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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. 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Wyld, H.W., "Mathematical Methods For Physics", Addison-Wesley, 1993. 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)