Hyperfine Interact (2009) 194:381–389DOI 10.1007/s10751-009-9987-7

Calculation of the Debye–Waller factor of crystalsusing the n-dimensional Debye function involvingbinomial coefficients and incomplete gamma functions

Erhan Eser · I. M. Askerov · B. A. Mamedov

Published online: 9 July 2009© Springer Science + Business Media B.V. 2009

Abstract The objective of this work was to obtain a simpler and more accurateanalytical expression to determine the Debye–Waller factor of crystals using an n-dimensional Debye approximation involving binomial coefficients and incompletegamma functions. The results obtained were compared with the correspondingexperimental and theoretical results and, the calculated values were shown to bein excellent agreement with those obtained in the experimental and other previousstudies.

Keywords Mössbauer spectroscopy · Debye–Waller factor · Debye temperature ·Debye functions

PACS 61.18.Fs · 61.18.-j · 61.66.Bi

1 Introduction

The Mössbauer effect not only elucidates the recoilless emission and the resonantabsorption of gamma rays in a solid [1, 2], but also is a very useful technique forinvestigating certain problems concerning the dynamics of crystal lattices [3, 4]. Thephenomenon was discovered by Mössbauer [5, 6].

The main parameter related to the Mössbauer effect—the Debye–Waller factor(DWF), is important for determining the intensity of X-ray diffraction lines [7, 8]and for crystal-structure determinations [9]. In addition, knowledge of the DWF isrequired in both neutron inelastic scattering [10] and dynamical electron diffraction[11]. In spite of its importance in various branches of physics and chemistry, the DWFis not accurately known for many crystals.

E. Eser (B) · I. M. Askerov · B. A. MamedovFaculty of Science and Arts, Department of Physics,Gaziosmanpasa University, Tokat 60250, Turkeye-mail: erhaneser@gop.edu.tr

382 E. Eser et al.

Experimentally, the DWF can be measured using different techniques such as X-ray diffraction [12–18], neutron diffraction [10], Mössbauer spectroscopy [19–23],and radioactive ion-beam technique [24]. The results obtained for the DWF areexpressed, however, only at fixed temperatures, such as room temperature (293 K)[25, 26], and hence these results are difficult to interpolate into different temperatureregions and do not correspond completely to the experimental situation.

Since the discovery of the Mössbauer effect [5, 6], many theoretical studies [4, 9–11, 27–34] dealing with the DWF of a crystal in addition to various experimentalstudies have been reported. However, these theoretical calculations are based on twoexpansions under low-temperature (T = θD) and high-temperature (T � 1/2 θD)

regimes. For example, Heberle [32] showed that the DWF involving Debye integralscan be separately expressed in terms of the Bernoulli numbers at low and hightemperatures. Mahesh [9] calculated the DWF with the two different infinite-seriesexpansions of the Debye integral at low and high temperatures. At low temperature,the temperature variation of DWF is remarkably small and can be defined usingthe small temperature variation of the Debye temperature, θD. Contrarily, at hightemperature the DWF diminishes relatively sharply with increasing temperature.Moreover, the expansions stated are not accurate over a wide range of temperatures.

In this study, a simple analytical expression for the DWF is obtained using an n-dimensional Debye function involving binomial coefficients and incomplete gammafunctions [35]. Subsequently, the validity and reliability of the analytical relationobtained is tested by applying it to the DWFs of aluminum and copper crystals.

2 Theory and method of calculation

In the Debye approximation, the recoilless fraction f for gamma rays emitted orabsorbed by a nucleus in a crystal lattice can be written in the following form [5, 6,34, 36]:

f = e−2W(T), (1)

where W(T) is the DWF and is represented by expression

W(T) = ER A(T)/2 (2)

ER, the recoil energy, and A(T) are defined, respectively, as:

ER = E2γ

2mc2(3)

A(T) = 3

2kθD

[1 + 4

(TθD

)D1 (1,θD/T)

](4)

In Eqs. 3 and 4, θD is the Debye temperature, Eγ is the energy of gamma transition,m is the mass of the nucleus, c is the velocity of light, and Dn(β, x) is the generalizedDebye function, which is defined as [37, 38]:

Dn (β,x) = nxn

x∫0

tn

(et − 1)βdt (5)

Calculation of the Debye–Waller factor of crystals 383

The accurate and quick calculation of the n-dimensional Debye functions is ofconsiderable importance, as can be noted from Ref. [37], in various branches ofphysics and chemistry. Kaufman and Lipkin [39] have worked out the detailedderivation of Eq. 4. Later, Hardy and coworkers [40] found a simple method toevaluate the Debye integral in Eq. 4.

Recently, using the binomial expansion theorem (for details, refer [35, 37]), a newformula has been introduced for arriving at the n-dimensional Debye functions for awide range of parameters in the following form [37]:

Dn (β,x) = nxn

limN→∞

N∑i=0

(−1)i Fi (−β)γ (n + 1, (i + β) x)

(i + β)n+1 (6)

where n, β are integer and noninteger values, Fm (n) = n!/[m! (n − m)!] is the bino-mial coefficient for the integer n, and γ (α, x) is the well-known incomplete gammafunction defined by the following expression [35]

γ (α, x) =x∫

0

tα−1e−tdt (7)

The index N occurring in Eq. 6 is the upper limit of sum. The detailed derivationof the above mentioned formulae can be found in a previous report by the currentauthors [37]. Finally, considering the n-dimensional Debye function, Dn(β, x), thegeneral case of recoilless fractions f in Eq. 1 can be obtained from the followingequation:

f = exp

{− 3ER

2kθD

[1 + 4

(TθD

)D1(1,θD/T)

]}(8)

In this article, the temperature-dependence of DWF and A(T) for aluminum andcopper crystals has been calculated using the formulas provided in Eqs. 2 and 4.The following formula has been used by the authors for calculating the Debyetemperatures provided in Table 1 [4]:

θD(T) = 413 (1 − 0.000461T)1/2 K (9)

For the recoil energy, values obtained by fitting the experimental results of Jameset al. were used [41].

3 Results and discussion

We used the Mathematica software system (version 5.0) to calculate the Debyefunction using the formula presented in this article. The calculated and the compiledvalues [4, 18, 28–31, 42–48] of the DWF and A(T) for aluminum and copper crystalsare shown in Tables 1–7.

The calculated and compiled values of DWF for the aluminum crystal in thetemperature range of 80–860 K are shown in Table 1. As can be observed fromTable 1, the values compiled by Killean [4], and McDonald [42] express excellentagreement with current results. The largest discrepancy (4.6%) compared to theexperimental values occurred at the temperature 375 K.

384 E. Eser et al.

Tab

le1

The

expe

rim

enta

l-,t

heor

etic

al-,

and

calc

ulat

edre

sult

sof

the

Deb

ye–W

alle

rfa

ctor

sfo

ral

umin

um

T(K

)80

295

300

375

475

485

552

588

655

700

730

810

830

860

θD

(K)

405

386

383

375

365

361

355

353

345

340

336

326

325

320

Ref

.[4]

0.32

0.89

0.90

1.16

1.54

1.58

1.87

2.03

2.36

2.59

2.77

3.25

3.37

3.57

Exp

erim

enta

l0.

325

0.90

00.

890

1.22

1.56

1.60

1.89

2.07

2.42

2.62

2.77

3.24

3.34

3.64

Cal

cula

ted,

Eq.

40.

327

0.88

20.

900

1.16

31.

534

1.59

91.

876

2.01

72.

375

2.57

72.

766

3.23

63.

375

3.56

2

The

expe

rim

enta

lval

ues

are

obta

ined

from

the

neut

ron-

diff

ract

ion

data

ofR

ef.[

42]

apar

tfr

omth

ose

at80

Kan

d30

0K

,whi

chw

ere

obta

ined

from

Ref

.[43

]an

dth

eph

onon

-dis

pers

ion

calc

ulat

ions

usin

gth

eda

taof

Sted

man

and

Nils

son

[44]

,res

pect

ivel

y

Calculation of the Debye–Waller factor of crystals 385

Table 2 The values of A(T)

for aluminum (×102 eV−1)T (K) θD (K) Walker [32] Calculated Eq. 4

4 – 0.471 –20 418 0.478 0.42280 392 0.598 0.562300 382 1.54 1.493400 – 2.02 –500 – – –

Table 3 The calculated andcompiled results of DWF foraluminum at the roomtemperature (293 K)

Authors θD (K) Other studies Calculated

Dingle and Medlin [13] 393 0.84 0.8431Flinn and McManus [18] 410 0.79 0.7778

390 – 0.8555McDonald [42] 386 0.89 0.8725

390 0.87 0.8555Butt et al. [26] 394 0.86 0.8390Owen and Williams [45] 395 0.84 0.8350Chipman [46] 390 0.87 0.8555

407 0.79 0.7887Mothersole and Owen [47] 397 0.84 0.8270De Marco [48] 387 0.89 0.8683

Table 4 Values of A(T) for copper (×102 eV−1)

T (K) Flinn et al. [18] C. F. model Jacobsen White A-S cal. [29] Calculatedexperimental [18] [28] [30] (axially Eq. 4values symmetric)

θD A(T)

4 320 0.544 0.552 0.579 0.537 0.57 0.54420 320 0.588 0.566 0.593 0.548 0.582 0.55780 320 0.755 0.762 0.808 0.733 0.779 0.754300 315 2.17 2.10 2.29 2.03 2.18 2.166400 300 3.14 2.77 3.02 2.67 2.87 3.137

Table 2 shows the calculated temperature-dependence of A(T) for aluminum. Forcomparison, the experimental and theoretical results available in literature are alsoshown. From Table 2, it is clear that the current calculated results are approximatelyin agreement with the results reported by Walker [31].

The calculated and compiled results of DWF for aluminum crystals at roomtemperature (293 K) are provided in Table 3. The results calculated in this studyare very close to the data obtained by Dingle et al. [13], Flinn et al. [18], and Owenet al. [45]. For example, the calculated and the other reported values of the DWFare, respectively, 0.8431 and 0.840 [13] for 393 K, 0.778 and 0.79 [18] for 410 K, and0.835 and 0.840 [45] for 395 K. In addition, Table 3 shows that the current resultsare in agreement with the results of Butt et al. [26], McDonald [42], Chipman [46],Mothersole and Owen [47], and De Marco [48].

In Table 4, the experimental and theoretical results available in literature areillustrated using the values of A(T) calculated for copper. As is clear from Table 3,the agreement between the herein-presented calculated results and the experimental

386 E. Eser et al.

Table 5 Values of Debye function Dn(β,x) for N = 600

n β x Eq. 6 Eq. 5 in Ref. [49]

5 1 5 7.6863425441801433915281E−02 7.6863425441801921613944E−028 1 8.5 5.6687155559824027864627E−03 5.6687155559824027864628E−0318 1 45 2.0127382194209472048073E−13 2.0127382194209472048073E−1312 1 14.6 4.2773491950712724740958E−05 4.2773491950712724740958E−0523 1 23.4 9.1549162252123059203204E−09 9.1549162252123059203204E−0928 1 31.4 7.1784942037477506356873E−12 7.1784942037477506356873E−1232 1 4.5 5.6573490148599273944602E−02 5.6573490148599273944602E−023.5 4 5.5 9.5330710049633372642951E−03 –7.3 3.4 2.8 1.9538993040469541344579E−03 –13.4 8.3 14.1 6.5516025950104737049663E−17 –17.7 8.4 7.6 2.4375463342646329557567E−18 –21.1 0.8 2.6 3.6566607629500999424915E−01 –

Table 6 Convergence ofderived expression forD16.2(3.7,13.2) as a function ofsummation limit N

N Eq. 6

100 7.510614468047612882207382E−14120 7.510614468047612882575749E−14140 7.510614468047612882608908E−14160 7.510614468047612882613146E−14180 7.510614468047612882613848E−14200 7.510614468047612882613990E−14220 7.510614468047612882614024E−14240 7.510614468047612882614033E−14260 7.510614468047612882614036E−14280 7.510614468047612882614037E−14300 7.510614468047612882614037E−14320 7.510614468047612882614037E−14340 7.510614468047612882614037E−14

Table 7 Convergence ofderived expression forD6(1,5.4) as a function ofsummation limit N

N Eq. 6

100 0.05331329256979465120 0.05331329256980074140 0.05331329256980614160 0.05331329256980812180 0.05331329256980895200 0.05331329256980933220 0.05331329256980951240 0.05331329256980962260 0.05331329256980967280 0.05331329256980970300 0.05331329256980972320 0.05331329256980974340 0.05331329256980974

results obtained by Flinn et al. [18] are in good agreement with the other compiledresults.

The examples of computer calculation of Debye function are shown in Table 5.As can be seen from Table 5, the accuracy of computer calculations obtained in thepresent algorithm is satisfactory. Tables 6 and 7 list the partial summation in Eq. 6

Calculation of the Debye–Waller factor of crystals 387

corresponding to progressively increasing upper limit by N. We see from Tables 6and 7 that Eq. 6 displays the most rapid convergence to the numerical result with 17-digit precision when the 340 terms in the infinite summation are taken into account.

In the literature, as far as we know, there are no studies on the evaluation of Debyefunction with noninteger values n and β. Equation 6 is in progress for the calculationof the Debye function for noninteger values n and β.

Finally, a new simple analytical expression for the DWF of crystals which avoidstwo separate expansions for the low and high temperature regimes is presented byusing the Debye approximation based on binomial coefficients and the incompletegamma functions. In our study we have performed calculations on N = 600. The useof the simple analytical expression for modeling and simulation can prove usefulin the determination of the properties of crystals. Moreover, using the expressionobtained in this article, numerous problems in various branches of physics andchemistry can be solved quickly, and accurately.

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