Download 1. The primitive translation vectors of the hexagonal space lattice

HW1
1. The primitive translation vectors of the hexagonal space lattice may be taken as
3
a
3
a
axˆ  ayˆ , a2  
axˆ  ayˆ , a 2  czˆ
2
2
2
2
(a) Derive the volume of the primitive cell.
(b) Derive the primitive translations of the reciprocal lattice. Show that the lattice
is its own reciprocal, but with a rotation of axes.
(c) Sketch the first Brillouin zone of the hexagonal space lattice.
a1 
2. Show that the volume of the first Brillouin zone is
 2 
Vc
3
, where Vc is the
volume of a crystal primitive cell. (Hint: The volume of a Brillouin zone is equal
to the volume of the primitive cell parallelepiped in Fourier space. Recall the
vector identity
 c  a  a  b  c  a  b a )
3. For the crystal structure of diamond, the basis consists of 8 atoms if the cell is
1 1 1
 , , ,
4 4 4
1 1   3 3 1  1 1 1 3 3 1 1  3 1 3
 , ,0  ,  , ,  ,  0, ,  ,  , ,  ,  ,0,  ,  , ,  .
2 2  4 4 4  2 2 4 4 4 2 2 4 4 4
(a) Find the structure factor S of this basis.
(b) Discuss the allowed and forbidden reflections, i.e. the values of S in all cases.
taken as the conventional cube. Carbon atoms locate at
 0,0,0 ,
4. NaCl crystallizes in a FCC lattice with a basis of Na and Cl ions separated by half
the body diagonal of the cube. The atomic numbers of Na and Cl are 11 and 17,
respectively.
(a) Determine which X-ray reflections will be observed. Index them for the
conventional cubic unit cell.
(b) Of these which group will be strong and which group weak?
HW2
1. Consider a (i) 3-D (crystal) (ii) 2-D (quantum well) (iii) 1-D (quantum wire) free
electron gas in which the electrons are restricted to move freely within its
boundary (3-D: L3, 2-D: L2, 1-D: L).
(a) Show that the density of states, g  E  , for the 3-D, 2-D and 1-D systems are
(i)  E (ii) constant (iii) 1
E , respectively.
(b) Draw the above result of g  E  versus E.
2. Metallic sodium crystallizes in body-centered cubic form, the length of the cube
being 4.25  108 cm.
(a) Find the concentration of conduction electrons. Assume one conduction
electron.
(b) What is the Fermi energy?
3. For metal,
(a) Please show that the average kinetic energy of a conduction electron at 0 K is
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given by E  E f .
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(b) The Fermi energy of copper at 0K is 7.05 eV. At what temperature would the
average energy a molecule of an ideal gas equal the average energy of a
conduction electron in copper at 0K?
4. Consider the general expression for the conductivity of metals in terms of density
1
of states g  E f  at E f given by   e2v f 2 g  E f  . Show that within the free
3
ne 2
, the Drude expression. Explain what
m*
the main difference is between quantum and classical theory.
electron theory, this reduces to  