- 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. 10. Use elementary vector analysis to find the value
- 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 are the indices of these places when referred to the
- 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 value, the crystal structure may be thought of as
- 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 plane.(b) Prove that the distance between two
- 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 the vector identity (e x a) x (a x b) = (e ∙ a x
- 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 is an integer. By analogy with (20), the total
- 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 conventional cube.? (a) Find the structure factor S
- 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 Bohr radius. Show that the form factor is fG = 16/
- 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. The incident beam of x-rays is perpendicular to
- 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 crystalline He4 with each he atom confined to a line
- 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 structures (Ans, 0.958). The lattice sums for the bcc
- 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 the cohesive energy in kJ per mole of H2; do the
- 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 atom or molecule R. This is believed to occur
- Linear ionic crystal Consider a line of 2N ions of alternating charge ± q with a repulsive potential energy A/Rn between nearest neighbors. (a) Show that at the equilibrium separation (CGS) U
- 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 structure described in Chapter 1. Compare with the
- Divalent ionic crystals Barium oxide has the NaC1 structure. Estimate the cohesive energies per molecule of the hypothetical crystals Ba+ O– and Ba++ O– – referred to separated
- 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 nesses for Young’s modulus and Poisson’s ratio
- 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 + 4C44)/p]1/2.
- 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 + C44/p]1/2.
- 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 equal to zero, as in Fig. 22.
- 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 and the zero occurring R – 1 times. If all
- General propagation direction (a) By substitution in (57) Ibid the determinantal equation which expresses the condition that the displacement R(r) = [u0x + v0y + w0z] exp [i(K∙ r – wt)] be a
- Stability criteria the criterion that a cubic crystal with one atom in the primitive cell he stable against small homogeneous deformations is that the energy density (43) be positive for all
- Monatomic linear lattice consider a longitudinal wave us = u cos (wt ? sKa) which propagates in a monatomic linear lattice of atoms of mass M, spacing a, and nearest-neighbor interaction C. (a) Show
- Continuum ware equation show that for long wavelengths the equation of motion (2) reduces to the continuum elastic wave equation where v is the velocity of sound
- 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 this value of K the two lattices act as if
- 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 constants and p runs over all integers. Such a form is
- 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 the masses he equal, and let the nearest-neighbor
- 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 be in stable equilibrium when at regular lattice
- 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 potential is the sum of two contributions (1) a
- Singularity in density of states (a) From the dispersion relation derived for a monatomic linear lattice of N atoms with nearest-neighbor interactions, show the density of modes is D(w) = 2N/π
- 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 the bulk modulus as 7 x 1010 erg cm-3. Note that the
- Zero point lattice displacement and strain (a) In the Debye approximation, show that the mean square displacement of an atom at absolute zero is R) = 3hw2D/8π2 pv3, where v is the velocity of
- 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 the atoms is restricted to the plane of the layer,
- 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 zero-point energy ½hw to obtain this result. (b)
- 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 bulk modulus of an electron gas at 0 K is
- 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 be written as p = 2/3(U0/V). (b) Show that
- 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) – 1] for n electrons per unit area. The density
- 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 star this number of electrons may be ionized and
- 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 and the Fermi temperature TF.
- Frequency dependence of the electrical conductivity, use the equation m(dv/dt + v/?) = ? ?E for the electron drift velocity v to show that the conductivity at frequency w is where ?(0) = ne2?/m.
- 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 B. The electric current density in the xy plane
- 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 electron, and aH is the bohr radius h2/e2m. (a)
- 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 the high magnetic field limit of wc? >> 1,
- Maximum surface resistance considers a square sheet of side L, thickness d, and electrical resistivity p. The resistance measured between opposite edges of the sheet is called the surface resistance:
- Square lattice, free electron energies (a) Show for a simple square lattice (two dimensions) that the kinetic energy of a free electron at a corner of the first zone is higher than that of an
- Free electron energies in reduced zone. Consider the free electron energy bands of an fcc crystal lattice in the approximation of an empty lattice, hut in the reduced zone scheme in which all k’ s
- 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 problem find the band gap at k = π/a.
- Potential energy in the diamond structure(a) Show that for the diamond structure the Fourier component UG of the crystal potential seen by an electron is equal to zero for G = 2A, where A is a basis
- 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 Brillouin zone, in the approximation that led to Eq.
- 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 equation to find approximately the energy gap at
- Impurity orbits indium antimonidc has Eg = 0.23eV; dielectric constant ε = 18; electron effective mass me = 0.015m. Calculate (a) The donor ionization energy; (b) The radius of the ground
- 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) Estimate the concentration of conduction electrons
- 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 drift velocity approximations is (CGS) RH =
- Cyclotron resonance for a spheroidal energy surface considers the energy surface where mt is the transverse mass parameter and m1 is the longitudinal mass parameter. A surface on which ?(k) is
- Magneto resistance with two carrier types, Problem 6.9 shows that in the drift velocity approximation the motion of charge carriers in electric and magnetic fields does not lead to transverse magneto
- 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 = 3a.
- 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 A. (a) Draw the first Brillouin zone give its
- Hexagonal emit-packed structure Consider first Brillouin zone of a crystal with a simple hexagonal lattice in three dimensions with lattice constants a and c. Let Gc denote the shortest reciprocal
- 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 the almost free electron approximation, sketch
- 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 = 2 x 108 cm1. A magnetic field B = 103 gauss =
- 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 well of depth U0 and width a. (This Is the standard
- 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 space of the external orbit, for B 10kG = 1 T?
- 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 second-order perturbation theory to find the result where the
- 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) Prove that Wannier functions about different lattice
- Open orbit and magneto resistance. We considered the transverse magneto resistance of free electrons in Problem 6.9 and of electrons and holes in Problem 8.5. In some crystals the magneto resistance
- 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) inside a super conducting plate perpendicular to the
- 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 thickness δ in an external magnetic field Ba is
- 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 density may be written as the sum of the
- 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 coordinates, we want a solution of B ? ?2?2B = 0
- London penetration depth (a) Take the time derivative of the London equation (10) to show that ∂j/∂t = (c2/4πλ2L) E. (b) If mdv/dt = qE, as for free carriers
- Diffraction effect of Josephson junction, consider a junction of rectangular cross-section with a magnetic field B applied in the plane of the junction, normal to an edge of width w. Let 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 effective magnetization M within the sphere is
- 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), where a0, = h2/me2 = 0.529 x 10-8 cm. The charge
- 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 5s2 p6; (b) Yb3+; (c) Tb3+. The
- 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) Find an expression for the magnetic moment (μ) in
- Heat capacity from internal degrees of freedom (a) Consider a two-level system with an energy splitting kB? between upper and lower states; the splitting may arise from a magnetic field or in other
- Pauli spin susceptibility the spin susceptibility of a conduction electron gas at alsolute zero may be approached by another method. Let N+ = ½N (1 + ζ); N– = ½ N (1 - ζ) be the concentrations
- Conduction electron ferromagnetism we approximate the effect of exchange interactions among the conduction electrons if we assume that electrons with parallel spins interact with each other 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 and heat capacity are U = - Δ tanh (Δ/kBT); C =
- 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, moment μ and concentration n. (b) Show that in
- 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 entropy (8) is called the Configurational entropy
- 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 atom is at p and the six nearest neighbors arc
- Heat capacity of magnons use the approximate magnon dispersion relation w = Ak2 find the leading term in the heat capacity of a three-dimensional Ferro magnet at low temperatures kBT << J. The
- Neel temperature taking the effective field on the two-sub-lattice model of an anti Ferromagnetic as BA = Ba – μMB – εMA; BB = Ba – μMA – μMB, show that θ/TN = μ + ε/μ – ε.
- Coercive force of a small particle (a) Consider a small spherical single-domain particle of a uniaxial Ferro-magnet. Show that the reverse field along the axis required to reverse the
- Saturation magnetization near Tc shows that in the mean field approximation the saturation magnetization just below the Curie temperature has the dominant temperature dependence (Tc – T)1/2. Assume
- Giant magneto resistance in a ferromagnetic metal, the conductivity ?p for electrons whose magnetic moments are oriented parallel to the magnetization is typically larger than ?a for those anti
- 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 material of negligible crystalline anisotropy
- 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 spin system characterized by the susceptibility
- 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 an instantaneous angular velocity Ω, so that
- 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 hyperfine interaction of the electron spin with 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 sin2 θ, where θ is the angle between the
- 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 spin system is at rest. The gyro magnetic ratios
- 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 is φi (x, z) = A cos kx e – kz whence Esi = kA
- 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 in both metals are those or free-electron gases.
- 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 in a compensated semiconductor. Place the solid in
- 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 w << wh = eB/mhc. Show that ε(w) =
- 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, where the polarization P = –ner, with r as the

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