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Physics 463 Problem Set 2: Diffraction and Crystal Lattices - Prof. David A. Reis, Assignments of Solid State Physics

University of Michigan (UM) - Ann ArborSolid State Physics
Prof. David A. Reis

Problem set 2 for physics 463, winter 2004, with four questions covering topics such as 1d diffraction, interplaner separation, structure factor of bcc lattice, and electron, neutron, and x-ray diffraction. The set includes calculations and proofs related to scattering amplitude, reciprocal lattice vector, interplaner separation, and structure factor for bcc lattice. It also covers the comparison of energy and velocity of longest wavelength radiation for electron, neutron, and photon diffraction in na metal.

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Uploaded on 09/17/2009

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Physics 463, Winter 2004 Problem Set 2, due Thursday January 31, in class page 1 of 3
1. 1D diffraction (similar to Kittel 2.4) Consider a finite 1-dimensional chain of equally spaced atoms
at positions ~rm=maˆx
(a) Show that the magnitude square of the scattering amplitude
|F|2=sin21
2N~a ·∆~
k
sin21
2~a ·∆~
k
hint: m=N−1
X
m=0
xm=1−xN
1−x
(b) The diffraction condition occurs when ~a ·∆~
k= 2πp (pinteger). Show that the peak |F|2scales
as N2and the width scales as 1/N (hint, consider what happens if a∆k= 2π+, 1).
2. Interplaner separation (based on Kittel 2.1) Consider a plane hkl in a crystal lattice
(a) Prove that the reciprocal lattice vector ~
G=h~
b1+k~
b2+l~
b3is perpendicular to this plane.
(b) Prove that the distance between two adjacent parallel planes of the lattice is
d(hkl) = 2π
|~
G|
(c) Show that for a simple cubic lattice that
d2=a2
h2+k2+l2
(this is true for the fcc & bcc lattices as well, when we use the conventional unit cell).
3. Structure Factor of bcc Lattice The geometrical structure factor is given by
S~
G=X
j
fjei~
G·~r
where ~rjis the position of jth atom in the unit cell, and fjis the atomic form factor of the jth atom
(the scattering amplitude of a single atom, in units of single electron scattering).
(a) Show that for the bcc structure with a single atom per primitive cell, that the structure factor
reduces to
S= 0 when h+k+l=an odd integer,
S= 2fwhen h+k+l=an even integer
when referred to in the conventional unit cell (i.e, two identical atoms, one at r= 0~a1+ 0~a2+
0~a3, and another at r= (1/2)~a1+ (1/2)~a2+ (1/2)~a3)
(b) What is the physical significance of the missing diffraction peaks when h+k+lis an odd
integer i.e, (100), (111), (300), etc.?
4. Electron, neutron and x-ray (photon) diffraction Diffraction from solids is not limited to x-ray
radiation. Electron and neutrons may also be used, so long as their deBroglie wavelength (h/p) is
short enough. Compare the energy and velocity of the longest wavelength radiation of diffraction for
electrons, neutrons and photons from Na metal, given that it forms a bcc lattice with a cubic lattice
constant of a=4.23 ˚
A.
pf3

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Download Physics 463 Problem Set 2: Diffraction and Crystal Lattices - Prof. David A. Reis and more Assignments Solid State Physics in PDF only on Docsity!

  1. 1D diffraction (similar to Kittel 2.4) Consider a finite 1-dimensional chain of equally spaced atoms at positions ~rm = maxˆ

(a) Show that the magnitude square of the scattering amplitude

|F |^2 =

sin2 1 2 N~a · ∆~k sin2 1 2 ~a · ∆~k

hint: m=∑N − 1

m=

xm^ =

1 − xN 1 − x

(b) The diffraction condition occurs when ~a · ∆~k = 2πp (p integer). Show that the peak |F |^2 scales as N 2 and the width scales as 1 /N (hint, consider what happens if a∆k = 2π + ,   1 ).

  1. Interplaner separation (based on Kittel 2.1) Consider a plane hkl in a crystal lattice

(a) Prove that the reciprocal lattice vector G~ = h~b 1 + k~b 2 + l~b 3 is perpendicular to this plane. (b) Prove that the distance between two adjacent parallel planes of the lattice is

d(hkl) =

2 π | G~|

(c) Show that for a simple cubic lattice that

d^2 =

a^2 h^2 + k^2 + l^2 (this is true for the fcc & bcc lattices as well, when we use the conventional unit cell).

  1. Structure Factor of bcc Lattice The geometrical structure factor is given by

S (^) G~ =

j

fj ei G~·~r

where r~j is the position of jth atom in the unit cell, and fj is the atomic form factor of the jth atom (the scattering amplitude of a single atom, in units of single electron scattering).

(a) Show that for the bcc structure with a single atom per primitive cell, that the structure factor reduces to

S = 0 when h + k + l = an odd integer, S = 2f when h + k + l = an even integer

when referred to in the conventional unit cell (i.e, two identical atoms, one at r = 0~a 1 + 0~a 2 + 0 ~a 3 , and another at r = (1/2)~a 1 + (1/2)~a 2 + (1/2)~a 3 ) (b) What is the physical significance of the missing diffraction peaks when h + k + l is an odd integer i.e, (100), (111), (300), etc.?

  1. Electron, neutron and x-ray (photon) diffraction Diffraction from solids is not limited to x-ray radiation. Electron and neutrons may also be used, so long as their deBroglie wavelength (h/p) is short enough. Compare the energy and velocity of the longest wavelength radiation of diffraction for electrons, neutrons and photons from Na metal, given that it forms a bcc lattice with a cubic lattice constant of a =4.23 ˚A.

100 200

8 220 NaCI CuKa

7

5

(^3 222 ) (^2 400 ) I 333 3 511 30 40 50 9029

Figure 1: Powder diffraction pattern for NaCl, from Warren

  1. Powder Diffraction

Figure 1 shows powder diffraction data for NaCl for λ = 1. 542 A (Cu K˚ α radiation). In table 1, the angles, a geometrical polarization factor, and the area of each peak are tabulated. Answer the following (filling in the table where appropriate):

(a) Identify the peaks in terms of their miller indices. What is the degeneracy, m, of the particular peak (i.e., how many possible permutations of hkl will give the same Bragg condition). Hint, this may take some trial and error. It may help to start with low orders of hkl and consider what the ratios would be for a cubic crystal. (b) Verify that NaCl is face-centered cubic and determine the cubic lattice constant a (of the con- ventional cell)? (c) The density of NaCl is 2.17g/cm^3 , show that there are 4 Na and 4 Cl ions in the unit cell. (d) Given that the structure is fcc, with 4 Na and 4 Cl ions, it turns out that the only two ways of placing the Na+^ and Cl−^ ions in the unit cell correspond to the zinc-blende and the rock salt structure. By considering the strength of the diffraction peaks, verify that NaCl forms the rock-salt structure (aka NaCl). (e) The area under the diffraction peak is expected to be proportional to the square of the structure factor, S^2 , multiplied by the degeneracy, m, and a geometrical correction due to polarization effects, P (values of which are tabulated in table 1). Complicating matters is that the atomic form factors are functions of sin θ/λ (see Kittel for explanation and my figure 2 for approximate values). When all of these factors are taken into account, how well do we reproduce the peaks in the data?