- Tetrahedral angles the angles between the tetrahedral bonds of diamond are the same as the angles between the body diagonals of a cube, as in Fig.
- Indices of planes consider the planes with indices (100) and (001); the lattice is fee, and the indices refer to the conventional cubic cell. What
- Hcp structure show that the c/a ratio for an ideal hexagonal close-packed structure is (8/3)1/2 = 1.633. If c/a is significantly larger than this
- Inter planar separation consider a plane hkl in a crystal lattice.(a) Prove that the reciprocal lattice vector G = hb1 + kb3 is perpendicular to this
- Hexagonal space lattice the primitive translation vectors of the hexagonal space lattice may be taken asa1 = (31/2 a/2)x + (a/2)y
- Volume of Brillouin zone show that the volume of the first Brillouin zone is (2π)3/Vc, where Vc is the volume of a crystal primitive cell. Recall
- Width of diffraction maximum we suppose that in a linear crystal there are identical point scattering centers at every lattice point pm = ma, where m
- Structure factor of diamond the crystal structure of diamond is described Chapter 1. The basis consists of eight atoms if the cell is taken as the
- Form factor of atomic hydrogen for the hydrogen atom is its ground state, the number density is n(r) = (πa03)–1 exp (– 2r/a0), where a0 is the
- Diatomic line, consider a line of atoms ABAB. . . AB, with an A––B bond length of 1/2a, the form factors are fA, fB for atoms A, B, respectively.
- Quantum solid in a quantum solid the dominant repulsive energy is the zero-point energy of the atoms. Consider a crude one-dimensional model of
- Cohesive energy of bcc and fcc neon. Using the Lenard-Jones potential, calculate the ratio of the cohesive energies of neon in the bcc and fcc
- Solid molecular hydrogen for H2 one finds from measurements on the gas that the Lenard-Jones parameters are ε = 50 x 10-16 erg and σ = 2.96 A. Find
- Possibility of ionic crystals R+R– Imagine a crystal that exploits for binding the coulomb attraction of the positive and negative ions of the same
- Linear ionic crystal Consider a line of 2N ions of alternating charge ± q with a repulsive potential energy A/Rn between nearest neighbors. (a)
- Cubic ZnS structure using λ and p from Table 7 and the Madelung constants given in the text, calculate the cohesive energy of KCI in the cubic ZnS
- Divalent ionic crystals Barium oxide has the NaC1 structure. Estimate the cohesive energies per molecule of the hypothetical crystals Ba+ O–
- Young’s modulus and Poisson’s ratio A cubic crystal is subject to tension in the [100] direction. Find expressions in terms of the elastic stiff
- Longitudinal wave velocity, show that the velocity of a longitudinal wave in the [111] direction of a cubic crystal is given by vs = [1/3(C11 + 2C12
- Transverse wave velocity show that the velocity of transverse waves in the [111] direction of a cubic crystal is given by vs = [1/3(C11 – C12 +
- Effective shear constant show that the shear constant 1/2 (C11 ? C12) in a cubic crystal is defined by setting exx = ? eyy = ?e and all other strains
- Determinantal approach it is known that an R-dimensional square matrix with all elements equal to unity has roots R and 0, with the R occurring once
- General propagation direction (a) By substitution in (57) Ibid the determinantal equation which expresses the condition that the displacement
- Stability criteria the criterion that a cubic crystal with one atom in the primitive cell he stable against small homogeneous deformations is that
- Monatomic linear lattice consider a longitudinal wave us = u cos (wt ? sKa) which propagates in a monatomic linear lattice of atoms of mass M,
- Continuum ware equation show that for long wavelengths the equation of motion (2) reduces to the continuum elastic wave equation where v is the
- Basis of two unlike atoms for the problem treated by (18) to (26), find the amplitude ratios u/v for the two branches at K max = π/a. Show that at
- Kohn anomaly we suppose that the inter planar force constant C between planes sands + p is of the form Cp = A sin pk0a/pa, where A and k( are
- Diatomic chain Consider the normal modes of a linear chain in which the force constants between nearest-neighbor atoms are alternately C and 10C. Let
- Atomic vibrations in a metal consider point ions of mass M and charge e immersed in a uniform sea of conduction electrons. The ions are imagined to
- Soft phonon mode consider a line of ions of equal mass but alternating in charge, with ep = e(?1)p as the charge on the pth ion the inter atomic
- Singularity in density of states (a) From the dispersion relation derived for a monatomic linear lattice of N atoms with nearest-neighbor
- Rms thermal dilation of crystal cell (a) Estimate for 300 K the root mean square thermal dilation ΔV/V for a primitive cell of sodium. Take
- Zero point lattice displacement and strain (a) In the Debye approximation, show that the mean square displacement of an atom at absolute zero is
- Heat capacity of layer lattice(a) Consider a dielectric crystal made up of layers of atoms, with rigid coupling between layers so that the motion of
- Gruneisen constant (a) Show that the free energy of a phonon mode of frequency w is kBT in [2sinh (hw/2kBT)]. It is necessary to retain the
- Kinetic energy of electron gas show that the kinetic energy of a three-dimensional gas of N free electrons at 0 K is U0 = 3/5NεF and prove that the
- Pressure and bulk modulus of an electron gas (a) Derive a relation connecting the pressure and volume of an electron gas at 0 K. The result may
- Chemical potential in two dimensions, show that the chemical potential of a Fermi gas in two dimensions is given by: μ(T) = kBT in [exp(πth2/mkBT)
- Fermi gases in oil astrophysics. (a) Given M = 2 x 1033 g for the mass of the Sun, estimate the number of electrons in the Sun. In a white dwarf
- Liquid He3 the atom He3 has spin ½ and is a fermions, the density of liquid He3 is 0.081 g cm–3 near absolute zero. Calculate the Fermi energy εF
- Frequency dependence of the electrical conductivity, use the equation m(dv/dt + v/?) = ? ?E for the electron drift velocity v to show that the
- Dynamic magneto conductivity tensor for free electrons a metal with a concentration n of free electrons of charge – e is in a static magnetic field
- Cohesive energy of free electron Fermi gas, we define the dimensionless length r1, as r0/aH, where r0 is the radius of a sphere that contains one
- Static magneto conductivity tensor for the drift velocity theory of (51), show that the static current density can be written in matrix form as in
- Maximum surface resistance considers a square sheet of side L, thickness d, and electrical resistivity p. The resistance measured between opposite
- Square lattice, free electron energies (a) Show for a simple square lattice (two dimensions) that the kinetic energy of a free electron at a
- Free electron energies in reduced zone. Consider the free electron energy bands of an fcc crystal lattice in the approximation of an empty lattice,
- Kronig-Penney model(a) For the delta-function potential and with P << 1, find at k = 0 the energy of the Lowest energy band.(b) Fur the same
- Potential energy in the diamond structure(a) Show that for the diamond structure the Fourier component UG of the crystal potential seen by an
- Complex wave vectors in the energy gap find an expression for the imaginary part of the wave vector in the energy gap at the boundary of the first
- Square lattice consider a square lattice in two dimensions with the crystal potential U(x, y) = – 4U cos (2πx/a) cos (2πy/a). Apply the central
- Impurity orbits indium antimonidc has Eg = 0.23eV; dielectric constant ε = 18; electron effective mass me = 0.015m. Calculate (a) The donor
- Ionization of donors in a particular semi conductor there are 1013 donors/cm3 with an ionization energy Ed of 1meV and an effective mass 0.01m.(a)
- Hall effect with two carrier types assuming concentration n, p; relaxation times τe, τh; and masses me, mh, show that the Hall coefficient in the
- Cyclotron resonance for a spheroidal energy surface considers the energy surface where mt is the transverse mass parameter and m1 is the longitudinal
- Magneto resistance with two carrier types, Problem 6.9 shows that in the drift velocity approximation the motion of charge carriers in electric and
- Brillouin zones of rectangular lattice make a plot of the first two Brillouin zones of a primitive rectangular two-dimensional lattice with axes a, b
- Brillouin zone, rectangular lattice a two-dimensional metal has one atom of valency one in a simple rectangular primitive cell a = 2 A; b = 4
- Hexagonal emit-packed structure Consider first Brillouin zone of a crystal with a simple hexagonal lattice in three dimensions with lattice constants
- Brillouin zones of two-dimensional divalent metal. A two-dimensional metal in the form of a square lattice has two conduction electrons per atom. In
- Open orbits an open orbit in monovalent tetragonal metal connects opposite faces of the boundary of a Brillouin zone. The faces are separated by G =
- Cohesive energy for a square well potential (a) Find an expression for the binding energy of an electron in one dimension in a single square
- De Haas-van Alphen period of potassium(a) Calculate the period Δ(1/B) expected for potassium on the free electron model.(b) What is the area in real
- Band edge structure on k • p perturbation theory consider a non-degenerate orbital ψnk at k = 0 in time band n of a cubic crystal. Use
- Wannier function the Wannier functions of a hand are defined In terms of the Bloch functions of the same baud by where rn is a lattice point. (a)
- Open orbit and magneto resistance. We considered the transverse magneto resistance of free electrons in Problem 6.9 and of electrons and holes in
- Magnetic field penetration in a plate the penetration equation may be written as ?2?2B?= B,?where ? is the penetration depth.? (a) Show that B(x)
- Critical field of thin films(a) Using the result of Problem lb, show that the free energy density at T = 0 K within a superconducting film of
- Two-fluid model of a superconductor on the two-fluid model of a super conductor we assume that at temperatures 0 < T < T0, the current
- Structure of a vortex? (a) Find a solution to the London equation that has cylindrical symmetry and applies outside a line core. In cylindrical polar
- London penetration depth (a) Take the time derivative of the London equation (10) to show that
- Diffraction effect of Josephson junction, consider a junction of rectangular cross-section with a magnetic field B applied in the plane of the
- Meissner effect in sphere considers a sphere of a type 1 super conductor with critical field Hc. (a) Show that in the Meissner regime the
- Diamagnetic susceptibility of atomic hydrogen the wave function of the hydrogen atom in its ground state (Is) is ψ = (πa30)-1/2 exp (– r/a0),
- Hund rules apply the Hund rules to find the ground state (the basic level in the notation of Table 1) of (a) Eu++, in the configuration 4f7
- Triplet excited states some organic molecules have a triplet (S = 1) excited state at an energy kBΔ above a singlet (S = 0) ground state. (a)
- Heat capacity from internal degrees of freedom (a) Consider a two-level system with an energy splitting kB? between upper and lower states; the
- Pauli spin susceptibility the spin susceptibility of a conduction electron gas at alsolute zero may be approached by another method. Let N+ = ½N (1
- Conduction electron ferromagnetism we approximate the effect of exchange interactions among the conduction electrons if we assume that electrons with
- Two-level system the result of Problerrl4 is often seen in another form. (a) If the two energy levels are at Δ and – Δ, show that the energy
- Para-magnetism of S = 1 system. (a) Find the magnetization as a function of magnetic field and temperature for a system of spins with S = 1,
- Configurational heat capacity derive an expression in terms of P(T) for the heat capacity associated with order/disorder effects in an AB alloy. [The
- Magnon dispersion relation derive the magnon dispersion relation (24) for a spin S on a simple cubic lattice, z = 6 is replaced by where the central
- Heat capacity of magnons use the approximate magnon dispersion relation w = Ak2 find the leading term in the heat capacity of a three-dimensional
- Neel temperature taking the effective field on the two-sub-lattice model of an anti Ferromagnetic as BA = Ba – μMB – εMA; BB = Ba – μMA –
- Coercive force of a small particle (a) Consider a small spherical single-domain particle of a uniaxial Ferro-magnet. Show that the reverse field
- Saturation magnetization near Tc shows that in the mean field approximation the saturation magnetization just below the Curie temperature has the
- Giant magneto resistance in a ferromagnetic metal, the conductivity ?p for electrons whose magnetic moments are oriented parallel to the
- Neel wall the direction of magnetization change in a domain wall goes from that of the Bloch wall to that of a Neel wall (Fig. 36) in thin films of
- Equivalent electrical circuit considers an empty coil of inductance L0 in a series with a resistance R0; show if the coil is completely filled with a
- Rotating coordinate system we define the vector F(t) = Fs(t)x + Fy (t) y + Fs(t)z. Let the coordinate system of the unit vectors x, y, z rotate with
- Hyperfine effects on ESR in metals we suppose that the electron spin of a conduction electron in a metal sees an effective magnetic field from the
- FMR in the anisotropy field consider a spherical specimen of a uniaxial ferromagnetic crystal with an anisotropy energy density of the form UK = K
- Exchange frequency resonance consider a Ferrimagnet with two sub-lattices A and B of magnetizations MA, and MB, where MB, is opposite to MA when the
- Surface Plasmon’s consider a semi-infinite plasma on the positive side of the plane z = 0. A solution of Laplace's equation Δ2φ = 0 in the plasma
- Interface Plasmon?s we consider the plane interface z = 0 between metal 1 at z > 0 and metal 2 at z p1; metal 2 has wp2. The dielectric constants
- Alfven waves consider a solid with an equal concentration n of electrons of mass m, and holes of mass mh. This situation may arise in a semimetal or
- Helicon wanes (a) Employ the method of Problem 3 to treat a specimen with only one carrier type, say holes in concentration p, and in the limit
- Plasmon mode of a sphere the frequency of the uniform Plasmon mode of a sphere is determined by the depolarization field E = – 4πP/3 of a sphere,