equations of state ronald cohen geophysical laboratory carnegie institution of washington
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2012 Summer School on Computational Materials Science Quantum Monte Carlo: Theory and Fundamentals July 23–-27, 2012 • University of Illinois at Urbana–Champaign http://www.mcc.uiuc.edu /summerschool/ 2012/. Equations of State Ronald Cohen Geophysical Laboratory - PowerPoint PPT PresentationTRANSCRIPT
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Equations of StateRonald Cohen
Geophysical LaboratoryCarnegie Institution of Washington
2012 Summer School on Computational Materials Science Quantum Monte Carlo: Theory and FundamentalsJuly 23–-27, 2012 • University of Illinois at Urbana–Champaignhttp://www.mcc.uiuc.edu/summerschool/2012/
QMC Summer School 2012 UIUC
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Need for Equations of State
• QMC gives us the energy at a set of points for different structures and volumes
• To predict phase stability and to compare with experiment we need the pressure
• The most stable phase has the lowest free energy, or at zero temperature, the lowest enthapy.
• The relationship among E, V, P, and T is the equation of state.
• Also enthalpy H=E+PV and free energy G=H-TS
CohenQMC Summer School 2012 UIUC 2
Cohen, R. E. & Gulseren, O. Thermal equation of state of tantalum. Phys. Rev. B 63, 224101-224111 (2000).
Ta Thermal equation of state
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Pressure vs. volumeTa isotherms
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Cohen 4
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Residuals for T=0 isotherm:Evidence for electronic transition
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Ta bands and DOS
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V=12.66 Å3 (5 GPa)
V=9.3 Å3 (460 GPa)
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Vinet parameters vs. temperature
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Thermal pressure vs. V
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Thermal pressure vs. T
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Average Thermal Pressure
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QMC Summer School 2012 UIUC
Simple Equation of State for Ta
• P (GPa) = P0K+Pth • P0K is Vinet equation:• x=(V/V0)1/3• P=3 K0 (1-x) exp (3/2 (K0’-1) (1-x))/x2• with V0=123.632 K0=190.95 K0'=3.98 • Pth = 0.00441 T• This should be good to better than ±5 GPa to 9000 K and for V>80 bohr3
(35% compression).
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QMC Summer School 2012 UIUC
An accurate high temperature global equation of state
• T=0 Vinet isotherm• V dependent Thermal Pressure• Heat Capacity
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3
112
10922
82
72
65
20
21
00
0
31
0
0
1123exp3135
12
14
GPa)in K and Rydin E(for 75614710
T+PTT+P)+PxTx+PT+PTT+P=V(PE
)))(x-(K'-(-x))-K'(x-+)()(K'-kV-(
K'-kP+=eE
./k=K
VVx
+EE=E
th
th
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Global free energy fit
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cBN Raman Frequencies
• Within harmonic approx. DFT frequency is reasonable
• But, cBN Raman mode is quite anharmonic
• With anharmonic corrections, DFT frequencies are not so good.
• Compute energy vs. displacement with DMC and do 4th-order fit. Solve 1D Schrodinger eq. to get frequency
• Anharmonic DMC frequency is correct to within statistical error
Cohen QMC Summer School 2012 UIUC 19
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Summary
• Fit your DFT and QMC results to equations of state, carefully.
• Much can be learned from the equation of state, and the parameterizations are very useful, particularly for comparing with experiments or input to other studies.
Cohen QMC Summer School 2012 UIUC 20