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  {\bf University of Kentucky, Physics 520 \\
    Homework \#1, Rev. A, due Friday, 2015-09-02 }
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{\bf 0.} Griffiths [2ed]
Ch.~1 \#1, %p12, std dev
      \#3. %p12, gaussian

{\bf 1. Planck's law}---The spectral energy density of thermal radiation
inside a black body is $u(\nu,T) = 8\pi \hbar \nu^3/c^3(e^{\hbar \nu/kT}-1)$.
You can solve the following problems using the functions {\tt D}, {\tt
  NSolve}, and {\tt Integrate} in Mathematica.

{\bf \quad a)} Show Wien's displacement law, that $u(\nu,T)/T^3$ depends only
on $x=\hbar \nu/kT$, not $\nu$ or $T$ individually.  Calculate the value of
$x$ for which $u(\nu)$ attains its maximum at temperature $T$.

{\bf \quad b)} Integrate $\int_0^\infty u(\nu,T) d\nu$ to obtain the total
energy density over all wavelengths.  Show that the average forward intensity
of radiation is $I=u c/4$ (see my notes) and calculate the constant
$\sigma_{SB}$ of the Stephan-Boltzman law $I=\sigma_{SB}T^4$.

{\bf 2. COW Experiment}---A beam of neutrons of de Broglie wavelength 2~\AA\
passes through a Mach-Zehnder interferometer, with lattice spacing
$d$=3.135~\AA\ between the $\langle 111\rangle$ planes of Si.  These planes
are alligned perpendicular to the blade faces, so that diffracted neutrons
pass through the blade and are bent in the opposite direction as undiffracted
neutrons, but at the same angle of incidence about the normal, as shown in the
figure.

{\bf \quad a)} Calculate the Bragg scattering angle $\theta$, which equals the
angle \mbox{of incidence and diffraction.}

{\bf \quad b)} As neutrons rise and lose kinetic energy to the gravitational
potential, their wavelength expands.  Assuming an $L=30$~mm separation between
each of the three blades, calculate the phase difference between two
inteferring paths.  (see
\href{http://www.phys.utk.edu/neutron-summer-school/lectures/wietfeldt.pdf}{Neutron
  interferometry}, page 41).

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