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This calculator applies the standard weak-field free-electron Landau diamagnetism model, with an optional low-temperature correction.
Example data table
These rows illustrate realistic starting values for metals, semiconductors, and heavier effective-mass systems under the same weak-field framework.
| Case | n (m⁻³) | B (T) | m*/me | T (K) | Volume | χL | M (A/m) |
|---|---|---|---|---|---|---|---|
| Copper-like electron gas | 8.5e28 | 1.0 | 1.0 | 300 | 1 cm³ | -4.067e-06 | -3.236 |
| Light-mass semiconductor | 1.0e24 | 0.5 | 0.067 | 50 | 1 cm³ | -1.373e-06 | -0.546 |
| Heavier effective-mass material | 5.0e27 | 3.0 | 2.0 | 100 | 5 cm³ | -7.907e-07 | -1.888 |
Formula used
The calculator follows the weak-field free-electron model. It first converts all inputs into SI units, then computes Fermi quantities and orbital susceptibility.
Core equations
- Fermi wavevector: kF = (3π²n)1/3
- Fermi energy: EF = ħ²kF² / (2m*)
- Fermi temperature: TF = EF / kB
- Pauli susceptibility: χP = 3nμ0μB*² / (2EF)
- Landau susceptibility: χL = −χP / 3
- Magnetization: M = χH ≈ χB / μ0
Advanced checks
- Effective Bohr magneton: μB* = eħ / (2m*)
- Cyclotron frequency: ωc = eB / m*
- Level spacing: ħωc
- Weak-field ratio: ħωc / EF
- Low-temperature correction: 1 − (π²/12)(T/TF)²
When the weak-field ratio grows, discrete Landau-level physics becomes more important and the linear response estimate should be treated cautiously.
How to use this calculator
- Enter the carrier density using the unit that matches your source data.
- Choose the magnetic field strength and unit for the applied field.
- Set the effective mass ratio m*/me for the material or band.
- Add temperature and sample volume to estimate finite-temperature response and total moment.
- Optionally include a background susceptibility to compare orbital response with a wider material contribution.
- Submit the form to show the result block above the form, then export the visible summary as CSV or PDF.
Frequently asked questions
1. What does this calculator estimate?
It estimates Landau diamagnetic susceptibility for a free-electron-like system, along with Fermi quantities, magnetization, magnetic moment, and weak-field validity checks.
2. Why is the Landau susceptibility negative?
Diamagnetic orbital currents oppose the applied magnetic field. That opposition makes the susceptibility negative in the standard sign convention.
3. When is the weak-field model appropriate?
It works best when ħωc is much smaller than the Fermi energy. If the reported ratio becomes sizable, full Landau-level quantization should be considered.
4. What does the effective mass ratio change?
It changes the Fermi energy, effective Bohr magneton, cyclotron frequency, and therefore the final susceptibility and magnetization values.
5. Does temperature always matter strongly here?
Not usually for dense metals, because T is often far below TF. The optional correction mainly matters when T/TF is no longer very small.
6. What is the background susceptibility input for?
It lets you add a separate material contribution, such as lattice or core response, so you can compare the orbital term with a broader total susceptibility.
7. Can I use carrier density in cm⁻³?
Yes. The calculator converts cm⁻³, nm⁻³, and m⁻³ into SI units before applying the formulas, so mixed data sources remain easy to use.
8. Why do I get zero magnetization at zero field?
Susceptibility is still a material property, but magnetization scales with the applied field in this linear-response model. No field means no induced moment.