Solid State Physics Flashcards
What is a lattice and a basis?
Lattice: Describes the underlying periodicity of the structure.
Basis: Group of atoms being repeated.
What is a Bravais Lattice?
A periodic array of points that look identical from every point.
What is the primitive unit cell?
The minimum volume that fills all space when translated by all lattice vectors.
What is the volume of the primitive unit cell in lattice vectors?
V = |a1 * (a2 x a3)|
How many lattice points does a primitive unit cell contain?
One.
How is the volume of the Weigner-Seitz cell defined?
The volume enclosed by planes that perpendicularly bisect the lines to the nearest lattice points.
How does Bragg reflection work?
Lattice planes reflect x-rays specularly, with constuctive interference when paths drift by n*2pi.
State Bragg’s Law.
2dsin(theta) = n*lamda
For Bragg’s Law, what is the use of using a neutron beam over X-rays?
Neutrons are scattered only by nuclei, not electrons, and so are more useful in studying light elements.
How do you find the Miller indices of a crystal plane?
- Take any lattice plane and find its intercepts, m1, m2, and m3, relative to the crystallographic axis.
- Find reciprocals 1/m1, 1/m2, 1/m3
- Scale all reciprocals by a common factor to find the smallest set of integers; h,k,l.
How is the reciprocal lattice defined?
The reciprocal lattice is the set of all wavevectors k that yield plane waves with the periodicity of a Bravais lattice.
How do reciprocal wavevectors G relate to lattice vectors R?
exp( i(G*R)) = 1
or GR = 2pin
How do lattice vectors a1, a2 and a3 relate to reciprocal lattice vectors b1, b2 and b3?
- b1 = 2pi/V * (a2 x a3)
- b2 = 2pi/V * (a3 x a1)
- b3 = 2pi/V * (a1 x a2)
where volume V = a1 * (a2 x a3)
What is the formula for the volume of a reciprocal unit cell?
V = b1 * (b2 x b3)
What is the formula for distance between reciprocal crystal planes?
d_hkl = 2pi / | G_hkl |
What is the formula for distance between reciprocal planes of a cubic crystal?
d = a / sqrt( h^2 + k^2 + l^2 )
where a = lattice constant
How is momentum transfer Q defined in Laue diffraction, with regard to wave vectors?
Q = k’ - k
What is the Laue condition for diffraction?
Q = G
How is the structure factor S_G defined?
S_G = sum[ f_j * exp( i(-G * r_j)) ]
where f_j = integral [ n_j (rho) * exp( -G * rho) ], the atomic form factor in the amplitude scattered by the electron.
How is the Brillouin zone defined?
The Brillouin zone is the Weigner-Seitz cell of the reciprocal lattice; i.e. it is formed by the volume enclosed of the perpendicular bisectors of lines to nearest atoms.
It has the same volume as the primitive unit cell of the reciprocal lattice.
How does the Brillouin zone play into the Laue condition for diffraction?
When G = Q = k’ - k is satisfied, k terminates of the Brillouin zone boundary. This is a sufficient condition for diffraction; i.e if a wavevector terminates on the BZ boundary, the wave will be diffracted.
What is the dispersion relation for monatomic atoms on a 1D chain?
w^2 = 4C/m * sin^2(ka/2)
where C is the spring constant, m is the atomic mass, and a is the lattice constant.
What does the dispersion relation describe?
It describes how frequency of an elastic wave (w) changes with wavevector k (where k = 2pi/wavelength).
In regards to atomic displacement, why is the first Brillouin zone boundary unique?
The first BZ, i.e -pi/a