- Email: [email protected]
The elastic constants of rubidium
J. Phys.
Chem. Solids.
Pergamon
Press 1966. Vol. 27, pp. 1401-1407.
THE ELASTIC
CONSTANTS
of Electrical
Engineering,
(Received
OF RUBIDIUM*
and R. MEISTER
C. A. ROBERTSt Department
Printed in Great Britain.
The Catholic University
21 February
1966; in reetisedform
of America, 24 March
Washington
17, D.C.
1966)
Abstract-The elastic constants of single crystal rubidium have been measured at 80 f 3°K using the ultrasonic pulse echo method at a frequency of 15 mc. The elastic constants (in units of lOlO dyn/cms) are clr = 2.96, crs = 2.44, ~44 = 1.60. Five wave velocities were measured, allowing the computation of the three independent elastic constants with two internal checks. Single crystal rubidium has an anisotropy indicated by the ratio (crr-crs)/2c44 = 0.14.
1. INTRODUCTION THIS study
presents the results of ultrasonic velocity measurements at 15 mc in single crystal rubidium at temperatures of 80” and 293°K. To the authors’ knowledge these measurements represent the first experimental determination of the elastic constants for this alkali metal. The measured constants are used to calculate the polycrystalline shear and bulk moduli. The measured constants are compared with the Fuchs theoretical values and are also used to calculate the Debye temperature. The calculated Debye temperatures are compared with values obtained from specific heat measurements. 2. SAMPLE
PREPARATION
#20 wire. The nucleating tip was also modified. The first + in. of the tip was drilled with a 45” taper. At the bottom of this tapered section a cylindrical hole & in. in dia. and 1 in. deep was
AND
ORIENTATION
Single crystals of rubidium were grown using a modified Bridgman technique’ developed at this laboratory by D. L. Shelley. The crucible used was similar to that described by HOYTE and MIELCZARE@ for growth of potassium. Because rubidium has a lower melting point than potassium, the crucible was redesigned with four sections of heating wire instead of three. Each of the four sections of this crucible are 1%in. in length with 15, 12, 8 and 4 turns per in. of nichrome *This research is sponsored by the U.S. A.E.C. Grant No. AT(40-l)-2861. t This work is based in part on a thesis submitted in partial fulfillment of the requirements for the degree of Master of Electrical Engineering, Department of Electrical Engineering, Catholic University of America. 1401
FIG. 1. Crucible
for growing single crystal (TPI-turns per in.).
rubidium
1402
C. A.
ROBERTS
drilled to form the bottom of the cup. (Refer to Fig. 1.) Rubidium of 99.9% purity was obtained from MSA Research Corporation. The metal was removed from the stainless steel shipping container and stored in paraffin oil until used. The paraffin oil had previously been exposed to rubidium chips to dry it. The crucible was filled with the dried paraffin oil and a voltage of 20 V was applied to the heater terminals. Rubidium was transferred to the crucible after removal of surface contaminants. The alkali metal displaced the oil and, when the crucible was filled to within an in. of the top, the voltage was raised to 30 V and the melt was stirred at 5 min. intervals for 45 min. This served to drive off any gases in the crucible, thus leaving the boule without trapped bubbles. The voltage was then lowered to approximately 10 V so that the temperature at the nucleating tip of the crucible could come to equilibrium slightly above the freezing point of the rubidium metal (312.65”K, FILBY and R/IARTIN.c2)After a period of 24 hr, the voltage was lowered at the rate of approximately 1 V/hr by means of a motor-driven variac transformer. Simultaneously a cooled, dry gas was pumped through the cooling coils of the crucible and a heat shield similar to that described by FOSTER(~) was placed over the top of the crucible. The boule was completely frozen in about 6 hr. The boule was etched in a mixture of xylene and secondary butyl alcohol (about 50 : 1) until a bright metallic luster resulted. The boule was then stored in paraffin oil. Examination of the boule showed that blaze planes were present. These regions of strong light reflection occur because the etching process is preferential in certain crystallographic directions causing microscopic pits in the surface. These etch pits have sides which are characteristic of the plane faces of the crystal.(l) Two such boules were judged to be single crystals. The blaze planes were examined and measured. One of the crystals showed four blaze planes spaced 90” apart; the other showed six blaze planes spaced 60” apart. Using stereographic projections of the various crystallographic directions, the crystals were found to be growing in the [loo] and [ill] directions respectively. The crystals were cut with a spark erosion machine, the Agietron universal machine tool, type
and
R.
MEISTER
AAO5. The Agietron has a servo driven vibrating head for precision control of the vertical motion of the arcing electrode and a precision mounted table upon which the crystal is cut.(s) Before the boule was cut, the Agietron was first used with a pencilpoint electrode to scribe a line along the length of the boule. This line was used as a reference marker in all subsequent work. Orientation of the crystal by standard X-ray techniques was attempted. Back reflection Laue patterns could not be obtained because of the large amount of thermal motion present at room temperature. Transmission Laue photographs were also attempted. Because the photoelectric absorption coefficient of rubidium is very high, a very thin piece of the metal was necessary for a transmission study. The cutting of samples sufficiently thin for transmission photographs by the spark erosion machine resulted in severe distortion of the crystalline structure of the samples. Since an orientation of the crystal by X-ray techniques was not obtainable, optical orientation techniques were used in determining crystallographic directions. The boules were cut into slices approx. 0.75 in. long, perpendicular to the growth axis, using the Agietron which insured that the faces were parallel. Cuts were also made perpendicular to two of the blaze planes on the crystal which grew in the [loo] direction, giving plane faces in the [llO] direction to within + 10”. This estimate of orientation error is pessimistic in view of the internal consistency of the velocity measurements. Sample thickness for this orientation was approx. 0.65 in. The crystals were placed into the etchant immediately after they were cut; after a few min. in the etchant the surfaces were cleaned gently with a flannel cloth. Sample thicknesses were measured at both room temperature and 77°K using a micrometer. The X-ray density of rubidium at 77°K is 1.608 g/cm3. At 20°C the density is l-531 g/cm3.c4) 3. EXPERIMENTAL
MEASUREMENTS sound velocities in the various crystallographic directions were determined by measuring the transit time of an ultrasonic pulse through a known length of sample. X cut and AC cut, 5 mc, 0.5 in. in dia., quartz transducers were used. A frequency of 15 mc was chosen for both the The
THE
ELASTIC
CONSTANTS
OF
1403
RUBIDIUM
Table 1. Measured velocities in rubidium*
Propagation direction
11001 11101 11111
1.255
1.360 1.645 1.710
1.460 1.535
* Velocitv units are in kmisec. +2..59/,. -
The precision of these measurements
longitudinal and shear measurements, thus exciting the third harmonic of the 5 mc transducers. A plane wave assumption was allowable since the minimum linear dimension of the faces was greater than 80 wavelengths. Time was measured by alternately displaying on the sweep of a Tektronix 541 oscilloscope in the alternate sweep mode the echo pattern and 1 psec marker pulses from a crystal controlled time marker generator. Polaroid photographs were taken of the display and time was estimated to approximately a psec. Errors arising from the electronic delays between the start of the r.f. transmitter pulse and the arrival of the first echo were neglected. In those cases where the echo trains contained several reflections, the time between successive echoes was, within our precision, the same as the time from the transmitter pulse to the first arrival. The uncertainty in the velocity measurements from transit times of about 20 psec and sample lengths of 0.7 in. is estimated to be 24%. Dow Corning 510 series silicone oil with viscosity of 60,000 cs provided a satisfactory acoustic bond from 293°K to 80°K for the longitudinal wave studies. However, for the shear measurements the bonds were not sufficiently rigid until the low temperature was reached. Shear measurements are reported at 80°K only. Velocity measurements were obtained in the [loo], [llO] and [ill] directions. The fast shear waves were studied in both the [loo] and [llO] directions. Attempts to excite the fast shear wave in the [ll l] direction by varying the direction of polarization of the AC cut quartz transducer were unsuccessful. Attempts to excite the slow shear wave in the [llO] direction were similarly unsuccessful. 2
1 .ooo 1 .OOOP01[001] -
4.
ULTRASONIC
is
VELOCITY AND CONSTANTS
ELASTIC
The velocities of plane sound waves in an anisotropic solid of infinite extent are expressed in terms of the Christoffel Stiffnesses. For crystals of the cubic system, the wave velocities for the appropriate crystallographic directions are given by equations (l)-(7). The notation v,[llO] Pol[llO] denotes the velocity of a transverse wave propagating along the [110] sample axis with the direction of vibration along [llO]. The notation v~[100] indicates the longitudinal wave velocity in the [loo] crystallographic direction. The density at 80°K is 1.607 g/ems. pv~z[loo] = Cl1
(1)
pv,2[100]
(2)
=
/-JVL2[110]=
c44 fr(C11+C12)+C44
= $(Cri-
(3)
pQ2[l
lO]P0i$o]
pa2[1
~qP01[0011 =
(744
(5)
/m2[111]
;(Cn+%h+‘+C44)
(6)
&(cll--cl2+c44)
(7)
=
/‘%2[111] =
C12)
(4)
Equations (2) and (7) are independent of polarization for the cubic case. The velocity measurements and their temperature dependence are summarized in Table 1. The values for cl1 and ~44 were obtained directly using equations (1) and (2). crs was computed from equation (3). Although it would have been more direct to obtain cls from equation (4) it was not possible to excite the slow shear wave. Equation (5) provided a check on the value of ~44. A second determination of cl2 was obtained from equation (6) by measuring the longitudinal velocity in the
1404
C.
A.
ROBERTS
and
R.
MEISTER
Table 2. Elastic constants of rubidium* ---r
Measured-293 “K Measured--80°K Extrapolated-0°K Theoreticalt 0°K
Cl1
Cl2
2.41 kO.12 2.96kO.15 3.16 kO.17 3.30
244 kO.37 2.57 kO.38 2.86
-
* All cg values are in units of lOlo dyn/cms. t Theoretical values of Fuchs modified by Bailyn at
FIG. 2. Velocity profile for rubidium at 80°K.
[ill] direction and substituting the values of cl1 and ~44 obtained from equations (1) and (2). The elastic constants and their most probable error are tabulated in Table 2. 5. DISCUSSION Using the calculated C’tj’s obtained from the 80°K data, velocity profiles were determined by computer program.@) The velocity profiles as a function of 0 for q5 = 0” (propagation in the (001) plane) for #J = 45” (propagation in the (110) plane) are shown in Figs. 2 and 3 ; the angles are defined by the inset in Fig. 2. The velocity profiles in the (001) plane for rubidium at 80°K and for potassium at 78°K are compared in Fig. 4.
c44
1.60+0.08 2.11 kO.20 1.96
0°K.
in the (001) plane
The potassium velocity profile was obtained from the data of MARQUARDTand TRIVISONNO.~~)The anisotropy factor, defined as (cu-cls)/2c44 is 0.14 for rubidium and 0.132 for potassium. This high degree of anisotropy is illustrated in both sets of velocity profiles. The measured C’ij’s are compared with theoretical values determined for rubidium by K. Fuchs and modified by BAILYN.(~) The Fuchs values are calculated from electrostatic and repulsive interactions and are given in Table 2. The theoretical Ctj values are given for 0°K. The measured ~11values in the table have been linearly extrapolated to 0” for comparison with the Fuchs value. The Fuchs value of cl1 equal to 3.33 x lOlo
THE
ELASTIC
CONSTANTS
dyn/cms is compared with the measured 3.16 x 101s dyn/cm2. Since the temperature dependence of cl2 and ~44 is not available because of the difficulty of obtaining these values at room temperature, a direct comparison with the Fuchs values at 0”
OF RUBIDIUM
1405
cannot be made. The temperature dependence of cl1 is available for both rubidium and potassium. Assuming that the ratio of the slopes, i.e. ~Ac~~A~]~~~~~~A~], for potassium and rubidium are approximately the same, the value of ~44 for
FIG. 3. Velocity profile for rubidium in the (110) at 80’K.
plane
FIG. 4. Velocity profile for rubidium and potassium in the (001) plane at 80°K.
1406
C.
A.
ROBERTS
rubidium is found to be 1 a85 x 1010 dyn/cms at 0°K. The Fuchs value for ~44is given as l-96 x 1010 dynlcms. The agreement between measured and theoretical values is reasonable. The alkali metal study supports the conclusion that electrostatic interaction contribution plays a major role in binding. It is interesting to note that the alkali metals studied have an anisotropy ratio factor between 3-7 times that of the f.c.c. metals. For the f.c.c. metals, i.e. copper, silver, aluminum, etc. the repulsive interaction plays the major role in binding. Mean sound velocities obtained from the computer program have been used to determine the Debye temperature 6~6. The equation determining the Debye temperature is (8)
where ?z/k is the ratio of Plan&s constant to Boltzmann’s constant, N is the Avogadro number, p is the density, M is the molecular weight, and ran&is the mean sound velocity. The velocity V~ given by BLACKMAN@) is
and R. MEISTER where the moduli are in units of 101s dyn/cma. The average ultrasonic shear velocity and longitudinal velocity as determined from these moduli are vS, = [Gli,:p]l’s = 705 m/see
(13)
and
Q%
= [(rc_+$G)/p]l~2 = 1520 m/xc
(14)
(the subscript N refers to the Hill value). The average velocity v, is then calculated from the relation
The velocity calculated by averaging the velocity profile, when substituted into equation (8), leads to tide of 50°K. The Anderson method leads toa &e temperature of 53”K.Thesecalculated Debye temperatures may be compared with the recent value of B&p by FILBY and MARTIN who have found from specific heat measurements a Debye temperature of 55*6”K.
6. CONCLUSIONS
where vi’s are the three velocities in the crystal which are to be averaged over all space and, da is the differential solid angle. The value of V~ obtained from the computer averaging process is found to be 755 m/set. A simplified method is suggested by PERSONS*} for obtaining v= from the bulk modulus (K) and the Voigt (Gv) and Reuss (GR) shear moduli for the cubic crystal. While there is only one bulk modulus for the cubic crystal, two shear moduli can be calculated. An average shear modulus, termed the Hill average GH, is given by the arithmetic mean of Gv and Ga,@a) The bulk and shear moduli are calculated below :
K = $(cu +2cis)
= 2.61
(10)
Gv = ~[C11-C~2+3c44]= 1.06 GR=
5@11- C12)C44 =
4c44+ 3(Cll
-c12)
0.523
The results of this study of the ultrasonic velocities confirm that the electrostatic interaction between atoms plays a major role in chemical binding in rubidium. This agrees with Fuchs theory and corresponds to conclusions reached from data for the other alkali metals whose properties have been measured and reported in the literature. Acknoz&edgments-The authors wish to thank Dr. D. L. SHELLEY for showing the technique of growing single crystal rubidium; Dr. RICHARD ROBIE for preparing the digital computer calculations of the veIocity profiles; Dr. LOUIS ~ELNICK for critically reading the manuscript; and Mr. FLOYD, Machinist. Gratitude is also expressed to the Atomic Energy Commission for its financial support. REFERENCES
(11) (12)
1. HOYTE A. F. and MIELCZAREK E. V., Appl. Mater. Res. 4, 121 (1965). 2. FILBY J.D. and MARTIN D. L., Proc. R. SW. A284, X3 (1965).
THE
ELASTIC
3. FOSTERH. J., NBS Report 8850 (1964). 4. PEARSON W. B., Handbook of Lattice
CONSTANTS
Spacings,
p. 128. Pergamon (1958). 5. ROBIE R., Computer Program 9139, U.S. Geological Survey, Washington, D.C. 6. MARQUARDT W. R. and TRIVISONNO J., J. Phys. Chem. Solids 26, 273 (1965).
OF
RUBIDIUM
1407
7. Bailyn’s values are tabulated by HUNTINGTONH. B., Solid St. Phys. (editors F. Seitz and D. Turnbull). Academic Press, New York (1958). 8. BLACKMAN M., Encyclopedia of Physics (editor S. Flugge) p. 341. Springer-Verlag, Berlin (1955). 9. ANDERSON0. L., J. Phys. Chem. Solids 24,909 (1963). 10. HILL R., Proc. Phys. Sot. Lond. A 65 349 (1952).
Chem. Solids.
Pergamon
Press 1966. Vol. 27, pp. 1401-1407.
THE ELASTIC
CONSTANTS
of Electrical
Engineering,
(Received
OF RUBIDIUM*
and R. MEISTER
C. A. ROBERTSt Department
Printed in Great Britain.
The Catholic University
21 February
1966; in reetisedform
of America, 24 March
Washington
17, D.C.
1966)
Abstract-The elastic constants of single crystal rubidium have been measured at 80 f 3°K using the ultrasonic pulse echo method at a frequency of 15 mc. The elastic constants (in units of lOlO dyn/cms) are clr = 2.96, crs = 2.44, ~44 = 1.60. Five wave velocities were measured, allowing the computation of the three independent elastic constants with two internal checks. Single crystal rubidium has an anisotropy indicated by the ratio (crr-crs)/2c44 = 0.14.
1. INTRODUCTION THIS study
presents the results of ultrasonic velocity measurements at 15 mc in single crystal rubidium at temperatures of 80” and 293°K. To the authors’ knowledge these measurements represent the first experimental determination of the elastic constants for this alkali metal. The measured constants are used to calculate the polycrystalline shear and bulk moduli. The measured constants are compared with the Fuchs theoretical values and are also used to calculate the Debye temperature. The calculated Debye temperatures are compared with values obtained from specific heat measurements. 2. SAMPLE
PREPARATION
#20 wire. The nucleating tip was also modified. The first + in. of the tip was drilled with a 45” taper. At the bottom of this tapered section a cylindrical hole & in. in dia. and 1 in. deep was
AND
ORIENTATION
Single crystals of rubidium were grown using a modified Bridgman technique’ developed at this laboratory by D. L. Shelley. The crucible used was similar to that described by HOYTE and MIELCZARE@ for growth of potassium. Because rubidium has a lower melting point than potassium, the crucible was redesigned with four sections of heating wire instead of three. Each of the four sections of this crucible are 1%in. in length with 15, 12, 8 and 4 turns per in. of nichrome *This research is sponsored by the U.S. A.E.C. Grant No. AT(40-l)-2861. t This work is based in part on a thesis submitted in partial fulfillment of the requirements for the degree of Master of Electrical Engineering, Department of Electrical Engineering, Catholic University of America. 1401
FIG. 1. Crucible
for growing single crystal (TPI-turns per in.).
rubidium
1402
C. A.
ROBERTS
drilled to form the bottom of the cup. (Refer to Fig. 1.) Rubidium of 99.9% purity was obtained from MSA Research Corporation. The metal was removed from the stainless steel shipping container and stored in paraffin oil until used. The paraffin oil had previously been exposed to rubidium chips to dry it. The crucible was filled with the dried paraffin oil and a voltage of 20 V was applied to the heater terminals. Rubidium was transferred to the crucible after removal of surface contaminants. The alkali metal displaced the oil and, when the crucible was filled to within an in. of the top, the voltage was raised to 30 V and the melt was stirred at 5 min. intervals for 45 min. This served to drive off any gases in the crucible, thus leaving the boule without trapped bubbles. The voltage was then lowered to approximately 10 V so that the temperature at the nucleating tip of the crucible could come to equilibrium slightly above the freezing point of the rubidium metal (312.65”K, FILBY and R/IARTIN.c2)After a period of 24 hr, the voltage was lowered at the rate of approximately 1 V/hr by means of a motor-driven variac transformer. Simultaneously a cooled, dry gas was pumped through the cooling coils of the crucible and a heat shield similar to that described by FOSTER(~) was placed over the top of the crucible. The boule was completely frozen in about 6 hr. The boule was etched in a mixture of xylene and secondary butyl alcohol (about 50 : 1) until a bright metallic luster resulted. The boule was then stored in paraffin oil. Examination of the boule showed that blaze planes were present. These regions of strong light reflection occur because the etching process is preferential in certain crystallographic directions causing microscopic pits in the surface. These etch pits have sides which are characteristic of the plane faces of the crystal.(l) Two such boules were judged to be single crystals. The blaze planes were examined and measured. One of the crystals showed four blaze planes spaced 90” apart; the other showed six blaze planes spaced 60” apart. Using stereographic projections of the various crystallographic directions, the crystals were found to be growing in the [loo] and [ill] directions respectively. The crystals were cut with a spark erosion machine, the Agietron universal machine tool, type
and
R.
MEISTER
AAO5. The Agietron has a servo driven vibrating head for precision control of the vertical motion of the arcing electrode and a precision mounted table upon which the crystal is cut.(s) Before the boule was cut, the Agietron was first used with a pencilpoint electrode to scribe a line along the length of the boule. This line was used as a reference marker in all subsequent work. Orientation of the crystal by standard X-ray techniques was attempted. Back reflection Laue patterns could not be obtained because of the large amount of thermal motion present at room temperature. Transmission Laue photographs were also attempted. Because the photoelectric absorption coefficient of rubidium is very high, a very thin piece of the metal was necessary for a transmission study. The cutting of samples sufficiently thin for transmission photographs by the spark erosion machine resulted in severe distortion of the crystalline structure of the samples. Since an orientation of the crystal by X-ray techniques was not obtainable, optical orientation techniques were used in determining crystallographic directions. The boules were cut into slices approx. 0.75 in. long, perpendicular to the growth axis, using the Agietron which insured that the faces were parallel. Cuts were also made perpendicular to two of the blaze planes on the crystal which grew in the [loo] direction, giving plane faces in the [llO] direction to within + 10”. This estimate of orientation error is pessimistic in view of the internal consistency of the velocity measurements. Sample thickness for this orientation was approx. 0.65 in. The crystals were placed into the etchant immediately after they were cut; after a few min. in the etchant the surfaces were cleaned gently with a flannel cloth. Sample thicknesses were measured at both room temperature and 77°K using a micrometer. The X-ray density of rubidium at 77°K is 1.608 g/cm3. At 20°C the density is l-531 g/cm3.c4) 3. EXPERIMENTAL
MEASUREMENTS sound velocities in the various crystallographic directions were determined by measuring the transit time of an ultrasonic pulse through a known length of sample. X cut and AC cut, 5 mc, 0.5 in. in dia., quartz transducers were used. A frequency of 15 mc was chosen for both the The
THE
ELASTIC
CONSTANTS
OF
1403
RUBIDIUM
Table 1. Measured velocities in rubidium*
Propagation direction
11001 11101 11111
1.255
1.360 1.645 1.710
1.460 1.535
* Velocitv units are in kmisec. +2..59/,. -
The precision of these measurements
longitudinal and shear measurements, thus exciting the third harmonic of the 5 mc transducers. A plane wave assumption was allowable since the minimum linear dimension of the faces was greater than 80 wavelengths. Time was measured by alternately displaying on the sweep of a Tektronix 541 oscilloscope in the alternate sweep mode the echo pattern and 1 psec marker pulses from a crystal controlled time marker generator. Polaroid photographs were taken of the display and time was estimated to approximately a psec. Errors arising from the electronic delays between the start of the r.f. transmitter pulse and the arrival of the first echo were neglected. In those cases where the echo trains contained several reflections, the time between successive echoes was, within our precision, the same as the time from the transmitter pulse to the first arrival. The uncertainty in the velocity measurements from transit times of about 20 psec and sample lengths of 0.7 in. is estimated to be 24%. Dow Corning 510 series silicone oil with viscosity of 60,000 cs provided a satisfactory acoustic bond from 293°K to 80°K for the longitudinal wave studies. However, for the shear measurements the bonds were not sufficiently rigid until the low temperature was reached. Shear measurements are reported at 80°K only. Velocity measurements were obtained in the [loo], [llO] and [ill] directions. The fast shear waves were studied in both the [loo] and [llO] directions. Attempts to excite the fast shear wave in the [ll l] direction by varying the direction of polarization of the AC cut quartz transducer were unsuccessful. Attempts to excite the slow shear wave in the [llO] direction were similarly unsuccessful. 2
1 .ooo 1 .OOOP01[001] -
4.
ULTRASONIC
is
VELOCITY AND CONSTANTS
ELASTIC
The velocities of plane sound waves in an anisotropic solid of infinite extent are expressed in terms of the Christoffel Stiffnesses. For crystals of the cubic system, the wave velocities for the appropriate crystallographic directions are given by equations (l)-(7). The notation v,[llO] Pol[llO] denotes the velocity of a transverse wave propagating along the [110] sample axis with the direction of vibration along [llO]. The notation v~[100] indicates the longitudinal wave velocity in the [loo] crystallographic direction. The density at 80°K is 1.607 g/ems. pv~z[loo] = Cl1
(1)
pv,2[100]
(2)
=
/-JVL2[110]=
c44 fr(C11+C12)+C44
= $(Cri-
(3)
pQ2[l
lO]P0i$o]
pa2[1
~qP01[0011 =
(744
(5)
/m2[111]
;(Cn+%h+‘+C44)
(6)
&(cll--cl2+c44)
(7)
=
/‘%2[111] =
C12)
(4)
Equations (2) and (7) are independent of polarization for the cubic case. The velocity measurements and their temperature dependence are summarized in Table 1. The values for cl1 and ~44 were obtained directly using equations (1) and (2). crs was computed from equation (3). Although it would have been more direct to obtain cls from equation (4) it was not possible to excite the slow shear wave. Equation (5) provided a check on the value of ~44. A second determination of cl2 was obtained from equation (6) by measuring the longitudinal velocity in the
1404
C.
A.
ROBERTS
and
R.
MEISTER
Table 2. Elastic constants of rubidium* ---r
Measured-293 “K Measured--80°K Extrapolated-0°K Theoreticalt 0°K
Cl1
Cl2
2.41 kO.12 2.96kO.15 3.16 kO.17 3.30
244 kO.37 2.57 kO.38 2.86
-
* All cg values are in units of lOlo dyn/cms. t Theoretical values of Fuchs modified by Bailyn at
FIG. 2. Velocity profile for rubidium at 80°K.
[ill] direction and substituting the values of cl1 and ~44 obtained from equations (1) and (2). The elastic constants and their most probable error are tabulated in Table 2. 5. DISCUSSION Using the calculated C’tj’s obtained from the 80°K data, velocity profiles were determined by computer program.@) The velocity profiles as a function of 0 for q5 = 0” (propagation in the (001) plane) for #J = 45” (propagation in the (110) plane) are shown in Figs. 2 and 3 ; the angles are defined by the inset in Fig. 2. The velocity profiles in the (001) plane for rubidium at 80°K and for potassium at 78°K are compared in Fig. 4.
c44
1.60+0.08 2.11 kO.20 1.96
0°K.
in the (001) plane
The potassium velocity profile was obtained from the data of MARQUARDTand TRIVISONNO.~~)The anisotropy factor, defined as (cu-cls)/2c44 is 0.14 for rubidium and 0.132 for potassium. This high degree of anisotropy is illustrated in both sets of velocity profiles. The measured C’ij’s are compared with theoretical values determined for rubidium by K. Fuchs and modified by BAILYN.(~) The Fuchs values are calculated from electrostatic and repulsive interactions and are given in Table 2. The theoretical Ctj values are given for 0°K. The measured ~11values in the table have been linearly extrapolated to 0” for comparison with the Fuchs value. The Fuchs value of cl1 equal to 3.33 x lOlo
THE
ELASTIC
CONSTANTS
dyn/cms is compared with the measured 3.16 x 101s dyn/cm2. Since the temperature dependence of cl2 and ~44 is not available because of the difficulty of obtaining these values at room temperature, a direct comparison with the Fuchs values at 0”
OF RUBIDIUM
1405
cannot be made. The temperature dependence of cl1 is available for both rubidium and potassium. Assuming that the ratio of the slopes, i.e. ~Ac~~A~]~~~~~~A~], for potassium and rubidium are approximately the same, the value of ~44 for
FIG. 3. Velocity profile for rubidium in the (110) at 80’K.
plane
FIG. 4. Velocity profile for rubidium and potassium in the (001) plane at 80°K.
1406
C.
A.
ROBERTS
rubidium is found to be 1 a85 x 1010 dyn/cms at 0°K. The Fuchs value for ~44is given as l-96 x 1010 dynlcms. The agreement between measured and theoretical values is reasonable. The alkali metal study supports the conclusion that electrostatic interaction contribution plays a major role in binding. It is interesting to note that the alkali metals studied have an anisotropy ratio factor between 3-7 times that of the f.c.c. metals. For the f.c.c. metals, i.e. copper, silver, aluminum, etc. the repulsive interaction plays the major role in binding. Mean sound velocities obtained from the computer program have been used to determine the Debye temperature 6~6. The equation determining the Debye temperature is (8)
where ?z/k is the ratio of Plan&s constant to Boltzmann’s constant, N is the Avogadro number, p is the density, M is the molecular weight, and ran&is the mean sound velocity. The velocity V~ given by BLACKMAN@) is
and R. MEISTER where the moduli are in units of 101s dyn/cma. The average ultrasonic shear velocity and longitudinal velocity as determined from these moduli are vS, = [Gli,:p]l’s = 705 m/see
(13)
and
Q%
= [(rc_+$G)/p]l~2 = 1520 m/xc
(14)
(the subscript N refers to the Hill value). The average velocity v, is then calculated from the relation
The velocity calculated by averaging the velocity profile, when substituted into equation (8), leads to tide of 50°K. The Anderson method leads toa &e temperature of 53”K.Thesecalculated Debye temperatures may be compared with the recent value of B&p by FILBY and MARTIN who have found from specific heat measurements a Debye temperature of 55*6”K.
6. CONCLUSIONS
where vi’s are the three velocities in the crystal which are to be averaged over all space and, da is the differential solid angle. The value of V~ obtained from the computer averaging process is found to be 755 m/set. A simplified method is suggested by PERSONS*} for obtaining v= from the bulk modulus (K) and the Voigt (Gv) and Reuss (GR) shear moduli for the cubic crystal. While there is only one bulk modulus for the cubic crystal, two shear moduli can be calculated. An average shear modulus, termed the Hill average GH, is given by the arithmetic mean of Gv and Ga,@a) The bulk and shear moduli are calculated below :
K = $(cu +2cis)
= 2.61
(10)
Gv = ~[C11-C~2+3c44]= 1.06 GR=
5@11- C12)C44 =
4c44+ 3(Cll
-c12)
0.523
The results of this study of the ultrasonic velocities confirm that the electrostatic interaction between atoms plays a major role in chemical binding in rubidium. This agrees with Fuchs theory and corresponds to conclusions reached from data for the other alkali metals whose properties have been measured and reported in the literature. Acknoz&edgments-The authors wish to thank Dr. D. L. SHELLEY for showing the technique of growing single crystal rubidium; Dr. RICHARD ROBIE for preparing the digital computer calculations of the veIocity profiles; Dr. LOUIS ~ELNICK for critically reading the manuscript; and Mr. FLOYD, Machinist. Gratitude is also expressed to the Atomic Energy Commission for its financial support. REFERENCES
(11) (12)
1. HOYTE A. F. and MIELCZAREK E. V., Appl. Mater. Res. 4, 121 (1965). 2. FILBY J.D. and MARTIN D. L., Proc. R. SW. A284, X3 (1965).
THE
ELASTIC
3. FOSTERH. J., NBS Report 8850 (1964). 4. PEARSON W. B., Handbook of Lattice
CONSTANTS
Spacings,
p. 128. Pergamon (1958). 5. ROBIE R., Computer Program 9139, U.S. Geological Survey, Washington, D.C. 6. MARQUARDT W. R. and TRIVISONNO J., J. Phys. Chem. Solids 26, 273 (1965).
OF
RUBIDIUM
1407
7. Bailyn’s values are tabulated by HUNTINGTONH. B., Solid St. Phys. (editors F. Seitz and D. Turnbull). Academic Press, New York (1958). 8. BLACKMAN M., Encyclopedia of Physics (editor S. Flugge) p. 341. Springer-Verlag, Berlin (1955). 9. ANDERSON0. L., J. Phys. Chem. Solids 24,909 (1963). 10. HILL R., Proc. Phys. Sot. Lond. A 65 349 (1952).