Direct parameterization of the volume on pressure dependence by using
an inverted approximative Vinet equation of state
Supplementary material
Martin EtterI and Robert E. Dinnebier
I
I∗
Max Planck Institute for Solid State Research, Stuttgart, Germany
∗
e-mail: [email protected]
Contents
1 Inversions of the
1.1 Solution 1 . .
1.2 Solution 2 . .
1.3 Solution 3 . .
1.4 Solution 4 . .
third order Vinet EoS approximation
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2
2
4
6
8
2 Validity of the third order Vinet EoS approximation for transition metals, diamond and osmium 10
3 Input files and macros for TOPAS
11
4 As2 O5 as an example for high B0′ values
22
5 Graphical overview of P(V) and V(P)
24
5.1 Graphical overview of P(V) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
5.2 Graphical overview of V(P) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1
1
Inversions of the third order Vinet EoS approximation
In the following all 4 different solutions of the inversion of the third order Vinet EoS approximation can be found.
Some plus and minus signs are colored in red to mark the differences.
1.1
Solution 1
With the specified range of positive parameters in the publication this gives an wrong physical behavior, with
increasing volume with increasing pressure.
"
4uv 3 + 3uv 2 1 1 (−4uv 3 − 3uv 2 )2
f (P ) =
+
4uv 3
2 4
u2 v 6
9uv 2 − 6P + 6uv 3 + 6uv
1
−
+ 3 36u2 v 3 P + 21u2 v 4 P
uv 3
uv
+ 8u3 v 3 − 8P 3 + 24uv 3 P 2 + 24uvP 2 − 6u2 v 2 P + 36uv 2 P 2
1
+ − −72u4v 3 − 720u3v 3 P − 756u2vP 2 − 432u3v 4 P
v
− 3060u2v 3 P 2 − 3024P 3uv − 2664u2v 4 P 2 − 1296u2v 2 P 2
− 7920uv 2P 3 + 288P 3u − 825u2 v 5 P 2 − 9576P 3uv 3 + 1152vP 4
− 6000uv 4P 3 + 576v 2 P 4 + 1152P 4 − 2304uv 5P 3 + 192P 4v 3
1 21 2 3
6 3
+ 2 −(u2 v 2 ) − 6uv 2 P
− 512uv P
uv
2
3
+ 2P − 4uv P − 4uvP
uv 3 36u2 v 3 P + 21u2 v 4 P
+ 8u3 v 3 − 8P 3 + 24uv 3 P 2 + 24uvP 2 − 6u2 v 2 P + 36uv 2 P 2
1
+ − (−72u4 v 3 − 720u3v 3 P − 756u2vP 2 − 432u3v 4 P
v
− 3060u2v 3 P 2 − 3024P 3uv − 2664u2v 4 P 2 − 1296u2v 2 P 2
− 7920uv 2P 3 + 288P 3u − 825u2 v 5 P 2 − 9576P 3uv 3 + 1152vP 4
− 6000uv 4P 3 + 576v 2 P 4 + 1152P 4 − 2304uv 5P 3 + 192P 4v 3
# 12
1 2 13
−3uv 2 + 2P − 2uv 3 − 2uv
6 3 2
− 512uv P ) uv
−
uv 3
"
9uv 2 − 6P + 6uv 3 + 6uv
1 1 (−4uv 3 − 3uv 2 )2
−
+
2 2
u2 v 6
uv 3
1
− 3 36u2 v 3 P + 21u2 v 4 P + 8u3 v 3 − 8P 3 + 24uv 3 P 2
uv
1
+ 24uvP 2 − 6u2 v 2 P + 36uv 2 P 2 + − −72u4v 3
v
− 720u3 v 3 P − 756u2 vP 2 − 432u3 v 4 P − 3060u2v 3 P 2
− 3024P 3uv − 2664u2v 4 P 2 − 1296u2v 2 P 2 − 7920uv 2P 3
+ 288P 3 u − 825u2v 5 P 2 − 9576P 3uv 3 + 1152vP 4
− 6000uv 4P 3 + 576v 2 P 4 + 1152P 4 − 2304uv 5P 3 + 192P 4v 3
1 21 2 3
6 3
uv
− 2 −(u2 v 2 ) − 6uv 2 P + 2P 2
− 512uv P
3
− 4uv P − 4uvP
uv 3 · 36u2 v 3 P + 21u2 v 4 P
+ 8u3 v 3 − 8P 3 + 24uv 3 P 2 + 24uvP 2 − 6u2 v 2 P
2
1
+ 36uv 2 P 2 + − (−72u4 v 3 − 720u3v 3 P − 756u2vP 2
v
− 432u3v 4 P − 3060u2v 3 P 2 − 3024P 3uv − 2664u2v 4 P 2
− 1296u2v 2 P 2 − 7920uv 2 P 3 + 288P 3 u − 825u2v 5 P 2
− 9576P 3uv 3 + 1152vP 4 − 6000uv 4P 3 + 576v 2 P 4 + 1152P 4
13 1
− 2304uv 5P 3 + 192P 4 v 3 − 512uv 6P 3 ) 2 uv 2
−3uv 2 + 2P − 2uv 3 − 2uv
+
uv 3
2
(9uv − 6P + 6uv 3 + 6uv)(−4uv 3 − 3uv 2 )
+
u2 v 6
3
2
2(−4uv − 9uv − 12uv − 6u) (−4uv 3 − 3uv 2 )3
−
−
uv 3
4u3 v 9
3
2 2
2
3
1 (−4uv − 3uv )
9uv − 6P + 6uv + 6uv
−
2
6
4
u v
uv 3
1
+ 3 36u2 v 3 P + 21u2 v 4 P + 8u3 v 3 − 8P 3 + 24uv 3 P 2
uv
1
+ 24uvP 2 − 6u2 v 2 P + 36uv 2 P 2 + − (−72u4 v 3
v
− 720u3v 3 P − 756u2vP 2 − 432u3v 4 P − 3060u2v 3 P 2
− 3024P 3uv − 2664u2v 4 P 2 − 1296u2v 2 P 2 − 7920uv 2P 3
+ 288P 3u − 825u2 v 5 P 2 − 9576P 3uv 3 + 1152vP 4
− 6000uv 4P 3 + 576v 2 P 4 + 1152P 4 − 2304uv 5P 3 + 192P 4 v 3
31 1
− 512uv 6P 3 ) 2 uv 2 + 2 − (u2 v 2 ) − 6uv 2 P + 2P 2
. 3
uv 36u2 v 3 P + 21u2 v 4 P
− 4uv 3 P − 4uvP
+ 8u3 v 3 − 8P 3 + 24uv 3 P 2 + 24uvP 2 − 6u2 v 2 P
1
+ 36uv 2 P 2 + (− (−72u4 v 3 − 720u3v 3 P − 756u2vP 2
v
− 432u3v 4 P − 3060u2v 3 P 2 − 3024P 3uv − 2664u2v 4 P 2
− 1296u2v 2 P 2 − 7920uv 2 P 3 + 288P 3 u − 825u2v 5 P 2
− 9576P 3uv 3 + 1152vP 4 − 6000uv 4P 3 + 576v 2 P 4 + 1152P 4
1 1
− 2304uv 5P 3 + 192P 4 v 3 − 512uv 6P 3 )) 2 uv 2 3
1 # 21
−3uv 2 + 2P − 2uv 3 − 2uv 2
−
uv 3
3
(1)
1.2
Solution 2
This is the solution given in the publication with the specified range of positive values.
"
4uv 3 + 3uv 2 1 1 (−4uv 3 − 3uv 2 )2
+
f (P ) =
4uv 3
2 4
u2 v 6
9uv 2 − 6P + 6uv 3 + 6uv
1
−
36u2 v 3 P + 21u2 v 4 P
+
uv 3
uv 3
+ 8u3 v 3 − 8P 3 + 24uv 3 P 2 + 24uvP 2 − 6u2 v 2 P + 36uv 2 P 2
1
+ − −72u4v 3 − 720u3v 3 P − 756u2vP 2 − 432u3v 4 P
v
− 3060u2v 3 P 2 − 3024P 3uv − 2664u2v 4 P 2 − 1296u2v 2 P 2
− 7920uv 2P 3 + 288P 3u − 825u2 v 5 P 2 − 9576P 3uv 3 + 1152vP 4
− 6000uv 4P 3 + 576v 2 P 4 + 1152P 4 − 2304uv 5P 3 + 192P 4v 3
1 21 2 3
uv
+ 2 −(u2 v 2 ) − 6uv 2 P
− 512uv 6 P 3
+ 2P 2 − 4uv 3 P − 4uvP
uv 3 36u2 v 3 P + 21u2 v 4 P
+ 8u3 v 3 − 8P 3 + 24uv 3 P 2 + 24uvP 2 − 6u2 v 2 P + 36uv 2 P 2
1
+ − (−72u4 v 3 − 720u3v 3 P − 756u2vP 2 − 432u3v 4 P
v
− 3060u2v 3 P 2 − 3024P 3uv − 2664u2v 4 P 2 − 1296u2v 2 P 2
− 7920uv 2P 3 + 288P 3u − 825u2 v 5 P 2 − 9576P 3uv 3 + 1152vP 4
− 6000uv 4P 3 + 576v 2 P 4 + 1152P 4 − 2304uv 5P 3 + 192P 4v 3
# 12
1 2 13
−3uv 2 + 2P − 2uv 3 − 2uv
6 3 2
− 512uv P ) uv
−
uv 3
"
1 1 (−4uv 3 − 3uv 2 )2
9uv 2 − 6P + 6uv 3 + 6uv
−
−
2
6
2 2
u v
uv 3
1
− 3 36u2 v 3 P + 21u2 v 4 P + 8u3 v 3 − 8P 3 + 24uv 3 P 2
uv
1
+ 24uvP 2 − 6u2 v 2 P + 36uv 2 P 2 + − −72u4v 3
v
− 720u3 v 3 P − 756u2 vP 2 − 432u3 v 4 P − 3060u2v 3 P 2
− 3024P 3uv − 2664u2v 4 P 2 − 1296u2v 2 P 2 − 7920uv 2P 3
+ 288P 3 u − 825u2v 5 P 2 − 9576P 3uv 3 + 1152vP 4
− 6000uv 4P 3 + 576v 2 P 4 + 1152P 4 − 2304uv 5P 3 + 192P 4v 3
1 21 2 3
6 3
− 512uv P
uv
− 2 −(u2 v 2 ) − 6uv 2 P + 2P 2
− 4uv 3 P − 4uvP
uv 3 · 36u2 v 3 P + 21u2 v 4 P
+ 8u3 v 3 − 8P 3 + 24uv 3 P 2 + 24uvP 2 − 6u2 v 2 P
4
1
+ 36uv 2 P 2 + − (−72u4 v 3 − 720u3v 3 P − 756u2vP 2
v
− 432u3v 4 P − 3060u2v 3 P 2 − 3024P 3uv − 2664u2v 4 P 2
− 1296u2v 2 P 2 − 7920uv 2 P 3 + 288P 3 u − 825u2v 5 P 2
− 9576P 3uv 3 + 1152vP 4 − 6000uv 4P 3 + 576v 2 P 4 + 1152P 4
13 1
− 2304uv 5P 3 + 192P 4 v 3 − 512uv 6P 3 ) 2 uv 2
−3uv 2 + 2P − 2uv 3 − 2uv
+
uv 3
2
(9uv − 6P + 6uv 3 + 6uv)(−4uv 3 − 3uv 2 )
+
u2 v 6
3
2
2(−4uv − 9uv − 12uv − 6u) (−4uv 3 − 3uv 2 )3
−
−
uv 3
4u3 v 9
3
2 2
2
3
1 (−4uv − 3uv )
9uv − 6P + 6uv + 6uv
−
2
6
4
u v
uv 3
1
+ 3 36u2 v 3 P + 21u2 v 4 P + 8u3 v 3 − 8P 3 + 24uv 3 P 2
uv
1
+ 24uvP 2 − 6u2 v 2 P + 36uv 2 P 2 + − (−72u4 v 3
v
− 720u3v 3 P − 756u2vP 2 − 432u3v 4 P − 3060u2v 3 P 2
− 3024P 3uv − 2664u2v 4 P 2 − 1296u2v 2 P 2 − 7920uv 2P 3
+ 288P 3u − 825u2 v 5 P 2 − 9576P 3uv 3 + 1152vP 4
− 6000uv 4P 3 + 576v 2 P 4 + 1152P 4 − 2304uv 5P 3 + 192P 4 v 3
31 1
− 512uv 6P 3 ) 2 uv 2 + 2 − (u2 v 2 ) − 6uv 2 P + 2P 2
. 3
uv 36u2 v 3 P + 21u2 v 4 P
− 4uv 3 P − 4uvP
+ 8u3 v 3 − 8P 3 + 24uv 3 P 2 + 24uvP 2 − 6u2 v 2 P
1
+ 36uv 2 P 2 + (− (−72u4 v 3 − 720u3v 3 P − 756u2vP 2
v
− 432u3v 4 P − 3060u2v 3 P 2 − 3024P 3uv − 2664u2v 4 P 2
− 1296u2v 2 P 2 − 7920uv 2 P 3 + 288P 3 u − 825u2v 5 P 2
− 9576P 3uv 3 + 1152vP 4 − 6000uv 4P 3 + 576v 2 P 4 + 1152P 4
1 1
− 2304uv 5P 3 + 192P 4 v 3 − 512uv 6P 3 )) 2 uv 2 3
1 # 21
−3uv 2 + 2P − 2uv 3 − 2uv 2
−
uv 3
5
(2)
1.3
Solution 3
This solution is needed in the case of a negative B0′ . For example in chapter 4. For positive values of B0′ this
equation can lead to complex solutions.
"
4uv 3 + 3uv 2 1 1 (−4uv 3 − 3uv 2 )2
−
f (P ) =
4uv 3
2 4
u2 v 6
9uv 2 − 6P + 6uv 3 + 6uv
1
−
36u2 v 3 P + 21u2 v 4 P
+
uv 3
uv 3
+ 8u3 v 3 − 8P 3 + 24uv 3 P 2 + 24uvP 2 − 6u2 v 2 P + 36uv 2 P 2
1
+ − −72u4v 3 − 720u3v 3 P − 756u2vP 2 − 432u3v 4 P
v
− 3060u2v 3 P 2 − 3024P 3uv − 2664u2v 4 P 2 − 1296u2v 2 P 2
− 7920uv 2P 3 + 288P 3u − 825u2 v 5 P 2 − 9576P 3uv 3 + 1152vP 4
− 6000uv 4P 3 + 576v 2 P 4 + 1152P 4 − 2304uv 5P 3 + 192P 4v 3
1 21 2 3
uv
+ 2 −(u2 v 2 ) − 6uv 2 P
− 512uv 6 P 3
+ 2P 2 − 4uv 3 P − 4uvP
uv 3 36u2 v 3 P + 21u2 v 4 P
+ 8u3 v 3 − 8P 3 + 24uv 3 P 2 + 24uvP 2 − 6u2 v 2 P + 36uv 2 P 2
1
+ − (−72u4 v 3 − 720u3v 3 P − 756u2vP 2 − 432u3v 4 P
v
− 3060u2v 3 P 2 − 3024P 3uv − 2664u2v 4 P 2 − 1296u2v 2 P 2
− 7920uv 2P 3 + 288P 3u − 825u2 v 5 P 2 − 9576P 3uv 3 + 1152vP 4
− 6000uv 4P 3 + 576v 2 P 4 + 1152P 4 − 2304uv 5P 3 + 192P 4v 3
# 12
1 2 13
−3uv 2 + 2P − 2uv 3 − 2uv
6 3 2
− 512uv P ) uv
−
uv 3
"
9uv 2 − 6P + 6uv 3 + 6uv
1 1 (−4uv 3 − 3uv 2 )2
−
+
2
6
2 2
u v
uv 3
1
− 3 36u2 v 3 P + 21u2 v 4 P + 8u3 v 3 − 8P 3 + 24uv 3 P 2
uv
1
+ 24uvP 2 − 6u2 v 2 P + 36uv 2 P 2 + − −72u4v 3
v
3 3
2
2
3 4
− 720u v P − 756u vP − 432u v P − 3060u2v 3 P 2
− 3024P 3uv − 2664u2v 4 P 2 − 1296u2v 2 P 2 − 7920uv 2P 3
+ 288P 3 u − 825u2v 5 P 2 − 9576P 3uv 3 + 1152vP 4
− 6000uv 4P 3 + 576v 2 P 4 + 1152P 4 − 2304uv 5P 3 + 192P 4v 3
1 21 2 3
6 3
− 512uv P
uv
− 2 −(u2 v 2 ) − 6uv 2 P + 2P 2
− 4uv 3 P − 4uvP
uv 3 · 36u2 v 3 P + 21u2 v 4 P
+ 8u3 v 3 − 8P 3 + 24uv 3 P 2 + 24uvP 2 − 6u2 v 2 P
6
1
+ 36uv 2 P 2 + − (−72u4 v 3 − 720u3v 3 P − 756u2vP 2
v
− 432u3v 4 P − 3060u2v 3 P 2 − 3024P 3uv − 2664u2v 4 P 2
− 1296u2v 2 P 2 − 7920uv 2 P 3 + 288P 3 u − 825u2v 5 P 2
− 9576P 3uv 3 + 1152vP 4 − 6000uv 4P 3 + 576v 2 P 4 + 1152P 4
13 1
− 2304uv 5P 3 + 192P 4 v 3 − 512uv 6P 3 ) 2 uv 2
−3uv 2 + 2P − 2uv 3 − 2uv
+
uv 3
2
(9uv − 6P + 6uv 3 + 6uv)(−4uv 3 − 3uv 2 )
−
u2 v 6
3
2
2(−4uv − 9uv − 12uv − 6u) (−4uv 3 − 3uv 2 )3
−
−
uv 3
4u3 v 9
3
2 2
2
3
1 (−4uv − 3uv )
9uv − 6P + 6uv + 6uv
−
2
6
4
u v
uv 3
1
+ 3 36u2 v 3 P + 21u2 v 4 P + 8u3 v 3 − 8P 3 + 24uv 3 P 2
uv
1
+ 24uvP 2 − 6u2 v 2 P + 36uv 2 P 2 + − (−72u4 v 3
v
− 720u3v 3 P − 756u2vP 2 − 432u3v 4 P − 3060u2v 3 P 2
− 3024P 3uv − 2664u2v 4 P 2 − 1296u2v 2 P 2 − 7920uv 2P 3
+ 288P 3u − 825u2 v 5 P 2 − 9576P 3uv 3 + 1152vP 4
− 6000uv 4P 3 + 576v 2 P 4 + 1152P 4 − 2304uv 5P 3 + 192P 4 v 3
31 1
− 512uv 6P 3 ) 2 uv 2 + 2 − (u2 v 2 ) − 6uv 2 P + 2P 2
. 3
uv 36u2 v 3 P + 21u2 v 4 P
− 4uv 3 P − 4uvP
+ 8u3 v 3 − 8P 3 + 24uv 3 P 2 + 24uvP 2 − 6u2 v 2 P
1
+ 36uv 2 P 2 + (− (−72u4 v 3 − 720u3v 3 P − 756u2vP 2
v
− 432u3v 4 P − 3060u2v 3 P 2 − 3024P 3uv − 2664u2v 4 P 2
− 1296u2v 2 P 2 − 7920uv 2 P 3 + 288P 3 u − 825u2v 5 P 2
− 9576P 3uv 3 + 1152vP 4 − 6000uv 4P 3 + 576v 2 P 4 + 1152P 4
1 1
− 2304uv 5P 3 + 192P 4 v 3 − 512uv 6P 3 )) 2 uv 2 3
1 # 21
−3uv 2 + 2P − 2uv 3 − 2uv 2
−
uv 3
7
(3)
1.4
Solution 4
Similar to solution 1, but only reliable for negative B0′ .
"
4uv 3 + 3uv 2 1 1 (−4uv 3 − 3uv 2 )2
−
f (P ) =
4uv 3
2 4
u2 v 6
9uv 2 − 6P + 6uv 3 + 6uv
1
−
36u2 v 3 P + 21u2 v 4 P
+
uv 3
uv 3
+ 8u3 v 3 − 8P 3 + 24uv 3 P 2 + 24uvP 2 − 6u2 v 2 P + 36uv 2 P 2
1
+ − −72u4v 3 − 720u3v 3 P − 756u2vP 2 − 432u3v 4 P
v
− 3060u2v 3 P 2 − 3024P 3uv − 2664u2v 4 P 2 − 1296u2v 2 P 2
− 7920uv 2P 3 + 288P 3u − 825u2 v 5 P 2 − 9576P 3uv 3 + 1152vP 4
− 6000uv 4P 3 + 576v 2 P 4 + 1152P 4 − 2304uv 5P 3 + 192P 4v 3
1 21 2 3
uv
+ 2 −(u2 v 2 ) − 6uv 2 P
− 512uv 6 P 3
+ 2P 2 − 4uv 3 P − 4uvP
uv 3 36u2 v 3 P + 21u2 v 4 P
+ 8u3 v 3 − 8P 3 + 24uv 3 P 2 + 24uvP 2 − 6u2 v 2 P + 36uv 2 P 2
1
+ − (−72u4 v 3 − 720u3v 3 P − 756u2vP 2 − 432u3v 4 P
v
− 3060u2v 3 P 2 − 3024P 3uv − 2664u2v 4 P 2 − 1296u2v 2 P 2
− 7920uv 2P 3 + 288P 3u − 825u2 v 5 P 2 − 9576P 3uv 3 + 1152vP 4
− 6000uv 4P 3 + 576v 2 P 4 + 1152P 4 − 2304uv 5P 3 + 192P 4v 3
# 12
1 2 13
−3uv 2 + 2P − 2uv 3 − 2uv
6 3 2
− 512uv P ) uv
−
uv 3
"
1 1 (−4uv 3 − 3uv 2 )2
9uv 2 − 6P + 6uv 3 + 6uv
−
−
2
6
2 2
u v
uv 3
1
− 3 36u2 v 3 P + 21u2 v 4 P + 8u3 v 3 − 8P 3 + 24uv 3 P 2
uv
1
+ 24uvP 2 − 6u2 v 2 P + 36uv 2 P 2 + − −72u4v 3
v
− 720u3 v 3 P − 756u2 vP 2 − 432u3 v 4 P − 3060u2v 3 P 2
− 3024P 3uv − 2664u2v 4 P 2 − 1296u2v 2 P 2 − 7920uv 2P 3
+ 288P 3 u − 825u2v 5 P 2 − 9576P 3uv 3 + 1152vP 4
− 6000uv 4P 3 + 576v 2 P 4 + 1152P 4 − 2304uv 5P 3 + 192P 4v 3
1 21 2 3
6 3
− 512uv P
uv
− 2 −(u2 v 2 ) − 6uv 2 P + 2P 2
− 4uv 3 P − 4uvP
uv 3 · 36u2 v 3 P + 21u2 v 4 P
+ 8u3 v 3 − 8P 3 + 24uv 3 P 2 + 24uvP 2 − 6u2 v 2 P
8
1
+ 36uv 2 P 2 + − (−72u4 v 3 − 720u3v 3 P − 756u2vP 2
v
− 432u3v 4 P − 3060u2v 3 P 2 − 3024P 3uv − 2664u2v 4 P 2
− 1296u2v 2 P 2 − 7920uv 2 P 3 + 288P 3 u − 825u2v 5 P 2
− 9576P 3uv 3 + 1152vP 4 − 6000uv 4P 3 + 576v 2 P 4 + 1152P 4
13 1
− 2304uv 5P 3 + 192P 4 v 3 − 512uv 6P 3 ) 2 uv 2
−3uv 2 + 2P − 2uv 3 − 2uv
+
uv 3
2
(9uv − 6P + 6uv 3 + 6uv)(−4uv 3 − 3uv 2 )
−
u2 v 6
3
2
2(−4uv − 9uv − 12uv − 6u) (−4uv 3 − 3uv 2 )3
−
−
uv 3
4u3 v 9
3
2 2
2
3
1 (−4uv − 3uv )
9uv − 6P + 6uv + 6uv
−
2
6
4
u v
uv 3
1
+ 3 36u2 v 3 P + 21u2 v 4 P + 8u3 v 3 − 8P 3 + 24uv 3 P 2
uv
1
+ 24uvP 2 − 6u2 v 2 P + 36uv 2 P 2 + − (−72u4 v 3
v
− 720u3v 3 P − 756u2vP 2 − 432u3v 4 P − 3060u2v 3 P 2
− 3024P 3uv − 2664u2v 4 P 2 − 1296u2v 2 P 2 − 7920uv 2P 3
+ 288P 3u − 825u2 v 5 P 2 − 9576P 3uv 3 + 1152vP 4
− 6000uv 4P 3 + 576v 2 P 4 + 1152P 4 − 2304uv 5P 3 + 192P 4 v 3
31 1
− 512uv 6P 3 ) 2 uv 2 + 2 − (u2 v 2 ) − 6uv 2 P + 2P 2
. 3
uv 36u2 v 3 P + 21u2 v 4 P
− 4uv 3 P − 4uvP
+ 8u3 v 3 − 8P 3 + 24uv 3 P 2 + 24uvP 2 − 6u2 v 2 P
1
+ 36uv 2 P 2 + (− (−72u4 v 3 − 720u3v 3 P − 756u2vP 2
v
− 432u3v 4 P − 3060u2v 3 P 2 − 3024P 3uv − 2664u2v 4 P 2
− 1296u2v 2 P 2 − 7920uv 2 P 3 + 288P 3 u − 825u2v 5 P 2
− 9576P 3uv 3 + 1152vP 4 − 6000uv 4P 3 + 576v 2 P 4 + 1152P 4
1 1
− 2304uv 5P 3 + 192P 4 v 3 − 512uv 6P 3 )) 2 uv 2 3
1 # 21
−3uv 2 + 2P − 2uv 3 − 2uv 2
−
uv 3
9
(4)
2
Validity of the third order Vinet EoS approximation for transition
metals, diamond and osmium
Table 1: Pressure difference between the Vinet EoS and the third order Vinet EoS approximation at a volume
compression ratio of 20%. Bulk modulus and pressure derivative values for the transitions metals are obtained
from Raju et al. (1997), whereas diamond osmium values are obtained from Cynn et al. (2002).
Element
Sc
Ti
V
Cr
Mn
Fe
Co
Ni
Cu
Y
Zr
Nb
Mo
Tc
Ru
Rh
Pd
Ag
La
Hf
Ta
W
Re
Os
Ir
Pt
Au
C
Os
B0 (GPa)
60
105
160
180
131
170
192
183
137
40
89
170
267
281
310
269
184
101
22.6
109
200
317
368
410
364
280
173
444
462
B0′
2
4.37
4.26
4.89
5
5.29
4.26
5.26
5.48
2
4.11
4.06
4.5
0
6.61
4.5
5.42
6.12
3.2
3.95
3.79
4.33
5.41
3.4
4.83
5.18
6.29
4
2.4
Vinet EoS (GPa)
16.672
37.644
56.688
68.244
50.257
67.285
68.026
72.197
55.343
11.115
31.028
58.950
97.071
62.972
141.407
97.798
73.851
43.707
7.145
37.353
67.368
113.162
147.544
132.433
137.116
109.519
76.245
152.972
134.016
third order Vinet EoS approximation (GPa)
16.672
37.624
56.661
68.181
50.206
67.197
67.993
72.105
55.258
11.115
31.016
58.927
97.011
62.972
140.923
97.738
73.743
43.599
7.144
37.340
67.350
113.103
147.330
132.413
136.998
109.389
76.033
152.919
134.014
10
Difference (GPa)
0
0.02
0.027
0.063
0.051
0.088
0.0323
0.092
0.085
0
0.012
0.023
0.06
0
0.484
0.06
0.108
0.108
0.001
0.013
0.018
0.059
0.214
0.02
0.118
0.13
0.212
0.053
0.002
3
Input files and macros for TOPAS
Listing 1: Input files and macros for TOPAS
’−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
’−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
’ Information
’−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
’
’ This exa mples shows how t o d e t e r m i n e t h e bulk modulus , t h e p r e s s u r e d e r i v a t i v e
’ and t h e volume and/ o r l a t t i c e p a r a m e t e r s a t z e r o p r e s s u r e with t h e i n v e r t e d
’ t h i r d o r d e r Vinet EoS a p p r o x i m a t i o n .
’
’ By d e f a u l t t h i s example d e t e r m i n e s t h e l i n e a r i z e d v a l u e s . P l e a s e check t h e
’ a d d i t i o n a l comments i n t h e code , i n o r d e r t o change t h i s example t o t h e
’ d e t e r m i n a t i o n o f t h e bulk v a l u e s .
’
’ Q u e s t i o n s and s u g g e s t i o n s can be submitted t o :
’
’ P r o f Dr . Robert E . D i n n e b i e r ( r . d i n n e b i e r @ f k f . mpg)
’
’ and / o r
’
’ Martin E t t e r (m. e t t e r @ f k f . mpg . de )
’
’
’
’−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
’−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
’−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
’ Glo ba l d e f i n i t i o n s
’−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
r e x p 2 . 6 6 5 6 1 6 8 0 7 r e x p d a s h 6 . 3 6 8 6 0 3 4 6 5 r wp 1 . 7 0 2 4 1 9 5 8 9 r wp da sh 4 . 0 6 7 3 6 4 5 4 6
r p 1 . 0 4 7 1 2 7 0 2 3 r p d a s h 4 . 6 5 6 1 1 9 2 6 weig hted Dur bin Wa tso n 0 . 8 1 2 7 1 3 2 9 2 8 g o f
0.6386587842
i t e r s 100000
do errors
conserve memory
’−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
’ Glo ba l p a r a m e t r i c p a r a m e t e r s
’−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
’ Parameter f o r bulk i n v e r t e d t h i r d o r d e r Vinet EoS a p p r o x i m a ti o n
prm V0
prm VK0
prm VKp0
242.875
200.0
5.0
min =100; max=400;
min =0; max=400;
min =0; max=100;
prm uuu = 3 * VK0 ;
:0
prm vvv = 3/2* (VKp0 − 1 ) ; : 0
11
’ Parameter f o r l i n e a r i z e d and i n v e r t e d t h i r d o r d e r Vinet EoS
approximation
prm A0 Ma
prm A0 Mb
prm A0 Mc
5.55904
5.56612
7.85481
prm AK0 Ma
prm AK0 Mb
prm AK0 Mc
206.88492
141.71096
174.72222
prm AKp0 Ma 9 . 2 3 1 2 8
prm AKp0 Mb 4 . 3 1 4 7 8
prm AKp0 Mc 1 . 7 9 8 6 3
prm
prm
prm
prm
prm
prm
uuu
uuu
uuu
vvv
vvv
vvv
Ma
Mb
Mc
Ma
Mb
Mc
=
=
=
=
=
=
min =5; max=6;
min =5; max=6;
min =6; max=9;
min =0; max=400;
min =0; max=400;
min =0; max=400;
min =−100; max=100;
min =−100; max=100;
min =−100; max=100;
3 * AK0 Ma ;
:0
3 * AK0 Mb ;
:0
3 * AK0 Mc ;
:0
3/2* (AKp0 Ma − 1 ) ; : 0
3/2* (AKp0 Mb − 1 ) ; : 0
3/2* (AKp0 Mc − 1 ) ; : 0
’−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
’ Macro f o r Header i n f o r m a t i o n
’−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
macro H e a d e r i n f o {
r e x p 2 . 6 6 5 6 1 6 8 0 7 r e x p d a s h 6 . 3 6 8 6 0 3 4 6 5 r wp 1 . 7 0 2 4 1 9 5 8 9 r wp da sh
4 . 0 6 7 3 6 4 5 4 6 r p 1 . 0 4 7 1 2 7 0 2 3 r p d a s h 4 . 6 5 6 1 1 9 2 6 weig hted Dur bin Wa tso n
0.8127132928 gof 0.6386587842
start X 5
finish X 20.8
LP Factor ( 9 0 )
Z e r o E r r o r ( , −0.00003)
convolution step 5
Rp 2 1 7 . 5
Rs 2 1 7 . 5
lam
ymin on ymax 0 . 0 0 0 1
l a 1 l o 0 . 4 5 5 8 7 l h 1 e −005
x c a lc ula tio n s te p 0.014
}
’−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
’ Macro f o r EoS + S t r u c t u r e pa r a meter
’−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
macro S t r u c t u r e {
str
loca l prss = pressure loc ;
:0
’−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
12
’ bulk i n v e r t e d t h i r d o r d e r Vinet EoS a p p r o x i m a t io n
’−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
l o c a l v i n e t f v o l u m e = −(1/4)*(−4*uuu*vvvˆ3−3*uuu*vvv ˆ 2 ) / ( uuu*
vvv ˆ 3 ) +(1/2)* ( ( 1 / 4 ) *(−4*uuu*vvvˆ3−3*uuu*vvv ˆ 2 ) ˆ 2 / ( uuuˆ2*
vvv ˆ 6 ) −(9*uuu*vvvˆ2−6* p r s s+6*uuu*vvvˆ3+6*uuu*vvv ) / ( uuu*
vvv ˆ 3 ) +(36*uuuˆ2*vvv ˆ3* p r s s +21*uuuˆ2*vvv ˆ4* p r s s+8*uuuˆ3*vvvˆ3−8*
p r s s ˆ3+24*uuu*vvv ˆ3* p r s s ˆ2+24*uuu*vvv* p r s s ˆ2−6*uuuˆ2*vvv ˆ2*
p r s s +36*uuu*vvv ˆ2* p r s s ˆ2+(−(−72*uuuˆ4*vvv ˆ3−720*uuuˆ3*vvv ˆ3*
p r s s −756*uuuˆ2*vvv* p r s s ˆ2−432*uuuˆ3*vvv ˆ4* p r s s −3060*uuuˆ2*vvv ˆ3*
p r s s ˆ2 −3024* p r s s ˆ3*uuu*vvv−2664*uuuˆ2*vvv ˆ4* p r s s ˆ2 −1296*uuuˆ2*
vvv ˆ2* p r s s ˆ2 −7920*uuu*vvv ˆ2* p r s s ˆ3+288*uuu* p r s s ˆ3−825*uuuˆ2*
vvv ˆ5* p r s s ˆ2 −9576* p r s s ˆ3*uuu*vvv ˆ3+1152*vvv* p r s s ˆ4 −6000*uuu*
vvv ˆ4* p r s s ˆ3+576*vvv ˆ2* p r s s ˆ4+1152* p r s s ˆ4 −2304*uuu*vvv ˆ5*
p r s s ˆ3+192* p r s s ˆ4*vvvˆ3−512*uuu*vvv ˆ6* p r s s ˆ 3 ) / vvv ) ˆ ( 1 / 2 )*uuu*
vvv ˆ 2 ) ˆ ( 1 / 3 ) / ( uuu*vvv ˆ 3 )+2*(−(uuuˆ2*vvv ˆ 2 )−6*uuu*vvv ˆ2* p r s s+2*
p r s s ˆ2−4*uuu*vvv ˆ3* p r s s −4*uuu*vvv* p r s s ) / ( uuu*vvv ˆ3* ( 3 6*uuuˆ2*
vvv ˆ3* p r s s +21*uuuˆ2*vvv ˆ4* p r s s+8*uuuˆ3*vvvˆ3−8* p r s s ˆ3+24*uuu*
vvv ˆ3* p r s s ˆ2+24*uuu*vvv* p r s s ˆ2−6*uuuˆ2*vvv ˆ2* p r s s +36*uuu*vvv ˆ2*
p r s s ˆ2+(−(−72*uuuˆ4*vvvˆ3−720*uuuˆ3*vvv ˆ3* p r s s −756*uuuˆ2*vvv*
p r s s ˆ2−432*uuuˆ3*vvv ˆ4* p r s s −3060*uuuˆ2*vvv ˆ3* p r s s ˆ2 −3024* p r s s ˆ3*
uuu*vvv−2664*uuuˆ2*vvv ˆ4* p r s s ˆ2 −1296*uuuˆ2*vvv ˆ2* p r s s ˆ2 −7920*uuu*
vvv ˆ2* p r s s ˆ3+288*uuu* p r s s ˆ3−825*uuuˆ2*vvv ˆ5* p r s s ˆ2 −9576* p r s s ˆ3*
uuu*vvv ˆ3+1152*vvv* p r s s ˆ4 −6000*uuu*vvv ˆ4* p r s s ˆ3+576*vvv ˆ2*
p r s s ˆ4+1152* p r s s ˆ4 −2304*uuu*vvv ˆ5* p r s s ˆ3+192* p r s s ˆ4*vvv ˆ3−512*
uuu*vvv ˆ6* p r s s ˆ 3 ) / vvv ) ˆ ( 1 / 2 )*uuu*vvv ˆ 2 ) ˆ ( 1 / 3 ) )−(−3*uuu*vvvˆ2+2*
p r s s −2*uuu*vvvˆ3−2*uuu*vvv ) / ( uuu*vvv ˆ 3 ) ) ˆ ( 1 / 2 ) −(1/2)* ( ( 1 / 2 ) *(−4*
uuu*vvvˆ3−3*uuu*vvv ˆ 2 ) ˆ 2 / ( uuu ˆ2*vvv ˆ 6 ) −(9*uuu*vvvˆ2−6* p r s s+6*uuu*
vvvˆ3+6*uuu*vvv ) / ( uuu*vvv ˆ 3 ) −(36*uuuˆ2*vvv ˆ3* p r s s +21*uuuˆ2*vvv ˆ4*
p r s s+8*uuuˆ3*vvvˆ3−8* p r s s ˆ3+24*uuu*vvv ˆ3* p r s s ˆ2+24*uuu*vvv*
p r s s ˆ2−6*uuuˆ2*vvv ˆ2* p r s s +36*uuu*vvv ˆ2* p r s s ˆ2+(−(−72*uuuˆ4*
vvvˆ3−720*uuuˆ3*vvv ˆ3* p r s s −756*uuuˆ2*vvv* p r s s ˆ2−432*uuuˆ3*vvv ˆ4*
p r s s −3060*uuuˆ2*vvv ˆ3* p r s s ˆ2 −3024* p r s s ˆ3*uuu*vvv−2664*uuuˆ2*
vvv ˆ4* p r s s ˆ2 −1296*uuuˆ2*vvv ˆ2* p r s s ˆ2 −7920*uuu*vvv ˆ2* p r s s ˆ3+288*
uuu* p r s s ˆ3−825*uuuˆ2*vvv ˆ5* p r s s ˆ2 −9576* p r s s ˆ3*uuu*vvv ˆ3+1152*vvv*
p r s s ˆ4 −6000*uuu*vvv ˆ4* p r s s ˆ3+576*vvv ˆ2* p r s s ˆ4+1152* p r s s ˆ4 −2304*
uuu*vvv ˆ5* p r s s ˆ3+192* p r s s ˆ4*vvv ˆ3−512*uuu*vvv ˆ6*
p r s s ˆ 3 ) / vvv ) ˆ ( 1 / 2 )*uuu*vvv ˆ 2 ) ˆ ( 1 / 3 ) / ( uuu*vvv ˆ 3 )−2*(−(uuuˆ2*
vvv ˆ 2 )−6*uuu*vvv ˆ2* p r s s+2* p r s s ˆ2−4*uuu*vvv ˆ3* p r s s −4*uuu*vvv*
p r s s ) / ( uuu*vvv ˆ3* ( 3 6*uuuˆ2*vvv ˆ3* p r s s +21*uuuˆ2*vvv ˆ4* p r s s+8*
uuuˆ3*vvvˆ3−8* p r s s ˆ3+24*uuu*vvv ˆ3* p r s s ˆ2+24*uuu*vvv* p r s s ˆ2−6*
uuuˆ2*vvv ˆ2* p r s s +36*uuu*vvv ˆ2* p r s s ˆ2+(−(−72*uuuˆ4*vvv ˆ3−720*
uuuˆ3*vvv ˆ3* p r s s −756*uuuˆ2*vvv* p r s s ˆ2−432*uuuˆ3*vvv ˆ4* p r s s −3060*
uuuˆ2*vvv ˆ3* p r s s ˆ2 −3024* p r s s ˆ3*uuu*vvv−2664*uuuˆ2*vvv ˆ4*
p r s s ˆ2 −1296*uuuˆ2*vvv ˆ2* p r s s ˆ2 −7920*uuu*vvv ˆ2* p r s s ˆ3+288*uuu*
p r s s ˆ3−825*uuuˆ2*vvv ˆ5* p r s s ˆ2 −9576* p r s s ˆ3*uuu*vvv ˆ3+1152*vvv*
p r s s ˆ4 −6000*uuu*vvv ˆ4* p r s s ˆ3+576*vvv ˆ2* p r s s ˆ4+1152* p r s s ˆ4 −2304*
uuu*vvv ˆ5* p r s s ˆ3+192* p r s s ˆ4*vvv ˆ3−512*uuu*vvv ˆ6*
p r s s ˆ 3 ) / vvv ) ˆ ( 1 / 2 )*uuu*vvv ˆ 2 ) ˆ ( 1 / 3 ) )+(−3*uuu*vvvˆ2+2* p r s s −2*uuu*
vvvˆ3−2*uuu*vvv ) / ( uuu*vvv ˆ 3 ) +((9*uuu*vvvˆ2−6* p r s s+6*uuu*vvvˆ3+6*
uuu*vvv ) *(−4*uuu*vvvˆ3−3*uuu*vvv ˆ 2 ) / ( uuu ˆ2*vvv ˆ 6 )−2*(−4*uuu*
vvvˆ3−9*uuu*vvvˆ2−12*uuu*vvv−6*uuu ) / ( uuu*vvv ˆ 3 ) −(1/4)*(−4*uuu*
vvvˆ3−3*uuu*vvv ˆ 2 ) ˆ 3 / ( uuu ˆ3*vvv ˆ 9 ) ) / ( ( 1 / 4 )*(−4*uuu*vvvˆ3−3*uuu*
vvv ˆ 2 ) ˆ 2 / ( uuu ˆ2*vvv ˆ 6 ) −(9*uuu*vvvˆ2−6* p r s s+6*uuu*vvvˆ3+6*uuu*
vvv ) / ( uuu*vvv ˆ 3 ) +(36*uuuˆ2*vvv ˆ3* p r s s +21*uuuˆ2*vvv ˆ4* p r s s+8*
uuuˆ3*vvvˆ3−8* p r s s ˆ3+24*uuu*vvv ˆ3* p r s s ˆ2+24*uuu*vvv* p r s s ˆ2−6*
uuuˆ2*vvv ˆ2* p r s s +36*uuu*vvv ˆ2* p r s s ˆ2+(−(−72*uuuˆ4*vvv ˆ3−720*
uuuˆ3*vvv ˆ3* p r s s −756*uuuˆ2*vvv* p r s s ˆ2−432*uuuˆ3*vvv ˆ4* p r s s −3060*
13
uuuˆ2*vvv ˆ3* p r s s ˆ2 −3024* p r s s ˆ3*uuu*vvv−2664*uuuˆ2*vvv ˆ4*
p r s s ˆ2 −1296*uuuˆ2*vvv ˆ2* p r s s ˆ2 −7920*uuu*vvv ˆ2* p r s s ˆ3+288*uuu*
p r s s ˆ3−825*uuuˆ2*vvv ˆ5* p r s s ˆ2 −9576* p r s s ˆ3*uuu*vvv ˆ3+1152*vvv*
p r s s ˆ4 −6000*uuu*vvv ˆ4* p r s s ˆ3+576*vvv ˆ2* p r s s ˆ4+1152* p r s s ˆ4 −2304*
uuu*vvv ˆ5* p r s s ˆ3+192* p r s s ˆ4*vvv ˆ3−512*uuu*vvv ˆ6*
p r s s ˆ 3 ) / vvv ) ˆ ( 1 / 2 )*uuu*vvv ˆ 2 ) ˆ ( 1 / 3 ) / ( uuu*vvv ˆ 3 )+2*(−(uuuˆ2*
vvv ˆ 2 )−6*uuu*vvv ˆ2* p r s s+2* p r s s ˆ2−4*uuu*vvv ˆ3* p r s s −4*uuu*vvv*
p r s s ) / ( uuu*vvv ˆ3* ( 3 6*uuuˆ2*vvv ˆ3* p r s s +21*uuuˆ2*vvv ˆ4* p r s s+8*
uuuˆ3*vvvˆ3−8* p r s s ˆ3+24*uuu*vvv ˆ3* p r s s ˆ2+24*uuu*vvv* p r s s ˆ2−6*
uuuˆ2*vvv ˆ2* p r s s +36*uuu*vvv ˆ2* p r s s ˆ2+(−(−72*uuuˆ4*vvv ˆ3−720*
uuuˆ3*vvv ˆ3* p r s s −756*uuuˆ2*vvv* p r s s ˆ2−432*uuuˆ3*vvv ˆ4* p r s s −3060*
uuuˆ2*vvv ˆ3* p r s s ˆ2 −3024* p r s s ˆ3*uuu*vvv−2664*uuuˆ2*vvv ˆ4*
p r s s ˆ2 −1296*uuuˆ2*vvv ˆ2* p r s s ˆ2 −7920*uuu*vvv ˆ2* p r s s ˆ3+288*uuu*
p r s s ˆ3−825*uuuˆ2*vvv ˆ5* p r s s ˆ2 −9576* p r s s ˆ3*uuu*vvv ˆ3+1152*vvv*
p r s s ˆ4 −6000*uuu*vvv ˆ4* p r s s ˆ3+576*vvv ˆ2* p r s s ˆ4+1152* p r s s ˆ4 −2304*
uuu*vvv ˆ5* p r s s ˆ3+192* p r s s ˆ4*vvv ˆ3−512*uuu*vvv ˆ6*
p r s s ˆ 3 ) / vvv ) ˆ ( 1 / 2 )*uuu*vvv ˆ 2 ) ˆ ( 1 / 3 ) )−(−3*uuu*vvvˆ2+2* p r s s −2*uuu*
vvvˆ3−2*uuu*vvv ) / ( uuu*vvv ˆ 3 ) ) ˆ ( 1 / 2 ) ) ˆ ( 1 / 2 ) ; : 0
l o c a l v o l u m e l o c = v i n e t f v o l u m e ˆ3*V0 ;
l o c a l multparam = ( l p a p r e d e t * l p b p r e d e t *
l p c p r e d e t ) /( volume loc ) ;
’−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
’ l i n e a r i z e d and i n v e r t e d t h i r d o r d e r Vinet EoS a p p r o x i m a t io n
’−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
l o c a l v i n e t f l a t t i c e a = −(1/4)*(−4*uuu Ma*vvv Maˆ3−3*uuu Ma*
vvv Ma ˆ 2 ) / ( uuu Ma*vvv Ma ˆ 3 ) +(1/2)* ( ( 1 / 4 ) *(−4*uuu Ma*vvv Maˆ3−3*
uuu Ma*vvv Ma ˆ 2 ) ˆ 2 / ( uuu Maˆ2*vvv Ma ˆ 6 ) −(9*uuu Ma*vvv Maˆ2−6*
p r s s+6*uuu Ma*vvv Maˆ3+6*uuu Ma*vvv Ma ) / ( uuu Ma*vvv Ma ˆ 3 ) +(36*
uuu Maˆ2*vvv Ma ˆ3* p r s s +21*uuu Maˆ2*vvv Ma ˆ4* p r s s+8*uuu Maˆ3*
vvv Maˆ3−8* p r s s ˆ3+24*uuu Ma*vvv Ma ˆ3* p r s s ˆ2+24*uuu Ma*vvv Ma*
p r s s ˆ2−6*uuu Maˆ2*vvv Ma ˆ2* p r s s +36*uuu Ma*vvv Maˆ2*
p r s s ˆ2+(−(−72*uuu Maˆ4*vvv Maˆ3−720*uuu Maˆ3*vvv Ma ˆ3* p r s s −756*
uuu Maˆ2*vvv Ma* p r s s ˆ2−432*uuu Maˆ3*vvv Ma ˆ4* p r s s −3060*uuu Maˆ2*
vvv Ma ˆ3* p r s s ˆ2 −3024* p r s s ˆ3*uuu Ma*vvv Ma −2664*uuu Maˆ2*
vvv Ma ˆ4* p r s s ˆ2 −1296*uuu Maˆ2*vvv Ma ˆ2* p r s s ˆ2 −7920*uuu Ma*
vvv Ma ˆ2* p r s s ˆ3+288*uuu Ma* p r s s ˆ3−825*uuu Maˆ2*vvv Ma ˆ5*
p r s s ˆ2 −9576* p r s s ˆ3*uuu Ma*vvv Maˆ3+1152*vvv Ma* p r s s ˆ4 −6000*
uuu Ma*vvv Ma ˆ4* p r s s ˆ3+576*vvv Ma ˆ2* p r s s ˆ4+1152* p r s s ˆ4 −2304*
uuu Ma*vvv Ma ˆ5* p r s s ˆ3+192* p r s s ˆ4*vvv Maˆ3−512*uuu Ma*vvv Ma ˆ6*
p r s s ˆ 3 ) /vvv Ma ) ˆ ( 1 / 2 )*uuu Ma*vvv Ma ˆ 2 ) ˆ ( 1 / 3 ) / ( uuu Ma*
vvv Ma ˆ 3 )+2*(−(uuu Maˆ2*vvv Ma ˆ 2 )−6*uuu Ma*vvv Ma ˆ2* p r s s+2*
p r s s ˆ2−4*uuu Ma*vvv Ma ˆ3* p r s s −4*uuu Ma*vvv Ma* p r s s ) / ( uuu Ma*
vvv Ma ˆ3* ( 3 6*uuu Maˆ2*vvv Ma ˆ3* p r s s +21*uuu Maˆ2*vvv Ma ˆ4* p r s s+8*
uuu Maˆ3*vvv Maˆ3−8* p r s s ˆ3+24*uuu Ma*vvv Ma ˆ3* p r s s ˆ2+24*uuu Ma*
vvv Ma* p r s s ˆ2−6*uuu Maˆ2*vvv Ma ˆ2* p r s s +36*uuu Ma*vvv Ma ˆ2*
p r s s ˆ2+(−(−72*uuu Maˆ4*vvv Maˆ3−720*uuu Maˆ3*vvv Ma ˆ3* p r s s −756*
uuu Maˆ2*vvv Ma* p r s s ˆ2−432*uuu Maˆ3*vvv Ma ˆ4* p r s s −3060*uuu Maˆ2*
vvv Ma ˆ3* p r s s ˆ2 −3024* p r s s ˆ3*uuu Ma*vvv Ma −2664*uuu Maˆ2*
vvv Ma ˆ4* p r s s ˆ2 −1296*uuu Maˆ2*vvv Ma ˆ2* p r s s ˆ2 −7920*uuu Ma*
vvv Ma ˆ2* p r s s ˆ3+288*uuu Ma* p r s s ˆ3−825*uuu Maˆ2*vvv Ma ˆ5*
p r s s ˆ2 −9576* p r s s ˆ3*uuu Ma*vvv Maˆ3+1152*vvv Ma* p r s s ˆ4 −6000*
uuu Ma*vvv Ma ˆ4* p r s s ˆ3+576*vvv Ma ˆ2* p r s s ˆ4+1152* p r s s ˆ4 −2304*
14
uuu Ma*vvv Ma ˆ5* p r s s ˆ3+192* p r s s ˆ4*vvv Maˆ3−512*uuu Ma*vvv Ma ˆ6*
p r s s ˆ 3 ) /vvv Ma ) ˆ ( 1 / 2 )*uuu Ma*vvv Ma ˆ 2 ) ˆ ( 1 / 3 ) )−(−3*uuu Ma*
vvv Maˆ2+2* p r s s −2*uuu Ma*vvv Maˆ3−2*uuu Ma*vvv Ma ) / ( uuu Ma*
vvv Ma ˆ 3 ) ) ˆ ( 1 / 2 ) −(1/2)* ( ( 1 / 2 ) *(−4*uuu Ma*vvv Maˆ3−3*uuu Ma*
vvv Ma ˆ 2 ) ˆ 2 / ( uuu Maˆ2*vvv Ma ˆ 6 ) −(9*uuu Ma*vvv Maˆ2−6* p r s s+6*
uuu Ma*vvv Maˆ3+6*uuu Ma*vvv Ma ) / ( uuu Ma*vvv Ma ˆ 3 ) −(36*uuu Maˆ2*
vvv Ma ˆ3* p r s s +21*uuu Maˆ2*vvv Ma ˆ4* p r s s+8*uuu Maˆ3*vvv Maˆ3−8*
p r s s ˆ3+24*uuu Ma*vvv Ma ˆ3* p r s s ˆ2+24*uuu Ma*vvv Ma* p r s s ˆ2−6*
uuu Maˆ2*vvv Ma ˆ2* p r s s +36*uuu Ma*vvv Maˆ2* p r s s ˆ2+(−(−72*
uuu Maˆ4*vvv Maˆ3−720*uuu Maˆ3*vvv Ma ˆ3* p r s s −756*uuu Maˆ2*
vvv Ma* p r s s ˆ2−432*uuu Maˆ3*vvv Ma ˆ4* p r s s −3060*uuu Maˆ2*vvv Ma ˆ3*
p r s s ˆ2 −3024* p r s s ˆ3*uuu Ma*vvv Ma −2664*uuu Maˆ2*vvv Ma ˆ4*
p r s s ˆ2 −1296*uuu Maˆ2*vvv Ma ˆ2* p r s s ˆ2 −7920*uuu Ma*vvv Ma ˆ2*
p r s s ˆ3+288*uuu Ma* p r s s ˆ3−825*uuu Maˆ2*vvv Ma ˆ5* p r s s ˆ2 −9576*
p r s s ˆ3*uuu Ma*vvv Maˆ3+1152*vvv Ma* p r s s ˆ4 −6000*uuu Ma*vvv Ma ˆ4*
p r s s ˆ3+576*vvv Maˆ2* p r s s ˆ4+1152* p r s s ˆ4 −2304*uuu Ma*vvv Ma ˆ5*
p r s s ˆ3+192* p r s s ˆ4*vvv Maˆ3−512*uuu Ma*vvv Ma ˆ6*
p r s s ˆ 3 ) /vvv Ma ) ˆ ( 1 / 2 )*uuu Ma*vvv Ma ˆ 2 ) ˆ ( 1 / 3 ) / ( uuu Ma*
vvv Ma ˆ 3 )−2*(−(uuu Maˆ2*vvv Ma ˆ 2 )−6*uuu Ma*vvv Ma ˆ2* p r s s+2*
p r s s ˆ2−4*uuu Ma*vvv Ma ˆ3* p r s s −4*uuu Ma*vvv Ma* p r s s ) / ( uuu Ma*
vvv Ma ˆ3* ( 3 6*uuu Maˆ2*vvv Ma ˆ3* p r s s +21*uuu Maˆ2*vvv Ma ˆ4* p r s s+8*
uuu Maˆ3*vvv Maˆ3−8* p r s s ˆ3+24*uuu Ma*vvv Ma ˆ3* p r s s ˆ2+24*uuu Ma*
vvv Ma* p r s s ˆ2−6*uuu Maˆ2*vvv Ma ˆ2* p r s s +36*uuu Ma*vvv Ma ˆ2*
p r s s ˆ2+(−(−72*uuu Maˆ4*vvv Maˆ3−720*uuu Maˆ3*vvv Ma ˆ3* p r s s −756*
uuu Maˆ2*vvv Ma* p r s s ˆ2−432*uuu Maˆ3*vvv Ma ˆ4* p r s s −3060*uuu Maˆ2*
vvv Ma ˆ3* p r s s ˆ2 −3024* p r s s ˆ3*uuu Ma*vvv Ma −2664*uuu Maˆ2*
vvv Ma ˆ4* p r s s ˆ2 −1296*uuu Maˆ2*vvv Ma ˆ2* p r s s ˆ2 −7920*uuu Ma*
vvv Ma ˆ2* p r s s ˆ3+288*uuu Ma* p r s s ˆ3−825*uuu Maˆ2*vvv Ma ˆ5*
p r s s ˆ2 −9576* p r s s ˆ3*uuu Ma*vvv Maˆ3+1152*vvv Ma* p r s s ˆ4 −6000*
uuu Ma*vvv Ma ˆ4* p r s s ˆ3+576*vvv Ma ˆ2* p r s s ˆ4+1152* p r s s ˆ4 −2304*
uuu Ma*vvv Ma ˆ5* p r s s ˆ3+192* p r s s ˆ4*vvv Maˆ3−512*uuu Ma*vvv Ma ˆ6*
p r s s ˆ 3 ) /vvv Ma ) ˆ ( 1 / 2 )*uuu Ma*vvv Ma ˆ 2 ) ˆ ( 1 / 3 ) ) +(−3*uuu Ma*
vvv Maˆ2+2* p r s s −2*uuu Ma*vvv Maˆ3−2*uuu Ma*vvv Ma ) / ( uuu Ma*
vvv Ma ˆ 3 ) +((9*uuu Ma*vvv Maˆ2−6* p r s s+6*uuu Ma*vvv Maˆ3+6*uuu Ma*
vvv Ma ) *(−4*uuu Ma*vvv Maˆ3−3*uuu Ma*vvv Ma ˆ 2 ) / ( uuu Maˆ2*
vvv Ma ˆ 6 )−2*(−4*uuu Ma*vvv Maˆ3−9*uuu Ma*vvv Maˆ2−12*uuu Ma*
vvv Ma−6*uuu Ma ) / ( uuu Ma*vvv Ma ˆ 3 ) −(1/4)*(−4*uuu Ma*vvv Maˆ3−3*
uuu Ma*vvv Ma ˆ 2 ) ˆ 3 / ( uuu Maˆ3*vvv Ma ˆ 9 ) ) / ( ( 1 / 4 )*(−4*uuu Ma*
vvv Maˆ3−3*uuu Ma*vvv Ma ˆ 2 ) ˆ 2 / ( uuu Maˆ2*vvv Ma ˆ 6 ) −(9*uuu Ma*
vvv Maˆ2−6* p r s s+6*uuu Ma*vvv Maˆ3+6*uuu Ma*vvv Ma ) / ( uuu Ma*
vvv Ma ˆ 3 ) +(36*uuu Maˆ2*vvv Ma ˆ3* p r s s +21*uuu Maˆ2*vvv Ma ˆ4*
p r s s+8*uuu Maˆ3*vvv Maˆ3−8* p r s s ˆ3+24*uuu Ma*vvv Ma ˆ3* p r s s ˆ2+24*
uuu Ma*vvv Ma* p r s s ˆ2−6*uuu Maˆ2*vvv Ma ˆ2* p r s s +36*uuu Ma*
vvv Ma ˆ2* p r s s ˆ2+(−(−72*uuu Maˆ4*vvv Maˆ3−720*uuu Maˆ3*vvv Ma ˆ3*
p r s s −756*uuu Maˆ2*vvv Ma* p r s s ˆ2−432*uuu Maˆ3*vvv Ma ˆ4* p r s s −3060*
uuu Maˆ2*vvv Ma ˆ3* p r s s ˆ2 −3024* p r s s ˆ3*uuu Ma*vvv Ma −2664*
uuu Maˆ2*vvv Ma ˆ4* p r s s ˆ2 −1296*uuu Maˆ2*vvv Ma ˆ2* p r s s ˆ2 −7920*
uuu Ma*vvv Ma ˆ2* p r s s ˆ3+288*uuu Ma* p r s s ˆ3−825*uuu Maˆ2*vvv Ma ˆ5*
p r s s ˆ2 −9576* p r s s ˆ3*uuu Ma*vvv Maˆ3+1152*vvv Ma* p r s s ˆ4 −6000*
uuu Ma*vvv Ma ˆ4* p r s s ˆ3+576*vvv Ma ˆ2* p r s s ˆ4+1152* p r s s ˆ4 −2304*
uuu Ma*vvv Ma ˆ5* p r s s ˆ3+192* p r s s ˆ4*vvv Maˆ3−512*uuu Ma*vvv Ma ˆ6*
p r s s ˆ 3 ) /vvv Ma ) ˆ ( 1 / 2 )*uuu Ma*vvv Ma ˆ 2 ) ˆ ( 1 / 3 ) / ( uuu Ma*
vvv Ma ˆ 3 )+2*(−(uuu Maˆ2*vvv Ma ˆ 2 )−6*uuu Ma*vvv Ma ˆ2* p r s s+2*
p r s s ˆ2−4*uuu Ma*vvv Ma ˆ3* p r s s −4*uuu Ma*vvv Ma* p r s s ) / ( uuu Ma*
vvv Ma ˆ3* ( 3 6*uuu Maˆ2*vvv Ma ˆ3* p r s s +21*uuu Maˆ2*vvv Ma ˆ4* p r s s+8*
uuu Maˆ3*vvv Maˆ3−8* p r s s ˆ3+24*uuu Ma*vvv Ma ˆ3* p r s s ˆ2+24*uuu Ma*
vvv Ma* p r s s ˆ2−6*uuu Maˆ2*vvv Ma ˆ2* p r s s +36*uuu Ma*vvv Ma ˆ2*
p r s s ˆ2+(−(−72*uuu Maˆ4*vvv Maˆ3−720*uuu Maˆ3*vvv Ma ˆ3* p r s s −756*
15
uuu Maˆ2*vvv Ma* p r s s ˆ2−432*uuu Maˆ3*vvv Ma ˆ4* p r s s −3060*uuu Maˆ2*
vvv Ma ˆ3* p r s s ˆ2 −3024* p r s s ˆ3*uuu Ma*vvv Ma −2664*uuu Maˆ2*
vvv Ma ˆ4* p r s s ˆ2 −1296*uuu Maˆ2*vvv Ma ˆ2* p r s s ˆ2 −7920*uuu Ma*
vvv Ma ˆ2* p r s s ˆ3+288*uuu Ma* p r s s ˆ3−825*uuu Maˆ2*vvv Ma ˆ5*
p r s s ˆ2 −9576* p r s s ˆ3*uuu Ma*vvv Maˆ3+1152*vvv Ma* p r s s ˆ4 −6000*
uuu Ma*vvv Ma ˆ4* p r s s ˆ3+576*vvv Ma ˆ2* p r s s ˆ4+1152* p r s s ˆ4 −2304*
uuu Ma*vvv Ma ˆ5* p r s s ˆ3+192* p r s s ˆ4*vvv Maˆ3−512*uuu Ma*vvv Ma ˆ6*
p r s s ˆ 3 ) /vvv Ma ) ˆ ( 1 / 2 )*uuu Ma*vvv Ma ˆ 2 ) ˆ ( 1 / 3 ) )−(−3*uuu Ma*
vvv Maˆ2+2* p r s s −2*uuu Ma*vvv Maˆ3−2*uuu Ma*vvv Ma ) / ( uuu Ma*
vvv Ma ˆ 3 ) ) ˆ ( 1 / 2 ) ) ˆ ( 1 / 2 ) ; : 0
l o c a l l p a = v i n e t f l a t t i c e a *A0 Ma ;
l o c a l v i n e t f l a t t i c e b = −(1/4)*(−4*uuu Mb*vvv Mbˆ3−3*uuu Mb*
vvv Mb ˆ 2 ) / ( uuu Mb*vvv Mb ˆ 3 ) +(1/2)* ( ( 1 / 4 ) *(−4*uuu Mb*vvv Mbˆ3−3*
uuu Mb*vvv Mb ˆ 2 ) ˆ 2 / ( uuu Mbˆ2*vvv Mb ˆ 6 ) −(9*uuu Mb*vvv Mbˆ2−6*
p r s s+6*uuu Mb*vvv Mbˆ3+6*uuu Mb*vvv Mb ) / ( uuu Mb*vvv Mb ˆ 3 ) +(36*
uuu Mbˆ2*vvv Mbˆ3* p r s s +21*uuu Mbˆ2*vvv Mbˆ4* p r s s+8*uuu Mbˆ3*
vvv Mbˆ3−8* p r s s ˆ3+24*uuu Mb*vvv Mbˆ3* p r s s ˆ2+24*uuu Mb*vvv Mb*
p r s s ˆ2−6*uuu Mbˆ2*vvv Mbˆ2* p r s s +36*uuu Mb*vvv Mbˆ2*
p r s s ˆ2+(−(−72*uuu Mbˆ4*vvv Mbˆ3−720*uuu Mbˆ3*vvv Mbˆ3* p r s s −756*
uuu Mbˆ2*vvv Mb* p r s s ˆ2−432*uuu Mbˆ3*vvv Mbˆ4* p r s s −3060*uuu Mbˆ2*
vvv Mbˆ3* p r s s ˆ2 −3024* p r s s ˆ3*uuu Mb*vvv Mb−2664*uuu Mbˆ2*
vvv Mbˆ4* p r s s ˆ2 −1296*uuu Mbˆ2*vvv Mbˆ2* p r s s ˆ2 −7920*uuu Mb*
vvv Mbˆ2* p r s s ˆ3+288*uuu Mb* p r s s ˆ3−825*uuu Mbˆ2*vvv Mb ˆ5*
p r s s ˆ2 −9576* p r s s ˆ3*uuu Mb*vvv Mbˆ3+1152*vvv Mb* p r s s ˆ4 −6000*
uuu Mb*vvv Mbˆ4* p r s s ˆ3+576*vvv Mbˆ2* p r s s ˆ4+1152* p r s s ˆ4 −2304*
uuu Mb*vvv Mbˆ5* p r s s ˆ3+192* p r s s ˆ4*vvv Mbˆ3−512*uuu Mb*vvv Mbˆ6*
p r s s ˆ 3 ) /vvv Mb ) ˆ ( 1 / 2 )*uuu Mb*vvv Mb ˆ 2 ) ˆ ( 1 / 3 ) / ( uuu Mb*
vvv Mb ˆ 3 )+2*(−(uuu Mbˆ2*vvv Mb ˆ 2 )−6*uuu Mb*vvv Mb ˆ2* p r s s+2*
p r s s ˆ2−4*uuu Mb*vvv Mbˆ3* p r s s −4*uuu Mb*vvv Mb* p r s s ) / ( uuu Mb*
vvv Mbˆ3* ( 3 6*uuu Mbˆ2*vvv Mbˆ3* p r s s +21*uuu Mbˆ2*vvv Mbˆ4* p r s s+8*
uuu Mbˆ3*vvv Mbˆ3−8* p r s s ˆ3+24*uuu Mb*vvv Mbˆ3* p r s s ˆ2+24*uuu Mb*
vvv Mb* p r s s ˆ2−6*uuu Mbˆ2*vvv Mbˆ2* p r s s +36*uuu Mb*vvv Mbˆ2*
p r s s ˆ2+(−(−72*uuu Mbˆ4*vvv Mbˆ3−720*uuu Mbˆ3*vvv Mbˆ3* p r s s −756*
uuu Mbˆ2*vvv Mb* p r s s ˆ2−432*uuu Mbˆ3*vvv Mbˆ4* p r s s −3060*uuu Mbˆ2*
vvv Mbˆ3* p r s s ˆ2 −3024* p r s s ˆ3*uuu Mb*vvv Mb−2664*uuu Mbˆ2*
vvv Mbˆ4* p r s s ˆ2 −1296*uuu Mbˆ2*vvv Mbˆ2* p r s s ˆ2 −7920*uuu Mb*
vvv Mbˆ2* p r s s ˆ3+288*uuu Mb* p r s s ˆ3−825*uuu Mbˆ2*vvv Mb ˆ5*
p r s s ˆ2 −9576* p r s s ˆ3*uuu Mb*vvv Mbˆ3+1152*vvv Mb* p r s s ˆ4 −6000*
uuu Mb*vvv Mbˆ4* p r s s ˆ3+576*vvv Mbˆ2* p r s s ˆ4+1152* p r s s ˆ4 −2304*
uuu Mb*vvv Mbˆ5* p r s s ˆ3+192* p r s s ˆ4*vvv Mbˆ3−512*uuu Mb*vvv Mbˆ6*
p r s s ˆ 3 ) /vvv Mb ) ˆ ( 1 / 2 )*uuu Mb*vvv Mb ˆ 2 ) ˆ ( 1 / 3 ) )−(−3*uuu Mb*
vvv Mbˆ2+2* p r s s −2*uuu Mb*vvv Mbˆ3−2*uuu Mb*vvv Mb ) / ( uuu Mb*
vvv Mb ˆ 3 ) ) ˆ ( 1 / 2 ) −(1/2)* ( ( 1 / 2 ) *(−4*uuu Mb*vvv Mbˆ3−3*uuu Mb*
vvv Mb ˆ 2 ) ˆ 2 / ( uuu Mbˆ2*vvv Mb ˆ 6 ) −(9*uuu Mb*vvv Mbˆ2−6* p r s s+6*
uuu Mb*vvv Mbˆ3+6*uuu Mb*vvv Mb ) / ( uuu Mb*vvv Mb ˆ 3 ) −(36*uuu Mbˆ2*
vvv Mbˆ3* p r s s +21*uuu Mbˆ2*vvv Mbˆ4* p r s s+8*uuu Mbˆ3*vvv Mbˆ3−8*
p r s s ˆ3+24*uuu Mb*vvv Mbˆ3* p r s s ˆ2+24*uuu Mb*vvv Mb* p r s s ˆ2−6*
uuu Mbˆ2*vvv Mbˆ2* p r s s +36*uuu Mb*vvv Mbˆ2* p r s s ˆ2+(−(−72*
uuu Mbˆ4*vvv Mbˆ3−720*uuu Mbˆ3*vvv Mbˆ3* p r s s −756*uuu Mbˆ2*
vvv Mb* p r s s ˆ2−432*uuu Mbˆ3*vvv Mbˆ4* p r s s −3060*uuu Mbˆ2*vvv Mbˆ3*
p r s s ˆ2 −3024* p r s s ˆ3*uuu Mb*vvv Mb−2664*uuu Mbˆ2*vvv Mb ˆ4*
p r s s ˆ2 −1296*uuu Mbˆ2*vvv Mbˆ2* p r s s ˆ2 −7920*uuu Mb*vvv Mbˆ2*
p r s s ˆ3+288*uuu Mb* p r s s ˆ3−825*uuu Mbˆ2*vvv Mb ˆ5* p r s s ˆ2 −9576*
p r s s ˆ3*uuu Mb*vvv Mbˆ3+1152*vvv Mb* p r s s ˆ4 −6000*uuu Mb*vvv Mbˆ4*
p r s s ˆ3+576*vvv Mbˆ2* p r s s ˆ4+1152* p r s s ˆ4 −2304*uuu Mb*vvv Mbˆ5*
p r s s ˆ3+192* p r s s ˆ4*vvv Mbˆ3−512*uuu Mb*vvv Mb ˆ6*
16
p r s s ˆ 3 ) /vvv Mb ) ˆ ( 1 / 2 )*uuu Mb*vvv Mb ˆ 2 ) ˆ ( 1 / 3 ) / ( uuu Mb*
vvv Mb ˆ 3 )−2*(−(uuu Mbˆ2*vvv Mb ˆ 2 )−6*uuu Mb*vvv Mb ˆ2* p r s s+2*
p r s s ˆ2−4*uuu Mb*vvv Mbˆ3* p r s s −4*uuu Mb*vvv Mb* p r s s ) / ( uuu Mb*
vvv Mbˆ3* ( 3 6*uuu Mbˆ2*vvv Mbˆ3* p r s s +21*uuu Mbˆ2*vvv Mbˆ4* p r s s+8*
uuu Mbˆ3*vvv Mbˆ3−8* p r s s ˆ3+24*uuu Mb*vvv Mbˆ3* p r s s ˆ2+24*uuu Mb*
vvv Mb* p r s s ˆ2−6*uuu Mbˆ2*vvv Mbˆ2* p r s s +36*uuu Mb*vvv Mbˆ2*
p r s s ˆ2+(−(−72*uuu Mbˆ4*vvv Mbˆ3−720*uuu Mbˆ3*vvv Mbˆ3* p r s s −756*
uuu Mbˆ2*vvv Mb* p r s s ˆ2−432*uuu Mbˆ3*vvv Mbˆ4* p r s s −3060*uuu Mbˆ2*
vvv Mbˆ3* p r s s ˆ2 −3024* p r s s ˆ3*uuu Mb*vvv Mb−2664*uuu Mbˆ2*
vvv Mbˆ4* p r s s ˆ2 −1296*uuu Mbˆ2*vvv Mbˆ2* p r s s ˆ2 −7920*uuu Mb*
vvv Mbˆ2* p r s s ˆ3+288*uuu Mb* p r s s ˆ3−825*uuu Mbˆ2*vvv Mb ˆ5*
p r s s ˆ2 −9576* p r s s ˆ3*uuu Mb*vvv Mbˆ3+1152*vvv Mb* p r s s ˆ4 −6000*
uuu Mb*vvv Mbˆ4* p r s s ˆ3+576*vvv Mbˆ2* p r s s ˆ4+1152* p r s s ˆ4 −2304*
uuu Mb*vvv Mbˆ5* p r s s ˆ3+192* p r s s ˆ4*vvv Mbˆ3−512*uuu Mb*vvv Mbˆ6*
p r s s ˆ 3 ) /vvv Mb ) ˆ ( 1 / 2 )*uuu Mb*vvv Mb ˆ 2 ) ˆ ( 1 / 3 ) ) +(−3*uuu Mb*
vvv Mbˆ2+2* p r s s −2*uuu Mb*vvv Mbˆ3−2*uuu Mb*vvv Mb ) / ( uuu Mb*
vvv Mb ˆ 3 ) +((9*uuu Mb*vvv Mbˆ2−6* p r s s+6*uuu Mb*vvv Mbˆ3+6*uuu Mb*
vvv Mb ) *(−4*uuu Mb*vvv Mbˆ3−3*uuu Mb*vvv Mb ˆ 2 ) / ( uuu Mbˆ2*
vvv Mb ˆ 6 )−2*(−4*uuu Mb*vvv Mbˆ3−9*uuu Mb*vvv Mbˆ2−12*uuu Mb*
vvv Mb−6*uuu Mb ) / ( uuu Mb*vvv Mb ˆ 3 ) −(1/4)*(−4*uuu Mb*vvv Mbˆ3−3*
uuu Mb*vvv Mb ˆ 2 ) ˆ 3 / ( uuu Mbˆ3*vvv Mb ˆ 9 ) ) / ( ( 1 / 4 )*(−4*uuu Mb*
vvv Mbˆ3−3*uuu Mb*vvv Mb ˆ 2 ) ˆ 2 / ( uuu Mbˆ2*vvv Mb ˆ 6 ) −(9*uuu Mb*
vvv Mbˆ2−6* p r s s+6*uuu Mb*vvv Mbˆ3+6*uuu Mb*vvv Mb ) / ( uuu Mb*
vvv Mb ˆ 3 ) +(36*uuu Mbˆ2*vvv Mbˆ3* p r s s +21*uuu Mbˆ2*vvv Mbˆ4*
p r s s+8*uuu Mbˆ3*vvv Mbˆ3−8* p r s s ˆ3+24*uuu Mb*vvv Mbˆ3* p r s s ˆ2+24*
uuu Mb*vvv Mb* p r s s ˆ2−6*uuu Mbˆ2*vvv Mbˆ2* p r s s +36*uuu Mb*
vvv Mbˆ2* p r s s ˆ2+(−(−72*uuu Mbˆ4*vvv Mbˆ3−720*uuu Mbˆ3*vvv Mbˆ3*
p r s s −756*uuu Mbˆ2*vvv Mb* p r s s ˆ2−432*uuu Mbˆ3*vvv Mbˆ4* p r s s −3060*
uuu Mbˆ2*vvv Mbˆ3* p r s s ˆ2 −3024* p r s s ˆ3*uuu Mb*vvv Mb−2664*
uuu Mbˆ2*vvv Mbˆ4* p r s s ˆ2 −1296*uuu Mbˆ2*vvv Mbˆ2* p r s s ˆ2 −7920*
uuu Mb*vvv Mbˆ2* p r s s ˆ3+288*uuu Mb* p r s s ˆ3−825*uuu Mbˆ2*vvv Mbˆ5*
p r s s ˆ2 −9576* p r s s ˆ3*uuu Mb*vvv Mbˆ3+1152*vvv Mb* p r s s ˆ4 −6000*
uuu Mb*vvv Mbˆ4* p r s s ˆ3+576*vvv Mbˆ2* p r s s ˆ4+1152* p r s s ˆ4 −2304*
uuu Mb*vvv Mbˆ5* p r s s ˆ3+192* p r s s ˆ4*vvv Mbˆ3−512*uuu Mb*vvv Mbˆ6*
p r s s ˆ 3 ) /vvv Mb ) ˆ ( 1 / 2 )*uuu Mb*vvv Mb ˆ 2 ) ˆ ( 1 / 3 ) / ( uuu Mb*
vvv Mb ˆ 3 )+2*(−(uuu Mbˆ2*vvv Mb ˆ 2 )−6*uuu Mb*vvv Mb ˆ2* p r s s+2*
p r s s ˆ2−4*uuu Mb*vvv Mbˆ3* p r s s −4*uuu Mb*vvv Mb* p r s s ) / ( uuu Mb*
vvv Mbˆ3* ( 3 6*uuu Mbˆ2*vvv Mbˆ3* p r s s +21*uuu Mbˆ2*vvv Mbˆ4* p r s s+8*
uuu Mbˆ3*vvv Mbˆ3−8* p r s s ˆ3+24*uuu Mb*vvv Mbˆ3* p r s s ˆ2+24*uuu Mb*
vvv Mb* p r s s ˆ2−6*uuu Mbˆ2*vvv Mbˆ2* p r s s +36*uuu Mb*vvv Mbˆ2*
p r s s ˆ2+(−(−72*uuu Mbˆ4*vvv Mbˆ3−720*uuu Mbˆ3*vvv Mbˆ3* p r s s −756*
uuu Mbˆ2*vvv Mb* p r s s ˆ2−432*uuu Mbˆ3*vvv Mbˆ4* p r s s −3060*uuu Mbˆ2*
vvv Mbˆ3* p r s s ˆ2 −3024* p r s s ˆ3*uuu Mb*vvv Mb−2664*uuu Mbˆ2*
vvv Mbˆ4* p r s s ˆ2 −1296*uuu Mbˆ2*vvv Mbˆ2* p r s s ˆ2 −7920*uuu Mb*
vvv Mbˆ2* p r s s ˆ3+288*uuu Mb* p r s s ˆ3−825*uuu Mbˆ2*vvv Mb ˆ5*
p r s s ˆ2 −9576* p r s s ˆ3*uuu Mb*vvv Mbˆ3+1152*vvv Mb* p r s s ˆ4 −6000*
uuu Mb*vvv Mbˆ4* p r s s ˆ3+576*vvv Mbˆ2* p r s s ˆ4+1152* p r s s ˆ4 −2304*
uuu Mb*vvv Mbˆ5* p r s s ˆ3+192* p r s s ˆ4*vvv Mbˆ3−512*uuu Mb*vvv Mbˆ6*
p r s s ˆ 3 ) /vvv Mb ) ˆ ( 1 / 2 )*uuu Mb*vvv Mb ˆ 2 ) ˆ ( 1 / 3 ) )−(−3*uuu Mb*
vvv Mbˆ2+2* p r s s −2*uuu Mb*vvv Mbˆ3−2*uuu Mb*vvv Mb ) / ( uuu Mb*
vvv Mb ˆ 3 ) ) ˆ ( 1 / 2 ) ) ˆ ( 1 / 2 ) ; : 0
l o c a l l p b = v i n e t f l a t t i c e b *A0 Mb ;
l o c a l v i n e t f l a t t i c e c = −(1/4)*(−4*uuu Mc*vvv Mcˆ3−3*uuu Mc*
vvv Mc ˆ 2 ) / ( uuu Mc*vvv Mc ˆ 3 ) +(1/2)* ( ( 1 / 4 ) *(−4*uuu Mc*vvv Mcˆ3−3*
uuu Mc*vvv Mc ˆ 2 ) ˆ 2 / ( uuu Mcˆ2*vvv Mc ˆ 6 ) −(9*uuu Mc*vvv Mcˆ2−6*
p r s s+6*uuu Mc*vvv Mcˆ3+6*uuu Mc*vvv Mc ) / ( uuu Mc*vvv Mc ˆ 3 ) +(36*
17
uuu Mcˆ2*vvv Mc ˆ3* p r s s +21*uuu Mcˆ2*vvv Mcˆ4* p r s s+8*uuu Mcˆ3*
vvv Mcˆ3−8* p r s s ˆ3+24*uuu Mc*vvv Mc ˆ3* p r s s ˆ2+24*uuu Mc*vvv Mc*
p r s s ˆ2−6*uuu Mcˆ2*vvv Mc ˆ2* p r s s +36*uuu Mc*vvv Mcˆ2*
p r s s ˆ2+(−(−72*uuu Mcˆ4*vvv Mcˆ3−720*uuu Mcˆ3*vvv Mcˆ3* p r s s −756*
uuu Mcˆ2*vvv Mc* p r s s ˆ2−432*uuu Mcˆ3*vvv Mc ˆ4* p r s s −3060*uuu Mcˆ2*
vvv Mc ˆ3* p r s s ˆ2 −3024* p r s s ˆ3*uuu Mc*vvv Mc −2664*uuu Mcˆ2*
vvv Mc ˆ4* p r s s ˆ2 −1296*uuu Mcˆ2*vvv Mcˆ2* p r s s ˆ2 −7920*uuu Mc*
vvv Mc ˆ2* p r s s ˆ3+288*uuu Mc* p r s s ˆ3−825*uuu Mcˆ2*vvv Mc ˆ5*
p r s s ˆ2 −9576* p r s s ˆ3*uuu Mc*vvv Mcˆ3+1152*vvv Mc* p r s s ˆ4 −6000*
uuu Mc*vvv Mcˆ4* p r s s ˆ3+576*vvv Mc ˆ2* p r s s ˆ4+1152* p r s s ˆ4 −2304*
uuu Mc*vvv Mcˆ5* p r s s ˆ3+192* p r s s ˆ4*vvv Mcˆ3−512*uuu Mc*vvv Mc ˆ6*
p r s s ˆ 3 ) /vvv Mc ) ˆ ( 1 / 2 )*uuu Mc*vvv Mc ˆ 2 ) ˆ ( 1 / 3 ) / ( uuu Mc*
vvv Mc ˆ 3 )+2*(−(uuu Mcˆ2*vvv Mc ˆ 2 )−6*uuu Mc*vvv Mc ˆ2* p r s s+2*
p r s s ˆ2−4*uuu Mc*vvv Mcˆ3* p r s s −4*uuu Mc*vvv Mc* p r s s ) / ( uuu Mc*
vvv Mc ˆ3* ( 3 6*uuu Mcˆ2*vvv Mc ˆ3* p r s s +21*uuu Mcˆ2*vvv Mcˆ4* p r s s+8*
uuu Mcˆ3*vvv Mcˆ3−8* p r s s ˆ3+24*uuu Mc*vvv Mc ˆ3* p r s s ˆ2+24*uuu Mc*
vvv Mc* p r s s ˆ2−6*uuu Mcˆ2*vvv Mcˆ2* p r s s +36*uuu Mc*vvv Mc ˆ2*
p r s s ˆ2+(−(−72*uuu Mcˆ4*vvv Mcˆ3−720*uuu Mcˆ3*vvv Mcˆ3* p r s s −756*
uuu Mcˆ2*vvv Mc* p r s s ˆ2−432*uuu Mcˆ3*vvv Mc ˆ4* p r s s −3060*uuu Mcˆ2*
vvv Mc ˆ3* p r s s ˆ2 −3024* p r s s ˆ3*uuu Mc*vvv Mc −2664*uuu Mcˆ2*
vvv Mc ˆ4* p r s s ˆ2 −1296*uuu Mcˆ2*vvv Mcˆ2* p r s s ˆ2 −7920*uuu Mc*
vvv Mc ˆ2* p r s s ˆ3+288*uuu Mc* p r s s ˆ3−825*uuu Mcˆ2*vvv Mc ˆ5*
p r s s ˆ2 −9576* p r s s ˆ3*uuu Mc*vvv Mcˆ3+1152*vvv Mc* p r s s ˆ4 −6000*
uuu Mc*vvv Mcˆ4* p r s s ˆ3+576*vvv Mc ˆ2* p r s s ˆ4+1152* p r s s ˆ4 −2304*
uuu Mc*vvv Mcˆ5* p r s s ˆ3+192* p r s s ˆ4*vvv Mcˆ3−512*uuu Mc*vvv Mc ˆ6*
p r s s ˆ 3 ) /vvv Mc ) ˆ ( 1 / 2 )*uuu Mc*vvv Mc ˆ 2 ) ˆ ( 1 / 3 ) )−(−3*uuu Mc*
vvv Mcˆ2+2* p r s s −2*uuu Mc*vvv Mcˆ3−2*uuu Mc*vvv Mc ) / ( uuu Mc*
vvv Mc ˆ 3 ) ) ˆ ( 1 / 2 ) −(1/2)* ( ( 1 / 2 ) *(−4*uuu Mc*vvv Mcˆ3−3*uuu Mc*
vvv Mc ˆ 2 ) ˆ 2 / ( uuu Mcˆ2*vvv Mc ˆ 6 ) −(9*uuu Mc*vvv Mcˆ2−6* p r s s+6*
uuu Mc*vvv Mcˆ3+6*uuu Mc*vvv Mc ) / ( uuu Mc*vvv Mc ˆ 3 ) −(36*uuu Mcˆ2*
vvv Mc ˆ3* p r s s +21*uuu Mcˆ2*vvv Mcˆ4* p r s s+8*uuu Mcˆ3*vvv Mcˆ3−8*
p r s s ˆ3+24*uuu Mc*vvv Mc ˆ3* p r s s ˆ2+24*uuu Mc*vvv Mc* p r s s ˆ2−6*
uuu Mcˆ2*vvv Mc ˆ2* p r s s +36*uuu Mc*vvv Mcˆ2* p r s s ˆ2+(−(−72*
uuu Mcˆ4*vvv Mcˆ3−720*uuu Mcˆ3*vvv Mcˆ3* p r s s −756*uuu Mcˆ2*
vvv Mc* p r s s ˆ2−432*uuu Mcˆ3*vvv Mc ˆ4* p r s s −3060*uuu Mcˆ2*vvv Mc ˆ3*
p r s s ˆ2 −3024* p r s s ˆ3*uuu Mc*vvv Mc −2664*uuu Mcˆ2*vvv Mc ˆ4*
p r s s ˆ2 −1296*uuu Mcˆ2*vvv Mcˆ2* p r s s ˆ2 −7920*uuu Mc*vvv Mcˆ2*
p r s s ˆ3+288*uuu Mc* p r s s ˆ3−825*uuu Mcˆ2*vvv Mc ˆ5* p r s s ˆ2 −9576*
p r s s ˆ3*uuu Mc*vvv Mcˆ3+1152*vvv Mc* p r s s ˆ4 −6000*uuu Mc*vvv Mc ˆ4*
p r s s ˆ3+576*vvv Mcˆ2* p r s s ˆ4+1152* p r s s ˆ4 −2304*uuu Mc*vvv Mc ˆ5*
p r s s ˆ3+192* p r s s ˆ4*vvv Mcˆ3−512*uuu Mc*vvv Mc ˆ6*
p r s s ˆ 3 ) /vvv Mc ) ˆ ( 1 / 2 )*uuu Mc*vvv Mc ˆ 2 ) ˆ ( 1 / 3 ) / ( uuu Mc*
vvv Mc ˆ 3 )−2*(−(uuu Mcˆ2*vvv Mc ˆ 2 )−6*uuu Mc*vvv Mc ˆ2* p r s s+2*
p r s s ˆ2−4*uuu Mc*vvv Mcˆ3* p r s s −4*uuu Mc*vvv Mc* p r s s ) / ( uuu Mc*
vvv Mc ˆ3* ( 3 6*uuu Mcˆ2*vvv Mc ˆ3* p r s s +21*uuu Mcˆ2*vvv Mcˆ4* p r s s+8*
uuu Mcˆ3*vvv Mcˆ3−8* p r s s ˆ3+24*uuu Mc*vvv Mc ˆ3* p r s s ˆ2+24*uuu Mc*
vvv Mc* p r s s ˆ2−6*uuu Mcˆ2*vvv Mcˆ2* p r s s +36*uuu Mc*vvv Mc ˆ2*
p r s s ˆ2+(−(−72*uuu Mcˆ4*vvv Mcˆ3−720*uuu Mcˆ3*vvv Mcˆ3* p r s s −756*
uuu Mcˆ2*vvv Mc* p r s s ˆ2−432*uuu Mcˆ3*vvv Mc ˆ4* p r s s −3060*uuu Mcˆ2*
vvv Mc ˆ3* p r s s ˆ2 −3024* p r s s ˆ3*uuu Mc*vvv Mc −2664*uuu Mcˆ2*
vvv Mc ˆ4* p r s s ˆ2 −1296*uuu Mcˆ2*vvv Mcˆ2* p r s s ˆ2 −7920*uuu Mc*
vvv Mc ˆ2* p r s s ˆ3+288*uuu Mc* p r s s ˆ3−825*uuu Mcˆ2*vvv Mc ˆ5*
p r s s ˆ2 −9576* p r s s ˆ3*uuu Mc*vvv Mcˆ3+1152*vvv Mc* p r s s ˆ4 −6000*
uuu Mc*vvv Mcˆ4* p r s s ˆ3+576*vvv Mc ˆ2* p r s s ˆ4+1152* p r s s ˆ4 −2304*
uuu Mc*vvv Mcˆ5* p r s s ˆ3+192* p r s s ˆ4*vvv Mcˆ3−512*uuu Mc*vvv Mc ˆ6*
p r s s ˆ 3 ) /vvv Mc ) ˆ ( 1 / 2 )*uuu Mc*vvv Mc ˆ 2 ) ˆ ( 1 / 3 ) ) +(−3*uuu Mc*
vvv Mcˆ2+2* p r s s −2*uuu Mc*vvv Mcˆ3−2*uuu Mc*vvv Mc ) / ( uuu Mc*
vvv Mc ˆ 3 ) +((9*uuu Mc*vvv Mcˆ2−6* p r s s+6*uuu Mc*vvv Mcˆ3+6*uuu Mc*
18
vvv Mc ) *(−4*uuu Mc*vvv Mcˆ3−3*uuu Mc*vvv Mc ˆ 2 ) / ( uuu Mcˆ2*
vvv Mc ˆ 6 )−2*(−4*uuu Mc*vvv Mcˆ3−9*uuu Mc*vvv Mcˆ2−12*uuu Mc*
vvv Mc−6*uuu Mc ) / ( uuu Mc*vvv Mc ˆ 3 ) −(1/4)*(−4*uuu Mc*vvv Mcˆ3−3*
uuu Mc*vvv Mc ˆ 2 ) ˆ 3 / ( uuu Mcˆ3*vvv Mc ˆ 9 ) ) / ( ( 1 / 4 )*(−4*uuu Mc*
vvv Mcˆ3−3*uuu Mc*vvv Mc ˆ 2 ) ˆ 2 / ( uuu Mcˆ2*vvv Mc ˆ 6 ) −(9*uuu Mc*
vvv Mcˆ2−6* p r s s+6*uuu Mc*vvv Mcˆ3+6*uuu Mc*vvv Mc ) / ( uuu Mc*
vvv Mc ˆ 3 ) +(36*uuu Mcˆ2*vvv Mc ˆ3* p r s s +21*uuu Mcˆ2*vvv Mcˆ4*
p r s s+8*uuu Mcˆ3*vvv Mcˆ3−8* p r s s ˆ3+24*uuu Mc*vvv Mcˆ3* p r s s ˆ2+24*
uuu Mc*vvv Mc* p r s s ˆ2−6*uuu Mcˆ2*vvv Mc ˆ2* p r s s +36*uuu Mc*
vvv Mc ˆ2* p r s s ˆ2+(−(−72*uuu Mcˆ4*vvv Mcˆ3−720*uuu Mcˆ3*vvv Mcˆ3*
p r s s −756*uuu Mcˆ2*vvv Mc* p r s s ˆ2−432*uuu Mcˆ3*vvv Mc ˆ4* p r s s −3060*
uuu Mcˆ2*vvv Mc ˆ3* p r s s ˆ2 −3024* p r s s ˆ3*uuu Mc*vvv Mc −2664*
uuu Mcˆ2*vvv Mc ˆ4* p r s s ˆ2 −1296*uuu Mcˆ2*vvv Mcˆ2* p r s s ˆ2 −7920*
uuu Mc*vvv Mcˆ2* p r s s ˆ3+288*uuu Mc* p r s s ˆ3−825*uuu Mcˆ2*vvv Mc ˆ5*
p r s s ˆ2 −9576* p r s s ˆ3*uuu Mc*vvv Mcˆ3+1152*vvv Mc* p r s s ˆ4 −6000*
uuu Mc*vvv Mcˆ4* p r s s ˆ3+576*vvv Mc ˆ2* p r s s ˆ4+1152* p r s s ˆ4 −2304*
uuu Mc*vvv Mcˆ5* p r s s ˆ3+192* p r s s ˆ4*vvv Mcˆ3−512*uuu Mc*vvv Mc ˆ6*
p r s s ˆ 3 ) /vvv Mc ) ˆ ( 1 / 2 )*uuu Mc*vvv Mc ˆ 2 ) ˆ ( 1 / 3 ) / ( uuu Mc*
vvv Mc ˆ 3 )+2*(−(uuu Mcˆ2*vvv Mc ˆ 2 )−6*uuu Mc*vvv Mc ˆ2* p r s s+2*
p r s s ˆ2−4*uuu Mc*vvv Mcˆ3* p r s s −4*uuu Mc*vvv Mc* p r s s ) / ( uuu Mc*
vvv Mc ˆ3* ( 3 6*uuu Mcˆ2*vvv Mc ˆ3* p r s s +21*uuu Mcˆ2*vvv Mcˆ4* p r s s+8*
uuu Mcˆ3*vvv Mcˆ3−8* p r s s ˆ3+24*uuu Mc*vvv Mc ˆ3* p r s s ˆ2+24*uuu Mc*
vvv Mc* p r s s ˆ2−6*uuu Mcˆ2*vvv Mcˆ2* p r s s +36*uuu Mc*vvv Mc ˆ2*
p r s s ˆ2+(−(−72*uuu Mcˆ4*vvv Mcˆ3−720*uuu Mcˆ3*vvv Mcˆ3* p r s s −756*
uuu Mcˆ2*vvv Mc* p r s s ˆ2−432*uuu Mcˆ3*vvv Mc ˆ4* p r s s −3060*uuu Mcˆ2*
vvv Mc ˆ3* p r s s ˆ2 −3024* p r s s ˆ3*uuu Mc*vvv Mc −2664*uuu Mcˆ2*
vvv Mc ˆ4* p r s s ˆ2 −1296*uuu Mcˆ2*vvv Mcˆ2* p r s s ˆ2 −7920*uuu Mc*
vvv Mc ˆ2* p r s s ˆ3+288*uuu Mc* p r s s ˆ3−825*uuu Mcˆ2*vvv Mc ˆ5*
p r s s ˆ2 −9576* p r s s ˆ3*uuu Mc*vvv Mcˆ3+1152*vvv Mc* p r s s ˆ4 −6000*
uuu Mc*vvv Mcˆ4* p r s s ˆ3+576*vvv Mc ˆ2* p r s s ˆ4+1152* p r s s ˆ4 −2304*
uuu Mc*vvv Mcˆ5* p r s s ˆ3+192* p r s s ˆ4*vvv Mcˆ3−512*uuu Mc*vvv Mc ˆ6*
p r s s ˆ 3 ) /vvv Mc ) ˆ ( 1 / 2 )*uuu Mc*vvv Mc ˆ 2 ) ˆ ( 1 / 3 ) )−(−3*uuu Mc*
vvv Mcˆ2+2* p r s s −2*uuu Mc*vvv Mcˆ3−2*uuu Mc*vvv Mc ) / ( uuu Mc*
vvv Mc ˆ 3 ) ) ˆ ( 1 / 2 ) ) ˆ ( 1 / 2 ) ; : 0
l o c a l l p c = v i n e t f l a t t i c e c *A0 Mc ;
’−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
’ Structure
’−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
r bragg 0.8278468831
MVW( 9 7 1 . 0 0 2 4 5 7 7 , 2 4 1 . 9 6 5 5 8 3 6 , 1 0 0 )
s c a l e @ 1 . 9 9 0 6 1 1 6 1 e −005
s p a c e g r o u p Pbnm
Phase LAC 1 on cm ( 8 2 . 7 3 3 4 6 4 3 1 )
Phase Density g on cm3 ( 6.663710563)
’ I f you want t o d e t e r m i n e t h e bulk i n v e r t e d t h i r d o r d e r Vinet
EoS a ppr o xima tio n ,
’ p l e a s e change t h e l a t t i c e p a r a m e t e r s t o t h e e q u a t i o n s which a r e
commented behind
’ the p a r t i c u l a r l i n e !
a = lp a ;
’ a =l p a b u l k ;
19
b = lp b ;
c = lp c ;
’ b =l p b b u l k ;
’ c =l p c b u l k ;
s i t e La1 num posns 4 x =La1 x ;
=L a 1 o c c ; beq =La1 beq ;
s i t e Fe1 num posns 4 x =Fe1 x ;
=F e 1 o c c ; beq =Fe1 beq ;
s i t e O1 num posns 4 x =O1 x ;
=O1 occ ; beq =O1 beq ;
s i t e O2 num posns 8 x =O2 x ;
=O2 occ ; beq =O2 beq ;
y =La1 y ; z =La 1 z ; o c c La+3
y =Fe1 y ; z =Fe1 z ; o c c Fe+3
y =O1 y ;
z =O1 z ;
o c c O−2
y =O2 y ;
z =O2 z ;
o c c O−2
}
’−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
’ l i s t o f a l l da ta f i l e s with c o r r e s p o n d i n g i n d i v i d u a l p a r a m e t e r s
’−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
xdd ” r 5 9 6 0 4 2 . xy”
l o c a l ! p r e s s u r e l o c 0 . 0 ’ l o c a l p r e s s u r e v a l u e ( i n GPa) f o r t h a t
measurement
Header info
bkg @ 1 5 5 1 . 7 4 2 2 8 ‘ 0 . 4 4 1 9 3 3 8 9 1 3 3 7 . 7 5 2 1 8 4 ‘ 0 . 7 2 3 6 9 1 2 7 8
− 1 5 4 .4 6 5511 ‘ 0 . 6 9 3 6 7 7 0 4 − 1 .6 1 7 9 3378 ‘ 0 . 6 6 2 1 0 4 4 8
7.96843609 ‘ 0 .648085549 1.64117618 ‘ 0 .620984183
− 1 0 .8 8 2 3511 ‘ 0 . 6 1 4 7 4 5 1 1 0 . 3 3 3 6 3 2 1 8 2 ‘ 0 . 5 8 2 6 4 0 9 4 2
1 0 . 2 2 4 0 3 8 ‘ 0 . 5 8 7 6 0 8 7 6 4 − 1 1 .7 7 7 764 ‘ 0 . 5 3 5 6 1 1 4 3 4
2.30890721 ‘ 0 .523207601
Structure
l o c a l c s l 5 7 9 . 1 0 7 2 7 ‘ 2 2 . 1 3 4 3 1 min = 0 . 3 ; max =10000;
l o c a l c s g 6 8 . 7 8 8 2 6 ‘ 0 . 1 9 3 5 9 min = 0 . 3 ; max =10000;
local stl
0 . 0 3 8 6 9 ‘ 0 . 0 0 1 4 7 min = 0 . 0 0 0 1 ; max =5;
CS L ( c s l )
CS G( c s g )
Strain G ( s t l )
Strain L ( s t l )
l o c a l lp a predet 5.55696 ‘ 0 .00007
l o c a l lp b predet 5.56405 ‘ 0 .00009
l o c a l lp c pr e de t 7.85508 ‘ 0 .00011
l o c a l l p a b u l k = multparam * l p a p r e d e t ;
l o c a l l p b b u l k = multparam * l p b p r e d e t ;
l o c a l l p c b u l k = multparam * l p c p r e d e t ;
l o c a l La1 x 0 . 0 0 6 0 2 ‘ 0 . 0 0 0 2 6
l o c a l La1 y 0 . 0 2 9 1 7
l o c a l ! La 1 z 0 . 2 5
l o c a l ! Fe1 x 0
l o c a l ! Fe1 y 0 . 5
l o c a l ! Fe1 z 0
l o c a l O1 x 0 . 0 7 6 0 8 ‘ 0 . 0 0 1 9 2
l o c a l O1 y 0 . 4 8 6 5 0 ‘ 0 . 0 0 0 7 6
l o c a l ! O1 z 0 . 2 5
20
l o c a l O2 x −0.27783 ‘ 0 . 0 0 1 3 4
l o c a l O2 y 0 . 2 8 3 4 4 ‘ 0 . 0 0 1 2 5
l o c a l O2 z 0 . 0 3 8 5 0 ‘ 0 . 0 0 0 7 6
’ Occ p a r a m e t e r s
local
local
local
local
! La1 occ 1
! Fe1 occ 1
! O1 occ 1
! O2 occ 1
’ Temperature / Displa cement F a c t o r s
local
local
local
local
La1 beq 0 . 3 min =0; max =10;
Fe1 beq =La1 beq ;
min =0; max =10;
O1 beq =La1 beq ;
min =0; max =10;
O2 beq =La1 beq ;
min =0; max =10;
phase name ” r 5 9 6 0 4 2 ”
xdd ”Z : \ Data \LaFeO3 High−P r e s s u r e \ Daten (XY) \ r 5 9 6 0 4 3 . xy”
...
21
4
As2O5 as an example for high B0′ values
The arsenic pentoxide (space group P 21 21 21 at ambient conditions) builds a framework of corner-sharing distorted
AsO6 -octahedra and AsO4 -tetrahedra (Jansen 1977, Locherer et al. 2010). As the empty channels of As2 O5 provide
much additional space for the framework structure to deform under pressure, it is a good candidate to investigate
channel framework structures which tend to have much smaller bulk moduli in contrast to perovskites which are
normally described as three-dimensional frameworks with higher moduli (Hazen & Finger 1979). Because nitrogen
as the used pressure medium has a very low hydrostatic limit (3.0 GPa according to Angel et al. (2007)), the
parameterized data of Locherer et al. (2010) were taken in the non-hydrostatic regime and therefore in a range of
2.74 GPa ≤ P ≤ 19.5 GPa.
Following the same procedure as for LaFeO3 , parametric refinements were carried out determining the lattice parameters in a first run with the Le Bail method (LeBail et al. 1988) and fixing them in a second run to fit equation
2. The determination of the linear modulus etc. were performed as described above. Fitted curves obtained by
the parametric refinement as well as the data points of the sequential refinement can be seen in figure 1. In table
2 the obtained values of the inverted approximative Vinet EoS can be found compared to the values of the pure
Vinet EoS determined by (Locherer et al. 2010) with the program EOSFIT 5.2 (Angel 2000).
Figure 1: Volume and lattice parameters (a) and the corresponding f-F plot (b) for As2 O5 (space group P 21 21 21 )
obtained by sequential Rietveld refinement (points) and by parametric Rietveld refinement with an inverted third
order Vinet EoS approximation. The dotted line for the lattice parameter a is fitted by another solution of the
inverted third order Vinet EoS approximation as one has to respect the negative value of B0′ .
Table 2: Bulk modulus, its pressure derivative and the volume at ambient conditions of As2 O5 determined by the
inverted third order Vinet EoS approximation as well as the linearized versions in the non-hydrostatic regime of
nitrogen which is used as pressure medium (2.74 GPa ≤ P ≤ 19.5 GPa). Due to reasons of comparison the values
obtained by Locherer et al. (2010) with a usual Vinet EoS are also listed. The values for the lattice direction a
are obtained by another solution of the inverted third order Vinet EoS approximation as one has to respect the
negative value of B0′ .
inv. approx. Vinet (3rd ord.)
V0
B0
B0′
a0
B0a
′
B0a
b0
B0b
′
B0b
c0
B0c
′
B0c
337.038(1)
83.6(1)
4.7(1)
8.674(1)
117.0(1)
-2.1(1)
8.513(1)
15.5(2)
14.0(1)
4.624(1)
207.1(13)
5.3(2)
Vinet
(Locherer et al. 2010)
337.93(5)
78.1(24)
5.29(45)
8.65(1)
125.06(167592)
-2.28(21060)
8.45(30)
23.60(6749)
11.95(3684)
4.626(2)
186.88(183733)
8.36(400.53)
In table 2, the values for the direction of the lattice parameter a cannot be directly obtained by the given
linearized inverted third order Vinet EoS approximation displayed in equation 2. This is due to one of the seldom
22
cases, where the pressure derivative becomes negative and this leads to a complex solution of equation 2. To
overcome this issue, it is possible to use another solution of the inverse function of the third order Vinet EoS approximation (chapter 1, equation 3) where a negative pressure derivative does not result in a complex solution. In
general this will lead to the correct results for the linear modulus, its pressure derivative and the lattice parameter
at ambient conditions.
Comparing the values of the linear moduli and the pressure derivatives it seems that the magnitude of the determined values is the same within three estimated standard deviations for both equations, bearing in mind that the
′
same values are not completely reached. This can be attributed to the large pressure derivatives of B0b
= 11.95 and
′
B0c = 8.36 and the total volume compression of 15% in the investigated pressure range, which is slightly beyond
the validity of the third order Vinet EoS approximation.
23
5
5.1
Graphical overview of P(V) and V(P)
Graphical overview of P(V)
The following graphics are obtained by an analytical investigation of the Vinet EoS and the third order Vinet EoS
approximation.
Figure 2: Selected fixed volume contractions and selected values of the bulk modulus showing the pressure differences between the Vinet EoS and the third order Vinet EoS approximation for the positive parameter space of the
pressure derivative of the bulk modulus B0′ .
Figure 3: 3D map showing the differences of the calculated pressure values between Vinet EoS and third order
Vinet EoS approximation at a fixed volume contraction of 5% for the positive parameter space of the bulk modulus
B0 and its derivative B0′ .
24
Figure 4: 3D map showing the differences of the calculated pressure values between Vinet EoS and third order
Vinet EoS approximation at a fixed volume contraction of 10% for the positive parameter space of the bulk modulus
B0 and its derivative B0′ .
Figure 5: 3D map showing the differences of the calculated pressure values between Vinet EoS and third order
Vinet EoS approximation at a fixed volume contraction of 20% for the positive parameter space of the bulk modulus
B0 and its derivative B0′ .
25
5.2
Graphical overview of V(P)
The following graphics are obtained by an investigation of the numerically inverted Vinet EoS and the inverted
third order Vinet EoS approximation.
Figure 6: Selected fixed pressures and selected values of the bulk modulus showing the volume differences between
the numerical inverted Vinet EoS and the inverted third order Vinet EoS approximation for the positive parameter
space of the pressure derivative of the bulk modulus B0′ .
Figure 7: 3D map showing the volume differences of the calculated volume values between numerical inverted Vinet
EoS and inverted third order Vinet EoS approximation at a fixed pressure of 10 GPa for the positive parameter
space of the bulk modulus B0 and its derivative B0′ .
26
References
Angel, R. J. (2000). Equations of state, in R. M. Hazen, R. T. Downs & P. H. Ribbe (eds), Reviews in Mineralogy
& Geochemistry Volume 41: High-Temperature and High-Pressure Crystal Chemistry, Vol. 41 of Reviews in
Mineralogy & Geochemistry Volume, Mineralogical Society of America, Geochemical Society, Washington D.C.,
chapter 2, pp. 35–59.
Angel, R. J., Bujak, M., Zhao, J., Gatta, G. D. & Jacobsen, S. D. (2007). Effective hydrostatic limits of pressure
media for highpressure crystallographic studies, Journal of Applied Crystallography 40: 26–32.
Cynn, H., Klepeis, J. E., Yoo, C.-S. & Young, D. A. (2002). Osmium has the lowest experimentally determined
compressibility, Physical Review Letters 88(13): 135701–1–4.
Hazen, R. M. & Finger, L. W. (1979). Bulk modulus-volume relationship for cation-anion polyhedra, Journal of
Geophysical Research 84(B12): 6723–6727.
Jansen, M. (1977). Crystal structure of as2 o5 , Angewandte Chemie International Edition 16(5): 314–315.
LeBail, A., Duroy, H. & Fourquet, J. L. . (1988). Ab-initio structure determination of lisbwo6 by x-ray powder
diffraction, Materials Research Bulletin 23: 447–452.
Locherer, T., Halasz, I., Dinnebier, R. & Jansen, M. (2010). Isothermal compressibility and anisotropic structural
response of as2 o5 under pressure, Solid State Communications 150: 201–204.
Raju, S., Mohandas, E. & Raghunathan, V. S. (1997). The pressure derivative of bulk modulus of transition metals:
An estimation using the method of model potentials and a study of the systematics, Journal of Physics and
Chemistry of Solids 58(9): 1367–1373.
27