Download Magnetic resonance measurements of hyperfine structure using optical pumping Contents

Magnetic resonance measurements of hyperfine
structure using optical pumping
Martin G. H. Gustavsson
January 15, 2003
Contents
1 Introduction
1.1 Spectroscopic notation . . . . . . . . . . . . . . . . . . . . . .
1.2 Selection rules and polarization . . . . . . . . . . . . . . . . .
1
2
2
2 Optical pumping
3
3 The applied magnetic field
4
4 Hyperfine structure
4.1 Hyperfine structure in an external magnetic field
4.2 Zeeman effect . . . . . . . . . . . . . . . . . . . .
4.3 Paschen-Back effect . . . . . . . . . . . . . . . .
4.4 Intermediate field . . . . . . . . . . . . . . . . . .
4.5 Selection rules . . . . . . . . . . . . . . . . . . . .
5 Concluding remarks
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Introduction
Optical pumping is a very useful experimental method, e.g. for accurate
frequency measurements, which can be applied to many different physical
systems. The idea is that large differences in population between the sublevels of the ground state in an atom can be obtained using polarized light,
which was shown by A. Kastler [1]. The first application of the method was
to do magnetic resonance experiments for determination of nuclear spins
and hyperfine-splitting constants. Optical pumping is still a useful accurate
method and is widely used, e.g. in atomic parity-violation measurements
which have been done by Vold et al. [2]. Moreover, optical pumping is essential in the large progress in laser cooling, which was awarded by the Nobel
prize in 1997 [3]. Another application is concerned with the first human lung
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images obtained by NMR with the use of 129 Xe, the gas was first polarized
by optical pumping and then inhaled and imaged [4].
Magnetic resonance measurements with the help of optical pumping is
one of several radio-frequency methods in atomic and molecular spectroscopy
and these methods are extremely accurate. Radio-frequency methods have,
among other things, been used for hyperfine-structure investigations. The
first motivation for doing theses investigations was to get a clearer understanding of fundamental physics, but such investigations have also resulted
in atomic clocks and the definition of the second (the second is presently
defined in the following way: “The second is the duration of 9 192 631 770
periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium-133 atom.”). Furthermore,
the Global positioning system, GPS, had not been possible without atomic
clocks.
1.1
Spectroscopic notation
The spectroscopic notation for an atomic state will be used here. A letter symbol is then indicating the sum L of the orbital angular momentum
numbers of the valence electrons, these symbols are
L=
symbol
0
S
1
P
2
D
3
F
4
G
5
H
...
...
A right subscript attached to the angular momentum indicates the total
angular momentum J. A left superscript indicates the spin multiplicity
2S + 1, where S is the sum of the spins of the valence electrons. The ground
state in e.g. hydrogen or an alkali atom is then denoted by a 2 S1/2 -term.
1.2
Selection rules and polarization
Transitions between different atomic energy-levels which proceed under absorption and emission of electric dipole radiation have the following selection
rules:
∆J
∆M
= 0, ±1 0 → 0 forbidden
= 0, ±1
where J is the resulting angular momentum and M is the projection of
J along an applied magnetic field (see for instance the textbooks [5–7]).
The different values of ∆M corresponds to different angular distributions
of the radiation and to different polarization conditions. For ∆M = 0 the
electrical vector of the radiation oscillates parallel to the magnetic field and
the intensity is zero in the direction of the magnetic field. For ∆M = ±1 the
electrical vector rotates perpendicular to the magnetic field, the radiation
is thus circular polarized in the direction of the magnetic field and linear
2
polarized in directions perpendicular to the magnetic field. In the case of
∆M = 0 the radiation is called π radiation and for ∆M = ±1 the term σ ±
radiation is used. The notation π and σ stands for parallel and perpendicular
(German “senkrecht”), this notation refer to the orientation of the electrical
vector with respect to the magnetic field. Furthermore σ + radiation is righthand circular polarized and σ − radiation is left-hand circular polarized along
the direction of the magnetic field.∗
2
Optical pumping
The idea of optical pumping is to change the internal population of the
ground state atomic sublevels by shining light onto a cell containing low
density gas (see for instance the textbooks [5–7]). To be able to transfer
momentum to the gas atoms the light must have a frequency that coincides
with the atomic optical resonance frequency. This is taken care of by using
a spectral lamp of the same kind as the gas under study. A interference
filter then selects one of the spectral lines in the light spectrum for pumping
of the system. The light must also be circularly polarized for the pumping
to occur. Depending on whether the light is left- or right-hand polarized
one speaks of negative and positive pumping respectively. In the case of an
alkali atom a 2 S1/2 -term is the ground state and a 2 P1/2 -term is the first
excited state. Both these levels are split into two sublevels with MJ = +1/2
and MJ = −1/2, respectively, when a weak magnetic field is applied. The
basic principle of optical pumping in this case is shown in Fig. 1. When
σ + radiation is shone upon the cell containing the gas, the ground state
ν
MJ
2
P
1/2
+1/2
−1/2
ν
σ+
2
π
σ−
S
B
1/2
ν
+1/2
−1/2
σ+
Figure 1: The basic principle of optical pumping is here shown for the case of alkali
atom in the ground state. The sign σ ± denotes the radiation for a transition with
∆M = ±1, π denotes the radiation for a transition with ∆M = 0 and ν is the
resonance frequency for a induced magnetic dipole transition.
∗
N.B. the notation right-hand and left-hand depends on the direction and the convention to denote σ + radiation left-hand circular polarized can also be found in the literature.
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atoms with MJ = −1/2 are transferred to the excited MJ = +1/2 state, i.e.
∆MJ = +1. ¿From this state the atoms decay back to the MJ = −1/2 and
MJ = +1/2 ground states within 10−8 s at a rate of about 2 to 1. Atoms
that now find themselves in the MJ = +1/2 ground state cannot be excited
any more according to the selection rules. This implies that after some time
of pumping all atoms have been transferred from the MJ = −1/2 to the
MJ = +1/2 ground state. Atoms may be transferred back to the MJ =
−1/2 state via collisions with the cell walls (relaxation) but this is often
taken care of by introducing an inert gas in the cell, thereby diminishing the
number of collisions with the walls. When the gas is pumped it becomes
totally transparent to the pumping light and a signal from a detection of
the transmitted light will be high. Magnetic dipole transitions ∆MJ = −1
will be induced if an radio frequency (rf) field is applied perpendicular to
the static, thereby enabling transfer of atoms into the MJ = −1/2 state.
This give rise to absorption of pumping light and thus a fall of transmitted
light intensity. After the excitation due to the pumping, fluorescence light
from σ and π transitions is obtained, which can be observed by a detector
placed perpendicular to the magnetic field.
3
The applied magnetic field
The relation between the applied B-field and the current i of a pair of
Helmholtz coils is, according to the Biot-Savart law,
8µ0 N
B= √ i
5 5r
where the number of windings N is 315 and the radius r is 18.1 cm in this
case. This relation can then be rewritten as B = ki, where k = 1.565 ×
10−3 T/A. However, this k-value is approximative since the Helmholtz couple is not perfect.
Note that the earth generates a magnetic field which will make the application of a magnetic field in a certain direction a bit tricky.
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Hyperfine structure
The hyperfine structure is treated in several textbooks, e.g. Refs. [5,6,9–11],
a brief review follows below. Hyperfine structure (hfs) indicates a coupling
between the electronic and nuclear spins. The angular momentum (spin)
of the nucleus, I, and the electron, J , respectively, couple to form a total
angular momentum F , given by
F =I +J
4
with the quantum numbers
F
MF
= I + J, I + J − 1, . . . , |I − J|
= F, F − 1, . . . , −F .
The hfs is then a splitting of an atomic energy level caused by interactions
between the electrons and electrodynamical moments of the nucleus. For
electronic s-states and other states with J = 1/2, only the interaction with
the nuclear magnetic dipole moment contributes. This interaction can be
described by the Hamiltonian
Hhfs = −µI · B J ,
where µI is the magnetic (dipole) moment of the nucleus with angular momentum I and B J is the magnetic field generated by the electrons, this
magnetic field is proportional to J . The Hamiltonian can be rewritten in
terms of an “effective” operator as
Hhfs =
A
I ·J ,
h̄2
(1)
where h̄ is the Planck constant divided by 2π. The magnetic dipole constant
A depends on I and J, but not on F , MF , MI or MJ . It is easily seen that
A has the dimension of energy, but A-factors are for various reasons most
often given as frequencies, e.g. the A-factor for 133 Cs which is defined to be
2298.157 942 5 MHz [13]. From perturbation theory, the first order energy
contribution can be expressed as
Ehfs =
i
Ah
F (F + 1) − I(I + 1) − J(J + 1) ,
2
(2)
if the hfs can be considered as a rather small perturbation to the atomic
energy level. An atomic energy level is, according to Eqs. (1) and (2), split
into (2I + 1) hfs-levels if I ≤ J and (2J + 1) hfs-levels if J ≤ I. Each level
has a (2F + 1)-degeneracy. The separation between the levels F and F − 1
is
∆Ehfs = AF ,
where F = I + 1/2 for s-states.
4.1
Hyperfine structure in an external magnetic field
An interaction between the atom and an external magnetic field B (along
the z-axis) can be described with the Hamiltonian
Hm = −µJ · B − µI · B
= µB gJ J · B − µN gI I · B
= B(µB gJ Jz − µN gI Iz ) ,
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(3)
where µJ is the magnetic moment of the electrons, µB and µN are the Bohr
magneton and the nuclear magneton, respectively and the g-factors are given
by the relations
µJ
gJ =
µB J
J(J + 1) − L(L + 1) + S(S + 1)
= 1+
2J(J + 1)
and
µI
.
µN I
gI =
4.2
Zeeman effect
When the interaction between the external field and the atom is smaller
than the hyperfine interaction, as in the case of a weak external field (not
appreciably exceeding a value of about 10−3 T), Hm can be considered as a
perturbation to the hfs. This is the Zeeman effect for the hyperfine structure
and the first order energy contribution is given by
"
Em = B gJ µB
F (F + 1) − I(I + 1) + J(J + 1)
2F (F + 1)
#
F (F + 1) + I(I + 1) − J(J + 1)
MF ,
− gI µN
2F (F + 1)
where the second term can be neglected since gI µN is about mp /me ≈ 1800
times smaller than gJ µB . The energy relation gives that every hyperfine
level will be split into a number of equidistant sublevels, which will have
a separation linear to the strength of the magnetic field (as long as the
field is weak). States with J = 1/2 are split into two hyperfine levels with
F = I + 1/2 and F = I − 1/2, respectively, giving the following relation:
Em = ±
gJ µB B
MF ,
2I + 1
where the (+) sign is used for the higher F -value.
The ground state in an alkali atom is a 2 S1/2 -term, giving gJ = 2. Furthermore, if an external field is applied the hyperfine levels will be split into
a number of sublevels, which will be separated by
∆Em =
2µB B
,
2I + 1
the resonance condition for a transition between two sublevels is then
ν=
2µB B
,
h(2I + 1)
this relation can be used to determine the nuclear spin I.
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4.3
Paschen-Back effect
When the interaction between the external field and the atom is stronger
than the hyperfine interaction, as in the case of a strong external field (in
the order of magnitude 1 T), Hm is applied as a perturbation before Hhfs .
This is the Paschen-Back effect for the hyperfine structure and the first order
energy contribution is given by
Em + EHFS = B (gJ µB MJ − gI µN MI ) + AMI MJ .
However, this energy relation is not valid if the external field is to strong,
since such a field will generate Zeemann effect for the fine structure (i.e. the
spin-orbit coupling for states with l > 0).
4.4
Intermediate field
In the case of an intermediate field, i.e. when the interaction between the
external field and the atom is of the same size as the hyperfine interaction,
the sum of Eqs. (1) and (3) can be considered as the Hamiltonian for the
perturbation of the atomic ground state:
H = AI · J + µB gJ J · B − µN gI I · B .
The energy-eigenvalues in the case of J = 1/2 can be determined exactly
and are given by the Breit-Rabi formula [12]:
s
A
4MF
A
x + x2
E(F, MF ) = − − gI µN BMF ± (2I + 1) 1 +
4
4
2I + 1
where
x=
(4)
2B(gJ µB + gI µN )
,
A(2I + 1)
furthermore, in the case when MF = ±(I + 1/2) one obtains the simpler
relation
’
“
A
1
E(F, ±(I + 1/2)) = I ± B
gJ µB − IgI µN .
2
2
The (+) sign in Eq. (4) is used for the higher F -value, the terms containing
gI µN can be neglected as discussed above. The resonance condition for a
transition between the sublevels MF and MF − 1 for the same F -state is
obtained by taking the difference between E(F, MF ) and E(F, MF − 1):
s
4MF
A
(2I + 1)  1 +
x + x2 −
ν=
4h
2I + 1
where
x=
s
2gJ µB B
,
A(2I + 1)
7

4(MF − 1)
x + x2 
1+
2I + 1
(5)
1000
800
600
MF=2
1
0
400
F=2
−1
E (MHz)
200
0
−2
−200
−1
F=1
−400
0
1
−600
−800
−1000
0
0.005
0.01
0.015
0.02
0.025
B (T)
0.03
Figure 2: Breit-Rabi diagram for the ground state of
0.035
39
0.04
0.045
0.05
K (J = 1/2 and I = 3/2).
this relation can be used to determine the hyperfine splitting constant A.
It is sometimes convenient to use a series expansion of Eq. (5) to second
order in x:
“
’
2MF − 1 2
A
ν=
x−
x .
(6)
2h
2I + 1
Furthermore, a relation for the B-field can then be obtained from Eq. (6),
this relation depends, however, still on MF . One therefore gets a shift in the
B-field for a certain resonance frequency ν if the transition between the two
highest MF -levels is compared with the transition between the two lowest
MF -levels (for the same F -state). The relation for the shift in the B-field is
∆B =
8Ih2 ν 2
,
gJ µB A
which can be used to determine A if it is more accurate to scan the B-field
instead of the resonance frequency.
4.5
Selection rules
Transitions between different hyperfine levels which proceed under the emission or absorption of electric dipole radiation have the following selection
rules:
∆F
= 0, ±1
8
∆MF
= 0
(π radiation)
∆MF
= ±1 (σ radiation)
when a weak magnetic field is applied, in analogy with the discussion in
Sec. 1.2.
5
Concluding remarks
Hyperfine structure has many details and aspects, both from a theoretical and experimental point of view. This brief overview contains hopefully
the most essential aspects of the hyperfine phenomenon and the method of
optical pumping.
References
[1] A. Kastler, J. Phys. Radium 11, 255 (1950); Physica 17, 191 (1951);
J. Opt. Soc. Am. 47, 460 (1947).
[2] T. G. Vold, F. J. Raab, B. Heckel and E. N. Fortson, Phys. Rev. Lett.
52, 2229 (1984).
[3] S. Chu, Rev. Mod. Phys. 70, 685 (1998); C. N. Cohen-Tannoudji, Rev.
Mod. Phys. 70, 707 (1998); W. D. Phillips, Rev. Mod. Phys. 70, 721
(1998).
[4] M. S. Albert et al., Nature 370, 199 (1994).
[5] I. Lindgren and S. Svanberg, Atomfysik (Universitetsförlaget, 1974).
[6] H. Haken and H. C. Wolf, The Physics of Atoms and Quanta (SpringerVerlag, 1996).
[7] S. Svanberg, Atomic and Molecular Spectroscopy (Springer-Verlag,
1992).
[8] R. Bernheim, Optical pumping (W. A. Benjamin, 1965).
[9] H. Kopfermann, Nuclear Moments (Academic Press, 1958).
[10] N. F. Ramsey, Molecular Beams (Oxford University Press, 1956).
[11] L. Armstrong, Theory of the Hyperfine Structure of Free Atoms (John
Wiley & Sons, 1971).
[12] G. Breit and I. I. Rabi, Phys. Rev. 38, 2002 (1930).
[13] E. Arimondo, M. Inguscio and P. Violino, Rev. Mod. Phys. 49, 31
(1977).
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