TEMPERATURE DEPENDENCE OF THE
REFLECTION LINES
Md. Readul Mahmud and Md. Obaidur Rahman
Department of Physics., Jahangirnagar University, Savar, Dhaka-1342, Bangladesh
Email: maileriad@gmail.com, rahmankek@hotmail.com
Abstract:
We have calculated the temperature dependence of reflection lines for two elements Al, Si and
for one compound NaCl. For the calculation, firstly we have considered the temperature factor
exp (-KBTG2/Mω2) (Einstein model) and secondly we have considered the Debye temperature
1.15  10 4 T
x2
x4
sin  2
(1 

)](
) (Debye model). The effect is same for both
factor exp[ 
2
36 3600

A
cases i.e. scattered intensity decreases with the increase of temperature as well as Miller indices.
Keywords: Temperature, reflection line, intensity, Debye temperature factor, Einstein model.
Introduction:
As the temperature of the crystal is increased, the intensity of the Bragg-reflected beams
decreases, but the angular width of the reflected line does not change [1]. We get a sharp x-ray
reflection from atoms undergoing large amplitude random thermal motion, with instantaneous
nearest-neighbor spacing differing by 10 percent at room temperature. Before the Laue
experiment was done [2], the object was made that the instantaneous positions of the atoms in a
crystal at room temperature are far from a periodic array, because of the large thermal
fluctuation. Therefore the argument went; one should not expect a well-defined diffracted beam
[2]. But such a beam is found. Thus it is of great importance to incorporate the temperature effect
in X-ray intensities. From this point of view, we have studied the temperature dependence of
reflection lines both for Einstein and Debye model.
1|Page
Temperature Dependence of the Reflection Lines (Einstein Approximation):
Consider the radiation amplitude scattered by a crystal; let the position of the atom normally at rj
contain a term a(t) fluctuating in time: r(t) = rj + u(t). [2]
We suppose each atom fluctuates independently about its own equilibrium position. Then the
thermal average of the structure factor contains terms
fj exp (-iG.rj ) < exp ( -iG.u ) >
(1)
Where <…….> denotes thermal average. The series expansion of the exponential is
< exp (-iG.u ) > = 1 – i< G.u > - ½ < ( G.u )2 > + --------
(2)
But < G.u > = 0, because u is a random thermal displacement uncorrected with the direction of
G.
Further,
< (G.u)2 > = G2 <u2 >< cos2 θ > = 1/3 < u2 > G2
The factor 1/3 arises as geometrical average of cos2θ over a sphere.
The function
exp ( -1/6 < u2 > G2 ) = 1 – 1/6 < u2 > G2 + -------
(3)
has the same series expansion as (2) for the first two terms shows here. For a harmonic oscillator
all terms in series (2) and (3) can be shown to be identical. Then the scattered intensity, which is
the square of the amplitude, is
I = I0 exp ( -1/3< u2 > G2)
(4)
Where, I0 is the scattered intensity from the rigid lattice. The exponential factor is the DebyeWaller factor.
Here < u2 > is the mean square displacement of an atom. The thermal average potential
energy <U> of a classical harmonic oscillator in three dimension is 3/2 KBT, whence
< U > = ½ C < u2 > = ½ Mω2 < u > = 3/2 KBT
(5)
2|Page
Where C is the force constant, M is the mass of the atom and ω is the frequency of the oscillator.
We have used the result ω2 = C/M. Thus the scattered intensity is
I ( hkl ) = I0 exp ( -KBTG2/Mω2 ) (Einstein approximation)
(6)
Where, (hkl) are the indices of the reciprocal lattice vector G. This classical result is a good
approximation at high temperatures.
Temperature Dependence of the Reflection Lines (Debye Approximation):
The thermal vibrational amplitude of the atom will have an affect on X-ray scattering [1]. We
need any quantitative information about the temperature factor e-M, but it is convenient to
describe the calculation. We have
I = I0 e-M
(7)
The quantity I0 is then the incident intensity. Because the intensity of any line depends on I0,
calculated intensities must be multiplied by e-M to allow for thermal vibration and the scattering
angle 2θ [1]. Where,
M  2
 
2 u2
d2
 sin  
 sin  
 8 u 
  B

  
  
2
2
2
2
Where u 2 is the mean square displacement of the atom in a direction normal to the reflecting
planes. The exact calculation of u 2 as a function of temperature is extremely difficult, which
means that M is hard to determine accurately. Debye has given the following expression [1]:
6h 2 T
M 
mk 2

  x  
x  sin  


4   
2
(Debye approximation)
(8)
Where h is the plank’s constant, T is the absolute temperature, m is the mass of the vibrating
atom, k is Boltzmann’s constant, Θ is the Debye characteristic temperature of the substance in ok,
x = Θ/T, and  (x) is a function, m = A/N, where A = atomic weight and N = Avogadro’s number,
the coefficient of the bracketed terms above becomes
3|Page
M
6h 2T
mk 2
(6)(6.02  10 26 )(6.63  10 34 )T
x sin  2
[ ( x)  ](
)
2
 23
 20
4 
A (1.38  10 ) 10

=


1.15 10 4 T
x sin  2
[ ( x)  ](
)
2
A
4 
(9)
Again,  (x) can be express as, [3]
Thus, M can be written as,
M 
1.15  10 4 T
x2
x 4 sin  2
[
1


](
)
36 3600 
A 2
(10)
Finally we get,
1.15  10 4 T
x2
x4
sin  2
I  I 0 exp[ 
(1 

)](
)
2
36 3600

A
(11)
Intensity calculations:
We have calculated the intensity of Al for different reflection planes such as (200), (400), (600),
(800) and (1000) for various temperature. For the calculation we have used equations (6), (11)
and calculated values are tabulated in table 1 and table 4.
From equation (6), we get
I ( hkl ) = I0 exp ( -KBTG2/Mω2 )
Here, Incident intensity, I0 = 1012 photons/sec
Boltzmann constant, KB = 1.38*10-16 erg/degree
Mass of Al, M = 4.48*10-23 gm
Angular frequency, ω = 1014 Hz
Reciprocal lattice vector, G = 1/d
4|Page
Again, dhkl =
Where, lattice parameter, a = 4.05*10-8 cm and (hkl) are the Miller indices.
Similarly, we have calculated the intensity of Si and NaCl for different planes, such as (200),
(400), (600), (800), (1000) and for various temperature using equations (6), (11) and calculated
values are tabulated in tables 2, table 3 and table 5.
Table 1: Data for Al (Einstein Approximation)
Temperature
in K
Scattered Intensity
Incident
Intensity
Photons/sec
Photons/sec
Plane(200)
Plane(400)
Plane(600)
Plane(800)
Plane(1000)
100
9.97E+11
9.88E+11
9.74E+11
9.54E+11
9.29E+11
150
9.96E+11
9.82E+11
9.61E+11
9.31E+11
8.95E+11
9.94E+11
9.77E+11
9.48E+11
9.10E+11
8.62E+11
250
9.93E+11
9.71E+11
9.35E+11
8.88E+11
8.31E+11
300
9.91E+11
9.65E+11
9.23E+11
8.68E+11
8.01E+11
200
1E+12
Fig.1 Temperature dependence of Al for different planes (Einstein Approximation)
5|Page
Table 2: Data for Si (Einstein Approximation)
Temperature
in K
Scattered Intensity
Incident
Intensity
Photons/sec
Photons/sec
Plane(200)
Plane(400)
Plane(600)
Plane(800)
Plane(1000)
100
9.98E+11
9.94E+11
9.86E+11
9.75E+11
9.60E+11
150
9.98E+11
9.91E+11
9.79E+11
9.63E+11
9.41E+11
9.97E+11
9.87E+11
9.72E+11
9.51E+11
9.22E+11
250
9.96E+11
9.84E+11
9.65E+11
9.39E+11
9.04E+11
300
9.95E+11
9.81E+11
9.58E+11
9.27E+11
8.86E+11
200
1E+12
Fig.2 Temperature dependence of Si for different planes (Einstein Approximation)
6|Page
Table 3: Data for NaCl (Einstein Approximation)
Temperature
in K
Scattered Intensity
Incident
Intensity
Photons/sec
Photons/sec
Plane(200)
Plane(400)
Plane(600)
Plane(800)
Plane(1000)
100
8.75E+11
5.87E+11
3.016E+11
1.198E+11
3.601E+10
150
8.192E+11
4.497E+11
1.656E+11
4.148E+10
6.833E+9
7.665E+11
3.445E+11
9.094E+10
1.436E+10
1.297E+9
250
7.172E+11
2.64E+11
2.742E+10
4.971E+9
2.461E+8
300
6.711E+11
2.022E+11
1.506E+10
1.721E+9
4.669E+7
200
1E+12
Fig.3 Temperature dependence of NaCl for different planes (Einstein Approximation)
7|Page
Table 4: Data for Al (Debye Approximation)
Temperature
in K
Scattered Intensity
Incident
Intensity
Photons/sec
Photons/sec
Plane(200)
Plane(400)
Plane(600)
Plane(800)
Plane(1000)
100
9.441E+11
8.914E+11
8.416E+11
7.946E+11
7.502E+11
150
9.174E+11
8.416E+11
7.721E+11
7.083E+11
6.498E+11
8.914E+11
7.946E+11
7.083E+11
6.314E+11
5.629E+11
250
8.662E+11
7.502E+11
6.498E+11
5.629E+11
4.875E+11
300
8.416E+11
7.083E+11
5.962E+11
5.017E+11
4.223E+11
200
1E+12
Fig.4 Temperature dependence of Al for different planes (Debye Approximation)
8|Page
Table 5: Data for Si (Debye Approximation)
Temperature
in K
Scattered Intensity
Incident
Intensity
Photons/sec
Photons/sec
Plane(200)
Plane(400)
Plane(600)
Plane(800)
Plane(1000)
100
9.821E+11
9.644E+11
9.47E+11
9.273E+11
9.134E+11
150
9.732E+11
9.471E+11
9.216E+11
8.93E+11
8.729E+11
9.644E+11
9.301E+11
8.969E+11
8.599E+11
8.342E+11
250
9.557E+11
9.134E+11
8.728E+11
8.28E+11
7.973E+11
300
9.471E+11
8.971E+11
8.494E+11
97.974E+11
7.619E+11
200
1E+12
Fig.5 Temperature dependence of Si for different planes (Debye Approximation)
9|Page
Results and Discussions
Intensity of reflection lines decreases with the increase of temperature. Intensity of reflection
lines also decreases with the increase of indices. Si is found to be the best element among three
as it has low temperature effect i.e. it absorbed less amount of incident light.
We have calculated the temperature dependence of reflection lines for two elements Al, Si and
for one compound NaCl. For the calculation, firstly we have considered the temperature factor
exp (-KBTG2/Mω2) (Einstein approximation) and obtained the fig.1, fig.2 and fig.3. These curves
show that intensity of reflection lines decreases with the increase of temperature. Secondly we
have considered the Debye temperature factor exp[ 
1.15  10 4 T
x2
x4
sin  2
(
1


)](
)
2
36 3600

A
(Debye approximation) and obtained fig.4 and fig.5. Again intensity of reflection lines decreases
with the increase of temperature. Interestingly, the effect is same for both cases i.e. scattered
intensity decreases with the increase of temperature as well as indices. Further, we found that
scattered intensity for (200), (400), (600), (800), (1000) planes are slightly higher in Einstein
model than that of Debye model. This comparison is done for the first time to the best of our
knowledge.
References:
[1] B.D. Cullity, “Element of X-ray Diffraction”, Addition-Wesley publishing company, Inc.
(1967)
[2] Charles Kittel, “Introduction to Solid State Physics”, Wiley Eastern Private Ltd (1974)
[3] Warren B.E., “X-ray Diffraction”, Addition-Wesley Publishing Company (1968)
[4] W.H. Zachariasen, “Theory of X-ray Diffraction in Crystals”, Dover Publication, Inc. New
York (1929)
[5] Saxena, Gupta, Saxena, “Fundamental of Solid State Physics”, Kedar nath Ram nath &
Co,(1988).
[6] Singhal R. L, “Solid State Physics” Kedar Nath Ram & Co (2003)
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