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Thermal expansion of zirconium—II
J. Phys. Chem. Solids. 1973,Vol.34, pp. 77-84. PergamonPress. Printedin Great Britain
THERMAL
EXPANSION
OF
ZIRCONIUM-II
C. S. MENON and R. RAMJI RAO Department of Physics, Indian Institute of Technology, Madras-36, India (Received 4 N o v e m b e r 1971; in revised f o r m 25 April 1972)
Abstract--The generalised G runeisen parameters 3" = - - a log to]ad and 3'"= - a log o~[ad' have been calculated for various normal mode frequencies using the force model for Zirconium obtained on Keating's approach. There is a general agreement between the normalised frequency distribution curve of Zirconium obtained on the present model and that of Bezdek et al. The temperature dependence of the effective G runeisen functions ~ (T) and ~ (T) has been calculated using the procedure of Blackman. The high temperature limits of ~±(T), Y~I(T) and "7,, are in good agreement with those obtained by Goldak et al. from an analysis of the thermal expansion data of Zirconium. 1. INTRODUCTION
employing five third order anharmonic parameters. Two and three body interactions were considered to write the potential energy of the Zirconium lattice. The atom 1 in the basis cell has three sets of neighbours of the same type (equivalent type) and three other sets of the non-equivalent type. The position coordinates of these sets of neighbours and the symbols used to designate them are given in the paper of Srinivasan and Ramji Rao [5] (a) Two body interaction involving nearest neighbour atoms of the same type along the unique axis: These neighbours are denoted by the symbol K.
ZIRCONIUM belongs to the hexagonal system with c/a ratio equal to 1.5926 and undergoes phase transformation to bcc structure at a high temperature of 1155°K. The dispersion relations in this metal were obtained by Bezdek et al. [1] using inelastic neutron scattering and the pressure derivatives of the SOE constants of Zirconium were reported by Fisher et al. [2]. The lattice parameters and the coefficients of thermal expansion of zirconium over temperature range of 4.2-1130°K were reported by Goldak et al. [3]. Using Keating's approach[4] the lattice dynamics, the expressions for SOE and T O E constants and the anisotropic thermal expansion in the quasi-harmonic approximation, of the hexagonal metals have been worked out by Srinivasan and Ramji Rao[5, 6] and applied successfully to investigate the anharmonic properties and the thermal expansion of Magnesium, Zinc and Beryllium [5, 6]. In an earlier paper[7], we worked out the lattice dynamics, third order elastic constants and low temperature lattice thermal expansion of Zirconium using the theory of Srinivasan and Ramji Rao[5,6] based on Keating's method [4]. Twelve second order parameters were used to calculate the SOE constants and to fit the dispersion relations of Zirconium while its ten T O E constants were evaluated
/-t=l K=I
•
,,,,.
_/LL+K~]
• %,. •
R(',,,.-,.'-.-.:":)
,.,. ,2
,.,. )J
:',:[R t,.,. ,.,--) --
•
k,/~ /~ IJ J" 77
1 F ,ILL+K'x
\i~
i•
I
78
c . S . MENON and R. RAMJI RAO
Here R( L
L+K~I~ /
is the equilibrium vector
•
,
distance between the particle/z in the cell L and its equivalent neighbour K along the unique axis.
R'(~L~ K)----
distance when the particles are displaced, r and X are the second and third order parameters respectively. T h e two body potentials involving the atoms of the sets I, N, J and M can be written similarly. Table 1 gives the second and third order parameters for the twobody interactions considered in Zirconium. (b) T h r e e body interaction involving the second neighbours of the same type in the basal plane: T h e s e neighbours are designated by N. qb'ta~ = ~ ~
/z
refers to the vector
R'
L+
/Z=l N=I
/j
L
\P,
/x/
,
H e r e N ' and N" are the two second neighbour atoms on either side of the neighbour N./3' is the second order parameter. T h e three-body potentials involving the atoms of the sets I, J, M and K are written in a similar manner. Table 2 lists the triplet of atoms involved in the various interactions considered in
Table 1. Second and third order parameters for the two-body interactions in Zirconium
No. 1
Neighbour involved Nearest neighbour of the same type in the basal plane (symbol 1)
Second order parameter a
Third order inharmonic parameter
Physicaldescription of the parameter related to the stretching of the bond between theatoms(Ol)and(ll)
Second neighbour of the same type in the basal plane (symbol N)
related to the stretching of the bond between theatoms(01)and(1N)
Third neighbour of the same type along the c-axis (symbol K)
related to the stretching of the bond between 0 K the atoms (1) and ( 1)
Nearest neighbour of the second type out of the basal plane (symbolJ )
related to the stretching of the bond between the atoms (0) and (J2)
Second nearest neighbour of the second type out of the basal plane (symbolM)
related to the stretching of the bond between the atoms (0) and ( M)
THERMAL EXPANSION OF Z I R C O N I U M - I I
79
Table 2. Three body interactions considered in Zirconium Second order No. Triplet of atoms involved parameter L~ (L+ 1~ ( L + / ' ) / \ ~/
13
5
Physical description of the parameter related to tbe distortion of the angle between two adjacent bonds
related to the distortion of the angle between two adjacent bonds /ON' (01N,
)
related to the distortion of the angle between the bonds (out of plane bending constant). related to the distortion of the angle between the bonds (0 1N) and (01K) (out of plane bending constanO related to the distortion of the angle between the two adjacent bonds OJ' OJ~
(,
related to the distortion of the angle between the bonds 12] (out of plane bending constan0
7
(L)(L+2M')(L+K')
8'
related to the distortion of the bonds M' (~ 2 )and \I/OK'~I] (out of plane bending constant)
80
c. s. MENON and R. RAMJI RAO
Zirconium and the corresponding second order parameters. The values of the second and third order parameters for the present model of Zirconium are given in Table 3.
2. THERMAL EXPANSION IN QUASI-HARMONIC APPROXIMATION
The above model was employed to work out the lattice dynamics, pressure derivatives of the second order elastic constants and the low temperature lattice thermal expansion of Zirconium[7]. In the present paper the anharmonic thermal expansion of Zirconium is investigated on the basis of quasi-harmonic approximation. A brief account of the quasi-harmonic theory of anisotropic thermal expansion was given in the paper of Srinivasan and Ramji Rao [6]. Uniaxial crystals are characterised by two linear thermal expansion coefficients designated by a~, and a±, parallel and perpendicular to the unique axis respectively. Effective Gruneisen functions y ( T ) are defined to study the temperature variation of these linear expansion coefficients. 7'(q,J) C v ( q , j ) y±(T) -- q'/ ~, C v ( q , j ) q,J
E y"(q,j)Cv(q,j) ~,(T) =
q,J
(2.1) E Cv(q,j) q.J
These effective Gruneisen functions are weighted averages of the generalised Gruneisen parameters (GPs) y(q,j). Here q is the wave vector and j is the polarisation index. Cv is the contribution of a single normal mode of frequency co, wave vector q and polarisation indexj to the specific heat of the lattice. y(q,j) are defined as follows: y,(q,j) = _ 0 log oJ(q,j) aE' y,,(q,j) = _ 0 log o~(q,j) de"
(2.2)
Cis a uniform longitudinal strain along the unique axis and e' is a uniform areal strain in the basal plane perpendicular to the unique axis. The effective Gruneisen functions approach the low temperature limits designated by y± (--3) and % ( - 3 ) . These low temperature limits can be calculated from a knowledge of SOE and T O E constants of the uniaxial crystal using the procedure given by Ramji Rao and Srinivasan [8, 9]. The high temperature limits of the effective Gruneisen functions
Table 3. Values o f the s e c o n d a n d third order p a r a m e t e r s f o r the present m o d e l o f Z i r c o n i u m Secondorder parameter
Value in 10n dynes/cmz
Third order parameter
Value in 10n dynes/cm2
(D4/Va)a (D4/Va) P (D4/Va) r (D4lVa)y (D41Va) tr (D4lVa)fl (D41Va)[3' (D4[Va ) e (D4IVa)K (D41Va)K' (D4/Va) 8 (D'/Va)8'
1.803 --0.034 --0.351 2-041 -0" 104 --0.324 O.108 --0.973 --0.113 +0-023 1.123 --0.187
(D6IVa)¢ (DSVa) ~
--7.67 --7.00
(D6/Va) v (D6/Va)x (De/Va)'q
+0"26 --0.42 --0"18
THERMAL
EXPANSION OF ZIRCONIUM-
,Y__, y'(q,j) ~--- q'J
81
0-40 0`38 0-56 0-54 C-32
are designated by ?±(0) and ~/,t(0):
~l(O)
II
0-50
N
0-28 0-26 024
~, y"(q,j) 3
~,(0) = ~,s N
(2.3)
0-22 0-20 0`18 0.16 0.14 0,12 0"10
where N is the total number of normal modes. In order to calculate the effective Gruneisen functions at different temperatures from a model of lattice dynamics, the procedure of Blackman[10] is adopted. A grid of equally spaced points in the Brillouin zone is chosen for the wave vectors q. The normal mode frequencies and the individual GPs, ~/'(q,j) and 3/'(q,j) have been calculated for 84 points evenly distributed over 1/24 of the volume of the BriUouin zone using a programme written for C D C 3600. The coarse mesh is considered good enough to yield the thermal properties to an accuracy of 3 per cent. The range of frequencies from 0 to tOmax is divided into small intervals (Ato = 0.4 × 101arad/sec) and the number of frequencies in each interval is counted. A histogram is drawn for g(to) vs to and is replaced by a smooth curve enclosing unit area with the to-axis. This gives the normalized frequency distribution curve for zirconium and is shown in Fig. 1. In the region of very low frequencies, where there are not enough frequencies to draw a reliable histogram, the parabolic equation g(to)= Coo2 is
0`08 OO6
I•1
0-04 O'O2
0-4
0"8
I 1.2
I 1.6
I 2"0
I 2"4
I 2"8
I 3"2
I 4-0 4.2
Fig. 1. Normalised frequency distribution curve for Zirconium.
used to get g(to). C is calculated from a knowl3
edge of the average value of E Vfa(O,d~) over J=l
all directions where V~ is the acoustic wave velocity for the j t h mode propagating in the direction (0, ~b). The y' and y" values for the different modes of vibration in each frequency interval of width Ato = 0"4 rad/sec are noted and the average values of ~' and ~" of these Gruneisen parameters are found for each interval. The plots of ~'(to) and ~/'(to) as a function of to are shown in Fig. 2 in which in the low frequency range ~/' (to) and ~"(to) tend to their low temperature limits 0.11 and 0.81 respectively obtained from the elastic constants data of Zirconium
1.4 I-2 I-0
~
51.6
=x,O 's, rad/sec
o.e 0.6 0-4 0.2
I
I
I
I
I
I
I
I
i
0"4
0"8
1"2
I-6
2"0
2"4
2-8
3-2
3'6
w x l O '3,
rad/sec
Fig. 2. y(to) vs to for Zirconium.
.:
I
I
4-0
4"2
82
C.S.
M E N O N and R . R A M J I R A O
[7] using the procedure of Ramji Rao and Srinivasan [8, 9]. The effective Gruneisen functions 7± (T) and % (T) are then evaluated from the formulae: ¢~max ¢
fo
y (oJ)g(o~)Cv(oJ,T)doJ
~ffttiee(T ) =
f~maXg(oJ)Cv(co,T)doJ =max
f0
.
~/(co)g(o~)Cv(oJ,T)d~o
~atUee ( T ) =
(2.4)
fT"Xg(o~)Cv(co,T)dco The variation of the effective Gruneisen functions y . (T) and % (T) with temperature is shown in Fig. 3. In metals, in addition to the lattice contribution, there is also electronic contribution to thermal expansion. At low temperatures, the electronic contribution varies linearly with absolute temperature T and plays an important role, while at high temperatures this becomes less significant in comparison to the lattice contribution. 3. DISCUSSION OF THE RESULTS
(a) Salient features of the normalised frequen-
cy distribution curve of Zirconium The frequency distribution curve of Zircon-
o-~~
o.a~"?
~±(T)
0-4 0-2
0
20
40
60 T,
80
I00
°K
Fig. 3. ~ ±(T) and ~f(T) vs. T°K for Zirconium.
I10
ium obtained on the present model does no~ exhibit the finer details shown in the one obtained by Bezdek et al. [1]. However there is general agreement between the two frequency distribution curves. There are two broad peaks centered around co = 1-7 × 101~ rad/sec and co = 2.7 × 1013 rad/sec in the present distribution curve for Zirconium. The frequency distribution curve of Bezdek et al. [1] also exhibits the peaks around the same regions. There is a deep minimum at co= 2-2 × 1013 rad/sec in between the two broad maxima and a shallow minimum at 3.4 × 1013rad/sec. The even moment /z2 of the frequency distribution function (calculated from the trace of the dynamical matrix) on the present model for Zirconium has the value 6.91 × 1026[seC and 0= has the value 259°K. (b) Low temperature limits %(--3) and %(-3) The low temperature limits y±(--3) and %(--3) of Zirconium which depend on the TOE constants of the crystal were calculated in the earlier paper [7] and these are 0-11 and 0.81 respectively. Using these values of 0.11 and 0-81, the low temperature limit of the volume GP was found to be equal to 0.3. This is in good agreement with value obtained by Cowan et al.[ll] from thermal expansion data. In the evaluation of y± (-3) and % ( - 3 ) , SOE and TOE constants are involved. The TOE constants are evaluated using five third order anharmonic parameters which, in their turn, are calculated from the pressure derivatives of the SOE constants of Zirconium. Fisher et al. [2] have not mentioned the errors in the measured pressure derivatives of the elastic moduli of this metal in their paper. Fisher and Renken[17] report an error of 0.2 per cent in the measured SOE constants. A precise estimation of the errors involved in the evaluation of the low temperature limits 0-11 and 0-81 for Zirconium cannot be done in the absence of a knowledge of the errors in the measured pressure derivatives. If the errors involved in the measured second order elastic constants only are taken into account,
THERMAL EXPANSION OF Z I R C O N I U M - I I
the estimated errors in the calculated values of Yz ( - 3 ) and ylt(-3) turn out to be insignificant.
83
Here it may be mentioned that Fisher and Manghnani [ 13] calculated the YH for Zirconium using their hydrostatic pressure deriva(c) Salient features of ~/'(w) and y"(oJ) vs oJ tives data[2] and the Gerlich computer curves program[14]. The value obtained by Fisher The high frequencies in Zirconium have and Manghnani[13] for y n = 0 " 3 7 and this values of y' ~ 0.7 while y" values are around does not agree with %/(av)= 1-01 obtained 0-8. Thus in Zirconium the values of ¢/' and by Goldak et al. [3] from their thermal expan~7" are very nearly the same for high frequen- sion data for Zirconium. Fisher and Manghnani cies. At low frequencies, the values of y' are [13] explain this difference between YM and lower than the values of y". The low tempera- YH(av) for Zirconium by-taking into account ture limit "~1(-3) is greater than 7 ± ( - 3 ) . The the effect of the change in c/a ratio of Zirconfrequencies in the interval oJ = 1.0 to 1-2 x 1013 ium with pressure. They show that the disrad/sec have large values of y' as well as y". crepancy between %/ calculated from the The frequencies in the interval co = 2-8 to hydrostatic pressure derivatives data and 3.2 x 1013radlsec have large values of ¢/' and 7n(av) can be removed by taking into account the difference in d(c/a)/dV term under hydrocomparatively smaller values of y'. static pressure and thermal expansion con(d) Temperature dependence of the effective ditions[13]. However, our present calculalattice Griineisen functions % ( T ) and tions do not reveal significant discrepancy 7H(T) between YH calculated from the pressure The temperature dependence of y± (T) and derivatives of the elastic moduli of zirconium ~ (T) in Zirconium calculated on the present and %/(a~) obtained by Goldak et al. [3] from model is shown in Fig. 3 where-in the low an analysis of their thermal expansion data temperature limits tend to y± (--3) and ~(--3) between 100 and l130°K. Goldak et al.[3] respectively. One finds that the variation in calculated the lattice Griineisen functions 7± (T) with temperature is more than that of "~±lattiee(T) and "~lllattiee(T) using the lattice 711(T) between 0 and 30°K. Beyond 30°K, specific heat Cp--Ce, with two different both the curves are flat and show little varia- electronic specific heat Ce corrections [ 15, 16]. tion with temperature. ~I(T) is found to be In Zirconium the electronic contribution to always greater than y± (T). The high tempera- the thermal expansion is negligible above ture limits of ~± (T) and ~,(T) are 0-89 and 0.300 and so they have not corrected their 1.04 respectively and ~, = 0.93. observed o~1 and a± values for electronic From a study of the high temperature contribution. The elastic constants of ZirGruneisen functions of the axial metals Munn conium at various temperatures, were taken [12] concludes that the larger of y± and Y~tis the from the work of Fisher and Renken[17]. one referring to the symmetry direction The y's were calculated from the formulae associated with the stronger forces. In ZirconV ium, the c/a ratio is less than the ideal value 7±lamee(T) = [ ( C u + Clz)Ot± + Cz3o~l] --7and the forces between the planes perpen~p dicular to the c-axis are stronger than those within such planes. So in Zirconium the c-axis V ~lllattice(T) = [2ClzoL±+ C3~o~I] Co" (3.1) is the symmetry direction associated with stronger forces and y~ is to be greater than y±. The present calculations of the high tempera- Values of Cp were obtained from the comture limits of y, lattice = 1.04 and y± lattice = bined data of Skinner and Johnston[18] and 0.89 vindicate this expectation of Munn[12]. Scott[19]. If the Ce correction is based upon
84
C.S.
M E N O N and R. RAMJI RAO
the w o r k o f Kneip et al. [15], then y,, = 1.01 __+ 0.08, '~alattice = ~.1.latuce = 0"94 ± 0"05 and T e lattice = '~1 lattice = 1 "20 ± 0" 16 b e t w e e n 100 and 1130°K. If it is based upon Shimizu and Kalsuki's data [16] then
% = 1.08 ± 0.15,
~ a lattice =
"~.1.lattice = 1 "00___ 0" 11
and yclatuce = ~1 lattice = 1.29 ± 0.25 in the same temperature range. Thus the high temperature limits of y± (T), ~l (T) and y~ for Zirconium obtained on the present model agree well with the above values obtained by Goldak et al. [3] from their analysis of the thermal expansion data. Acknowledgements-The authors wish to thank Dr. A. Ramachandran, Director, Indian Institute of Technology, Madras, for his egcouragement and Profs. R. Srinivasan and C. Ramasastry for their interest during this work. The authors thanks are also due to Mr S. Rangarajan for executing the programme on the C D C 3600 Computer at T.I.F.R., Bombay. REFERENCES
1. B E Z D E K H. F., S H M U N K R. E. and F I N E G O L D L., Phys. Status Solidi 4 2 , 275 (1970).
2. F I S H E R E. S., M A N G H N A N I M. H. and SOKC LOWSKI T. M.,J. appl. Phys 41, 2991 (1970). 3. G O L D A K J., L L O Y D L. T. and B A R R E T F C. S Phys. Rev. 144, 478 (1966). 4. K E A T I N G P. N., (a) Phys. Rev. 145, 637 (19661 (b) Phys. Rev. 149, 674 (1966). 5. S R I N I V A S A N R. and RAMJI RAO R., J. Phy. Chem. Solids 32, 1769 (1971). 6. S R I N I V A S A N R. and RAMJI RAO, R. J. Phy~ Chem. Solids 33, 491 (1972). 7. M E N O N C. S. and RAMJI RAO, R.,J. Phys. Chert Solids, 33, 1325 (1972). 8. RAMJI RAO, R. and S R I N I V A S A N , R., Phy: Status Solidi 29, 865 (1968). 9. RAMJI RAO, R. and S R I N I V A S A N , R., Pro~ Ind. NatI.Acad. Sci. 36A, 97 (1970). 10. B L A C K M A N , M., Proc. Phys. Soc. Lond. B70, 82 (1957). 11. COWAN, J. A., P A W L 6 W l C Z , A. T. and WHIT] G. K., Cryogenics8, 155 (1968). 12. M U N N R. W.,Adv. Phys. 18, 515 (1969). 13. F I S H E R E. S. and M A N G H N A N I M. H.,J. Phy., Chem. Solids32, 657 (1971). 14. G E R L I C H D.,J. Phys. Chem. Solids 30, 1638 (19691 15. K N E I P G. D., Jr., BE'I"FERTON J. O. Jr. an, S C A R B R O U G H J. O., Phys. Rev. 131, 2425 (1963', 16. M A S A O S H I M I Z U and A T S H U S H I KALSUKI J. phys. S oc. Japan 19, 1856 (1964). 17. F I S H E R E. S. and R E N K E N C. J., Phys. Rev. 13-~ A482 (1964). 18. S K I N N E R G. B. and J O H N S T O N H. L., J. Am Chem. Soc. 73, 4549 (1951). 19. SCOTT J. L., Oak Ridge National Laborator. Report No. ORNL-2328, 1957.
THERMAL
EXPANSION
OF
ZIRCONIUM-II
C. S. MENON and R. RAMJI RAO Department of Physics, Indian Institute of Technology, Madras-36, India (Received 4 N o v e m b e r 1971; in revised f o r m 25 April 1972)
Abstract--The generalised G runeisen parameters 3" = - - a log to]ad and 3'"= - a log o~[ad' have been calculated for various normal mode frequencies using the force model for Zirconium obtained on Keating's approach. There is a general agreement between the normalised frequency distribution curve of Zirconium obtained on the present model and that of Bezdek et al. The temperature dependence of the effective G runeisen functions ~ (T) and ~ (T) has been calculated using the procedure of Blackman. The high temperature limits of ~±(T), Y~I(T) and "7,, are in good agreement with those obtained by Goldak et al. from an analysis of the thermal expansion data of Zirconium. 1. INTRODUCTION
employing five third order anharmonic parameters. Two and three body interactions were considered to write the potential energy of the Zirconium lattice. The atom 1 in the basis cell has three sets of neighbours of the same type (equivalent type) and three other sets of the non-equivalent type. The position coordinates of these sets of neighbours and the symbols used to designate them are given in the paper of Srinivasan and Ramji Rao [5] (a) Two body interaction involving nearest neighbour atoms of the same type along the unique axis: These neighbours are denoted by the symbol K.
ZIRCONIUM belongs to the hexagonal system with c/a ratio equal to 1.5926 and undergoes phase transformation to bcc structure at a high temperature of 1155°K. The dispersion relations in this metal were obtained by Bezdek et al. [1] using inelastic neutron scattering and the pressure derivatives of the SOE constants of Zirconium were reported by Fisher et al. [2]. The lattice parameters and the coefficients of thermal expansion of zirconium over temperature range of 4.2-1130°K were reported by Goldak et al. [3]. Using Keating's approach[4] the lattice dynamics, the expressions for SOE and T O E constants and the anisotropic thermal expansion in the quasi-harmonic approximation, of the hexagonal metals have been worked out by Srinivasan and Ramji Rao[5, 6] and applied successfully to investigate the anharmonic properties and the thermal expansion of Magnesium, Zinc and Beryllium [5, 6]. In an earlier paper[7], we worked out the lattice dynamics, third order elastic constants and low temperature lattice thermal expansion of Zirconium using the theory of Srinivasan and Ramji Rao[5,6] based on Keating's method [4]. Twelve second order parameters were used to calculate the SOE constants and to fit the dispersion relations of Zirconium while its ten T O E constants were evaluated
/-t=l K=I
•
,,,,.
_/LL+K~]
• %,. •
R(',,,.-,.'-.-.:":)
,.,. ,2
,.,. )J
:',:[R t,.,. ,.,--) --
•
k,/~ /~ IJ J" 77
1 F ,ILL+K'x
\i~
i•
I
78
c . S . MENON and R. RAMJI RAO
Here R( L
L+K~I~ /
is the equilibrium vector
•
,
distance between the particle/z in the cell L and its equivalent neighbour K along the unique axis.
R'(~L~ K)----
distance when the particles are displaced, r and X are the second and third order parameters respectively. T h e two body potentials involving the atoms of the sets I, N, J and M can be written similarly. Table 1 gives the second and third order parameters for the twobody interactions considered in Zirconium. (b) T h r e e body interaction involving the second neighbours of the same type in the basal plane: T h e s e neighbours are designated by N. qb'ta~ = ~ ~
/z
refers to the vector
R'
L+
/Z=l N=I
/j
L
\P,
/x/
,
H e r e N ' and N" are the two second neighbour atoms on either side of the neighbour N./3' is the second order parameter. T h e three-body potentials involving the atoms of the sets I, J, M and K are written in a similar manner. Table 2 lists the triplet of atoms involved in the various interactions considered in
Table 1. Second and third order parameters for the two-body interactions in Zirconium
No. 1
Neighbour involved Nearest neighbour of the same type in the basal plane (symbol 1)
Second order parameter a
Third order inharmonic parameter
Physicaldescription of the parameter related to the stretching of the bond between theatoms(Ol)and(ll)
Second neighbour of the same type in the basal plane (symbol N)
related to the stretching of the bond between theatoms(01)and(1N)
Third neighbour of the same type along the c-axis (symbol K)
related to the stretching of the bond between 0 K the atoms (1) and ( 1)
Nearest neighbour of the second type out of the basal plane (symbolJ )
related to the stretching of the bond between the atoms (0) and (J2)
Second nearest neighbour of the second type out of the basal plane (symbolM)
related to the stretching of the bond between the atoms (0) and ( M)
THERMAL EXPANSION OF Z I R C O N I U M - I I
79
Table 2. Three body interactions considered in Zirconium Second order No. Triplet of atoms involved parameter L~ (L+ 1~ ( L + / ' ) / \ ~/
13
5
Physical description of the parameter related to tbe distortion of the angle between two adjacent bonds
related to the distortion of the angle between two adjacent bonds /ON' (01N,
)
related to the distortion of the angle between the bonds (out of plane bending constant). related to the distortion of the angle between the bonds (0 1N) and (01K) (out of plane bending constanO related to the distortion of the angle between the two adjacent bonds OJ' OJ~
(,
related to the distortion of the angle between the bonds 12] (out of plane bending constan0
7
(L)(L+2M')(L+K')
8'
related to the distortion of the bonds M' (~ 2 )and \I/OK'~I] (out of plane bending constant)
80
c. s. MENON and R. RAMJI RAO
Zirconium and the corresponding second order parameters. The values of the second and third order parameters for the present model of Zirconium are given in Table 3.
2. THERMAL EXPANSION IN QUASI-HARMONIC APPROXIMATION
The above model was employed to work out the lattice dynamics, pressure derivatives of the second order elastic constants and the low temperature lattice thermal expansion of Zirconium[7]. In the present paper the anharmonic thermal expansion of Zirconium is investigated on the basis of quasi-harmonic approximation. A brief account of the quasi-harmonic theory of anisotropic thermal expansion was given in the paper of Srinivasan and Ramji Rao [6]. Uniaxial crystals are characterised by two linear thermal expansion coefficients designated by a~, and a±, parallel and perpendicular to the unique axis respectively. Effective Gruneisen functions y ( T ) are defined to study the temperature variation of these linear expansion coefficients. 7'(q,J) C v ( q , j ) y±(T) -- q'/ ~, C v ( q , j ) q,J
E y"(q,j)Cv(q,j) ~,(T) =
q,J
(2.1) E Cv(q,j) q.J
These effective Gruneisen functions are weighted averages of the generalised Gruneisen parameters (GPs) y(q,j). Here q is the wave vector and j is the polarisation index. Cv is the contribution of a single normal mode of frequency co, wave vector q and polarisation indexj to the specific heat of the lattice. y(q,j) are defined as follows: y,(q,j) = _ 0 log oJ(q,j) aE' y,,(q,j) = _ 0 log o~(q,j) de"
(2.2)
Cis a uniform longitudinal strain along the unique axis and e' is a uniform areal strain in the basal plane perpendicular to the unique axis. The effective Gruneisen functions approach the low temperature limits designated by y± (--3) and % ( - 3 ) . These low temperature limits can be calculated from a knowledge of SOE and T O E constants of the uniaxial crystal using the procedure given by Ramji Rao and Srinivasan [8, 9]. The high temperature limits of the effective Gruneisen functions
Table 3. Values o f the s e c o n d a n d third order p a r a m e t e r s f o r the present m o d e l o f Z i r c o n i u m Secondorder parameter
Value in 10n dynes/cmz
Third order parameter
Value in 10n dynes/cm2
(D4/Va)a (D4/Va) P (D4/Va) r (D4lVa)y (D41Va) tr (D4lVa)fl (D41Va)[3' (D4[Va ) e (D4IVa)K (D41Va)K' (D4/Va) 8 (D'/Va)8'
1.803 --0.034 --0.351 2-041 -0" 104 --0.324 O.108 --0.973 --0.113 +0-023 1.123 --0.187
(D6IVa)¢ (DSVa) ~
--7.67 --7.00
(D6/Va) v (D6/Va)x (De/Va)'q
+0"26 --0.42 --0"18
THERMAL
EXPANSION OF ZIRCONIUM-
,Y__, y'(q,j) ~--- q'J
81
0-40 0`38 0-56 0-54 C-32
are designated by ?±(0) and ~/,t(0):
~l(O)
II
0-50
N
0-28 0-26 024
~, y"(q,j) 3
~,(0) = ~,s N
(2.3)
0-22 0-20 0`18 0.16 0.14 0,12 0"10
where N is the total number of normal modes. In order to calculate the effective Gruneisen functions at different temperatures from a model of lattice dynamics, the procedure of Blackman[10] is adopted. A grid of equally spaced points in the Brillouin zone is chosen for the wave vectors q. The normal mode frequencies and the individual GPs, ~/'(q,j) and 3/'(q,j) have been calculated for 84 points evenly distributed over 1/24 of the volume of the BriUouin zone using a programme written for C D C 3600. The coarse mesh is considered good enough to yield the thermal properties to an accuracy of 3 per cent. The range of frequencies from 0 to tOmax is divided into small intervals (Ato = 0.4 × 101arad/sec) and the number of frequencies in each interval is counted. A histogram is drawn for g(to) vs to and is replaced by a smooth curve enclosing unit area with the to-axis. This gives the normalized frequency distribution curve for zirconium and is shown in Fig. 1. In the region of very low frequencies, where there are not enough frequencies to draw a reliable histogram, the parabolic equation g(to)= Coo2 is
0`08 OO6
I•1
0-04 O'O2
0-4
0"8
I 1.2
I 1.6
I 2"0
I 2"4
I 2"8
I 3"2
I 4-0 4.2
Fig. 1. Normalised frequency distribution curve for Zirconium.
used to get g(to). C is calculated from a knowl3
edge of the average value of E Vfa(O,d~) over J=l
all directions where V~ is the acoustic wave velocity for the j t h mode propagating in the direction (0, ~b). The y' and y" values for the different modes of vibration in each frequency interval of width Ato = 0"4 rad/sec are noted and the average values of ~' and ~" of these Gruneisen parameters are found for each interval. The plots of ~'(to) and ~/'(to) as a function of to are shown in Fig. 2 in which in the low frequency range ~/' (to) and ~"(to) tend to their low temperature limits 0.11 and 0.81 respectively obtained from the elastic constants data of Zirconium
1.4 I-2 I-0
~
51.6
=x,O 's, rad/sec
o.e 0.6 0-4 0.2
I
I
I
I
I
I
I
I
i
0"4
0"8
1"2
I-6
2"0
2"4
2-8
3-2
3'6
w x l O '3,
rad/sec
Fig. 2. y(to) vs to for Zirconium.
.:
I
I
4-0
4"2
82
C.S.
M E N O N and R . R A M J I R A O
[7] using the procedure of Ramji Rao and Srinivasan [8, 9]. The effective Gruneisen functions 7± (T) and % (T) are then evaluated from the formulae: ¢~max ¢
fo
y (oJ)g(o~)Cv(oJ,T)doJ
~ffttiee(T ) =
f~maXg(oJ)Cv(co,T)doJ =max
f0
.
~/(co)g(o~)Cv(oJ,T)d~o
~atUee ( T ) =
(2.4)
fT"Xg(o~)Cv(co,T)dco The variation of the effective Gruneisen functions y . (T) and % (T) with temperature is shown in Fig. 3. In metals, in addition to the lattice contribution, there is also electronic contribution to thermal expansion. At low temperatures, the electronic contribution varies linearly with absolute temperature T and plays an important role, while at high temperatures this becomes less significant in comparison to the lattice contribution. 3. DISCUSSION OF THE RESULTS
(a) Salient features of the normalised frequen-
cy distribution curve of Zirconium The frequency distribution curve of Zircon-
o-~~
o.a~"?
~±(T)
0-4 0-2
0
20
40
60 T,
80
I00
°K
Fig. 3. ~ ±(T) and ~f(T) vs. T°K for Zirconium.
I10
ium obtained on the present model does no~ exhibit the finer details shown in the one obtained by Bezdek et al. [1]. However there is general agreement between the two frequency distribution curves. There are two broad peaks centered around co = 1-7 × 101~ rad/sec and co = 2.7 × 1013 rad/sec in the present distribution curve for Zirconium. The frequency distribution curve of Bezdek et al. [1] also exhibits the peaks around the same regions. There is a deep minimum at co= 2-2 × 1013 rad/sec in between the two broad maxima and a shallow minimum at 3.4 × 1013rad/sec. The even moment /z2 of the frequency distribution function (calculated from the trace of the dynamical matrix) on the present model for Zirconium has the value 6.91 × 1026[seC and 0= has the value 259°K. (b) Low temperature limits %(--3) and %(-3) The low temperature limits y±(--3) and %(--3) of Zirconium which depend on the TOE constants of the crystal were calculated in the earlier paper [7] and these are 0-11 and 0.81 respectively. Using these values of 0.11 and 0-81, the low temperature limit of the volume GP was found to be equal to 0.3. This is in good agreement with value obtained by Cowan et al.[ll] from thermal expansion data. In the evaluation of y± (-3) and % ( - 3 ) , SOE and TOE constants are involved. The TOE constants are evaluated using five third order anharmonic parameters which, in their turn, are calculated from the pressure derivatives of the SOE constants of Zirconium. Fisher et al. [2] have not mentioned the errors in the measured pressure derivatives of the elastic moduli of this metal in their paper. Fisher and Renken[17] report an error of 0.2 per cent in the measured SOE constants. A precise estimation of the errors involved in the evaluation of the low temperature limits 0-11 and 0-81 for Zirconium cannot be done in the absence of a knowledge of the errors in the measured pressure derivatives. If the errors involved in the measured second order elastic constants only are taken into account,
THERMAL EXPANSION OF Z I R C O N I U M - I I
the estimated errors in the calculated values of Yz ( - 3 ) and ylt(-3) turn out to be insignificant.
83
Here it may be mentioned that Fisher and Manghnani [ 13] calculated the YH for Zirconium using their hydrostatic pressure deriva(c) Salient features of ~/'(w) and y"(oJ) vs oJ tives data[2] and the Gerlich computer curves program[14]. The value obtained by Fisher The high frequencies in Zirconium have and Manghnani[13] for y n = 0 " 3 7 and this values of y' ~ 0.7 while y" values are around does not agree with %/(av)= 1-01 obtained 0-8. Thus in Zirconium the values of ¢/' and by Goldak et al. [3] from their thermal expan~7" are very nearly the same for high frequen- sion data for Zirconium. Fisher and Manghnani cies. At low frequencies, the values of y' are [13] explain this difference between YM and lower than the values of y". The low tempera- YH(av) for Zirconium by-taking into account ture limit "~1(-3) is greater than 7 ± ( - 3 ) . The the effect of the change in c/a ratio of Zirconfrequencies in the interval oJ = 1.0 to 1-2 x 1013 ium with pressure. They show that the disrad/sec have large values of y' as well as y". crepancy between %/ calculated from the The frequencies in the interval co = 2-8 to hydrostatic pressure derivatives data and 3.2 x 1013radlsec have large values of ¢/' and 7n(av) can be removed by taking into account the difference in d(c/a)/dV term under hydrocomparatively smaller values of y'. static pressure and thermal expansion con(d) Temperature dependence of the effective ditions[13]. However, our present calculalattice Griineisen functions % ( T ) and tions do not reveal significant discrepancy 7H(T) between YH calculated from the pressure The temperature dependence of y± (T) and derivatives of the elastic moduli of zirconium ~ (T) in Zirconium calculated on the present and %/(a~) obtained by Goldak et al. [3] from model is shown in Fig. 3 where-in the low an analysis of their thermal expansion data temperature limits tend to y± (--3) and ~(--3) between 100 and l130°K. Goldak et al.[3] respectively. One finds that the variation in calculated the lattice Griineisen functions 7± (T) with temperature is more than that of "~±lattiee(T) and "~lllattiee(T) using the lattice 711(T) between 0 and 30°K. Beyond 30°K, specific heat Cp--Ce, with two different both the curves are flat and show little varia- electronic specific heat Ce corrections [ 15, 16]. tion with temperature. ~I(T) is found to be In Zirconium the electronic contribution to always greater than y± (T). The high tempera- the thermal expansion is negligible above ture limits of ~± (T) and ~,(T) are 0-89 and 0.300 and so they have not corrected their 1.04 respectively and ~, = 0.93. observed o~1 and a± values for electronic From a study of the high temperature contribution. The elastic constants of ZirGruneisen functions of the axial metals Munn conium at various temperatures, were taken [12] concludes that the larger of y± and Y~tis the from the work of Fisher and Renken[17]. one referring to the symmetry direction The y's were calculated from the formulae associated with the stronger forces. In ZirconV ium, the c/a ratio is less than the ideal value 7±lamee(T) = [ ( C u + Clz)Ot± + Cz3o~l] --7and the forces between the planes perpen~p dicular to the c-axis are stronger than those within such planes. So in Zirconium the c-axis V ~lllattice(T) = [2ClzoL±+ C3~o~I] Co" (3.1) is the symmetry direction associated with stronger forces and y~ is to be greater than y±. The present calculations of the high tempera- Values of Cp were obtained from the comture limits of y, lattice = 1.04 and y± lattice = bined data of Skinner and Johnston[18] and 0.89 vindicate this expectation of Munn[12]. Scott[19]. If the Ce correction is based upon
84
C.S.
M E N O N and R. RAMJI RAO
the w o r k o f Kneip et al. [15], then y,, = 1.01 __+ 0.08, '~alattice = ~.1.latuce = 0"94 ± 0"05 and T e lattice = '~1 lattice = 1 "20 ± 0" 16 b e t w e e n 100 and 1130°K. If it is based upon Shimizu and Kalsuki's data [16] then
% = 1.08 ± 0.15,
~ a lattice =
"~.1.lattice = 1 "00___ 0" 11
and yclatuce = ~1 lattice = 1.29 ± 0.25 in the same temperature range. Thus the high temperature limits of y± (T), ~l (T) and y~ for Zirconium obtained on the present model agree well with the above values obtained by Goldak et al. [3] from their analysis of the thermal expansion data. Acknowledgements-The authors wish to thank Dr. A. Ramachandran, Director, Indian Institute of Technology, Madras, for his egcouragement and Profs. R. Srinivasan and C. Ramasastry for their interest during this work. The authors thanks are also due to Mr S. Rangarajan for executing the programme on the C D C 3600 Computer at T.I.F.R., Bombay. REFERENCES
1. B E Z D E K H. F., S H M U N K R. E. and F I N E G O L D L., Phys. Status Solidi 4 2 , 275 (1970).
2. F I S H E R E. S., M A N G H N A N I M. H. and SOKC LOWSKI T. M.,J. appl. Phys 41, 2991 (1970). 3. G O L D A K J., L L O Y D L. T. and B A R R E T F C. S Phys. Rev. 144, 478 (1966). 4. K E A T I N G P. N., (a) Phys. Rev. 145, 637 (19661 (b) Phys. Rev. 149, 674 (1966). 5. S R I N I V A S A N R. and RAMJI RAO R., J. Phy. Chem. Solids 32, 1769 (1971). 6. S R I N I V A S A N R. and RAMJI RAO, R. J. Phy~ Chem. Solids 33, 491 (1972). 7. M E N O N C. S. and RAMJI RAO, R.,J. Phys. Chert Solids, 33, 1325 (1972). 8. RAMJI RAO, R. and S R I N I V A S A N , R., Phy: Status Solidi 29, 865 (1968). 9. RAMJI RAO, R. and S R I N I V A S A N , R., Pro~ Ind. NatI.Acad. Sci. 36A, 97 (1970). 10. B L A C K M A N , M., Proc. Phys. Soc. Lond. B70, 82 (1957). 11. COWAN, J. A., P A W L 6 W l C Z , A. T. and WHIT] G. K., Cryogenics8, 155 (1968). 12. M U N N R. W.,Adv. Phys. 18, 515 (1969). 13. F I S H E R E. S. and M A N G H N A N I M. H.,J. Phy., Chem. Solids32, 657 (1971). 14. G E R L I C H D.,J. Phys. Chem. Solids 30, 1638 (19691 15. K N E I P G. D., Jr., BE'I"FERTON J. O. Jr. an, S C A R B R O U G H J. O., Phys. Rev. 131, 2425 (1963', 16. M A S A O S H I M I Z U and A T S H U S H I KALSUKI J. phys. S oc. Japan 19, 1856 (1964). 17. F I S H E R E. S. and R E N K E N C. J., Phys. Rev. 13-~ A482 (1964). 18. S K I N N E R G. B. and J O H N S T O N H. L., J. Am Chem. Soc. 73, 4549 (1951). 19. SCOTT J. L., Oak Ridge National Laborator. Report No. ORNL-2328, 1957.