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Formulas in crystallography
Metric
Basis vectors
a,b,c
Basis vectors in reciprocal space
*
a
*
b
*
c
Metric matrix
G=
(
)=
=
a |
b |
c |
a | b | c |
a·a | a·b | a·c |
a·b | b·b | b·c |
a·c | b·c | c·c |
2 a | abcosγ | accosβ |
abcosγ | 2 b | bccosα |
accosβ | bccosα | 2 c |
Metric matrix of reciprocal space (sometimes denoted by )
H
*
G
-1
G
Volume
V=
detG
Fractional coordinates (unitless)
x,y,z
Miller (or Laue) indices
h,k,l
Coordinate vector (in direct space)
r=r(x,y,z)
=
xa+yb+zcReciprocal lattice vector (, , or is sometimes used instead of )
Q
OH
G
*
r
hkl
H
H==H(h,k,l)=h+k+l
H
hkl
*
a
*
b
*
c
H=≠H
H
hkl
Node in reciprocal space/lattice
H=(
)
h | k | l |
Coordinate matrix (unitless)
X=
x |
y |
z |
Magnitude of coordinate vector
r
r=GX
T
X
Interatomic bond length
r
12
T
X
12
X
12
X
12
x 2 x 1 |
y 2 y 1 |
z 2 z 1 |
Interatomic bond angle (vertex at atom )
1
cosθ=G=GGG
T
X
12
X
13
r
12
r
13
T
X
12
X
13
T
X
12
X
12
T
X
13
X
13
Density
ρ=;
ZM
V
N
A
Z=formulaunitsperunitcell |
M=atomicmassofoneunit |
V=unitcellvolume |
N A |
Transformations
Transformation matrix
P
Transforming basis vectors
(
)=(
)P
a 2 | b 2 | c 2 |
a 1 | b 1 | c 1 |
Transforming coordinates
X
2
-1
P
X
1
X
1
X
2
Transforming metric matrices
G
2
T
P
G
1
Volume relation
detP=
V
2
V
1
Conversion from crystal coordinates to Cartesian coordinates (note: this is stored as )
$TransformationMatrices["CrystallographicToCartesian"]
P
C
a | bcosγ | ccosβ |
0 | bsinγ | c(cosα-cosγcosβ) sinγ |
0 | 0 | c 1- 2 cos 2 (cosα-cosγcosβ) 2 sin |
a | bcosγ | ccosβ |
0 | bsinγ | c C 1 |
0 | 0 | c C 2 |
C
1
cosα-cosγcosβ
sinγ
C
2
1-β-
2
cos
2
C
1
Interplanar spacing
From correspondence between crystallographic planes and reciprocal lattice points:
d
hkl
1
H
1
H
*
G
T
H
Cubic
1
2
d
hkl
2
h
2
k
2
l
2
a
Tetragonal
1
2
d
hkl
2
h
2
k
2
a
2
l
2
c
Orthorhombic
1
2
d
hkl
2
h
2
a
2
k
2
b
2
l
2
c
Hexagonal and trigonal ( centring)
P
1
2
d
hkl
4
3
2
a
2
h
2
k
2
l
2
c
Rhombohedral (trigonal centring)
R
1
2
d
hkl
1
2
a
(++)α+2(hk+hl+kl)(α-cosα)
2
h
2
k
2
l
2
sin
2
cos
1+2α-3α
3
cos
2
cos
Monoclinic
1
2
d
hkl
2
h
2
a
2
sin
2
k
2
b
2
l
2
c
2
sin
2hlcosβ
acβ
2
sin
2
h
2
a
2
k
2
sin
2
b
2
l
2
c
2hlcosβ
ac
2
csc
Triclinic
Reciprocal space
Definition of the reciprocal lattice
Accordingly relations
Angles in reciprocal space
Volume relations
Unit conversions
Scattering
Common absorption edges