def calc_orhomb(dhkl):
"""
Orthorhombic
Quadratic form:
1 / d^2 = h^2 / a^2 + k^2 / b^2 + l^2 / c^2
"""
d, h, k, el = dhkl
y = d**-2
bl = el**2
bk = k**2
bh = h**2
yh = aver(bh * y)
yk = aver(bk * y)
yl = aver(bl * y)
h2 = aver(bh**2)
k2 = aver(bk**2)
l2 = aver(bl**2)
hk = aver(bh * bk)
hl = aver(bh * bl)
kl = aver(bk * bl)
matrA = array([[h2, hk, hl], [hk, k2, kl], [hl, kl, l2]])
colB = array([yh, yk, yl])
ba, bb, bc = solve(matrA, colB)
a = ba**-.5
b = bb**-.5
c = bc**-.5
chi2 = ba ** 2 * h2 + 2 * ba * bb * hk + 2 * ba * bc * hl - \
2 * ba * yh + bb ** 2 * k2 + 2 * bb * bc * kl - 2 * bb * yk + \
bc ** 2 * l2 - 2 * bc * yl + aver(y ** 2)
return (a, b, c, None, None, None, chi2, None, None, None, None, None,
None)
def calc_cubic(dhkl):
"""
Cubic
Quadratic form:
1 / d^2 = (h^2 + k^2 + l^2) / a^2
"""
d, h, k, el = dhkl
y = d**-2
bm = h**2 + k**2 + el**2
ym = aver(y * bm)
m2 = aver(bm**2)
ba = ym / m2
a = ba**-.5
chi2 = ba**2 * m2 - 2 * ba * ym + aver(y**2)
return (a, None, None, None, None, None, chi2, None, None, None, None,
None, None)
def calc_tetra(dhkl):
"""
Tetrahonal
Quadratic form:
1 / d^2 = (h^2 + k^2) / a^2 + l^2 / c^2
"""
d, h, k, el = dhkl
y = d**-2
bl = el**2
bm = h**2 + k**2
yl = aver(y * bl)
ym = aver(y * bm)
lm = aver(bl * bm)
m2 = aver(bm**2)
l2 = aver(bl**2)
matrA = array([[lm, l2], [m2, lm]])
colB = array([yl, ym])
ba, bb = solve(matrA, colB)
a = ba**-.5
c = bb**-.5
chi2 = ba ** 2 * m2 + 2 * ba * bb * lm - 2 * ba * ym + bb ** 2 * l2 - \
2 * bb * yl + aver(y ** 2)
return (a, None, c, None, None, None, chi2, None, None, None, None, None,
None)
def calc_hex(dhkl):
"""
Hexagonal
Quadratic form:
1 / d^2 = 4/3 (h^2 + hk + k^2) / a^2 + l^2 / c^2
"""
d, h, k, el = dhkl
y = d**-2
bl = el**2
bm = h**2 + h * k + k**2
yl = aver(y * bl)
ym = aver(y * bm)
lm = aver(bl * bm)
m2 = aver(bm**2)
l2 = aver(bl**2)
matrA = array([[lm, l2], [m2, lm]])
colB = array([yl, ym])
ba, bb = solve(matrA, colB)
a = (4. / 3. / ba)**.5
c = bb**-.5
y2 = aver(y**2)
chi2 = ba ** 2 * m2 + 2. * ba * bb * lm - 2. * ba * ym + \
bb ** 2 * l2 - 2. * bb * yl + y2
delta = m2 * l2 - lm**2
sig2a = l2 / delta * chi2
sig2b = m2 / delta * chi2
sig2a /= 3 * ba**3
sig2b /= 4 * bb**3
return (a, None, c, None, None, 120., chi2, sig2a, None, sig2b, None, None,
None)
def calc_rhombohedral(dhkl):
"""
Rhombohedral
Quadratic form:
1/d^2 = {(1+cos alpha)((h^2+k^2+l^2)-
(1-tan^2(alpha/2))(hk+kl+lh))}/
{a^2(1+cos alpha - 2 cos^2 alpha)}
"""
d, h, k, el = dhkl
y = d**-2
bl = h**2 + k**2 + el**2
bm = h * k + k * el + el * h
yl = aver(y * bl)
ym = aver(y * bm)
lm = aver(bl * bm)
m2 = aver(bm**2)
l2 = aver(bl**2)
matrA = array([[l2, lm], [lm, m2]])
colB = array([yl, ym])
ba, bb = solve(matrA, colB)
a = sqrt((2 * ba + bb) / ((ba + bb) * (2 * ba - bb)))
alp = arccos(-bb / (2 * ba + bb))
chi2 = ba ** 2 * l2 + 2 * ba * bb * lm - 2 * ba * yl + \
bb ** 2 * m2 - 2 * bb * ym + aver(y ** 2)
return (a, None, None, alp, None, None, chi2, None, None, None, None, None,
None)
def chi2n(d1a, d2a, poss):
dev = array([((d2a**-2 - i)**2).min() for i in d1a**-2])
inds = [((d2a**2 - i)**2).argmin() for i in d1a**2]
iset = unique(inds)
ave = aver(poss.transpose()[inds])
return aver(dev) * var(dev)**2 * ((len(d1a) - len(iset))**2 + 1) * ave**2
def calc_monoclinic(dhkl):
"""
Monoclinic
Quadratic form:
1/d^2 = h^2 / (a^2 sin^2 beta) + k^2 / b^2 +
+ l^2 / (c^2 sin(beta)^2) - 2 h l cos(beta) / (a c sin(beta)^2)
"""
d, h, k, el = dhkl
y = d**-2
bh = h**2
bk = k**2
bl = el**2
bm = h * el
h2 = aver(bh**2)
hk = aver(bh * bk)
hl = aver(bh * bl)
hm = aver(bh * bm)
k2 = aver(bk**2)
kl = aver(bk * bl)
km = aver(bk * bm)
l2 = aver(bl**2)
lm = aver(bl * bm)
m2 = aver(bm**2)
yh = aver(y * bh)
yk = aver(y * bk)
yl = aver(y * bl)
ym = aver(y * bm)
matrA = array([[h2, hk, hl, -hm], [hk, k2, kl, -km], [hl, kl, l2, -lm],
[hm, km, lm, -m2]])
colB = array([yh, yk, yl, ym])
ba, bb, bc, bd = solve(matrA, colB)
b = sqrt(1 / bb)
c = 2 * sqrt(ba / (4 * ba * bc - bd**2))
a = bc * sqrt(1 / ba / bc) * c
bet = arccos(bd / sqrt(ba * bc) / 2)
chi2 = ba ** 2 * h2 + 2 * ba * bb * hk + 2 * ba * bc * hl - \
2 * ba * bd * hm - 2 * ba * yh + \
bb ** 2 * k2 + 2 * bb * bc * kl - 2 * bb * bd * km - \
2 * bb * yk + \
bc ** 2 * l2 - 2 * bc * bd * lm - 2 * bc * yl + \
bd ** 2 * m2 + 2 * bd * ym + aver(y ** 2)
return (a, b, c, None, bet, None, chi2, None, None, None, None, None, None)