A note on the elastic constants of sodium bromate from 350 K to 77 K
K R SRINIVASAN* and E S R. G O P A L
Physics Department, Indian Institute of Science, Bangalore 560012 MS received 8 October 1975
Abstract. The elastic constants Clt, Cxa and C,4 of sodium bromate single crystals have been evaluated using 10 MHz ultrasonic pulse echo superposition technique.
The values are Clt = 5"57s, Ctz = I "075, C44 = 1.51o (× 101°N/m 2) at 290K and 6 "35, 1 "98 and 1 "65 (× 101° N/m 2) at 77 K. The present room temperature values agree closely with the recent values of Gluyas et al. but the other earlier measure- ments show some scatter. A comparison between the elastic constants of sodium bromate and sodium chlorate is also made.
Keywords. Elastic constant; sodium bromate; ultrasonic velocity.
1. Introduction
Sodium bromate which has cubic symmetry belonging to the tetrahedral class 2 3, is a piezoelectric material. It is isomorphous with sodium chlorate. It has three independent elastic constants Ctl, CI~ and C44 and one independent piezo- electric constant. One of the reasons for conducting the measurement of ultra- sonic velocity was to evaluate more precisely the elastic constants of NaBrOa.
The earlier measurements carried out by various workers show a large amount of scatter. Hence it was more appropriate to evaluate the elastic constants more precisely using the pulse-superposition technique whicb is among the high pre- cision methods in vogue today. Moreover at the start of the present work no low temperature data were available for NaBrO3. Very recently Gluyas and others (Gluyas e t al. 1975) have evaluated the elastic constants of NaBrOa at low tem- peratures. However, their measurements extend only up to 150 ° K, whereas the present data extend down to liquid nitrogen temperatures.
It is quite interesting to compare the elastic constants C , of NaBrO3 with those of NaCIOa. The present investigation clearly establishes the higher elastic con- stants of the bromate over the chlorate of sodium. It should be noted with interest that in the case of alkali halides
C~j (fluorides) > C,j (chlorides) > C~j (bromides) > C~j (iodines).
2. Experimental procedure
The three elastic constants (711, Cx~ and C44 of NaBrO3 are evaluated by measure- ment of ultrasonic velocities along [100] and [111] directions. Single crystals o f sodium bromate grown by slow evaporation of the aqueous solution were used.
* C.E.D.T. ECE Department, Indian Institute of Science, Bar, galore 560012.
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20
The crystals cut along the [100] and [111] directions were lapped plane and parallel to better than 100 pp. The crystallographic directions were checked with x-ray back reflection photography. Longitudinal and shear wave velocities along [100]
direction yield Ca1 and C44 pVr. ~ [1001 = C1~, and
pVs 2 [100] "- C4,U.
Longitudinal and shear wave velocities along (111) direction yields pVL 2 [111] : (C1~ + 2C~ + 4C44)/3.
pVs 2 [111] = (C~, --C~z ÷ Ca4)/3.
Hence knowirtg C~ and C4a, C~z can be evaluated from either the longitudinal or shear wave velocity along [111] direction. Longitudinal, wave velocity is used in the present investigation for determining the elastic constant C1~ of NaBrOa and was cross checked by measuring shear wave velocity.
A 10 MHz ultrasonic pulse interferometer (Srinivasar~ et al. 1974) using pulse superposition technique was used for the velocity measurements. Using a simple cryostat, velocities were measured down to liquid nitrogen temperatures, with the specimen cooled at the rate of about 10 K/hour, in order to avoid cracking of the specimen due to thermal shocks. It was found experimerttalty that no one seal material could be used over the entire temperature region. Four seal materials, namely dihydroxy Benzene, Salol, Nonaq stopcock grease and an organic mixture of 5 parts by volume of ethyl ether, 6 parts of isopentane and 2 parts of ethyl alcohol were used to bond the transducer with the specimen at different temperatures.
In experiments using pulse-echo interferorneter, great care has to be taken in the preparatiort of the sample and the selection of proper seal materials if accu- racies of 0.1X or better are aimed at. These and other details are reported in detail elsewhere (Srinivasan et al. 1975).
3. Results and discussions
Room temperature values
Assuming the density of sodium bromate to be 3,335 kg/m 3 at 298 K (American Handbook 1972), the elastic constants are calculated from the velocity measurements.
Table 1 summarises the present values of the three elastic constants Cll, C1~ and C44, together with the results of the earlier workers. As could be seer~ from the table, the room temperature values, especially of Cll and Ca2, of different workers show some scatter. The C44 values of all the workers show broad agreement. The present values agree on the whole closely with that of Bechmann and the recent measurement of Gluyas etal.
[t is to be noted with interest that the present values and the data of Gluyas et al. agree to better than 1% even in Ca2. In comparing the present values with those of Bechmann (Bechmann 1951), both C~1 and C12 differ by about 3.5~.
Radha's (Radha et at. 1968) values of Clt and C~4 are within 2% of the present measurement, whereas the variation in C1~ is about 20% which is quite high, espe- cially for the pulse-echo method. Mason's (Mason 1946) values are quite different
Table 1. Comparative study of the elastic constants of NaBrO3 at room temperature
Author Year Method Elastic constants in 10 TM n/m ~
Mason .. 1946 Resonance 6.16 2 -36 1.54 3.80
Sundara Rao .. 1949 Wedge 5.45 1 -91 1.50 3 -54
Bechmann .. 1951 Resonance 5.73 1 -77 1.52 3.96
Haussuhl .. 1964 Shaeffer-Bergmann 5 "478 1.628 l "506 3 "850 Radha and Gopal 1968 Pulse-echo 5 "648 2.038 1.54~ 3 '618 Gluyas et al. .. 1975 Sing-around 5 "58e 1 "682 1 "518 3 "904 Present .. 1975 Pulse-super-position 5 '578 l '705 1 "510 3 "873
from the present values except in the case of C 4 4 , where the difference is only about 0" 3~. The present values are quite close to that of Gluyas et al. who have used sing-around technique (Forgacs 1960) at room temperature. The variation in Cll and C12 are 0 . 1 4 ~ and 1 . 3 ~ and the agreement in C44 is very good. The slightly larger difference in the values of C~2 is due to the fact that the elastic constant C1~, which is evaluated as the difference between two large quantitieg, is liable to a greater error. However, it is seen that the value ( C l l - Cj~) agrees among most of the workers. This combination (Ci1 - - C12) rather than CI~ alone occurs in the wave stiffnesses along many directions. F o r the shear waves the transit time is larger, yielding perhaps better accuracies, for this combination.
Possibly no proper explanation could be given for the large spread in the C~j values of the earlier workers. The wedge method is not credited for accuracies better than 5 to 107o in the values of even Cll and C44. The Shaeffer-Bergmann values depend a little on the sharpness of the pattern and could perhaps have an error of the order of 2 - 3 ~ in the elastic constants, although Haussuhl's (Haussuhl 1964) measurements have been p e r f o r m e d with great care. However, the reso- nance methods used by Mason and Bechmann are correct to within 1.07o. Hence one is unable to advance a satisfactory explanation for the scatter in the earlier studies. Moreover Mason had evaluated the elastic constants of NaC103 with the same set up and at the same time as NaBrO3 and one finds excellent agreement with Mason in the case of NaC1Os (Srinivasan et al 1975). Hence this may suggest that this deviation could arise from the chemical purity and crystal perfection o f the samples used, where specific studies of these effects have been made, differences of 5 to 107o have been reported in the main constants, which would give even larger difference in the case of off-diagonal constants like C12.
It is to be noted that even though, in the pulse-superposition technique, one could resolve the transiit time to better than i0 pp, the ultimate accuracy that is obtained in the evaluation of the elastic constants C~j is not better than 0.17o.
It is because of the fact that the density values, the thickness of the specimen and thermal expansion coefficients are generally not known to better than 0.17o,
22
z.c
E _o~ 6C
2
1'00[
70 I00 Figure 1.
NoBrO 3.
o Prellent
• GtuyO~ et all
o O o o o : • • • CI2
C4~l
Temperature (K)
Variation of elastic constant with temperature
3;0 3~o 370
Low temperature experiment
The variation of the adiabatic elastic constants Cll, C~z and C44 with tempera- ture is given in figure 1. In the same figure, the values of Gluyas, et al. reported up to 150 K are indicated. Correction for thermal expansion at various tempera- tures was also done by making use of the thermal expnsion data provided by Ganesan (Ganesan 1959). From figure I, it is seen that the present d a t a agree very well in the case of C44, but show a smaller increase of Cxl at low temperature, being smaller by about 1.5% at 150 K. In the case of C~, the present values are higher by about 6% at 150 K. In going from 300 to 77 K, the percentage increase in Clx, Ca2 and C44 are respectively 14, 18 and 10. Even though the rate of increase of C1~ in the present data is higher as compared to that of Gluyas et al. it is also seen that the curves for Cll, C~4 are nearly linear while that of Cx~ has a change of slope at around 230 K in the case of Gluyas et al. and at about 180 K in the present case. It is to be noted that measurements of Gluyas et al. were carried out using sing-around technique at all temperatures except at ice point where pulse superposition technique was used.
The present experiment was carried out with the pulse superposition technique at all temperatures.
Comparison of the elastic constants of sodium bromate and sodium chlorate
A comparative study of the elestic constants of sodium bromate with those of sodium chlorate is quite interesting. In table 2 the elastic constants of NaCIO8 and NaBrO8 are compared. From the table it is clear that the elastic constants of the bromate are higher than that of the chlorate. It should be noted that in the case of alkali halides,
C. (fluorides) > C 0 (chlorides) > C~ (bromides) > C o (iodides).
Table 2. Ela4tie constants of NaBrOs and NaClOa
i i
300 ° K 77 ° K
c~ c~, c,, c~ c~, c,,
N a B r O s .. 5.58 1.71 1.51 6.35 1.98 1.65
N a C I O 3 .. 4.90 I .39 1.17 6.15 2.16 1.32
C o in units o f 101° N / m s
X-ray studies show that the unit cell of the isomorphous NaC103 and NaBrOs contains four molecules and has the edges a 0 - - 6 . 5 7 0 A for NaCIOa (Aravin- dakshan 1959) and a0 = 6.689 A for NaBrO3 (Hamilton 1938). The densities are 2,482 and 3,335 kg/rn 3 respectively. The ClOa and Br03 groups are pyra- midal and belong to the C3~ point group. The oxygen atoms are at the comers of an equilateral triangle, but the cation is displaced from the triangular base, the displacement of the central cation from the basal plane being greater for Br than for C1. Hence Na-Br distance of 3.72 A is shorter than the Na-C1 distance of 4.02 A.
Therefore, the Bragg contact law (Bragg 1920, 1926) for ionic distances is not obeyed in this case. There has been proof from other properties that sodium bromate has stronger interatomic forces. The melting point of NaBrO3 (381 ° C) is higher than that of NaC108 (255 ° C). Thermal expansion studies of Ganesan and Sharma (Sharma 1950) show that the coefficient of thermal expansion as well as its temperature variation is much smaller for the bromate than for the chlorate.
Raman spectra and the low frequency lattice lines (Couture 1948) show that the bromato frequencies are only slightly less than the chlorate frequencies. Taking into account the substantially heavier mass of the bromate this also suggests stronger interatomic forces in sodium bromate. This could be related to the shorter Na-Br bond distance which is only 3.72 A as compared with the Na-CI distance of 4.02 A in sodium chlorate.
The relationship between the bond distance and the elastic constants is different in the case of alkeli halides. In them the unit cell size is directly determined by the interatomic distance and the contact law implies larger lattice constants for the bromides than for the chlorides. Thus the larger ion size and the larger inter- atomic distances result in weaker bonding and consequently lower elastic con- stants and lower melting point. Hence we would say that the shorter Na-Br distance could be responsible for the larger elastic constants of NaBrOa, in spite of the fact that the unit cell and the Br ion are larger. From the above, it is evident that a complete lattice dynamics of NaBrOa deserves further consideration.
Acknowledgement
The financial support of PL-480 is acknowledged.
References
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