20 W Power
Figure 8.25 Spectra of the Homogeneous Coil for Pulse Lengths, 2gs, 5gs, 10gs &
20gs, with different RF Powers, 20W & 2W. (a) to (h) are in kHz. (i) and (j) are in gs.
1
0 8
0.8
0.6
0.6
0.4
0.4
0 2
0.2
f.
r
,
CrL
-r'
w
50
40
30
20
10
14Y
u
5
10
15
20
25
(b) 10 pgs Length with 2 W Power
(a) 10 pgs Length with 20 W Power
1
0.8
0.8
0.6
0.6
0.4
0.2
L
10
5
.WA
-
0.4
0.2
MjA! LI
15
k
r
2
I
(c) 25 lgs Length with 20 W Power
""
4
6
8
(d) 25 gs Length with 2 W Power
148
10
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
4
6
8
0
-0.2
-
1
2
3
4
5
(f) 50 gs Length with 2 W Power
(e) 50 gs Length with 20 W Power
1
1
0.8
0.8
0.6
0.6
0.40.4
0..
0.0.2
00.2
-0.2
1
25 us
-' ..
50 us
10
us
25
us
50 us
(h) Different Lengths with 2 W Power
(g) Different Lengths with 20 W Power
Figure 8.26 Nutation Spectra of the Quadrupole Coil for Pulse Lengths, 10gs, 25gs &
50gs, with different RF Powers, 20W & 2W. (a) to (f) are in kHz. (g) and (h) are in gs.
Figure 8.26 presents the different nutations of the quadrupole coil by varying the
pulse length and the RF power. The (a), (c) and (e) show the different nutation spectra
with different nutation pulse lengths, 10 gs, 25 gs and 50 gs under 20W power. Since
the frequencies vary from 0 kHz to 7.5 kHz, the quadrupole field strength is zero Gauss at
the center and 1.76 Gauss at the edge of a radial symmetric sample. The field strength for
2W RF power in the (b), (d), and (f) vary from 0 Gauss to 0.564 Gauss with respect to the
0 kHz at the center and 2.4 kHz at the sample edge. The RF field strength is reduced by
3.12 times while the RF power is decreased by 10dB. This result is the same as the
homogeneous coil. The horizontal axes in the figure 8.26 from (a) to (f) are frequency in
kHz. The (g) and (h) show the spectra at different nutation pulse lengths in gs.
8.3.3 Spatial Orthogonality of Two Fields
Since pure NQR imaging requires two separate coils with their RF circuit, one of
which is on while the other is off, the fields of them have to be decoupled. To determine
this decoupling efficiency (spatial orthogonality) quantitatively, we may use sample
spinning technique, which could be described by quantum mechanics or classically.
The spin spinning about the center of a sample is represented by its orbital angular
momentum, L. The exponential operator of the orbital angular momentum provides the
spin dynamics at time, t, and position, r,
149
((8.27)
= e-'itL ,I(,F)eeWstL
where L = F x p, is the cross product of the radial vector, F, from the origin to the spin
and the spin linear momentum, P, and w, is the spinning angular frequency.
In the coil laboratory frame, the spin experiences two rotations. One of them is a
rotation at the Larmor frequency, oo, about the z axis of the spin laboratory frame, and the
other is the spinning at the o s about the z axis along the sample center. Therefore, the total
motion of the spin is the superimposition of those two rotations. Since the Lz and Iz are
commute, the two superimposed rotations can be decomposed into two separate rotation,
wol e i stLz
e )
I(Wt)j +=,t)
for
e
_
R(t) = e(
SeiotLz eiW
0otiz
(8.28)
-zzi]=0.
The spin dynamics within the coil can be calculated from its initial states of the spin
in equation (8.29) and (8.30). For the homogeneous coil, the spinning in equation (9.29b)
has no effect because the initial spin state is position independent in equation (8.29a).
Thus, the spectra of the sample spinning are the same at different spinning frequencies.
The experimental results show this phenomenon in figure 8.28 and figure 8.30(a).
H(O)= (0
sin(Blr,
/2)
(8.29a)
0)
IH(t)= R-1 (t)IH(O)R(t)= sin(Bjr,/12 )(-sin(ot) cos( 0ot)
0)
(8.29b)
However, the rotation of spins in a quadrupolar coil in equation (8.30b) is
influenced by the spinning since the initial spin state is position dependent in equation
(8.30a). The central frequencies of the spectra are shifted with the spinning frequencies.
The experimental results are shown in figure 8.30(c).
IQ(0,)
= e-'Lz (0
sin(,Grr,,/2) O)eiOLz
(8.30a)
(-sin(Coot + ost + 0)
ia(t,F) = R-'(t)Ia(o,F)R(t) = sin(Grr,,/ 2 ) cos(oot
st + 0
0
150
(8.30b)
In order to study the sample spinning in general and quantitatively, a simple
calculation is based on the Bloch equation and EMF theory. In the LAS, after it is excited
by either a homogeneous or quadrupole RF pulse, x,/2, from the Z axis, the spin in the
transverse plane will remain the same at any time and position, as represented,
I(t)= iH(t) = sin(2GBjr/ 2 )(-sin(ot) cos(ot)
o)
=lQ(t) = sin(Grr,,/2)(-sin(aozt+O) cos(cot+O)
(8.31)
o)
The sample spinning is equivalent to a reverse rotation of the RF coil about the
LAS. Therefore, the position, 7(t) = (r(t),O(t),z(t)) = (x(t),y(t),z(t)), of the RF coil/field
at time, t, is changed with respect to this spinning frequency, ws, calculated from its initial
position, T = (r, , z) = (x, y, z), in equation (8.32)'s,
(x(t) y(t)
'1
z(t)
1) = T-'(F,)Rz(Wt)T(7F)(rcosO rsinO
0 0
T( 0 1 0
001
000
r,cosc
0
r,sin 0c
0
1
1) (8.32a)
(8.32b)
Lcos
Cst
-sin Ost
sin st cos Cst
Rz(cst) =
0
0
0
0
z
0
0
0
O
0
I
(8.32c)
I
where T,= (rc,Oc,0) is the spinning center. The result described in equation (8.33) is the
form of their Taylor expansion and becomes simple when the center is at the LAS origin,
r(t)=
(t)
Talor
Expension
r + Ar((Ost) F'=O
r
(Ost))
(8.33)
st
The RF fields, Bf, at time, t, and position, 7(t), are represented in equation (8.34)
for the homogeneous and quadrupole RF fields. Since the
151
o.
is much larger than the os,
the derivative of the RF field to time with the m, as a constant is approximated to zero.
Thus, the induced EMF, 71, in the coil is calculated in equation (3.35).
Since the
homogeneous RF field, !BH, is position independent in equation (8.34), the induced EMF
is Os constant in equation (8.35a). The spectra of it are the same as experimental data in the
figure 8.28 and 8.30(a), which have no difference when the o s is changed.
B =
H=B,(1
0
0)
+ A,(t)) 0) (8.34)
=B= G,(r + Ar(st))(cos(O + A(ost)) sin(8
7
-J
(8.35a)
dF = M o cos(wot)
6H
_dt
>>
Do
0o
0
oc Mo
Jcos(wot + A (cost))dOj r(r + A,(O t))sin(yGrrr,1 )dr
oc Mo
cos(wot + COst)
2
While the quadrupole RF field, Bf,
(8.35b)
for F = 0
in equation (8.34) is position dependent, the
induced EMF in equation (8.35b) varies with the o,. In a special case, where the field and
sample are symmetric about the Z axis in the LAS, i, = 0, the induced signal is a simple
sinusoidal function with the sum of the two angular frequencies, oo+ms, by neglecting the
T 2 decay. Figure 8.27(a) indicates the FID simulations of equation (8.35b) with different
spinning frequencies, OHz, 10Hz and 20Hz, about the center. The spectra of simulation
and experiment are presented in figure 8.27(c) and 8.30(c) respectively. The peaks of them
are shifted with the spinning frequencies from OHz to 20Hz linearly.
In general, the z axes of a sample and a radial RF field are not the same. There is a
position offset of these two origins, Fc = (rc,0). When the sample is spinning about the
sample center, the field strength it experiences is given in equation (8.34). As an example
of the spinning center at (R,j,0), a FID calculation is displayed in figure 8.27(b) from
equation (8.35b). By comparing it with the spinning about the center, the difference can be
represented by a very small negative peak at zero frequency in figure 8.27(c)
From the analysis and experiment of the sample spinning, the frequency of the
spectral peak either remains the same or varies linearly with respect to the homogeneous
coil or the quadrupole coil when the spinning frequency is changed. If the radial RF field
152
is not symmetric of the sample, a small ne Egative peak will appear at zero frequency, the
amplitude of which grows as the spinning frequency increases.
FID with Rotation 1 OHz about Center
FID without Rotation
.
250
300nn
200...
200
150
100
......................
S100..
. ................
50.....
0
-
-100
-200
FID with Rotation 20Hz about Cent er
300
1 0 00
500
0
t ms
Comparasion:O--,10---,20-300
200
200
1000
500
t ms
0
...
........
.
100
. ...
.
1
0
100
-100
:
'"
-100
-200
1000
500
O
t ms
--200
0
500
10( O0
t ms
8.27 (a) FID with Sample Spinning at 0, 10 & 20 Hz about Sample Cen ter
FID with Rotation 1 OHz off Cent er
FID without Rotation
250
300
200
200
. ....... ...
150
...... . . .
10 0
50
-100
0
1000
500
t ms
FID with Rotation 20Hz off Center
400
0
-200
500
0
t ms
Comparasion:O= -,1 0= -400
101
00
,20=-
200
-200
-200
-400
500
0
t ms
1000
1000
500
0
t ms
8.27 (b) FID with Sample Spinning at 0, 10 & 20 Hz out-off Sample Center
Coilppfison of Rotation at 0, 10 & 20 Hz about Center(+) & Off-Center(o)
....................
4
3
..............................................
-
-
3
0.
-
-----
-
...........
. .
-1
-20
-10
10
Frequency Hz
20
A0
4O
8.27 (c) Spectral Comparison of Sample Spinning at 0, 10 & 20 Hz about/off Center
Figure 8.27 Spectra of Sample Spinning at Different Frequencies about/off Sample Center
in the RF Quadrupole Coil.
153
The following experiments demonstrate the decoupling between the homogeneous
and quadrupole coils and the symmetry of the radial RF field, that is, spatial orthogonality.
The FID's after a ninety degree excitation pulse is collected by the same coil in a Bruker
AMX400 Spectrometer. The 900 pulse lengths for the homogeneous coil, the coupled and
decoupled quadrupole coil are 35gs, 40gs and 50gs respectively after the magnet is
shimmed to 27Hz, 10Hz and 2Hz. The sampling time are 100 gs, 300 gs and 300 gs.
4
10
3
10
83
10
-100
50
-50
-100
100
5
10
5
10
4
10
4
10
3
10
3
10
2
10
1
10
-100
100
50
-50
100
50
-50
(c) Spinning at 20 Hz
(b) Spinning at 10 Hz
(a) Spinning at 0 Hz
Figure 8.28 FID Spectra of the Homogeneous Coil for Sample Spinning at Different
Rates. There is no frequency shift due to sample spinning.
The three spectra in figure 8.28 indicate that the homogeneous coil is well
decoupled from the quadrupole coil since there is no additional peak shifted away from
those peaks at the zero frequency as the spinning frequency is increased.
8
3
8
10
2 5 10
2
8
4.
10
3.
10
2
10
1
1
8
10
1 5 10
I
10
-10
20
-20
nz
20
8
8
10
3
10 8
2.5
10
2.
10
IL
(b) Decoupled Spinning at 0 Hz
(a) Coupled Spinning at 0 Hz
3.5
40
4.
10
3.
10
2
10
1
10
8
1.5 10
-10
1.
10
5
10
10
20
-20
Hz
(c) Coupled Spinning at 10 Hz
20
40
(d) Decoupled Spinning at 10 Hz
154
I
8
3 5 11
3
3 5 10
3
1(
2.5 11
2.
10
8
2.5 10
2
10
8
1 5 108
1I
1 5 11
1.
1I
1
10
5.
11
5.
10
10
-10
20
7
20
-20
Iz
40
H
(f) Decoupled Spinning at 20 Hz
(e) Coupled Spinning at 20 Hz
Figure 8.29 Spectra of Quadrupole Coil for Sample Spinning at Different Rates with
Coupled and Decoupled Effects Respectively. The peaks from the quadrupole coil are
shifted according to the spinning frequencies while the peaks at zero frequency in (c) & (e)
are the coupling effect from the homogeneous coil and remain at the same position.
Figure 8.29 (a), (c) and (e) describe the quadrupole coil coupled with the
homogeneous coil, an unsymmetric radial RF field generated by this coil or both of them
because there is a split peak at the zero frequency as the spinning rate varies. The (b), (d)
and (f) represent the well decoupled quadrupole coil with a radially symmetric RF field
since the spectral peak is linearly increased with the changes of the spinning frequency.
As a conclusion of the spatial orthogonality of two fields according to the
simulation and experiment, figure 8.30(a) indicates that the homogeneous RF coil is
spatially orthogonal of the quadrupole coil. The (b) shows that the quadrupole RF coil is
coupled with the homogeneous coil, that the radial RF field is not symmetric of the sample,
or both of these. Figure 8.30(c) presents a spatially orthogonal quadrupole field with the
homogeneous field after the probe is redeveloped.
-100
,-O Hz
5.
10
4.
10
3.
10
2.
10
1.
10
10 Hz
'20 Hz
50
-50
Hz
(a) No any Differences on Homogeneous Coil Spectra at Different Sample Spinning Rates.
155
8
3.5
3.
2.5
10
10 Hz
0-Hz
20 Hz
8
10 Hz
1 O8
10
2.
10
1.5
1
1.
Hz
20
10
-10
(b) Quadrupole Coil Spectra with Coupled Effect at Different Sample Spinning Rates. The
gradient parks are shifted by spinning while the homogeneous peaks remain at the 0 Hz.
10 Hz
4.
20 Hz
10
8
3.
10
2.
107
1.
1
20
-20
40
Hz
(c) Quadrupole Coil Spectra without Coupled Effect at Different Sample Spinning Rates.
The spectra are shifted by the spinning frequencies, 10Hz and 20 Hz.
Figure 8.30 Spinning Spectral Comparison of Homogeneous and Quadrupole Coils and of
Coupling Effects between Homogeneous and Quadrupole Coils
8.3.4 Coupling Efficiency between Fields and Spins
The efficiency of coupling between an RF field and spins can be described by the
length of a 7r/2 pulse. Under the same situation, such as the same RF power, the shorter
the RF pulse length, the better the coupling between the RF field and spins. The following
experiment gives a set of spectrum of FID with respect to different pulse length. When the
pulse reaches 900, the transverse magnetization becomes the maximum so does the spectral
peak. The t/2 pulse lengths are 10s for the homogeneous coil with 20W RF power in
figure 8.31(a), 30gs for the homogeneous coil with 2W RF power in the (b), 50gs for the
quadrupole coil with 20W RF power in the (c) and 140gs for the quadrupole coil with 2W
RF power in the (d).
156
8
8
3.
10
3.
10
2.
88
108
2.
10
-1.
10
-2.
10
-3.
10
-2.
3.
10
2.
10
10
(b) Spectra of Homogeneous Coil with 2W
4.
10
3.
10
2.
10
8
8
8
8
1.
60 [
8
8
10
40
108
8
(a) Spectra of Homogeneous Coil with 20W
4.
20
108.
-1.-1.
10
20
40
60
*1.
80
10
1
1
01
50
100
150
200
250
(d) Spectra of Quadrupole Coil with 2W
(c) Spectra of Quadrupole Coil with 20W
Figure 8.31 Spectra of Different Pulse Lengths and Different RF Powers in Different
Coils. The r/2 pulse lengths (Horizontal axis), 10gs, 30 gs, 50 jis and 140gs,
correspond to the maximum peaks for different coils and different RF powers.
By comparing the change, -10dB, of the RF power from 20W to 2W, its strength
decreases three times, thus the pulse length for 900 flip angle increases almost 3 times. The
result is the same as figure 8.25 and 8.26.
For the same RF power in figure 8.31 (a) and (c) or in the (b) and (d), the ratio of
the t/2 pulse length between the homogeneous coil and the quadrupole coil is close to
20%. Therefore, it tells us that the coupling efficiency between the homogeneous coil and
spins is five times higher than between the quadrupole coil and spins. The reason of this
difference is that the geometry designs are different, which is discussed in section 8.2.5.
The quadrupole coil is located outside the homogeneous coil, while the sample is placed in
the center of the coils. Also since the quadrupole coil provides a radial field, the t/2 pulse
length is the averaged length. Overall, the coupling efficiency of the quadrupole is much
lower than the homogenous coil.
8.3.5 Isolation: Electrical Coupling between two Coils
To determine the isolation between the homogeneous and quadrupole coils, a series
of RF pulses, such as a three pulse sequence: 50gs-1000gs-50gs, with relatively low
157
powers are inputted to a small coil , which is placed inside the RF coils and provides an RF
signal as an excited sample. Either of the RF coil, which is on resonance but the other is
not, receives this signal. The output of it is fed back to an RF receiver. Firstly, by
opening the ADC before the pulsing, the shapes of the pulses are collected into computer,
as shown in figure 8.32 (a) or (b). Then, by controlling the RF channel selector to make
this coil off-resonance and the other on-resonance during the second pulse, the shapes of
the first and third pulses are obtained, as shown in the (c) or (d). The middle part during
the second pulse should be zero if it is well isolated with the other coil. By comparing the
figure (a) and (c), the isolation is almost 100% for the homogeneous coil, while the
Since the
quadrupole coil has only 62% by comparing the figure (b) and (d).
homogeneous coil sits inside of quadrupole coil, this geometry makes that the quadrupole
coil can "see" the homogeneous coil better than that it is "seen".
4.
7.
10
6.
10
10
8
10
3.
8
10
2.
88
10
1.
500
1000
1500
2000
2500
5.
10
4.
3
10 8
8
10
2
1.
10
10
3000
500
1000
1500
2000
2500
3000
(b) Quadrupole Coil on-on-on Resonance
(a) Homogeneous Coil on-on-on Resonance
8
4.
3.
2.
8
5.
10
8
4.
10
3.
10
2.
10
1.
10
10
8
8
10
8
10
8
1.
1.
10
10
500
1000
1500
2000
2500
3000
8
500
1000
1500
2000
2500
3000
(c) Homogeneous Coil on-off-on Resonance (d) Quadrupole Coil on-off-on Resonance
Figure 8.32 Coil Isolation Test (Vertical axis : Signal intensity) vs. Their On-Off
Responses Time (Horizontal axis : gs)
The Homogeneous coil isolation is about 100% while the quadrupole is about 62%.
8.3.6 Switching Time between Two Coils
Using the same setup, three pulse experiment, Pl-100ps-P2, is carried out to
determine the switching time of the coils by adjusting the first and third pulse lengths, P1
and P2. The only difference is that the coil is only on resonance during the P1 and P2.
When the P1 is less than 15gs, the transient part between the first and second pulses has
158
ringing phenomena. After the P1 is greater than 15gs, the ringing vanishes, as shown in
figure 8.33. It is the same for the P2. This result indicates that the switching transience is
about 15gs for both of the coils. Therefore, this period of time is going to be reserved
during experiments. There is no difference for the switching transience under different RF
power by comparing the (a) with the (c) or the (b) with the (d) in the figure 8.33.
8
8
4.
10
3
10
8
100
200
300
3.
10
2.5
10
2.
10
8
100
400
200
300
400
(b) Quadrupole Coil with High RF Power
(a) Homogeneous Coil with High RF Power
8
8
5.
10
4.
10
3.
10
2.
10
1.
10
8
10
8
4.
5.
10
3.
10
2
10
1.
10
8
8
100
200
300
400
8
100
200
300
400
(c) Homogeneous Coil with Low RF Power (d) Quadrupole Coil with High Low Power
Figure 8.33 Switching between Homogeneous Coil and Quadrupole Coil during the
Pulsing. By adjusting the first and third pulse lengths, the switching transience vanishes
when they are greater than 15gs. There is no switching transient effect in these
experiments, where the pulse lengths of the first and the third are 20 gs. The horizontal
axis is gs and the vertical axis is signal intensity.
The general characteristics of the RF imaging probe for NMR and pure NQR has
been discussed for switching between a cylindrical quadrupole B, gradient and a
homogeneous B1 . Such probe can normally generate the gradient fields simultaneously in
two orthogonal directions, and with a spatially varying spin state dependence. The
complications arise from this superposition and new sets of composite pulses for creating a
B1 gradient with a spatially uniform spin dependence have been introduced.
159
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164
Appendix
A.Basic Equations for Calculation
Coordinate Transformation:
Rotation Frames, (x', y'), rotating about Laboratory, (x, y), at o has a relation
with rotation matrix R,
=(R=-sin
cosctcot
sin Cott I
cos
(A.1)
(A.1)
(x' y')= Rz (Ot)(x y).
(A.2)
Angle Transformation Equations:
sin(a + ) = sin acos ± cos asin
cos(a + /) = cos a cos T sin a sin
(A.3)
sin a sin f = [cos(a - 3) - cos(a + /)]/2
cosacosp = [cos(a - 3) + cos(a + /)]/2
sin a cos / = [sin(a - /) + sin(a + 0)]/2
(A.4)
Angle Rotation Transformation:
Let (X,Y,Z) rotating about x,y and z axis in (x,y,z), there is the following relation:
(x
y
z) = RRotaton -about_(xyz)
(1
0
(X
Y
R-0
0
(A.5)
0
(A.6)
Rx_ = 0 cosP -sin
0 sin/3 cosp3)
cos 6
Z)
0 sin O
1
(A.7)
,-sin 8 0 cos 0
cos a
Rz-a = sin a
0
-sin a
O
cos a
0
0
1)
(A.8)
165
B.Homogeneous and Quadrupole Field: Lab -> Rot Frame
Homogeneous Field: B 1,Lab(tp)
->
B 1,Rot()
Boo = 2B, cos(ot + 9)q
= B,[cos(ot + 9)x + sin(ct + 9)^] + B, [cos(ct + q9)i - sin(wt + 9p)q]
(B.1)
Bx = 2B, cos(t + 9)
B, =0
(B.2)
Bx, = Bx cos(wt) + B, sin(wt) = B, [cos(p) + cos(2ot + ()]
(B.3)
B,. = -B x sin(ot) + B, cos(cot) = B, [sin(p) - sin(2t + 9)]
(B.4)
Bx, = B,[cos(p() +cos(2ct + p)] =B, cos(T)
B,. = B,[sin() )-sin(2cot + )] SB, sin(p)
(B.5)
(B.6)
,to ° q)= B[cos(p)i' +sin(q)5' ]
Quadrupole Field: B 1,Lab (t,)
LabI=
-> B ,Rot(0)
G,r cos(ot)re -i = G,r cos(ot)r[cos(O6) - sin(0)]
Bx = Grrcos(O)cos(wt)
(B.7)
(B.8)
B, = -Grrsin(O) cos(cot)
Bx. = Bx cos(wt) + B, sin(wt) = Gr/2[cos(O) + cos(2ot + 0)]
B, = -B x sin(ct) + B, cos(ot) = -Gr[sin(2ot + 0) + sin(0)]/2
Bx,=
B,
Gr [cos(O) + cos(2ot + 0)]
2
=
BR
1Qu(
Gr [sin(O) + sin(2ot + 8)]
2
= Gr
2 [cos()i'
Grrcos(8)
S
2
S Grrsin(O)
2
-sin(O)^']= Gr re2
(B.9)
(B.10)
Quadrupole Field: B ,Lab (t,,p) -> B1,Rot(O,)
B,
d
= Gr cos(ot +
O)re- '0
= Gr cos(wt + 9)r[cos(6)^ - sin(e0)5]
166
(B.11)
Bx = G,rcos(O)cos(t+ 9p)
B, = -Grsin(O)cos(cot + 9)
(B.12)
Bx. = B cos(cot) + B, sin(wt) = GC,[cos(O - 9) + cos(2wot + q + 0)]
2
(B.12a)
B,. = -B x sin(ct) + B, cos(ot) =
)+cos(2cot + p + 0)]
Gr[cos(6 -
B x,
=
B,.
=
Grr[sin(6 - 9) + sin(2t + (p+ 0)]
2
2
Gr [sin(o- 0) + sin(2cot + p + 0)]
2
Bo(0,0) = Grr [cos(O 2
=
- 9)'] = G re-z'
2
)' -sin(
cos(O
- (p)
2
= -rsin(- cp)
2
-
(B.12b)
(B.13)
(B.14)
C. Spin Nutation by a Homogeneous RF Pulse
Representations of Spin and RF Pulse:
I =Ix+ I + z
1(0) = I
=
1,RLat
1,Rot
(C.1)
2B, cos(ot + 9p)X
(C.2)
SB,[cos(qp)X +sin(p)Y']
Spin Nutation by Bloch Equation Method:
dlI
dt
dt y
x1
Lab
XB, Rot
dt
at
dx I/
t
d2Z
dt 2
t
(C.3)
in Rotating - Frame
(C.4)
(C.5)
-yB sin((p)Iz
i /at
/
in Lab - Frame
yB, cos()
yB sin()ixyB-
= yB, sin(p)
aIx
-
x -yB
cc
cos(p)
=-
167
(C.6)
dx/dt
(C.7)
-yB, sin(p)Icos(yB1t)
i/t = B,cos(T)Icos(,t)
i/
(,
sin(aBt,)
()=(-Isin
I
If let yBt, =
) = (-Isin(T)
,(IX Ix
Icos(yB,tw))
I cos p sin(yBlt, )
Icos(p) 0)
(C.8)
(C.9)
Exponential-Operator Method:
Spin Nutation by
(C.10)
HILab = -*
(C.12)
--_= B cos(q)x + sin(),.]
(C.13)
=[Ho, ]
^o,^
dt
dIx
dt
(C.11)
-Ah2B, cos(wt + p)1x
=
0 B1,Rot
Rot
=-
-ih
Bl,Lab
-iB 1{cos(T)[I,,]
+ sin(P)[i~,,
I I
[f-Ro,x]=
-
-
I
Io'x
]}
(C.14)
= -yB, sin(Tp)I = -yB1 sin(T)Icos(yBt)
dI
=
dt
tf'io,,(
dt = h
yBI cos(p)i = yB cos(p)Icos(yBt)
=
lRotIz]
Ro Iz
1 sin(p) x
- y cos(p),
T)^ = -yIsin(yB 1t)
(C.15)
(C.16)
D.Spin Motion Transformation from Rotating to Lab Frame
Transformation from Rotation Frame to Laboratory Frame by rotation Matrix Rz,
cos Ot -sin
Rz
Lab
=
t 0
sin Ct
cos Ot
0
0
0
1)
z
(D.1)
(D.2)
Rot
168
R-1
' ,-Rot
z+j
S' Lab
= R
S Lab
1
z+ o
,1,,
Rot
R - { -sA,s'B,s cos(yBt w)}
= RZ+o
{s'A,-sB,s' cos(yBt,w)}
-
(D.3)
(-sAcos(w't) - s'Bsin(t)
Sx
(D.4)
-sA sin('t)+ s'Bcos(wo't)
,.=
S z)
s' cos(yBt,)
Sx
rs
s"A cos(°"t) + s"Bsin(a"t)
=
' sAsin(o't)- s"Bcos(o"t)
s cos(Bt,)
j,
X=
(D.5)
- sB sin(WQt) - sin(-,Qt)]
s x + J x = sA -cos(OQt) + cos -OQt)
(D.6)
-IB sin(Qt) = -I cos(q) sin(yBt,) sin(cQt)
,.
= S' + S, = -IA
sin(wQt) = -Isin(p)sin(yBlt)sin(cQt)
(D.7)
(D.8)
Iz= sz + "z= (s + s")cos(BtBt) = Icos(yBtw)
s= s =s and I=s +s =2s.
where A = sin(q)sin(yBt,w), B = cos(T9)sin(yBtw)
Spin motion in Lab Frame:
Ixb
-Icos(
) sin (
yBt,)
sin ( O
t)
(
(D.9)
= -Isin(Tp)sin(B, tw)sin(wpt)
Iz
Icos(yBtw)
ILab =
E. RF Transformation from LAS to PAS
Transformation Matrix Rt, Ry_(-p ), Ry-0
and Rz-a
are given as, where
BX=B 1,LASX and By=B 1,LASY:
(0
R,
=
(E.1)
0
1l
169
R-
RI,
=
I
±Bx
0 -By'
1 01
0
cos 0
0
sin O
0
1
0
\-sin0
(E.2)
(E.3)
0 cos0)
cos a
-sina
O'
Ra = sin a
cos a
0.
0
1)
0
(E.4)
Since the LAS is transformed by assuming that first the PAS and LAS are identical,
by transforming by Rt, by rotating -P angle about y axis, by rotating 0 angle about y axis,
then rotating a angle about z axis.
The transformation relation is calculated by the
following equations:
BI,PAS-Lab
B1,PAS-Lab
=Rz-a R , ORv(_ f)RBl_ , LLab
Rz-a-R,_
-(-)RB1,LAS-Lab
(E.5)
Rz-aR,-oR(_
fl)R (,x
By
O)
(E.6)
2AS-ab,X
+ BlAS-LbY
(cos a sin
sin a sin
cos 0)
From the definition of the LAS and PAS in figure 3.2, there is a relation between
angle 3 and RF field component B1,LASX and B1,LASY in the rotation matrix Ry_(_):
cos
sinf
B2 lSXB
s
B
Y
s.(E.7)
LASX+B
B1,
2
B ASY
=
BSx
+ BI LASY
F. RF Transformation from PAS to RAS
Since the RF field B 1,PAS in the PAS has three components in x, y and z direction,
it has to be decomposed into three RF fields, then they are again decomposed into counterclockwise and clockwise terms for rotation transformation as below:
170
B,PAS
= 2B cos(ot+ p)(cosasinO sinasin0
=
+
,PASx+ B1,PAS
= B,PASxL
BI,PASL
B,PASxR
BI,PASR
B1,PASz
(F.1)
B1,PASyR +
B,PASy = 2B cos(cot +p)(0
=2Bcos (0
B,PASxL
=
B,PASxR
=
B,PAS,,L
=
,PAS,,
=
0
B,PASyL +
B,PASz
B,PASz
B,PASx = 2B, cos(wt + p)(cos a sin0
B,PASz
cos8)
0
O)=
sinasin0
,,PASxL +
(F.2)
l,PASxR
(F.3)
0) = hI,PASyR + AIPASyL
(F.4)
cos(ot + ))
B, sin 8 cos a(cos(ot + p) sin(wt + 9)
0)
-sin(cot + 9) 0)
B, sin0sina(-sin(ot+p) cos(ot + 9) 0)
B, sin 8 sin a(sin(t + q) cos(cot + p) 0)
Bsin0cosa(cos(ot + p)
(F.5)
After the decomposition of RF field in PAS, the components are transformed from
the principle axis system to rotation axis system, the details of which could be found in
Fictitious Spin section:
B1,RASL
= RZ+wQB,PASL
=
B, sin 0(cos(a + p)
sin(a + T) 0)
(F.6)
BI,RASR = Rz-,Bl,PASR = Rz-o[Bl,PASxR + B1,PASyR]
(cos(-Opt)
= B, sin 0 -sin(-cot)
sin(-cot)
x
cos(-W t)
0
0
Scos(coQt - 9)
sin(oQt - 9)
(F.7)
cosa -sin(wQ t-9) +sina cos(wQt -P)
0
= B, sin
BI,RAsz
= Rz±,QB1,PASz =Bcos0(
0
cos(oQtp
171
))
(0
0
0).
(F.8)
G. Wave Function Derivation
Time-Dependent Schrodinger Equation:
Using the wave function to the derivative and calculation of eigen energy
HQ~m=Emm,
ih
t
= ih Of
Et
,Y2
("
t
Cmme--
I
=
m=-
m= /2
Et
+3/2
(HQ+ H1,PAS-Lab f)t
(G.1)
me-
(ih dt +cmEm
Cme
+ HIPAS-Lab )Om
(
m=-3/2
tE
+3/2
SEcme
mt
h
(Em +
HI,PAS-Lb)Om
(G.2)
.
m=-3/2
HQy
i
Xcme
t
+3/2
_ Emt
+3/2
=
I HQm =
m=-3/2
E,n
jcme
h Em
m
m=-3/2
Based on the time-dependent Schrodinger equation in equation (3.56), there is the
following results with respect to 0 < t < t, and t > t,:
+3/2
I
iErn
ih
e' im
=
Ent
~Cme
Hl,PAS-LabOm
0
By using the orthogonality properties of On
Vnm =
nH1,PAS-Labom
for
0<
t < t
0m=-3/2
(G.3)
t t,
for
m =6nm
and letting
)nm
= (E, -E,)Ih and
after On is multiplied on both sides, the above equation becomes the
equation (3.57).
Calculation of Matrix Elements:
The calculation of matrix elements Vnm
=
nH1,PAS-LabOm
is based on the basic
principle of Quantum Postulates, where om=lm> for i1=0 and m=-3/2,-1/2,1/2 &3/2:
172
I(I+l)m)
Irm) =
i+m)
(G.4)
I(I + 1)- m(m ± 1)m1
=
(i±±i)/2
A
S- I=
Therefore, the matrix elements becomes equation (G.5) and have the results in Table G. 1,
where D= -2ho cos(wt + T), a=cosoxsinO, b=sin(sinO, c=cose:
Vm = (nHI,PAS- b m)
=-2ho cos(ot + p) (n (ix,i,)m)-(cos sinG sinoasin0
Table G. 1 Expectation value of the Matrix Elements
m=
-3/2
-1/2
1/2
n
II
1/2
3/2
-Dc3/2
DV /2(a+ib)
-Dc/2
D 1I2(a-ib)
0
D(a-ib)
0
0
By letting onm = (E, - Em)
0
D(a+ib)
Dc/2
DV /2(a-ib)
0
0
D /2(a+ib)
Dc3/2
= +oa,O ,the coefficient equations are given as,
/I
ih -3/2L
ihc-1/ 2
e-imD(a - ib)V/ 2c-3 2 - Dc/2c-1/2 + D(a + ib)cl /2
ih+ 1/2
D(a - ib)c-
-Dc3/2c
ihC+3/2
let's
3/2
Vnm
-3/2
If
c)
m),(n|, m), (nim)) (a b
=D((n
(G.5)
cos6).
assume
e
that
/2
t
+
3/ 2
+e
Dc/2c
D(a + ib) / 2c_-1/2
/2 +
(G.6)
t
e-iQ D(a + ib) /312c
3/2
D(a - ib)3,,/2cl/2 + Dc3/2c3 / 2
0o=oQ
and
exp(inoQt)=0
with
n
0
and
D=--ho,(e'( ') + e-'(It+')), (a+ib)=sin0exp(io) and (a-ib)=sin0exp(-ia), we have,
173
using
c-1/2
(G.7)
sin e-(-a)C-12
/2),
i(
-3~~-
i(-/2)a, sin ei,-a)c3 2
(G.7
i( /2)o, sin 8e '(+a)C3, 2
c+1/
+3/2
3/2), sin e'(2+a)c,/2
i(
In order to solve the first order derivative equations, first let's make the second order
derivative then combine the first order resulting in:
2
c 3 / 2 +((V/2)o, sin ) c±
+l
/2 +
((-/2)o, sin 0)2 ci
by assuming the initial values c
C-3 /
=0
2
=0
(G.8)
=-3/20, c,/2 =1/ -
and
Ic,m(t) 2 = 1, we have,
(G.9)
(l//2)cos(-/2)co, sin(0)t)
cos((V/2)0, sin(O)t)
Cs(1/-2)
C+1 / 2
for 0
2
(i/2)e-i(p-a)sin((-3/2)O, sin(8)t
C-1 / 2
c+32)
3
(ii2)e
-
i((+a) sin((/-3/2)0,
sin(0)t)
t < tw. From the equation (G.3) for t > t_, since the first order derivative is equal
to zero, the solution is a constant as below:
C-3/2(t))
C-3/2(t)
c- (t)
/2
- / 2(w)
+1 / 2
C+(tw)
C+32())
.c+312())
(il
)e-i(p-a) sin((V-3/2) , sin(O)tw)
(G.9a)
(/-COs
cos((-3/2), sin(O)tw)
(1/
)cos((
/2)o sin(O)tw)
(il/2)e-(p+a)sin((/-/2)o, sin(8)t,)
Calculation of Expectation Values of Spin System:
The calculation of expectation values of matrix elements of spin Ix, ly and Iz based
on the equation (G.4):
174
Table G.2 Expectation value of the Matrix Elements of Spin
n
m=
-3/2
-1/2
1/2
3/2
II (nl"xy, zm)
-3/2
3
0
0
00, -,
,i
0
2
2
2
-1/2
0, 1
i, i,~
0
1/2
1 iO
3/2
0
, 0, 1
2
0
.0
0
33
2
2
2
h
'
By neglecting e+iw",
the expectation values of spin system are given as:
3/2
3/2
)
" "t( n
C(tt)cm(tw)e'
'
=
(x
l
x
m)
n=-3/2m=-3/2
10-3
C3C2 2
2
=
-i
It V3
22
2
2
sin(
+i
+ CC 3 e
2
3t
22
2
- i
3/2
-
\
*
+C3Cle
22
C1 c 3 e
sin
+i
+ i L3sin( V_3Vf )et((+a)e-iQt -i
= cosasin(V
13t
+
2
)e
fnl
ItW
3
i(-l
2 2
2
si
)e-,(P-a)e-
sin(i
10311 V
22
2 2
2
iOQt
(G. 10)
(p+a) 1)e't
,e
)sin((ot + )
3/2
)=
c (t )Cm (tw )e' "mt'(n i 1m)
n=-3/2m=-3/2
10 3 It
c
I
e
1
2 2
3 c 22+icc
2
2
3C
ic*
2
= sinasin(
2
2
3/2
22
2
2 2
)e '- ( a
+ 8 sin(
fln) sin(t
o)31t
101 3t
3e
3
2
SiC3C l e
2 2
)e
n)e+)
sin( /-3ln )ei(+aesin(-
-
(i 2 )=
2
)e'(a)e
Ssin(,
c3e
ic
3t
22
22
3
2
(G.11)
e
+ p)
3/2
c:(tw)cm(tw)eImt(nizm)= OlTime-Consiant
n=-3/2 m=-3/2
175
(G.12)
H. Spin Nutation by a Quadrupole RF Pulse
Spin Nutation by Bloch Equation Method:
I =x +IV +
(H.1)
i(0) = I
= ri B a.
(H.2)
dt
y(Gr/2)sin(6 - go)Iz
y(Gr/2)cos(O - ))^
-2'( G,r/2)sin(O - T)I x - y(Gr/2) cos(O-q9)J
di, /dt
dI,/dt
Ix
(H.3)
q)sin(O
cos(0-- T)sin(y(Gr/2)tw)
T)sin(y(Gr/2)t,w )
I,=I
)
Icos(y(Gr/2)t,)
iz
(H.4)
Spin Nutation by Exponential-Operator Method:
p=A =
i"uad = _ *Ba
fLab
--
Quad
uf
(H.5)
+ I,+-(Gx
iz)
•
"lLab
[
Quad
(H.6)
_hGrrCOS(Ot + p)[cos(0)1x - sin(0)i
Grr [cos(6
X
-
(H.7)
T)I - sin( - T)I ]
(H.8)
dt
OdIx _r f
dt
dI
dtOSt
uad
hRo
X
Gr
=
y
sin( 1
L
-- 5
Gr
jcos(
i2
)Iz =
- g)[Ixx - Sin(O
(H.9)
Gr .(
2sn(-)Icos
r
uad
fot
'
cos
ua d
°
=di
i Rot
ht[,ua
, x
-
)Icos(y
r
r
t
(H.10)
22
d I
Gr2 1sin(7 Gr t )
176
(H.11)