Journal

rssN 1022-8594JUJS

O 2010 Jahangirnagar University Journal of ScienceVol.33, No.2, pp.23-32

Temperature Dependence Of The Reflection Lines

Md. Readul Mahmud and Md. Obaidur RahmanDepartment of Physics., Jahangirnagar University, Savar, Dhaka,Bangladesh

Abstract:

We have calculated the temperature dependence of reflection lines for twoelements Al, Si and for one compound NaCl. For the calculation, firstly we haveconsidered the temperature factor using Einstein model and secondly we haveconsidered the Debye temperature factor using Debye model. The effect is samefor both cases i.e. scattered intensity decreases with the increase of temperatureas well as with the Miller indices. Further, we found that scattered intensity for(200), (400), (600), (800), (1000) planes are slightly higher in Einstein moderthan that of Debye model

Keywords: Reflection line, Temperature factor, scattered intensity, Debyetemperature, Einstein model, Debye model.

Introduction;, ', .

As the temperature of the crystal is increased, the intensity of the Bragg-reflectedbeams decreases, but the angular width of the reflected line does not change tl].We,get a Sharp x-ray reflection from atoms undergoing large amplitude random

. !fie1mal motion, with instantaneous nearest-neighbor spacing differing by 10

,*,";-,.:.,.rpercent 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 roomtemperature are far from a periodic unay, because of the large thermal fluctuation.Therefore the argument went; one should not expect a well-defined diffractedbeam [2]. But such a beam is found. Thus it is of great importance to incorporatethe temperature effect in X-ray intensities. From this point of view, we havestudied the temperature dependence of reflection lines both for Einstein andDebye model.

Address for Correspondece: Professor Md. Obaidur Rahman, Department of Physics.,Jahangirnagar University, Savar, Dhaka, Email: rahmankek@hotrnail,com

24 Mahmud and Rahman,

Temperature Dependence of the Reflection Lines (BinsteinApproximation):Consider the radiation amplitude scattered by a crystal; let the position of theatom normally at r, contain a term u(t) fluctuating in time: r(t) = 11 + u(t). 12)We suppose each atom fluctuates independently about its own equilibriumposition. Then the thermal average of the structure factor contains terms

fi ,rp (-iG.r1 ) < exp ( -iG.u ) > (1)

Where<exp(-iG.u)> =l- i<G.u

But < G.u > = 0, because u is a random thermal displacement uncorrected withthe direction of G.

Further, < (G.u)z > = G2 <u2 >< ror2 0 > = I/3 < ut > G' The factor Il3 arisesas geometrical average of cos20 over a sphere.

The function

exp ( -I/6 <ut > G2 ) = I - I/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 = Io exp ( -I/3< u2 > G2) @)

Where, Io is the scattered intensity from the rigid lattice. The exponential factor isthe Debve-Waller factor.

Here < u'> is the meanaverage potential energy <U>dimension is 3/2 KsT, whence

square displacement of an atom. The thermalof a classical harmonic oscillator in three

<U> = %C<u2> = ]/zMa]<ut> = 3/2KnT (s)

Where C is the force constant, M is the mass of the atom and ro is the frequencyof the oscillator. We have used the result art = Cn4. Thus the scattered intensity is

I ( hkl ) = Io exp ( -KBTG2/Ma)2 ) (Einstein approximation) (Ol

Where, (hkl) are the indices of the reciprocal lattice vector G. This classical resultis a good approximation at high temperatures.

Temperature Dependence Of The Reflection Lines

Temperature Dependence of the Reflection Lines (DebyeApproximation):

The thermal vibrational amplitude of the atom will have an affect on X-rayscattering []. We need any quantitative information about the temperature factore''",bnt it is convenient to describe the calculation. We have

I = Ioe-M

25

The quantity Is is then the incident intensity. Because the intensity of apy linedepends on Is, calculated intensities must be multiplied by ,'' to allow for thermalvibration and the scattering angle 20 tll. Where,

- , /-z \ ,-,( sin d)' ^( sin d)tM_2n.ffi)=8n.E,l ^ )=ul ^

)Where rz2is the mean square displacement of the atom in a direction normal tothe reflecting planes. The exact calculation of i2 as a function of temperature is

extremely difficult, which means that M is hard to determine accurately. Debyehas given the following expression [1]:

ltrl -SU [" t ' *](sin4\2*k@rl4(n. ;)l; ) (Debve approximation)

(1)

(8)

(e)

Where h is the plank's constant, Z is the absolute temperature, m is the mass ofthe vibrating atom, k is Boltzmann's constant, @ is the Debye characteristictemperature of the substance in ok, r = @/7, and Q(x) is a functiofl, ffi = AlN,

where A = atomic weight and N = Avogadro's number, Inserting the numericalvalues of the constants in equation (8), We get

6h2T - ,. .,* "l(rin9,,z_ (6X6.0?x1026)(0.e3x10-34)'T - ,. . ,r- ,sin?.,M -

mtorLa(D ' or )" , Ao2(1.sg*ro-tf0-fr)-LQ\x)+ ol\ ^

)-

- l ' l5x]-o4z wG) 1

r11sm ey'A@"47

Again, Q$) canbe express as, [3]

d(x) + { : t + d - r*r\ 4 3'S ts,Sfr'S

Thus, M can be written as,

1.15xr047 2 in0,2M_ ) x lu ^{ -. x- x' _,sln a.

A@'z Lt*G-**r ^t

( 10)

26 Mahmud and Rahmaru

(t t)

Finally we get,

r =ro"*pt_l:llxt^o? o*t_frlrr jrIntensity 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 calculationwe have used equations (6), (11) and calculated values are tabulated in table 1 andtable 4.From equation (6), we get

I ( hkl ) = Io exp ( -KsTG2/Mcrr2;Here, Incident intensity, Io = l0l2photons/sec

Boltzmann constant, Kn = 1.38* 10-16 ergldegreeMass of Al, M = 4.48* 10-23 gmAngular frequency, cD = l}ta HzReciprocal lattice vector, G = 1/d.

Again, dnn = '.jp6s+lTWhere, 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 usingequations (6), (11) and calculated values are tabulated in tables 2, table 3 andtable 5.

Table 1. Data for A1 (Einstein Approximation)

femperaturenK

Incidentlntensity

Scattered IntensityPhotons/sec

Hnotons/sec Plane(200) Plane(400) Plane(600) Plane(800) Plane(1000)100

lE+12

9.97E+ll 9.88E+11 9.74E+ll 9.54E+l l 9.298+ll

1509.968+l I 9.82E+11 9.61E+l I 9.31E+11 8.95E+l I

2009.948+11 9.778+ll 9.48E+11 9,108+11 8.62E+l I

2509.93E+11 9.7lB+ll 9.35E+l I 8.88E+11 8.3lE+11

3009.93E+11 9.65E+11 9.23E+11 8.68E+11 8.01E+11

Temperature Dependence Of The Reftection Lines

1.05 t+ I2

1.008 r 12

!"5ULr lll9.00tttli

8.50[r11

I .0clIf 1t

7"50[r 11

/,00 L+.1.I

6, 50 t+1.1

6.00trl.1

27

Temperature dependance of Intensity

(v

oosg

'6

a

*** pl.inri i200i

*N* plunr {400}

..,..',r',,".,. plA rtg (6 00)

*"-q* plone {U0U)

',',,'i',i,'",* plit rltl ( 1000)

200

Ternpe,rature In K

Figure 1. Temperature dependence of Al for different planes (Einstein Approximation)

, t.: .,.,,,

Table 2.Data for Si (Einstein Approximation)

emperatureinK

IncidentIntensity

Photons/sec

Scattered IntensityPhotons/sec

Plane(200) Plane(400) Plane(600) Plane(800) Plane( 1 000)-: Ii

100

lE+12

ii i -t.ri

9.98E+l I 9.94E+11 9.86E+l I 9.75E+l I 9.60E+11

t::r50 9.98E+11 9.91E+11 9.19E+11 9.638+11 9.4lE+11

9.978+11 9.87E+11 9,72F+ll 9.51E+11 9.22F+ll

250 9.96E+11 9.848+l l 9.65E+11 9.39E+11 9.04E+11

300 9,95E+ll 9.81E+l I 9.58E+l I 9.278+11 8.86E+11

28 Mahmud and Rahmnn,

Figure 2.Temperature dependence of Si for different planes (Einstein Approximation)

Table 3. Data for NaCl (Einstein Approximation)

Temperature dependa nce of I ntensity

IoVr

ao6

|,}

ac

1.058r1i

i.00E+12

9.508+11

9,00Er11

8.508+1tr

8,008+11

7.508+1X

7,008+11

6,50E+l1

6.008{11

*-6* pl6ne {2001

*.S*plane {400)

,,"rrit*.' pla n e{600}

,*"r* plane {800}

'*.+"* plane t1000)

100 150 200 t50

Temperature in K

300

Temperatureiir K

Incident'Intensity

Photons/sec

Scattered IntensityPhotons/sec

Plane(200) Plane(400) Plane(600) Plane(800) Plane(1000)

r00

lE+12

8.75E+l I 5.87E+l I 3.016E+l I l. I 98E+1 I 3.601E+10

150 8.192E+11 4,4978+ll 1.656E+11 4.148E+10 6.833E+9

200 7.665E+11 3.4458+11 9.094E+10 1.4368+10 |,297E+9

250 7.172F+ll 2.648+ll 2.7428+10 4.9718+9 2.461E+8

300 6.71lE+l l 2.022E+ll 1.5068+10 l,72lE+9 4.6698+7

Temperature Dependence Of The Reflection Lines

Temperature Dependance of Intensity

tu

{gooso.).itg{u

c

1.00t+12

9,008+11.

8.008+11

7.00E+l1

5.00F+11

5,00[ r 11

4,00E.r"U

3,008+11

2.00E+11

1.00E+11

0.00[+00

*"+* plsne {200}

*#*pliure {4001

-"'ir*- olane{600)

*** plane {800}

"".*,*. plonrr {1000}

100 150 200 250

Temperature in K

300

Figure 3.Temperature dependence of NaCl for different planes (Einstein Approximation)

Temperature Dependence of lntensity

1,00E+1?

g.00L,nf t

, 8.00E+1.1{,{. 7.008r'116E g.ooEnrtof 5,oo[*xl

F +.ooe*rtcg J.uuL+l1

1.008+11

1.QS[+11

0.008+00

*-pin Plan0{200}

*.S*Plancia00l

"1s1"*plane{6Cr0}

***xPhns{$fl$} :

****plane{1000}

100 150 t00 250 300 '

Temperatuer In K

z9

t'

Figure 4. Temperature,dependence of Al for different planes (Debye Approximation)

30 Mahmud and.,Rahman,

Table 4.Data for Al (Debye Approximation)

Table 5. Data for Si (Debye Approximation)

TemperatureinK

IncidentIntensity

Photons/sec

Scattered IntensityPhotons/sec

Plane(200) Plane(400) Plane(600) Plane(800) Plane(1000)

100

lE+12

9.4418+11 8.914E+11 8.416E+11 7.946F+11 7.5028+11

150 9.174E+ll 8.4168+ll 7.7218+11 7.083E+l l 6.498E+l I

200 8.914E+11 7.946E+ll 7.0838+l I 6.314E+l1 5.6298+ll

250 8.662E+11 ,7.5028+Il 6.498E+ I I 5.6298+ll 4.8758+ll

300 8.416E+l,l 7.083E+11 5.962E+ll 5.017E+l I 4.223E+11

TemperatureinK

IncidentIntensity

Photons/sec

Scattered IntensityPhotons/sec

Plane(200) Plane(400) Plane(600) Plane(800) Plane(1000)

t00

lE+12

UJ

9.8218+l I 9.644E+ll 9.47E+ll 9.2'l3E+11 9.1348+l r

150 9.732E+ll 9.47lE+|1 9.2168+11 8.93E+l l 8.729E+ll

200 9.644E+ll 9.301E+l I 8.969E+l I 8.5998+11 8.3428+l I

250 9.557E+l l 9.1348+l I

),

8.728E+l I 8.28E+11 7.973E+rt i

300 9.a7rc+ll s.ezreli i*.tr :

8.4948+11 97.974E+11 7.619E+\1

Temiterature Dependence Of The Reflection Lines

Temperature Dependenc,e of lntensity Ir i

1.J0E+12

' i 1.00E+12

uotA

i' , ?,'8.00E+11oo

d 6.00E+It

+,'g 4.ooE+rr

2.00E+i I

0.008+00

*@-*Plonet200)

*S*Plane(400|,

""',rrr",*planei600)

*+*Plane(8001

$ds."Pl'ijne(1000)

100 1s0 200 't50

Temperature in K

300

Figure 5. Temperature dependence of Si for different planes(Debye Approximation)

Results and Discussions:Intensity of reflection lines decreases with the increase of temperature. Intensityof reflection lines also decreases with the increase of indices. Si is found to be thebest element among three as it has low temperature effect i.e. it absorbed lessamount of incident light. We have calculated the temperature dependence ofreflection lines for two elements Al, Si and for one compound NaCl. For thecalculation, firstly we have considered the temperature factor exp (-KsTG'/Mrt 1

(Einstein approximation) and obtained the figurel, figure 2 and tigure 3. Thesecurves show that intensity of reflection lines decreases with the increase oftemperature. Secondly we have considered the Debye temperature factor

31

expI

figure 4 and figure 5. Again intensity of reflection lines decreases with theincrease of temperature. lnterestingly, the effect is same for both cases i.e.scattered intensity decreases with the increase of temperature as well as with theindices. Further, we found that scartered intensity for (200), (400), (600), (800),(1000) planes are slightly higher in Einstein rnodel than that of Debye model andthis is done for the first time.

%g o*t-S,r, fff (Debye approximation) and obtained

32 Mahmud and Rahman,

References:

tU Cullity, 8.D., (1967). F,lement of X-ray Diffraction, Addition-Wesleypublishing company, [nc.

12) Kittle, C., (1974). Introduction to Solid State Physics, Wiley Eastern PrivateLrd.

t3l Warren, 8.8., (1968). X-ray Diffraction, Addition-Wesley PublishingCompany.

t4l Zachartasen, W.H. , (1929). Theory of X-ray Diffraction in Crystals, DoverPublication, fnc. New York

t5l Saxena, Gupta, Saxena (1988). Fundamental of Solid State Physics, Kedarnath Ram nath & Co.

t6l Singhal, R. L., (2003).Solid State Physics, Kedar Nath Ram & Co.

. , .' i i