ESPS

volume of sphere

4/3 pi r³

electrostatic contribution to energy for a lattice

-(N_A*A*Z+*Z-*e²) / (4*pi*ε0*a)
- e is a constant in the DB
- A is Madelung const.

total energy at equilibrium separation for lattice

E0 = -(N_A*A*Z+*Z-*e²) / (4*pi*ε0*a) (1-1/n)

∆U(0) =

-E0

∆_LH =

∆U(0)

area of 2D cell

|a1 x a2|

volume of 3D cell

a1 . (a2 x a3)

H for electron in FEG model

-(h-)² / 2m_e d²/dx²

psi for FEG model

exp(ikx)

lambda = (k)

2pi/k

wavevector K components

Kx, Ky, Kz

what boundary positions are applied to the FEG model to allow quantisation

Born von Karmen (BvK)

what are the BvK BCs?

• The wavefunction, as it leaves one edge of the cell, must be continuous with the wavefunction at the other edge (cyclic)

edge-edge distance (BvK)

L1 . a1

phase difference between wavefunctions (BvK)

k x L1a1

what does the BvK BCs require (eqn)

kL1a1 = n1 . 2pi

what values of k are permitted under BvK

discrete, therefore quantisation

density of BvK allowed states

(L/A/V) / 2pi

L =

L1a1

A =

L1L2Ac

V =

L1L2L3Vc

why is the boltzmann distribution not acceptable for FEG model

doesnt take into account the restrictions on how electrons can occupy energy levels

what distribution is acceptable for the FEG model

fermi-dirac distribution

what is the F-D distribution mathematically

1 / exp(E-Ef / KT)

Ef = fermi energy, a good approx. for chemical potential here

how many electrons can occupy each state in F-D distribution

2

wavenumber corresponding to fermi energy

kf

number of BvK e- in states in k sphere

W(E) = k³V / 3pi²

k = (E)

(2m_eE / (h-)²)^(3/2)
- comes from energy eigenvalue

W(Ef) =

n_eV

n_e =

x*ro*N_A / M *10^6
10^6 = cm-3 to m-3

x = # of VE

D(E) =

d(W(E)) / dE

how to find average E, <E>

∫E*D(E) dE / ∫D(E) dE [Ef, 0]

what does integrating D(E) find

total number of occupied states

what e- are thermally excited

e- within kT of Ef

rough estimate of # of excited e-

N_A * (kT/Ef)

thermal energy, U_T of the excited e-

N_A(kT)²/Ef

fermi temperature, Tf =

Ef / k

what is the bulk modulus

how a solid changes in volume when subject to hydrostatic stress (i.e. solid in a fluid then fluid compressed)

B = (Ef, n_e)

2/3*Ef*n_e