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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