Quantum Mechanics and the SemiClassical Description of Relaxation

1
Quantum Mechanics and the Semi-Classical
Description of Relaxation

• John C. Gore
• Vanderbilt University
• Institute of Imaging Science

ISMRM May 2005
2
Outline

• Review classical viewpoint - equations of motion
  of spins in a magnetic field
• Simple overview of quantum picture
• Review of simple quantum mechanics
• Effects of time varying magnetic fields -
  perturbation theory
• Calculation of transition probabilities -
  spectral densities
• Dipole-dipole relaxation in water - the effects
  of motions of the spins

3
Caveats

• Biological media are usually complex
• Heterogeneous at different levels
• Many different environments, types of interaction
• Multiple processes / mechanisms may contribute
• Discuss only principles of relaxation and
  formalism to describe relaxation
• Restrict to homogeneous system spin
• e.g. protons in water
• Focus on spin-lattice relaxation
• But ideas approach generalize to other types of
  interaction

4
What is relaxation?

• Relaxation the recovery of a disturbed nuclear
  magnetization back to equilibrium
• In equilibrium, M (the magnetization that results
  from aligning nuclear spins) lines up with the
  field Bo
• After an RF (B1) pulse at the resonant frequency,
  M points in some other direction
• Relaxation all those processes that speed the
  recovery back to equilibrium

5
RF pulses tip the magnetization

• We know that RF pulses on resonance can alter the
  directions of alignment of the nuclear spins
  (e.g. 90, 180 degree pulses)
• RF pulses alternating magnetic fields of
  amplitude 0.1 gauss, oscillating at the NMR
  resonant frequency (64 Mhz at 1.5T) applied
  orthogonal to the main field

6
RF pulses magnetic fields rotating at resonance
frequency for short times
Bo
B1 rotates at wo
B1
7
What causes relaxation?

• Relaxation is also a re-alignment of nuclear
  spins
• Will occur if transverse magnetic fields rotating
  at the correct (resonant) frequency are present
• When there are lots of these, relaxation will be
  fast a classical view of relaxation

8
Classical picture

• Spins have magnetic moments
• Equation of motion rate of change of angular
  momentum torque

mz changes if Bx,y present, rotating at w0
In rotating frame
Integrate
9
Simple quantum view of RF excitation
DE
DE hw0

• RF pulses whose frequency energy difference
  between levels can excite nuclei - this
  corresponds to changing their directions

10
Excited states are unstable

• excited nuclei will recover back to equilibrium
  - thereby re-establishing magnetization along
  field
• But takes very long time to happen spontaneously
  (1013 years)
• Needs some other process to bring it to 1second!

11
Relaxation by Stimulated Emission

• Requires a source of quanta of the same energy
  (frequency) as the difference in E levels of the
  nuclei
• Relaxation fast when there is a supply of
  suitable energy

12
Dipole field from nucleus
Each proton exerts a field 5 gauss at its
neighbour As molecule rotates or moves this
field fluctuates at a rate that depends on how
fast the molecule moves the needed source of
stimulating energy
13
The local magnetic field fluctuates
The time scale over which the field fluctuates
depends on the motion responsible. The average
time is called the correlation time. It varies
with e.g. temperature, viscosity, mobility
14
Relaxation in pure water
translation
rotation
exchange
Overall rate of change of field experienced by
nucleus may depend on several processes
15
Fourier Transforms
t
J(w)
F.T.
1/t
frequency
Local field contains multiple frequencies
Local field is time dependent
16
Frequency content (spectral density) of local
field variations
Slow fluctuations (low frequency motions)
Power at resonant frequency determines rate of
relaxation
Medium
Amount of each frequency present
Fast fluctuations (high frequency)
J(w0)
frequency
64Mhz
17
Relaxation rates depend on power at w0
J(w0)
1/T1
Solid (slow)
Liquid (fast)
Timescale of fluctuations
R1 depends directly on the power in the local
field at the NMR resonant frequency, J(w0)
18
How do we describe this quantitatively using QM?

• A single spin system can be described by wave
  functions corresponding to different spin states
  in Diracs notation
• Observeable physical properties are calculated by
  applying appropriate operators to the
  wavefunction, integrating the expectation value

19
Energy levels Eigenvalues

• An operator (the Hamiltonian) acts on and
  the expectation value is the energy of the spin.
• Time independent Hamiltonian dominated by Zeeman
  term

• , b denote z component of Ang. Momentum Iz
• Eigenvalues have units of angular frequency

20
General state
are the relevant orthogonal base
states (eigenfunctions) before relaxation occurs
(i.e. time independent) A general state a
linear combination
The amplitudes C correspond to relative
probabilities of each state
21
Relaxation

• Occurs when there is a suitable time dependent
  term in the Hamiltonian able to induce transition
  between the spin states
• Time-dependent term can cause
  states
• to interconvert
• Rate of relaxation rate at which an initially
  pure state is converted to a mixture of
• Found using perturbation theory in which
  is treated as a small perturbation of

22
Time Dependent Schrodinger Equation
Time dependent wave functions

• General solution a superposition of possible
  states

solutions to time independent Schrodinger
Equation Complex time dependent coefficients
23
Using orthogonal properties

• The coefficients (probabilities) vary with time
  - so the overall states are changing
• This equation expresses the fact that the
  perturbation is causing transitions between
  

24
Time dependent perturbations

• Suppose probability of being in state at
  any moment is
• Then, rate at which a spin initially in goes
  to is
• is the probability per sec for 1 spin
• Observeable transition probability is the
  ensemble average

25
Just algebra!!!
Ensemble average
26
Relaxation rate
Ensemble average over all spins for randomly
varying field Convenient to separate the time
dependence from the spin elements
a selection rule H1(t) can convert states only
if it connects
27
Correlation Functions
If H1(t) (and therefore F(t)) are stationary
random functions
Spectral Density
If
then
28
Relaxation
Probability of changing state for whole
ensemble Relaxation rate
selection rule determines which transitions
occur
ensemble average (snapshot) measure of
strength of interaction
Power within the local H field at the frequency
of transition frequency w
29
Specific example - water dipolar relaxation

• Consider 2 spins, I and S (no J coupling)
• Predict 4 allowed configurations for the energy
  levels

30
Populations balance
Add all transitions
ETC.
Cross relaxation
Direct relaxation
31
For water (2 identical spins)
32
Transition Probabilities - Water

• Require explicit expressions for transition
  probabilities (W)
• These are determined by the explicit nature of
  the interaction that gives rise to the perturbing
  local field e.g. dipolar, chemical shift
  anisotropy, quadrupolar etc
• For water, dipolar interaction - requires a
  geometrical model of the field and an explicit
  description of the motion of the molecules
• Hence not so easy for more complex systems

33
Classical

• Energy of interaction between two dipoles is

• is vector connecting two dipoles

34
Quantum View
is the angular momentum operator
varies with time if molecule rotates or
translates to describe rotation effects (most
important) then convert to polar coordinates
35
Use simplified operators
Raising () and Lowering operators (-) simplify
effects of I operators on the z angular momentum
The interaction caused by a dipolar field is
equivalent to applying combinations of these
angular momentum operators to the spin
wavefunctions of the system
36
In polar coordinates
37
Interpretation

• A causes no spin flips
• B causes a flip-flop
• C causes a single spin flip
• D causes a single spin flip
• E causes the double flip
• F causes the double flip
• All terms are of the form

38
Calculate the average values
Use
Type of motion Distance
Frequency gt Rate
39
Finally
40
Transverse relaxation
41
Spectral density v frequency effect of
correlation time
Motional averaging
42
Effect of correlation time
43
Summary

• Fluctuating local fields at the resonance
  frequency (and twice that value) promote spin
  lattice relaxation
• Correlation times describe their time dependence
  spectral densities provide equivalent insight
  into their bandwidth
• Quantum theory leads to same physical picture as
  simple classical view

44
Thank you

• Slide presentation will be available at
• www.vuiis.vanderbilt.edu

45
T1 Dispersion with Frequency
46
Paramagnetic materials

• Unpaired electron spins e.g. in transition metal
  ions generate very strong local magnetic fields
• Electron magnetic moment 700 x proton
• If these fields contain fluctuations at the NMR
  frequency they promote relaxation
• If the fields fluctuate too fast then they may
  not be very effective

47
Electron dipolar field
Water molecule in hydration layer around
paramagnetic ion undergoes enhanced relaxation
from unpaired electron spins
H
O
H
Gd
48
Local fields

• Effectiveness of a paramagnetic species depends
  on
• Number of unpaired electrons
• How close water can approach to species
• How rapidly the field from the electron
  fluctuates
• Theory Solomon, Bloembergen and Morgan equations

49
Paramagnetic relaxivity

• T1m relaxation time of protons in the
  coordination shell of the metal ion
• S electron spin of ion
• g , gN electron and proton g factors
• , ?N Bohr and nuclear magneton
• a hyperfine coupling constant
• r ion-nucleus distance
• wI ,ws Larmor frequencies for the proton and
  electron spins
• tc, te correlation times describing the rate of
  change of the local dipolar fields experienced by
  the nuclei

50
Relaxation rate R2 depends on power at w0 AND w0
1/T2
Rate
1/T1
Solid (slow)
Liquid (fast)
Timescale of fluctuations
51
T1 /T2 depends on time scale of fluctuations
100
Solid (slow)
T1 /T2
Tissue10
1
Liquid (fast)
Timescale of fluctuations