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
To find solutions of 1D chain problems, what steps are needed?
- Use equations of motion (F=ma, F=kx) on displacements of atoms, u_n and v_n.
- Try travelling wave solutions: u_n = u_0exp[ i(kna - wt)]
- Substitute.
- If necessary, express equations in the form of a matrix, and find the determinant for solutions.
What is a phonon?
A phonon is a quantum of energy of a mechanical excitation, similar to photons.
They are bosons with spin 0, and so obey Bose-Einstein statistics: when a mode is excited to quantum number n, it is said to be occupied by n phonons:
n(w) = 1 / [exp( hbar * w / k_b * T) - 1]
Phonons carry energy hbarw and momentum hbark, known as crystal momentum.
How would you express the components of total heat capacity of a generic crystal?
Cv = Cv_phonons + Cv_electrons + Cv_magnetic
Cv_phonons is found in all solids.
Cv_electrons is only found in metals.
Cv_magnetic is only found in magnets.
How does heat capacity Cv relate to energy E and temperature T?
Cv = dE/dT
For a crystal of N atoms, what is its classical heat capacity?
Cv = 3Nk_b
where k_b is the Boltzmann constant. This formula breaks down at low temperatures.
In Debye Theory, what is the formula for total lattice energy E?
E = integral [0 -> w_D] [hbar*w * g(w) * u(w)] dw per branch.
Where g(w) is the density of states and u(w) is the bose-einstein factor, 1/(e^(hbar*w/kt)).
What assumptions does Debye Theory make?
- The crystal is harmonic (independent modes)
- Elastic waves are non-dispersive (w = k*v_s)
- Crystal is isotropic
- There is a high-frequency cutoff, w_D.
In Debye theory, how many unit cells are the there in a crystal?
N = integral [0 -> w_D] {g(w)*dw]
What happens in a ‘Normal’ and an ‘Umklapp’ event? How does it affect heat flow?
- Phonons interact and scatter off each other (k1 + k2 = k3), which defines the thermal conductivity of a crystal.
- In normal events, k3 is within the 1st BZB.
- In Umklapp events, k3 lies outside the 1st BZB, but can be translated into it by reciprocal lattice vector G, producing a phonon sometimes going the opposite direction.
- Heat flow carried by phonons is unaffect by normal (N) events, but is impeded by Umklapp scattering, so U events contribute to thermal resistivity.
What assumptions does the Free Electron Theory make?
- A fixed background of fixed positive charges doe to nuclei
- Valence electrons propagate freely without interacting with ion cores or other electrons.
- Zero potential energy.
