The electrons in a metallic solid may be considered to be a...

The electrons in a metallic solid may be considered to be a threedimensional free electron gas. For this case: (a) Obtain the allowed values of $k$, and sketch the appropriate Fermi sphere in $k$-space. (Use periodic boundary conditions with length $L$ ). (b) Obtain the maximum value of $k$ for a system of $N$ electrons, and hence an expression for the Fermi energy at $T=0 \mathrm{~K}$. (c) Using a simple argument show that the contribution the electrons make to the specific heat is proportional to $T$.

The electrons in a metallic solid may be considered to be a threedimensional free electron gas. For this case:
(a) Obtain the allowed values of $k$, and sketch the appropriate Fermi sphere in $k$-space. (Use periodic boundary conditions with length $L$ ).
(b) Obtain the maximum value of $k$ for a system of $N$ electrons, and hence an expression for the Fermi energy at $T=0 \mathrm{~K}$.
(c) Using a simple argument show that the contribution the electrons make to the specific heat is proportional to $T$.

Key Concepts

Free Electron Model

The free electron model approximates conduction electrons in a metal as non-interacting particles that move in a potential-free space, neglecting the periodic ionic lattice. This simplification allows for using quantum mechanics to derive properties like energy levels and density of states, making it a fundamental tool in understanding electronic behavior in metals.

Periodic Boundary Conditions

Periodic boundary conditions are used to simplify calculations in solid state physics by treating the finite system as if it were infinite and repeating. This assumption leads to the quantization of electron wavevectors, as the wavefunction must be continuous and periodic across the boundaries, resulting in discrete allowed values in momentum or k-space.

k-Space Quantization

The quantization of the electron wavevector (k-space quantization) arises from applying boundary conditions to the electron's wavefunction in a finite-sized system. The allowed k-values become discretized, typically in units dependent on the system size (for example, 2?/L), which is a crucial step in deriving the density of states and understanding the electronic properties of the system.

Fermi Sphere and Fermi Wavevector

At absolute zero, electrons fill available energy states up to a maximum energy level, forming a sphere in k-space known as the Fermi sphere. The radius of this sphere is the Fermi wavevector (k_F), which represents the highest occupied momentum state, and is directly related to the electron density in the material.

Fermi Energy

Fermi energy is the energy corresponding to the highest occupied electron state at absolute zero. It provides a reference point for the energies of electrons in a metal and is essential for calculating thermodynamic properties. The relationship between the Fermi wavevector and Fermi energy links the electron density to the energetic scale of the electrons in a solid.

Electronic Specific Heat and Temperature Dependence

The contribution of electrons to the specific heat in metals is determined by the excitation of electrons near the Fermi surface. Since only a small fraction of electrons at the Fermi surface can be thermally excited at low temperatures, the specific heat contribution is proportional to temperature (T). This linear dependence is a key result of Fermi-Dirac statistics and contrasts with the classical equipartition prediction.

Transcript

0:00 Hi there.

00:01 So for this problem, we are told in a one -dimensional system, the number of energy states per unit length is equal to l divided by blams constant, and this times the square root of two times the mass divided by the energy.

00:26 So with that said, l is the length of the sample, and m is the mass of the mass of the mass.

00:34 Of the electron.

00:36 So there are n electrons in the sample, and each estate can be occupied by two electrons.

00:46 So for part a of this problem, we need to determine the fermi energy at a temperature of zero kelvin degrees.

00:57 Zero kelvin degrees.

01:00 So to solve this problem, we are given that this expression in here, is the number of electrons and that depends on the energy.

01:17 Let's just put like a normal e in this case.

01:23 So we know that then also the value of e can take the value of two electrons or zero.

01:35 Now, the value of 2 is when the energy is less or equal to the fermi energy.

01:48 And it is zero when the energy is greater than the fermi energy.

01:57 Now, with that said, for part a, the number of particles is given by the following integral.

02:09 That is the integral from zero to infinity of the number of electrons that depends on the energy times the number and function times the differential in the energy...

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