Geometry of crystal
Cell constants a, b, c, α, β, γ
σ11= b2 c2 sin2α
σ22= c2 a2 sin2β
σ33= a2 b2 sin2γ
σ23= a2 b c (cosβ cosγ – cosα )
σ31= a b2 c (cosγ cosα – cosβ )
σ12= a b c2 (cosα cosβ – cosγ)
Volume V
V2 = a2 b2 c2 (1 – cos2α – cos2β – cos2γ + 2 cosα cosβ cosγ)
(hkl) d -spacing
d2 = V2/(h2 σ11 + k2 σ22 + l2 σ33 + 2 k l σ23 + 2 l h σ31 + 2 h k σ12)
Angle θ between (h1k1l1) and (h2k2l2)
cosθ = d1 d2 (h1 h2 σ11+ k1 k2 σ22 + l1 l2 σ33 + (k1 l2 + k2 l1) σ23 + (l1 h2 + l2 h1 ) σ31+ (h1 k2 + h2 k1 ) σ12) / V2
Length ｒ of [uvw]
r2 = u2 a2 + v2 b2 + w2 c2 + 2 v w b c cosα + 2 w u c a cosβ + 2 u v a b cosγ
Angle ψ between [u1v1w1] and [u2v2w2]
cos ψ = (u1 u2 a2 + v1 v2 b2 + w1 w2 c2 + (v1 w2 + v2 w1) b c cosα + (w1 u2 + w2 u1) c a cosβ +(u1 v2 + u2 v1) a b cosγ) / (r1 r2)
Angle φ between (hkl) and [uvw]
cos φ = (h u + k v + l w) d / r