1. Electrons in metals ++++++++ Energy E Spatial coordinate x Nucleus with localized core electrons Jellium model: electrons shield potential to a large extent Electron “sees” effective smeared potential  

2. Electron in a box In one dimension: In three dimensions: where and

3.  Fixed boundary conditions: ++++++++ x 0 L + + + + + + + + Periodic boundary conditions: and kxkx “free electron parabola” density of states Remember the concept of # of states in

4. 1. approach use the technique already applied for phonon density of states where Density of states per unit volume Because I copy this part of the lecture from my solid state slides, I use E as the single particle energy. In our stat. phys. lecture we labeled the single particle energy  to distinguish it from the total energy of the N-particle system. Please don’t be confused due to this inconsistency.

5. 1/ Volume occupied by a state in k-space kxkx kyky kzkz Volume( )

6. Free electron gas: Independent from and  Independent from and  2 Each k-state can be occupied with 2 electrons of spin up/down k2k2 dk

7. 2. approach calculate the volume in k-space enclosed by the spheres and kxkx kyky # of states between spheres with k and k+dk : with 2 2 spin states

8. E D(E) E’E’+dE D(E)dE =# of states in dE / Volume

9. The Fermi gas at T=0 E f(E,T=0) EFEF 1 E D(E) EF0EF0 Electron density #of states in [E,E+dE]/volume Fermi energy depends on T Probability that state is occupied T=0

10. Energy of the electron gas @ T=0: there is an average energy of per electron without thermal stimulation with electron densitywe obtain Energy of the electron gas :

11. Specific Heat of a Degenerate Electron Gas here: strong deviation from classical value only a few electrons in the vicinity of E F can be scattered by thermal energy into free states Specific heat much smaller than expected from classical consideration D(E) Density of occupied states E EFEF energy of electron state #states in [E,E+dE] probability of occupation, average occupation # 2k B T Before we calculate U let us estimate: These # of electrons increase energy from to

12. π2π2 3 subsequent more precise calculation Calculation of C el from Trick: Significant contributions only in the vicinity of E F

13. with and E D(E) EFEF decreases rapidly to zero for

14. withand in comparison with for relevant temperatures W.H. Lien and N.E. Phillips, Phys. Rev. 133, A1370 (1964) Heat capacity of a metal: electronic contribution lattice contribution @ T<<Ө D