Topic 3.2: Fluctuating Magnetic fields

Question 1

• To get relaxation need …, to get … need … fields at … of transition. Get these fields from moving molecules, changing their … changes the magnitude of … interactions (e.g. dipolar coupling/… ) and gives an … field.

Answer 1

• To get relaxation need transitions, to get transitions need oscillating fields at frequency of transition. Get these fields from moving molecules, changing their orientation changes the magnitude of anisotropic interactions (e.g. dipolar coupling/CSA) and gives an oscillating field.

Question 2

• How can the effects of incoherent fields in a system be described in more detail? State any assumptions made along the way

Answer 2

• Must translate random fluctuations into a measure of how much each motion at the right frequency can be attained
• Liked to transition probability;
• higher prob –> transition more often –> relaxes quicker
• lower prob –> transition less often –> relaxes slower
• oscillating field average over long time is 0 (stochastic) so mean square used to define fluctuation magnitude
• assumed process is ergodic – average over long time for one spin is the same as average over many spins at any particular moment.

Question 3

• What is a correlation function in NMR?

Answer 3

• Correlation function, G(τ) is used to characterise how rapidly a random field fluctuates.
• G(τ) is defined as an ensemble average over a product of the fluctuating field at two different time points separated by τ

Question 4

• How does the correlation function change with the speed of a motion and its associated correlation time?

Answer 4

• Probability average of product being a positive number decreases as τ increases as greater chance of crossing 0
• G(τ) is therefore a decaying function

Question 5

(IMP) Motions are random, therefore involve fluctuation with a distribution of frequencies, how can one describe this distribution and why is it required?

Answer 5

• Need to know what fluctuations at the correct (Larmor) frequency are present and their strength as will influence transition probability
• To learn the frequencies of motions that make up these fluctuations take a Fourier transform (FT) of the correlation function.
• An FT of a G(τ) is called spectral density (J(ω))

Question 6

• How can a spectral density tell us how much a specific frequency contributes to motion?

Answer 6

Question 7

(IMP) What is the problem of attaining the spectral density of a correlation function?

Answer 7

• Only thing known about correlation function if that it is a decaying function
• Taking a Fourier transform of this can be difficult, so it is assumed G(τ) is a decaying exponential or sum of decaying exponentials, which can be analysed easier
• An FT of this is a Lorentzian (other side decays below 0)

Question 8

(IMP) How does a spectral density profile change with respect to the motion it is describing?

Answer 8

• Follows same rule as T₂ peak width

Question 9

(IMP) Explain the relationship between spectral density and correlation time. describe the features of the J(ω) plot

Answer 9

• Area under curve is independent of τ_c
• A shorter τ_c will have a higher frequency present in motion (faster motion)
• J(ω) is always largest at ω = 0 with J(0) larger for slower motion (lower frequency motion)
• Transition at certain frequency leads to different contribution of motion

Question 10

• For relaxation need … fields via changes of … of molecules/modulation of … interactions
• Need … at given frequency for a system; interested in frequency associated with … /… levels
• … motion described by correlation function describes fluctuations in …
• … … (…) gives spectral density Lorentzian of how much frequencies contribute to …
• Need a method to connect this to relaxation/transition probability

Answer 10

• For relaxation need oscillating fields via changes of orientations of molecules/modulation of anisotropic interactions
• Need oscillations at given frequency for a system; interested in frequency associated with α/β levels
• Random motion described by correlation function describes fluctuations in time
• FT gives spectral density Lorentzian of how much frequencies contribute to motion
• Need a method to connect this to relaxation/transition probability

Question 11

• Derive an equation that would allow understanding of what frequency fluctuations are important for spin-lattice relaxation

Answer 11

• Shows how magnetisation changes with time
• R₁ measures using inversion recovery

Question 12

• Relaxation rates are related to the probability between the … … levels at their given … frequency
• Probabilities for transitions are related to spectral density J(ω) at frequencies for appropriate transitions

Answer 12

• Relaxation rates are related to the probability between the spin energy levels at their given Larmor frequency
• Probabilities for transitions are related to spectral density J(ω) at frequencies for appropriate transitions

Question 13

(PPQ) Describe how longtidinal relaxation -T₁ is measured

Answer 13

• Spin-lattice relaxation measured using inversion recovery
• Perform 10 experiments varying τ, the signal from each will depend on how much equilibrium magnetisation has been rebuilt
• Measure intensity and fit resulting curve to equation

Question 14

• Lecture summary
  
  • Correlation function, G(τ) describes how a … magnetic field … in time (due to … )
  • G(τ) = 〈B_x(t)B_x(t + τ)〉 ≠ 0
  • … average or product of … field probing at two different time points in system
  • The further we sample, probability of points having the same sign … , so average to 0
  • … decay approximated
  • … density, J(ω) is a FT of a correlation function and tells us how much motion there is at different … (specifically those that induce … of interest)
  • Relaxation rates are related to the probability of transitions between the … … levels, i.e. transitions related to … frequency
  • Probabilities for transitions are related to spectral density J(ω) at frequencies for appropriate transitions
  • … recovery is a popular method to measure … … relaxation

Answer 14

• Lecture summary
  
  • Correlation function, G(τ) describes how a random magnetic field oscillates in time (due to motion)
  • G(τ) = 〈B_x(t)B_x(t + τ)〉 ≠ 0
  • Ensemble average or product of fluctuating field probing at two different time points in system
  • The further we sample, probability of points having the same sign decreases, so average to 0
  • Exponential decay approximated
  • Spectral density, J(ω) is a FT of a correlation function and tells us how much motion there is at different frequencies (specifically those that induce transition of interest)
  • Relaxation rates are related to the probability of transitions between the spin energy levels, i.e. transitions related to Larmor frequency
  • Probabilities for transitions are related to spectral density J(ω) at frequencies for appropriate transitions
  • Inversion recovery is a popular method to measure spin lattice relaxation