6.6: Metric tensor
A metric tensor is used to measure distances in a space. In crystallography the spaces considered are vector spaces with Euclidean metrics, i.e. ones for which the rules of Euclidean geometry apply. In that case, given a basis e i of a Euclidean space , E n , the metric tensor is a rank 2 tensor the components of which are:
g ij = e i . e j = e j . e i = g ji .
It is a symmetrical tensor. Using the metric tensor, the scalar product of two vectors, x = x i e i and y = y j e j is written:
x . y = x i e i . y j e j = g ij x i y j .
In a three-dimensional space with basis vectors a , b , c , the coefficients g ij of the metric tensor are:
g
11
=
a
2
;
g
12
=
a . b
;
g
13
=
a . c
;
g
21
=
b . a
;
g
22
=
b
2
;
g
23
=
b . c
;
g
31
=
c . a
;
g
32
=
c . b
;
g
33
=
c
2
;
Because the metric tensor is symmetric, g 12 = g 21 , g 13 = g 31 , and g 13 = g 31 . Thus there are only six unique elements, often presented as
g
11
g
22
g
33
g
23
g
13
g
12
or, multiplying the second row by 2, as a so-called G 6 ("G" for Gruber) vector
( a 2 , b 2 , c 2 , 2 b . c , 2 a . c , 2 a . b )
The inverse matrix of g ij , g ij , ( g ik g kj = δ k j , Kronecker symbol, = 0 if i ≠ j , = 1 if i = j ) relates the dual basis, or reciprocal space vectors e i to the direct basis vectors e i through the relations:
e j = g ij e j
In three-dimensional space, the dual basis vectors are identical to the reciprocal space vectors and the components of g ij are:
g
11
=
a*
2
;
g
12
=
a* . b*
;
g
13
=
a* . c*
;
g
21
=
b* . a*
;
g
22
=
b*
2
;
g
23
=
b* . c*
;
g
31
=
c* . a*
;
g
32
=
c* . b*
;
g
33
=
c*
2
;
with:
g 11 = b 2 c 2 sin 2 α/ V 2 ; g 22 = c 2 a 2 sin 2 β/ V 2 ; g 33 = a 2 b 2 sin 2 γ/ V 2 ;
g 12 = g 21 = ( abc 2 / V 2 )(cos α cos β - cos γ); g 23 = g 32 = ( a 2 bc / V 2 )(cos β cos γ - cos α); g 31 = g 13 = ( ab 2 c / V 2 )(cos γ cos α - cos β)
where V is the volume of the unit cell ( a , b , c ).
Change of basis
In a change of basis the direct basis vectors and coordinates transform like:
e' j = A j i e i ; x' j = B i j x i ,
where A j i and B i j are transformation matrices, transpose of one another. According to their definition, the components g ij , of the metric tensor transform like products of basis vectors:
g' kl = A k i A l j g ij .
They are the doubly covariant components of the metric tensor.
The dual basis vectors and coordinates transform in the change of basis according to:
e' j = B i j e i ; x' j = A j i x i ,
and the components g ij transform like products of dual basis vectors:
g' kl = A i k A j l g ij .
They are the doubly contravariant components of the metric tensor.
The mixed components, g i j = δ i j , are once covariant and once contravariant and are invariant.
Properties of the metric tensor
- The tensor nature of the metric tensor is demonstrated by the behaviour of its components in a change of basis. The components g ij and g ij are the components of a unique tensor.
- The squares of the volumes V and V* of the direct space and reciprocal space unit cells are respectively equal to the determinants of the g ij 's and the g ij 's:
V 2 = Δ ( g ij ) = abc (1 - cos 2 α - cos 2 β - cos 2 γ + 2 cos α cos α cos α)
V* 2 = Δ ( g ij ) = 1/ V 2 .
- One changes the variance of a tensor by taking the contracted tensor product of the tensor by the suitable form of the metric tensor. For instance:
g im t ij.. kl.. = t j.. klm..
Multiplying by the doubly covariant form of the metric tensor increases the covariance by one, multiplying by the doubly contravariant form increases the contravariance by one.
See also
- Section 1.1.3 of International Tables of Crystallography, Volume B
- Section 1.1.2 of International Tables of Crystallography, Volume D