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Frequency distribution for graphite based on the unfolding technique
Volume 78A, number 2
PHYSICS LETTERS
21 July 1980
FREQUENCY DISTRIBUTION FOR GRAPHITE BASED ON THE UNFOLDING TECHNIQUE Usha MALIK ’ and L.S. KOTHARl Department of Physics and Astrophysics,
University of Delhi, Delhi 110007,
India
Received 25 April 1980
We have extended the “unfolding technique”, developed in Reactor Physics, to calculate the frequency distribution function for graphite, starting from the measured values of its specific heat at different temperatures. The advantages and limitations of this approach are discussed.
Lattice dynamics is an extensively pursued area of research in Solid State Physics: a very large number of theoretical models exist to explain the experimental data that are rapidly accumulating. And yet the situation with regard to anisotropic materials is very different. One such material, which is of great interest both from the point of view of Solid State Physics as well as Reactor Physics, is graphite which is a layered structure. Although a number of models have been proposed [l31 to explain the observed temperature variation of the specific heat of graphite any one of them can explain the experimental data only in a very limited temperature range (fig. 1). Furthermore, none of these models is able to explain satisfactorily its neutron scattering properties. To understand the basic nature of the thermal vibrations in a solid calculations based on first principles as well as phenomenological models are essential. However, for studies like neutron scattering and the temperature variation of the specific heat and thermal conductivity it is sufficient to have a good model for the frequency distribution function; and this can be deduced directly from measurements of the temperature variation of the specific heat at constant volume of a solid. We intend to discuss here this aspect of the problem. Recently a lot of work has been reported in Reactor Physics on the so called “unfolding technique” [4-61. This technique has been developed to determine neutron 1 On leave of absence from Miranda House, University of Delhi, Delhi.
178
20x18
I5Xd t Y B 'i J
Id
5
5X10'
0
200
400 11
600
800
IOOC
IO-
Fig. 1. Variation of the specific heat with temperature for graphite: calculated values (solid curve), Krumhansl and Brooks model [l] (- - -), Yoshimori and Kitano model [2] (-. - .), Young and Koppel model [ 31 (- - --). Dots represent experimental [9-121 values.
spectra from measured activities of different neutron irradiated samples. A few attempts [7,8] were made earlier to use a similar technique for calculating the frequency distribution function,g(g) (g = ?zo/kB is the phonon energy measured in Kelvin, ti is Planck’s constant divided by 2r1, o is the angular frequency and k, is the Boltzmann constant), starting from the known
Volume 78A, number 2
PHYSICS LETTERS
21 July 1980
temperature variation of the specific heat. However these were not very satisfactory. In view of the recent developments in the unfolding technique [5,6], we propose to examine here the problem of determining the frequency distribution function for materials using this technique. The integral equation to be solved for g(t) is 3Nk, *D ~2~t/T -.I-
T2
o (et/T-1)2
g(5) dg = C, 0’) ,
where N is the Avogadro number, 0~ is the Debye temperature of the material and C&T) is the measured specific heat (per gm-mole) of the material at temperature T. Here g(t) is normalized to unity. We consider a linear integral equation b s a
W,
where in the given. matrix
s)K(s)ds
=A4
,
(2)
the kernel K(t, s), assumed to be well behaved domain of integration, and the function y(t) are We replace the above integral equation by the equation
KX=Y ,
(3)
where K is an (n X m) matrix and X and Y are column matrices of order m and n respectively. In brief, the method works as follows. We start with a trial solutionXo. Let Xf be the solution to be calculated, Using the variational principle (for details see e.g. Fischer [4]) one obtains the following equation as a result of one iteration: Xfl = (CTFCW2 + G)-l(CTFlnW2
+ GXo) ,
(4)
where C is an (n X m) matrix obtained by dividing each row of K by the corresponding element of Y ,F is an (n X n) diagonal matrix consisting of the weights Wi for measurements of Y, G is an (m X m) diagonal matrix consisting of the inverse squares of trial solution, W is a multiplier constant determining the speed of convergence, 1 n is a unit column matrix of order n, andCT denotes the transpose of C. If Xfl does not satisfy eq. (2) within the experimental accuracy the calculations are repeated using Xfl as the trial function in place of X0 in eq. (4). This procedure is continued till the required accuracy in Y is achieved (details will be reported elsewhere).
Fig. 2. Frequency distribution function for graphite: calculated values (solid curves a and b for upper cut-offs at 1500 K and 2500 K, respectively; peakg(E) value for curve a is 53.5 and for curve b it is 19.4), Krumhansl and Brooks model [l] (- - -) Young and Koppel model [ 31 (---). Experimental values [ 131 are shown by the (- . - .) curve.
Using the experimental data [9-121 in the temperature range 0.44-1033 K we have calculated the function g(t) using the method outlined above. We find that one can determine functions g(t) with upper cut-offs ranging from 1500 K to 2500 K which all give very good agreement between the calculated and the observed specific heat data. The g(t) functions, marked a and b for the two cut-offs 1500 K and 2500 K, respectively, are shown in fig. 2. Also plotted are the g(t) functions proposed by Krumhansl and Brooks [ 1 ] and Young and Koppel [3], along with the values obtained experimentally by Page [ 131 using the neutron scattering technique. Specific heat values based on our calculated g(t) functions are shown in fig. 1 by the solid curve. (The specific heat values based on our two frequency distribution functions a and b do not differ from each other by more than 1% at any temperature.) The agreement between the calculated and the observed data is very good and within 2% in the entire temperature range 179
Volume 78A, number 2
PHYSICS LETTERS
considered here. The calculated specific heat values based on other models are also shown in fig. 1. The values given by Krumhansl and Brooks differ from the measured values by almost 50% in the temperature range 0.44-2 K, by ~10% in the range lo-100 K (this is not very clear from fig. 1 but shows up if C&T) versus T is plotted on a log-log scale), by 4% in the range 100-300 K while they are within 2% in the range 300- 1000 K. On the other hand, Yoshimori and Kitano [2] have calculated specific heat values in the range 45-300 K and good agreement is obtained only for temperatures below 150 K. Young and Koppel [3] have calculated the values in the range 100-1000 K giving good agreement only in the range 100-300 K. We propose to evaluate neutron scattering cross sections for graphite on the basis of the models proposed here. These studies, which will be reported in due course, should help to fix more precisely the upper limit on g(t) in defining the frequency distribution function. The authors are thankful to Drs. Feroz Ahmed and A.N. Verma for many useful discussions. Part of this work was supported by an NSF Grant.
180
21 July 1980
References [l] J. Krumhansl and H. Brooks, Chem. Phys. 21 (1953) 1663. [ 21 A. Yoshimori and Y. Kitano, J. Phys. Sot. Japan 11 (1956) 352. [3] J.A. Young and J.U. Koppel, J. Chem. Phys. 42 (1965) 357. [4] A. Fischer, KFA report RFSP-JUL-1475 (Dec. 1977). [5] H. Sekimoto, Nucl. Sci. Eng. 68 (1978) 351. [6] R.H. Johnson, B.W. Wehring and J.J. Doming, Nucl. Sci. Eng. 73 (1980) 93. [ 71 V.A. Korshunov and V.P. Tanana, Sov. Phys. Solid State 18 (1976) 378. [ 81 V .A. Korshunov and V.P. Tanana, Sov. Phys. Dokl. 21 (1976) 714. [9] C.F. Lucks, H.W. Deem and W.D. Wood, Am. Ceram. Sot. Bull. 39 (1960) 313. [lo] W. DeSorbo and G.E. Nichols, J. Phys. Chem. Solids 6 (1958) 352. [ 1l] B.J.C. VanderHoeven Jr. and R.H. Keesom, Phys. Rev. 130 (1963) 1318. [12] W. DeSorbo and W.W. Tyler, J. Chem. Phys. 21 (1953) 1650. [ 131 D.I. Page, Proc. Symp. Neutron inelastic scattering, Vol. 1 (Vienna, 1968) p. 325.
PHYSICS LETTERS
21 July 1980
FREQUENCY DISTRIBUTION FOR GRAPHITE BASED ON THE UNFOLDING TECHNIQUE Usha MALIK ’ and L.S. KOTHARl Department of Physics and Astrophysics,
University of Delhi, Delhi 110007,
India
Received 25 April 1980
We have extended the “unfolding technique”, developed in Reactor Physics, to calculate the frequency distribution function for graphite, starting from the measured values of its specific heat at different temperatures. The advantages and limitations of this approach are discussed.
Lattice dynamics is an extensively pursued area of research in Solid State Physics: a very large number of theoretical models exist to explain the experimental data that are rapidly accumulating. And yet the situation with regard to anisotropic materials is very different. One such material, which is of great interest both from the point of view of Solid State Physics as well as Reactor Physics, is graphite which is a layered structure. Although a number of models have been proposed [l31 to explain the observed temperature variation of the specific heat of graphite any one of them can explain the experimental data only in a very limited temperature range (fig. 1). Furthermore, none of these models is able to explain satisfactorily its neutron scattering properties. To understand the basic nature of the thermal vibrations in a solid calculations based on first principles as well as phenomenological models are essential. However, for studies like neutron scattering and the temperature variation of the specific heat and thermal conductivity it is sufficient to have a good model for the frequency distribution function; and this can be deduced directly from measurements of the temperature variation of the specific heat at constant volume of a solid. We intend to discuss here this aspect of the problem. Recently a lot of work has been reported in Reactor Physics on the so called “unfolding technique” [4-61. This technique has been developed to determine neutron 1 On leave of absence from Miranda House, University of Delhi, Delhi.
178
20x18
I5Xd t Y B 'i J
Id
5
5X10'
0
200
400 11
600
800
IOOC
IO-
Fig. 1. Variation of the specific heat with temperature for graphite: calculated values (solid curve), Krumhansl and Brooks model [l] (- - -), Yoshimori and Kitano model [2] (-. - .), Young and Koppel model [ 31 (- - --). Dots represent experimental [9-121 values.
spectra from measured activities of different neutron irradiated samples. A few attempts [7,8] were made earlier to use a similar technique for calculating the frequency distribution function,g(g) (g = ?zo/kB is the phonon energy measured in Kelvin, ti is Planck’s constant divided by 2r1, o is the angular frequency and k, is the Boltzmann constant), starting from the known
Volume 78A, number 2
PHYSICS LETTERS
21 July 1980
temperature variation of the specific heat. However these were not very satisfactory. In view of the recent developments in the unfolding technique [5,6], we propose to examine here the problem of determining the frequency distribution function for materials using this technique. The integral equation to be solved for g(t) is 3Nk, *D ~2~t/T -.I-
T2
o (et/T-1)2
g(5) dg = C, 0’) ,
where N is the Avogadro number, 0~ is the Debye temperature of the material and C&T) is the measured specific heat (per gm-mole) of the material at temperature T. Here g(t) is normalized to unity. We consider a linear integral equation b s a
W,
where in the given. matrix
s)K(s)ds
=A4
,
(2)
the kernel K(t, s), assumed to be well behaved domain of integration, and the function y(t) are We replace the above integral equation by the equation
KX=Y ,
(3)
where K is an (n X m) matrix and X and Y are column matrices of order m and n respectively. In brief, the method works as follows. We start with a trial solutionXo. Let Xf be the solution to be calculated, Using the variational principle (for details see e.g. Fischer [4]) one obtains the following equation as a result of one iteration: Xfl = (CTFCW2 + G)-l(CTFlnW2
+ GXo) ,
(4)
where C is an (n X m) matrix obtained by dividing each row of K by the corresponding element of Y ,F is an (n X n) diagonal matrix consisting of the weights Wi for measurements of Y, G is an (m X m) diagonal matrix consisting of the inverse squares of trial solution, W is a multiplier constant determining the speed of convergence, 1 n is a unit column matrix of order n, andCT denotes the transpose of C. If Xfl does not satisfy eq. (2) within the experimental accuracy the calculations are repeated using Xfl as the trial function in place of X0 in eq. (4). This procedure is continued till the required accuracy in Y is achieved (details will be reported elsewhere).
Fig. 2. Frequency distribution function for graphite: calculated values (solid curves a and b for upper cut-offs at 1500 K and 2500 K, respectively; peakg(E) value for curve a is 53.5 and for curve b it is 19.4), Krumhansl and Brooks model [l] (- - -) Young and Koppel model [ 31 (---). Experimental values [ 131 are shown by the (- . - .) curve.
Using the experimental data [9-121 in the temperature range 0.44-1033 K we have calculated the function g(t) using the method outlined above. We find that one can determine functions g(t) with upper cut-offs ranging from 1500 K to 2500 K which all give very good agreement between the calculated and the observed specific heat data. The g(t) functions, marked a and b for the two cut-offs 1500 K and 2500 K, respectively, are shown in fig. 2. Also plotted are the g(t) functions proposed by Krumhansl and Brooks [ 1 ] and Young and Koppel [3], along with the values obtained experimentally by Page [ 131 using the neutron scattering technique. Specific heat values based on our calculated g(t) functions are shown in fig. 1 by the solid curve. (The specific heat values based on our two frequency distribution functions a and b do not differ from each other by more than 1% at any temperature.) The agreement between the calculated and the observed data is very good and within 2% in the entire temperature range 179
Volume 78A, number 2
PHYSICS LETTERS
considered here. The calculated specific heat values based on other models are also shown in fig. 1. The values given by Krumhansl and Brooks differ from the measured values by almost 50% in the temperature range 0.44-2 K, by ~10% in the range lo-100 K (this is not very clear from fig. 1 but shows up if C&T) versus T is plotted on a log-log scale), by 4% in the range 100-300 K while they are within 2% in the range 300- 1000 K. On the other hand, Yoshimori and Kitano [2] have calculated specific heat values in the range 45-300 K and good agreement is obtained only for temperatures below 150 K. Young and Koppel [3] have calculated the values in the range 100-1000 K giving good agreement only in the range 100-300 K. We propose to evaluate neutron scattering cross sections for graphite on the basis of the models proposed here. These studies, which will be reported in due course, should help to fix more precisely the upper limit on g(t) in defining the frequency distribution function. The authors are thankful to Drs. Feroz Ahmed and A.N. Verma for many useful discussions. Part of this work was supported by an NSF Grant.
180
21 July 1980
References [l] J. Krumhansl and H. Brooks, Chem. Phys. 21 (1953) 1663. [ 21 A. Yoshimori and Y. Kitano, J. Phys. Sot. Japan 11 (1956) 352. [3] J.A. Young and J.U. Koppel, J. Chem. Phys. 42 (1965) 357. [4] A. Fischer, KFA report RFSP-JUL-1475 (Dec. 1977). [5] H. Sekimoto, Nucl. Sci. Eng. 68 (1978) 351. [6] R.H. Johnson, B.W. Wehring and J.J. Doming, Nucl. Sci. Eng. 73 (1980) 93. [ 71 V.A. Korshunov and V.P. Tanana, Sov. Phys. Solid State 18 (1976) 378. [ 81 V .A. Korshunov and V.P. Tanana, Sov. Phys. Dokl. 21 (1976) 714. [9] C.F. Lucks, H.W. Deem and W.D. Wood, Am. Ceram. Sot. Bull. 39 (1960) 313. [lo] W. DeSorbo and G.E. Nichols, J. Phys. Chem. Solids 6 (1958) 352. [ 1l] B.J.C. VanderHoeven Jr. and R.H. Keesom, Phys. Rev. 130 (1963) 1318. [12] W. DeSorbo and W.W. Tyler, J. Chem. Phys. 21 (1953) 1650. [ 131 D.I. Page, Proc. Symp. Neutron inelastic scattering, Vol. 1 (Vienna, 1968) p. 325.