def get_integer_formula_and_factor(self, max_denominator=10000):
"""
Calculates an integer formula and factor.
Args:
max_denominator (int): all amounts in the el:amt dict are
first converted to a Fraction with this maximum denominator
Returns:
A pretty normalized formula and a multiplicative factor, i.e.,
Li0.5O0.25 returns (Li2O, 0.25). O0.25 returns (O2, 0.125)
"""
vals = [
Fraction(v).limit_denominator(max_denominator)
for v in self.values()
]
denom = lcm(*(f.denominator for f in vals))
mul = gcd(*[int(v * denom) for v in vals])
d = {
k: round(v / mul * denom)
for k, v in self.get_el_amt_dict().items()
}
(formula, factor) = reduce_formula(d)
if formula in Composition.special_formulas:
formula = Composition.special_formulas[formula]
factor /= 2
return formula, factor * mul / denom
def reduce_formula(sym_amt):
syms = sorted(sym_amt.keys(), key=lambda s: Element(s).X)
syms = list(filter(
lambda s: abs(sym_amt[s]) > Composition.amount_tolerance, syms
))
num_el = len(syms)
contains_polyanion = (
num_el >= 3 and
Element(syms[num_el-1]).X - Element(syms[num_el-2]).X < 1.65
)
factor = abs(gcd(*sym_amt.values()))
reduced_form = []
n = num_el-2 if contains_polyanion else num_el
for i in range(0, n):
s = syms[i]
normamt = sym_amt[s] * 1.0 / factor
reduced_form.append(s)
reduced_form.append(formula_double_format(normamt))
if contains_polyanion:
poly_sym_amt = {
syms[i]: sym_amt[syms[i]] / factor for i in range(n, num_el)
}
(poly_form, poly_factor) = reduce_formula(poly_sym_amt)
if poly_factor != 1:
reduced_form.append("({}){}".format(poly_form, int(poly_factor)))
else:
reduced_form.append(poly_form)
reduced_form = "".join(reduced_form)
return reduced_form, factor
def reduce_formula(sym_amt):
syms = sorted(sym_amt.keys(), key=lambda s: Element(s).X)
syms = list(
filter(lambda s: abs(sym_amt[s]) > Composition.amount_tolerance, syms))
num_el = len(syms)
contains_polyanion = (
num_el >= 3
and Element(syms[num_el - 1]).X - Element(syms[num_el - 2]).X < 1.65)
factor = abs(gcd(*sym_amt.values()))
reduced_form = []
n = num_el - 2 if contains_polyanion else num_el
for i in range(0, n):
s = syms[i]
normamt = sym_amt[s] * 1.0 / factor
reduced_form.append(s)
reduced_form.append(formula_double_format(normamt))
if contains_polyanion:
poly_sym_amt = {
syms[i]: sym_amt[syms[i]] / factor
for i in range(n, num_el)
}
(poly_form, poly_factor) = reduce_formula(poly_sym_amt)
if poly_factor != 1:
reduced_form.append("({}){}".format(poly_form, int(poly_factor)))
else:
reduced_form.append(poly_form)
reduced_form = "".join(reduced_form)
return reduced_form, factor
def get_integer_formula_and_factor(self, max_denominator=10000):
mul = gcd(*[
Fraction(v).limit_denominator(max_denominator)
for v in self._elmap.values()
])
sym_amt = self.get_el_amt_dict()
d = { k: round(v/mul) for k,v in sym_amt.items() }
(formula, factor) = reduce_formula(d)
if formula in Composition.special_formulas:
formula = Composition.special_formulas[formula]
factor /= 2
return formula, factor * mul
def reduce_formula(sym_amt, iupac_ordering=False):
"""
Helper method to reduce a sym_amt dict to a reduced formula and factor.
Args:
sym_amt (dict): {symbol: amount}.
iupac_ordering (bool, optional): Whether to order the
formula by the iupac "electronegativity" series, defined in
Table VI of "Nomenclature of Inorganic Chemistry (IUPAC
Recommendations 2005)". This ordering effectively follows
the groups and rows of the periodic table, except the
Lanthanides, Actanides and hydrogen. Note that polyanions
will still be determined based on the true electronegativity of
the elements.
Returns:
(reduced_formula, factor).
"""
syms = sorted(sym_amt.keys(), key=lambda x: [get_el_sp(x).X, x])
syms = list(filter(
lambda x: abs(sym_amt[x]) > Composition.amount_tolerance, syms))
factor = 1
# Enforce integers for doing gcd.
if all((int(i) == i for i in sym_amt.values())):
factor = abs(gcd(*(int(i) for i in sym_amt.values())))
polyanion = []
# if the composition contains a poly anion
if len(syms) >= 3 and get_el_sp(syms[-1]).X - get_el_sp(syms[-2]).X < 1.65:
poly_sym_amt = {syms[i]: sym_amt[syms[i]] / factor
for i in [-2, -1]}
(poly_form, poly_factor) = reduce_formula(
poly_sym_amt, iupac_ordering=iupac_ordering)
if poly_factor != 1:
polyanion.append("({}){}".format(poly_form, int(poly_factor)))
syms = syms[:len(syms) - 2 if polyanion else len(syms)]
if iupac_ordering:
syms = sorted(syms,
key=lambda x: [get_el_sp(x).iupac_ordering, x])
reduced_form = []
for s in syms:
normamt = sym_amt[s] * 1.0 / factor
reduced_form.append(s)
reduced_form.append(formula_double_format(normamt))
reduced_form = "".join(reduced_form + polyanion)
return reduced_form, factor
def reduce_formula(sym_amt, iupac_ordering=False):
"""
Helper method to reduce a sym_amt dict to a reduced formula and factor.
Args:
sym_amt (dict): {symbol: amount}.
iupac_ordering (bool, optional): Whether to order the
formula by the iupac "electronegativity" series, defined in
Table VI of "Nomenclature of Inorganic Chemistry (IUPAC
Recommendations 2005)". This ordering effectively follows
the groups and rows of the periodic table, except the
Lanthanides, Actanides and hydrogen. Note that polyanions
will still be determined based on the true electronegativity of
the elements.
Returns:
(reduced_formula, factor).
"""
syms = sorted(sym_amt.keys(), key=lambda x: [get_el_sp(x).X, x])
syms = list(filter(
lambda x: abs(sym_amt[x]) > Composition.amount_tolerance, syms))
factor = 1
# Enforce integers for doing gcd.
if all((int(i) == i for i in sym_amt.values())):
factor = abs(gcd(*(int(i) for i in sym_amt.values())))
polyanion = []
# if the composition contains a poly anion
if len(syms) >= 3 and get_el_sp(syms[-1]).X - get_el_sp(syms[-2]).X < 1.65:
poly_sym_amt = {syms[i]: sym_amt[syms[i]] / factor
for i in [-2, -1]}
(poly_form, poly_factor) = reduce_formula(
poly_sym_amt, iupac_ordering=iupac_ordering)
if poly_factor != 1:
polyanion.append("({}){}".format(poly_form, int(poly_factor)))
syms = syms[:len(syms) - 2 if polyanion else len(syms)]
if iupac_ordering:
syms = sorted(syms,
key=lambda x: [get_el_sp(x).iupac_ordering, x])
reduced_form = []
for s in syms:
normamt = sym_amt[s] * 1.0 / factor
reduced_form.append(s)
reduced_form.append(formula_double_format(normamt))
reduced_form = "".join(reduced_form + polyanion)
return reduced_form, factor
def get_integer_formula_and_factor(self, max_denominator=10000):
mul = gcd(*[
Fraction(v).limit_denominator(max_denominator)
for v in self._elmap.values()
])
sym_amt = self.get_el_amt_dict()
d = {k: round(v / mul) for k, v in sym_amt.items()}
(formula, factor) = reduce_formula(d)
if formula in Composition.special_formulas:
formula = Composition.special_formulas[formula]
factor /= 2
return formula, factor * mul
def reduce_formula(sym_amt):
"""
Helper method to reduce a sym_amt dict to a reduced formula and factor.
Args:
sym_amt (dict): {symbol: amount}.
Returns:
(reduced_formula, factor).
"""
syms = sorted(sym_amt.keys(), key=lambda s: get_el_sp(s).X)
syms = list(
filter(lambda s: abs(sym_amt[s]) > Composition.amount_tolerance, syms))
num_el = len(syms)
contains_polyanion = (
num_el >= 3 and
get_el_sp(syms[num_el - 1]).X - get_el_sp(syms[num_el - 2]).X < 1.65)
factor = 1
# Enforce integers for doing gcd.
if all((int(i) == i for i in sym_amt.values())):
factor = abs(gcd(*(int(i) for i in sym_amt.values())))
reduced_form = []
n = num_el - 2 if contains_polyanion else num_el
for i in range(0, n):
s = syms[i]
normamt = sym_amt[s] * 1.0 / factor
reduced_form.append(s)
reduced_form.append(formula_double_format(normamt))
if contains_polyanion:
poly_sym_amt = {
syms[i]: sym_amt[syms[i]] / factor
for i in range(n, num_el)
}
(poly_form, poly_factor) = reduce_formula(poly_sym_amt)
if poly_factor != 1:
reduced_form.append("({}){}".format(poly_form, int(poly_factor)))
else:
reduced_form.append(poly_form)
reduced_form = "".join(reduced_form)
return reduced_form, factor
def reduce_formula(sym_amt):
"""
Helper method to reduce a sym_amt dict to a reduced formula and factor.
Args:
sym_amt (dict): {symbol: amount}.
Returns:
(reduced_formula, factor).
"""
syms = sorted(sym_amt.keys(),
key=lambda s: get_el_sp(s).X)
syms = list(filter(lambda s: abs(sym_amt[s]) >
Composition.amount_tolerance, syms))
num_el = len(syms)
contains_polyanion = (num_el >= 3 and
get_el_sp(syms[num_el - 1]).X
- get_el_sp(syms[num_el - 2]).X < 1.65)
factor = 1
# Enforce integers for doing gcd.
if all((int(i) == i for i in sym_amt.values())):
factor = abs(gcd(*(int(i) for i in sym_amt.values())))
reduced_form = []
n = num_el - 2 if contains_polyanion else num_el
for i in range(0, n):
s = syms[i]
normamt = sym_amt[s] * 1.0 / factor
reduced_form.append(s)
reduced_form.append(formula_double_format(normamt))
if contains_polyanion:
poly_sym_amt = {syms[i]: sym_amt[syms[i]] / factor
for i in range(n, num_el)}
(poly_form, poly_factor) = reduce_formula(poly_sym_amt)
if poly_factor != 1:
reduced_form.append("({}){}".format(poly_form, int(poly_factor)))
else:
reduced_form.append(poly_form)
reduced_form = "".join(reduced_form)
return reduced_form, factor
def get_integer_formula_and_factor(self, max_denominator=10000):
"""
Calculates an integer formula and factor.
Args:
max_denominator (int): all amounts in the el:amt dict are
first converted to a Fraction with this maximum denominator
Returns:
A pretty normalized formula and a multiplicative factor, i.e.,
Li0.5O0.25 returns (Li2O, 0.25). O0.25 returns (O2, 0.125)
"""
mul = gcd(*[Fraction(v).limit_denominator(max_denominator) for v
in self.values()])
d = {k: round(v / mul) for k, v in self.get_el_amt_dict().items()}
(formula, factor) = reduce_formula(d)
if formula in Composition.special_formulas:
formula = Composition.special_formulas[formula]
factor /= 2
return formula, factor * mul
def anonymized_formula(self) -> str:
"""
An anonymized formula. Unique species are arranged in ordering of
increasing amounts and assigned ascending alphabets. Useful for
prototyping formulas. For example, all stoichiometric perovskites have
anonymized_formula ABC3.
"""
reduced = self.element_composition
if all(x == int(x) for x in self.values()):
reduced /= gcd(*(int(i) for i in self.values()))
anon = ""
for e, amt in zip(string.ascii_uppercase, sorted(reduced.values())):
if amt == 1:
amt_str = ""
elif abs(amt % 1) < 1e-8:
amt_str = str(int(amt))
else:
amt_str = str(amt)
anon += f"{e}{amt_str}"
return anon
def anonymized_formula(self):
"""
An anonymized formula. Unique species are arranged in ordering of
increasing amounts and assigned ascending alphabets. Useful for
prototyping formulas. For example, all stoichiometric perovskites have
anonymized_formula ABC3.
"""
reduced = self.element_composition
if all(x == int(x) for x in self._elmap.values()):
reduced /= gcd(*self._elmap.values())
anon = ""
for e, amt in zip(string.ascii_uppercase, sorted(reduced.values())):
if amt == 1:
amt_str = ""
elif abs(amt % 1) < 1e-8:
amt_str = str(int(amt))
else:
amt_str = str(amt)
anon += ("{}{}".format(e, amt_str))
return anon
def test_gcd(self):
self.assertEqual(gcd(7, 14, 63), 7)
def solute_site_preference_finder(
structure, e0, T, vac_defs, antisite_defs, solute_defs,
solute_concen=0.01, trial_chem_pot = None):
"""
Compute the solute defect densities using dilute solution model.
Args:
structure: pymatgen.core.structure.Structure object representing the
primitive or unitcell of the crystal.
e0: The total energy of the undefected system.
This is E0 from VASP calculation.
T: Temperature in Kelvin
vac_defs: List of vacancy defect parameters in the dictionary format.
The keys of the dict associated with each vacancy defect are
1) site_index, 2) site_specie, 3) site_multiplicity, and
4) energy. 1-3 can be obtained from
pymatgen.analysis.defects.point_defects.Vacancy class.
Site index is expected to start with 1 (fortran index).
antisite_defs: List of antisite defect parameters in the dictionary
format. The keys of the dict associated with each antisite
defect are 1) site_index, 2) site_specie, 3) site_multiplicity,
4) substitution_specie, and 5) energy. 1-3 can be obtained
from pymatgen.analysis.defects.point_defects.Vacancy class.
solute_defs: List of solute defect parameters in the dictionary
format. Similary to that of antisite defs, wtih solute specie
specified in substitution_specie
solute_concen: Solute concentration (in fractional value)
trial_chem_pot: Trial chemical potentials to speedup the plot
generation. Format is {el1:mu1,...}
Returns:
plot_data: The data for plotting the solute defect concentration.
"""
if not check_input(vac_defs):
raise ValueError('Vacancy energy is not defined')
if not check_input(antisite_defs):
raise ValueError('Antisite energy is not defined')
formation_energies = {}
formation_energies['vacancies'] = copy.deepcopy(vac_defs)
formation_energies['antisites'] = copy.deepcopy(antisite_defs)
formation_energies['solute'] = copy.deepcopy(solute_defs)
for vac in formation_energies['vacancies']:
del vac['energy']
for asite in formation_energies['antisites']:
del asite['energy']
for solute in formation_energies['solute']:
del solute['energy']
# Setup the system
site_species = [vac_def['site_specie'] for vac_def in vac_defs]
solute_specie = solute_defs[0]['substitution_specie']
site_species.append(solute_specie)
multiplicity = [vac_def['site_multiplicity'] for vac_def in vac_defs]
m = len(set(site_species)) # distinct species
n = len(vac_defs) # inequivalent sites
# Reduce the system and associated parameters such that only distinctive
# atoms are retained
comm_div = gcd(*tuple(multiplicity))
multiplicity = [val/comm_div for val in multiplicity]
multiplicity.append(0)
e0 = e0/comm_div
T = Integer(T)
c0 = np.diag(multiplicity)
#print(('c0', c0))
mu = [Symbol('mu'+str(i)) for i in range(m)]
# Generate maps for hashing
# Generate specie->mu map and use it for site->mu map
specie_order = [] # Contains hash for site->mu map Eg: [Al, Ni]
site_specie_set = set() # Eg: {Ni, Al}
for i in range(len(site_species)):
site_specie = site_species[i]
if site_specie not in site_specie_set:
site_specie_set.add(site_specie)
specie_order.append(site_specie)
site_mu_map = [] # Eg: [mu0,mu0,mu0,mu1] where mu0->Al, and mu1->Ni
for i in range(len(site_species)):
site_specie = site_species[i]
j = specie_order.index(site_specie)
site_mu_map.append(j)
specie_site_index_map = [] # Eg: [(0,3),(3,4)] for Al & Ni
for i in range(m):
low_ind = site_species.index(specie_order[i])
if i < m-1:
hgh_ind = site_species.index(specie_order[i+1])
else:
hgh_ind = len(site_species)
specie_site_index_map.append((low_ind,hgh_ind))
"""
dC: delta concentration matrix:
dC[i,j,k]: Concentration change of atom i, due to presence of atom
j on lattice site k
Special case is [i,i,i] which is considered as vacancy
Few cases: dC[i,i,i] = -1 due to being vacancy special case
dC[k,k,i] = +1 due to increment in k at i lattice if i
lattice type is of different element
dC[i,k,i] = -1 due to decrement of ith type atom due to
presence of kth type atom on ith sublattice and kth type
atom specie is different from ith sublattice atom specie
dC[i,k,k] = 0 due to no effect on ith type atom
dC[i,j,k] = 0 if i!=j!=k
"""
dC = np.zeros((n+1,n+1,n), dtype=np.int)
for i in range(n):
for j in range(n):
for k in range(n):
if i == j and site_species[j] != site_species[k] and \
site_species[i] != site_species:
dC[i,j,k] = 1
for j in range(n+1):
for k in range(n):
if i == k:
#if j == k or site_species[j] != site_species[k]:
dC[i,j,k] = -1
for k in range(n):
dC[n,n,k] = 1
for k in range(n):
for j in range(n):
if i != j:
if site_species[i] == site_species[k]:
dC[i,j,k] = 0
for ind_map in specie_site_index_map:
if ind_map[1]-ind_map[0] > 1:
for index1 in range(ind_map[0]+1,ind_map[1]):
for index2 in range(ind_map[0]):
for i in range(n):
print (i, index1, index2)
dC[i,index1,index2] = 0
for index2 in range(ind_map[1],n):
for i in range(n):
print (i, index1, index2)
dC[i,index1,index2] = 0
print ('dC', dC)
# dE matrix: Flip energies (or raw defect energies)
els = [vac_def['site_specie'] for vac_def in vac_defs]
dE = []
for i in range(n+1):
dE.append([])
for i in range(n+1):
for j in range(n):
dE[i].append(0)
for j in range(n):
for i in range(n):
if i == j:
dE[i][j] = vac_defs[i]['energy']
else:
sub_specie = vac_defs[i]['site_specie']
site_specie = vac_defs[j]['site_specie']
if site_specie == sub_specie:
dE[i][j] = 0
else:
for as_def in antisite_defs:
if int(as_def['site_index']) == j+1 and \
sub_specie == as_def['substitution_specie']:
dE[i][j] = as_def['energy']
break
# Solute
site_specie = vac_defs[j]['site_specie']
for solute_def in solute_defs:
def_site_ind = int(solute_def['site_index'])
def_site_specie = solute_def['site_specie']
#print((def_site_specie, site_specie))
#print((def_site_ind, j+1))
if def_site_specie == site_specie and def_site_ind == j+1:
#print(('se', solute_def['energy']))
dE[n][j] = solute_def['energy']
break
dE = np.array(dE)
#np.where(dE == np.array(None), 0, dE)
# Initialization for concentrations
# c(i,p) == presence of ith type atom on pth type site
c = Matrix(n+1,n,[0]*n*(n+1))
for i in range(n+1):
for p in range(n):
c[i,p] = Integer(c0[i,p])
#print((c[i,p]))
for epi in range(n+1):
sum_mu = sum([mu[site_mu_map[j]]*Integer(
dC[j,epi,p]) for j in range(n+1)])
#print((sum_mu))
#print((multiplicity[p], dC[i,epi,p], dE[epi,p]))
c[i,p] += Integer(multiplicity[p]*dC[i,epi,p]) * \
exp(-(dE[epi,p]-sum_mu)/(k_B*T))
#print(("--------c---------"))
#for i in range(n+1):
# print((c[i,:]))
#print(("--------c---------"))
#specie_concen = [sum(mult[ind[0]:ind[1]]) for ind in specie_site_index_map]
#total_c = [sum(c[ind[0]:ind[1]]) for ind in specie_site_index_map]
total_c = []
for ind in specie_site_index_map:
total_c.append(sum([sum(c[i,:]) for i in range(*ind)]))
#total_c = [sum(c[i,:]) for i in range(n)]
c_ratio = [total_c[i]/sum(total_c) for i in range(m)]
print(('-------c_ratio-------------'))
for i in range(m):
print((c_ratio[i]))
# Expression for Omega, the Grand Potential
omega = e0 - sum([mu[site_mu_map[i]]*sum(c0[i,:]) for i in range(n+1)])
for p_r in range(n):
for epi in range(n):
sum_mu = sum([mu[site_mu_map[j]]*Integer(
dC[j,epi,p_r]) for j in range(n+1)])
omega -= k_B*T*multiplicity[p_r]*exp(-(dE[epi,p_r]-sum_mu)/(k_B*T))
print ('omega')
print (omega)
def compute_mus():
def reduce_mu():
host_concen = 1-solute_concen
new_c0 = c0.astype(float)
for i in range(n):
new_c0[i,i] = host_concen*c0[i,i]
new_c0[n,n] = 2*solute_concen
omega = [
e0-sum([mu[site_mu_map[i]]*sum(new_c0[i,:])
for i in range(n+1)])]
x = solve(omega)
return x
# Compute trial mu
mu_red = reduce_mu()
#print(('mu_red', mu_red))
mult = multiplicity
#for ind in specie_site_index_map:
# print(ind[0], ind[1])
specie_concen = [
sum(mult[ind[0]:ind[1]]) for ind in specie_site_index_map]
#print('specie_concen', specie_concen)
max_host_specie_concen = 1-solute_concen
host_specie_concen_ratio = [specie_concen[i]/sum(specie_concen)* \
max_host_specie_concen for i in range(m)]
host_specie_concen_ratio[-1] = solute_concen
#print('hostspecie_concen_rat', host_specie_concen_ratio)
y_vect = host_specie_concen_ratio
#print(('y_vect', y_vect))
vector_func = [y_vect[i]-c_ratio[i] for i in range(m)]
vector_func.append(omega)
#print((vector_func))
mu_vals = None
c_val = None
m_min = -15.0
if e0 > 0:
m_max = 10 # Search space needs to be modified
else:
m_max = 0
for m1 in np.arange(m_min,m_max,0.3):
for m2 in np.arange(m_min,m_max,0.3):
m0 = mu_red[mu[0]].subs([(mu[1],m1),(mu[2],m2)])
try:
#print(m1,m2)
mu_vals = nsolve(vector_func,mu,[m0,m1,m2],module="numpy")
#mu_vals = nsolve(vector_func,mu,[m0,m1,m2])
# Line needs to be modified to include all mus when n > 2
except:
continue
break
if mu_vals:
mu_vals = [float(mu_val) for mu_val in mu_vals]
break
else:
raise ValueError("Couldn't find mus")
#print (('mu_vals', mu_vals))
return mu_vals
if not trial_chem_pot:
mu_vals = compute_mus()
else:
try:
mu_vals = [trial_chem_pot[element] for element in specie_order]
except:
mu_vals = compute_mus()
#print(('mu_vals', mu_vals))
# Compute ymax
max_host_specie_concen = 1-solute_concen
mult = multiplicity
specie_concen = [
sum(mult[ind[0]:ind[1]]) for ind in specie_site_index_map]
host_specie_concen_ratio = [specie_concen[i]/sum(specie_concen)* \
max_host_specie_concen for i in range(m)]
host_specie_concen_ratio[-1] = solute_concen
#print('hostspecie_concen_rat', host_specie_concen_ratio)
li = specie_site_index_map[0][0]
hi = specie_site_index_map[0][1]
comp1_min = sum(multiplicity[li:hi])/sum(multiplicity)* \
max_host_specie_concen - 0.01
comp1_max = sum(multiplicity[li:hi])/sum(multiplicity)* \
max_host_specie_concen + 0.01
#print(ymin, ymax)
delta = (comp1_max - comp1_min)/50.0
#for i in range(len(mu)):
# print(mu[i], mu_vals[i])
# Compile mu's for all composition ratios in the range
#+/- 1% from the stoichiometry
result = {}
for y in np.arange(comp1_min,comp1_max+delta,delta):
result[y] = []
y_vect = []
y_vect.append(y)
y2 = max_host_specie_concen - y
y_vect.append(y2)
y_vect.append(solute_concen)
#print ('y_vect', y_vect)
vector_func = [y_vect[i]-c_ratio[i] for i in range(1,m)]
vector_func.append(omega)
#try:
#print (vector_func)
#print (mu)
#print (mu_vals)
x = nsolve(vector_func,mu,mu_vals,module="numpy")
#x = nsolve(vector_func,mu,mu_vals)
#except:
# del result[y]
# continue
result[y].append(x[0])
result[y].append(x[1])
result[y].append(x[2])
res = []
#print ('result', result.keys())
# Compute the concentrations for all the compositions
for key in sorted(result.keys()):
mu_val = result[key]
total_c_val = [total_c[i].subs(dict(zip(mu,mu_val))) \
for i in range(len(total_c))]
c_val = c.subs(dict(zip(mu,mu_val)))
#print ('c_val', c_val)
# Concentration of first element/over total concen
res1 = []
res1.append(float(total_c_val[0]/sum(total_c_val)))
sum_c0 = sum([c0[i,i] for i in range(n)])
#print ('c_val_n_0', c_val[n,0])
#print ('c_val_n_1', c_val[n,1])
for i in range(n+1):
for j in range(n):
if i == j: # Vacancy
#res1.append(float((c0[i,i]-sum(c_val[:,i]))/c0[i,i]))
vac_conc = float(exp(-(mu_val[site_mu_map[i]]+dE[i,i])/(k_B*T)))
res1.append(vac_conc)
else: # Antisite
res1.append(float(c_val[i,j]/c0[j,j]))
res.append(res1)
#print ('c0', c0)
#print ('res', res)
res = np.array(res)
#print ('res', res)
dtype = [(str('x'),np.float64)]+[(str('y%d%d' % (i, j)), np.float64) \
for i in range(n+1) for j in range(n)]
res1 = np.sort(res.view(dtype),order=[str('x')],axis=0)
conc = []
for i in range(n+1):
conc.append([])
for j in range(n):
conc[i].append([])
#print(conc)
for i in range(n+1): # Append vacancies
for j in range(n):
y1 = [dat[0][i*n+j+1] for dat in res1]
conc[i][j] = y1
#print(type(conc[i][j]))
plot_data = {}
"""Because all the plots have identical x-points storing it in a
single array"""
#for dat in res1:
# print dat
plot_data['x'] = [dat[0][0] for dat in res1] # x-axis data
# Element whose composition is varied. For x-label
plot_data['x_label'] = els[0]+ "_mole_fraction"
plot_data['y_label'] = "Fraction of {} at {} sites".format(
solute_specie,els[0])
y_data = []
#for i in range(n): # Vacancy plots
inds = specie_site_index_map[m-1]
#print ('inds', inds)
data1 = np.sum([conc[ind][0] for ind in range(*inds)],axis=0)
#print ('data1', data1)
data2 = np.sum([conc[ind][1] for ind in range(*inds)],axis=0)
#print ('data2', data2)
frac_data = data1/(data1+data2)
#print ('frac_data', frac_data)
frac_data = frac_data.tolist()
y_data.append({'data':frac_data})
plot_data['y'] = y_data
return plot_data
def dilute_solution_model(structure, e0, vac_defs, antisite_defs, T,
trial_chem_pot = None, generate='plot'):
"""
Compute the defect densities using dilute solution model.
Args:
structure: pymatgen.core.structure.Structure object representing the
primitive or unitcell of the crystal.
e0: The total energy of the undefected system.
This is E0 from VASP calculation.
vac_defs: List of vacancy defect parameters in the dictionary format.
The keys of the dict associated with each vacancy defect are
1) site_index, 2) site_specie, 3) site_multiplicity, and
4) energy. 1-3 can be obtained from
pymatgen.analysis.defects.point_defects.Vacancy class.
Site index is expected to start with 1 (fortran index).
antisite_defs: List of antisite defect parameters in the dictionary
format. The keys of the dict associated with each antisite defect
are 1) site_index, 2) site_specie, 3) site_multiplicity,
4) substitution_specie, and 5) energy. 1-3 can be obtained
from pymatgen.analysis.defects.point_defects.Vacancy class.
T: Temperature in Kelvin
trial_chem_pot (optional): Trial chemical potentials to speedup
the plot generation. Format is {el1:mu1,...}
generate (string): Options are plot or energy
Chemical potentials are also returned with energy option.
If energy option is not chosen, plot is generated.
Returns:
If generate=plot, the plot data is generated and returned in
HighCharts format.
If generate=energy, defect formation enthalpies and chemical
potentials are returned.
"""
if not check_input(vac_defs):
raise ValueError('Vacancy energy is not defined')
if not check_input(antisite_defs):
raise ValueError('Antisite energy is not defined')
formation_energies = {}
formation_energies['vacancies'] = copy.deepcopy(vac_defs)
formation_energies['antisites'] = copy.deepcopy(antisite_defs)
for vac in formation_energies['vacancies']:
del vac['energy']
for asite in formation_energies['antisites']:
del asite['energy']
# Setup the system
site_species = [vac_def['site_specie'] for vac_def in vac_defs]
multiplicity = [vac_def['site_multiplicity'] for vac_def in vac_defs]
m = len(set(site_species)) # distinct species
n = len(vac_defs) # inequivalent sites
# Reduce the system and associated parameters such that only distinctive
# atoms are retained
comm_div = gcd(*tuple(multiplicity))
multiplicity = [val/comm_div for val in multiplicity]
e0 = e0/comm_div
T = Integer(T)
c0 = np.diag(multiplicity)
#print ('c0',c0)
mu = [Symbol('mu'+i.__str__()) for i in range(m)]
# Generate maps for hashing
# Generate specie->mu map and use it for site->mu map
specie_order = [] # Contains hash for site->mu map Eg: [Al, Ni]
site_specie_set = set() # Eg: {Ni, Al}
for i in range(n):
site_specie = site_species[i]
if site_specie not in site_specie_set:
site_specie_set.add(site_specie)
specie_order.append(site_specie)
site_mu_map = [] # Eg: [mu0,mu0,mu0,mu1] where mu0->Al, and mu1->Ni
for i in range(n):
site_specie = site_species[i]
j = specie_order.index(site_specie)
site_mu_map.append(j)
specie_site_index_map = [] # Eg: [(0,3),(3,4)] for Al & Ni
for i in range(m):
low_ind = site_species.index(specie_order[i])
if i < m-1:
hgh_ind = site_species.index(specie_order[i+1])
else:
hgh_ind = n
specie_site_index_map.append((low_ind,hgh_ind))
"""
dC: delta concentration matrix:
dC[i,j,k]: Concentration change of atom i, due to presence of atom
j on lattice site k
Special case is [i,i,i] which is considered as vacancy
Few cases: dC[i,i,i] = -1 due to being vacancy special case
dC[k,k,i] = +1 due to increment in k at i lattice if i
lattice type is of different element
dC[i,k,i] = -1 due to decrement of ith type atom due to
presence of kth type atom on ith sublattice and kth type
atom specie is different from ith sublattice atom specie
dC[i,k,k] = 0 due to no effect on ith type atom
dC[i,j,k] = 0 if i!=j!=k
"""
dC = np.zeros((n,n,n), dtype=np.int)
for i in range(n):
for j in range(n):
for k in range(n):
if i == j and site_species[j] != site_species[k] and \
site_species[i] != site_species[k]:
dC[i,j,k] = 1
for j in range(n):
for k in range(n):
if i == k:
#if j == k or site_species[j] != site_species[k]:
dC[i,j,k] = -1
for k in range(n):
for j in range(n):
for i in range(n):
if i != j:
if site_species[j] == site_species[k]:
dC[i,j,k] = 0
#for j in range(n):
# if k != j:
# for i in range(n):
# if i == j:
# if abs(i-j) <= 1 and abs(j-k) <= 1 and abs(i-k) <= 1:
# dC[i,j,k] = 0
for ind_map in specie_site_index_map:
if ind_map[1]-ind_map[0] > 1:
for index1 in range(ind_map[0]+1,ind_map[1]):
for index2 in range(ind_map[0]):
for i in range(n):
#print (i, index1, index2)
dC[i,index1,index2] = 0
for index2 in range(ind_map[1],n):
for i in range(n):
#print (i, index1, index2)
dC[i,index1,index2] = 0
#print ('dC', dC)
# dE matrix: Flip energies (or raw defect energies)
els = [vac_def['site_specie'] for vac_def in vac_defs]
dE = []
for i in range(n):
dE.append([])
for i in range(n):
for j in range(n):
dE[i].append(0)
for j in range(n):
for i in range(n):
if i == j:
dE[i][j] = vac_defs[i]['energy']
else:
sub_specie = vac_defs[i]['site_specie']
site_specie = vac_defs[j]['site_specie']
if site_specie == sub_specie:
dE[i][j] = 0
else:
for as_def in antisite_defs:
if int(as_def['site_index']) == j+1 and \
sub_specie == as_def['substitution_specie']:
dE[i][j] = as_def['energy']
break
dE = np.array(dE)
#np.where(dE is None, dE, 0)
#print ('dE', dE)
# Initialization for concentrations
# c(i,p) == presence of ith type atom on pth type site
c = Matrix(n,n,[0]*n**2)
for i in range(n):
for p in range(n):
c[i,p] = Integer(c0[i,p])
site_flip_contribs = []
for epi in range(n):
sum_mu = sum([mu[site_mu_map[j]]*Integer(dC[j,epi,p]) \
for j in range(n)])
#print (i, epi, p, dC[i,epi, p], sum_mu)
#c[i,p] += Integer(multiplicity[p]*dC[i,epi,p]) * \
# exp(-(dE[epi,p]-sum_mu)/(k_B*T))
flip = Integer(multiplicity[p]*dC[i,epi,p]) * \
exp(-(dE[epi,p]-sum_mu)/(k_B*T))
if flip not in site_flip_contribs:
site_flip_contribs.append(flip)
c[i,p] += flip
#else:
#print (i, epi, p)
#print ('flip already present in site_flips')
#print ('c', c)
total_c = []
for ind in specie_site_index_map:
total_c.append(sum([sum(c[i,:]) for i in range(*ind)]))
c_ratio = [total_c[-1]/total_c[i] for i in range(m)]
#print ('c_ratio')
#for i in range(len(c_ratio)):
# print(c_ratio[i])
# Expression for Omega, the Grand Potential
omega = e0 - sum([mu[site_mu_map[i]]*sum(c0[i,:]) for i in range(n)])
for p_r in range(n):
for epi in range(n):
sum_mu = sum([mu[site_mu_map[j]]*Float(
dC[j,epi,p_r]) for j in range(n)])
omega -= k_B*T*multiplicity[p_r]*exp(-(dE[epi,p_r]-sum_mu)/(k_B*T))
def compute_mus():
def reduce_mu():
omega = [e0 - sum([mu[site_mu_map[i]]*sum(c0[i,:]) for i in range(n)])]
x = solve(omega)
return x
# Compute trial mu
mu_red = reduce_mu()
mult = multiplicity
specie_concen = [sum(mult[ind[0]:ind[1]]) for ind in specie_site_index_map]
y_vect = [specie_concen[-1]/specie_concen[i] for i in range(m)]
vector_func = [y_vect[i]-c_ratio[i] for i in range(m-1)]
vector_func.append(omega)
min_diff = 1e10
mu_vals = None
c_val = None
m1_min = -20.0
if e0 > 0:
m1_max = 10 # Search space needs to be modified
else:
m1_max = 0
for m1 in np.arange(m1_min,m1_max,0.1):
m0 = mu_red[mu[0]].subs(mu[-1],m1)
try:
x = nsolve(vector_func,mu,[m0,m1],module="numpy")
except:
continue
c_val = c.subs(dict(zip(mu,x)))
#if all(x >= 0 for x in c_val):
specie_concen = []
for ind in specie_site_index_map:
specie_concen.append(sum([sum(c_val[i,:]) for i in range(*ind)]))
y_comp = [specie_concen[-1]/specie_concen[i] for i in range(m)]
diff = math.sqrt(sum([pow(abs(y_comp[i]-y_vect[i]),2) for i in range(m)]))
if diff < min_diff:
min_diff = diff
mu_vals = x
if mu_vals:
mu_vals = [float(mu_val) for mu_val in mu_vals]
else:
raise ValueError()
return mu_vals
def compute_def_formation_energies():
i = 0
for vac_def in vac_defs:
site_specie = vac_def['site_specie']
ind = specie_order.index(site_specie)
uncor_energy = vac_def['energy']
formation_energy = uncor_energy + mu_vals[ind]
formation_energies['vacancies'][i]['formation_energy'] = formation_energy
specie_ind = site_mu_map[i]
indices = specie_site_index_map[specie_ind]
specie_ind_del = indices[1]-indices[0]
cur_ind = i - indices[0] + 1
if not specie_ind_del-1:
label = '$V_{'+site_specie+'}$'
else:
label = '$V_{'+site_specie+'_'+str(cur_ind)+'}$'
formation_energies['vacancies'][i]['label'] = label
i += 1
i = 0
for as_def in antisite_defs:
site_specie = as_def['site_specie']
sub_specie = as_def['substitution_specie']
ind1 = specie_order.index(site_specie)
ind2 = specie_order.index(sub_specie)
uncor_energy = as_def['energy']
formation_energy = uncor_energy + mu_vals[ind1] - mu_vals[ind2]
formation_energies['antisites'][i]['formation_energy'] = formation_energy
specie_ind = site_mu_map[i]
indices = specie_site_index_map[specie_ind]
specie_ind_del = indices[1]-indices[0]
cur_ind = i - indices[0] + 1
if not specie_ind_del-1:
label = '$'+sub_specie+'_{'+site_specie+'}$'
else:
label = '$'+sub_specie+'_{'+site_specie+'_'+str(cur_ind)+'}$'
formation_energies['antisites'][i]['label'] = label
i += 1
return formation_energies
if not trial_chem_pot:
mu_vals = compute_mus()
else:
try:
mu_vals = [trial_chem_pot[element] for element in specie_order]
except:
mu_vals = compute_mus()
if generate == 'energy':
formation_energies = compute_def_formation_energies()
mu_dict = dict(zip(specie_order,mu_vals))
return formation_energies, mu_dict
#sys.exit()
# Compute ymax
li = specie_site_index_map[0][0]
hi = specie_site_index_map[0][1]
comp1_min = int(sum(multiplicity[li:hi])/sum(multiplicity)*100)-1
comp1_max = int(sum(multiplicity[li:hi])/sum(multiplicity)*100)+1
delta = float(comp1_max-comp1_min)/120.0
yvals = []
for comp1 in np.arange(comp1_min,comp1_max+delta,delta):
comp2 = 100-comp1
y = comp2/comp1
yvals.append(y)
#ymin = comp2_min/comp1_max
#print comp1_min, comp1_max
#print comp2_min, comp2_max
#print ymax, ymin
# Compile mu's for all composition ratios in the range
#+/- 1% from the stoichiometry
result = {}
#for y in np.arange(ymin,ymax+delta,delta):
for y in yvals:
result[y] = []
vector_func = [y-c_ratio[0]]
vector_func.append(omega)
x = nsolve(vector_func,mu,mu_vals,module="numpy")
result[y].append(x[0])
result[y].append(x[1])
res = []
new_mu_dict = {}
# Compute the concentrations for all the compositions
for key in sorted(result.keys()):
mu_val = result[key]
total_c_val = [total_c[i].subs(dict(zip(mu,mu_val))) \
for i in range(len(total_c))]
c_val = c.subs(dict(zip(mu,mu_val)))
res1 = []
# Concentration of first element/over total concen
res1.append(float(total_c_val[0]/sum(total_c_val)))
new_mu_dict[res1[0]] = mu_val
sum_c0 = sum([c0[i,i] for i in range(n)])
#print res1[0]
for i in range(n):
for j in range(n):
if i == j: # Vacancy
#print (i, c_val[:,i])
# Consider numerical accuracy
#res1.append(float((c0[i,i]-sum(c_val[:,i]))/c0[i,i]))
#print ((mu_val[site_mu_map[i]]-dE[i,i])/(k_B*T))
vac_conc = float(exp(-(mu_val[site_mu_map[i]]+dE[i,i])/(k_B*T)))
#print vac_conc
res1.append(vac_conc)
else: # Antisite
res1.append(float(c_val[i,j]/c0[j,j]))
res.append(res1)
res = np.array(res)
dtype = [(str('x'),np.float64)]+[(str('y%d%d' % (i, j)), np.float64) \
for i in range(n) for j in range(n)]
res1 = np.sort(res.view(dtype), order=[str('x')],axis=0)
conc_data = {}
"""Because all the plots have identical x-points storing it in a
single array"""
conc_data['x'] = [dat[0][0] for dat in res1] # x-axis data
# Element whose composition is varied. For x-label
conc_data['x_label'] = els[0]+ " mole fraction"
conc_data['y_label'] = "Point defect concentration"
conc = []
for i in range(n):
conc.append([])
for j in range(n):
conc[i].append([])
for i in range(n):
for j in range(n):
y1 = [dat[0][i*n+j+1] for dat in res1]
conc[i][j] = y1
y_data = []
for i in range(n):
data = conc[i][i]
specie = els[i]
specie_ind = site_mu_map[i]
indices = specie_site_index_map[specie_ind]
specie_ind_del = indices[1]-indices[0]
cur_ind = i - indices[0] + 1
#if 'V' not in els:
# vac_string = "$V_{"
#else:
vac_string = "$Vac_{"
if not specie_ind_del-1:
label = vac_string+specie+'}$'
else:
label = vac_string+specie+'_'+str(cur_ind)+'}$'
# Plot data and legend info
y_data.append({'data':data,'name':label})
for i in range(n):
site_specie = els[i]
specie_ind = site_mu_map[i]
indices = specie_site_index_map[specie_ind]
specie_ind_del = indices[1]-indices[0]
cur_ind = i - indices[0] + 1
for j in range(m): # Antisite plot dat
sub_specie = specie_order[j]
if sub_specie == site_specie:
continue
if not specie_ind_del-1:
label = '$'+sub_specie+'_{'+site_specie+'}$'
else:
label = '$'+sub_specie+'_{'+site_specie+'_'+str(cur_ind)+'}$'
inds = specie_site_index_map[j]
data = np.sum([conc[ind][i] for ind in range(*inds)],axis=0)
data = data.tolist()
y_data.append({'data':data,'name':label})
conc_data['y'] = y_data
# Compute the formation energies
def compute_vac_formation_energies(mu_vals):
en = []
for vac_def in vac_defs:
site_specie = vac_def['site_specie']
ind = specie_order.index(site_specie)
uncor_energy = vac_def['energy']
formation_energy = uncor_energy + mu_vals[ind]
en.append(float(formation_energy))
return en
en_res = []
for key in sorted(new_mu_dict.keys()):
mu_val = new_mu_dict[key]
en_res.append(compute_vac_formation_energies(mu_val))
en_data = {'x_label':els[0]+' mole fraction', 'x':[]}
en_data['x'] = [dat[0][0] for dat in res1] # x-axis data
i = 0
y_data = []
for vac_def in vac_defs:
data = [data[i] for data in en_res]
site_specie = vac_def['site_specie']
ind = specie_order.index(site_specie)
specie_ind = site_mu_map[i]
indices = specie_site_index_map[specie_ind]
specie_ind_del = indices[1]-indices[0]
cur_ind = i - indices[0] + 1
vac_string = "$Vac_{"
if not specie_ind_del-1:
label = vac_string+site_specie+'}$'
else:
label = vac_string+site_specie+'_'+str(cur_ind)+'}$'
y_data.append({'data':data,'name':label})
i += 1
def compute_as_formation_energies(mu_vals):
en = []
for as_def in antisite_defs:
site_specie = as_def['site_specie']
sub_specie = as_def['substitution_specie']
ind1 = specie_order.index(site_specie)
ind2 = specie_order.index(sub_specie)
uncor_energy = as_def['energy']
form_en = uncor_energy + mu_vals[ind1] - mu_vals[ind2]
en.append(form_en)
return en
en_res = []
for key in sorted(new_mu_dict.keys()):
mu_val = new_mu_dict[key]
en_res.append(compute_as_formation_energies(mu_val))
i = 0
for as_def in antisite_defs:
data = [data[i] for data in en_res]
site_specie = as_def['site_specie']
sub_specie = as_def['substitution_specie']
ind1 = specie_order.index(site_specie)
ind2 = specie_order.index(sub_specie)
specie_ind = site_mu_map[i]
indices = specie_site_index_map[specie_ind]
specie_ind_del = indices[1]-indices[0]
cur_ind = i - indices[0] + 1
if not specie_ind_del-1:
label = '$'+sub_specie+'_{'+site_specie+'}$'
else:
label = '$'+sub_specie+'_{'+site_specie+'_'+str(cur_ind)+'}$'
y_data.append({'data':data,'name':label})
i += 1
en_data['y'] = y_data
# Return chem potential as well
mu_data = {'x_label':els[0]+' mole fraction', 'x':[]}
mu_data['x'] = [dat[0][0] for dat in res1] # x-axis data
y_data = []
for j in range(m):
specie = specie_order[j]
mus = [new_mu_dict[key][j] for key in sorted(new_mu_dict.keys())]
y_data.append({'data':mus, 'name':specie})
mu_data['y'] = y_data
return conc_data, en_data, mu_data
def get_formula_from_components(components: List[Component],
molecules_first: bool = False,
use_iupac_formula: bool = True,
use_common_formulas: bool = True) -> str:
"""Reconstructs a chemical formula from structure components.
The chemical formulas for the individual components will be grouped
together. If two components share the same composition, they will be
treated as equivalent.
Args:
components: A list of structure components, generated using
:obj:`pymatgen.analysis.dimensionality.get_structure_components`.
molecules_first: Whether to put any molecules (zero-dimensional
components) at the beginning of the formula.
use_iupac_formula (bool, optional): Whether to order formulas by the
iupac "electronegativity" series, defined in Table VI of
"Nomenclature of Inorganic Chemistry (IUPAC Recommendations 2005)".
This ordering effectively follows the groups and rows of the
periodic table, except the Lanthanides, Actanides and hydrogen. If
set to ``False``, the elements will be ordered according to the
electronegativity values.
use_common_formulas: Whether to use the database of common formulas.
The common formula will be used preferentially to the iupac or
reduced formula.
Returns:
The chemical formula.
"""
def order(comp_formula):
composition = Composition(comp_formula)
if use_iupac_formula:
return (sum(
[get_el_sp(s).iupac_ordering
for s in composition.elements]) / len(composition.elements))
else:
return composition.average_electroneg
components = get_formula_inequiv_components(
components,
use_iupac_formula=use_iupac_formula,
use_common_formulas=use_common_formulas)
if molecules_first:
mol_comps, other_comps = filter_molecular_components(components)
else:
mol_comps = []
other_comps = components
formulas = (sorted([c['formula'] for c in mol_comps], key=order) +
sorted([c['formula'] for c in other_comps], key=order))
# if components include special formulas, then the count can be 0.5
# therefore if any non integer amounts we can just use a factor of 2
all_int = all(v['count'] % 1 == 0 for v in components)
prefactor = 1 if all_int else 2
form_count_dict = {
c['formula']: int(c['count'] * prefactor)
for c in components
}
# the following is based on ``pymatgen.core.composition.reduce_formula``
num_comps = len(formulas)
factor = abs(gcd(*(form_count_dict.values())))
reduced_form = []
for i in range(0, num_comps):
formula = formulas[i]
normamt = form_count_dict[formula] * 1.0 / factor
formatted_formula = formula if normamt == 1 else "({})".format(formula)
reduced_form.append(formatted_formula)
reduced_form.append(formula_double_format(normamt))
reduced_form = "".join(reduced_form)
return reduced_form
def test_gcd(self):
self.assertEqual(gcd(7, 14, 63), 7)
def dilute_solution_model(structure,
e0,
vac_defs,
antisite_defs,
T,
trial_chem_pot=None,
generate='plot'):
"""
Compute the defect densities for a structure based on the input parameters
using dilute solution model.
Args:
structure:
pymatgen.core.structure.Structure object representing the
primitive or unitcell of the crystal.
e0:
The total energy of the undefected system.
This is E0 from VASP calculation.
vac_defs:
List of vacancy defect parameters in the dictionary format.
The keys of the dict associated with each vacancy defect are
1) site_index, 2) site_specie, 3) site_multiplicity, and
4) energy. 1-3 can be obtained from
pymatgen.analysis.defects.point_defects.Vacancy class.
Site index is expected to start with 1 (fortran index).
antisite_defs:
List of antisite defect parameters in the dictionary format.
The keys of the dict associated with each antisite defect are
1) site_index, 2) site_specie, 3) site_multiplicity,
4) substitution_specie, and 5) energy. 1-3 can be obtained
from pymatgen.analysis.defects.point_defects.Vacancy class.
T:
Temperature in Kelvin
trial_chem_pot:
Trial chemical potentials to speedup the plot generation
Format is {el1:mu1,...}
generate (string): Options are plot or energy
Chemical potentials are also returned with energy option.
If energy option is not chosen, plot is generated.
"""
if not check_input(vac_defs):
raise ValueError('Vacancy energy is not defined')
if not check_input(antisite_defs):
raise ValueError('Antisite energy is not defined')
formation_energies = {}
formation_energies['vacancies'] = copy.deepcopy(vac_defs)
formation_energies['antisites'] = copy.deepcopy(antisite_defs)
for vac in formation_energies['vacancies']:
del vac['energy']
for asite in formation_energies['antisites']:
del asite['energy']
# Setup the system
site_species = [vac_def['site_specie'] for vac_def in vac_defs]
multiplicity = [vac_def['site_multiplicity'] for vac_def in vac_defs]
#print multiplicity
m = len(set(site_species)) # distinct species
n = len(vac_defs) # inequivalent sites
# Reduce the system and associated parameters such that only distinctive
# atoms are retained
comm_div = gcd(*tuple(multiplicity))
multiplicity = [val / comm_div for val in multiplicity]
e0 = e0 / comm_div
T = Integer(T)
c0 = np.diag(multiplicity)
#print 'c0', c0
mu = [Symbol('mu' + str(i)) for i in range(m)]
# Generate maps for hashing
# Generate specie->mu map and use it for site->mu map
specie_order = [] # Contains hash for site->mu map Eg: [Al, Ni]
site_specie_set = set() # Eg: {Ni, Al}
for i in range(n):
site_specie = site_species[i]
if site_specie not in site_specie_set:
site_specie_set.add(site_specie)
specie_order.append(site_specie)
site_mu_map = [] # Eg: [mu0,mu0,mu0,mu1] where mu0->Al, and mu1->Ni
for i in range(n):
site_specie = site_species[i]
j = specie_order.index(site_specie)
site_mu_map.append(j)
specie_site_index_map = [] # Eg: [(0,3),(3,4)] for Al & Ni
for i in range(m):
low_ind = site_species.index(specie_order[i])
if i < m - 1:
hgh_ind = site_species.index(specie_order[i + 1])
else:
hgh_ind = n
specie_site_index_map.append((low_ind, hgh_ind))
#print 'specie_site_index_map', specie_site_index_map
#for el in specie_site_index_map:
# print range(*el)
#print site_species
#print 'site_mu_map', site_mu_map
#print specie_order
"""
dC: delta concentration matrix:
dC[i,j,k]: Concentration change of atom i, due to presence of atom
j on lattice site k
Special case is [i,i,i] which is considered as vacancy
Few cases: dC[i,i,i] = -1 due to being vacancy special case
dC[k,k,i] = +1 due to increment in k at i lattice if i
lattice type is of different element
dC[i,k,i] = -1 due to decrement of ith type atom due to
presence of kth type atom on ith sublattice and kth type
atom specie is different from ith sublattice atom specie
dC[i,k,k] = 0 due to no effect on ith type atom
dC[i,j,k] = 0 if i!=j!=k
"""
dC = np.zeros((n, n, n), dtype=np.int)
for i in range(n):
for j in range(n):
for k in range(n):
if i == j: # and site_species[j] != site_species[k]:
dC[i, j, k] = 1
for j in range(n):
for k in range(n):
if i == k:
#if j == k or site_species[j] != site_species[k]:
dC[i, j, k] = -1
#print dC
# dE matrix: Flip energies (or raw defect energies)
els = [vac_def['site_specie'] for vac_def in vac_defs]
dE = []
for i in range(n):
dE.append([])
for i in range(n):
for j in range(n):
dE[i].append(None)
for j in range(n):
for i in range(n):
if i == j:
dE[i][j] = vac_defs[i]['energy']
else:
sub_specie = vac_defs[i]['site_specie']
site_specie = vac_defs[j]['site_specie']
if site_specie == sub_specie:
dE[i][j] = 0
else:
for as_def in antisite_defs:
if as_def['site_index'] == j+1 and \
sub_specie == as_def['substitution_specie']:
dE[i][j] = as_def['energy']
break
dE = np.array(dE)
np.where(dE == None, dE, 0)
#print dE
# Initialization for concentrations
# c(i,p) == presence of ith type atom on pth type site
c = Matrix(n, n, [0] * n**2)
for i in range(n):
for p in range(n):
c[i, p] = Integer(c0[i, p])
for epi in range(n):
sum_mu = sum([
mu[site_mu_map[j]] * Integer(dC[j, epi, p])
for j in range(n)
])
c[i,p] += Integer(multiplicity[p]*dC[i,epi,p]) * \
exp(-(dE[epi,p]-sum_mu)/(k_B*T))
#specie_concen = [sum(mult[ind[0]:ind[1]]) for ind in specie_site_index_map]
#total_c = [sum(c[ind[0]:ind[1]]) for ind in specie_site_index_map]
total_c = []
for ind in specie_site_index_map:
total_c.append(sum([sum(c[i, :]) for i in range(*ind)]))
#total_c = [sum(c[i,:]) for i in range(n)]
c_ratio = [total_c[-1] / total_c[i] for i in range(m)]
#print 'c_ratio'
#for i in range(len(c_ratio)):
#print c_ratio[i]
# Expression for Omega, the Grand Potential
omega = e0 - sum([mu[site_mu_map[i]] * sum(c0[i, :]) for i in range(n)])
for p_r in range(n):
for epi in range(n):
sum_mu = sum([
mu[site_mu_map[j]] * Integer(dC[j, epi, p_r]) for j in range(n)
])
omega -= k_B * T * multiplicity[p_r] * exp(
-(dE[epi, p_r] - sum_mu) / (k_B * T))
def compute_mus():
def reduce_mu():
omega = [
e0 -
sum([mu[site_mu_map[i]] * sum(c0[i, :]) for i in range(n)])
]
x = solve(omega)
return x
# Compute trial mu
mu_red = reduce_mu()
#print mu_red
mult = multiplicity
#for ind in specie_site_index_map:
# print ind[0], ind[1]
specie_concen = [
sum(mult[ind[0]:ind[1]]) for ind in specie_site_index_map
]
#print 'specie_concent', specie_concen
y_vect = [specie_concen[-1] / specie_concen[i] for i in range(m)]
#print 'y_vect', y_vect
vector_func = [y_vect[i] - c_ratio[i] for i in range(m - 1)]
vector_func.append(omega)
#print vector_func
#vector_func.append(mu_equalities)
#print 'y0', y0
min_diff = 1e10
mu_vals = None
c_val = None
m1_min = -20.0
if e0 > 0:
m1_max = 10 # Search space needs to be modified
else:
m1_max = 0
for m1 in np.arange(m1_min, m1_max, 0.1):
m0 = mu_red[mu[0]].subs(mu[-1], m1)
try:
x = nsolve(vector_func, mu, [m0, m1], module="numpy")
# Line needs to be modified to include all mus when n > 2
except:
continue
c_val = c.subs(dict(zip(mu, x)))
#print c_val
#if all(x >= 0 for x in c_val):
specie_concen = []
for ind in specie_site_index_map:
specie_concen.append(
sum([sum(c_val[i, :]) for i in range(*ind)]))
y_comp = [specie_concen[-1] / specie_concen[i] for i in range(m)]
diff = math.sqrt(
sum([pow(abs(y_comp[i] - y_vect[i]), 2) for i in range(m)]))
if diff < min_diff:
min_diff = diff
mu_vals = x
if mu_vals:
mu_vals = [float(mu_val) for mu_val in mu_vals]
else:
raise ValueError()
print mu_vals
return mu_vals
#print els
def compute_def_formation_energies():
i = 0
for vac_def in vac_defs:
site_specie = vac_def['site_specie']
ind = specie_order.index(site_specie)
uncor_energy = vac_def['energy']
formation_energy = uncor_energy + mu_vals[ind]
print site_specie, 'vancancy formation_energy', formation_energy
formation_energies['vacancies'][i][
'formation_energy'] = formation_energy
specie_ind = site_mu_map[i]
indices = specie_site_index_map[specie_ind]
specie_ind_del = indices[1] - indices[0]
cur_ind = i - indices[0] + 1
if not specie_ind_del - 1:
label = '$V_{' + site_specie + '}$'
else:
label = '$V_{' + site_specie + '_' + str(cur_ind) + '}$'
formation_energies['vacancies'][i]['label'] = label
i += 1
i = 0
for as_def in antisite_defs:
site_specie = as_def['site_specie']
sub_specie = as_def['substitution_specie']
ind1 = specie_order.index(site_specie)
ind2 = specie_order.index(sub_specie)
uncor_energy = as_def['energy']
formation_energy = uncor_energy + mu_vals[ind1] - mu_vals[ind2]
print site_specie, sub_specie, 'antisite ', formation_energy
formation_energies['antisites'][i][
'formation_energy'] = formation_energy
specie_ind = site_mu_map[i]
indices = specie_site_index_map[specie_ind]
specie_ind_del = indices[1] - indices[0]
cur_ind = i - indices[0] + 1
if not specie_ind_del - 1:
label = '$' + sub_specie + '_{' + site_specie + '}$'
else:
label = '$' + sub_specie + '_{' + site_specie + '_' + str(
cur_ind) + '}$'
formation_energies['antisites'][i]['label'] = label
i += 1
return formation_energies
if not trial_chem_pot:
mu_vals = compute_mus()
else:
try:
mu_vals = [trail_chem_pot[element] for element in specie_ordger]
except:
mu_vals = compute_mus()
if generate == 'energy':
formation_energies = compute_def_formation_energies()
mu_dict = dict(zip(specie_order, mu_vals))
return formation_energies, mu_dict
# Compute ymax
li = specie_site_index_map[0][0]
hi = specie_site_index_map[0][1]
comp1_min = int(sum(multiplicity[li:hi]) / sum(multiplicity) * 100) - 1
comp2_max = 100 - comp1_min
ymax = comp2_max / comp1_min
comp1_max = int(sum(multiplicity[li:hi]) / sum(multiplicity) * 100) + 1
comp2_min = 100 - comp1_max
ymin = comp2_min / comp1_max
#print ymin, ymax
delta = (ymax - ymin) / 40.0
#for i in range(len(mu)):
# print mu[i], mu_vals[i]
# Compile mu's for all composition ratios in the range
#+/- 1% from the stoichiometry
result = {}
for y in np.arange(ymin, ymax, delta):
result[y] = []
vector_func = [y - c_ratio[0]]
vector_func.append(omega)
x = nsolve(vector_func, mu, mu_vals, module="numpy")
result[y].append(x[0])
result[y].append(x[1])
res = []
# Compute the concentrations for all the compositions
for key in result:
mu_val = result[key]
total_c_val = [total_c[i].subs(dict(zip(mu,mu_val))) \
for i in range(len(total_c))]
c_val = c.subs(dict(zip(mu, mu_val)))
res1 = []
# Concentration of first element/over total concen
res1.append(float(total_c_val[0] / sum(total_c_val)))
sum_c0 = sum([c0[i, i] for i in range(n)])
for i in range(n):
for j in range(n):
if i == j: # Vacancy
res1.append(float(
(c0[i, i] - sum(c_val[:, i])) / c0[i, i]))
else: # Antisite
res1.append(float(c_val[i, j] / c0[i, i]))
res.append(res1)
res = np.array(res)
dtype = [('x',np.float64)]+[('y'+str(i)+str(j),np.float64) \
for i in range(n) for j in range(n)]
res1 = np.sort(res.view(dtype), order=['x'], axis=0)
plot_data = {}
"""Because all the plots have identical x-points storing it in a
single array"""
plot_data['x'] = [dat[0][0] for dat in res1] # x-axis data
# Element whose composition is varied. For x-label
plot_data['x_label'] = els[0] + " mole fraction"
plot_data['y_label'] = "Point defect concentration"
conc = []
for i in range(n):
conc.append([])
for j in range(n):
conc[i].append([])
#print conc
for i in range(n): # Append vacancies
for j in range(n):
y1 = [dat[0][i * n + j + 1] for dat in res1]
conc[i][j] = y1
#print type(conc[i][j])
y_data = []
for i in range(n): # Vacancy plots
data = conc[i][i]
specie = els[i]
specie_ind = site_mu_map[i]
indices = specie_site_index_map[specie_ind]
specie_ind_del = indices[1] - indices[0]
cur_ind = i - indices[0] + 1
if 'V' not in els:
vac_string = "$V_{"
else:
vac_string = "$Vac_{"
if not specie_ind_del - 1:
label = vac_string + specie + '}$'
else:
label = vac_string + specie + '_' + str(cur_ind) + '}$'
# Plot data and legend info
y_data.append({'data': data, 'name': label})
site_specie = els[i]
for j in range(m): # Antisite plot dat
sub_specie = specie_order[j]
if sub_specie == site_specie:
continue
if not specie_ind_del - 1:
label = '$' + sub_specie + '_{' + specie + '}$'
else:
label = '$' + sub_specie + '_{' + specie + '_' + str(
cur_ind) + '}$'
inds = specie_site_index_map[j]
data = np.sum([conc[ind][i] for ind in range(*inds)], axis=0)
data = data.tolist()
y_data.append({'data': data, 'name': label})
plot_data['y'] = y_data
return plot_data