Introduction To Quantum Mechanics
3rd Edition
ISBN: 9781107189638
Author: Griffiths, David J., Schroeter, Darrell F.
Publisher: Cambridge University Press
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Chapter 10, Problem 10.19P
To determine
The relation between the total scattering cross-section and the forward scattering amplitude.
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1.1 The conventional unit cell for an fcc lattice is a cube with side length a.
(a) Assuming that each lattice point is associated with a sphere of radius r, show that
a² = 8r².
(Hint: consider a single face of the cube, and that certain atoms will be touching each other.)
(b) What is the total volume per unit cell taken up by spheres?
(c) Hence show that the packing fraction of this structure is
2π
3√8
~74%.
Problem 1.6.5. A magnetic moment µ in a magnetic field h has energy E+ = Fµh
when it is parallel (antiparallel) to the field. Its lowest energy state is when
it is aligned with h.
probabilities for being parallel or antiparallel given by P(par)/P(antipar) =
exp(-E+/T]/ exp[-E-/T] where T is the absolute temperature. Using the fact
that the total probability must add up to 1, evaluate the absolute probabilities for
the two orientations. Using this show that the average magnetic moment along
the field h is m = µ tanh(uh/T) Sketch this as a function of temperature at fixed
h. Notice that if h = 0, m vanishes since the moment points up and down with
%3D
However at any finite temperature, it has a nonzero
%3D
equal probability. Thus h is the cause of a nonzero m. Calculate the susceptibility,
dm
lh=0 as a function of T.
For KMnF 3 shown in Fig. 9.9, it becomes an antifoerromagnet at low temperature.Namely the magnetic moment of the Mn in the sublattice A and B pointing to the oppositedirection. Now assign the angular momentum on the A and B sublattice as J and –J respectively,a) Derive the expression of the cross-section for magnetic diffractionb) Find out the first three non-zero magnetic Bragg peaksc) Derive the expression of the cross-section for nuclear (crystalline) diffraction
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- Problem 9.9 Write down the (real) electric and magnetic fields for a monochro- matic plane wave of amplitude Eo, frequency w, and phase angle zero that is (a) traveling in the negative x direction and polarized in the z direction; (b) traveling in the direction from the origin to the point (1, 1, 1), with polarization parallel to the xz plane. In each case, sketch the wave, and give the explicit Cartesian components of k and în.arrow_forwardProblem 3.17. The Fermi level in a silicon region at thermal equilibrium at room temperature is 0.220 eV above the intrinsic Fermi level E. (a) Calculate the thermal-equilibrium concentration of electrons nº. concentration of hole pº. (b) Calculate the thermal-equilibriumarrow_forward1.7 A crystal has a basis of one atom per lattice point and a set of primitive translation vectors (in A): c = 1.5(i + j+ k), a = 3i, where i, j and k are unit vectors in the x, y and z directions of a Cartesian coordinate system. What is the Bravais lattice type of this crystal, and what are the Miller indices of the set of planes most densely populated with atoms? Calculate the volumes of the primitive unit cell and the conventional unit cell. b= 3j.arrow_forward
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- A. at pattern. Let's ssuming that the e angular spatial poral frequencies w, correspond to e is then W, 1) ir) [7.33] avelength of the quals the group eing amplitude- e waves of fre- of modulating and sum over is called the wer sideband. 7.19 Given the dispersion relation w = ak', compute both the phase and group velocities. -7.20* Using the relation 1/v = dk/dv. prove that 1 Vg I V₂ 7.21* In the case of lightwaves, show that V = 1 Ve v dv v² dv V 7.22 The speed of propagation of a surface wave in a liquid of depth much greater than λ is given by v dn c dv 11 -+- C solve 7.25 only where g = acceleration of gravity, λ = wavelength, p = density. Y = surface tension. Compute the group velocity of a pulse in the long wavelength limit (these are called gravity waves). 7.23* Show that the group velocity can be written as dv dλ 7.24 Show that the group velocity can be written as gλ 2πΥ + 2πT pλ Vg = V-A ( n + w(dn/dw) 7.25 With the previous problem in mind prove that dn (v) dv n₂ = n(v) + v.…arrow_forwardVerify that the two eigenvectors in (11.8) are perpendicular, and that C in (11.10) satisfies the condition (7.9) for an orthogonal matrix.arrow_forwardThere is a one-dimensional solid lattice with two basis atoms of mass m and M, respectively (lattice constant = a). Let the interaction between atoms act only on adjacent atoms, and let the bonding force coefficient be C (Fig.). M C C a a) Draw the Brillouin zone and the approximate phonon dispersion w=w(K), and write the name of each phonon branch b) When K = TT/a, write down the normal mode energy and discuss the relative motion of adjacent atoms. c) If two atoms are of the same kind (m = M), plot the Brillouin %3D zone and phonon dispersion w=w(K).arrow_forward
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