1. Random field fluctuations Introduction
… are random
… have zero mean
… and small amplitude
𝐵 𝑥,𝑦,𝑧 2 = 0
𝐵 𝑥,𝑦,𝑧 ≪ 𝐵 0
Arrow representation taken from
SpinDynamica (M. Levitt)

2. Random field fluctuations Introduction
Random fluctuations …
… can be slow … and fast
How to measure the rapidness of fluctuations?
 correlation function

3. Random field fluctuations Correlation function: Auto-correlation
… defined as sliding integral with itself
… is called Auto-correlation function
𝑮 𝝉 =𝑭 𝟎 𝑭 𝝉
= 𝑭 𝟎 𝑭 𝝉

4. Random field fluctuations Correlation function: Cross-correlation
… defined as sliding integral with another function
… is called Cross-correlation function
𝑮 𝝉 =𝑭 𝟎 𝑯 𝝉
= 𝑭 𝟎 𝑯 𝝉

5. Random field fluctuations Auto-correlation
Use sliding integral to judge rapidness of random field fluctuations
… slow
fluctiations
… rapid
fluctiations

6. Random field fluctuations Auto-correlation
Auto correlation for random field fluctuations
~ exponential decay
Only valid for isotropic tumbling
other motions have different (more complex) correlation function
𝑮 𝝉 =
… slow
fluctiations
… rapid
fluctiations

7. Random field fluctuations From Auto-correlation to Spectral density
random fluctuations
… are oscillatory (like an FID)
which frequencies contribute to the oscillations?
 perform Fourier Analysis to obtain spectral density
Lorentzian (= Cauchy distribution) FT FT FT
Intensity scaled x 10

8. Random field fluctuations Spectral density
Transition probabilites ~ Spectral density ~ ~
600 MHz