Questions and Answers of Solid State

Magneto plasma frequency use the method of Problem 5 to find the frequency of the uniform plasma mode of a sphere placed in a constant uniform magnetic field B. Let B he along the z axis. The
Photon branch at low wave vector (a) Find what (56) becomes when ε(∞) is taken into account. (b) Show that there is a solution of (55) which at low wave vector is w = cK/√ε(0), as
Plasma frequency and electrical conductivity an organic conductor has been found by optical studies to have wp = 1.80 x 1015 s-1 for the plasma frequency, and τ = 2.83 x 10-15 s for the electron
Bulk modulus of the Femi gas show that the contribution of the kinetic energy to the bulk modulus of the electron gas at absolute zero is B = 1/3nmv2/F. It is convenient to use (6.60). We can use our
Response of electron gas it is sometimes stated erroneously in books on electromagnetism that the static conductivity σ, which in Gaussian units has the dimensions of a frequency, measures the
Gap Plasmon’s and the van der Waals interaction consider two semi-infinite media with plane surfaces z = 0, d. The dielectric function of the identical media is ε(w). Show that for surface
Causality and the response function, the Kramer?s-Kronig relations are consistent with the principle that an effect not precedes its cause. Consider a delta function force applied at time t = 0:
Dissipation sum rule by comparison of a' (w) from (9) and from (11a) in the limit w ? ?, show that the following sum rule for the oscillator strengths must hold:
Reflection at normal incidence consider an electromagnetic wave in vacuum, with field components of the form let the wave be incident upon a medium of dielectric constant E and permeability ? = 1
Conductivity sum rule and super conductivity we write the electrical conductivity as ?(w) = ?' (w) + i?'' (w) where ?', ?" are real.? (a) Show by a Kramer?s- Kronig relation that This result is used
Dielectric constant and the semiconductor energy gap the effect on ε'' (w) of an energy gap wg in a semiconductor may be approximated very roughly by substituting ½ δ(w – wg) for δ (w) in the
Polarizability of atomic hydrogen consider a semi classical model of the ground state of the hydrogen atom in an electric field normal to the plane of the orbit (Fig. 25), and show that for this
Polarization of sphere a sphere of dielectric constant E is placed in a uniform external electric field E0. (a) What is the volume average electric field E in the sphere?(b) Show that the
Saturation polarization at Curie point in a first-order transition the equilibrium condition (43) with T set equal to T, gives one equation for the polarization Ps (Tc). A further condition at the
Dielectric constant below transition temperature in terms of the parameters in the Landau free energy expansion, show that for a second-order phase transition the dielectric constant below the
Soft modes and lattice transformations sketch a monatomic linear lattice of lattice constant a. (a) Add to each of six atoms a vector to indicate the direction of the displacement at a given
Ferroelectric linear array consider a line of atoms of Polarizability a and separation a. Show that the array can polarize spontaneously if a > a3/4Σn–3, where the sum is over all positive
Diffraction from a linear array and a square array the diffraction pattern of a linear structure of lattice constant a is explained in Fig. 19. Somewhat similar structures are important in molecular
Surface sub bands in electric quantum limit consider the contact plane between an insulator and a semiconductor, as in a metal-oxide-semiconductor transistor or MOSFET. With a strong electric field
Properties of the two-dimensional electron gas consider a two-dimensional electron gas (2DEG) with twofold spin degeneracy but no valley degeneracy. (a) Show that the number of orbital’s per
Filling sub bands for electrons in a square GaAs wire of width 20 nm, find the linear electron density at which the nx, = 2, ny, = 2 sub band is first populated in equilibrium at T = 0. Assume an
Breit-Wigner form of a transmission resonance the purpose of this problem is to derive (33) from (29). (a) By expanding the cosine for small phase differences away from resonance, δφ = φ –
Barriers in series and Ohm's Law (a) Derive (36) from (35). (b) Show that the 1D Drude conductivity σ1D, = n1De2τ/m can be written as σ1D = (2e2/h)ℓB. (The momentum relaxation rate and
Energies of a spherical quantum dot (a) Derive the formula (63) for the charging energy. (b) Show that, for d << B. the result is the same as that obtained using the parallel plate
Thermal properties in ID (a) Derive the formula (77) for the low temperature heat capacity of a single 1D phonon mode within the Debye approximation. (b) Derive the relation for the thermal
Frenkel defects show that the number n of interstitial atoms in equilibrium with n lattice vacancies in a crystal having N lattice points and N' possible interstitial positions is given by the
Schottky vacancies suppose that the energy required to remove a sodium atom from the inside of a sodium crystal to the boundary is 1eV. Calculate the concentration of Schottky vacancies at 300K.
F center(a) Treat an F center as a free electron of mass m moving in the field of a point charge e in a medium of dielectric constant ε = n2; what is the 1s-2p energy difference of F centers in
Lines of closset packing show that the lines of closest atomic packing are (110) in fee structures and (111) in bee structures.
Dislocation pairs(a) Find a pair of dislocations equivalent to a row of lattice vacancies;(b) Find a pair of dislocations equivalent to a row of interstitial atoms.
Force on dislocation considers a crystal in the form of a cube of side L containing an edge dislocation of Burgers vector h if the crystal is subjected to a shear stress σ on the upper and lower
Super lattice lines in Cu3Au, Cu3Au alloy (75% Cu, 25% Au) has an ordered state below 400oC, in which the gold atoms occupy the 000 positions and the copper atoms the 11/22, 1/201/2, and 0 11/22
Suppose g(U) = CU3N/2, where C is a constant and N is the number of particles (a) Show that U = 3/2Nt.(b) Show that (∂2σ/∂U2)N is negative. This form of g(U) actually applies to an ideal gas.
Find the equilibrium value at temperature τ of the fractional magnetization M Nm = 2< s >/NOf the system of N spins each of magnetic moment m in a magnetic field B. The spin excess is 2s. take the
(a) Find the entropy of a set of N oscillators of frequency ω as a function of the total quantum number n. Use the multiplicity function (1.55) and make the Stirling approximation log N! ≈ N log N
It has been said* that “set to strum unintelligently on the typewrites for millions of years, would be bound in time to write all the books in the British Museum.” This statement is nonsense, for
Given two systems of N1 ≈ N2 = 1022 spins with multiplicity functions g1(N1, S1) an g2(N2, s – s1) and g2(N2, s – s1), the product g1g2 as a function of s1 is relatively sharply peaked at s1 =
For the example that gave the result (17), calculate approximately the probability that the fractional deviation from equilibrium δ/N1 is 10–10 or larger. Take N1 = N2 = 1022. You will find it
(a) Find an expression for the free energy as a function as a function of τ of a system with two states, one at energy 0 and one at energy ε.(b) From the free energy, find expressions for the
(a) Use the partition function to find an exact expression for the magnetization M and the susceptibility X = dM/dB as a function of temperature and magnetic field for the model system of magnetic nm
A one-dimensional harmonic oscillator has an infinite series of equally spaced energy states, with Ex = shω, where s is a positive integer or zero, and ω is the classical frequency of the
Consider a system of fixed volume I thermal contact with a reservoir. Show that the mean square fluctuation in the energy of the system is)2> = τ2(∂U/∂τ)V
Suppose that by a suitable external mechanical or electrical arrangement one can add αε to the energy of the heat reservoir whenever the reservoir passes to the system the quantum of energy e. The
In our first look at the ideal gas we considered only the translational energy of the particles. But molecules can rotate, with kinetic energy. The rotational motion is quantized: and the energy
Zipper has N links: each link has a state in which it is closed with energy 0 and a state in which it is open with energy ε. We require, however, that the zipper can only unzip from the left end,
Consider one particle confined to a cube of side L; the concentration in effect is n = l/L3. Find the kinetic energy of the particle when in the ground orbit. There will be a value of ht
Show that the partition function Z(1 + 2) of two independent systems 1 and 2 in thermal contact at a common temperature τ is equal to the product of the partition functions of the separate
The thermodynamic identify for a one-dimensional system is τdσ = dU – fdl When f is the external force exerted on the line and dl is the extension of line. By analog with (32) we form the
Consider an ideal gas of N particles, each of mass M, confined to a one-dimensional line of Length L. find the entropy at temperature τ. The particles have spin zero
Show that the number of photons ∑< sn > in equilibrium at temperature τ in a cavity of volume V isN = 2.40π-2V(τ/hc)3From (23) the entropy is σ (4π2V/45)(τ/hc)3, whence σ/N ( 3.602. It is
The value of the total radiant energy flux density at the Earth from the sun normal to the incident rays is called the solar constant of the earth. The observed value integrated over all emission
(a) Estimate by a dimensional argument or otherwise the order of magnitude of the gravitational self-energy of the Sun, with Mθ = 2 × 1033g and Rθ = 7 × 1010 cm. The gravitational constant G is
Sun. Suppose 4 × 1026Js-1 is the (oral rate at which the Sun radiates energy at the present time. (a) Find the total energy of the Sun available for radiation on the rough assumptions that the
Calculate the temperature of the surface of the Earth on the assumption that as a black body in thermal equilibrium it reradiates as much thermal radiation as it receive from the Sun. Assume also
Show for a photon gas that:(a)(b) dωj/dV = –ωj/3V (c) p = U/3VThus the radiation pressure is equal to 1/3 × (energy density).(d) Compare the pressure of thermal radiation with the kinetic
(a) Show that the partition function of a photon gas is given byWhere the product is over the modes n(b) The Helmholtz free energy is found directly from (53) asTransform the sum to an integral;
A black (nonreflective) plane at temperature Tu is parallel to a black plane at temperature Tl. The net energy flux density in vacuum between the two planes is Ju = σB(T4u – T4u), where σB is the
Consider a transmission line of length L on which electromagnetic waves satisfy the one-dimensional wave equation = v2∂2E/∂x2 = ∂2E/∂t2, where E is an electric field component. Find the heat
Intergalactic space is believed to be occupied by hydrogen atoms in a concentration ≈ 1 atom m-3. The space is that the ration of the heat capacity of matter to that of radiation is ( 10–9.
Show that in the limit T >> 0 the heat capacity of a solid goes towards the limit Cv → 3NkB, in conventional units. To obtain higher accuracy when T is only moderately larger than 0, the heat
Consider a dielectric solid with a Debye temperature equal to 100 K and with 1022 atoms cm-3. Estimate the temperature at which the photon contribution to the heat capacity would be equal to the
Consider a solid of N atoms in the temperature region in which the Debye T3 law is valid. The solid is in thermal contact with a heat reservoir. Use the results on energy fluctuations from Chapter 3
The velocity of longitudinal sound waves in liquid 4Heat temperatures below 0.6 K is 2.383 × 104 cm s–1. There are no transverse sound waves in the liquid. The density is 0.145gcm-3.(a) Calculate
(a) Show that the spectral density of the radiant energy flux the arrives in the solid angle dΩ is cuω cosθ · dΩ/4π, where θ is the angle the normal to the unit area makes with the incident
Let a lens image the hole in a cavity of area AH on a black object of area A0. Use an equilibrium argument to related the product AHΩH to A0Ω0 where ΩH and Ω0are the solid angles subtended by the
We argued in this chapter that the entropy of the cosmic black body radiation has not changed with time because the number of photons in each mode has not changed with time, although the frequency of
Consider the gas of photons of the thermal equilibrium radiation in a cube of volume V at temperature τ. Let the cavity volume increase; the radiation pressure performs work during the expansion and
Consider a plane sheet of material or absorptivity a, emissivity e, and reflectivity r = 1- a. Let the sheet be suspended between and parallel with two black sheets maintained at temperatures τu and
A circular cylinder of radius R rotates about the long axis with angular velocity ω. The cylinder contains an ideal gas of atoms of mass M at temperature τ. Find an expression for the dependence of
If n is the concentration of molecules at the surface of the Earth, M the mass of a molecule, and g the gravitational acceleration at the surface, show that at constant temperature the total number
Consider a column of atoms each of mass M at temperature τ in a uniform gravitational field g. find the thermal average potential energy per atom. The thermal average kinetic energy density is
The concentration of potassium K+ ions in the internal sap of a plant cell (for example) a fresh water alga) may exceed by a factor of l04 the concentration of K+ ions in the pond water in which the
Determine the ratio m/τ for the figure is drawn. If T = 300 K, how many Bohr magnetons μB ≡ eh/2me would the particles contains to give a magnetic concentration effect of the magnitude shown?
(a) Consider a system that may be unoccupied with energy zero or occupied with energy zero or occupied by one particle in either of two states, one of energy zero and one of energy ε. Show that the
Consider a lattice of fixed hydrogen atoms; suppose that each atom can exist in four states: Find the condition that the average number of electrons per atom be unity. The condition will involve
In carbon monoxide poisoning the CO replaces the O2 absorbed on hemoglobin (Nb) molecules in the blood. To show the effect, consider a model for which each adsorption site on a heme may be vacant or
Suppose that at most one O2 can be bound to a heme group (see problem 8), and that when λ(O2) = 10-5 we have 90 percent of ht hemes occupied by O2. Consider O2 as having a spin of 1 and a magnetic
The chemical potential was defined by (5) as (∂F/∂N)a,V. an equivalent expression listed in table 5.1 isμ = (∂U/∂N)a,VProve that this relation, which was used by Gibbs to define μ, is
Find the maximum height to which water may rise in a tree the assumption that the roots stand in a pool of water and the uppermost leaves are in air containing water vapor at a relative humidity r =
(a) Show that the entropy of an ideal gas can be expressed as a function only of the orbit occupancies (b) From this result show that τV2/3 is constant in an isentropic expansion of an ideal
A hemoglobin molecule can bind four O2 molecules. Assume that ε is the energy of each bound O2, relative to O2 at rest at infinite distance. Let λ denote the absolute activity exp (μ/τ) of the
Consider a system at temperature τ, with N atoms of mass M in volume. Let μ(0) denote the value of the chemical potential at the surface of the earth (a) Prove carefully and honestly that the value
Show that - ∂f/∂c evaluated at the Fermi level ε = μ has the value (4t)–1. Thus the lower the temperature, the steeper the slope of the Fermi-Direct function.
Let ε = μ + δ, so that f(ε) appears as f(μ + δ). Show thatf(μ + δ) = 1 – f(μ + δ).Thus the probability that an orbital δ above the Fermi level is occupied is equal to the probability an
Let us imagine a new mechanics in which the allowed occupancies of an orbital are 0, 1, and 2. The values of the energy associated with these occupancies are assumed to be 0, ε, and 2ε,
Extreme relativistic particles have momenta p such that pc >> Mc2, where M is the rest mass of the particle. The de Broglie relation λ = h/p for the quantum wavelength continues to apply. Show that
From the thermodynamic identity at constant number of particles we haveShow by integration that for an ideal gas the entropy isσ = Cv log τ + N log V + σ1,where σ1 is constant, independent
Suppose that a system of N atoms of type A is placed in diffusive contact with a system of N atoms of type B at the same temperature and volume. Show that after diffusive equilibrium is reached the
Consider an ideal monatomic gas, but one for which the atom has two internal energy states, one an energy ∆ above the other. There are N atoms in volume V at temperature τ. Find the (a) Chemical
The lower 10-15 km of the atmosphere-the troposphere-is often in a convective steady state at constant entropy, not constant temperature. In such equilibrium pVy is independent of altitude, where γ
(a) Find the chemical potential of an ideal monatomic gas in two dimension, with N atoms confined to a square of are A = L2. The spin is zero, (b) Find an expression for the energy U of the gas.(c)
Consider one mole of an ideal monatomic gas at 300K and 1 atm. First, let the gas expand isothermally and reversibly to twice the initial volume, second, let this be followed by an isentropic
A diesel engine is an internal combustion engine in which fuel is sprayed into the cylinders after the air charge has been so highly compressed that it has attained a temperature sufficient to ignite
Density of orbitals in one and two dimensions. (a) Show that the density of orbitals of a free electron in one dimension isD1(ε) = (L/π) (2m/h2ε)τ 2,where L is the length of the line. (b) Show
For electrons with an energy ε >> mc2, where m is the rest mass of the election, the energy is given by ε ≈ pc, where p is the momentum. For electrons in a cube of volume V = L3 the momentum is
Pressure and entropy of degenerate Fermi gas.(a) Show that a Fermi electron gas in the ground state exerts a pressureIn a uniform decrease of the volume of a cube every orbital has its energy raised:
Explain graphically why the initial curvature of μ versus τ is upward for a fermion gas in one dimension and downward in three dimensions (Figure).
The atom 3He has spin I = ½ and is a fermion.(a) Calculate as in Table 7.1 the Fermi sphere parameters vF, εF, and TF for 3He at absolute zero, viewed as a gas of non-interacting fermions. The
Consider a white dwarf of mass M and radius R. Let the electrons be degenerate but non- relativistic; the protons are non-degenerate.(a) Show that the order of magnitude of the gravitational
Consider a science fiction universe in which the number of photons N is constant, at a concentration of 1020 cm–3. The number of thermally excited photons we assume is given by the result of

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