The Reciprocal Lattice, Reciprocal Space Flashcards
How can the structure of crystals be found?
From X-Ray diffraction measurements which probe the Reciprocal Lattice.
What is a periodic function?
e.g. f(x) = f(x+a) = f(x+2a)
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?
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)
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?
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
What equation do we choose for G(h,k,l)?
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
What do A,B and C represent in the new equation for G?
The basis vectors for a new lattice for the G’s.
What is the equation for the potential describing an electrons interaction with nuclei and other electrons?
Must have symmetry of the lattice: V(r) = sum over h,k,l of V(h,k,l)*exp(iG(h,k,l).r)
How do we check for periodicity in V?
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.
What can we learn from the orientation of the G(h,k,l)’s?
They define planes in the original real lattice, e.g. cubic crystal R(n1,n2,n3) = n1ax(hat)+n2ay(hat)+n3az(hat)
For R(n1,n2,n3) = n1ax(hat)+n2ay(hat)+n3az(hat), what do we fine G(h,k,l) equals?
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
What would G(0,0,1) be for the example above?
G(0,0,1) = 2π/a z(hat), which is a vector which specifies a plane perpendicular to the z-axis.
What defines the first Brillouin Zone?
The Wigner-Seitz unit cell for the reciprocal lattice is called the first Brillouin zone and is constructed in the same way.
What is the free electron model of a metal?
Loosely bound valence electrons modelled as non-interacting fermions moving in effective potential set up by nuclei and other electrons.
In the free electrons model of a metal, what is the potential modelled as?
Potential modelled as a uniform positively charged background, which ensures that the system is overall charge neutral.
What does each electron move between in a box of dimensions Lx, Ly, Lz?
Each electron moves between 0<=x<=L
What is the TISE for this electron in box problem?
-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)
What boundary conditions do we use in the electrons in box problem?
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.
What is the equation for the amount of fermions in the box?
Fill up the states with N fermions, 2 per state (up and down), so N = 2n(F)
What is the Fermi wavevector?
The wavevector k(F) for the highest occupied state.
What is the equation for k(F)?
k(F) = n(F)*π/L = Nπ/2L = π/2 *ρ, where ρ = N/L and is the density of electrons.
What is the equation for the fermi energy E(F)?
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
What does the graph of E against k look like?
Increasing gradient curve starting at 0 with distance between dots equal to π/L
What equation for Ф do we use for running waves?
Ф(x) = A*exp(ikx)
What do we do to the equation for Ф for the running waves?
Impose periodic boundary conditions Ф(x+L) = Ф(x), so Aexp(ik(x+L)) = Aexp(ikx), so exp(ikL) = 1 and kL = 2nπ
What does the graph of E against k look like for running wave?
Same as for standing wave but with 2π/L gaps between points and goes negative too.
What is the equation for N for the running wave?
N = 2(2*|n(F)|+1) ~ 4|n(F)| = 2Lk(F)/π
What is the equation for the fermi wavevector and the fermi energy for the running wave?
Same as the standing wave equations.
What is the equation for N in terms of E?
N(E) = 2L/π *(2m/ћ^2)^1/2 *E^1/2
If we look at L-> inf what happens?
There is a continuum of states (the E-k graph becomes a straight line)
What is the equation for how the states are distributed with respect to energy N(E+ΔE)?
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)
What is the density of states and what is its equation?
n(E) = 2/π *(2m/ћ^2)^1/2 *L/2 *1/sqrt(E)
What can the density of states be used for?
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
How can we use the density of states to work out the total energy of the N electron system?
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
How do we expand the box theory to 3D instead of just 1D?
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
What is k equal to for the 3D problem?
k(x) = n(x)2π/Lx, k(y) = n(y)2π/Ly, k(z) = n(z)*2π/Lz, with nx, ny, nz integers
How do we determine how many states are in each thingy in k-space?
The electron states can be occupied with 2 electrons, so there is 1 state in every (2π)^3/LxLyLz = (2π)^3/V
What do we get for the equation for N in 3D?
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.
What is the equation for k(F)^3 in 3D?
k(F)^3 = 3π^2 *(N/V) = 3π^2 *ρ, where ρ is the electron density.
What is the equation for E(F) in the 3D problem?
E(F) = ћ^2/2m k(F)^2 = ћ^2/2m(3π^2 *ρ)^2/3
