Chapter 6 (part 1) Ideal Gas: Fermions and Bosons

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1 Last Time LECTURE 10 Gibbs Factor (Boltzmann factor with N) Gibbs Sum (Partition function with N) Adsorption (O 2 - myoglobin ) Semiconductor Impurity Sites Today LECTURE 10 Chapter 6 (part 1) Ideal Gas: Fermions and Bosons Fermions and bosons Fermi-Dirac Distribution (fermions) Bose-Einstein Distribution (bosons) Density of states 1

2 Matter No two pieces of matter may occupy the same space at the same time Half-true Two kinds of particles Fermions (spin 1/2, 3/2, 5/2, etc.) Cannot occupy the same state at the same time Pauli exclusion principle Bosons (spin 0, 1, 2, etc.) Can occupy the same state at the same time Antisocial All Follow the Crowd The difference between a gas of fermions and a gas of bosons becomes apparent at low temperature. 2

3 Electrons are Fermions Pauli exclusion principle Only One Fermion Allowed Why most matter cannot occupy the same space at the same time Bosons Can occupy the same space at the same time Photons are bosons lasers Helium is a boson superfluidity 3

4 Orbitals Orbital: a state of Schrödinger equation for one particle Thermal average occupancy f(ε,τ,µ) distribution function For weak interaction: approximate exact quantum state of a system by assigning particles to orbitals, each being a solution of single particle Schrödinger equation Different f for fermions and bosons 1 or 0 particles on each orbital Different Gibbs Sums for each orbital Any number of particles on each orbital f(ε,τ,µ)<<1: distribution for fermions and bosons should be similar Fermions Occupancy of any orbital may be 0 or 1. Look at one state or orbital : Only One Fermion Allowed Only two terms for a single fermion state Average occupancy of the orbital Fermi-Dirac Distribution Function 4

5 Fermions and the Reservoirs Around Them Only One Fermion Allowed Reservoir Reservoir Reservoir and Many Fermions 1, 2, 3 represent different orbitals in the same system. ε = N ε tot n n Only One Fermion Allowed Each one can have 0 or 1 fermions in it. Reservoir OR Reservoir Remember we assume they don t interact. Each orbital may as well be a different system. 5

6 Reservoir and Many Fermions Only One Fermion Allowed System N 6 =0,1; ε 6 =0, ε Consider many independent orbitals, then treat one orbital as The System, and the rest of the orbitals as The Reservoir. Reservoir Fermi Level Chemical Potential: is often called Fermi Level for fermions. Chemical Potential at T=0: Fermi Energy Fermi level can change; Fermi Energy is a number. 6

7 Fermi-Dirac Distribution Function Orbitals below the Fermi level are surely occupied at low T: Orbitals above the Fermi level are surely not occupied at low T: Fermi-Dirac Distribution Function At Fermi level: ε = µ f ( µ ) = 1 2 Becomes a step function at T=0. Low E: f ~ 1. High E: f ~ 0. 7

8 Classical Limit Classical distribution function This is the exponential tail above the Fermi level (chemical potential.) Classical Limit 8

9 Classical Limit: Chemical Potential To find Chemical Potential (Fermi level): Require thermal average of N to equal the number of particles in the system. I.e., Set <N> = N. Ideal Gas Result! Classical Limit The classical limit is the dilute limit. 9

10 Classical or Quantum? Ideal Gas: Dilute. Quantum effects matter. Crossover is at: Classical above this temperature; quantum below. Quantum regime: sometimes called degenerate gas, and statistics (whether fermion or boson) will matter. Quantum Limit Nernst noticed: For classical gas, as, the entropy diverges: This violates the 3rd law of thermodynamics, which is that as, entropy approaches a finite constant. Quantum Theory: there is a unique ground state as and the entropy goes to zero! Entropy is squeezed out upon cooling. 10

11 Fermi Gas Occupancies Ideal Gas L One atom in a box. Wavefunction Energy Partition Function This was for one particle. Sum over all combinations of n x, n y, and n z Lecture 5 11

12 Fermions in a Box L Get out of my way! Now fill up the states, Pauli-like, starting with the lowest energies. Lowest energies are lowest n. n is a vector in 3D-space. At every n, there is a whole surface area of ways to do this. For a given ε, there is a particular n, and there are of order 4πn 2 states available. Fermions in a Box L Now fill up the states, Pauli-like, starting with the lowest energies. Lowest energies are lowest n. At every n, there is a whole surface area of ways to do this. Get out of my way! (Think zero temperature...) Fill up the states, two by two, until you ve put in all the particles you have. (N) We call the highest energy the Fermi Energy. 12

13 Fermions in a Box L Get out of my way! Fermi Temperature (Notice in our units it is the Fermi Energy) Fine print: Note that this n is the particle density, not the n-vector associated with quantizing wavefunctions in the box, as in the rest of lecture. Internal Energy at T=0 Two for spin 1/8 for one octant of space Internal energy is proportional to the Fermi Energy, and the total number of particles. 13

14 Changing variables Sum to integral Sometimes we want an integral over energies: Density of States Density of States is a Change of Variables. Tells how many states are at a given energy. Density of States Calculate the Density of States. Since the point is to be able to do an integral with energy as the variable, at the end we ll have to express the DOS as a function of energy. Play Derivative Games... 14

15 Partial Derivatives Example: It always works. Density of States Use the partial derivative trick. This was for three dimensions. How would it change for 2 dimensions? 15

16 Density of States This was for three dimensions. How would it change for 2 dimensions? Follow the same steps but in 2D 3D: Bosons Occupancy of any orbital may be 0 or 1 or 2 or 3 or Look at one state or orbital : Infinite number of terms for a single boson state Always Room For More Bosons What are the conditions necessary for the sum to converge? 16

17 Bosons Always Room For More Bosons Average occupancy of the orbital Bosons Always Room For More Bosons Bose-Einstein Distribution Function Average occupancy of the orbital 17

18 Fermions and Bosons Fermions Fermi-Dirac Distribution Function Bosons Bose-Einstein Distribution Function Classical Limit In classical limit distribution of bosons and fermions look alike 18

19 Fermions and Bosons What We Did Today Fermions: Pauli exclusion principle Fermi-Dirac Distribution Function Bosons: no limitation of occupancy Bose-Einstein Distribution Fermi level chemical potential µ(τ) Fermi energy µ(0) Classical limit gives ideal gas result ε µ >> τ 19

20 What Else We Did Today Density of States is a Change of Variables In 3D. Partial Derivative Trick: 20

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