Calculator Inputs
Formula Used
Self intermediate scattering function:
Fs(q,t) = (1 / N) Σ exp[i q · (rj(t) - rj(0))]
Directional real part: Re Fs(q,t) = average cos(qx dx + qy dy + qz dz)
Isotropic average: Fs(q,t) = average sin(q |Δr|) / (q |Δr|)
Normalized decay: Fnorm(q,t) = Fs(q,t) / Fs(q,0)
Relaxation time: find τ where Fnorm(q,τ) = 1 / e.
Effective diffusion estimate: Deff = -ln(Fnorm) / (q²t), when Fnorm is positive.
Collective estimate used here: F(q,t) ≈ S(q) × Re Fs(q,t).
How to Use This Calculator
- Choose a wavevector mode. Use magnitude and angles, or enter q components directly.
- Select directional averaging for a known q vector. Select isotropic averaging for random directions.
- Paste displacement rows as time, dx, dy, dz. Use repeated times for many particles.
- Set S(q), units, normalization, and precision.
- Press the calculate button. Review the result cards, graph, and table.
- Use CSV or PDF buttons to save the computed report.
Example Data Table
The sample below shows the expected input style for two particles at each time.
| Time | dx | dy | dz | Meaning |
|---|---|---|---|---|
| 0.0 | 0.000 | 0.000 | 0.000 | Initial reference displacement |
| 0.5 | 0.150 | 0.041 | -0.015 | Particle sample at later time |
| 1.0 | 0.260 | 0.085 | -0.026 | Larger displacement lowers correlation |
| 2.0 | 0.430 | 0.145 | -0.045 | Longer lag time observation |
Understanding the Intermediate Scattering Function
What the Value Shows
The intermediate scattering function measures how particle positions lose memory over time. It links microscopic motion with scattering observations. A value near one means the structure is still similar to the starting state. A value near zero means the selected wavevector has decorrelated.
Why Wavevector Matters
The wavevector q sets the length scale of the test. Large q values observe short distance motion. Small q values observe broader structural change. The same trajectory can therefore show fast decay at high q and slower decay at low q.
Self and Collective Views
The self function uses single particle displacement. It is useful for diffusion, cage motion, and relaxation checks. A collective function also needs density correlations between particles. This calculator gives a practical collective estimate by scaling the self result with S(q).
Directional and Isotropic Averages
Use directional mode when the q vector has a fixed direction. The calculator evaluates the cosine and sine phase terms. Use isotropic mode when the system is rotationally averaged. It then uses the sinc form based on displacement magnitude.
Relaxation and Fitting
The 1/e time is a common relaxation marker. It is found by interpolation between computed points. The stretched exponential fit estimates beta and tau from the normalized decay. A beta below one often suggests broad relaxation behavior.
Data Quality Notes
Good inputs need many particles, many time origins, and steady units. Noisy data can create negative values or oscillations. That does not always mean the calculation is wrong. It can reflect limited sampling, strong structure, or an unsuitable q value.
FAQs
1. What does this calculator compute?
It computes the self intermediate scattering function from displacement data. It also estimates normalized decay, relaxation time, effective diffusion, and a simple collective scaled value.
2. What input format should I use?
Use one row per particle sample. Each row should contain time, dx, dy, and dz. Commas, spaces, or semicolons are accepted.
3. Should I use directional or isotropic mode?
Use directional mode when q has a known direction. Use isotropic mode when directions are averaged or when only displacement magnitudes matter.
4. Why can values become negative?
Negative values can occur from oscillatory phase factors, limited sampling, or strong ordering. Check q, trajectory size, and averaging before interpreting them.
5. What is the 1/e relaxation time?
It is the time where normalized correlation first drops to about 0.3679. The calculator estimates it by linear interpolation.
6. Is the collective estimate exact?
No. Exact collective scattering needs density correlations from particle positions. This tool provides a useful scaled estimate using S(q).
7. What does fitted beta mean?
Beta comes from a stretched exponential decay model. Values below one indicate broader relaxation. Values near one resemble simple exponential decay.
8. Can I export the results?
Yes. Use the CSV button for spreadsheet work. Use the PDF button for a formatted report with the calculated table.