Theoretical Studies of ThermodynamicProperties of Condensed Matter underHigh Temperature and High Pressure
Yi Wang
Stockhom 2004
Doctoral DissertationRoyal Institute of Technology
Department of Materials Science and Engineering
KTHISRN KTH/MSE—04/07—SE+AMFY/EX SE-10044 StockholmISBN 91-7283-699-7 SWEDEN
Akademisk avhandling som med tillstånd av Kungl Tekniska Högskolan fram-lägges till offentlig granskning för avläggande av filosofic doktorsexamen ons-dagen den 24 March 2004 kl. 10.00 i B1, MSE, Kungl Tekniska Högskolan,Brinellvägen 23, Stockholm.
c° Yi Wang, February 2004
Tryck: Universitetsservice US AB
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AcknowledgmentThe above presented work has been done under the supervision of DocentRajeev Ahuja and Prof. Börje Johansson. Many other colleagues in our grouphave greatly contributed to my work and enriched my life. Words are notenough to express my cherishing of all the joyful moments of my stay in Swe-den. I would like to say thanks to all those that have given me a helpful handduring the past years.
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AbstractBased on first-principles calculations coupled with the classical mean-fieldpotential approach. A wide range of thermodynamic properties have beenstudied. This includes ambient condition properties, 300-K equation of state,shock-wave Hugoniot, reduction of shock-wave data, high pressure melting,anharmonic effects and etc. The elements Na, Al, Cu, Th, Be, U, Au and Wtogether with the compound MgO have been taken as prototype systems.
The approach is also attempted to go beyond the standard phonon theory bymaking a precise solution for high anomalous phonon modes for some transi-tion metals. The H point phonon of Mo and the T1 N point and the ω pointoscillations of Zr are studied in detail.
ISBN 91-7283-699-7 . ISRN KTH/MSE—04/07—SE+AMFY/EX
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List of Publications
Papers included in this thesis
I) Yi Wang, Rajeev Ahuja, and Börje Johansson, ”A Model for Phase Coex-istence in Phase Transitions”, Int. J. Quantum. Chem. (Proof).
II) Yi Wang, Rajeev Ahuja, and Börje Johansson, ”Mean-Field PotentialApproach to the Quasiharmonic Theory of Solids”, Int. J. Quantum.Chem. 96, 501 (2004); Int. J. Quantum. Chem. 97, 700 (2004).
III) Yi Wang, Rajeev Ahuja, and Börje Johansson, ”LiH under High Pressureand High Temperature: A First-principles Study”, Phys. Stat. Sol (b)235, 470 (2003).
IV) Yi Wang, Rajeev Ahuja, and Börje Johansson, ”Calculated HugoniotCurves of Porous Metal: Copper, Nickel, and Molybdenum”, in ShockCompression of Condensed Matter-2001, edited by M. D. Furnish, N.N. Thadhani, and Y. Horie (American Institute of Physics, New York2002), p67.
V) Yi Wang, Rajeev Ahuja, and Börje Johansson, ”Reduction of Shock-Wave Data with a First-Principles Approach”, J. Appl. Phys. 14, 6616(2002).
VI) Yi Wang, Rajeev Ahuja, and Börje Johansson, ”Thermodynamic Prop-erties of MgO, Be, and W: A Simplified Computational Approach”, J.Phys.: Conden. Matter 14, 10895 (2002).
VII) Y. Wang, R. Ahuja, M. C. Qian, and B. Johansson, ”Accurate QuantumMechanical Treatment of Phonon Instabilty: bcc Zirconium”, J. Phys.:Conden. Matter (Letters to the Editor) 14, L695 (2002).
VIII) Yi Wang, Rajeev Ahuja, and Börje Johansson, ”Thermodynamic Prop-erties at the Earth’s core Conditions and Shock-reduced Isotherm of Iron:A first-principles Study”, J. Phys.: Conden. Matter 14, 7321 (2002).
IX) Y. Wang, R. Ahuja, O. Eriksson, B. Johansson, and G. Grimvall, ”PreciseSolution of H-point Oscillation: Na, Mo, and Fe”, J. Phys.: Conden.Matter (Letters to the Editor) 14, L453 (2002).
X) Yi Wang, Rajeev Ahuja, and Börje Johansson, ”Melting of Iron andOther Metals at Earth’s Core Conditions: A Simplified ComputationalApproach”, Phys. Rev. B. 65, 014104 (2002).
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XI) Yi Wang, Rajeev Ahuja, and Börje Johansson, ”Going to 10 TPa: TheCalculated Hugoniots for Cu, Ta, and Mo”, High Pressure Res. 22, 485(2002).
XII) Yi Wang, Rajeev Ahuja, and Börje Johansson, ”Ab Initio Reductionsof Shock-Wave Data: NaCl of B1 and B2 Structures” (manuscript).
Papers not included in this thesis
1) H. Zheng, Y. Wang and G. Ma, ”Electronic structure of LaNi5 and itshydride LaNi5H7”, Eur. Phys. J. B 29, 61 (2002).
2) W Luo, Y Z Ma, G T Zou, H K Mao, Z C Wang, and Y Wang, ”High-pressure synchrotron studies on TiO2-II nanocrystallite doped with SnO2”,J. Phys.: Conden. Matter 14, 11069 (2002).
3) J. M. Osorio.Guillen, S. I. Simak, Y. Wang, B. Johansson, and R. Ahuja,”Bonding and Elastic Properties of Superconducting MgB2”, Solid StateCommun. 123, 257 (2002).
4) X. Lu and Y. Wang, ”Calculation of the Vibrational Contribution to theGibs Energy of Formation for Al3Sc”, CALPHAD 26, 555 (2002).
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Contents
1 Introduction 1
2 Thermodynamic model based on first-principles calculations 32.1 Phonon theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Debye model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Many-body expansion . . . . . . . . . . . . . . . . . . . . . . . 62.4 Mean-field approximation . . . . . . . . . . . . . . . . . . . . . 7
3 Mean-field potential approach 83.1 Free energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.2 Mean-field potential . . . . . . . . . . . . . . . . . . . . . . . . 83.3 Second order approximation . . . . . . . . . . . . . . . . . . . . 83.4 Hard-sphere potential . . . . . . . . . . . . . . . . . . . . . . . 93.5 Some thermodynamic quantities . . . . . . . . . . . . . . . . . 10
4 Application of the mean-field potential 114.1 Calculational details . . . . . . . . . . . . . . . . . . . . . . . . 114.2 Demonstration of the Helmholtz free-energy: MgO . . . . . . . 124.3 Properties at ambient pressure . . . . . . . . . . . . . . . . . . 134.4 300-K static EOS . . . . . . . . . . . . . . . . . . . . . . . . . . 174.5 Hugoniot state . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.6 Grüneisen gamma along the principal Hugoniot . . . . . . . . . 204.7 Temperature along the principal Hugoniot . . . . . . . . . . . . 204.8 Reduction of shock-wave data . . . . . . . . . . . . . . . . . . . 234.9 Porous Hugoniot . . . . . . . . . . . . . . . . . . . . . . . . . . 254.10 Isentropic release . . . . . . . . . . . . . . . . . . . . . . . . . . 274.11 High pressure melting . . . . . . . . . . . . . . . . . . . . . . . 274.12 Specific heat for Iron under the Earth’s core condition . . . . . 294.13 Temperature dependence of the thermal pressure for MgO . . . 29
5 Beyond the harmonic theory: precise solution for a specialphonon mode 32
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6 Concluding remarks 39
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Chapter 1
Introduction
The study of thermodynamic propertis of materials is of great importance inorder to extend our knowledge about their specific behaviors when put undersevere constraints such as high pressure and high temperature environment.In order to proceed the hydrodynamic calculations for solving problems in geo-physics, astrophysics, particles accelerator, fission and fusion reactor, and etc.one needs an accurate knowledge of the equation-of-state (EOS) and relatedthermodynamic properties. This is particularly true since the developmentof modern technologies always depend on new advances and innovations inmaterials science to reach higher performances. The current pressure frontierlies in the pressure range 100 to 1000 GPa. This range opens up new excit-ing possibilities to study matter under controlled conditions at superextremecompressions.
Experimentally, the thermodynamic property of condensed matter at megabarpressures can now be probed in both static, low-temperature diamond-anvil-cell (DAC) experiments [1, 2, 3, 4, 5, 6] and in dynamic, high-temperatureshock-wave experiments [7, 8, 9, 10, 11, 12]. The DAC technique is suitablefor the ordinary laboratory in which samples can be studied under controlledconditions. Using laser heating and refinements in instrumentation, temper-atures of about 3000 K can be reached up to pressure of the order of 200- 300 GPa, while at room temperature Ruoff et al [2] reported an ultrahighDAC pressure of 560 GPa. Even higher temperatures and higher pressureconditions can be achieved by shock-wave methods. Accordingly, there is anurgent need to develop theoretical methods which can be used to facilitate theinterpretation of the new data at extreme experimental conditions.
Theoretically, the study of the temperature dependence of the propertiesof materials requires a proper account of nuclear motions and thermal exci-tation of electrons. Over the past decades, density-functional theory[13, 14]has successfully provided a framework within which ground-state properties
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of many physical systems can be calculated.[15, 16, 17, 18, 19, 20, 21, 22]However, while a high accuracy can very often be obtained in the 0-K ab ini-tio calculations,[17, 18] the ab initio thermodynamic calculation still remainsa great challenge to us. The basic difficulty in the systematic theoreticalcalculation of the thermodynamic properties of a substance by means of sta-tistical physics is how to incorporate correctly the structurally complicatedinterparticle interaction of the many-body problem. In this regard, some the-oretical methods have been developed, such as the Debye-Grüneisen theory byMoruzzi, Janak, and Schwarz,[19] the elaborate generalized pseudopotentialtheory (GPT) by Moriarty [23], the delicate tight-binding total-energy classi-cal cell model by Wasserman, Stixrude, and Cohen [24], and the well-knownfree-volume theory by Kirkwood [25] and by Vashchenko and Zubarev [26].
The subject of this work is divided into two parts. The major one is usingthe classical Mean-field potential (MFP) [27] approach to calculate a widerange of thermodynamic properties. The second part turns into quantumtheory, which is focused on the anharmonic effect which can not be accountedfor by the traditional phonon theory.
The rest of this work is organized as follows. In Chapter 2 we present a briefintroduction to some of the thermodynamic model based on first-principlescalculations, including the phonon theory, the Debye model, the many bodytheory, and the mean-field method. In Chapter 3 the MFP approach is sum-marized. Chapter 4 illustrates the application of the MFP approach. Chapter5 discusses how to derive a precise quantum solution for certain anomalousphonon modes for which the anharmonic effect is important
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Chapter 2
Thermodynamic model basedon first-principles calculations
If the Helmholtz free-energy for a substance is given as an explicit functionof volume and temperature, we can calculate all other thermodynamic para-meters. Let us consider a system with a given average atomic volume V andtemperature T . The Helmholtz free-energy F(V, T) per ion can be written as[28]
F (V, T ) = Ec(V ) + Fion(V, T ) + Fel(V, T ), (2.1)
where Ec represents the 0-K total energy, Fion the vibrational free-energy ofthe lattice ions, and Fel the free-energy due to the thermal excitations ofelectrons.
The calculation of Fel is relatively simple using the Mermin statistics. As agood approximation, one can adopt Fel = Eel−TSel, where the bare electronicentropy Sel takes the form [29]
Sel(V, T ) = −kBZn(², V )[f ln f + (1− f) ln(1− f)]d², (2.2)
where n(², V ) is the electronic density of states (DOS) and f is the Fermidistribution. With respect to Eq. (2.2), the energy Eel due to the electronexcitations can be expressed as
Eel(V, T ) =
Zn(², V )f²d²−
Z ²Fn(², V )²d², (2.3)
where ²F is the Fermi energy.The most difficult task is how to treat the term Fion in Eq. (2.1). It is
known that the vibrational contribution to the partition function takes theform [30]
Zion = exp(−NFion/kBT ), (2.4)
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where N is the total number of lattice ions. A number of approaches tocalculate Zion or Fion have been developed by many other workers. In thefollowing sections, we will briefly describe some selected methods.
2.1 Phonon theory
Theoretically, perhaps the harmonic approximation [31] is the most reasonableone. In this approximation, the potential of the system is truncated to thesecond order in the Taylor’s expansion. Consider a system consisting of Natoms. Le mi be the mass of atom i, the Hamiltonian of the system can thenbe expressed as [32]:
H = −12
Xi
h2
mi∇2 + 1
2
Xi,j
Xα,β
Φαβ(i, j)uα(i)uβ(j), (2.5)
where h is the Plank constant, uα(i) (α = 1, 2, 3) is the rectangular displace-ment of atom i from its static position, and Φαβ(i, j) is the so-called forceconstant in the form
Φαβ(i, j) =∂2U
∂uα(i)∂uβ(j), (2.6)
where U is the total potential of the system. In practice, it is more convenientto eliminate the mass mi from the equation by defining
wα(i) = (mi)1/2uα(i), (2.7)
andXαβ(i, j) =
1
(mimj)1/2Φαβ(i, j), (2.8)
From this one obtains
H = −12
Xi
XαPα2(i) +
1
2
Xi,j
Xα,β
Xαβ(i, j)wα(i)wβ(j), (2.9)
where
Pα(i) = −ih ∂
∂wα(i)(2.10)
represents the canonical momentum corresponding to wα(i).The free energy of the system can now be calculated through[33]
Fion =kBT
N
Xm
ln
·2 sinh
µhωm2kBT
¶¸, (2.11)
where m = 1, 2, ..., 3N , with ω being the phonon frequency calculated by¯Xαβ(i, j)− ω2δαβδij
¯= 0. (2.12)
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At present, the implementations of the harmonic approximation in theframework of first-principles calculations are branched into two major ways:a) the linear-response theory and b) the frozen phonon method. Many cal-culations [34, 35, 36, 37] have appeared with very good agreement with theobserved phonon-dispersion curves for the solid at low temperatures.
It is generally argued that the harmonic approximation can provide rea-sonable results for many simple materials from 0 K to approximately half oftheir Debye temperatures [38]. However, in the case of Ag the results are stillgood even when the temperature is as high as the melting points [39] whilein the case of MgO the results are much worse for high temperature [40]. Inthe case of the frozen phonon approximation, to calculate the force constantΦαβ(i, j), one generally express the total energy of the system as
U = Ec(V ) + Φαβ(i, j)u2 + bu3 + cu4. (2.13)
It is a common practice [42, 43, 44] to only consider the displacement u for arange of less than 4% of the lattice constant when calculating the Φαβ(i, j). Bycomparison, according to the Lindemann law for in melting, the displacementof an atom can reach up to 10% of the lattice constant.
However, in the real physical world the systems do not always behave inthe same simple manner. Exceptional cases are for example the transitionmetals of group 3 (Sc, Y, La) and 4 (Ti, Zr, Hf) and their alloys where thehigh temperature bcc phase exhibits intrinsic phonon anomalies towards thehcp and ω structures. Inelastic neutron scattering measurements [45, 46, 47,48] on bcc Zr and Zr alloys show that: a) the transverse [110] phonon at(1/2,1/2,0) (T1 N point) as well as the longitudinal phonon at (2/3,2/3,2/3){the so-called ω point, equivalent to the TA2 [111] phonon at (2/3,1/3,1/3)}have very low frequencies, very scattering line shapes, and strong quasielasticintensities; b) the measured phonon spectrum and the quasielastic scatteringindicate a strong asymmetry around the (2,2,2) bcc Bragg peak along the[112] direction which seems to violate the expected bcc symmetry and whichwas assumed to be persistent ω fluctuations [48, 49] in the bcc phase. Theseresults demonstrate a number of peculiarities of lattice vibrations, which donot agree with the conventional phonon picture.
2.2 Debye model
The Debye model originates from the continuous medium theory. From thespecific quantity- the Debye temperature ΘD- the function Fion in Eq. (2.1)can be written as [30, 19]
Fion(V, T ) =9
8kBΘD − kBT
hD(ΘD/T )− 3 ln(1− eΘD/T )
i, (2.14)
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where D(x) is the standard Debye function defined by [30]
D(x) =3
x3
Z x
0
z3dz
ez − 1 . (2.15)
In the framework of first-principles calculation, the difficulty lies in howto calculate ΘD. It can be shown that [19] under specific assumptions ΘD isgiven by
ΘD = 67.88
µRB
m
¶1/2, (2.16)
where R is the average Wigner-Seitz atomic radii in the unit of a.u., m theaverage atomic weight, and B the bulk modulus in the unit of kbar. Moruzziet al. found this value to be too large, and reduced it by a factor of 0.617,resulting in
ΘD = 41.63
µRB
m
¶1/2. (2.17)
We note that in Eq. (2.17), ΘD is already dependent on R. In other words,we can directly derive the expression for the Grüneisen parameter followingSlater [51];
γ =∂ lnΘD∂ lnV
. (2.18)
However, it is found the expression for the Grüneisen parameter by Slater[51]gives rise to too large thermal expansions. Therefore Moruzzi et al. [19] usedthe expression for the Grüneisen parameter due to Dugdale and MacDonald[50]. This makes an inselfconsistancy between the two expressions of Debyetemperature and the Grüneisen parameter.
To overcome this shortcoming, we instead define
ΘD(λ) =1
kB
½1
2m
1
R2λ∂
∂R
·R2λ
∂Ec(R)
∂R
¸¾1/2, (2.19)
where Ec(R) is the 0 K total energy. In addition to the one (λ = −1) used byMoruzzi et al [19], two new expressions in accordance with the thermodynamictheory of Dugdale and MacDonald [50] (λ = 0) and with the free volume theory[26] (λ = 1), are derived. The Grüneisen law is now implied explicitly in theDebye temperature expression and therefore no additional conditions for theGrüneisen constant are needed for realistic thermodynamic calculations.
2.3 Many-body expansion
In this form, the total potential of the system is expressed in a series ofmany-body interactions. Moriarty[23] developed a first-principles generalized
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pseudopotential theory (GPT), by which
U = v0(V )+1
2N
Xi6=jv2(ij)+
1
6N
Xi6=j 6=k
v3(ijk)+1
24N
Xi6=j 6=k 6=l
v4(ijkl), (2.20)
where v0, v3, v3, and v4 represent, respectively, the static, the two-, three-,and four-body interactions. Only a limited number of applications [41] of thismethod have been published.
2.4 Mean-field approximation
Within the so called mean-field approximation, the classical form of Zion canbe expressed as[24, 25]
Zion =
µmkBT
2πh2
¶3N/2 µZexp(−g(r, V )/kBT )dr
¶N, (2.21)
where m is the mass of the lattice ion. Notice that g(r, V ) in Eq. (2.21) isreferred as the mean-field potential (MFP).
The central issue of the mean-field theory is to find appropriate ways ofhow to calculate the MFP g(r, V ). In this regard, the free-volume theory [25]can be chosen to calculate the MFP g(r, V ) from the average of the empiricallyderived pair-wise potentials, while the tight-binding total-energy classical cellmodel [24] can be used to calculate the MFP g(r, V ) by the tight-binding total-energy method for which all the parameters are determined by first-principlesLAPW calculations.
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Chapter 3
Mean-field potential approach
3.1 Free energy
Following the previous work [26, 27], Fion is given by
Fion(V, T ) = −kBT·3
2lnmkBT
2πh2+ ln vf (V, T )
¸, (3.1)
where
vf (V, T ) = 4π
Zexp(−g(r, V )
kBT)r2dr. (3.2)
3.2 Mean-field potential
Inspired by the three commonly used expressions for the Grüneisen parameter[26, 50, 51], we have simply constructed the MFP in terms of the ab initio 0-Ktotal energy Ec as follows [27]
g(r, V ) =1
2[Ec(R+ r) +Ec(R− r)− 2Ec(R)]+λ
2
r
R[Ec(R+ r)−Ec(R− r)] ,
(3.3)where r is the distance that the lattice ion deviates from its equilibrium posi-tion and R is the lattice constant with respect to V .
3.3 Second order approximation
Let us make a Taylor expansion of Eq. (3.3), we have
g(r, V ) = k(V )r2 +O(r4), (3.4)
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where
k(V ) =1
R2λ∂
∂R
µR2λ
∂Ec(R)
∂R
¶, (3.5)
and O(r4), which represents the term higher than the third order, may inpart account for the so-called softening of phonons. We note that Eq. (3.4)does not contain a third (or an odd) order term. Furthermore, the Grüneisenparameter γion(V ) from Debye-Grüneisen theory [19] can be estimated byγion(V ) = −∂ ln ν/∂ lnV , where ν stands for the phonon frequency. Sinceν ' (k(V )/m)1/2, we obtain
γion(V ) =1
3(λ− 1)− V
2
∂2(PcV23 (λ+1))/∂V 2
∂(PcV23 (λ+1))/∂V
, (3.6)
where Pc(V ) =−∂Ec(V )/∂V is the so-called cold pressure.One can now understand the physical significance of λ. In particular, if λ =
-1, Eq. (3.6) is reduced to the Slater expression for the Grüneisen parameter[51]. If λ = 0, Eq. (3.6) is reduced to the Dugdale and MacDonald [50]expression for the Grüneisen parameter, and if λ = 1, Eq. (3.6) is reduced tothe free-volume theory [26] expression for the Grüneisen parameter.
It should be mentioned that Eqs (3.4)-(3.6) are just used to demonstratethe physical basis of the MFP. In realistic calculations, Eq. (3.6) is neverused since Fion in Eq. (3.1) can be easily evaluated employing the MFP,which is more general, via one-dimensional numerical integration (see nextsubsection).
3.4 Hard-sphere potential
We will now check the asymptotic behavior of g(r, V ) in Eq. (3.3). If Ec(R)is a type of a hard-sphere potential as
Ec(R) = 0, if R > b; =∞, if R ≤ b, (3.7)
thus
g(r, V ) = 0, if r < R− b; =∞, if r ≥ R− b. (3.8)
Then, vf in Eq. (3.3) equals (R − b)34π/3. Straightforwardly, from P =−(∂F/∂V )T , we get the EOS for the hard-sphere model [25]
P =R
R− bkBT
V. (3.9)
We note that Eq. (3.9) will be exactly reduced to that of an ideal gas whenb equals zero.
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3.5 Some thermodynamic quantities
By now we have derived the Helmholtz free-energy F (V, T ) as an explicitfunction of atomic volume V and temperature T. Other thermodynamic func-tions can be obtained in the usual way from F (V, T ); specifically, the entropy isS = −(∂F/∂T )V , internal energy is E = F+TS, pressure is P = −(∂F/∂V )T ,isothermal bulk modulus is BT = −V (∂P/∂V )T , and Gibbs free-energy isG = F + PV .
The thermal energy due to the lattice ion is
Eion(V, T ) = kBT ξ(T, V ), (3.10)
with
ξ(V, T ) =3
2+
·∂ ln vf (V, T )
∂ lnT
¸V. (3.11)
Using this function ξ(T, V ), the specific heat due to the lattice ions at constantvolume is then given by
CionV (V, T ) = kB
½ξ(V, T ) + T
·∂ξ(V, T )
∂T
¸V.
¾(3.12)
Through Eq. (2.3) and Eq. (3.10), the total internal energy can be ex-pressed by
E(V, T ) = Eion(V, T ) +Eel(V, T ), (3.13)
and the total constant-volume heat capacity is given by
CV (V, T ) = CionV (V, T ) +
·∂Eel(V, T )
∂T V
¸. (3.14)
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Chapter 4
Application of the mean-fieldpotential
4.1 Calculational details
To calculate the 0-K total energy Ec(V ) in Eq. (2.1), the full-potential lin-earized augmented plane wave (LAPW) method [15] is employed. As regardsthe choice of the exchange-correlational functional,i.e. between the local den-sity approximation (LDA) and the generalized gradient approximation (GGA)[16], we use the one which can produce the best 300-K EOS. As a matter offact, only for MgO (periclase), the LDA results are used in this thesis. In orderto obtain high accuracy, the plane wave cutoff Kcut is generally determined byRmt×Kcut ≥ 9.0, and more than 2000 k points in the full zone are used forreciprocal space integrations.
We note that we do not make any attempts to analytically fit the LAPWcalculated points since the fitting procedure might thereby alter the originalLAPW results. In all the thermodynamic calculations, the LAPW calculatednumerical points are directly taken as the input. Then more dense points inthe lattice constant step of 0.005 a.u. are derived by cubic spline interpola-tion. Away from the lattice constant region of the LAPW calculations, theLennard-Jones function is extrapolated towards zero and the Morse functionis extrapolated towards infinity in order to get the 0-K energy curve.
To examine the effects of the different choices of the MFP in Eq. (3.3) (orequivalently λ) on the calculated results, all the three MFP, namely λ = -1,λ = 0, and λ = 1, have been tested. The MFP in our previous work [52, 53]corresponded to the special case of λ = 0 in Eq. (3.3). The Choice of λ remindsus about the choices among the three expressions [26, 50, 51] of the Grüneisenparameter in analyzing and reducing [54, 55] to the experimental shock-wavedata. One finds that the different choices of λ do not have a too much impact
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-4000
-3000
-2000
-1000
0
1000
2000
3000
4000
0.6 0.7 0.8 0.9 1 1.1 1.2 1.3Helmholtz free-energy F(T,V/V0) (J/g)
V/V0
MgO
300 K, Anderson1000 K, Anderson1500 K, Anderson2000 K, Anderson
Figure 4.1: The Helmholtz free-energy of MgO. The solid lines represent thepresently calculated results at T = 300, 1000, 1500, and 2000 K, the dot-ted line marks the local minimum at the calculated curves, and the plusses,circles, uptriangle, and the downtriangles represent the results of empiricalthermodynamic calculations by Anderson and Zou [57].
on the P − V curve. Similar results had also been derived by Moriarty asregards the different choices of the expression of Grüneisen parameter, seereference [56] and references therein.
4.2 Demonstration of the Helmholtz free-energy: MgO
For this particular case, the LDA 0-K total energy is better than that of GGA.Enormous theoretical and experimental effort has been devoted to obtainingthe thermodynamic parameters of MgO. One major attraction is that MgO isone of the most important constituents of the Earth’s lower mantle. Knowledgeof its thermal properties at simultaneously high temperatures and pressures isa most necessary input for a better understanding of many geophysical phe-nomena. Another positive feature is that MgO has also been considered as the
12
potential internal pressure standard since no pressure induced phase transitionis observed at pressures up to 200 GPa. The presently calculated Helmholtzfree-energy of MgO is compared with those of the empirical thermodynamiccalculation by Anderson and Zou [57] in Fig. 4.1. Aside from the excellentagreement, one may note the small deviation on the small volume (high pres-sure) side. At present, it is difficult to judge which one is more preferablewhen comparing the present calculation and those of Anderson and Zou, dueto that in the calculation of Anderson and Zou an extra empirical assumptionregarding the Grueneisen parameter was used, while the present calculationcan give more accurate high temperature EOS.
4.3 Properties at ambient pressure
To check the ab initio thermodynamic model, calculation of ground state prop-erties is an important step. Knowing that we have calculated the Helmholtzfree-energy F (V, T ) as an explicit function of the atomic volume V and tem-perature T , we can easily calculate the equilibrium volume V (T ) at a givenT by solving ∂F/∂V = 0. With the equilibrium V (T ) in hand, the volumethermal-expansion coefficient βP can be calculated from
βP (V, T ) =1
V
µ∂V (T )
∂T
¶P. (4.1)
Similarly the isothermal bulk modulus can be calculated from the expression
BT (V, T ) =1
V
̶2F (V, T )
∂V 2
!T
, (4.2)
In figure 4.2 we show the calculated and measured curves for the thermalexpansion at zero-pressure for Na. From the comparison between calculationand experiment one can observe that the quantum effects [19] (≤ 150 K,neglected in present work) are important at low temperature, but that theyare explicitly cancelled at higher temperatures (≥ 150 K). Our calculatedatomic volume (V0) at ambient conditions is 39.53 Å3(23.80 cm3/mole), whichalmost coincides with the experimental value of 23.76 cm3/mole quoted byAleksandrov et al. [58].
Figure 4.3 shows the calculated and measured [59] volume thermal ex-pansion as a function of temperature at zero-pressure for Al. The agreementbetween theory and experiment is good.
As a further test of the present approach to zero-pressure properties, thecalculated curve for the isothermal bulk modulus versus temperature for Cu iscompared with experimental values [60] in Fig. 4.4. Note that the agreementbetween theory and experiment falls within 10% for this second-derivativequantity.
13
37.5
38
38.5
39
39.5
40
100 150 200 250 300 350
Atomic volume (angstrom3)
T (K)
Na
CalcAleksandrov
Figure 4.2: The calculated (solid line) and measured (open circles, from Ref.[59]) atomic volumes as a function of temperature at ambient pressure for Na.
14
1
1.01
1.02
1.03
1.04
1.05
1.06
300 400 500 600 700 800 900
V/V0
T (K)
Al
CalcTouloukian
Figure 4.3: The calculated (solid line) and measured (open circles, from Ref.[59]) relative atomic volume (V/V0) as a function of temperature at ambientconditions for Al.
15
105
110
115
120
125
130
135
300 400 500 600 700 800 900 1000
Isothermal bulk modulus (GPa)
T (K)
Cu
CalcChang
Figure 4.4: The the calculated isothermal bulk modulus (solid line) as a func-tion of temperature at ambient conditions for Cu. The circles show the exper-imental values by Chang and Himmel [60].
16
0.4
0.5
0.6
0.7
0.8
0.9
1
0 50 100 150 200 250 300
V/V0
P (GPa)
Th
Figure 4.5: The 300-K static EOS for Th. The solid line represents the calcu-lated result using the fcc crystal structure and the dotted line represents thecalculated result using the bct crystal structure (c/a = 1.655). The solid cir-cles are from Kennedy and Keeler (Ref. [55]), the open circles are from Vohraand Holzapfel (Ref. [61]), and the diamonds are the shock-reduced data byMcQueen and Marsh (Ref. [62]).
4.4 300-K static EOS
The calculation of the 300-K static EOS can serve as a good check of theaccuracy of the 0-K calculation since the 300-K static EOS is dominated bythe T = 0 energetics and the thermal contribution is very small. Therefore forthe 300-K static EOS, we will only discuss the calculated results using λ = 0in Eq. (3.3) for the actinide Th which is unique in being a transition metalwith s-d hybridization and on the threshold of being a regular 5f band lightactinide element.
For Th, Vohra and coworkers [61] have published measured data up to300 GPa, McQueen and Marsh [62] have given shock-reduced 300-K isothermup to 150 GPa, and Kennedy and Keeler [55] have presented a shock-reduced298-K isotherm up to 100 GPa in the AIP Handbook.
17
Note that in Fig. 4.5, our calculated results exactly go through the shock-reduced 298-K isotherm by Kennedy and Keeler [55] which have been claimedto have an accuracy as high as 5% in pressure. In addition we may notethat the traditional ways by McQueen and Marsh [62] for the reduction ofshock-wave data of Th might not be suitable for high compressions where thethermal electronic contribution were neglected.
4.5 Hugoniot state
The Hugoniot-equation-of-states, which are derived by the conventional shock-wave technique [7], are characterized by using measurements of shock velocity(D) and particle velocity (u) with VH/V0 = (D−u)/D and PH = ρ0Du wherePH is the pressure and ρ0 is the initial density. Through the Rankine-Hugoniotrelations, these data define a compression curve (volume (VH) versus pressure(PH)) as a function of the known Hugoniot energy (EH).
1
2PH(V0 − VH) = EH −E0, (4.3)
where V0 and E0 refer to the atomic volume and energy at ambient conditionsrespectively.
Unlike the static EOS, the temperature along the Hugoniot can undergo achange from room temperature to several tens of thousands of degrees. Thusthe calculations of the Hugoniot-equation-of-state could serve as a good checkof the theoretical method for the thermodynamic calculation. The MFP ap-proach is especially suitable for calculating the Hugoniot state. As long as the0-K ab initio calculation produces an accurate 0-K curve, the MFP will givethe Hugoniot with the same precision.
Beryllium is one of the key materials in nuclear reactors. The accuratedetermination of the high pressure and temperature EOS for beryllium playsa critical role in the field of inertial confinement fusion (ICF). The major mo-tivations for us for using Be as one of the prototypes lie in the fact that: i)Very recently, Nellis et al [64] published the EOS data of Be at shock pres-sures of 400-1000 GPa and Caube et al. [65] gave the absolute EOS data inthe TPa (1 TPa = 1000 GPa) regime; ii) When the shock wave pressure isincreased from 0.1 to 5 TPa, matter will undergo a continuous transition fromcondensed matter to a dense plasma and the theoretical understanding of thisis a long outstanding and interesting scientific issue; iii) It has been Be repre-sents a substance which when shock-compressed to P > a few Mbar (1 Mbar= 100 GPa) is a strongly coupled, partially ionized, with an unusually strongelectron-ion interaction, and a warm condensed matter/fluid that is extremelydifficult to model.
18
10
15
20
25
30
35
40
0 5 10 15 20 25
D (km/s)
u (km/s)
Be
CaclMcQueenNellis
Ragan IIICauble
Figure 4.6: The principal Hugoniot of Be. The solid line represents the presentcalculations and the circles, open diamonds, solid diamonds, and the trianglesrepresent the experiments of McQueen et al. [54], Nellis et al. [64], Ragan III[66], and Cauble et al. [65], respectively.
19
The presently calculated Hugoniot EOS at pressures up to 2000 GPa isplotted in Fig. 4.6 using the D-u (shock velocity against the mass velocity)representation together with the experimental data. Again, our calculatedresult falls well within the experimental uncertainties.
4.6 Grüneisen gamma along the principal Hugoniot
Since we have explicitly calculated the Helmholtz free-energy F(V, T) as afunction of V and T , all other thermodynamic parameters can be calculated.In particular, one can evaluate the thermodynamic Grüneisen gamma by
γth(T, V ) =V BT (V, T )αP (V, T )
CV (V, T ), (4.4)
where BT is the isothermal bulk modulus, βP the thermal volume expansioncoefficient, and CV is the constant volume heat capacity which should includeboth the lattice and the thermal electron contributions.
For demonstrating the effects of the choices of the MFP in Eq. (3.3) onthe calculated results, the calculated γths with λ = -1, 0, and 1 for α-U are de-picted in Fig. 4.7. One can immediately notice that the differences among thecalculated γths using the three MFP’s are decreasing rapidly as the pressureis increasing, and only when the pressure reaches 100 GPa have these differ-ences become smaller than 10% with respect to the value of γth. For pressuressmaller than 100 GPa, the thermal pressure is very small compared with thecold pressure. As pointed out by Mitchell et al. [67], metals shocked from am-bient conditions remain in the solid state up to pressures of typically 100-200GPa in which the EOS is dominated by the T=0 energetics. Altogether, thisdemonstrates that the calculated Hugoniot EOS is rather insensitive to thechoices of the MFP (see also the calculated results on the Hugoniot EOS andthe temperature along the principal Hugoniot).
4.7 Temperature along the principal Hugoniot
In the traditional reduction of the Hugoniot data [54, 55, 62], the temperatureestimate remains less certain since it requires an accurate knowledge of thespecific heat and the Grüneisen parameter values. In the MFP frame, all thesequantities can be calculated straightforwardly. Figure 4.8 exemplify, for MgO,a comparison between the calculated Hugoniot temperature, the measuredresults [68] and the model calculation by Svendsen and Ahrens [69].
20
1
1.5
2
0 200 400 600 800 1000
Thermodynamic Gamma
P (GPa)
U
Figure 4.7: The calculated thermodynamic Grüneisen gamma along the prin-cipal Hugoniot for U. The dashed, solid, and the dot-dashed lines representthe calculated results for the α-U structure using λ = -1, 0, and 1, respectively.
21
0
1000
2000
3000
4000
5000
6000
7000
0 50 100 150 200 250 300
T (K)
P (GPa)
MgO
CalcDuffy
Svendsen
Figure 4.8: The calculated temperature along the principal Hugoniot for MgO.The solid line represents the calculated result. The open circles are the mea-sured data by Duffy and Ahrens (Ref. [68]). The filled circles are from themodel calculation by Svendsen and Ahrens (Ref. [69]).
22
4.8 Reduction of shock-wave data
When analyzing and interpreting both the DAC and the shock-wave EOSexperiments, an important distinction has to be made between relative andabsolute measurements. In the static DAC experiments, only the volumeis measured from the X-ray line positions whereas the pressure is inferredindirectly from the diffraction line shifts of a pressure marker or referencematerial, which is mixed with the sample and whose P − V relation is wellknown. In the dynamic shock-wave experiments, where the deduced P − Vcurve is called Hugoniot, the shock velocity (D) and the particle velocity (u)are the measured quantities. Therefore, both P and V are direct measuredquantities [see Eq. (3.3)] in the shock-wave experiment.
When dealing with the pressure dependence of the volume it is an advan-tage to use the reduced atomic volume X = V/V0, where V0 is the theoreticalequilibrium volume, and then consider P = P (X). The experimental shock-wave data Xex
sw can be expressed as:
Xexsw(P ) = X
ex300(P ) +X
exht (P ), (4.5)
whereXex300 represents the experimental 300 K static EOS, andX
exht the volume
expansion of the principal Hugoniot relative to the 300 K isotherm. We canalso write the calculated results of the MFP approach in the equivalent form:
XMFPsw (P ) = XMFP
300 (P ) +XMFPht (P ). (4.6)
In fact, the terms XMFPsw and XMFP
300 in Eq. (4.6) can be easily calculatedwith the MFP approach. If we replace Xex
ht in Eq. (4.5) with XMFPht in Eq.
(4.6), then we can obtain the reduced EOS Xrd300 ' Xex
300 from shock-wave databy
Xrd300(P ) = X
exsw(P )− (XMFP
sw (P )−XMFP300 (P )). (4.7)
In Fig. 4.9 we show the results for 0-200 GPa of the present reductionsof shock-wave data for Cu and Au. For comparison, we also show the DACexperimental results, the empirical reductions by McQueen et al. [54], theempirical reductions by Kennedy and Keeler [55], and some of the calculatedresults by other authors [56]. In Fig. 4.9a we show the data for Cu. Cu is anexcellent prototype for both theoretical modelling and experimental calibra-tion. As can be seen, the presently reduced result (solid line) fit exactly to thewell calibrated experimental points by Bell et al. [3]. In Fig. 4.9b data forgold are collected. The present reduced results are virtually indistinguishablefrom the experimental data obtained by Bell et al [3] as well as the suggestedstandard by Heitz and Jeanloz [6].
23
0.7
0.8
0.9
1
0 50 100 150 200
V/V0
Cu
(a)
0.7
0.8
0.9
1
0 50 100 150 200
V/V0
Pressure (GPa)
Au
(b)
Figure 4.9: The 300 K EOS for a pressure of in 0 - 200 GPa. Solid line: thepresent reduction. (a) Cu. Solid diamonds: DAC of Bell et al. (see Ref.[3]), plusses: empirical reduction of McQueen et al. (see Ref. [54]), opendiamonds: empirical reduction of Kennedy and Keeler (see Ref. [55]), opensquares: LMTO-ASA calculation of Moriarty (see Ref. [56]); (b) Au. Opendiamonds: empirical reduction of Kennedy and Keeler (see Ref. [55]), soliddiamonds: DAC of Bell et al. (see Ref. [3]), plusses: suggested standard byHeitz and Jeanloz (see Ref. [90]).
24
4.9 Porous Hugoniot
The main purpose of measuring the Hugoniot of porous materials is due toa higher temperature can be reached with lower pressure. A variety of semi-empirical models [71, 72, 73] has been proposed to describe the shock-wavebehavior of porous materials during the past decades. Among these, the MieGrüneisen equation of state EOS has extensively been adopted. However,the success of the Mie Grüneisen EOS depends on the determination of theGrüneisen coefficient, which is usually taken as a function of volume only. Incontrast, the MFP approach does not invoke the Grüneisen approximation,therefore, the difficulties in determining the Grüneisen coefficient have beencircumvented.
To the best of our knowledge, there has been no first-principles calculationsfor the Hugoniot of porous materials. The present work is unique in thisregard.
We will only consider the complete compacted state of the porous material.In this case, one can simply imagine that the E0 in Eq. (4.3) for the porouscase is exactly the same as that for the nonporous one. The only new thingfor the porous case is that now the initial volume is V0 = mV n0 , where m isthe initial porosity and V n0 is the ambient volume of the nonporous material.
The calculations are done at pressures up to 300 GPa. We only showthe results for Cu with the initial porosity of m = 1.0, 1.13, 1.22, 1.41, 2.0,3.0, 3.5, and 4.0. These initial porosities are from the existing experimentaldata [7, 9, 74, 75]. Note that here m = 1.0 means the nonporous material.The presently calculated EOS’s are plotted in Fig. 4.10 for Cu (by pressureagainst the reduced density d/d0, where d0 represents the ambient density ofthe nonporous material).
Traditionally, the experimental data are partitioned into the normal com-pressibility and the anomalous compressibility depending on m. For the nor-mal compressibility, the pressure is increasing when the volume decreases, asusual. The most intriguing case is the anomalous compressibility where, in-stead of decreasing with increasing pressure, the volume increases, and then,when the pressure is high enough the compressibility becomes normal.
By the MFP approach in which the Grüneisen parameter is no longerinvoked, the so-called normal and anomalous behaviors have been reproduced.The interesting point is that the method of setting the initial parametersE0 and V0 in the present work is exactly the same as that for the normalcompressibility of the previous empirical work by Dijken and De Hosson [73].The reason why the work of Dijken and De Hosson had to treat the normal andanomalous compression differently may now be that: (i) their method neededthe Grüneisen parameter whose dependences on the volume and temperaturewas very difficult to determine and (ii) they neglected the thermal electronic
25
0
50
100
150
200
250
300
0.8 1 1.2 1.4 1.6
P (GPa)
d/d0 (normalized density)
Cu
Figure 4.10: The Hugoniot EOS for porous Cu. The solid lines, from theright to the left, represent the presently calculated results by initial porositym = 1.0, 1.13, 1.22, 1.41, 2.0, 3.0, 3.5, and 4.0 respectively. The uptrianglesare shock-wave data of the nonporous material by Mitchell and Nellis [9], thecrosses are from the LASL compilation [7], the squares are from Trunin et al.(1989) [74], and the solid circles are the new data published by Trunin (1994)[75].
26
contribution.
4.10 Isentropic release
After the shock-wave front passes the substance, it will undergo a processcalled isentropic release which is very important in the real application. Thecalculated P-T curves along the isentropic release and principal Hugoniot forW are plotted in Fig. 4.11 together with those from the empirically derivedtemperatures of McQueen et al [54]. Shown in Fig. 4.11 are also the veryrecently measured high pressure melting data of W by Errandonea et al. [76].The agreement between our calculation and those of McQueen et al. is verygood except for the very high pressure region where the thermal electroniccontribution has been neglected by McQueen et al.
One serious fact that one can not neglect from Fig. 11 is that the presentlycalculated curves of isentropic release cross the DAC melting data by Erran-donea et al. in an abnormal way, i.e. when the pressure decreases along theisentropic curves the metal W will transform into a solid from a liquid.
4.11 High pressure melting
As a part of our continuing program to investigate the thermodynamic prop-erties of materials by means of the MFP approach, the purpose of this sectionis to work out a relatively feasible way to compute high-pressure melting. Wepropose the following melting law
Tm = CR2∂
2F (Tm, V )
∂R2, (4.8)
where C is a constant for a given material, R is the Wigner-Seitz radius ofthe atom with V = R34π/3, and F (T, V ) is the Helmholtz free-energy as thefunctions of temperature T and atomic volume V .
By means of the idea of the MFP approach, Eq. (4.8) can be thought asa generalization of the Lindemann law [77], if we view the second derivativeterm in the right hand of Eq. (4.8) as being a ”Force Constant”. The majorimprovements are that now the ”Force Constant” is temperature dependentand the requirement for an extra assumption regarding the Grüneisen coeffi-cient γG has been cancelled. And since the Helmholtz free-energy F with theMFP approach can be computed free from parameters, where both the latticeand the thermal electron contribution have implicitly been considered, onlyone constant C is left to be determined. In general, there are two methodsto determine the constant C. For a simple metal, where no solid-solid phasetransition from the zero-pressure melting point to the considered pressure take
27
0
2000
4000
6000
8000
10000
12000
0 100 200 300 400 500
T (K)
P (GPa)
W
HugoniotIsentropic
Errandonea, MeltingMcQueenMcQueen
Figure 4.11: The P-T EOS of W. The solid lines represent the presently cal-culated curves of isentropic release at Hugoniot pressure 450, 350, 290, 270,250, 150, and 50 from the top to the bottom, respectively. The dotted line isthe presently calculated temperature along the principal Hugoniot. The solidtriangles are the empirically derived Hugoniot temperatures[54] and the opentriangles are the empirically derived temperatures[54] after isentropic releaseat the Hugoniot pressure 270, 250, 150, and 50 GPa, respectively. The crossesare the DAC melting data by Errandonea et al.[76].
28
place, one can simply determine C by just using the zero-pressure melting tem-perature, noting that with the MFP approach the Helmholtz free-energy F canbe computed at any temperature T and atomic volume V parameter-freely. Fora complex metal such as iron, where some solid-solid phase transitions occurfrom the zero-pressure melting point to the considered pressure, we can alsoeasily determine C by using the shock-wave melting pressure since with theMFP approach the temperature along the Hugoniot can be computed ratheraccurately. Once the constant C is determined, the melting temperature atall pressures can be computed through Eq. (4.8) self-consistently.
Based on first-principles calculations, we find that the new method canmodel the melting properties of metals very well. In particular, Cu is predictedto melt at a shock pressure of 238 GPa with a temperature of 6360 K and Tais predicted to melt at a shock pressure of 295 GPa with a temperature of10091 K. And at the most interesting Earth’s core pressures of 330 GPa and360 GPa, Fe is predicted to melt at 6645 K and 6928 K, respectively.
4.12 Specific heat for Iron under the Earth’s corecondition
The thermodynamic properties of Fe under the condition of the Earth’s corepressure ranges (136 to 364 GPa) are undoubtedly of importance for the un-derstanding of the core behavior, typically including the convective dynamicsand heat transport.
For reference we also calculated the specific heat with respect to our cal-culated temperature profile of the Earth’s core, and the results are shown inFig. 4.12. Notice that the calculated constant-volume heat capacities almostremain a constant at 40 J mol−1 K−1, which is considerably larger than thevalue of 24.9 J mol−1 K−1 (3kB) obtained from the Dulong-Petit approxima-tion, due to the large contribution from the thermal excitation of electrons.
4.13 Temperature dependence of the thermal pres-sure for MgO
Finally we will deal with a very interesting issue: namely the temperaturedependence of the thermal pressure βPBT for MgO. Experiments [57, 78] showthat when the temperature reaches about 1000 K, βPBT will decrease withincreasing temperature. No previous calculation except for this thesis canreproduce this behavior (See Fig. 4.13).
29
0
10
20
30
40
50
150 200 250 300 350
Heat capacity (J mol-1 K-1)
P (GPa)
Fe
Figure 4.12: The calculated heat capacities for hcp Fe under the Earth’s coreconditions. The solid, dashed, and dot-dashed lines represent the calculatedconstant-pressure specific heat CP , constant-volume specific heat CV , and thelattice ion only constant-volume specific heat CionV , respectively.
30
3
4
5
6
7
0 500 1000 1500 2000 2500 3000
beta*Bt (1.e-3 1/K GPa)
T (K)
MgOMFP
QuasiharmonicAnderson
Dubrovinsky
Figure 4.13: Temperature dependence of the thermal pressure βPBT for MgO.Solid line: MFP results; dashed-line: results of quasiharmonic approximation;open circles: empirical thermodynamic calculation by Anderson and Zou [57];plusses: measured data by Dubrovinsky and Saxena [78].
31
Chapter 5
Beyond the harmonic theory:precise solution for a specialphonon mode
In the present theory of lattice dynamics, the central roles are played by theharmonic and the quasi-harmonic approximations [32, 79]. The lattice poten-tial of the system is truncated at terms of second-order in the atomic displace-ments, by which the normal mode transformation allows the Hamiltonian ofthe system to be expressed as a sum of non-interacting one-dimensional har-monic Hamiltonians. The physical concept of ”phonon” is then introducednaturally since the energy eigenvalue spectra for a parabola-like potential con-sists of equally spaced energy levels which makes the energy excitation looklike a simple increase in the number of particles.
Nowadays the rapid developments of both the computer speed and the com-putational methods have made it routine to perform accurate first-principlescalculations of the lattice potential — total energy changes associated withatomic displacements at zero temperature [80, 81, 82]. For certain phononmodes, the harmonic approximation then yields an imaginary frequency [83,34] which is somewhat difficult to understand in the proper sense of energetics.It is therefore natural to ask whether there is any interesting physics underly-ing these features. It seems to be necessary to go beyond the simple phononpicture and to give a precise solution for certain systems. This was first ac-complished by Ho, Tao and Zhu [85] in their study of vibrations of hydrogenisotopes in NbH and later applied by Elsässer et al. [86] for weakly anhar-monic vibrations of H in bcc NbH and for strongly anharmonic vibrations ofH in fcc PdH.
In a bcc metal [44], the H-point oscillation is formed by two equivalentatoms from the primitive simple cubic unit cell moving in opposite directions
32
-400
0
400
800
1200
1600
-0.5 0 0.5 1 1.5
V(x) (meV)
x/b
ZrT1N
ω
Figure 5.1: The calculated lattice potential V (x) for bcc Zr as a func-tion of the displacements corresponding to the longitudinal phonon withq=(2π/a)(2/3,2/3,2/3) (solid line) and the transversal T1 phonon withq=(2π/a)(1/2,1/2,0) (dashed line). x is measured in units of the correspond-ing periodic lengths of displacements .
33
0
100
200
300
400
-0.2 0 0.2
Energy (meV)
x/b
Mo
(a)
H
0 10 20 30 40
Level space (meV)
(b)
Figure 5.2: Mo H point oscillation. (a) The calculated energy eigenvaluespectra; (b) The corresponding energy space (shown as width of the horizontallines) between the adjacent energy levels as a function of the energy eigenvalue.The lowest one corresponds to the quantum zero-point energy.
34
along the <001> direction. In another language, the so-called H-point oscil-lation can be viewed as if the atom in the body center position in the bcc unitcell is uniformly distorted towards the face center position. The T1 N pointoscillation [87] is formed by the shuffling of the two neighboring <110> planesalong the [110] direction. The ω point oscillation is formed by the oppositemovements of the two neighboring <111> planes and with every third planestaying at rest. Notice that the ’moving atom’ has a periodic length of b =
√2a
(a is the bcc lattice constant) for the T1 N point oscillation and b =√3a for
the ω point oscillation. This forms a straightforward physical picture of anatom in a one-dimensional periodic lattice potential of V (x) where x is therelative displacement between the moving adjacent planes. We can naturallysolve this problem since it is analogous to an one-dimensional electron energyband problem [84] in the form"
h2G2
2µ− ²
#C(G) +
XG0U(G−G0)C(G0) = 0, (5.1)
where h is the Planck constant divided by 2π, µ is the effective mass of a twoatom system, U(G) is the coefficient of the Fourier transformation of V (x),and C(G) is the coefficient of the linear combination of the ’one-atom’ wavefunction with the plane wave basis exp(iGx).
In order to obtain V (x), the projector augmented-wave (PAW) methodwithin the generalized gradient approximation (GGA) is employed. We usethe Vienna ab initio simulation package (VASP) [82] with a high precisionchoice and Monkhost 15×15×15 k points. V (x) is calculated by a latticedisplacement step of 0.01b (meaning 51 points in the half periodic length!)for the corresponding oscillation. Then more dense points corresponding tolattice displacement steps of 0.001b are derived by a cubic spline interpolationas the input to produce the coefficients of the fast-Fourier-transformation. Theconvergence was tested by varying the number of plane waves that are needed.We found that 500 (|b/2πGmax| = 250) plane waves were more than enough togive fully convergent results within the entire energy range examined in thisthesis.
Figure 5.1 displays V (x) for Zr, calculated at the measured [88] 1423 K bcclattice constant of 6.8445 a.u., for the T1 N point and ω point oscillations as afunction of the atomic displacements away from their static bcc crystal posi-tions. From its behavior, V (x) is now too anomalous to be approximated by aparabola. Instead of being a local minimum, the atomic equilibrium positionis located at the local peak top for the T1 N point oscillation and at a shoul-der for ω point oscillation. It is widely accepted that the occurrence of theseanomalies is closely connected to the detailed topology of the electronic bandsnear the Fermi energy — the electronic response to the ionic displacements.
35
0
100
200
300
400
-0.3 0 0.3
Energy (meV)
x/b
Zr
(a)
T1N
0 4 8 12
Level space (meV)
(b)
Figure 5.3: Zr T1 N point oscillation. (a) The calculated energy eigenvaluespectrum at the measured[88] 1423 K bcc lattice constant of 6.8445 a.u.. (b)The corresponding energy space (shown as width of the horizontal lines) be-tween the adjacent energy levels as a function of the energy eigenvalue withthe lowest one indicating the quantum zero-point energy.
36
0
100
200
300
400
-0.3 0 0.3
Energy (meV)
x/b
Zr
(a)
ω0 4 8 12
Level space (meV)
(b)
Figure 5.4: Zr ω point oscillation. (a) The calculated energy eigenvalue spec-trum at the measured[88] 1423 K bcc lattice constant of 6.8445 a.u.. (b) Thecorresponding energy space (shown as width of the horizontal lines) betweenthe adjacent energy levels as a function of the energy eigenvalue. The lowestone corresponds to the quantum zero-point energy.
37
Let us first focus on the energy eigenvalue spectra as the essential aspectfor understanding the present approach relative to that of the quasi-harmonicapproximation. Fig. 5.2 shows the energy eigenvalue spectra for the Mo H-point mode. In Figs. 5.3 and 5.4 we have plotted the energy eigenvalue spectrafor Zr solved for the T1 N point and ω point oscillations, respectively. It isamazing to notice the energy spacing (ES) between the adjacent energy levels.In the right column of Figs. 5.2, 5.3, and 5.4 the energy eigenvalue spectra(ESs) are also plotted with the lowest one indicating the quantum zero-pointenergy. If the system was harmonic all the ESs would be exactly equal. Inthe case of the T1 N point, the ESs first decrease rapidly for only a few levelsto a certain threshold and then increase. In the case of the ω point, the ESsshow a slight increase for a certain number of levels, then they decrease for acertain number of levels to a certain threshold, and then increase again. Inthe case of Mo, the increase of the ESs with increasing energy can be used toexplain the reason why the H point phonon initially seems to stiffen [89] attemperature up to 1200 K.
38
Chapter 6
Concluding remarks
Based on the first-principles calculations coupled with the classical mean-fieldpotential (MFP) approach. We have studied the ambient condition thermo-dynamic properties, 300-K equation of state, shock-wave Hugoniot, reductionof the shock-wave data, high pressure melting, anharmonic effect. The generalagreement with experiment is good.
An approach is also attemptted to go beyond the standard phonon theoryby making a precise solution for the anomalous phonon modes. We are ableto reproduce, for the first time, the anomalous temperature dependence ofthe H-point phonon energy for Mo, and the T1 N point and ω point phononanomalies in bcc Zr.
39
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