Lifshitz–Kosevich Quantum Oscillation Calculator
Introduction to Lifshitz–Kosevich damping and dHvA visibility
In de Haas–van Alphen analysis, the hard question is often not how to write an oscillation formula but whether a particular branch should remain measurable after temperature and scattering have reduced its amplitude. This Lifshitz–Kosevich calculator estimates the thermal factor R_T, the Dingle factor R_D, their combined attenuation, and the inverse-field period associated with a chosen frequency F. It is therefore a quick way to connect a proposed field window with the signal strength you might realistically expect.
The five inputs describe one branch at one evaluation point: frequency F, magnetic field B, temperature T, effective cyclotron mass m*/mₑ, and Dingle temperature T_D. They do not all do the same job. The frequency controls how tightly oscillations are packed in 1/B, while field, temperature, mass, and disorder determine the damping estimate. Keeping those roles separate makes a weak signal easier to diagnose.
What this Lifshitz–Kosevich calculator reports for one orbit
The Lifshitz–Kosevich calculator answers a practical visibility question: what fraction of an ideal oscillation amplitude survives the standard thermal and Dingle terms for this orbit? R_T is the thermal survival factor. R_D is the disorder or scattering survival factor. Their product is a compact relative attenuation estimate, not an absolute magnetization or voltage prediction.
The final result, Δ(1/B), answers a different question. It gives the period in inverse tesla, so it indicates whether peaks will be widely separated or densely packed when your data are plotted against 1/B. A high-frequency branch can have an excellent damping product yet still require sufficiently fine field sampling to resolve its closely spaced periods.
How to use the Lifshitz–Kosevich calculator for a dHvA branch
Start by selecting one clearly identified quantum-oscillation branch. Enter its dHvA frequency F in tesla, then enter the magnetic field B at which you want the estimate. Add the sample temperature, the effective mass ratio obtained or expected for that orbit, and the Dingle temperature describing its broadening. Click Compute Damping Factors to update the result panel.
For a useful comparison, change only one quantity between runs. For example, hold F, m*/mₑ, and T_D fixed while testing two fields, or hold B fixed while testing two plausible temperatures. This turns the page into a small sensitivity check rather than a collection of unrelated values. All entered quantities should be positive, and the mass ratio must remain dimensionless.
Lifshitz–Kosevich inputs: choosing F, B, T, m*/mₑ, and T_D
dHvA frequency F, in tesla, identifies the orbit and sets the inverse-field period. It is not a damping parameter in the calculation below. Magnetic field B, also in tesla, appears in the denominator of both damping arguments. At higher field, the arguments become smaller and both damping factors move closer to 1, all else being equal.
Temperature T is the sample or measurement temperature in kelvin. Raising it broadens the Fermi occupation and lowers R_T, especially for a heavy cyclotron mass. Effective mass m*/mₑ is the orbit’s cyclotron mass divided by the free-electron mass. Although it is written with an asterisk, enter only its numerical ratio; a value of 1.2 means 1.2 mₑ.
Dingle temperature T_D, in kelvin, summarizes scattering broadening in the usual Dingle treatment. A larger T_D gives a smaller R_D. Use values that belong to the same sample, orientation, and orbit where possible. Mixing a frequency from one branch with a mass or Dingle temperature from another can produce an arithmetically correct number that has little physical meaning.
A sensible first pass is to check units before interpreting a surprising result. B and F are both measured in tesla but are not interchangeable: B is the field where damping is evaluated, whereas F determines periodicity in inverse field. Likewise, T and T_D are both kelvin quantities, but one describes thermal broadening and the other scattering-related broadening.
Formulas behind the Lifshitz–Kosevich damping estimate
The calculator uses α = 14.69 when B is in tesla, temperatures are in kelvin, and m* is expressed as m*/mₑ. First it forms the thermal argument X and evaluates X/sinh(X). It then forms the Dingle argument X_D and evaluates its exponential suppression. Finally, it calculates the inverse-field spacing directly from the frequency.
Because B is below the fraction bar in both arguments, increasing field reduces thermal and Dingle damping. Increasing T lowers R_T, while increasing T_D lowers R_D. The frequency F does not enter either damping expression used by this page. Instead, the spacing relationship Δ(1/B) = 1/F tells you how much inverse-field range corresponds to one oscillation cycle.
Worked example: a 700 T dHvA branch at 15 T
Consider a branch with F = 700 T, evaluated at B = 15 T and T = 0.8 K, with m*/mₑ = 1.2 and T_D = 1.5 K. The thermal argument is about 0.940, giving R_T of roughly 0.867. The Dingle argument is about 1.763, giving R_D of roughly 0.172. Their product is therefore about 0.149, indicating that disorder damping is the stronger limitation in this particular example.
The same 700 T frequency gives Δ(1/B) ≈ 1.429 × 10⁻³ T⁻¹. If the temperature is lowered while all other values remain unchanged, R_T rises, but the spacing does not move because F has not changed. If T_D is lowered instead, R_D rises. This contrast is why it helps to read the three reported quantities separately before drawing a conclusion about detectability.
Interpreting R_T, R_D, combined attenuation, and inverse-field spacing
Values of R_T and R_D lie near 1 when their respective damping mechanisms are weak and fall toward zero when damping is strong. Their product is often the fastest screen for relative visibility: a small product means the ideal oscillation amplitude has been heavily reduced before other experimental effects are considered. It should not be interpreted as a probability, a signal-to-noise ratio, or a fitted amplitude by itself.
Read Δ(1/B) alongside your actual acquisition window and sampling density. Larger F means a smaller inverse-field period, which means more cycles fit in the same 1/B interval. If the damping product is favorable but the period is extremely short, resolution and background removal may still become the practical challenge. Conversely, a well-spaced low-frequency signal can remain hard to observe if either damping factor is very small.
If a result looks counterintuitive, first confirm that the entered values describe the same orbit and that the units are correct. Next, rerun the calculation after changing one major input slightly. A physically consistent LK estimate should improve with larger B and worsen with larger T, m*/mₑ, or T_D. That directional check catches many transcription errors quickly.
Limitations and assumptions of this Lifshitz–Kosevich estimate
This page is a deliberately focused first-pass calculation. It applies standard thermal and Dingle damping factors to one branch at one field rather than fitting a complete field sweep. It does not calculate spin-splitting factors, magnetic-breakdown effects, angular dependence, chemical-potential oscillations, multiband interference, sample inhomogeneity, or instrument background. Any of those can matter in a real quantum-oscillation experiment.
The Dingle temperature is treated as a single effective parameter, an assumption that can be less reliable when scattering changes strongly with field, angle, or orbit. The effective mass is likewise assumed to be the appropriate cyclotron mass for the selected branch. Rounded display values are intended for comparison and planning; retain more precision in a formal analysis and use the conventions appropriate to the source experiment.
Use the calculator as a decision aid: it can help identify promising fields, temperatures, and branches before a fuller fit. For publication-quality interpretation, compare the result with the original measurement geometry, the relevant Lifshitz–Kosevich model conventions, and the analysis method used for the raw data.