General case
Let us consider a system of periodic structure in space and use
a
{\displaystyle {\mathbf {a} }}
b
{\displaystyle \mathbf {b} }
c
{\displaystyle \mathbf {c} }
r
{\displaystyle \mathbf {r} }
r
=
u
a
+
v
b
+
w
c
.
{\displaystyle {\mathbf {r} }=u{\mathbf {a} }+v{\mathbf {b} }+w{\mathbf {c} }.}
Our task is to calculate the scalar coefficients known as fractional coordinates
u
{\displaystyle u}
v
{\displaystyle v}
w
{\displaystyle w}
r
{\displaystyle \mathbf {r} }
a
{\displaystyle \mathbf {a} }
b
{\displaystyle \mathbf {b} }
c
{\displaystyle \mathbf {c} }
For this purpose, let us calculate the following cell surface area vector
σ
a
=
b
×
c
,
{\displaystyle \mathbf {\sigma } _{\mathbf {a} }={\mathbf {b} }\times {\mathbf {c} },}
then
b
⋅
σ
a
=
0
,
c
⋅
σ
a
=
0
,
{\displaystyle {\mathbf {b} }\cdot \mathbf {\sigma } _{\mathbf {a} }=0,{\mathbf {c} }\cdot \mathbf {\sigma } _{\mathbf {a} }=0,}
and the volume of the cell is
Ω
=
a
⋅
σ
a
.
{\displaystyle \Omega ={\mathbf {a} }\cdot \mathbf {\sigma } _{\mathbf {a} }.}
If we do a vector inner (dot) product as follows
r
⋅
σ
a
=
u
a
⋅
σ
a
+
v
b
⋅
σ
a
+
w
c
⋅
σ
a
=
u
a
⋅
σ
a
=
u
Ω
,
{\displaystyle {\begin{aligned}{\mathbf {r} }\cdot \mathbf {\sigma } _{\mathbf {a} }&=u{\mathbf {a} }\cdot \mathbf {\sigma } _{\mathbf {a} }+v{\mathbf {b} }\cdot \mathbf {\sigma } _{\mathbf {a} }+w{\mathbf {c} }\cdot \mathbf {\sigma } _{\mathbf {a} }\\&=u{\mathbf {a} }\cdot \mathbf {\sigma } _{\mathbf {a} }\\&=u\Omega ,\end{aligned}}}
then we get
u
=
1
Ω
r
⋅
σ
a
.
{\displaystyle u={\frac {1}{\Omega }}{{\mathbf {r} }\cdot \mathbf {\sigma } _{\mathbf {a} }}.}
Similarly,
σ
b
=
c
×
a
,
c
⋅
σ
b
=
0
,
a
⋅
σ
b
=
0
,
b
⋅
σ
b
=
Ω
,
{\displaystyle \mathbf {\sigma } _{\mathbf {b} }={\mathbf {c} }\times {\mathbf {a} },{\mathbf {c} }\cdot \mathbf {\sigma } _{\mathbf {b} }=0,{\mathbf {a} }\cdot \mathbf {\sigma } _{\mathbf {b} }=0,{\mathbf {b} }\cdot \mathbf {\sigma } _{\mathbf {b} }=\Omega ,}
r
⋅
σ
b
=
u
a
⋅
σ
b
+
v
b
⋅
σ
b
+
w
c
⋅
σ
b
=
v
b
⋅
σ
b
=
v
Ω
,
{\displaystyle {\mathbf {r} }\cdot \mathbf {\sigma } _{\mathbf {b} }=u{\mathbf {a} }\cdot \mathbf {\sigma } _{\mathbf {b} }+v{\mathbf {b} }\cdot \mathbf {\sigma } _{\mathbf {b} }+w{\mathbf {c} }\cdot \mathbf {\sigma } _{\mathbf {b} }=v{\mathbf {b} }\cdot \mathbf {\sigma } _{\mathbf {b} }=v\Omega ,}
we arrive at
v
=
1
Ω
r
⋅
σ
b
,
{\displaystyle v={\frac {1}{\Omega }}{{\mathbf {r} }\cdot \mathbf {\sigma } _{\mathbf {b} }},}
and
σ
c
=
a
×
b
,
a
⋅
σ
c
=
0
,
b
⋅
σ
c
=
0
,
c
⋅
σ
c
=
Ω
,
{\displaystyle \mathbf {\sigma } _{\mathbf {c} }={\mathbf {a} }\times {\mathbf {b} },{\mathbf {a} }\cdot \mathbf {\sigma } _{\mathbf {c} }=0,{\mathbf {b} }\cdot \mathbf {\sigma } _{\mathbf {c} }=0,{\mathbf {c} }\cdot \mathbf {\sigma } _{\mathbf {c} }=\Omega ,}
r
⋅
σ
c
=
u
a
⋅
σ
c
+
v
b
⋅
σ
c
+
w
c
⋅
σ
c
=
w
c
⋅
σ
c
=
w
Ω
,
{\displaystyle {\mathbf {r} }\cdot \mathbf {\sigma } _{\mathbf {c} }=u{\mathbf {a} }\cdot \mathbf {\sigma } _{\mathbf {c} }+v{\mathbf {b} }\cdot \mathbf {\sigma } _{\mathbf {c} }+w{\mathbf {c} }\cdot \mathbf {\sigma } _{\mathbf {c} }=w{\mathbf {c} }\cdot \mathbf {\sigma } _{\mathbf {c} }=w\Omega ,}
w
=
1
Ω
r
⋅
σ
c
.
{\displaystyle w={\frac {1}{\Omega }}{{\mathbf {r} }\cdot \mathbf {\sigma } _{\mathbf {c} }}.}
If there are many
r
{\displaystyle \mathbf {r} }
u
=
r
⋅
σ
a
′
,
v
=
r
⋅
σ
b
′
,
w
=
r
⋅
σ
c
′
,
{\displaystyle {\begin{aligned}u&={{\mathbf {r} }\cdot \mathbf {\sigma } _{\mathbf {a} }^{\prime }},\\v&={{\mathbf {r} }\cdot \mathbf {\sigma } _{\mathbf {b} }^{\prime }},\\w&={{\mathbf {r} }\cdot \mathbf {\sigma } _{\mathbf {c} }^{\prime }},\end{aligned}}}
where
σ
a
′
=
1
Ω
σ
a
,
σ
b
′
=
1
Ω
σ
b
,
σ
c
′
=
1
Ω
σ
c
.
{\displaystyle {\begin{aligned}\mathbf {\sigma } _{\mathbf {a} }^{\prime }={\frac {1}{\Omega }}{\mathbf {\sigma } _{\mathbf {a} }},\\\mathbf {\sigma } _{\mathbf {b} }^{\prime }={\frac {1}{\Omega }}{\mathbf {\sigma } _{\mathbf {b} }},\\\mathbf {\sigma } _{\mathbf {c} }^{\prime }={\frac {1}{\Omega }}{\mathbf {\sigma } _{\mathbf {c} }}.\end{aligned}}}
In Crystallography
In crystallography , the lengths (
a
{\displaystyle a}
b
{\displaystyle b}
c
{\displaystyle c}
α
{\displaystyle \alpha }
β
{\displaystyle \beta }
γ
{\displaystyle \gamma }
a
{\displaystyle \mathbf {a} }
b
{\displaystyle \mathbf {b} }
c
{\displaystyle \mathbf {c} }
parallelepiped unit cell are known. For simplicty, it is chosen so that edge vector
a
{\displaystyle \mathbf {a} }
x
{\displaystyle x}
b
{\displaystyle \mathbf {b} }
x
−
y
{\displaystyle x-y}
y
{\displaystyle y}
c
{\displaystyle \mathbf {c} }
z
{\displaystyle z}
Unit cell definition using parallelepiped with lengths
a
{\displaystyle a}
b
{\displaystyle b}
c
{\displaystyle c}
α
{\displaystyle \alpha }
β
{\displaystyle \beta }
γ
{\displaystyle \gamma }
[ 1] Then the edge vectors can be written as
a
=
(
a
,
0
,
0
)
,
b
=
(
b
cos
(
γ
)
,
b
sin
(
γ
)
,
0
)
,
c
=
(
c
x
,
c
y
,
c
z
)
,
{\displaystyle {\begin{aligned}{\mathbf {a} }&=(a,0,0),\\{\mathbf {b} }&=(b\cos(\gamma ),b\sin(\gamma ),0),\\{\mathbf {c} }&=(c_{x},c_{y},c_{z}),\end{aligned}}}
where all
a
{\displaystyle a}
b
{\displaystyle b}
c
{\displaystyle c}
sin
(
γ
)
{\displaystyle \sin(\gamma )}
c
z
{\displaystyle c_{z}}
c
{\displaystyle \mathbf {c} }
c
⋅
a
=
a
c
cos
(
β
)
=
c
x
a
,
c
⋅
b
=
b
c
cos
(
α
)
=
c
x
b
cos
(
γ
)
+
c
y
b
sin
(
γ
)
,
c
⋅
c
=
c
2
=
c
x
2
+
c
y
2
+
c
z
2
.
{\displaystyle {\begin{aligned}{\mathbf {c} }\cdot {\mathbf {a} }&=ac\cos(\beta )=c_{x}a,\\{\mathbf {c} }\cdot {\mathbf {b} }&=bc\cos(\alpha )=c_{x}b\cos(\gamma )+c_{y}b\sin(\gamma ),\\{\mathbf {c} }\cdot {\mathbf {c} }&=c^{2}=c_{x}^{2}+c_{y}^{2}+c_{z}^{2}.\end{aligned}}}
Then
c
x
=
c
cos
(
β
)
,
c
y
=
c
cos
(
α
)
−
cos
(
γ
)
cos
(
β
)
sin
(
γ
)
,
c
z
2
=
c
2
−
c
x
2
−
c
y
2
=
c
2
{
1
−
cos
2
(
β
)
−
[
cos
(
α
)
−
cos
(
γ
)
cos
(
β
)
]
2
sin
2
(
γ
)
}
.
{\displaystyle {\begin{aligned}c_{x}&=c\cos(\beta ),\\c_{y}&=c{\frac {\cos(\alpha )-\cos(\gamma )\cos(\beta )}{\sin(\gamma )}},\\c_{z}^{2}&=c^{2}-c_{x}^{2}-c_{y}^{2}=c^{2}\left\{1-\cos ^{2}(\beta )-{\frac {[\cos(\alpha )-\cos(\gamma )\cos(\beta )]^{2}}{\sin ^{2}(\gamma )}}\right\}.\end{aligned}}}
The last one continues
c
z
2
=
c
2
sin
2
(
γ
)
−
sin
2
(
γ
)
cos
2
(
β
)
−
[
cos
(
α
)
−
cos
(
γ
)
cos
(
β
)
]
2
sin
2
(
γ
)
=
c
2
sin
2
(
γ
)
{
sin
2
(
γ
)
−
sin
2
(
γ
)
cos
2
(
β
)
−
[
cos
(
α
)
−
cos
(
γ
)
cos
(
β
)
]
2
}
{\displaystyle {\begin{aligned}c_{z}^{2}&=c^{2}{\frac {\sin ^{2}(\gamma )-\sin ^{2}(\gamma )\cos ^{2}(\beta )-[\cos(\alpha )-\cos(\gamma )\cos(\beta )]^{2}}{\sin ^{2}(\gamma )}}\\&={\frac {c^{2}}{\sin ^{2}(\gamma )}}\left\{\sin ^{2}(\gamma )-\sin ^{2}(\gamma )\cos ^{2}(\beta )-[\cos(\alpha )-\cos(\gamma )\cos(\beta )]^{2}\right\}\end{aligned}}}
where
sin
2
(
γ
)
−
sin
2
(
γ
)
cos
2
(
β
)
−
[
cos
(
α
)
−
cos
(
γ
)
cos
(
β
)
]
2
=
sin
2
(
γ
)
−
sin
2
(
γ
)
cos
2
(
β
)
−
cos
2
(
α
)
−
cos
2
(
γ
)
cos
2
(
β
)
+
2
cos
(
α
)
cos
(
γ
)
cos
(
β
)
=
sin
2
(
γ
)
−
cos
2
(
α
)
−
sin
2
(
γ
)
cos
2
(
β
)
−
cos
2
(
γ
)
cos
2
(
β
)
+
2
cos
(
α
)
cos
(
β
)
cos
(
γ
)
=
sin
2
(
γ
)
−
cos
2
(
α
)
−
[
sin
2
(
γ
)
+
cos
2
(
γ
)
]
cos
2
(
β
)
+
2
cos
(
α
)
cos
(
β
)
cos
(
γ
)
=
sin
2
(
γ
)
−
cos
2
(
α
)
−
cos
2
(
β
)
+
2
cos
(
α
)
cos
(
β
)
cos
(
γ
)
=
1
−
cos
2
(
α
)
−
cos
2
(
β
)
−
cos
2
(
γ
)
+
2
cos
(
α
)
cos
(
β
)
cos
(
γ
)
.
{\displaystyle {\begin{aligned}&\sin ^{2}(\gamma )-\sin ^{2}(\gamma )\cos ^{2}(\beta )-[\cos(\alpha )-\cos(\gamma )\cos(\beta )]^{2}\\&=\sin ^{2}(\gamma )-\sin ^{2}(\gamma )\cos ^{2}(\beta )-\cos ^{2}(\alpha )-\cos ^{2}(\gamma )\cos ^{2}(\beta )+2\cos(\alpha )\cos(\gamma )\cos(\beta )\\&=\sin ^{2}(\gamma )-\cos ^{2}(\alpha )-\sin ^{2}(\gamma )\cos ^{2}(\beta )-\cos ^{2}(\gamma )\cos ^{2}(\beta )+2\cos(\alpha )\cos(\beta )\cos(\gamma )\\&=\sin ^{2}(\gamma )-\cos ^{2}(\alpha )-[\sin ^{2}(\gamma )+\cos ^{2}(\gamma )]\cos ^{2}(\beta )+2\cos(\alpha )\cos(\beta )\cos(\gamma )\\&=\sin ^{2}(\gamma )-\cos ^{2}(\alpha )-\cos ^{2}(\beta )+2\cos(\alpha )\cos(\beta )\cos(\gamma )\\&=1-\cos ^{2}(\alpha )-\cos ^{2}(\beta )-\cos ^{2}(\gamma )+2\cos(\alpha )\cos(\beta )\cos(\gamma ).\end{aligned}}}
Remembering
c
z
{\displaystyle c_{z}}
c
{\displaystyle c}
sin
(
γ
)
{\displaystyle \sin(\gamma )}
c
z
=
c
sin
(
γ
)
1
−
cos
2
(
α
)
−
cos
2
(
β
)
−
cos
2
(
γ
)
+
2
cos
(
α
)
cos
(
β
)
cos
(
γ
)
.
{\displaystyle c_{z}={\frac {c}{\sin(\gamma )}}{\sqrt {1-\cos ^{2}(\alpha )-\cos ^{2}(\beta )-\cos ^{2}(\gamma )+2\cos(\alpha )\cos(\beta )\cos(\gamma )}}.}
Since the absolute value of the bottom surface area of the cell is
|
σ
c
|
=
a
b
sin
(
γ
)
,
{\displaystyle \left|\mathbf {\sigma } _{\mathbf {c} }\right|=ab\sin(\gamma ),}
the volume of the parallelepiped cell can also be expressed as
Ω
=
c
z
|
σ
c
|
=
a
b
c
1
−
cos
2
(
α
)
−
cos
2
(
β
)
−
cos
2
(
γ
)
+
2
cos
(
α
)
cos
(
β
)
cos
(
γ
)
{\displaystyle \Omega =c_{z}\left|\mathbf {\sigma } _{\mathbf {c} }\right|=abc{\sqrt {1-\cos ^{2}(\alpha )-\cos ^{2}(\beta )-\cos ^{2}(\gamma )+2\cos(\alpha )\cos(\beta )\cos(\gamma )}}}
[ 2] Once the volume is calculated as above, one has
c
z
=
Ω
a
b
sin
(
γ
)
.
{\displaystyle c_{z}={\frac {\Omega }{ab\sin(\gamma )}}.}
Now let us summarize the expression of the edge (period) vectors
a
=
(
a
x
,
a
y
,
a
z
)
=
(
a
,
0
,
0
)
,
b
=
(
b
x
,
b
y
,
b
z
)
=
(
b
cos
(
γ
)
,
b
sin
(
γ
)
,
0
)
,
c
=
(
c
x
,
c
y
,
c
z
)
=
(
c
cos
(
β
)
,
c
cos
(
α
)
−
cos
(
β
)
cos
(
γ
)
sin
(
γ
)
,
Ω
a
b
sin
(
γ
)
)
.
{\displaystyle {\begin{aligned}{\mathbf {a} }&=({a}_{x},{a}_{y},{a}_{z})=(a,0,0),\\{\mathbf {b} }&=({b}_{x},{b}_{y},{b}_{z})=(b\cos(\gamma ),b\sin(\gamma ),0),\\{\mathbf {c} }&=({c}_{x},{c}_{y},{c}_{z})=(c\cos(\beta ),c{\frac {\cos(\alpha )-\cos(\beta )\cos(\gamma )}{\sin(\gamma )}},{\frac {\Omega }{ab\sin(\gamma )}}).\end{aligned}}}
Conversion from cartesian coordinates
Let us calculate the following surface area vector of the cell first
σ
a
=
(
σ
a
,
x
,
σ
a
,
y
,
σ
a
,
z
)
=
b
×
c
,
{\displaystyle \mathbf {\sigma } _{\mathbf {a} }=(\mathbf {\sigma } _{\mathbf {a} ,x},\mathbf {\sigma } _{\mathbf {a} ,y},\mathbf {\sigma } _{\mathbf {a} ,z})={\mathbf {b} }\times {\mathbf {c} },}
where
σ
a
,
x
=
b
y
c
z
−
b
z
c
y
=
b
sin
(
γ
)
Ω
a
b
sin
(
γ
)
=
Ω
a
,
σ
a
,
y
=
b
z
c
x
−
b
x
c
z
=
−
b
cos
(
γ
)
Ω
a
b
sin
(
γ
)
=
−
Ω
cos
(
γ
)
a
sin
(
γ
)
,
σ
a
,
z
=
b
x
c
y
−
b
y
c
x
=
b
cos
(
γ
)
c
cos
(
α
)
−
cos
(
β
)
cos
(
γ
)
sin
(
γ
)
−
b
sin
(
γ
)
c
cos
(
β
)
=
b
c
{
cos
(
γ
)
cos
(
α
)
−
cos
(
β
)
cos
(
γ
)
sin
(
γ
)
−
sin
(
γ
)
cos
(
β
)
}
=
b
c
sin
(
γ
)
{
cos
(
γ
)
[
cos
(
α
)
−
cos
(
β
)
cos
(
γ
)
]
−
sin
2
(
γ
)
cos
(
β
)
}
=
b
c
sin
(
γ
)
{
cos
(
γ
)
cos
(
α
)
−
cos
(
β
)
cos
2
(
γ
)
−
sin
2
(
γ
)
cos
(
β
)
}
=
b
c
sin
(
γ
)
{
cos
(
α
)
cos
(
γ
)
−
cos
(
β
)
}
.
{\displaystyle {\begin{aligned}\mathbf {\sigma } _{\mathbf {a} ,x}&={b}_{y}{c}_{z}-{b}_{z}{c}_{y}=b\sin(\gamma ){\frac {\Omega }{ab\sin(\gamma )}}={\frac {\Omega }{a}},\\\mathbf {\sigma } _{\mathbf {a} ,y}&={b}_{z}{c}_{x}-{b}_{x}{c}_{z}=-b\cos(\gamma ){\frac {\Omega }{ab\sin(\gamma )}}=-{\frac {\Omega \cos(\gamma )}{a\sin(\gamma )}},\\\mathbf {\sigma } _{\mathbf {a} ,z}&={b}_{x}{c}_{y}-{b}_{y}{c}_{x}=b\cos(\gamma )c{\frac {\cos(\alpha )-\cos(\beta )\cos(\gamma )}{\sin(\gamma )}}-b\sin(\gamma )c\cos(\beta )\\&=bc\left\{\cos(\gamma ){\frac {\cos(\alpha )-\cos(\beta )\cos(\gamma )}{\sin(\gamma )}}-\sin(\gamma )\cos(\beta )\right\}\\&={\frac {bc}{\sin(\gamma )}}\left\{\cos(\gamma )[\cos(\alpha )-\cos(\beta )\cos(\gamma )]-\sin ^{2}(\gamma )\cos(\beta )\right\}\\&={\frac {bc}{\sin(\gamma )}}\left\{\cos(\gamma )\cos(\alpha )-\cos(\beta )\cos ^{2}(\gamma )-\sin ^{2}(\gamma )\cos(\beta )\right\}\\&={\frac {bc}{\sin(\gamma )}}\left\{\cos(\alpha )\cos(\gamma )-\cos(\beta )\right\}.\\\end{aligned}}}
Another surface area vector of the cell
σ
b
=
(
σ
b
,
x
,
σ
b
,
y
,
σ
b
,
z
)
=
c
×
a
,
{\displaystyle \mathbf {\sigma } _{\mathbf {b} }=(\mathbf {\sigma } _{\mathbf {b} ,x},\mathbf {\sigma } _{\mathbf {b} ,y},\mathbf {\sigma } _{\mathbf {b} ,z})={\mathbf {c} }\times {\mathbf {a} },}
where
σ
b
,
x
=
c
y
a
z
−
c
z
a
y
=
0
,
σ
b
,
y
=
c
z
a
x
−
c
x
a
z
=
a
Ω
a
b
sin
(
γ
)
=
Ω
b
sin
(
γ
)
,
σ
b
,
z
=
c
x
a
y
−
c
y
a
x
=
−
a
c
cos
(
α
)
−
cos
(
β
)
cos
(
γ
)
sin
(
γ
)
=
a
c
sin
(
γ
)
{
cos
(
β
)
cos
(
γ
)
−
cos
(
α
)
}
.
{\displaystyle {\begin{aligned}\mathbf {\sigma } _{\mathbf {b} ,x}&={c}_{y}{a}_{z}-{c}_{z}{a}_{y}=0,\\\mathbf {\sigma } _{\mathbf {b} ,y}&={c}_{z}{a}_{x}-{c}_{x}{a}_{z}=a{\frac {\Omega }{ab\sin(\gamma )}}={\frac {\Omega }{b\sin(\gamma )}},\\\mathbf {\sigma } _{\mathbf {b} ,z}&={c}_{x}{a}_{y}-{c}_{y}{a}_{x}=-ac{\frac {\cos(\alpha )-\cos(\beta )\cos(\gamma )}{\sin(\gamma )}}\\&={\frac {ac}{\sin(\gamma )}}\left\{\cos(\beta )\cos(\gamma )-\cos(\alpha )\right\}.\end{aligned}}}
The last surface area vector of the cell
σ
c
=
(
σ
c
,
x
,
σ
c
,
y
,
σ
c
,
z
)
=
a
×
b
,
{\displaystyle \mathbf {\sigma } _{\mathbf {c} }=(\mathbf {\sigma } _{\mathbf {c} ,x},\mathbf {\sigma } _{\mathbf {c} ,y},\mathbf {\sigma } _{\mathbf {c} ,z})={\mathbf {a} }\times {\mathbf {b} },}
where
σ
c
,
x
=
a
y
b
z
−
a
z
b
y
=
0
,
σ
c
,
y
=
a
z
b
x
−
a
x
b
z
=
0
,
σ
c
,
z
=
a
x
b
y
−
a
y
b
x
=
a
b
sin
(
γ
)
.
{\displaystyle {\begin{aligned}\mathbf {\sigma } _{\mathbf {c} ,x}&={a}_{y}{b}_{z}-{a}_{z}{b}_{y}=0,\\\mathbf {\sigma } _{\mathbf {c} ,y}&={a}_{z}{b}_{x}-{a}_{x}{b}_{z}=0,\\\mathbf {\sigma } _{\mathbf {c} ,z}&={a}_{x}{b}_{y}-{a}_{y}{b}_{x}=ab\sin(\gamma ).\end{aligned}}}
Summarize
σ
a
′
=
1
Ω
σ
a
=
(
1
a
,
−
cos
(
γ
)
a
sin
(
γ
)
,
b
c
cos
(
α
)
cos
(
γ
)
−
cos
(
β
)
Ω
sin
(
γ
)
)
,
σ
b
′
=
1
Ω
σ
b
=
(
0
,
1
b
sin
(
γ
)
,
a
c
cos
(
β
)
cos
(
γ
)
−
cos
(
α
)
Ω
sin
(
γ
)
)
,
σ
c
′
=
1
Ω
σ
c
=
(
0
,
0
,
a
b
sin
(
γ
)
Ω
)
.
{\displaystyle {\begin{aligned}\mathbf {\sigma } _{\mathbf {a} }^{\prime }&={\frac {1}{\Omega }}{\mathbf {\sigma } _{\mathbf {a} }}=\left({\frac {1}{a}},-{\frac {\cos(\gamma )}{a\sin(\gamma )}},bc{\frac {\cos(\alpha )\cos(\gamma )-\cos(\beta )}{\Omega \sin(\gamma )}}\right),\\\mathbf {\sigma } _{\mathbf {b} }^{\prime }&={\frac {1}{\Omega }}{\mathbf {\sigma } _{\mathbf {b} }}=\left(0,{\frac {1}{b\sin(\gamma )}},ac{\frac {\cos(\beta )\cos(\gamma )-\cos(\alpha )}{\Omega \sin(\gamma )}}\right),\\\mathbf {\sigma } _{\mathbf {c} }^{\prime }&={\frac {1}{\Omega }}{\mathbf {\sigma } _{\mathbf {c} }}=\left(0,0,{\frac {ab\sin(\gamma )}{\Omega }}\right).\end{aligned}}}
As a result[ 3]
[
u
v
w
]
=
[
1
a
−
cos
(
γ
)
a
sin
(
γ
)
b
c
cos
(
α
)
cos
(
γ
)
−
cos
(
β
)
Ω
sin
(
γ
)
0
1
b
sin
(
γ
)
a
c
cos
(
β
)
cos
(
γ
)
−
cos
(
α
)
Ω
sin
(
γ
)
0
0
a
b
sin
(
γ
)
Ω
]
[
x
y
z
]
{\displaystyle \left[{\begin{matrix}u\\v\\w\end{matrix}}\right]=\left[{\begin{matrix}{\frac {1}{a}}&-{\frac {\cos(\gamma )}{a\sin(\gamma )}}&bc{\frac {\cos(\alpha )\cos(\gamma )-\cos(\beta )}{\Omega \sin(\gamma )}}\\0&{\frac {1}{b\sin(\gamma )}}&ac{\frac {\cos(\beta )\cos(\gamma )-\cos(\alpha )}{\Omega \sin(\gamma )}}\\0&0&{\frac {ab\sin(\gamma )}{\Omega }}\end{matrix}}\right]\left[{\begin{matrix}x\\y\\z\end{matrix}}\right]}
where
(
a
{\displaystyle (a}
b
{\displaystyle b}
c
)
{\displaystyle c)}
r
{\displaystyle \mathbf {r} }
Conversion to cartesian coordinates
To return the orthogonal coordinates in ångströms from fractional coordinates, one can employ the first equation on top and the expression of the edge (period) vectors[ 4] [ 5]
[
x
y
z
]
=
[
a
b
cos
(
γ
)
c
cos
(
β
)
0
b
sin
(
γ
)
c
cos
(
α
)
−
cos
(
β
)
cos
(
γ
)
sin
(
γ
)
0
0
Ω
a
b
sin
(
γ
)
]
[
u
v
w
]
.
{\displaystyle \left[{\begin{matrix}x\\y\\z\end{matrix}}\right]=\left[{\begin{matrix}a&b\cos(\gamma )&c\cos(\beta )\\0&b\sin(\gamma )&c{\frac {\cos(\alpha )-\cos(\beta )\cos(\gamma )}{\sin(\gamma )}}\\0&0&{\frac {\Omega }{ab\sin(\gamma )}}\end{matrix}}\right]\left[{\begin{matrix}u\\v\\w\end{matrix}}\right].}
For the special case of a monoclinic cell (a common case) where
α
=
γ
=
90
∘
{\displaystyle \alpha =\gamma =90^{\circ }}
β
>
90
∘
{\displaystyle \beta >90^{\circ }}
x
=
a
u
+
c
w
cos
(
β
)
,
y
=
b
v
,
z
=
Ω
a
b
w
=
c
w
sin
(
β
)
.
{\displaystyle {\begin{aligned}x&=au+cw\cos(\beta ),\\y&=bv,\\z&={\frac {\Omega }{ab}}w=cw\sin(\beta ).\end{aligned}}}
References
^ "Unit cell definition using parallelepiped with lengths a , b , c and angles between the edges given by α , β , γ " . Ccdc.cam.ac.uk . Archived from the original on 2008-10-04. Retrieved 2016-08-17 .^ "Coordinate system transformation" . www.ruppweb.org . Retrieved 2016-10-19 .^ "Coordinate system transformation" . Ruppweb.org . Retrieved 2016-10-19 .^ Sussman, J.; Holbrook, S.; Church, G.; Kim, S (1977). "A Structure-Factor Least-Squares Refinement Procedure For Macromolecular Structures Using Constrained And Restrained Parameters". Acta Crystallogr. A . 33 : 800–804. doi :10.1107/S0567739477001958 . ^ Rossmann, M.; Blow, D. (1962). "The Detection Of Sub-Units Within The Crystallographic Asymmetric Unit". Acta Crystallogr . 15 : 24–31. doi :10.1107/S0365110X62000067 .
![{\displaystyle {\begin{aligned}c_{x}&=c\cos(\beta ),\\c_{y}&=c{\frac {\cos(\alpha )-\cos(\gamma )\cos(\beta )}{\sin(\gamma )}},\\c_{z}^{2}&=c^{2}-c_{x}^{2}-c_{y}^{2}=c^{2}\left\{1-\cos ^{2}(\beta )-{\frac {[\cos(\alpha )-\cos(\gamma )\cos(\beta )]^{2}}{\sin ^{2}(\gamma )}}\right\}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68f4b2b245ad322aa239174a1c015a928cde0a45)
![{\displaystyle {\begin{aligned}c_{z}^{2}&=c^{2}{\frac {\sin ^{2}(\gamma )-\sin ^{2}(\gamma )\cos ^{2}(\beta )-[\cos(\alpha )-\cos(\gamma )\cos(\beta )]^{2}}{\sin ^{2}(\gamma )}}\\&={\frac {c^{2}}{\sin ^{2}(\gamma )}}\left\{\sin ^{2}(\gamma )-\sin ^{2}(\gamma )\cos ^{2}(\beta )-[\cos(\alpha )-\cos(\gamma )\cos(\beta )]^{2}\right\}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/935e638c35a037b78b7b2bd379351fb7cbaf8691)
![{\displaystyle {\begin{aligned}&\sin ^{2}(\gamma )-\sin ^{2}(\gamma )\cos ^{2}(\beta )-[\cos(\alpha )-\cos(\gamma )\cos(\beta )]^{2}\\&=\sin ^{2}(\gamma )-\sin ^{2}(\gamma )\cos ^{2}(\beta )-\cos ^{2}(\alpha )-\cos ^{2}(\gamma )\cos ^{2}(\beta )+2\cos(\alpha )\cos(\gamma )\cos(\beta )\\&=\sin ^{2}(\gamma )-\cos ^{2}(\alpha )-\sin ^{2}(\gamma )\cos ^{2}(\beta )-\cos ^{2}(\gamma )\cos ^{2}(\beta )+2\cos(\alpha )\cos(\beta )\cos(\gamma )\\&=\sin ^{2}(\gamma )-\cos ^{2}(\alpha )-[\sin ^{2}(\gamma )+\cos ^{2}(\gamma )]\cos ^{2}(\beta )+2\cos(\alpha )\cos(\beta )\cos(\gamma )\\&=\sin ^{2}(\gamma )-\cos ^{2}(\alpha )-\cos ^{2}(\beta )+2\cos(\alpha )\cos(\beta )\cos(\gamma )\\&=1-\cos ^{2}(\alpha )-\cos ^{2}(\beta )-\cos ^{2}(\gamma )+2\cos(\alpha )\cos(\beta )\cos(\gamma ).\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/84639c5dd72fa894c1b432deb4a37048ec16689c)
![{\displaystyle {\begin{aligned}\mathbf {\sigma } _{\mathbf {a} ,x}&={b}_{y}{c}_{z}-{b}_{z}{c}_{y}=b\sin(\gamma ){\frac {\Omega }{ab\sin(\gamma )}}={\frac {\Omega }{a}},\\\mathbf {\sigma } _{\mathbf {a} ,y}&={b}_{z}{c}_{x}-{b}_{x}{c}_{z}=-b\cos(\gamma ){\frac {\Omega }{ab\sin(\gamma )}}=-{\frac {\Omega \cos(\gamma )}{a\sin(\gamma )}},\\\mathbf {\sigma } _{\mathbf {a} ,z}&={b}_{x}{c}_{y}-{b}_{y}{c}_{x}=b\cos(\gamma )c{\frac {\cos(\alpha )-\cos(\beta )\cos(\gamma )}{\sin(\gamma )}}-b\sin(\gamma )c\cos(\beta )\\&=bc\left\{\cos(\gamma ){\frac {\cos(\alpha )-\cos(\beta )\cos(\gamma )}{\sin(\gamma )}}-\sin(\gamma )\cos(\beta )\right\}\\&={\frac {bc}{\sin(\gamma )}}\left\{\cos(\gamma )[\cos(\alpha )-\cos(\beta )\cos(\gamma )]-\sin ^{2}(\gamma )\cos(\beta )\right\}\\&={\frac {bc}{\sin(\gamma )}}\left\{\cos(\gamma )\cos(\alpha )-\cos(\beta )\cos ^{2}(\gamma )-\sin ^{2}(\gamma )\cos(\beta )\right\}\\&={\frac {bc}{\sin(\gamma )}}\left\{\cos(\alpha )\cos(\gamma )-\cos(\beta )\right\}.\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eadbb81a1bf8feb74e2a08963ebe9b631ef39bcf)
![{\displaystyle \left[{\begin{matrix}u\\v\\w\end{matrix}}\right]=\left[{\begin{matrix}{\frac {1}{a}}&-{\frac {\cos(\gamma )}{a\sin(\gamma )}}&bc{\frac {\cos(\alpha )\cos(\gamma )-\cos(\beta )}{\Omega \sin(\gamma )}}\\0&{\frac {1}{b\sin(\gamma )}}&ac{\frac {\cos(\beta )\cos(\gamma )-\cos(\alpha )}{\Omega \sin(\gamma )}}\\0&0&{\frac {ab\sin(\gamma )}{\Omega }}\end{matrix}}\right]\left[{\begin{matrix}x\\y\\z\end{matrix}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ccbf2aa3407bf0ee76088a2a068ba2fcd067d7cd)
![{\displaystyle \left[{\begin{matrix}x\\y\\z\end{matrix}}\right]=\left[{\begin{matrix}a&b\cos(\gamma )&c\cos(\beta )\\0&b\sin(\gamma )&c{\frac {\cos(\alpha )-\cos(\beta )\cos(\gamma )}{\sin(\gamma )}}\\0&0&{\frac {\Omega }{ab\sin(\gamma )}}\end{matrix}}\right]\left[{\begin{matrix}u\\v\\w\end{matrix}}\right].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1e44ae0b4bf199429eb0763ac90ea96872c37690)
