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Bose Einstein distribution

In quantum statistics, Bose–Einstein statistics (or more colloquially B–E statistics) is one of two possible ways in which a collection of indistinguishable particles may occupy a set of available discrete energy state . The aggregation of particles in the same state, which is a characteristic of particles obeying Bose–Einstein statistics .who recognized that a collection of identical and indistinguishable particles can be distributed in this way.

The Bose–Einstein statistics apply only to those particles not limited to single occupancy of the same state—that is, particles that do not obey the Pauli exclusion restrictions.

Such particles have integer values of spin and are named boson , after the statistics that

correctly describe their behavior.

The wave function of particle will overlap to each other because mean separation of particles is less than the thermal wavelength, which is identified by λ.

(where is defined as the thermal wavelength )

Suppose there are l states with energies, E₁,E₂,E₃,...E_l and degeneracy g₁,g₂,g₃...g_l. Respectively, in which the particles are distributed. If there is N numbers of indistinguishable boson particles out of these n₁,n₂,n₃.....nl particles is adjusted in energy level E₁,E₂,E₃,...E_l respectively.

It is given

The total no. of arrangements of the particles in the given distributions is given by

W = , W =

If n_i and g_i are large numbers, we can omit 1 in comparison to them,

So we have, W =

The Bose Einstein distribution of the particle among various states n_i =

The Bose-Einstein energy distribution is,f(E) =

Where, β =  and

Here, μ is chemical potential which is generally a function of T . when A >> 1, Bose- Einstein gas reduces to the Maxwell-Boltzmann gas. The chemical potential for Bose gas

is negative, but for photon gas is zero.

Example 11: Two indistinguishable boson particles have to be adjusted in a state whose degeneracy is three.

(a) How many ways the particles can be adjusted?

(b) Show all arrangement.

(a) n_i = 2, g_i = 3, W_i =  = 6 ways

(b) Total number of arrangement for 2 indistinguishable boson particles in state whose degeneracy is 3.

Examples 12: (a) Write down distribution function of photon at temperature T , if average energy in each state is given by ε= hv.

(b) What is density of state of photon gas between frequency v to v + dv

(c) Write down expression of no of particle for photon gas at temperature T .

(d) Write down expression of average energy for photon gas at temperature T.

(a) The Bose Einstein distribution is given by f(E) =  where

For boson μ = 0 so

For if average energy in each state is given by ε= hv then  f(E) = .

(b) If j is quantum number associate with frequency v then total no of frequency between v to v + dv is same as the number of points between j to j + dj . The volume of spherical shell of radius j and thickness dj is 4πj²dj

Hence all three component of j is positive (same as particle in box) and there are two direction of polarization so degeneracy g = 2 .

So number of standing wave g(j)dj =

It is given,

So density of standing wave in cavity is given by g(v)dv =

(c) N =

N =  put

The integral have value, N =

(d) U =

U =

U =

Example 13: A system consisting of two boson particles each of which can be any one of three quantum state of respective energies o, ,3 is in equilibrium at temperature T. write the expression of partition function.

Two boson can be distributed in three given state with their respective energy level shown in table

Z =

Fermi Dirac Distribution

In quantum statistics, Fermi–Dirac statistics describes distribution of particles in a system comprising many identical particles that obey the Pauli Exclusion Principle (The Pauli Exclusion Principle is the quantum mechanical principle that no two identical fermions (particles with half-integer spin) may occupy the same quantum state simultaneously.)

Fermi–Dirac (F–D) statistics applies to identical particles with half odd integer spin in a system in thermal equilibrium. Additionally, the particles in this system are assumed to have negligible mutual interaction. This allows the many-particle system to be described

in terms of single-particle energy states. The result is the F-D distribution of particles over these states and includes the condition that no two particles can occupy the same state, which has a considerable effect on the properties of the system. Since F-D statistics applies to particles with half-integer spin, these particles have come to be called fermions. It is most commonly applied to electrons, which are fermions with spin 1/2.

No. of ways W in which n_i indistinguishable particles to place in level with the

condition that only one particle or no particle can be placed in g_i each level i.e. identical

particles that obey the Pauli exclusion principle. (It is given )

It is given by, W =

Fermi-Dirac distribution of the particles among various states is given by

So Fermi Dirac distribution f(E) =

where β =  and A =

when A >>1, Fermi-Dirac gas reduces to the Maxwell-Boltzmann gas. Fermi-Dirac gas is said to be weakly degenerate when A > 1 , degenerate when A<1 and strongly degenerate when A = 0 . Strongly degenerate Fermi gas A < 1

The Fermi-Dirac energy distribution is, f(E)=

Where, μ is the chemical potential which is a function of T , i.e., μ = μ (T ) . The gas is

strongly degenerate (A= 0) at T = 0 , where μ = μ(0) = E_F . The limiting chemical potential is known as the Fermi energy EF of the gas and the distribution function can be

written as

Strongly degenerate Fermi gas at T = 0

At T = 0 , when E<E_F we have f(E) =

At T = 0 , when E>E_F we have f(E) =

Fermi function f (E) versus E at T

The number of energy states in the energy range from E to E + dE is . Here, g is the spin degeneracy, g = (2s +1) , where s is the spin quantum number of a particle.

The number of particles in the energy range from E to E + dE at T = 0 is

 for E<E_F

N =  =

Thus, the Fermi energy is

and the Fermi temperature T_F is defined as T_F =

The Fermi momentum p_F is given by p_F =

Total energy of the gas at T = 0 is

U =  =

Thus, at T = 0 we have U/N = 3/5E_F

Example 14: What is no. of ways if two fermions have to adjust in energy state whose degeneracy is three.

g_i = 3, n_i = 2 , W =

Two indistinguishable particles is shown by A, A

Example 15: Fermions of mass m are kept in two dimensional box of area A at temperature T = 0

(a) What is total number of particle if is Fermi energy E_F. 

(b) What is the energy of the system if is Fermi energy E_F.

(c) Write expression of energy in term of E_F and N

For two dimensional systems, density of state

And distribution function at temperature T = 0 , is given by

(a) N =

(b) E =  ,

(c) E =

Example 16: (a) If Fermi gas is at temperature T > 0 what will f(E_F) .

(b) At E = E_F + x , find the fraction of occupied levels.

(c) At E = E_F - x , find fraction of unoccupied levels.

(a) It is also interesting to note that at T = 0 , when  E = E_F we have

(b) At T > 0, fraction of levels above E_F are occupied and a fraction of levels below E_F are vacant. The fraction of occupied levels at the energy E is

At E = E_F + x , find the fraction of occupied levels is

(c) The fraction of unoccupied levels at the energy E is

At E = E_F - x , find the fraction of occupied levels is