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5 Solid State Physics
According to the BCS theory superconductivity is due to a weak binding of two
electrons of equal and opposite momenta and spin to form the so-called a Cooper
pair which behaves as a single particle, a Boson.
An energy E g , called the superconducting energy gap, is required to break the
Cooper pair. At T = 0,
E g = 3.53 kT c
(5.20)
where k is the Boltzmann constant.
5.2 Problems
5.2.1 Crystal Structure
5.1 Show that π/6 of the available volume is occupied by hard spheres in contact
in a simple cubic arrangement.
5.2 Show that
√
3π/8 of the available volume is occupied by hard spheres in contact in a body-centered cubic arrangement.
5.3 Calculate the separations of the sets of planes which produce strong x-ray
diffractions beams at angles 4
◦ and 8
◦ in the first order, given that the x-ray
wavelength is 0.1 nm.
5.4 At what angle will a diffracted beam emerge from the (111) planes of a face
centered cubic crystal of unit cell length 0.4 nm? Assume diffraction occurs in
the first order and that the x-ray wavelength is 0.3 nm.
5.5 An x-ray beam of wavelength 0.16 nm is incident on a set of planes of a certain
crystal. The first Bragg reflection is observed for an incidence angle of 36
◦ .
What is the plane separation? Will there be any higher order reflections?
5.6 In the historical experiment of Davisson and Germer electrons of 54 eV at normal incidence on a crystal showed a peak at reflection angle θ r = 40
0 . At what
energy neutrons would also show a peak at θ r = 40
0 for the same order.
5.7 Write down the atomic radii r in terms of the lattice constant a, for (a) Simple
cubic structure (b) FCC structure (c) BCC structure (d) Diamond structure.
5.2.2 Crystal Properties
5.8 Show that the Madelung constant for a one-dimensional array of ions of alternating sign with equal distance between successive ions is equal to 2 ln 2.
5.9 Write down the first five terms for the Madelung constant corresponding to the
NaCl crystal.
5 Solid State Physics
According to the BCS theory superconductivity is due to a weak binding of two
electrons of equal and opposite momenta and spin to form the so-called a Cooper
pair which behaves as a single particle, a Boson.
An energy E g , called the superconducting energy gap, is required to break the
Cooper pair. At T = 0,
E g = 3.53 kT c
(5.20)
where k is the Boltzmann constant.
5.2 Problems
5.2.1 Crystal Structure
5.1 Show that π/6 of the available volume is occupied by hard spheres in contact
in a simple cubic arrangement.
5.2 Show that
√
3π/8 of the available volume is occupied by hard spheres in contact in a body-centered cubic arrangement.
5.3 Calculate the separations of the sets of planes which produce strong x-ray
diffractions beams at angles 4
◦ and 8
◦ in the first order, given that the x-ray
wavelength is 0.1 nm.
5.4 At what angle will a diffracted beam emerge from the (111) planes of a face
centered cubic crystal of unit cell length 0.4 nm? Assume diffraction occurs in
the first order and that the x-ray wavelength is 0.3 nm.
5.5 An x-ray beam of wavelength 0.16 nm is incident on a set of planes of a certain
crystal. The first Bragg reflection is observed for an incidence angle of 36
◦ .
What is the plane separation? Will there be any higher order reflections?
5.6 In the historical experiment of Davisson and Germer electrons of 54 eV at normal incidence on a crystal showed a peak at reflection angle θ r = 40
0 . At what
energy neutrons would also show a peak at θ r = 40
0 for the same order.
5.7 Write down the atomic radii r in terms of the lattice constant a, for (a) Simple
cubic structure (b) FCC structure (c) BCC structure (d) Diamond structure.
5.2.2 Crystal Properties
5.8 Show that the Madelung constant for a one-dimensional array of ions of alternating sign with equal distance between successive ions is equal to 2 ln 2.
5.9 Write down the first five terms for the Madelung constant corresponding to the
NaCl crystal.