Some remarks on algebraic and scaling properties of 3-dimensional bravais lattices

Vol. 34 (1494)

SOME

REPORTS

ON

MATHEMATICAL

No. 1

PHYSICS

REMARKS ON ALGEBRAIC AND SCALING PROPERTIES OF 3-DIMENSIONAL BRAVAIS LATTICES R DIRLt, B L DAVIESS tInstitut fiir Theoretische Physik, Technische Universitat Wien, A-1040 Wien, Wiedner Hauptstrai3e 8 - 10; Austria $School of Mathematics, University College of North Wales, Bangor, Gwynedd LL57 lUT, UK (Received

September 27, 1999)

The algebraic properties and some scaling properties of S-dimensional Bravais lattices are revisited. Bravais lattices are determined by elements of GL+(3,W) which can be factorized into scaling matrices and structure matrices. The scaling matrices contain the lattice constants and form Abelian Lie groups. They determine together with their corresponding structure matrices the point group symmetry of the Bravais lattices. The deformation and accidental degeneracy of Bravais types of lattices is also discussed.

1. Algebraic properties of Brads

lattices

It is well-known [4], [2] that lattices in arbitrary dimensions can be classified algebraically by classifying their symmetry groups. An alternative approach consists of studying the topology of the spaces of lattices and their natural compactifications which we however do not discuss in this short paper. For more details the reader is referred to Ref.[2]. Here we restrict our discussion to 3-dimensional lattices. A 3-dimensional lattice 5 forms a countable (discrete) subgroup of R3 which is an Abelianlocally compact 3parameter Lie group. Every % determines uniquely its maximal symmetry group p(T) which is a finite subgroup of the orthogonal group O(3,R) in three dimensions. Such groups are usually called by crystallographers holohedral point groups of lattices. To summarize (t=Cjnj

=

{R(w) E 0(3, R) :

tj

=

Ch DQ(G) ek,

D(G) E GL+(3,R),

(3)

ej

=

Ck

ID(w-l)

(4

p(T)

R(w)

Dkj(w)

tj Z

nj EZ),

=

Z

ek,

R(w) t E 2,

(1) vtcx},

= D (Wy )

(2)

42

R DIRL, B L DAVIES

R(w)tj

= XI, &j(w) t/t ,

S(w)

=

D(G)-l

s(w)

E

GL(3, z) -

(5) (6)

D(w) D(G), R(w) E Y(T),

(7)

where the vectors tj : j = 1,2,3 are called primitive lattice vectors (plw’s) which for the sake of distinction henceforward always refer to the universal reference system of mutually orthonormal basis vectors ej : j = 1,2,3 respectively. Note, the matrices D(w) are orthogonal matrices, whereas S(w) need not be necessarily orthogonal ones, since in general the tj are not mutually orthogonal. Condition (7) guarantees that R(w) belongs to the holohedral point group of 2. Two lattices, say 21 and 22, are said to determine the same crystal system if their symmetry groups ‘$(Tr) and p(T 2) are conjugate with respect to O(3, R). This requires that there must exist a linear isomorphism R(w) : IR3 H !R3 such that

I

= R(w)*y(G) * W-l,

R(w) E G(3,R)

Two lattices, say 21 and 22, define the same Bravais lattice (Bravais type of lattices) if their corresponding group actions (p(Tr), 2,) and (T(Ts), 2s) are conjugate. This requires that there must exist a linear isomorphism R(4) : IIt3 JR3such that

‘43G) = R(~)*~(TI)*R(~)-~, 22 = R(~)o~I,

(9) R(4) c G(3,R)

>

(10)

where R(4) t(l) = tc2) for all t(l) E 21 with t c2) E 22, respectively. Thus crystal systems and Bravais lattices do not refer to single lattices but to equivalence classes of lattices. For n = 3 there are 7 crystal systems and 14 Bravais types of lattices. Equation (10) shows that infinitely many different orientations exist for every fixed lattice 2. Another type of freedom which gives rise to infinitely many different lattices that belong to the same Bravais type of lattices comes from all A(S) E GL+(3,IW) which commute with the elements R(w) of p(T). Let % b e a fixed lattice and let us assume [A(S), R(w)] = 0 for al1 R(w) E y(T) which leads to

Z(S) = A(S) o %,

(11)

This yields in addition to the orientational ambiguity infinitely many lattices which belong to the same Bravais class. Loosely speaking, the lattice constants of 5 are modified by scale transformations that leave the point group symmetry of the lattices unchanged. Schur’s Lemma applied to real representations of T(T) yields for the various crystal systems the following types of commuting matrices:

SOME REMARKS

ON ALGEBRAIC

Cubic:

A(S) -

Tetragonal, Trigonal, Hexagonal:

Orthorhombic:

43

AND SCALING PROPERTIES

A(6) =

A(6) -

A(6) =

A(S) H

A(6) =

1 x00 [ 1 x 0 0X0) i 0 0

0

(12)

x

OYO

3

(13)

(14)

z

Monoclinic:

A(6) -

A(S) =

All

Al2

0

A21

A22

0

loo 0

0

z I

(15)

Here we assume not only that x, y, z > 0 in order to define proper scale transformations but also that the 2-dimensional submatrix has positive determinant. Note, in the case of Triclinic systems we may take A(S) E GL+(3, IIB)but arbitrary otherwise. The commuting matrices A(6) are defined with respect to the universal basis t?j, their actual form is determined by adopting implicitly some specific conventions. Thus we have A(6)

ej

=

Ck

akj(b)

A(6)

tj

=

Ck

{D-l(G)

B(6)

=

D-i(G)

S(w)

=

B(6) S(w) B-1(6)

06)

ek,

a(s)

D(G)}kj

tk = Ce

{a(b)

D(G>)ej

A(6) D(G),

ee,

(17) (18)

)

(1%

where the plv’s tj are assumed to refer to primitive Bravais lattices. The forms of the commuting matrices for non-primitive lattices (like base-, face-, or body-centred) are obtained by corresponding equivalence transformations. Thus if A(c) E GLf(3,1W) denotes the mapping from a primitive Bravais lattice % to an admissible non-primitive Bravais lattice z(c) = A(c) o ‘% wi‘th’m a fixed geometric class, then it follows from R(w) o 5 = 2, R(w) o Z(S) = Z(S), and finally R(w) o T(c) = Z(c) that tj(c>

=

A(c)

tj

=

xk

akj(C)

tk

R(w)

tj(c)

=

Ck

{a-‘(C)

s(w)

‘(‘)}kj

tk(C>)

A(‘)

tj(c>

=

Ck

{a-1(C)

B(6)

h(c))kj

tk(c>

1

(20)

(21)

7

(22)

where the first equation defines the plv’s tj(c) of the corresponding non-primitive Bravais lattices. Notice that the correlations S(w) E GL(3, Z) w A-‘(c) S(w) A(c) E GL(3, Z) must hold for each geometric class. By virtue of the general definition of plv’s given by (3), we may set tj(C) = ck Dkj(G(c)) e k where JfD(G(c)) refers to the centred and D(G) to the corresponding primitive Bravais lattice. This implies that D(G(c)) = D(G) A(c) when using the defining equation (20).

R DIRL, B L DAVIES

44 2.

Factorization of Bravais lattice matrices

Recall that plv’s of Bravais lattices are determined by (3) where the actual form of the lattice matrix D(G) E GL+(3, W) depends on the strategy of how the lattice should be fixed. Either one starts from a fixed D(G) E GLf(3, R) and determines the lattice type by deducing its holohedral point group p(T), or one fixes the holohedral point group ‘33 and deduces which D(G) E GL+(3, R) are compatible with the given p. Note that the restrictive condition D(G) E GL+(3, IR) guarantees that the volume of the corresponding primitive cell - which is equal to det D(G) - is positive. We know that for each Bravais type of lattices there exist infinitely many lattices, a fact which is mainly due to the infinite range of variation of the lattice constants and in addition in the case of Triclinic and Monoclinic systems due to the compact range of variation of the angles between the axes. This suggests the factorization of D(G) into a diagonal matrix that contains the lattice constants times a structure matrix whose determinant does not depend on the lattice constants in question. D(G) D(L)

= =

D(L) D(S) a 0 i 0

0 b 0

(23) 0 0 c I

(24)

The basic idea of this factorization is to separate the scaling transformation D(L) from the remaing part of D(G). The actual form of D(L), namely whether it generates a three-, two-, or one-parameter Abelian Lie group (a, b, c > 0), preselects the compatible point group operations of 2. Next in (23-36) we give the factorized D(G)-matrices by adopting implicitly some specific conventions where the symbol * shall simply indicate ordinary matrix multiplication.

Triclinic

P:

(25)

Monoclinic

P:

(26)

Monoclinic

C:

(27)

Orthorhombic

P:

(28)

Orthorhombic

C:

(29)

SOME REMARKS

ON ALGEBRAIC

AND SCALING PROPERTIES

Orthorhombic F:

(30) a

Orthorhombic I:

[

0

0

ObO 0.0

*

-l/2 l/2

i/2 -l/2

l/2

l/2

c I[

l/2 l/2 -l/2

(31) 1

Tetragonal P:

(32)

OaO

Tetragonal I:

[a 0

*

0 0 0 c I[

l/2 -l/2 i/2

-l/2 l/2

i/2 -l/2 112 1

Trigonal R:

(33)

(34)

Hexagonal P: [

Cubic P:

45

a 0 0 OaO * 0 0 c1

1

-l/2

0 0 1

J3/2 0 0

0 0 1 1

(35)

0

[ a 0 0 OaO 0 a 01*

(36)

0 0 Cubic F:

[ a 0 0 OaO

a 01*

a 0 OaO [ 0 0

0

Cubic I:

I[ *

a

0 1 01 1 0 l/2 l/2 i/2 0 l/2 l/2 l/2 0 -l/2 l/2 l/2

l/2 -l/2 l/2

(37) l/2 l/2 -l/2

(33)

Here it should be noticed that we take the same Bravais lattice matrices llD(G) as in Ref.[5], except for the Trigonal system where we follow the conventions of Ref.[lO]. First we comment on the form of resealed Bravais lattice matrices D(G(S)) = A(6) I[D(G) which are defined by (17) and where D(G) is assumed to be fixed within a Bravais class. It follows immediately from the preceding list of standardized Bravais lattice matrices that for all Bravais type of lattices we may write m(G(6)) = D(SL) ILD(S).

(3%

This implies that, except for Monoclinic and Triclinic systems, the lattice constants are simply resealed by the diagonal elements of A(6) whereas the structure matrices remain unaffected. Thus within a fixed Bravais class, except for Monoclinic and Triclinic systems, all lattices up to orthogonal equivalence (defined by (10)) are obtained by applying the corresponding scaling operations A(6) which form an Abelian locally compact group.

46

R DIRL, B L DAVIES

Rather different is the situation for Monoclinic (and Triclinic) discussion, let A(6) = diag (x, y, z) be the commuting matrix.

Monoclinic

P:

D(G(6))

ax 0

= [

0 ax

Monoclinic

C:

D(G(S))

0

= [

0

bxcosy

0

bysiny

0 0

cz

systems. To simplify the Then one simply deduces

, 1

bx cos y/2

-bx cos y/2

bysiny/2 CZ/2

-bysiny/2 CZ/2

(40) 1. (41)

This illustrates that the scaling transformation A(6) diag (x, y, z) produces more than a mere resealing of the lattice constants since b is resealed by x in the er-direction and by y in the ez-direction respectively. This shows that for Monoclinic systems the standardized Bravais lattice matrices D(G) do not cover the most general situation.

Next we comment on the commuting matrices B(S) which are defined by (18). Since these matrices refer to plv’s which are not necessarily mutually orthogonal, they transform into rather complicated matrices (42-48). Only for Orthorhombic P, Tetragonal P, Hexagonal P, and all Cubic systems do they remain unchanged whereas in all other cases they change their form. Monoclinic

P:

(42)

Monoclinic

C:

(43)

Orthorhombic

C:

(44)

Orthorhombic

F:

(45)

Orthorhombic

I:

(46)

Tetragonal

I:

(47)

Trigonal

R:

(48)

SOME REMARKS

ON ALGEBRAIC

AND SCALING PROPERTIES

47

Moreover note again, for Monoclinic systems we have chosen A(S) = diag (2, y, .z) since the most general of the scaling matrices A(6) would lead to correspondingly complicated transformed matrices which still satisfy (19). 3.

Deformation

and accidental degeneracy of lattices

Here we briefly comment on the possibility of deforming lattices in such manner that they become invariant with respect to distinct holohedral point groups. We state a simple criterion which allows one to check how lattice constants (and/or angles between axes) have to be modified in such manner that the deformed lattices show different symmetries. Such Bravais types of lattices are sometimes called accidentally degenerate lattices. Two lattices, say % and 2’ which are uniquely defined by tj = xk Dkj(G) ek and ti = XI, Dkj (G’) ek, are compatible if there exists an orthogonal transformation R(4) E SO(3,JW such that R(4) ti = tj respectively. Thus D(G) = D(4) %G’)

>

(49)

where D (4) refers to the universal reference basis {ej}. Here we assume that % is in standard setting whereas 2’ that is characterized by D(G’) in general must be re-oriented by some appropriate orthogonal transformation as otherwise compatibility cannot be achieved. Thus the basic criterion reads IID(d)= D(G) W’(G’)

E SO(3,R),

(50)

where the lattice type 2’ is prefixed by llD(G’) with unknown lattice parameters which have to be adjusted. The method consists of varying the lattice parameters D(G) and by adjusting D(G’) in such manner that D (4) g iven by (50) becomes an orthogonal transformation, i.e. D(4) @%)

= lW4)

D(4) = E 7

(51)

where E symbolizes the 3-dimensional- unit matrix. If ID(4) E A(3,R) \ O(3,Iw) for all possible lattice parameter values of Z and of 2’, then the two corresponding Bravais types cannot be deformed in a compatible manner. Next we give in tabular form a few examples to illustrate the procedure for finding solutions of (51) and that for fundamental reasons in some cases there cannot exist solutions of (51). Trigonal Trigonal Trigonal Trigonal Trigonal Trigonal

R R R R R R

c) c---f -

Cubic P: Cubic F: Cubic I: Hexagonal P: Tetragonal P: Tetragonal I:

c = a/a, c=aJZ,

c = no c= c =

a’ = a

312

a’=aJG

a/2&, a’ = a&@ solution a/&, a’ = c’ = a@ a/2JZ, a’ = c’ = am

R DIRL, B L DAVIES

48

Orthorhombic Orthorhombic Orthorhombic Orthorhombic Orthorhombic Orthorhombic Orthorhombic

C C C C C C C

+-+ I-+ +-+ -

Tetragonal P: Tetragonal I: Trigonal R: Hexagonal P: Cubic P: Cubic F: Cubic I:

a = b, a’ = a/d2, no solution a=b=cfi, b=a&,

c’ = c

a’=a/&,

c’=a’/fi

a’ = a, c’ = c

a=b=c&,

a’=a/&

no solution no solution

It is worth noting that correlations of compatible lattice deformations: % H %’ are reflexive. This implies that if % can be deformed into %‘, then 2’ can be deformed into % respectively. In mathematical terms KD(d) = D(G) IID-l(G’) _

D-‘(4)

= llD(G’) D-‘(G)

E S0(3,R)

(52)

ssswhich implies that if a deformation % 2’ is ruled out, then its inverse 2’ % is likewise ruled out. One also recognizes from the preceding examples that in some cases the images of the deformed lattices can also be accidentally degenerate. For instance ZTrigonal R P illustrates that the images of the deformed z Tetragonal P E hbic lattices are in fact Cubic P and therefore do not form generic Tetragonal P lattices. Acknowledgements: One of us (RD) is very grateful to Professor Michel not only for illuminating discussions but also for drawing his attention to Schwarzenberger’s paper. REFERENCES [l] Hahn, T., International

Tables for Crystallography

Reidel, Dordrecht 1983.

[2] Schwarzenberger, R. L. E.: Proc. Camb. Phil. Sot., 72, 325 - 349 (1972). [3] Cracknell, A. P., Davies, B. L., Miller, S. C., Love, W. F.: Kroneclcer Product

Tables,

vol. 1, Plenum, New York 1979. [4] Bradley, C. J., C racknell, A. P.: The Mathematical Clarendon, Oxford 1972.

Theory

of Symmetry

in Solids,