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Pressure variation of the elastic moduli of the sodium halides
5011d 5tate C0mmun1cat10n5, V01. 46, N0.2, pp. 161-164, 1983. Pr1nted 1n 6reat 8r1ta1n.
003 8 - 1 0 9 8/83/140161 --04503.00/0 Per9am0n Pre55 Ltd.
PRE55URE VAR1A710N 0 F 7HE ELA571C M0DUL1 0 F 7HE 50D1UM HAL1DE5 5. Hart and P.H. 6reenw00d* Nat10na1 Phy51ca1 Re5earch La60rat0ry, P . 0 . 8 0 x 395, Pret0r1a, 50uth Afr1ca
(Rece1ved 16 Au9u5t 1982 6y C. W. McC0m61e) An u1tra50n1c 5tudy ha5 6een made 0f the var1at10n 0f the e1a5t1c m0du11 0f 51n91e cry5ta15 0f the 50d1um ha11de5 t0 1.5 6Pa. A 4uadrat1c f1t 15 re4u1red f0r the data. 7he 5y5temat1c5 and 1n part1cu1ar the 5ec0nd der1vat1ve 0f the 6u1k m0du1u5 ha5 6een f0und t0 6e 1n 9enera1 a9reement w1th pred1ct10n5 0f the 80rn m0de1 0f the e4uat10n 0f 5tate.
7HE EVALUA710N 0f the pre55ure der1vat1ve5 0f the e1a5t1c c0n5tant5 0f 51n91e cry5ta15 6y u1tra50n1c techn14ue5 ena61e5 the P - V e4uat10n 0f 5tate t0 6e determ1ned t0 h19her accuracy than 6y 0ther techn14ue5 5uch a5 X-ray 0r v01umetr1c and hence p1ay5 an 1mp0rtant r01e 1n the deve10pment 0f e4uat10n 0f 5tate m0de15.7h15 w0rk wa5 p10neered 6y La2aru5 [1 ] and the meth0d ha5 6een put t0 1ncrea51n9 u5e dur1n9 the pa5t year5. Ru0ff and Chha611da5 [2] have p01nted 0ut that the a9reement 6etween exper1ment and the0ry 15 n0t yet 5at15fact0ry and that the exper1ment5 5h0u1d 6e carr1ed 0ut at h19her pre55ure 50 that, f0r examp1e, n0t 0n1y the f1r5t 6ut the 5ec0nd der1vat1ve5 0f e1a5t1c m0du11 c0u1d 6e 4uant1f1ed. 1t 15 kn0wn that the 6u1k m0du1u5 a5 a funct10n 0f pre55ure f0r the a1ka11ha11de5 [3] and 0ther mater1a15 [4] d0e5 5h0w curvature and attempt5 have 6een made t0 determ1ne th15 curvature 1n 1501ated 1n5tance5 [5 ]. A 5y5temat1c 5tudy ha5 theref0re 6een carr1ed 0ut 0n the 50d1um ha11de5 t0 1.5 6Pa t0 determ1ne, 1n part1cu1ar, the f1r5t and 5ec0nd pre55ure der1vat1ve5 0f the e1a5t1c m0du11 and hence t0 c0mpare the va11d1ty 0f the0ret1ca1 e4uat10n5 0f 5tate. An exten51ve u1tra50n1c 1nve5t19at10n 0n the 50d1um and p0ta551um ha11de5 wa5 carr1ed 0ut 6y R06ert5 and 5m1th [6] 6ut 0n1y t0 0.1 6Pa and a1th0u9h n0n-11near1ty 0f 50me m0du11 wa5 n0ted, 1t wa5 n0t 4uant1f1ed. 7he1r 1nve5t19at10n 5h0wed pr0n0unced 5y5temat1c5 acr055 the 5er1e5 0f ha11de5 w1th the f1r5t 0rder der1vat1ve5 5h0w1n9 an 1ncrea5e fr0m the f1u0r1de t0 the 10d1de. 1t wa5 a150 5h0wn that the m0re prec15e techn14ue rem0ved 50me an0ma11e5 wh1ch had prev10u51y ex15ted 1n the 5er1e5 when u51n9 6u1k c0mpre5510n data. F0r the pre5ent 5tudy 51n91e cry5ta15 0f the re4u1red ha11de5 were 9r0wn fr0m the me1t 6y the 5t0ck6ar9er meth0d and were 0f 51m11ar 4ua11ty t0 th05e rep0rted prev10u51y [7]. 5pec1men5 0f the c0rrect 0r1entat10n were * Pre5ent addre55: Nuc1ear Deve10pment C0rp0rat10n 0f 5A, Pe11nda6a, Pret0r1a, 50uth Afr1ca.
then e1ther c1eaved 0r 5awn 6y a d1am0nd-1mpre9nated w1re 5aw and 5u65e4uent1y 1apped and p0115hed w1th 0pp051te face5 p1ane and para11e1 t0 0.01 mm. Laue 6ack-ref1ect10n X-ray ph0t09raph5 were u5ed t0 en5ure that c0rrect 0r1entat10n wa5 ma1nta1ned t0 6etter than 0.5 °, 1.e. t0 the prec1510n ava11a61e 1n r0ut1ne Laue u5e. 7he 5pec1men5 were phy51ca11y 1 cm 10n9, 1 cm d1ameter cy11nder5 w1th the u1tra50und pr0pa9at10n a10n9 the ax15. 10 MH2 X and Y cut 4uart2 tran5ducer5 were wrun9 0nt0 the 5pec1men5 u51n9 a m1xture 0f 91ycer1ne and ptha11c anhydr1de a5 60nd1n9 a9ent (th15 91ue d0e5 n0t d15501ve 1n the pre55ure tran5m1tt1n9 f1u1d u5ed). 7he accurate determ1nat10n 0f wave tran51t t1me5 wa5 carr1ed 0ut u51n9 the pu15e ech0 0ver1ap meth0d w1th c0mmerc1a1 Panametr1c5 pu151n9 e4u1pment. 7he pre55ure 5y5tem u5ed wa5 a p15t0n-cy11nder apparatu5 [8] w1th 12.7 mm d1ameter 60re. 7he pre55ure p1ate wa5 end-10aded w1th 100 t0n5 t0 91ve further 5upp0rt1n9 10n91tud1na1 c0mpre5510n t0 the tun95ten car61de cy11nder and the car61de p15t0n wa5 dr1ven 1n 6y a 5ec0nd ram. 7he ce11 5y5tem 15 5h0wn 1n F19.1 and u5e5 tw0 5ea11n9 p1u95, 0ne 61ank and 0ne w1th e1ectr1c 1ead-thr0u9h5, t0 c0nta1n the pre55ure tran5m1tt1n9 f1u1d wh1ch, 1n th15 ca5e, wa5 a 50:50 m1xture 0f n-pentane and 150pentane. A man9an1n re515tance 9au9e wa5 1nc0rp0rated 1n the p1u9 a55em61y. A5 a1way5 w1th 5uch h19h pre55ure 5y5tem5 the determ1nat10n 0f pre55ure 15 c0mp11cated 6y fr1ct10na1 effect5 1n the ma1n ram and, t0 a 1e55er extent, 1n the 5ea11n9 p1u95. Here the unc0rrected pre55ure wa5 determ1ned 6y the 9e0metr1ca1 ma9n1f1cat10n 0f area5 (ram t0 p15t0n) and the ram hydrau11c pre55ure. 7he man9an1n 9au9e (and/0r a he19ht 9au9e t0 m0n1t0r p15t0n penetrat10n) wa5 then u5ed t0 c0rrect f0r fr1ct10n a55um1n9 1t t0 6e 5ymmetr1c. 7he up5tr0ke re515tance va1ue5 were a1way5 h19h and the d0wn5tr0ke va1ue5 10w due t0 fr1ct10n and the ar1thmet1c mean 9ave the c0rrected and true ce11 pre55ure. 7h15 meth0d ha5 6een checked a9a1n5t the mercury free21n9 p01nt [9] and f0und accurate t0 w1th1n 1%.
161
162
PRESSURE VARIATION
SPECIMEN AND TRANSCUCER
Table 1. Power law parameters
ELECTRICAL LEAD-THRWGH
HARD STEELS
’
Vol. 46, No. 2
OF THE ELASTIC MODULI OF THE SODIUM HALIDES
TUNGSFEN
CARBIDE
A (1 O-i9 J mol-‘)
n
2.14 1.5 1 1.34 1.15
7.68 8.73 9.21 9.81
NaF NaCl NaBr NaI
ti TUNGSTEN CARBIDE PISTDN
Fig. 1. High pressure cell system. The acoustic modes investigated were the usual ones used for cubic crystals, i.e. longitudinal and shear in the [ 10 0] and [ 1 lo] directions and the standard relationships relating velocities to the moduli were used [lo]. It is normal practice to use simply the wave transit time, t = l/f, through the specimen of ambient length Zeeither in the form of the “natural velocity” W = lo/t or the “natural modulus” poW2 (p,, = the ambient condition density) and to correct for length change under pressure by differentiation of the true modulus M = pv2 = pl 2f ‘. The required relationship for the first order derivative is dn!f z-=
4PoW2) M -+jgdP
atP=O
T
where BT is the isothermal bulk modulus. The first term is the slope of the experimental results and the second is the correction for length change. It is of passing interest that the experimental curve of the natural modulus for [ 10 0] shear in NaF shows a negative slope but the correction term lifts this to the positive value of 0.09 shown in Table 2 and this mode does not in reality soften. Similarly the second order equation is given by d2M dp2
d2(p,W2)+ = dP2
2 d(poW’)+ 3B,y
M -___ 9B;
where A, n, D and d are empirical fitting parameters. By noting that the bulk modulus is the volume derivative of pressure and pressure is the volume derivative of the free energy, the relationships An=D$
and
+
EidV)
where the first term is the Madelung attractive energy and the second term is the repulsive energy. The power and exponential forms of ER are in common use, i.e.
n+l
may be derived. Values of the parameters have been evaluated from thermodynamic and mechanical data by other workers [6] and are listed in Table 1. Further differentiation of the bulk modulus equation gives
=-
(power law)
(exponential x=--j-
law)
do
and
M dB, 3B; dp
also evaluated at P = 0. These two relationships allow the pressure variations of the moduli to be obtained at the ambient condition limit but do not give the values of the moduli at high pressure. Iterative methods to do this have been developed and that of Dandekar [ 1 l] has been adopted in the present study. The Born model for ionic solids expresses the lattice energy in volume dependent terms and the energy, E, can be written as [6]
+=
= -4;+3) P=O
(power law) T
=
- 5X3 + 16X2 + 2X - 32 9(X - 2)2B, (exponential
’
law)
It is perhaps noteworthy at this stage that both laws predict a negative value of the second derivative, i.e. that the values of the modulus with respect to pressure will be concave to the pressure axis which is what is observed in practice. Experimental results for the second order elastic constants are in very good agreement with other ultrasonic data published in recent years and are not reproduced here. The pressure variation of the moduli showed curvature and was fitted with a quadratic function from which the derivates were obtained. The results
V01.46, N0.2
PRE55URE VAR1A710N 0 F 7HE ELA571C M0DUL1 0 F 7HE 50D1UM HAL1DE5
163
7a61e 2. F1r5t-0rderpre55ure der1vat1ve5 0f the e1a5t1c m0du11 0f the 50d1um ha11de5
7a61e 4. 5ec0nd 0rder pre55ure der1vat1ve5 0f the e1a5t1c m0du11 0f the 50d1um ha11de5
F1r5t pre55ure der1vat1ve 0f
NaF
NaC1
Na8r
Na1
5ec0nd pre55ure der1vat1ve 0f
NaF
NaC1 Na8r (Un1t5 0f 6Pa -1)
Na1
cj1 c44 c12
11.29 0.09 1.81 4.97 5.03
11.48 0.36 1.82 5.04 5.10
11.80 0.43 2.03 5.29 5.33
11.89 0.55 1.92 5.25 5.46
c11 c44 c12
--0.72 -- 0.02 + 0.08 -- 0.18 -- 0.12
--0.94 -- 0.06 -- 0.04 -- 0.34 -- 0.14
------
85 87
85 87
--0.62 -- 0.08 -- 0.34 -- 0.42 -- 0.20
1.04 0.10 0.34 0.58 0.26
7a61e 3. C0mpar150n 0f exper1menta1 and m0de1 va1ue5 0f the f1r5t pre55ure der1vat1ve 0f the 6u1k m0du1u5 f0r 50d1um ha11de5
7a61e 5. C0mpar150n 0f exper1menta1 and m0de1 va1ue5 0f the 5ec0nd 0rder pre55ure der1vat1ve 0f the 6u1k m0du1u5 f0r 50d1um ha11de5
d87
d287
dP
NaF NaC1 Na8r Na1
dP 2 (6Pa-1)
P0wer
Exp0nent1a1
Exper1ment
4.90 5.24 5.40 5.60
4.46 4.82 4.99 5.19
5.03 5.05 5.33 5.46
f0r the f1r5t 0rder pre55ure der1vat1ve5 are 115ted 1n 7a61e 2. Fr0m the 5pread 0f re5u1t51n repeated run5 0n the 5ame cry5ta1 1t 15 c0nc1uded that the prec1510n 1n the va1ue5 0f 7a61e 2 15 a60ut 2%. Data pu6115hed 6y var10u5 auth0r5 u51n9 d1fferent techn14ue5 5h0w a 5pread 0f va1ue5 f0r d 8 7 / d P 0f 50me 20% w1th a mean f0r 50d1um ch10r1de, f0r examp1e, 0f 5.5 w1th the u1tra50n1c data c1u5tered m0re c105e1y at 5.3. H0wever, the Decker [ 12] e4uat10n 0f 5tate f0r NaC1 wh1ch ha5 6een exten51ve1y u5ed f0r pre55ure ca116rat10n f0r a num6er 0f year5 1nd1cate5 a va1ue 0f 4.9. Va1ue5 f0r d87/dP ca1cu1ated f0r the tw0 m0de15 fr0m the data 0f 7a61e 1 are c0ntra5ted w1th the f1tted va1ue5 fr0m the pre5ent 5tudy 1n 7a61e 3. A5 ha5 6een p01nted 0ut [6] exper1ment fav0ur5 the p0wer 1aw repre5entat10n. 7he 5ec0nd 0rder der1vat1ve5 are 91ven 1n 7a61e 4 and are expected t0 6e accurate t0 n0 6etter than 10% due t0 the 5ma11 065erved n0n-11near1ty and the c0n5e4uent d1ff1cu1ty 0f accurate f1tt1n9. 1t 15 un11ke1y that the p051t1ve 519n f0r d2c12/dP2 f0r NaF 15 rea1 6ut pr06a61y 1nd1cate5 51mp1y a va1ue near 2er0 1n ma9n1tude. V15ua11y there 15 n0 d0u6t that curvature 1n 9enera1 d0e5 ex15t, that 1t 15 6199er f0r 10n91tud1na1 m0de5 than f0r 5hear m0de5 and that 1t 15 pr09re551ve1y 1ar9er thr0u9h the 5er1e5 fr0m the f1u0r1de t0 the 10d1de. 7a61e 5 then 115t5the pred1ct10n5 0f the m0de15 and exper1menta1 va1ue5. 1t 15 unf0rtunate that the m0de15 91ve va1ue5 wh1ch
NaF NaC1 Na8r Na1
P0wer
Exp0nent1a1
Exper1ment
-----
-----
-----
0.10 0.22 0.27 0.38
0.11 0.24 0.30 0.43
0.12 0.14 0.20 0.26
are t00 c105e t09ether t0 d15t1n9u15h exper1menta11y. 0ther pu6115hed w0rk 15 very 5par5e w1th 0n1y 0ne va1ue [13] f0r NaC1 appear1n9 0 f - - 1.0 6Pa -1. 7h15 wa5 der1ved fr0m 1en9th chan9e mea5urement5 0f a 1 m 10n9 cry5ta1 5u6jected t0 pre55ure5 up t0 0.7 6Pa. H0wever, the a550c1ated va1ue 0f d87,/dP 0f 5.7 15 h19h c0mpared w1th m05t 0ther pu6115hed w0rk. An0ther va1ue 0 f - - 0.9 6Pa -1 15 rep0rted1y [2] der1va61e fr0m the u1tra50n1c data 0f 5pet21er [14]. 7he pre5ent va1ue 15 c0n51dera61y 1e55 than the5e 6ut 1t 5h0u1d 6e empha512ed that c0n51dera61e err0r5 ex15t 0n a11 rep0rted data 0f the5e 5ec0nd 0rder der1vat1ve5.7he pre5ent re5u1t5 5h0w a rea50na61y we11-deve10ped pattern f0r the ha11de5 wh1ch 151n 900d 4ua11tat1ve and 9enera1 4uant1tat1ve a9reement w1th the pred1ct10n5 0f the 80rn m0de1. REFERENCE5
1. 2. 3. 4. 5. 6. 7.
D. La2aru5,Phy5. Rev. 7 6 , 5 4 5 (1949). A.L. Ru0ff • L.C. Chha611da5, H19h Pre55ure 5c1ence and 7echn0109y, 6th A1rapt C0nference Pr0ceed1n95, p. 19. P1enum Pre55 (1979). J. Franke1, F.J. R1ch • C.6. H0man, J. 6e0phy5. Re5. 81, 6357 (1976). C. W0n9 • D.E. 5chue1e, J. Phy5. Chem. 5011d5 29, 1309 (1968). 5.N. Va1dya • 6.C. Kennedy, J. App1. Phy5. 32, 961 (1971). R.W. R06ert5 • C.5.5m1th, J. Phy5. Chem. 5011d5 3 1 , 6 1 9 (1970). 5. Hart,J. Phy5. D1, 1277 (1968).
164
PRE55URE VAR1A710N 0F 7HE ELA571C M0DUL1 0F 7HE 50D1UM HAL1DE5
C.W.F.7. P15t0r1u5• J.8. C1ark,H19h 7emp. H19h Pre55. 1,561 (1969). 9. 5. Hart,5. Afr. J. Phy5. 4,103 (1981). 10. R.F.5. Hearm0n,App11ed An150tr0p1c E1a5t1c1ty, 8.
11.
0xf0rd Un1ver51tyPre55, p. 82 (1961). D.P. Dandekar,J. App1. Phy5. 41,667 (1970).
12. 13. 14.
V01. 46, N0. 2
D.L. 1)ecker, J. App1. Phy~ 42, 3239 (1971). L.C. Chha611da5 • A.L. Ru0ff, J. AppL Phy5~ 47, 4182 (1976). H. 5pet21er, C.6.5amm15 • R.J. 0•C0nne11,2 Phy5. Chem. 5011d533, 1727 (1972).
003 8 - 1 0 9 8/83/140161 --04503.00/0 Per9am0n Pre55 Ltd.
PRE55URE VAR1A710N 0 F 7HE ELA571C M0DUL1 0 F 7HE 50D1UM HAL1DE5 5. Hart and P.H. 6reenw00d* Nat10na1 Phy51ca1 Re5earch La60rat0ry, P . 0 . 8 0 x 395, Pret0r1a, 50uth Afr1ca
(Rece1ved 16 Au9u5t 1982 6y C. W. McC0m61e) An u1tra50n1c 5tudy ha5 6een made 0f the var1at10n 0f the e1a5t1c m0du11 0f 51n91e cry5ta15 0f the 50d1um ha11de5 t0 1.5 6Pa. A 4uadrat1c f1t 15 re4u1red f0r the data. 7he 5y5temat1c5 and 1n part1cu1ar the 5ec0nd der1vat1ve 0f the 6u1k m0du1u5 ha5 6een f0und t0 6e 1n 9enera1 a9reement w1th pred1ct10n5 0f the 80rn m0de1 0f the e4uat10n 0f 5tate.
7HE EVALUA710N 0f the pre55ure der1vat1ve5 0f the e1a5t1c c0n5tant5 0f 51n91e cry5ta15 6y u1tra50n1c techn14ue5 ena61e5 the P - V e4uat10n 0f 5tate t0 6e determ1ned t0 h19her accuracy than 6y 0ther techn14ue5 5uch a5 X-ray 0r v01umetr1c and hence p1ay5 an 1mp0rtant r01e 1n the deve10pment 0f e4uat10n 0f 5tate m0de15.7h15 w0rk wa5 p10neered 6y La2aru5 [1 ] and the meth0d ha5 6een put t0 1ncrea51n9 u5e dur1n9 the pa5t year5. Ru0ff and Chha611da5 [2] have p01nted 0ut that the a9reement 6etween exper1ment and the0ry 15 n0t yet 5at15fact0ry and that the exper1ment5 5h0u1d 6e carr1ed 0ut at h19her pre55ure 50 that, f0r examp1e, n0t 0n1y the f1r5t 6ut the 5ec0nd der1vat1ve5 0f e1a5t1c m0du11 c0u1d 6e 4uant1f1ed. 1t 15 kn0wn that the 6u1k m0du1u5 a5 a funct10n 0f pre55ure f0r the a1ka11ha11de5 [3] and 0ther mater1a15 [4] d0e5 5h0w curvature and attempt5 have 6een made t0 determ1ne th15 curvature 1n 1501ated 1n5tance5 [5 ]. A 5y5temat1c 5tudy ha5 theref0re 6een carr1ed 0ut 0n the 50d1um ha11de5 t0 1.5 6Pa t0 determ1ne, 1n part1cu1ar, the f1r5t and 5ec0nd pre55ure der1vat1ve5 0f the e1a5t1c m0du11 and hence t0 c0mpare the va11d1ty 0f the0ret1ca1 e4uat10n5 0f 5tate. An exten51ve u1tra50n1c 1nve5t19at10n 0n the 50d1um and p0ta551um ha11de5 wa5 carr1ed 0ut 6y R06ert5 and 5m1th [6] 6ut 0n1y t0 0.1 6Pa and a1th0u9h n0n-11near1ty 0f 50me m0du11 wa5 n0ted, 1t wa5 n0t 4uant1f1ed. 7he1r 1nve5t19at10n 5h0wed pr0n0unced 5y5temat1c5 acr055 the 5er1e5 0f ha11de5 w1th the f1r5t 0rder der1vat1ve5 5h0w1n9 an 1ncrea5e fr0m the f1u0r1de t0 the 10d1de. 1t wa5 a150 5h0wn that the m0re prec15e techn14ue rem0ved 50me an0ma11e5 wh1ch had prev10u51y ex15ted 1n the 5er1e5 when u51n9 6u1k c0mpre5510n data. F0r the pre5ent 5tudy 51n91e cry5ta15 0f the re4u1red ha11de5 were 9r0wn fr0m the me1t 6y the 5t0ck6ar9er meth0d and were 0f 51m11ar 4ua11ty t0 th05e rep0rted prev10u51y [7]. 5pec1men5 0f the c0rrect 0r1entat10n were * Pre5ent addre55: Nuc1ear Deve10pment C0rp0rat10n 0f 5A, Pe11nda6a, Pret0r1a, 50uth Afr1ca.
then e1ther c1eaved 0r 5awn 6y a d1am0nd-1mpre9nated w1re 5aw and 5u65e4uent1y 1apped and p0115hed w1th 0pp051te face5 p1ane and para11e1 t0 0.01 mm. Laue 6ack-ref1ect10n X-ray ph0t09raph5 were u5ed t0 en5ure that c0rrect 0r1entat10n wa5 ma1nta1ned t0 6etter than 0.5 °, 1.e. t0 the prec1510n ava11a61e 1n r0ut1ne Laue u5e. 7he 5pec1men5 were phy51ca11y 1 cm 10n9, 1 cm d1ameter cy11nder5 w1th the u1tra50und pr0pa9at10n a10n9 the ax15. 10 MH2 X and Y cut 4uart2 tran5ducer5 were wrun9 0nt0 the 5pec1men5 u51n9 a m1xture 0f 91ycer1ne and ptha11c anhydr1de a5 60nd1n9 a9ent (th15 91ue d0e5 n0t d15501ve 1n the pre55ure tran5m1tt1n9 f1u1d u5ed). 7he accurate determ1nat10n 0f wave tran51t t1me5 wa5 carr1ed 0ut u51n9 the pu15e ech0 0ver1ap meth0d w1th c0mmerc1a1 Panametr1c5 pu151n9 e4u1pment. 7he pre55ure 5y5tem u5ed wa5 a p15t0n-cy11nder apparatu5 [8] w1th 12.7 mm d1ameter 60re. 7he pre55ure p1ate wa5 end-10aded w1th 100 t0n5 t0 91ve further 5upp0rt1n9 10n91tud1na1 c0mpre5510n t0 the tun95ten car61de cy11nder and the car61de p15t0n wa5 dr1ven 1n 6y a 5ec0nd ram. 7he ce11 5y5tem 15 5h0wn 1n F19.1 and u5e5 tw0 5ea11n9 p1u95, 0ne 61ank and 0ne w1th e1ectr1c 1ead-thr0u9h5, t0 c0nta1n the pre55ure tran5m1tt1n9 f1u1d wh1ch, 1n th15 ca5e, wa5 a 50:50 m1xture 0f n-pentane and 150pentane. A man9an1n re515tance 9au9e wa5 1nc0rp0rated 1n the p1u9 a55em61y. A5 a1way5 w1th 5uch h19h pre55ure 5y5tem5 the determ1nat10n 0f pre55ure 15 c0mp11cated 6y fr1ct10na1 effect5 1n the ma1n ram and, t0 a 1e55er extent, 1n the 5ea11n9 p1u95. Here the unc0rrected pre55ure wa5 determ1ned 6y the 9e0metr1ca1 ma9n1f1cat10n 0f area5 (ram t0 p15t0n) and the ram hydrau11c pre55ure. 7he man9an1n 9au9e (and/0r a he19ht 9au9e t0 m0n1t0r p15t0n penetrat10n) wa5 then u5ed t0 c0rrect f0r fr1ct10n a55um1n9 1t t0 6e 5ymmetr1c. 7he up5tr0ke re515tance va1ue5 were a1way5 h19h and the d0wn5tr0ke va1ue5 10w due t0 fr1ct10n and the ar1thmet1c mean 9ave the c0rrected and true ce11 pre55ure. 7h15 meth0d ha5 6een checked a9a1n5t the mercury free21n9 p01nt [9] and f0und accurate t0 w1th1n 1%.
161
162
PRESSURE VARIATION
SPECIMEN AND TRANSCUCER
Table 1. Power law parameters
ELECTRICAL LEAD-THRWGH
HARD STEELS
’
Vol. 46, No. 2
OF THE ELASTIC MODULI OF THE SODIUM HALIDES
TUNGSFEN
CARBIDE
A (1 O-i9 J mol-‘)
n
2.14 1.5 1 1.34 1.15
7.68 8.73 9.21 9.81
NaF NaCl NaBr NaI
ti TUNGSTEN CARBIDE PISTDN
Fig. 1. High pressure cell system. The acoustic modes investigated were the usual ones used for cubic crystals, i.e. longitudinal and shear in the [ 10 0] and [ 1 lo] directions and the standard relationships relating velocities to the moduli were used [lo]. It is normal practice to use simply the wave transit time, t = l/f, through the specimen of ambient length Zeeither in the form of the “natural velocity” W = lo/t or the “natural modulus” poW2 (p,, = the ambient condition density) and to correct for length change under pressure by differentiation of the true modulus M = pv2 = pl 2f ‘. The required relationship for the first order derivative is dn!f z-=
4PoW2) M -+jgdP
atP=O
T
where BT is the isothermal bulk modulus. The first term is the slope of the experimental results and the second is the correction for length change. It is of passing interest that the experimental curve of the natural modulus for [ 10 0] shear in NaF shows a negative slope but the correction term lifts this to the positive value of 0.09 shown in Table 2 and this mode does not in reality soften. Similarly the second order equation is given by d2M dp2
d2(p,W2)+ = dP2
2 d(poW’)+ 3B,y
M -___ 9B;
where A, n, D and d are empirical fitting parameters. By noting that the bulk modulus is the volume derivative of pressure and pressure is the volume derivative of the free energy, the relationships An=D$
and
+
EidV)
where the first term is the Madelung attractive energy and the second term is the repulsive energy. The power and exponential forms of ER are in common use, i.e.
n+l
may be derived. Values of the parameters have been evaluated from thermodynamic and mechanical data by other workers [6] and are listed in Table 1. Further differentiation of the bulk modulus equation gives
=-
(power law)
(exponential x=--j-
law)
do
and
M dB, 3B; dp
also evaluated at P = 0. These two relationships allow the pressure variations of the moduli to be obtained at the ambient condition limit but do not give the values of the moduli at high pressure. Iterative methods to do this have been developed and that of Dandekar [ 1 l] has been adopted in the present study. The Born model for ionic solids expresses the lattice energy in volume dependent terms and the energy, E, can be written as [6]
+=
= -4;+3) P=O
(power law) T
=
- 5X3 + 16X2 + 2X - 32 9(X - 2)2B, (exponential
’
law)
It is perhaps noteworthy at this stage that both laws predict a negative value of the second derivative, i.e. that the values of the modulus with respect to pressure will be concave to the pressure axis which is what is observed in practice. Experimental results for the second order elastic constants are in very good agreement with other ultrasonic data published in recent years and are not reproduced here. The pressure variation of the moduli showed curvature and was fitted with a quadratic function from which the derivates were obtained. The results
V01.46, N0.2
PRE55URE VAR1A710N 0 F 7HE ELA571C M0DUL1 0 F 7HE 50D1UM HAL1DE5
163
7a61e 2. F1r5t-0rderpre55ure der1vat1ve5 0f the e1a5t1c m0du11 0f the 50d1um ha11de5
7a61e 4. 5ec0nd 0rder pre55ure der1vat1ve5 0f the e1a5t1c m0du11 0f the 50d1um ha11de5
F1r5t pre55ure der1vat1ve 0f
NaF
NaC1
Na8r
Na1
5ec0nd pre55ure der1vat1ve 0f
NaF
NaC1 Na8r (Un1t5 0f 6Pa -1)
Na1
cj1 c44 c12
11.29 0.09 1.81 4.97 5.03
11.48 0.36 1.82 5.04 5.10
11.80 0.43 2.03 5.29 5.33
11.89 0.55 1.92 5.25 5.46
c11 c44 c12
--0.72 -- 0.02 + 0.08 -- 0.18 -- 0.12
--0.94 -- 0.06 -- 0.04 -- 0.34 -- 0.14
------
85 87
85 87
--0.62 -- 0.08 -- 0.34 -- 0.42 -- 0.20
1.04 0.10 0.34 0.58 0.26
7a61e 3. C0mpar150n 0f exper1menta1 and m0de1 va1ue5 0f the f1r5t pre55ure der1vat1ve 0f the 6u1k m0du1u5 f0r 50d1um ha11de5
7a61e 5. C0mpar150n 0f exper1menta1 and m0de1 va1ue5 0f the 5ec0nd 0rder pre55ure der1vat1ve 0f the 6u1k m0du1u5 f0r 50d1um ha11de5
d87
d287
dP
NaF NaC1 Na8r Na1
dP 2 (6Pa-1)
P0wer
Exp0nent1a1
Exper1ment
4.90 5.24 5.40 5.60
4.46 4.82 4.99 5.19
5.03 5.05 5.33 5.46
f0r the f1r5t 0rder pre55ure der1vat1ve5 are 115ted 1n 7a61e 2. Fr0m the 5pread 0f re5u1t51n repeated run5 0n the 5ame cry5ta1 1t 15 c0nc1uded that the prec1510n 1n the va1ue5 0f 7a61e 2 15 a60ut 2%. Data pu6115hed 6y var10u5 auth0r5 u51n9 d1fferent techn14ue5 5h0w a 5pread 0f va1ue5 f0r d 8 7 / d P 0f 50me 20% w1th a mean f0r 50d1um ch10r1de, f0r examp1e, 0f 5.5 w1th the u1tra50n1c data c1u5tered m0re c105e1y at 5.3. H0wever, the Decker [ 12] e4uat10n 0f 5tate f0r NaC1 wh1ch ha5 6een exten51ve1y u5ed f0r pre55ure ca116rat10n f0r a num6er 0f year5 1nd1cate5 a va1ue 0f 4.9. Va1ue5 f0r d87/dP ca1cu1ated f0r the tw0 m0de15 fr0m the data 0f 7a61e 1 are c0ntra5ted w1th the f1tted va1ue5 fr0m the pre5ent 5tudy 1n 7a61e 3. A5 ha5 6een p01nted 0ut [6] exper1ment fav0ur5 the p0wer 1aw repre5entat10n. 7he 5ec0nd 0rder der1vat1ve5 are 91ven 1n 7a61e 4 and are expected t0 6e accurate t0 n0 6etter than 10% due t0 the 5ma11 065erved n0n-11near1ty and the c0n5e4uent d1ff1cu1ty 0f accurate f1tt1n9. 1t 15 un11ke1y that the p051t1ve 519n f0r d2c12/dP2 f0r NaF 15 rea1 6ut pr06a61y 1nd1cate5 51mp1y a va1ue near 2er0 1n ma9n1tude. V15ua11y there 15 n0 d0u6t that curvature 1n 9enera1 d0e5 ex15t, that 1t 15 6199er f0r 10n91tud1na1 m0de5 than f0r 5hear m0de5 and that 1t 15 pr09re551ve1y 1ar9er thr0u9h the 5er1e5 fr0m the f1u0r1de t0 the 10d1de. 7a61e 5 then 115t5the pred1ct10n5 0f the m0de15 and exper1menta1 va1ue5. 1t 15 unf0rtunate that the m0de15 91ve va1ue5 wh1ch
NaF NaC1 Na8r Na1
P0wer
Exp0nent1a1
Exper1ment
-----
-----
-----
0.10 0.22 0.27 0.38
0.11 0.24 0.30 0.43
0.12 0.14 0.20 0.26
are t00 c105e t09ether t0 d15t1n9u15h exper1menta11y. 0ther pu6115hed w0rk 15 very 5par5e w1th 0n1y 0ne va1ue [13] f0r NaC1 appear1n9 0 f - - 1.0 6Pa -1. 7h15 wa5 der1ved fr0m 1en9th chan9e mea5urement5 0f a 1 m 10n9 cry5ta1 5u6jected t0 pre55ure5 up t0 0.7 6Pa. H0wever, the a550c1ated va1ue 0f d87,/dP 0f 5.7 15 h19h c0mpared w1th m05t 0ther pu6115hed w0rk. An0ther va1ue 0 f - - 0.9 6Pa -1 15 rep0rted1y [2] der1va61e fr0m the u1tra50n1c data 0f 5pet21er [14]. 7he pre5ent va1ue 15 c0n51dera61y 1e55 than the5e 6ut 1t 5h0u1d 6e empha512ed that c0n51dera61e err0r5 ex15t 0n a11 rep0rted data 0f the5e 5ec0nd 0rder der1vat1ve5.7he pre5ent re5u1t5 5h0w a rea50na61y we11-deve10ped pattern f0r the ha11de5 wh1ch 151n 900d 4ua11tat1ve and 9enera1 4uant1tat1ve a9reement w1th the pred1ct10n5 0f the 80rn m0de1. REFERENCE5
1. 2. 3. 4. 5. 6. 7.
D. La2aru5,Phy5. Rev. 7 6 , 5 4 5 (1949). A.L. Ru0ff • L.C. Chha611da5, H19h Pre55ure 5c1ence and 7echn0109y, 6th A1rapt C0nference Pr0ceed1n95, p. 19. P1enum Pre55 (1979). J. Franke1, F.J. R1ch • C.6. H0man, J. 6e0phy5. Re5. 81, 6357 (1976). C. W0n9 • D.E. 5chue1e, J. Phy5. Chem. 5011d5 29, 1309 (1968). 5.N. Va1dya • 6.C. Kennedy, J. App1. Phy5. 32, 961 (1971). R.W. R06ert5 • C.5.5m1th, J. Phy5. Chem. 5011d5 3 1 , 6 1 9 (1970). 5. Hart,J. Phy5. D1, 1277 (1968).
164
PRE55URE VAR1A710N 0F 7HE ELA571C M0DUL1 0F 7HE 50D1UM HAL1DE5
C.W.F.7. P15t0r1u5• J.8. C1ark,H19h 7emp. H19h Pre55. 1,561 (1969). 9. 5. Hart,5. Afr. J. Phy5. 4,103 (1981). 10. R.F.5. Hearm0n,App11ed An150tr0p1c E1a5t1c1ty, 8.
11.
0xf0rd Un1ver51tyPre55, p. 82 (1961). D.P. Dandekar,J. App1. Phy5. 41,667 (1970).
12. 13. 14.
V01. 46, N0. 2
D.L. 1)ecker, J. App1. Phy~ 42, 3239 (1971). L.C. Chha611da5 • A.L. Ru0ff, J. AppL Phy5~ 47, 4182 (1976). H. 5pet21er, C.6.5amm15 • R.J. 0•C0nne11,2 Phy5. Chem. 5011d533, 1727 (1972).