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The bulk modulus and its pressure derivative for 18 metals
J. Phys. Chem. Solids Vol. 46, No. 8, pp. 925-921, Printed in the U.S.A.
THE BULK
1985
0022-3697/85 $3.00 + .OO 0 1985 Per@mon Press Ltd.
MODULUS AND ITS PRESSURE FOR 18 METALS
DERIVATIVE
L. GERWARD Laboratory of Applied Physics III, Technical University of Denmark, DK-2800 Lyngby, Denmark (Received 5 November 1984; accepted 17 January 1985)
Abstract-Volume compression data of Vaidya and Kennedy [J. Whys.C/rem. Solids 31, 2329 (1970)] have been used to recalculate the isothermal bulk moduli and their pressure derivatives and to compare various equations of state.
1. INTRODUCTION
and the Birch equation
An equation of state for a solid makes it possible to describe its pressure-volume relationship to very high pressures using only a few parameters, for example the isothermal bulk modulus at ambient temperature and its pressure derivative. Vaidya and Kennedy [l] determined these parameters for 18 metals in a series of careful volume compression measurements up to 4.5 GPa. However, in a few cases their values for the pressure derivative of the bulk modulus deviate considerably from the expected value around 4. One of the purposes of the present work has been to recalculate the equations of state using powerful computing procedures in order to find whether the observed discrepancies have been due to computational problems or can be explained by experimental difficulties. Another purpose has been to use the high precision compression data to compare various equations of state in wide use.
P = jB,,[( VO/V)“’ - ( VO/V)s’3] X (1 + ;(Bb - 4)[(VO/V)2’3 - 11).
(4)
2. EQUATIONS OF STATE
The experimental V(P) data can be fitted to eqn (1) using polynomial regression, and to eqns (3) and (4) using a method of nonlinear least squares. Computer programmes for these purposes have been developed on the basis of the statistical analysis system (SAS). They have been successfully applied in current highpressure X-ray diffraction work using diamond anvil cells and synchrotron radiation. Table 1 shows the resulting values of BO and Bb when the data of Vaidya and Kennedy [l] are fitted to eqns (I), (3) and (4). Generally a second-order polynomial is sufficient in eqn (l), higher-order terms being statistically insignificant. A third-order polynomial is necessary only for the aluminium alloy Al 606 1T6 and Ba, Ca, In and La.
Anderson [2] derived the equation of a solid of the form
3. DISCUSSION
v/v, = 1 - (P/B,) + m(P/&)* - n(P/B#
+ --*
(1)
with m = f(1 + Bb), n = d[l + 3Bb + 2(&)* - B&l, etc. where & is the isothermal bulk modulus at ambient pressure, Bb and Bz its first- and secondorder pressure derivative, respectively, and V is the volume at pressure P. The Mumaghan [3] and Birch [4] first-order equations are both based on the assumption that the isothermal bulk modulus B varies linearly with pressure B = B,, + Bbp. The Mumaghan
equation
(2)
has the form
P = (Bo/Bb)[( V,/Vf’
- l]
(3)
The BO and Bb values from the polynomial fits (Table 1) agree perfectly with the corresponding values of Vaidya and Kennedy [ 11. The average percentage deviation for Bo is only 0.08% and the average deviation for Bb 0.03 units. Vaidya and Kennedy [I] have also calculated BO and Bb from the Mumaghan eqn (3). Generally, the agreement with the corresponding values of the present work is good as can be seen in Table 2. A striking deviation is found for Cu. In their original paper Vaidya and Kennedy [I] noted that they could not understand their unusually small value 0.43 for Bb. However, it is obvious from Table 1 that the value 4.79 found in the present work is very reasonable. Therefore, we conclude that the discrepancy for Cu in the original work of Vaidya and Kennedy [l] has been caused by some computational problems in the nonlinear regression analysis.
925
926
L. GERWARD
Table 1. Compressibility parameters calculated in the present work using tbe experimental data of Vaidya and Kennedy [ 1I. The bulk modulus B0 is given in units of GPa (multiply by 10 for corresponding value in kbar). The uncertainties, given in parentheses, are tbe standard deviations of tbe least squares fit to tbe equation of state Polynomial fit Element &
Al606lT6 Al Ba Bi(I) % cu Fe In La MO Ni Pb Sn Ta Tl(I) Zn
&I
BO GW
110.77(17) 81.62(21) 78.83(10) 9.46(O) 30.94(4) 18.26(2) 47.35(3) 150.88(28) 171.20(28) 38.86(6) 24.63(2) 250.9(1.1) 180.39(28) 42.38(3) 55.12(6) 201.83(49) 37.45(2) 6 1.46(4)
4.70( 10) 10.55(29) 3.17(5) 1.72(O) 4.1 l(4) 2.62(2) 4.07(2) 4.07(2) 7.67(2) 5.12(8) 2.84(3) 14.21(64) 16.48( 14) 3.74(2) 2.70(3) 3.82(25) 3.15(2) 3.29(2)
109.79(31) 87.8( 1.2) 78.12(11) 9.50(12) 29.84(35) 18.59(13) 45.05(48) 150.37(30) 169.71(38) 39.09( 13) 24.60(5) 247.7(1.1) 173.2(1.4) 40.1 l(44) 54.27( 17) 201.51(50) 36.37(25) 60.41(21)
Table 2. Difference between compressibility parameters Bi“ and (Bo)vk calculated by Vaidya and Kennedy [l] and the corresponding parameters B. and B. calculated in the present work. All parameters have been calculated from the Mumaghan equation
Ag Al606 1T6 Al Ba Bi(1) Ca Cd cu Fe In La :ic Pb Sn Ta TW) Zn
Wa)
Bb
109.69(33) 87.7(1.2) 78.06( 11) 9.14(6) 29.70(42) 18.38(10) 44.72(59) 150.33(31) 169.40(43) 38.91(11) 24.47(3) 246.5( 1.2) 170.2(2.2) 39.80(54) 54.20( 19) 201.48(50) 36.27(29) 60.32(23)
6.23(20) 3.18(67) 4.25(7) 2.78(4) 7.22(53) 2.88(6) 7.89(46) 4.86(18) 10.25(28) 5.54(8) 3.35(2) 20.78(85) 32.5( 1.9) 7.48(43) 4.00( 11) 4.32(28) 5.25(24) 4.89( 15)
Bb
BO @Pa)
The deviations found for Ag in Table 2 are more difficult to explain because. the Bb values of Vaidya and Kennedy [1] and the present work are lying on both sides of the expected value around 4 (2.48 and 6.05, respectively). Other notable deviations in Table 2 are found for MO and Ni. The Bb values for these two elements are anomalously large (Table 1) and this may he an indication of some systematic error in the expetimental data for MO and Ni.
Element
Birch equation
Mumagban equation B.
6.05(18) 3.10(68) 4.15(6) 2.14(8) 6.75(39) 2.53(8) 7.22(31) 4.79( 17) 9.75(22) 5.24(8) 3.1 l(3) 18.81(67) 26.49(92) 6.85(29) 3.87(10) 4.28(28) 5.03(20) 4.74(12)
The Mumaghan
and Birch equations
are widely
used in the literature, and therefore it should be of interest to compare them using the same experimental V(p) data. Table 3 shows that the two equations are in a close agreement for most elements in the present work. The mean percentage difference for B. is 0.6% and the mean difference for Bb is 0.3 units. Notable exceptions are again MO and Ni, which show larger deviations, in particular for Bb. This may be another indication of experimental problems for these two elements.
Table 3. Difference between compressibility parameters calculated in the present work using the Birch equation (Bt and (Bb)‘) and the Mumagban equation (Bf and (Bb)“), respectively
100. (BiK - Eo)/B, %
(Bb)“k - @I
Element
lOO.(@ - Bp)/BF %
10.1 -0.2 -0.3 -0.7 -0.5 0.5 -0.6 8.1 3.6 -0.03 -0.5 7.4 10.0 -0.3 1.2 2.0 0.6 -1.0
-3.57 0.11 0.11 0.05 -0.005 -0.0 1 0.08 -4.36 -2.08 0.003 0.04 -6.85 -6.33 -0.09 -0.22 -1.52 -0.24 0.14
Ag Al606 IT6 Al Ba Bi(1) Ca Cd cu Fe In La MO Ni Pb Sn Ta Tl(1) Zn
-0.10 -0.05 -0.08 -3.7 -0.5 -1.1 -0.7 -0.03 -0.2 -0.5 -0.6 -0.5 -1.7 -0.8 -0.14 -0.01 -0.3 -0.15
(W
- (BY 0.18 0.08 0.10 0.65 0.48 0.35 0.67 0.07 0.50 0.30 0.23 1.97 5.97 0.63 0.12 0.04 0.22 0.15
Bulk modulus and its pressure derivative
Table 3 also shows that the two equations of state di#er in a systematic way, the Birch equation resulting in a slightly smaller value of B0 and a correspondingly larger value of Bb as compared with the Mumaghan equation. In conclusion. it has been found that the exoerimental data generally is well d&bed by the Murnaghan equation as well as the Biih equation, although with some systematic differences with respect to Bo and &. Large differences, in particularly for
927
B6, between the two equations may be an indication of a systematic error in the experimental data.
RETESENCES
I. Vaidya S. N. and Kennedy G. C., J. Phys. Chem. Soids 31,2329 (1970). 2. Anderson 0. I., J. Phys. Chem. Sol& 27, 547 (1966). 3. Murnaghan F. D., Proc. Natl.Acad. Sci. 30,244 ( 1944).
4. Birch F., J. Geophys.Res. 57,227 (1952); J. Appl. Phys.
9,279 (1938).
THE BULK
1985
0022-3697/85 $3.00 + .OO 0 1985 Per@mon Press Ltd.
MODULUS AND ITS PRESSURE FOR 18 METALS
DERIVATIVE
L. GERWARD Laboratory of Applied Physics III, Technical University of Denmark, DK-2800 Lyngby, Denmark (Received 5 November 1984; accepted 17 January 1985)
Abstract-Volume compression data of Vaidya and Kennedy [J. Whys.C/rem. Solids 31, 2329 (1970)] have been used to recalculate the isothermal bulk moduli and their pressure derivatives and to compare various equations of state.
1. INTRODUCTION
and the Birch equation
An equation of state for a solid makes it possible to describe its pressure-volume relationship to very high pressures using only a few parameters, for example the isothermal bulk modulus at ambient temperature and its pressure derivative. Vaidya and Kennedy [l] determined these parameters for 18 metals in a series of careful volume compression measurements up to 4.5 GPa. However, in a few cases their values for the pressure derivative of the bulk modulus deviate considerably from the expected value around 4. One of the purposes of the present work has been to recalculate the equations of state using powerful computing procedures in order to find whether the observed discrepancies have been due to computational problems or can be explained by experimental difficulties. Another purpose has been to use the high precision compression data to compare various equations of state in wide use.
P = jB,,[( VO/V)“’ - ( VO/V)s’3] X (1 + ;(Bb - 4)[(VO/V)2’3 - 11).
(4)
2. EQUATIONS OF STATE
The experimental V(P) data can be fitted to eqn (1) using polynomial regression, and to eqns (3) and (4) using a method of nonlinear least squares. Computer programmes for these purposes have been developed on the basis of the statistical analysis system (SAS). They have been successfully applied in current highpressure X-ray diffraction work using diamond anvil cells and synchrotron radiation. Table 1 shows the resulting values of BO and Bb when the data of Vaidya and Kennedy [l] are fitted to eqns (I), (3) and (4). Generally a second-order polynomial is sufficient in eqn (l), higher-order terms being statistically insignificant. A third-order polynomial is necessary only for the aluminium alloy Al 606 1T6 and Ba, Ca, In and La.
Anderson [2] derived the equation of a solid of the form
3. DISCUSSION
v/v, = 1 - (P/B,) + m(P/&)* - n(P/B#
+ --*
(1)
with m = f(1 + Bb), n = d[l + 3Bb + 2(&)* - B&l, etc. where & is the isothermal bulk modulus at ambient pressure, Bb and Bz its first- and secondorder pressure derivative, respectively, and V is the volume at pressure P. The Mumaghan [3] and Birch [4] first-order equations are both based on the assumption that the isothermal bulk modulus B varies linearly with pressure B = B,, + Bbp. The Mumaghan
equation
(2)
has the form
P = (Bo/Bb)[( V,/Vf’
- l]
(3)
The BO and Bb values from the polynomial fits (Table 1) agree perfectly with the corresponding values of Vaidya and Kennedy [ 11. The average percentage deviation for Bo is only 0.08% and the average deviation for Bb 0.03 units. Vaidya and Kennedy [I] have also calculated BO and Bb from the Mumaghan eqn (3). Generally, the agreement with the corresponding values of the present work is good as can be seen in Table 2. A striking deviation is found for Cu. In their original paper Vaidya and Kennedy [I] noted that they could not understand their unusually small value 0.43 for Bb. However, it is obvious from Table 1 that the value 4.79 found in the present work is very reasonable. Therefore, we conclude that the discrepancy for Cu in the original work of Vaidya and Kennedy [l] has been caused by some computational problems in the nonlinear regression analysis.
925
926
L. GERWARD
Table 1. Compressibility parameters calculated in the present work using tbe experimental data of Vaidya and Kennedy [ 1I. The bulk modulus B0 is given in units of GPa (multiply by 10 for corresponding value in kbar). The uncertainties, given in parentheses, are tbe standard deviations of tbe least squares fit to tbe equation of state Polynomial fit Element &
Al606lT6 Al Ba Bi(I) % cu Fe In La MO Ni Pb Sn Ta Tl(I) Zn
&I
BO GW
110.77(17) 81.62(21) 78.83(10) 9.46(O) 30.94(4) 18.26(2) 47.35(3) 150.88(28) 171.20(28) 38.86(6) 24.63(2) 250.9(1.1) 180.39(28) 42.38(3) 55.12(6) 201.83(49) 37.45(2) 6 1.46(4)
4.70( 10) 10.55(29) 3.17(5) 1.72(O) 4.1 l(4) 2.62(2) 4.07(2) 4.07(2) 7.67(2) 5.12(8) 2.84(3) 14.21(64) 16.48( 14) 3.74(2) 2.70(3) 3.82(25) 3.15(2) 3.29(2)
109.79(31) 87.8( 1.2) 78.12(11) 9.50(12) 29.84(35) 18.59(13) 45.05(48) 150.37(30) 169.71(38) 39.09( 13) 24.60(5) 247.7(1.1) 173.2(1.4) 40.1 l(44) 54.27( 17) 201.51(50) 36.37(25) 60.41(21)
Table 2. Difference between compressibility parameters Bi“ and (Bo)vk calculated by Vaidya and Kennedy [l] and the corresponding parameters B. and B. calculated in the present work. All parameters have been calculated from the Mumaghan equation
Ag Al606 1T6 Al Ba Bi(1) Ca Cd cu Fe In La :ic Pb Sn Ta TW) Zn
Wa)
Bb
109.69(33) 87.7(1.2) 78.06( 11) 9.14(6) 29.70(42) 18.38(10) 44.72(59) 150.33(31) 169.40(43) 38.91(11) 24.47(3) 246.5( 1.2) 170.2(2.2) 39.80(54) 54.20( 19) 201.48(50) 36.27(29) 60.32(23)
6.23(20) 3.18(67) 4.25(7) 2.78(4) 7.22(53) 2.88(6) 7.89(46) 4.86(18) 10.25(28) 5.54(8) 3.35(2) 20.78(85) 32.5( 1.9) 7.48(43) 4.00( 11) 4.32(28) 5.25(24) 4.89( 15)
Bb
BO @Pa)
The deviations found for Ag in Table 2 are more difficult to explain because. the Bb values of Vaidya and Kennedy [1] and the present work are lying on both sides of the expected value around 4 (2.48 and 6.05, respectively). Other notable deviations in Table 2 are found for MO and Ni. The Bb values for these two elements are anomalously large (Table 1) and this may he an indication of some systematic error in the expetimental data for MO and Ni.
Element
Birch equation
Mumagban equation B.
6.05(18) 3.10(68) 4.15(6) 2.14(8) 6.75(39) 2.53(8) 7.22(31) 4.79( 17) 9.75(22) 5.24(8) 3.1 l(3) 18.81(67) 26.49(92) 6.85(29) 3.87(10) 4.28(28) 5.03(20) 4.74(12)
The Mumaghan
and Birch equations
are widely
used in the literature, and therefore it should be of interest to compare them using the same experimental V(p) data. Table 3 shows that the two equations are in a close agreement for most elements in the present work. The mean percentage difference for B. is 0.6% and the mean difference for Bb is 0.3 units. Notable exceptions are again MO and Ni, which show larger deviations, in particular for Bb. This may be another indication of experimental problems for these two elements.
Table 3. Difference between compressibility parameters calculated in the present work using the Birch equation (Bt and (Bb)‘) and the Mumagban equation (Bf and (Bb)“), respectively
100. (BiK - Eo)/B, %
(Bb)“k - @I
Element
lOO.(@ - Bp)/BF %
10.1 -0.2 -0.3 -0.7 -0.5 0.5 -0.6 8.1 3.6 -0.03 -0.5 7.4 10.0 -0.3 1.2 2.0 0.6 -1.0
-3.57 0.11 0.11 0.05 -0.005 -0.0 1 0.08 -4.36 -2.08 0.003 0.04 -6.85 -6.33 -0.09 -0.22 -1.52 -0.24 0.14
Ag Al606 IT6 Al Ba Bi(1) Ca Cd cu Fe In La MO Ni Pb Sn Ta Tl(1) Zn
-0.10 -0.05 -0.08 -3.7 -0.5 -1.1 -0.7 -0.03 -0.2 -0.5 -0.6 -0.5 -1.7 -0.8 -0.14 -0.01 -0.3 -0.15
(W
- (BY 0.18 0.08 0.10 0.65 0.48 0.35 0.67 0.07 0.50 0.30 0.23 1.97 5.97 0.63 0.12 0.04 0.22 0.15
Bulk modulus and its pressure derivative
Table 3 also shows that the two equations of state di#er in a systematic way, the Birch equation resulting in a slightly smaller value of B0 and a correspondingly larger value of Bb as compared with the Mumaghan equation. In conclusion. it has been found that the exoerimental data generally is well d&bed by the Murnaghan equation as well as the Biih equation, although with some systematic differences with respect to Bo and &. Large differences, in particularly for
927
B6, between the two equations may be an indication of a systematic error in the experimental data.
RETESENCES
I. Vaidya S. N. and Kennedy G. C., J. Phys. Chem. Soids 31,2329 (1970). 2. Anderson 0. I., J. Phys. Chem. Sol& 27, 547 (1966). 3. Murnaghan F. D., Proc. Natl.Acad. Sci. 30,244 ( 1944).
4. Birch F., J. Geophys.Res. 57,227 (1952); J. Appl. Phys.
9,279 (1938).