7th lecture: Summary 6th lecture: Spin manipulations
π:
M=0
Angular distributions of emitted radiation, with or without polarization analysis,
can be derived from the elements of the corresponding rotation matrices.
σ+ + σ−:
1st example: Emission of photons from polarized atoms:
Massless photons have s = 1, and only m = +1 or m = −1 i.e. are always circularly polarized.
Linearly polarized light is a coherent superposition of m = +1 and m = −1 photons.
π-light has angular distribution of dipole antenna: W(θ) = ½sin2θ.
σ-light has angular distribution of ring antenna: W(θ) = ½ (1+cos2θ).
M=±1
2nd example: parity violating asymmetries W (θ )  1  A cos θ .
a) Spin rotation with stored particles:
1st example: μSR = muon spin rotation in condensed matter:
p + A → A' + π's, π± → μ±↑ + νμ↓, μ± → e± + νμ + νe, τ = 2.2 μs.
P-violating β-asymmetry precesses about inner crystalline fields.
F.-T. gives the distribution of Larmor precession frequencies in the sample.
2nd example: Perturbed γγ-Angular Correlations.
The second γ has an anisotropic distribution with respect to the first γ.
The intermediate state precesses about internal crystalline fields.
3rd example: atomic Hanle effect, for lifetime measurements of atomic states.
γ1
θ
M=±1
γ2
b) Spin rotation 'in-beam':
1st example: Atomic beam spin precession.
Pz ( B0 ) is the Fourier transform of the wavelength spectrum Φ(λ) of the beam.
2nd example: Magnetic moment of hyperons
The magnetic moment of the Λ0 is measured in-flight by passage through a strong magnetic field.
There is a small spin precession seen in the angular distribution of the emitted protons.
3rd example: g − 2 of the muon.
Dubbers: Quantum physics in a nutshell WS 2008/09
7.1
4.2 The spin resonance method
B
a) Larmor precession from the Bloch equation:
  γB  M , with the macroscopic magnetisation M  N  μ   N  s :
M
M
  M  0,
  M follows M
1. |M| is a constant of the motion: from M
2
2

hence dM / dt  2 M  M  0 , and M  const .
M
  B  0,
  B , follows M
2. Mz is a constant of the motion: from M
hence, for B along z: M z Bz  0 , and M z  0 .
3. The transverse components M  performs a rotary motion:
  ω 2 M , M
  ω 2 M ,
With ω0  γBz then M x  ω0 M y , M y  ω0 M x , or M
x
0
x
y
0
y
M x (t )  M x 0 cos ω0 t , M y (t )  M x 0 sin ω0 t for M x (0)  M x 0 , M y (0)  0 .
S
b) Effect of an oscillatory magnetic field along x:
For B1 (t )  2B1 (0) cos ωt , the Hamiltonian is, with ω1  γB x (0) :
H (t ) 
1
2
ω0


 2ω1 cos ωt
2ω1 cos ωt 
.
 ω0 
Rotating field approximation: Instead of trying to solve the
Schrödinger equation with time-dependent Hamiltonian,
we decompose the oscillating field into
two rotating components B1+ and B1− in the x-y plane.
One component (B1 ) rotates, with frequency ω, in the same sense
as the precession of M with Larmor frequency ω0.
the other component (B1−) rotates in the countersense.
+
N
2B1cosωt = B1+(t) + B1−(t)
←
=
→
+
Dubbers: Quantum physics in a nutshell WS 2008/09
7.2
Go into the coordinate frame that rotates about the z-axis in phase with B1+.
In this frame:
1. the field B1+ stands still,
2. the field B1− counter rotates with double frequency −2ω,
3. the Larmor precession about B0 slows down to ω0 − ω
(and changes sign for ω > ω0).
0
Therefore, in the frame rotating with frequency ω,
  (ω  ω)  M , or.
M
the Bloch equations read
0

M  γBeff  M .
with the effective field Beff  B  ω/γ :
In the rotating frame, M then precesses about Beff.
From p. 4.11, the polarization along z is: Pz  c1 (t )  c 2 (t )  2 c1 (t )  1 ,
2
2
2
ω12
sin 2 ( ω02  ω12 t ) ,
with c 2 (t )  2
2
ω0  ω1
2
Dubbers: Quantum physics in a nutshell WS 2008/09
7.3
When the frequency ω of the rotating frame equals the Larmor frequency: ω  ω0 ,
then, in the rotating frame, M stands still, i.e. there is no field acting on M,
that is, the magnetic field B0 is transformed away.
M
Hence, at resonance ω  ω0 :
In the rotating frame, M precesses solely about B1:
In the laboratory frame, M spirals up and down:
In the quantum picture the system performs Rabi oscillations
between the upper and the lower energy states E+ and E−:
E+
M
N+
ħω0
E−
N−
c) Effect of the counter-rotating component
Seen from the magnetization M, the counter-rotating component B1−
wiggles so fast that it does not really disturb the magnetisation,
except for a minute ('Bloch-Siegert'-) shift of the resonance frequency,
which we shall discuss later in the context of dressed atoms.
The counter-rotating component can be avoided altogether
by installing a rotating rf-field (rf = radio-frequency),
by using two coils at right angles with π/2 phase shift.
Dubbers: Quantum physics in a nutshell WS 2008/09
7.4
4.3 Basics of Nuclear Magnetic Resonance (NMR)
a) Continuous wave NMR (CW-NMR)
1. Nuclear polarization: Nuclei with spin in a magnetic field B have magnetization (cf. p. 5.2):
μB
μ2B
, with nuclear magnetic moment μ  g I μ N ,
M  Nμ Pz  Nμ tanh
N
2kT
2kT
and nuclear magneton μ N  μ B / 1836  3.2  10 8 eV/Tesla (m p /m e  1836 , proton: gI = 5.6).
3  10 7
At room temperature in field B = 10 Tesla, Pz  g I
 g I 10 5 <<1.
2
2.5  10
2. Energy dissipation: In resonance, nuclei in the ground state E− absorb rf-photons of energy ħω,
nuclei in the excited state E+ emit rf-photons of energy ħω.
The transition probability is ν1 = 2πω1 = 2π γB1 transitions per second.
If thermal equilibrium is maintained at all times: N  /N   exp(  μB/kT)  const ,
then the power absorbed is dE/dt  ν1 ( N   N  )ω  ν1 NPz ω , N  N   N  .
As N is of order 10−22 for a ~ cm diam. probe, NMR is detectable in spite of the smallness of Pz.
The time constant to install equilibrium is called the longitudinal relaxation time T1.
It is the rate at which energy flows from the magnetized nuclei into the condensed matter system.
For our purposes, the system is always in equilibrium if the relaxation rate T1−1
is higher than the spin flip probability ν1.
For the proton 1H, the gyromagnetic ratio is 2πγ  ν/B  42 MHz/Tesla .
At typical rf-field strength B1 ~ 0.1 mT, the spin flip probability is ν1 = 2π γB1 ~ 40 kHz.
If the relaxation rate T1−1 is much slower than this, then dE/dt and with it the CW-NMR signal vanishes.
3. Resonance signal:
At resonance the rf-power dissipated in the sample changes the quality factor Q  ν/ν  L/C  R
of the electronic resonance circuit. This is recorded electronically as the NMR-signal.
Dubbers: Quantum physics in a nutshell WS 2008/09
7.5
b) Relaxation
Mz
The longitudinal magnetization Mz returns to its equilibrium value M z with the spin-lattice relaxation time T1:
M z (t )  M z  (M z  M z (0)) exp( t/T1 ) . Example: M z (0)  M z : M z (t )  M z (1  2 exp( t/T1 ))
The precessing transverse magnetization M   M x2  M y2 vanishes with the spin-spin relaxation time T2:
1. Spin-lattice relaxation time T1:
t(s)
T1
M  (t )  M  (0) exp( t/T2 ) .
M
In condensed matter, equilibrium between the spin states E+ and E− is established by
time varying crystalline B -fields, whose random fluctuation spectrum includes the resonance frequency ω0
and hence can induce transition between the states E+ and E−, just like the rf-field B1, and thus changes Bz.
T1 varies strongly with temperature and external magnetic field T1  T1 (T , B0 ) . The reason:
The magnetic fluctuations are due to phonons which lead to thermal motion of neighbouring spins.
With increasing B0, the position of the Larmor frequency in the fluctuation spectrum shifts to higher values
where, in general, the spectral density of fluctuations becomes lower.
T2
t(s)
2. Spin-spin relaxation time T2:
'Flip-flop' spin-spin interactions with neighboring spins: ↑↓ → ↓↑ are energy and angular momentum conserving, hence they
do not change Mz, but destroy the transverse magnetization M  . T2 is at most weakly temperature or field dependent.
T2 is the time-constant of decoherence, because M  is off-diagonal in the density operator, which is the definition of coherence.
T1 is always longer than T2. Proof: In thermal equilibrium, the density matrix has only diagonal elements (p. 5.10).
Assume we had T2 > T1, then the off-diagonal terms of the density matrix
would persist after thermal equilibrium has been reached, in contradiction to the above.
Dephasing of M  can also happen by spatial variation of the magnetic field over the sample volume.
This relaxation is called inhomogeneous, because the precession frequency ω0 is different for each subgroup of spins in the mixed ensemble.
1
1
 T2inhom
Spin-spin relaxation is called homogeneous, it shortens the life time of a pure ensemble. Altogether, T21  T2hom
.
Dubbers: Quantum physics in a nutshell WS 2008/09
7.6
c) NMR signal shapes
Spin relaxation can, in a non-rigorous way, be taken care of in the Bloch equations in the form:
M
M
M x  γ( B  M ) x  x
M x  (ω0  ω) M y  x
T2
T2
My
M
. In the NMR case B  ( B1 ,0, B0  ω/γ ) this is: M y  (ω0  ω) M x  ω1 M z  y .
M y  γ( B  M ) y 
T2
T2
M Mz
M Mz
M z  γ( B  M ) z  z
M z  ω1 M y  z
T1
T1
If relaxation rate T11 is fast compared to the transition rate ω1, then Mz is not changed considerably and can be replaced by M0.
We then find from first two equations:
M x  iM y  (ω0  ω)( M y  iM x )  iω1 M 0  (M y  iM y ) / T2 .
With M y  iM x  i (iM y  M x )  iM  , and setting M   M x  iM y ,
this becomes M   [i (ω0  ω)  1/T2 ]M   iω1 M 0 .
The equation y   ay  b has the solution y  A exp( t/a)  b/a  b/a in the steady state.
iω1 M 0
iω1 M 0

(1 / T2  i(ω  ω0 )) , and
1 / T2  i(ω  ω0 ) 1 / T22  (ω  ω0 ) 2
ω  ω0
M x  Re M   γM 0
B1 .
2
1 / T2  (ω  ω0 ) 2
Hence M  
Using M  χB , we find the magnetic susceptibility for excitation along x:
ω  ω0
, a 'dispersive' Lorentz curve, and, with M y  Im M  , for excitation along y:
χ'  χ0
1 / T22  (ω  ω0 ) 2
1
χ"  χ 0
, a 'resonant' Lorentz curve.
2
1 / T2  (ω  ω0 ) 2
The half-width at half maximum (HWHM) of the signal ω  1/T2 is determined by the transverse relaxation time T2.
Dubbers: Quantum physics in a nutshell WS 2008/09
7.7
Qmoment
*
*
*
*
2
H
He
3
1
½
*
natural
abundance/%
100
20
80
1
99.6
0.4
0.04
100
5
100
0.015
~0
NB: even-even nuclei like 4He, 12C, 16O all have I = 0, i.e. no NMR.
In solids, the neighboring spins have random orientations, and each spin configuration
produces a different local magnetic field at the probe nucleus.
For a neighboring spin in one direction, the field B0 is shifted upwards,
while for a neighboring spin in the other direction, the field B0 is shifted downwards.
↑↑↑↑
↓↑↑↑, ↑↓↑↑, ↑↑↓↑, ↑↑↑↓
↓↓↑↑, ↓↑↓↑, ↓↑↑↓, ↑↓↓↑, ↑↓↑↓, ↑↑↓↓
↑↓↓↓, ↓↑↓↓, ↓↓↑↓, ↓↓↓↑,
↓↓↓↓
For N neighboring spins, the weight of a given up-down configuration is given by the binomial law,
and the resulting local field and hence the NMR line-shape has a Gaussian distribution.
There are various methods how to narrow this line-shape, which we will not discuss in detail,
the catchwords: magic angle spinning, multi-pulse NMR, saturation of neighbor-spin transitions.
In liquids, the local field varies so rapidly that that it averages our, because the nuclear spin cannot follow.
This effect is called motional narrowing. It leads to extremely narrow lines for NMR with liquid samples.
Dubbers: Quantum physics in a nutshell WS 2008/09
7.8
z, B0
d) Pulsed NMR
Mz
CW-NMR is not very efficient, because only one frequency at a time can be measured.
This is different in pulsed NMR, which measures free induction decay:
B1
A π/2-flip is induced by an rf pulse of duration t, given by ω0 t  π/ 2 ,
which rotates the magnetization from the z-axis into the x-axis.
Mx
y
x
After the rf-pulse is switched off, the magnetization precesses about B0,
and the time varying magnetic field induces an induction signal in a pick-up coil.
B1(t)
t
The transverse magnetization signal decays with a
time constant T2, called the spin-spin relaxation time.
All precession frequencies are present in the signal.
U(t)
The time dependent signal then is Fourier analyzed,
and all frequencies in the NMR spectrum are measured simultaneously.
f(t)
t
F-T →
g(ω)
U(t)
B1(t)
t
ω
Dubbers: Quantum physics in a nutshell WS 2008/09
7.9
e) Chemical shifts
In molecules in solution, the chemical environment of a nucleus
produces measureable shifts of the NMR line.
Neighboring nuclei (mostly protons) produce via spin coupling
a characteristic pattern of these shifted lines.
This pattern can be used as a finger-print of the
molecule or subunit of the molecule in question.
5 basic types of H present in the ratio of 5 : 2 : 2 : 2 : 3.
The peaks at 4.4 and 2.8 ppm are a CH2-CH2 unit.
The peaks at 2.1 and 0.9 ppm are a CH2-CH3 unit.
Dubbers: Quantum physics in a nutshell WS 2008/09
7.10