Density of States Calculator
Find the density of states g(E) for a free electron gas using g(E) = (1/2π²)(2m/ħ²)^(3/2)√E.
📊 What is the Density of States Calculator?
The density of states calculator finds how many quantum states are available at a given energy, per unit energy per unit volume, for a free electron gas. Enter an energy and a particle mass, and it returns g(E) in both the practical states-per-eV-per-cm³ unit and SI units.
Density of states is a foundational concept in solid-state physics: it tells you how "crowded" a given energy level is with possible quantum states, independent of whether those states are actually occupied. Combined with the Fermi-Dirac occupation probability, it determines how electrons distribute themselves across energy levels in a metal, which in turn governs electronic heat capacity, conductivity, and optical response.
The key relationship, g(E) = (1/2π²)(2m/ħ²)^(3/2)√E, shows that in three dimensions the density of states grows with the square root of energy: higher energies have more available states, not fewer, a direct geometric consequence of counting momentum states within an expanding sphere in momentum space. This same free-electron model underlies the Fermi energy calculation for simple metals.
This calculator is useful for solid-state physics and materials science students exploring electronic structure, heat capacity, and conductivity, since it handles the physical constants and unit conversions automatically.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 - At copper's Fermi energy (7.05 eV)
Example 2 - At a low energy (1 eV)
Example 3 - At sodium's Fermi energy (3.24 eV)
❓ Frequently Asked Questions
🔗 Related Calculators
What is density of states?
Density of states, g(E), is the number of available quantum energy states per unit energy interval per unit volume, at a given energy E. It answers the question 'how crowded is energy level E with possible states', which is essential for working out how electrons fill up a material and how they respond to heat, light, or electric fields.
What is the formula for the free-electron-gas density of states?
g(E) = (1 / 2π²) × (2m / ħ²)^(3/2) × √E, where m is the particle's mass, ħ is the reduced Planck constant, and E is the energy. This applies to a 3D free electron gas, the same model used for the Fermi energy of simple metals, and already includes the factor of 2 for the two spin states.
Why does density of states increase with the square root of energy?
In 3D, the number of possible momentum states within a sphere of radius k grows as the volume of that sphere, proportional to k³, or E^(3/2) since energy scales as k². Differentiating with respect to energy to get states per unit energy interval brings this down by one power, giving the characteristic √E growth of the 3D free-electron density of states.
How is density of states related to the Fermi energy?
The Fermi energy is the energy up to which all states are filled at absolute zero; the density of states describes how many states exist at each energy along the way. Integrating g(E) from 0 up to the Fermi energy and multiplying by the occupation (1 at T=0, below EF) recovers the total electron density n, connecting the two concepts directly.
What units is density of states usually expressed in?
SI units are states per joule per cubic metre, but this is an inconveniently large, awkward number for everyday materials science, so density of states is far more commonly quoted in states per electronvolt per cubic centimetre (eV⁻¹cm⁻³), which gives values in the more readable 10^21 to 10^23 range for typical metals.
Does density of states depend on temperature?
No, g(E) itself is a purely geometric property of the allowed quantum states and does not depend on temperature. What does depend on temperature is the Fermi-Dirac occupation function, which determines what fraction of those available states are actually filled by electrons at a given temperature.
Why is density of states important for electronic properties of materials?
Many key properties, electronic heat capacity, electrical conductivity, and optical absorption, depend on how many electron states are available near the Fermi energy, precisely what g(EF) measures. A large density of states at the Fermi energy generally means a material responds strongly to small perturbations, contributing to high conductivity or a large electronic heat capacity.
Does this formula work for semiconductors and insulators?
Not directly. This √E formula assumes a simple free-electron (parabolic) band with no energy gap, which describes metals reasonably well. Semiconductors and insulators have a band gap and more complex band structure near their band edges, requiring an effective-mass version of the formula or full band-structure calculations for an accurate density of states.
How does density of states differ in 1D, 2D, and 3D?
The energy dependence changes with dimensionality: g(E) scales as 1/√E in 1D (diverging at low energy), as a constant independent of E in 2D, and as √E in 3D, the case covered by this calculator. These distinct signatures are used experimentally to identify whether a nanostructure behaves like a quantum wire, well, or dot.
Why is the factor of 2 for spin included in this formula?
Each spatial quantum state (each allowed momentum) can hold two electrons, one with spin up and one with spin down, by the Pauli exclusion principle. The standard density of states formula bakes in this factor of 2 automatically, so g(E) counts spin-up and spin-down states together rather than requiring a separate multiplication.