import numpy as np
import scipy.interpolate
from ase.atoms import Atoms
from scipy.constants import value as scipy_constants
from directupsampling.data import kB
from directupsampling.eos import EOS
from directupsampling.fecontrib import FreeEnergyContrib
from directupsampling.fesurface import FreeEnergySurface
GPa_to_kbar = 10
Ang3_to_Bohr3 = 6.7483345
# np.testing.assert_allclose(
# Ang3_to_Bohr3,
# (scipy_constants('Bohr radius')/1e-10)**3)
convert = 67.48 # conversion factor, Moruzzi Eq. (4)
empirical = 0.617 # empirical factor, Moruzzi Eq. (6)
[docs]
class Debye:
def __init__(self, atoms: Atoms, eos: EOS, gamma=2 / 3):
"""
Parameters
----------
atoms: Atoms
Atoms object for masses
eos: EOS
EOS or derived object
gamma: float
Grueneisen constant for the Moruzzi model
"""
self.atoms = atoms
self.eos = eos
self.gruneisen_constant = gamma
self.mass = np.mean(self.atoms.get_masses())
# gammalow = 1 # low T gamma (Gruneisen constant)
# gammahigh = 2/3 # high T gamma (Gruneisen constant)
@property
def free_energy_surface(self):
fec = FreeEnergyContrib(
"debye",
bounds=[self.eos.volume_bounds, [1, 1500]],
)
fec.parametrization = self
return FreeEnergySurface([fec])
[docs]
def get_free_energy(self, volume, temperature):
"""Calculates vibrational free energies in the Debye-Gueneisen
approximation as function of temperature at given volume.
According to Moruzzi, PRB 37 (1988) 790.
Arguments include eos parameters and effective mass (a.u.).
Fqhs are created for the low T gamma=1 and for high T gamma=2/3.
"""
t_debye = self.calculate_debye_temperature(volume)
free_energy = fdebye(temperature, t_debye)
return free_energy
[docs]
def calculate_debye_temperature(self, volume):
v0 = self.eos.eq_volume
b0 = self.eos.get_bulk_modulus(v0, unit="GPa")
bp = self.eos.eq_bulk_mod_deriv
gamma = self.gruneisen_constant
ws_radius0 = np.cbrt(3 / (4 * np.pi) * v0 * Ang3_to_Bohr3)
debye0 = (
empirical * convert * np.sqrt(ws_radius0 * b0 * GPa_to_kbar / self.mass)
)
debye_temperature = debye0 * (v0 / volume) ** (-gamma + (1 + bp) / 2)
return debye_temperature
[docs]
def fdebye(t, t_debye):
"""Debye free energy.
Corresponds to Eq. (19) from Moruzzi, but NOTE that there is an error in
Moruzzi, the 3ln term must be positive in the end (i.e., when the brackets
are eliminated) so inside the brackets the 3ln term should be negative to
cancel the minus in front of the brackets.
"""
x = t_debye / t
d = 3 / x**3 * debye_function(x)
f = 9 / 8 * kB * t_debye + kB * t * (3 * np.log(1 - np.exp(-x)) - d)
return f
[docs]
def debye_function(x):
# interpolate tabulated Debye function
return debye_interp(x)
# 0 0.007 5001 xStart xStep nx
nx = 5001
xx = np.linspace(0, 0.007 * (nx - 1), nx)
debye_function_table = [
0.0,
1.1403348844983693e-7,
9.098736303790828e-7,
3.0627579429926487e-6,
7.2407881701228895e-6,
0.000014104963893484593,
0.000024309216141459616,
0.00003850044099458789,
0.000057318533186942476,
0.0000813964197025638,
0.00011136009336613458,
0.00014782864642706904,
0.00019141430413619804,
0.00024272245831422765,
0.00030235170091115054,
0.00037089385755579007,
0.00044893402109466075,
0.0005370505851193198,
0.0006358152774813993,
0.0007457931937944957,
0.000867542830922108,
0.0010016161204507996,
0.0011485584621477752,
0.0013089087574020635,
0.0014831994426484805,
0.0016719565227735792,
0.001875699604502761,
0.002094941929767745,
0.002330190409053598,
0.0025819456547244947,
0.002850702014327429,
0.0031369476038730706,
0.0034411643410929304,
0.003763827978672086,
0.004105408137456632,
0.004466368339635081,
0.00484716604189288,
0.005248252668539321,
0.0056700736446059635,
0.0061130684289158604,
0.006577670547122745,
0.007064307624719427,
0.007573401420014584,
0.00810536785707717,
0.008660617058647726,
0.009239553379015683,
0.009842575436862015,
0.010470076148066401,
0.011122442758478126,
0.011800056876649952,
0.01250329450653424,
0.013232526080140498,
0.01398811649015361,
0.014770425122511985,
0.015579805888944939,
0.016416607259468436,
0.017281172294838474,
0.0181738386789615,
0.019094938751260886,
0.02004479953899886,
0.021023742789553216,
0.022032085002647812,
0.02307013746253646,
0.024138206270139156,
0.02523659237513021,
0.026365591607977253,
0.02752549471193071,
0.028716587374962743,
0.02993915026165517,
0.03119345904503547,
0.03247978443836025,
0.03379839222684551,
0.035149543299342965,
0.036533493679961594,
0.037950494559634014,
0.039400792327626645,
0.04088462860299327,
0.04240224026597114,
0.04395385948931892,
0.04553971376959596,
0.04716002595838208,
0.0488150142934372,
0.050504892429800295,
0.05222986947082667,
0.05399014999916338,
0.055785934107661785,
0.05761741743022664,
0.05948479117260123,
0.06138824214308764,
0.0633279527832019,
0.06530410119826292,
0.06731686118791498,
0.0693664022765829,
0.07145288974385934,
0.0735764846548237,
0.07573734389029177,
0.0779356201769959,
0.08017146211769458,
0.08244501422121132,
0.08475641693240192,
0.08710580666204977,
0.08949331581668808,
0.09191907282834934,
0.09438320218424048,
0.09688582445634392,
0.09942705633094337,
0.1020070106380742,
0.10462579638089763,
0.10728351876499816,
0.1099802792276039,
0.11271617546672896,
0.11549130147023762,
0.11830574754482943,
0.12115960034494523,
0.1240529429015928,
0.12698585465109236,
0.12995841146374118,
0.13297068567239612,
0.13602274610097492,
0.1391146580928743,
0.1422464835393053,
0.14541828090754558,
0.14863010526910667,
0.15188200832781773,
0.15517403844782376,
0.15850624068149854,
0.16187865679727179,
0.1652913253073695,
0.168744281495468,
0.17223755744426017,
0.17577118206293443,
0.179345181114565,
0.18295957724341394,
0.18661439000214383,
0.1903096358789411,
0.19404532832454946,
0.1978214777792127,
0.20163809169952696,
0.20549517458520197,
0.2093927280057301,
0.2133307506269645,
0.2173092382376032,
0.22132818377558253,
0.2253875773543751,
0.2294874062891966,
0.23362765512311653,
0.23780830565307637,
0.24202933695581166,
0.24629072541368072,
0.25059244474039577,
0.25493446600666125,
0.2593167576657131,
0.26373928557876414,
0.2682020130403506,
0.27270490080358406,
0.2772479071053033,
0.2818309876911308,
0.28645409584042936,
0.29111718239116263,
0.29582019576465446,
0.30056308199025145,
0.30534578472988505,
0.31016824530253556,
0.31503040270859417,
0.3199321936541275,
0.3248735525750403,
0.32985441166113766,
0.3348747008800874,
0.3399343480012799,
0.3450332786195886,
0.35017141617902636,
0.3553486819963034,
0.3605649952842794,
0.3658202731753166,
0.37111443074452866,
0.3764473810329273,
0.38181903507046605,
0.3872293018989815,
0.3926780885950295,
0.3981653002926203,
0.40369084020584817,
0.40925460965141747,
0.4148565080710661,
0.4204964330538818,
0.42617428035851834,
0.43188994393530183,
0.43764331594823813,
0.44343428679690994,
0.44926274513827325,
0.4551285779083459,
0.461031670343792,
0.46697190600340077,
0.47294916678946036,
0.4789633329690238,
0.48501428319507295,
0.4911018945275724,
0.49722604245442154,
0.5033866009122957,
0.5095834423073867,
0.5158164375360321,
0.5220854560052408,
0.5283903656531119,
0.5347310329691456,
0.5411073230144505,
0.5475190994418404,
0.5539662245158268,
0.5604485591325049,
0.5669659628393313,
0.5735182938547949,
0.5801054090879829,
0.5867271641580364,
0.5933834134135032,
0.6000740099515793,
0.606798805637247,
0.6135576511223025,
0.6203503958642806,
0.6271768881452682,
0.6340369750906126,
0.6409305026875233,
0.6478573158035652,
0.654817258205047,
0.6618101725752981,
0.668835900532845,
0.6758942826494743,
0.6829851584681934,
0.6901083665210807,
0.6972637443470316,
0.7044511285093967,
0.7116703546135126,
0.7189212573241258,
0.7262036703827116,
0.7335174266246843,
0.7408623579965028,
0.7482382955726664,
0.7556450695726089,
0.7630825093774827,
0.7705504435468358,
0.7780486998351881,
0.7855771052084942,
0.7931354858605072,
0.80072366722903,
0.8083414740120675,
0.8159887301838663,
0.8236652590108546,
0.8313708830674728,
0.8391054242519012,
0.8468687038016793,
0.854660542309225,
0.8624807597372427,
0.8703291754340327,
0.8782056081486895,
0.8861098760462023,
0.8940417967224457,
0.902001187219068,
0.9099878640382768,
0.9180016431575173,
0.9260423400440517,
0.9341097696694293,
0.9422037465238577,
0.9503240846304674,
0.9584705975594775,
0.9666430984422523,
0.9748413999852621,
0.9830653144839347,
0.9913146538364116,
0.9995892295571941,
1.007888852790695,
1.016213334324683,
1.0245624846036292,
1.0329361137419495,
1.0413340315371467,
1.049756047482854,
1.0582019707817731,
1.0666716103585174,
1.0751647748723487,
1.0836812727298204,
1.0922209120973139,
1.1007835009134848,
1.109368846901597,
1.1179767575817705,
1.1266070402831223,
1.135259502155811,
1.1439339501829828,
1.1526301911926224,
1.1613480318692972,
1.170087278765818,
1.1788477383147864,
1.1876292168400597,
1.1964315205681082,
1.2052544556392804,
1.2140978281189734,
1.2229614440087053,
1.23184510925709,
1.240748629770721,
1.249671811424955,
1.2586144600746072,
1.2675763815645444,
1.2765573817401894,
1.2855572664579278,
1.2945758415954267,
1.3036129130618506,
1.3126682868079935,
1.3217417688363122,
1.3308331652108716,
1.3399422820671933,
1.3490689256220127,
1.3582129021829528,
1.3673740181580916,
1.3765520800654527,
1.3857468945423979,
1.3949582683549306,
1.404186008406911,
1.4134299217491793,
1.4226898155885894,
1.4319654972969595,
1.4412567744199227,
1.4505634546857002,
1.4598853460137795,
1.4692222565235111,
1.478573994542609,
1.4879403686155728,
1.4973211875120178,
1.506716260234923,
1.5161253960287862,
1.5255484043877023,
1.5349850950633488,
1.5444352780728912,
1.5538987637067991,
1.5633753625365836,
1.5728648854224485,
1.5823671435208528,
1.5918819482919992,
1.6014091115072322,
1.6109484452563596,
1.6204997619548844,
1.6300628743511638,
1.639637595533478,
1.6492237389370272,
1.6588211183508372,
1.6684295479245954,
1.6780488421753978,
1.6876788159944256,
1.6973192846535292,
1.7069700638117489,
1.7166309695217483,
1.7263018182361665,
1.735982426813902,
1.7456726125263113,
1.7553721930633364,
1.765080986539548,
1.7747988115001216,
1.7845254869267295,
1.7942608322433657,
1.8040046673220864,
1.8137568124886814,
1.8235170885282714,
1.8332853166908303,
1.843061318696632,
1.8528449167416257,
1.86263593350274,
1.8724341921431094,
1.8822395163172365,
1.892051730176073,
1.9018706583720404,
1.9116961260639682,
1.921527958921971,
1.9313659831322485,
1.941210025401824,
1.9510599129632022,
1.9609154735789678,
1.9707765355463116,
1.9806429277014908,
1.9905144794242133,
2.000391020641965,
2.010272381834267,
2.0201583940368617,
2.0300488888458377,
2.039943698421683,
2.0498426554932845,
2.059745593361846,
2.0696523459047538,
2.0795627475793745,
2.08947663342679,
2.0993938390754607,
2.1093142007448398,
2.1192375552489127,
2.129163739999682,
2.1390925930105826,
2.149023952899845,
2.1589576588937853,
2.168893550830051,
2.1788314691607846,
2.1887712549557494,
2.198712749905378,
2.2086557963237747,
2.218600237151648,
2.2285459159591925,
2.2384926769489097,
2.2484403649583697,
2.258388825462921,
2.268337904578333,
2.2782874490633955,
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2.298187324407872,
2.308137352022496,
2.3180872385219855,
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2.337985988876196,
2.3479345547269532,
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2.536652533589259,
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2.6256727531019006,
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2.6651232074137585,
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2.684818258164331,
2.694657800300816,
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3.5446667434897745,
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3.6944041206111655,
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3.7890399412129154,
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3.806049675780318,
3.814531515001833,
3.822997943530368,
3.8314489180808473,
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3.85670869340347,
3.865097430362598,
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3.906805393554042,
3.9150995698034072,
3.923377852427825,
3.9316402047186707,
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3.9563313204919233,
3.9645295949975745,
3.972711763583176,
3.9808777927068357,
3.9890276493464683,
3.9971613009974987,
4.0052787156705625,
4.013379861889213,
4.021464708687608,
4.029533225608207,
4.037585382699461,
4.045621150513499,
4.053640500103801,
4.061643403022891,
4.069629831320005,
4.077599757538773,
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]
debye_interp = scipy.interpolate.interp1d(
xx,
debye_function_table,
kind=3,
bounds_error=False,
fill_value=debye_function_table[-1],
)