The Reciprocal Lattice, Reciprocal Space

Question 1

How can the structure of crystals be found?

Answer 1

From X-Ray diffraction measurements which probe the Reciprocal Lattice.

Question 2

What is a periodic function?

Answer 2

e.g. f(x) = f(x+a) = f(x+2a)

Question 3

For the fourier series where sum over n of fnexp(i2πnx/a) = sum over n of fnexp(iGnx), where Gn = 2πn/a, what do we find from plotting Gn?

Answer 3

If we plot Gn, we find that they sit on a lattice with spacing 2π/a, and in 3D the Gn become vector (Gh,k,l)

Question 4

For a lattice where each point is located by 3 integers n1,n2,n3, what does R(n1,n2,n3) equal and what do we get from this?

Answer 4

R(n1,n2,n3) = n1a+n2b+n3c, we need to look for exp(iG(hkl)*R(n1,n2,n3)) = 1 to define the G’s

Question 5

What equation do we choose for G(h,k,l)?

Answer 5

G(h,k,l) = hA+kB+lC, where A = 2π(bXc)/(a.(bXc)), B = 2π(cX)/(a.(bXc)), C = 2π(aXb)/(a.(bXc)), and h,k,l are integers

Question 6

What do A,B and C represent in the new equation for G?

Answer 6

The basis vectors for a new lattice for the G’s.

Question 7

What is the equation for the potential describing an electrons interaction with nuclei and other electrons?

Answer 7

Must have symmetry of the lattice: V(r) = sum over h,k,l of V(h,k,l)*exp(iG(h,k,l).r)

Question 8

How do we check for periodicity in V?

Answer 8

V(r+R(n1,n2,n3)) = sum over h,k,l of V(h,k,l)exp(iG(h,k,l).(r+R(n1,n2,n3))) which is equal to the original equation since exp(iG(hkl)R(n1,n2,n3)) = 1.

Question 9

What can we learn from the orientation of the G(h,k,l)’s?

Answer 9

They define planes in the original real lattice, e.g. cubic crystal R(n1,n2,n3) = n1ax(hat)+n2ay(hat)+n3az(hat)

Question 10

For R(n1,n2,n3) = n1ax(hat)+n2ay(hat)+n3az(hat), what do we fine G(h,k,l) equals?

Answer 10

G(h,k,l) = hA+kB+lC, where A = 2π(y(hat) X z(hat))a^2/(a^3(x(hat).(y(hat) X z(hat))) = 2π/a x(hat), B = 2π/a y(hat), C = 2π/a z(hat), so is a cubic lattice with spacing 2π/a

Question 11

What would G(0,0,1) be for the example above?

Answer 11

G(0,0,1) = 2π/a z(hat), which is a vector which specifies a plane perpendicular to the z-axis.

Question 12

What defines the first Brillouin Zone?

Answer 12

The Wigner-Seitz unit cell for the reciprocal lattice is called the first Brillouin zone and is constructed in the same way.

Question 13

What is the free electron model of a metal?

Answer 13

Loosely bound valence electrons modelled as non-interacting fermions moving in effective potential set up by nuclei and other electrons.

Question 14

In the free electrons model of a metal, what is the potential modelled as?

Answer 14

Potential modelled as a uniform positively charged background, which ensures that the system is overall charge neutral.

Question 15

What does each electron move between in a box of dimensions Lx, Ly, Lz?

Answer 15

Each electron moves between 0<=x<=L

Question 16

What is the TISE for this electron in box problem?

Answer 16

-ћ^2/2m d^2/dx^2 *Ф(x) = EФ(x), where Ф(x) = Aexp(ikx)+Bexp(-ikx) = Ccoskx + Dsinkx, E = ћ^2k^2/2m (for standing wave)

Question 17

What boundary conditions do we use in the electrons in box problem?

Answer 17

Trap electron in the box, so Ф(0)=Ф(L)=0, so sub these into the equation for Ф to get equations for k and therefore En.

Question 18

What is the equation for the amount of fermions in the box?

Answer 18

Fill up the states with N fermions, 2 per state (up and down), so N = 2n(F)

Question 19

What is the Fermi wavevector?

Answer 19

The wavevector k(F) for the highest occupied state.

Question 20

What is the equation for k(F)?

Answer 20

k(F) = n(F)*π/L = Nπ/2L = π/2 *ρ, where ρ = N/L and is the density of electrons.

Question 21

What is the equation for the fermi energy E(F)?

Answer 21

The same energy equation but with k(F) subbed in: E(F) = ћ^2/2m *(Nπ/2L)^2 = ћ^2 *π^2/8m *ρ^2

Question 22

What does the graph of E against k look like?

Answer 22

Increasing gradient curve starting at 0 with distance between dots equal to π/L

Question 23

What equation for Ф do we use for running waves?

Answer 23

Ф(x) = A*exp(ikx)

Question 24

What do we do to the equation for Ф for the running waves?

Answer 24

Impose periodic boundary conditions Ф(x+L) = Ф(x), so Aexp(ik(x+L)) = Aexp(ikx), so exp(ikL) = 1 and kL = 2nπ

Question 25

What does the graph of E against k look like for running wave?

Answer 25

Same as for standing wave but with 2π/L gaps between points and goes negative too.

Question 26

What is the equation for N for the running wave?

Answer 26

N = 2(2*|n(F)|+1) ~ 4|n(F)| = 2Lk(F)/π

Question 27

What is the equation for the fermi wavevector and the fermi energy for the running wave?

Answer 27

Same as the standing wave equations.

Question 28

What is the equation for N in terms of E?

Answer 28

N(E) = 2L/π *(2m/ћ^2)^1/2 *E^1/2

Question 29

If we look at L-> inf what happens?

Answer 29

There is a continuum of states (the E-k graph becomes a straight line)

Question 30

What is the equation for how the states are distributed with respect to energy N(E+ΔE)?

Answer 30

N(E+ΔE) = N(E) + dN/dE *ΔE = N(E)+n(E)ΔE = 2L/π *(2m/ћ^2)^1/2 *(E^1/2 + E^(-1/2)/2 *ΔE)

Question 31

What is the density of states and what is its equation?

Answer 31

n(E) = 2/π *(2m/ћ^2)^1/2 *L/2 *1/sqrt(E)

Question 32

What can the density of states be used for?

Answer 32

Can be used to define useful quantities e.g. the total number of electrons in the system: N = integral from 0 to E(F) of n(E) dE

Question 33

How can we use the density of states to work out the total energy of the N electron system?

Answer 33

Etot = integral from 0 to E(F) of E*n(E) dE = 1/π *(2m/ћ^2)^1/2 *L *2/3 *E(F)^3/2

Question 34

How do we expand the box theory to 3D instead of just 1D?

Answer 34

Change Ф(x) to Ф(x,y,z) = Aexp(ik.r) = Aexp(ik(x)x)exp(ik(y)y)exp(ik(z)z), and use the periodic boundary conditions in each direction separately, so Ф(x+Lx, y, z), Ф(x, y+Ly, z), etc

Question 35

What is k equal to for the 3D problem?

Answer 35

k(x) = n(x)2π/Lx, k(y) = n(y)2π/Ly, k(z) = n(z)*2π/Lz, with nx, ny, nz integers

Question 36

How do we determine how many states are in each thingy in k-space?

Answer 36

The electron states can be occupied with 2 electrons, so there is 1 state in every (2π)^3/LxLyLz = (2π)^3/V

Question 37

What do we get for the equation for N in 3D?

Answer 37

N = 2*(4π/3 *k(F)^3)/((2π)^3/V) = Vk(F)^3/3π^2 -> bottom part is volume per state and top part is the spin and the shere.

Question 38

What is the equation for k(F)^3 in 3D?

Answer 38

k(F)^3 = 3π^2 *(N/V) = 3π^2 *ρ, where ρ is the electron density.

Question 39

What is the equation for E(F) in the 3D problem?

Answer 39

E(F) = ћ^2/2m k(F)^2 = ћ^2/2m(3π^2 *ρ)^2/3