1D Diffusion Analysis

The diffusion analysis module in NMRAnalysis.jl provides a basic tool for analyzing diffusion-ordered spectroscopy (DOSY) experiments. This analysis determines the translational diffusion coefficient of molecules and estimates their hydrodynamic radius using the Stokes-Einstein relation.

Launching Diffusion Analysis

To launch the analysis of a diffusion experiment, specify the experiment folder number:

using NMRAnalysis
diffusion("106")

If no folder is given, you will be prompted to enter a path.

Analysis Workflow

1. Parameter Confirmation

When you launch the analysis, the program will parse the experiment parameters and ask you to confirm or correct them:

Parsing experiment parameters...
Gradient pulse length δ = 4000.0 μs (2*p30). Press enter to confirm or type correct value (in μs): 
Diffusion delay Δ = 0.1 s (d20). Press enter to confirm or type correct value (in s): 
Gradient shape factor σ = 0.9 (gpnam6 = SMSQ10.100). Press enter to confirm or type correct value: 
Max. gradient strength Gmax = 0.55 T m⁻¹ (typical value for Bruker systems). Press enter to confirm or type correct value (in T m⁻¹): 
Enter initial gradient strength (%): 5
Enter final gradient strength (%): 95
Enter gradient ramp type ('l' / 'q' / 'e'): l

2. Integration Region Selection

A spectrum plot will be displayed, and you'll be asked to define the integration region and noise estimation area:

Defining integration region - please enter first chemical shift: 7.5
Defining integration region - please enter second chemical shift: 9
Enter a chemical shift in the center of the noise region: -1

3. Visual Confirmation

The program displays the selected integration and noise regions for confirmation:

Displaying integration and noise regions. Press enter to continue.

4. Fitting and Results

After pressing enter, the fit runs automatically using the Stejskal-Tanner equation. Results are displayed in the terminal while a fit plot is shown:

[ Info: Viscosity: calculation based on Cho et al, J Phys Chem B (1999) 103 1991-1994

┌ Info: diffusion results
│ 
│ Current directory: /Users/chris/NMR/crick-702/kleopatra_231201_CRT_GSG_C163S
│ Experiment: 106/pdata/1
│ 
│ Integration region: 7.5 - 9.0 ppm
│ Noise region: -1.75 - -0.25 ppm
│ 
│ Solvent: h2o
│ Temperature: 298.1992 K
│ Expected viscosity: 0.8892568047202338 mPa s
│ 
│ Fitted diffusion coefficient: 1.248e-10 ± 4.0e-12 m² s⁻¹
└ Calculated hydrodynamic radius: 19.67 ± 0.62 Å

5. Saving Results

Finally, you can save the fit figure:

Enter a filename to save figure (press enter to skip): diffusion-fit.png
Figure saved to diffusion-fit.png.

The file format is automatically chosen based on the extension (e.g., .png or .pdf).

Theoretical Background

Stejskal-Tanner Equation

Experiments are fitted to the Stejskal-Tanner equation with finite gradient length correction:

\[I(g) = I_0 \cdot \exp \left( -\left[ \gamma\delta\sigma g G_\mathrm{max} \right] ^2 \left[\Delta - \delta/3 \right] D \right)\]

Where:

• $I(g)$ is the signal intensity at gradient strength $g$
• $I_0$ is the initial signal intensity
• $\gamma$ is the gyromagnetic ratio
• $\delta$ is the gradient pulse length
• $\sigma$ is the gradient shape factor (0.9 for trapezoidal gradients)
• $G_\mathrm{max}$ is the maximum gradient strength
• $\Delta$ is the diffusion delay
• $D$ is the diffusion coefficient

Hydrodynamic Radius Calculation

The hydrodynamic radius $r_h$ is calculated using the Stokes-Einstein relation:

\[D = \frac{kT}{6\pi \eta r_h}\]

Where:

• $k$ is the Boltzmann constant
• $T$ is the temperature
• $\eta$ is the dynamic viscosity

The viscosity is estimated based on solvent type and temperature using the formula from Cho et al., J. Phys. Chem. B (1999) 103, 1991-1994.

Noise Estimation

Noise levels for peak integrals are calculated by integrating a matching region of noise and taking the standard deviation across diffusion gradient strengths. This approach relies on good quality baselines for accurate noise estimation.