Bravais classes of two-dimensional relativistic lattices

Physica 45 (1969) 33-66

0 North-Holland Publishing Co., Amsterdam

BRAVAIS TWO-DIMENSIONAL

CLASSES

OF

RELATIVISTIC

LATTICES

A. JANNER Instituut VOOY theoretische Fysica, Katholieke Universiteit, Nijmegen, Nederland and E. ASCHER Battelle Institute, Advanced Studies Center, Carouge-Genive, Suisse Received

14 February 1969

synopsis Repetition of motifs in space and time gives rise to regular patterns with syrnmetries described by relativistic crystallographic groups. This leads to natural generalizations of concepts familiar in Euclidean crystallography. Here only the twodimensional relativistic case is considered. Conditions are derived for a Lorentz transformation to be crystallographic, i.e. to leave invariant a lattice in the twodimensional Minkowskian space. The introduction of crystallographic transformations that change the sign of the indefinite metric tensor appears to be a necessary step in relativistic crystallography. The corresponding concepts of the theory of binary quadratic forms and of real quadratic fields are briefly discussed. A classification of all possible relativistic two-dimensional lattices is given and the corresponding Bravais classes are derived (at least in principle, as there are an infinite number of them). Isotropic lattices (i.e. with lattice points on the light cone) and incommensurable lattices (i.e. with incommensurable metric tensor) have as holohedry a point group of finite order. The other ones, which are described essentially by metric tensors g(B) = (a, b, c) with relatively prime rational integers a, b, c and discriminant d = b2 - 4ac not a square, always have a holohedry of infinite order. A number of lattice representatives of Bravais classes ordered according to the kinematical interpretation of the Lorentz transformation is given in the appendix. 1. Introduction. Periodic repetition of an arbitrary (finite) motif in space gives rise to patterns having crystallographic space groups as symmetry groupslls). H omo g eneity of empty space, or in other words, covariancy of physical laws with respect to the Euclidean translation group, is a necessary condition for the existence of crystals. This property, which makes possible the existence of the same motif of matter (e.g. molecule) at different places, is of course not sufficient for ensuring the existence of crystals, but makes this existence not at all surprising. 33

34

A. JANNER

AND

E. ASCHER

Actually physical laws are covariant with respect to the inhomogeneous Lorentz group, which includes the Euclidean translations as a subgroup. In the same way as above, this fundamental property makes very plausible the existence in nature of periodic repetition of some (finite) motif in space and time. We call such four-dimensional patterns space-time crystals, which then have the symmetry of relativistic space-time groups 3). Periodic motions on the one hand, and static (three-dimensional) crystals on the other, are degenerate cases of space-time crystals, as they give rise to infinite space-time unit cells. To get a finite unit cell, one may consider the combination of the two previous examples in the form of some crystal vibrating in a given mode, which then has the symmetry of a space-time group. From a general point of view, however, this example may represent too simple a case. Further analysis of the problem of finding out the possible physical manifestations of space-time crystals shows that one has to look into kinematic symmetries in distributions of matter (like e.g. those of conduction electrons in metals). It is then clear that if the typical velocities occurring in such distributions with respect to the centre of mass are small compared to the velocity of light, one has to consider the non-relativistic limit, i.e. what we call Galilean space-time groups. These groups have very peculiar features, and will form the subject of a subsequent paper. Our lack of a synthetic intuition of physical phenomena in space and time together, and the incredible richness of relativistic crystallography (as compared with the Euclidean one) make it advisable not to start thinking about possible space-time crystals before having some basic knowledge of space-time groups334). In our opinion this is also true if one is interested in another fundamental question, namely that of the existence of a fundamental length in nature. If such a length does exist and space-time is homogeneous no more, then one expects that some type of crystallographic description of matter in space-time is not only possible but even necessary. This is in fact what can be observed already in the current literature where a lattice is associated to the fundamental lengths-g). However, we should like to underline the fact that matter can very well be arranged according to the symmetry of space-time groups even if there is no fundamental length. We begin now to have a first view of relativistic crystallography and this paper is the first one devoted to the two-dimensional case (one timedimension, one space-dimension). In two dimensions everthing can be done explicitly. This is no more the case in higher dimensions, in which (for the moment) only some families of groups can be derived. The interesting point is that even in two dimensions and because of the richness of the indefinite metric case, it is much more easy to grasp general laws than in the positive definite case. More than that, a two-dimensional crystallography appears which may be formulated completely in terms of

TWO-DIMENSIONAL

RELATIVISTIC

LATTICES

35

arithmetic functions independently of the character of the metric. We consider this as a main result. Even the case where the determinant of the metric tensor vanishes, which arises in the limiting case of Galilean spacetime, is correctly described in such an approach. In this paper, however, we consider only the relativistic case. The comparison between the relativistic and the Euclidean case will be made elsewhere. Elsewhere also we hope to be able to discuss the interesting relations between relativistic crystallographic groups and Lie groups. In the Euclidean case such relations play an important role in the classification of complex semi-simple Lie algebras and in the representation theory of linear Lie groups lo-14). 2. Relativistic lattices and crystallographic Lorentz transformations. Given the two-dimensional Minkowskian vector space V with metric tensor 0

( >

g(B)= ;

__1

(we put for the velocity of light: c = l), and orthonormal We consider a basis B’ = (ei, ei) = BS defined by

4 =

we1

+

s2le2

eH = s12el + s22e2

basis B = (ei, es).

with

(2.1)

and the lattice generated by B’ A(B’)

= {zlei + zzee 1.~1,.z2 E Z}.

The lattice A(B’) is a set of vectors in V. If T is a unimodular transformation, T E GL(2, Z), then A(B’T) = A(B’). We shall write simply A when it is not necessary to specify the basis. GL(2, R) and GL(2, 2) are the groups of real and integer non-singular two-by-two matrices, respectively. The metric tensor corresponding to the basis B’ is given by g(B’) = Stg(B) S.

(2.2)

One finds:

a

2 Sll tw')

=

4

( SllS12 - SZlS22

SllS12 - s21.722 De* XI= 2 s12 - G2

ii) and of course det g(B’) -C 0, which means that d !?F?b2 -

4ac > 0.

The quantity d is called the discriminant of (a, b, c). Those Lorentz transformation are crystallographic

b 5

b z

c

!!Zf(a, b,c)

(2.3) (2.4)

that are unimodular

36

A. JANNER

with respect to a certain L’ = s-lL.s,

AND E. ASCHER

basis B’, i.e. :

with

L E O(l, l),

for some S E GL(2, R). Since by definition: L’tg(B’)

(2.5)

Ltg(B) L = g(B) it follows that

L’ = g(B’).

The unimodular generated

L’ E GL(2,Z)

(24

character

of L’ ensures

by B’ under this Lorentz

the invariance

transformation,

of the lattice

n

so that we may write:

L’A = A.

(2.7)

Any set of transformations (Li, LB, . . ., Li} satisfying the above three relations generates a relativistic point group K which is a subgroup of GL(2, 2) and leaves n invariant. The largest point group H leaving a given lattice invariant is called its holohedry. Two metric tensors g(Bi) and g(Bz) are arithmetically equivalent g(Bl) A g(Bz) and belong to the same arithmetic class if g(Bi)

= SWz)

S

whereas the arithmetic KIZKK~

if

for some equivalence S-lKIS

S E GL(2, Z),

(2.8)

of point groups is given by

= KZ

for some

S E GL(2, 2).

(2.9)

Relation (2.9) means that K1 and Kz are two conjugate subgroups of GL(2, 2). To a given lattice there corresponds the whole set {B’S /S E GL(2, 2)) of its possible bases, and thus an arithmetic class {g(B’)} of metric tensors and also an arithmetic class {K] of point groups. The concept of Bravais classes arises from the fact that whereas a lattice II determines the arithmetic class class determines a Bravais class of of its holohedry {H), an arithmetic latticesrs). Two lattices belong to the same Bravais class if they have arithmetically equivalent holohedries : n,E.&

if

HI & HZ.

As the dimension of the space is even, it is possible to consider lattices

(2.10) with

orientation, and oriented Bravais classes. In this case conjugation occurs with respect to an element of SL(2, Z), the subgroup of proper elements of GL(2, 2) 15). In this paper we shall consider oriented lattices. If the holohedry of such a lattice is a subgroup of SL(2, Z), then its arithmetic class splits into two proper arithmetic classes and two enantiomorphic lattices A and ,4’ belong to the same (non-oriented) Bravais class, where rl’ arises from n by changing orientation. If the holohedry of II is not a subgroup dSL(2,Z) then ,4 = II and the lattice is called ambiguous. Further on we shall discuss this in greater detail and in a more general frame. The holohedry H of a lattice is determined by the metric tensor of any

TWO-DIMENSIONAL

lattice

basis B’, because

L’@I’)

RELATIVISTIC

LATTICES

37

the relation with

L’ = g(B’)

L’ E GL(2,Z)

(2.11)

is a necessary and sufficient condition for L’ to belong to H. But a metric tensor does not determine a single lattice, because it simply fixes the length and the relative position of the two basis vectors. Thus bases Bi and Bz (and therefore lattices rli and As) obtained one from the other by a Lorentz transformation: Bz = BlL, g(Bz) = Ltg(B1) L = g(B1)

with

L E GL(2,Z)

(2.12)

determine the same metric tensor. One says that B1 and BB, and the respective lattices Ai and As, are relativistic-geometrically equivalent (B1 25 Bz; Al 25 A,). Two lattices then belong to the same relativistic geometric class if they differ simply in the frame of reference. From (2.11) and (2.12) one arrives at the conclusion that relativistic-geometrically equivalent lattices belong to the same Bravais class. Consider now any lattice. It contains a space-like vector which can be taken as first basis vector ei. Then &ills = a > 0. In other words a lattice determines an arithmetic class of metric tensors; in this class there is always a metric tensor g(B’) = (a, b, c) with a > 0. This means that if a is negative, one has to go over to an arithmetically equivalent metric tensor in order to meet the requirement. By taking another representative of the Bravais class of the lattice considered above, it is always possible to have the first basis vector ei in the space direction ei. This fixes the basis B’ (and the lattice A) according to: ei = Ja

er;

b 42 = z~d.

a +

d 2Ja

(2.13) e2.

By this procedure, to each lattice A a metric tensor (a, b, c) with a > 0 is associated. Henceforth (if nothing else is explicitly stated) we shall always adhere to the choice (2.13). The condition (2.5) for L’ to be a proper Lorentz transformation leaving A invariant becomes : S-_lLS

=

L’

=

0 au

w

where

L=&

cash x

sinh x

sinh x

cash x ’

-0.4

Hn + bu)>

s=

E GL(2, 2)

(2.14)

38

A. JANNER

AND E. ASCHER

with n = f2 cash x and u = f ,/{(ns - 4)/d} with sign u = sign x. This means that ti and u are solutions of the equation ns -

du2 = 4

(2.15)

expressing the condition det L’ = + 1. One easily verifies that in fact L’ of (2.14) leaves invariant g(R’) = (a, b, c). Let us remark that the only transformations L’ of finite order are those associated with the trivial solutions u = 0 and n = 12, i.e. the identity E and the total inversion Improper Lorentz transformations are of the form cash x

ikiT=* therefore

S-lMS

-sinh

x

sinh x -cash

traceless. Referred

=

n/irl

=

-E.

to the lattice basis B’ of (2.13), they become:

i Hn+ w \-CXU

x ’

-cu+$(n+bu) -&(n

+ bu)

J

Note that M’ is always of order two and can be written as:

(2.17)

with L’ as in (2.14). Therefore if a divides b and L’ leaves A invariant, then so does A!f’, and A is ambiguous. A metric tensor g(B’) = (a, b, c ) is called integral when a, b, c are rational integers. If furthermore a, b, c are relatively prime, then g(B’) is called primitive. The lattices corresponding by (2.13) to these tensors also are called integral or primitive lattices. We now formulate a property that is of fundamental importance in relativistic crystallography. Proposition 1. Each Bravais class of lattices that have holohedries of infinite order contains a primitive lattice; at most two primitive lattices are contained in such a class. Proof. We first show that there is a primitive lattice. According to the hypothesis, there is in the Bravais class a lattice II, with g(B’) = (a, b, c) with some a, b, c E R, left invariant by a L’, as in (2.14), with u # 0. We may suppose u > 0; otherwise we consider L’-1 instead of L’. Note that lattices differing only by a positive scaling factor are invariant with respect to the same point groups and belong to the same Bravais class. Therefore g(uB’) = (au, bu, cu) defines by (2.13) an integral lattice ZUI that is in the same Bravais class as A. Consider the greatest positive divisor k of au, bu and cu. Then uA/k is primitive and is in that same Bravais class. For g(uB’/k) we write again (a, b, c) but now a, b, c E 2 are relatively prime.

TWO-DIMENSIONAL We

show that class. Let

now

Bravais

L’ =

B

there

RELATIVISTIC

LATTICES

are at most two primitive

39

lattices

in a same

( > OL

6

ESL(2, -q

Y

be a proper Lorentz transformation of infinite order leaving primitive lattice with g(B) = (a, b, c). Then by (2.14) one has S -

y = au,

-p

01 = bu,

= cu

invariant

a! + 6 = n

and

a

(2.18)

implying /U/= gcd(y, 6 -

(Y, +3)

E2

i.e. u is the greatest common a > 0, one has to take:

(u # O),

(2.19)

divisor of y, S -

ill and -/3. In order to have

sign U = sign y.

(2.20)

By (2.18), (2.19) and (2.20), together with (2.13), a primitive lattice with metric tensor g(B) = (a, b, c) is uniquely determined. L’ leaves invariant also --g(B) = (- a, -b, -c) which defines an inverse lattice d of A (see section 4) and possibly A + A. Corollary 1.1. Every element L’ E SL(2,Z) crystallographic Lorentz transformation referred primitive lattice. Corollary transformation L,

=

1.2. The general is :

:(a -

w

with (tr L’I > 2 is a to the basis (2.13) of a

form of a proper crystallographic

-cu

a26

Lorentz

(2.21a)

3(n + bu) >

for any a, b, c E 2 such that gcd(a, b, c) = 1 and a > 0, and for (1z, U) any integral solution of the Pell (plus) equation: nz -

d~z = 4 9

where

d = b2 -

4ac > 0.

(2.21b)

In this way we arrive at the theory of indefinite binary quadratic forms and of real quadratic fields, which of course are nothing else than other aspects of the same thing. The formulation of this matter in crystallographic terms is precisely the aim of this paper. From the theory of binary forms and quadratic fields we therefore mention only a minimum number of essential points. Consider the indefinite quadratic form : f(x, y) = ax2 + bxy +

cy2 Ef

[a,b, cl,

VX,YEZ a, b, c E R

(2.22)

40

A. JANNER

AND

E. ASCHER

with discriminant d = b2 -

4ac > 0.

One says that

the real numbers

f(x, y) are represented

by the

quadratic

form [a, b, c]. Clearly, one may make a lattice rl in the hyperbolic plane associated with the metric tensor g(B’) = (a, b, c) correspond to the set of numbers represented by [a, b, c]. The number f(x, y) then expresses the distance between the origin and a lattice point P with coordinates x and y with respect to the basis B’ of (2.13). We may say that the lattice il is represented by the quadratic form [a, b, c]. One has the following properties. Proposition 2. equivalent quadratic forms (i.e. conjugated in 4 Two arithmetically GL(2, 2)) represent the same set of numbers. b) To quadratic forms that are conjugated in SL(2, 2) there corresponds one single (oriented) lattice. 4 To quadratic forms that are conjugated by an improper element of GL(2,Z) there correspond two lattices, one of which is the mirror image of the other (with respect to the first basisvector). The two enantiomorphic lattices may coincide (ambiguous case). Proof. See Bachmannle), pp. 100-l 17. Conjugate of quadratic forms occurs as for metric tensors after identification of [a, b, c] with (a, b, c). Suppose now a

At

b

5

b

z

a

A=

c

b

-5

i- 2

(2.23)

1,

b c

then any A E GL(2, R) satisfying (2.23) defines an automorphism of the quadratic form (a, b, c), and any A E GL(2, 2) satisfying (2.23) an automorph of the same form 17).Comparison with (2.12) and (2.11) shows that Lorentz transformations are automorphisms, and the crystallographic ones automorphs of the quadratic form defined by the metric of the basis vectors (and vice versa). Actually, in order to obtain the relativistic Bravais classes one needs a generalization of these concepts, as is done in section 3. But before doing this, we add a few remarks on real quadratic fieldsIs? 19). Consider the real quadratic field Q(,/D) generated by JD over the rationals Q, where D is a positive rational integer (D > 0). We assume D is not a square, but it may have square divisors. The square-free part is noted by Do. One has:

QW)

= Q(JDo)

where

D = q2Do,

q EZ.

(2.24)

TWO-DIMENSIONAL

Any element 1=

RELATIVISTIC

3, E Q(,/D) can be written

41

LATTICES

as

zl+zzJD

(2.25)

v.21, 22, 23 EL-.

> z3

The uniquely 1’ =

defined element :

21 -

ZdD

(2.26)

23

is called the conjugate of il. A norm N and a trace S are defined by N(A) !kf Lil’

and

(2.27)

s(n) !zf il + 2’

with properties N(ilIit2) = N(1r) N(jZ2) ;

N(A) = N(A’);

S(A1 + 12) = S(Al) + S(&);

S(M)

N(A) = 0 o ;z = 0

= kS(1)

for

k E Q.

(2.28)

Quadratic integers v are defined as those elements of Q(JD) which have rational integral norm and trace. One showsI*) that they form a ring and are of the form : v=

.a + a& a, 22 E.z-

2

where d is the discriminant d=

(2.29)

'

of Q(JD)

given by

40

if

D = 2 or 3 (mod 4) ;

i D

if

D = 1 (mod4).

(2.30)

The quadratic field Q(JD) will also be denoted by Q(Jd). The quadratic integers of Q(JD):that have an integer inverse are called units of the quadratic field. They have norm f 1 and form a group under multiplication, called the group of units. Accordingly units are given by &=

eu,ld

where m, u E 2 are solution 122 -

(2.31a)

2

dus = &4.

of the (plus, minus) Pell equation: (2.31b)

It is now possible to indicate the relation between quadratic fields and twodimensional relativistic crystallography. Relativistic Bravais lattices having holohedries of infinite order can be brought in correspondence with Z-modules J of quadratic integers. Consider the Bravais lattice II defined by the basis B’ of (2.13) with primitive metric d = b2 - 4ac > 0. The tensor g(X) = (a, b, c), a > 0 and discriminant correspondence is then indicated in the following table (2.32).

A. JANNER

42

AND E. ASCHER

TABLE (2.32) Correspondence between relativistic Bravais lattices and Z-modules in real quadratic fields Minkowskian vector space V basis

Real quadratic field Q(JD)

Discriminant

el, JD e2 2Ja A

1, w Dz,/D

d = 40

lattice or module

A! = [2a, b + co]

d = 4q2Do

basis

el, Hqel + JD es)

l,o%Afi(q+,/‘D)

d=D

lattice or module

Ju A

d = q2Do

In the case d = D one has a = b (mod 2) = q (mod 2). The correspondence is such that if I E Q(,/D) corresponds to I E V, then N(A) = 11Jl12;

S(n) = 2Z.ei.

(2.33)

Note that 2Ja rl, as well as Ju n are lattices in the same Bravais class as A. Comparison of (2.31) with (2.21) shows that for a given primitive lattice (a, 6, c), crystallographic proper Lorentz transformations L =

Q(n - ba) a24

-cu Hfl + bu) >

are in one-to-one correspondence Q( JO) given by : & = j&z -

with the positive

units

(N(E) = + 1) of

uJd).

In this way we see that proper crystallographic Lorentz transformations and relativistic Bravais lattices (in the primitive case) appear as elements of one and the same quadratic field. The interpretation of negative units (N(E) = - 1) will b e indicated in the following section. 3. The g-uutomor$hs. We first consider the case of real (a, b, c), later on that of primitive (a, b, c), and we identify quadratic forms with corresponding metric tensors. An automorphism X of a real indefinite c is an element of GL(2, R) such that g(B) = (a, b, ) Definition. XW)

X = g(B),

X E GL(2, R).

Definition. A negautomorphism X of a real indefinite g(B) is an element of GL(2, R) such that: XW)

X = -g(B),

metric

X E GL(2, R).

tensor (3.1)

metric

tensor (3.2)

TWO-DIMENSIONAL

If no distinction

RELATIVISTIC

is made between

43

LATTICES

automorphism

and negautomorphism

we

speak of generalized automorphisms (or simply of g-automorphisms). In the case of g(B) = (1, 0, - 1) we have the following possible general parametrization of g-automorphisms for I# E R: cash 4

(

zt L(4) = f f

M(4)

= +

*

A(&

=&

sinh

(

proper automorphism

sinh + cash I#>

d

cash 4 _ sinh #

sinh q5 -cash

r$>

transformation)

;

improper

automorphism

(Lorentz

mirror) ;

improper negautomorphism;

y;;)

:;;

(Lorentz

(3.3a) (3.3b) (3.3~)

( cash r$ -sinh

$>

proper negautomorphism.

(3.3d)

In what follows we shall use the letters L, Ikl, A, N, for the above gautomorphisms. One easily verifies the following relations : A(h)

A(42)

=

A(42)

A(h)

=

L(f$l +

52);

Wl)

L(42)

=

W2)

L(h)

=

q+1+

+2j;

M(h)

Jf(42)

=

W(#2)

Jf(41))-1

=

L($z

-

41);

N(h)

N(42)

=

W(42)

N(h))-1

=

--L($z

-

41);

A-l@)

=

A(-$),

M-W

=

J,f(#f

therefore

N-l(+)

= -IV($),

therefore

L-l($)

=

q-4);

AP(#

=

1;

W(4)

= -1.

(3.4)

The result of conjugation of a g-automorphism Xi by another g-automorphism X2, i.e. the element X;lXiX2 is indicated in the following table. TABLE (3.5) Conjugations of g-automorphisms conjugated by

We

I

LZ

A2

M2

N2

Ll

Ll

Ll

Lil

Lil

Al

Al

AI

-Ai1

-Ai1

MI

ML.;

-MIA;

MzMlMz

-Ml

NI

NIL:

-NlA;

-NI

- NzNlNz

metric

tensors.

g-automorphism

now go over to primitive

44

A. JANNER

AND

E. ASCHER

Definition. An automorph Y of a primitive metric tensor g(B) = (a, b, c) with a > 0, is an element of GL(2, 2) such that YW)

Y E GL(2, 2).

Y = g(B),

Definition.

A negautomorph

(3.6)

Y of a primitive

metric tensor g(B) as

above is an element of GL(2, 2) such that: Y%(B) Elements

Y E GL(2, 2).

Y = -g(B), of GL(2, 2) satisfying

(3.7)

(3.6) or (3.7) are called g-automorphs

g(B)* Before discussing the basic properties of automorphs we introduce metic functions pk(a) defined by the recurrence relation: fit+&)

=

qbc(fin) -

Vrt, kE.Z

Pk-l(fi),

of

arith-

(3.8a)

and the initial values:

$0(n) = 0, Furthermore A$&)

Pl(%) =

VrtEZ.

1,

(3.8b)

we define : =‘fifi+l(fi)

also obeying

-

the recurrence and

A$o(%) = 2

Vn, keZ

$,-I(@,

(3.9)

relation (3.8a) but now with the initial values: Afir

vnez.

= 1~7

(3.10)

The detailed properties of these functions will be discussed elsewhere. Let us here mention only a useful parametrization in term of trigonometric and hyperbolic functions. For any n E Z, consider + E R with + > 0 such that (sign n) 2 cash + 2 cos I#

if

InI 2 2;

if

lfil I 2.

(3.11)

Then 2 cos k+ = A$k(n);

for

In’ L 2 2 sin k$ = J(4 -

(3.12a)

ns) fik(n)

and 2 2 cash k+ = Apk(n);

n,

for

2 sinh k+ = J(d

-

with $k(-n)

= (-

-

l)k+rfik[n) and dfir(--~)

(3.12b)

4) p,(n), = (-

l)kdfk(n).

Note that Afik(n) and *k(n) are solutions of the Pell (plus) equation of discriminant d = ns - 4: A&n)

-

Proposition

(nz -

4) p:(n)

(3.13)

= 4.

3. A proper automorph

L of a primitive

g(B) = (a, b, c)

TWO-DIMENSIONAL

can always be written L=&L;=*

RELATIVISTIC

LATTICES

45

as: #zl-bul)

-cu

i&k--b%k)

-c%k

au1

&l+bw)

a%k

@k+b%)

>’

(3.14) where

(121, ur) is the least positive

discriminant

d = b2 -

ns -

dus = 4

nk

&k(Q),

solution

of the Pell (plus) equation

of

4ac > 0:

and =

uk =

(3.15)

fh$k@l).

We call LO the fundamental Lorentz transformation (of g(B)). Proof. For any given primitive g(B) = (a, b, c) there is a one-to-one correspondence between positive units E of the quadratic field Q(,/d), where d = b2 - 4ac, and the proper automorphs L of g(B) : L =

t@ au (

4

--GsfT 3(a + bg) >

0 E = $(n + uJd).

(3.16)

We now show that Lk is given by Lk =

&(nk

-

buk)

--cuk +k

auk

+

buk)

>

where nk

=

and

&k(n)

uk

=

f@k(fi).

For k = 1 the statement is trivially true. Suppose it is true for k for Lk one finds the above expression with %k

=

$(flk-1

n

uk

=

i(uk-ln

where, according ns -

uk-1

ud),

+

uk-1

+

flk-1%))

to the induction

dzG = 4,

n;_r

-

hypothesis: = 4,

du;_l

nk-1

=

A$k--l(B)

Zdpk-l(n).

=

Using the relations : +k,+d’d A#‘k,+k,(n)

= +&,(@

Ai’h(rt)

=

A+,@)

+ AZk(“d

=

i(A+k,(n)

A#k,(n)

+

(n2

=

+(bk,(n)

A#k,(n)

+

f’k,@)

which are a simple consequence nk

and

1, then

@k

of (3.12), =

@‘k(n).

#k,(n))t -

4) !‘k,(@ f’k,(fi)

@d),

one finds :

#k,(@))

=

and

A. JANNER

46

In exactly

AND E. ASCHER

the same way one proves that

&k = &(Yzk+ UkJd). It follows that Lk corresponds Any positive unit of Q(,/d)

to $. can be written as E = &.$, where ~1 is the

fundamental

obtained

(positive)

unit

(tir, nr) of the Pell equation morph of g(B) is given by:

from

the

least

positive

solution

(see e.g. ref. 18), so that any proper

auto-

L = fLt with LO the fundamental Lorentz transformation (of g(B)). Note that in our case lrtil > 2 if d is not a square. Lorentz transformations LO and Corollary 3.1. The crystallographic L = fLi of (a, b, c) are in correspondence uniquely to ~1 = +(+zr + ulJd) the fundamental (positive) unit of Q(Jd), and to E = &E!, respectively. Corollary 3.2. An improper automorph can always be written as &(fik +

M=+M,,L;=f

buk)

--auk

M of a primitive g(B) = (a, b, c)

--cuk

+

-&(fik

& +

(wk +

buk)

(3.17)

buk)

where

and ?&k,%&uk: as above. Proof. Consider (2.16) and (2.17). The negautomorphs can be treated in the same way, with an important restriction, however; we have to consider only those primitive g(B) for which there are negautomorphs. The explicit conditions therefore are discussed further on. Let us, for the time being, merely underline the fact that there are (oriented) lattices that admit proper negautomorphs but no improper ones. (See section 4.) We first introduce new arithmetic functions qk(??%)defined by the recurrent relation : qk+l(m)

=

mqk(m)

+

qk-l(m),

Vm,kcZ

(3.18a)

V9nE.z.

(3.18b)

and the initial values: 40(m) = 0, Furthermore &r(m)

41(m)

=

1,

we define sf qk+l(m)

+

qk-l(m),

Vk,mEZ

(3.19)

TWO-DIMENSIONAL

also obeying

RELATIVISTIC

LATTICES

47

the recurrent relation (3.18a) but with the initial values: and

Aqo(m) = 2

VW%EZ.

Aqr(m) = m,

(3.20)

Note that q&(l) are the numbers of Fibonacci and Aqk(l) those of Lucas. These arithmetic functions too can be parametrized in terms of hyperbolic functions. Consider for any m E Z a 4 E R such that 2 sinh 4 = m.

(3.21)

Then for any k E 2: 2 sinh 2k+ = J(m2 and

+ 4) 4sk(m),

i 2 cash 2k+ = A42+2) 2 sinh (2k + i 2 cash (2k +

(3.22)

1) 4 = A4sk+r(m), 1) + = ,/(m2 + 4)

42k+1(4

where Aqk(m) and qk(m) are solutions of the Pell equation discriminant d = ms + 4: A&m)

-

(m2 + ‘4) q;(m) = (-

(plus, minus) of

1)” 4.

(3.23)

Proposition 4. Suppose g(W) = (a, b, c), a > 0, a primitive metric tensor admitting as in (3.7) an improper negautomorph A. Then A can always be written as: *(ml---bvr)

A=*Af$=&

( au1

k

-c’or ii(ml+h)

>

=

* (

i(mk-bvk)

-cvk

@vk

i(mk+

bvk) >

(3.24) with k = 1 (mod 2) and (ml, VI) the least positive solution of the Pell (minus) equation of discriminant d = b2 - 4ac > 0: m2 -

dv2 = -4,

(3.25a)

furthermore mk

=

Aqk(ml),

vk =

"lqk(ml).

(3.25b)

Proof. One may regard an improper negautomorph A of g(W) as an improper negautomorphism of ( 1, 0, - I), which becomes unimodular when referred to the basis B’ of (2.13). Therefore, using (3.3~) one gets:

= f

+(7iz av

bv)

-cv i(m + bv) >

(3.26)

A. JANNER

48

AND

E. ASCHER

where m = 2 sinh x E 2, and v = J((ms solutions of the Pell (minus) equation: ms -

+ 4)/d) E 2.

(m, v) are integral

dvs = -4.

Therefore E = j--(m + vJd) is a negative unit of Q(Jd) and is uniquely determined by A. Let Aa be the improper negautomorph that corresponds to the fundamental (negative) unit ~1 of Q(Jd) given by: ~1 = *(ml + v~,/d). Then E = &E: for some k E 1 (mod 2) (see e.g. ref. 18, p. 303) and correspondingly A = &At. Defining now #I by : mr=2sinh+

(3.27)

it follows that 2 sinh k+ = rnk = Aqk(mr) and vk = vrqk(mr), then, is a simple consequence of (3.23). Corollary 4.1. The improper negautomorphs A = *.4ik+’ of g(B) = (a, b, c) are in one-to-one negative units E = &@+’ of Q(Jd), for k E 2.

of infinite order correspondence with the

Remark. The restriction in corrollary 4.1 to negautomorphs of infiniteorder, i.e. to the case ml > 0, is due to the fact that in the case of negautomorph of finite order the correspondence with negative units breaks down. Fo(m = 0 the Pell (minus) equation becomes dv2 = 4 with positive solutions : v=2

for

a=

v=l

for

d = 4.

1,

In both cases Q(Jd) = Q and the only units are & 1, thus positive. For d = 1, no primitive metric tensor is possible (see ref. 17 theorem 75, p. 1Ol), but for d = 4, the metric tensor g(B) = (1, 0, - 1) admits improper negautomorphs of finite order:

A=*&=*

0 1

I

( >

(3.28)

o.

Note also that the group of g-automorphs of a given primitive g(B), is in general not isomorphic with the group of units of Q(,,/d) for d being the discriminant of g(B). In fact, no correspondence has been formulated between units of Q(Jd) and improper automorphs or proper negautomorphs of g(B) * The one-to-one correspondence broadly summarized as follows :

between

units and g-automorphs

positive units t--tproper automorphs (also of finite order), negative units ts improper negautomorphs (only of infinite order).

can be

TWO-DIMENSIONAL

A table indicating

real quadratic

RELATIVISTIC

fields Q(,/De)

LATTICES

49

having negative units can

be found in ref. 19, pp. 271-274 for Da up to 97. We now go over to proper negautomorphs. In the same way as in (3.26) and using (3.13) and (3.3d) one sees that the general form of a proper negautomorph of a primitive g(B) = (a, b, c), if it exists, is given by:

(3.29)

with (m, V) solutions of the Pell (minus) equation (3.25a). Note that these solutions need be integral (and therefore of the form (3.25b)). Comparing with (2.16) and (2.17) one finds that N can be written as:

(3.30)

Me is an improper automorphism and A an improper negautomorphism as in (3.26). The point is that only if a divides b, i.e. if g(B) is ambiguous, are Me and A necessarily unimodular, i.e. (m, V) are integral solutions of (3.25). In general this is not the case. As an example let us consider the lattice defined by g(B) = (5, 11, -5). One verifies that g(B) is not ambiguous and does not admit improper negautomorphs. In fact the discriminant is d = D = 221 and the fundamental unit of Q(J221) is ~1 = a(15 + J221), thus of norm + 1. But g(B) d oes admit proper negautomorphs, namely, for example : No

=

(3.3 1)

which may indeed be obtained from (3.29) for m = 1 l/5 and v = l/5, a non-integral solution of m2 - 221~2 = -4. We think that in the case of enantiomorphic lattices, proper negautomorphs can always be put into the form (possibly after a suitable basis transformation) : N = NoL

(3.32)

where No is as in (3.31) and L is an arbitrary proper automorph of g(B). This is indeed the case in the example mentioned above, although in general, we have not been able to prove it. Clearly proper negautomorphs need further investigation. 4. Classification of primitive lattices. Let II be an oriented lattice defined by primitive g(B) = (a, 6, c ) , a > 0, according to (2.13). By rl’ we indicate

50

A.

the mirror lattice il’=X/l We know that

JANNER

AND

E.

ASCHER

X E GL(2, 2) -

SL(2,Z).

of n given by: with

if A’ = A, the lattice

form an enantiomorphic

(4.1)

is ambiguous,

otherwise

A and A’

pair.

We now define A, the inverse of lattice A, is the lattice of {-g(B)}+. Here we have to consider the proper arithmetic class of --g(B) = (-a, --b, -c), because of the convention (2.13). We have the following correspondences A 0

(g(B)}+

and

J + {-g(B))+.

(4.2)

In the case where A = d we speak of a stable are called unstable.

lattice,

otherwise

A and R

Proposition 5. A lattice A defined by primitive g(B) according (2.13) is stable if and only if it is left invariant by proper negautomorphs

to of

g(B). Proof. Suppose A = 3, this means that -g(B) E {g(B)}+; thus there is a N E SL(2,Z) such that Ntg(B) N = -g(B). Clearly N is a proper negautomorph. The converse is also true, as can be seen by reversing the arguments. Corollary 5.1. A lattice A of a primitive nant d > 0, which is ambiguous and stable TABLE Classification Transformations

(4.3)

of relativistic

leaving

n

g(B) # (1, 0, - 1) of discrimiis left invariant by improper

primitive

Characterization

invariant

of

Type

the lattice

negautomorphs improper

lattices

ambiguous

_4 = n’

stable negative

‘4=/I /I = -A’

ambiguous

A = A’

unstable

A#/i

positive

‘4 # ?i’

proper negautomorphs A==‘%

enantiomorphic

A # A’

stable positive

A=fi

improper negautomorphs A = 2’

enantiomorphic

A # A’

unstable

Afji A = A

A=ii

I

automorphs n = n’

automorphs

only

Aitd

no improper automorphs A # A’

proper automorphs only (no negautomorphs)

negative

A # ;i’

enantiomorphic

A # A’

unstable positive

A#A A # ;i’

II -. III ____ IV

V

TWO-DIMENSIONAL

negautomorphs

of g(B).

fundamental unit. We call such a lattice

The

RELATIVISTIC

real

quadratic

field

51

LATTICES

Q(,/d) has a negative

with (il)’ = (d’) = ?i’ a negative

lattice.

If ,4 # /I

the lattice

is said to be positive. In this case there are no improper negautomorphs of g(B) and the units of Q(,/d) are all positive. Using these various definitions relativistic primitive lattices can be divided into five different types as indicated in the table (4.3). Explicit examples are indicated in the appendix showing that lattices of the five different types effectively occur. There are infinitely many primitive lattices and it is therefore worthwhile to group some of them in families of lattices having some peculiar properties. Given a value d of the discriminant, there is always a principal quadratic form with this discriminant (see e.g. ref. 16, p. 119). Accordingly we call principal lattices Pa,those lattices which are defined by the correspondence pd

(l,l,G) o

(I,o,-G) I Principal lattices are always pd one finds:

if

dEl(mod4),

(4.4a)

if

d=O(mod4).

(4.4b)

ambiguous.

if if

For the inverse

principal

lattices

d = 1 (mod 4),

(4.5a)

d F 0 (mod4).

(4.5b)

The concept of natural lattice plays an important ralein crystallography, even if not yet generally recognized. The name “natural” has been suggested by W. Opechowski. In order to explain the idea, consider a g-automorph X of a lattice rl. One has VXEA Xx=yEfl. (4.6) In general (x, y) is not a basis of ,4 but only of a sublattice. Suppose now that there is an element ei E ~4 and a g-automorph X of n such that (ei, Xei) forms a basis of rl. In this case we speak of a natural lattice generated by X. If X is an improper negautomorph

Xf

0

1

1

0

( >

we speak of a negative

natural

lattice;

and if X is a proper automorph

52

A. JANNER

we speak of a positive

natural

AND

E. ASCHER

lattice.

Proposition 6. For any n E 2, there are two positive natural lattices, M, and its inverse ii?!,. The lattice M, is the principal lattice of discriminant d = 12s -

4. One has the following correspondences

Ms,

o(l,O,

MB+1

-3

M2Y

e(v2

M2v+1

0

-

1 -V2)

(1, 1, 1 - v(v+ 1)) -l,O,

-1)

vv E z

(4.7)

-

(v(v

+

1) -1,

1, -1).

Proof. Consider a primitive lattice corresponding general form of a proper automorph L being (2.21), a vector x = xiei + xse6 E A and its transformed with yi = &i(1z - bu) - X~CU and ys = X~UZC + lattice of index d in _4, where d = (&

by (2.13) to (a, b, c). The the lattice generated by y = Lx = ylei + yzei +xs(n. + bu) is a sub-

+ bxlx2 + cx;) U.

(4.8)

For II natural: d = f 1; consequently may suppose 24 = 1. Then for d = 1 ax; + bxlx2 + cx; =

1

and

u = & 1. Without d = ns -

restriction

we

4.

According to a theorem of Lagrange (see ref. 17, p. 111) there is in the proper arithmetic class of (a, b, c) a quadratic form (a’, b’, c’) with a’ = 1. For 1z = 2v + 1 (odd n), b’ is also odd and, indicating proper arithmetic equivalence by 2, one has: (a,b,c)~(l,b’,c’)~(l,l,c”)=

(l,l,

For even n, n = 2v, b’ is also even and, analogously,

now A = - 1 and

1)).

one finds:

(u,b,c)i(l,b’,c.)i(l,O,c”)-(1,0.-;)=(I,O,l Suppose

1 --y(v+

-v2).

ax; + bxlx2 + cxi = - 1. As above,

in the

proper arithmetic class there is an (a’, b’, c’) with c’ = - 1, and one finds the inverse lattices i@, in the same way. Corollary 6.1. For getting all positive natural to consider all integers n 2 0, since M_, = M,.

lattices

it is sufficient

Proposition 7. The value vz = 3 is the only one for which i@, = M,. The lattice MS is of type I. Proof. For 1z = 3 the discriminant of MS is d = 5 and the fundamental unit of Q(J5) is negativeia). Being ambiguous and negative, Ms is stable.

TWO-DIMENSIONAL

Quadratic

fields

RELATIVISTIC

of discriminant

d = ns -

LATTICES

4, for

53

InI > 3, have

positive units. To prove this, suppose that there are integral

only

solutions of the

corresponding Pell (minus) equation. Take the least positive one (ml, ~1). It follows that the least positive solution of the Pell (plus) equation for same discriminant is given by and

ni = m; + 2

Ui = mi211.

(4.9)

This can be seen by squaring the improper negautomorph Aa of (3.24). But for natural lattices, ztr = 1, and this implies ml = zli = 1, thus ni = 3, which is contrary to the hypothesis. Proposition 8. For any m E 2 there is a negative natural lattice which is the principal lattice of discriminant d = ~2 + 4. One has: Nzfi

+ (1, 0, -(l

+ p2))

Nzfi+l

0 (1, 1, 1 -

p(p +

Proof. The generated by with (m, v) = one arrives at

p # 0,

VllEZ,

proof is obtained exactly as in proposition 7. One finds Nm the improper negautomorph A (3.26) of a primitive (a, b, c) (m, l), so that d = ms + 4. Applying Lagrange’s theorem the desired results.

Corollary 8.1. For getting all negative natural to consider integral positive m (since N-, = N,). Corollary

(4. IO)

V/.&EZ.

1))

Nnz,

8.2.

Any negative

natural

lattice

lattices,

Nm is stable

it is sufficient

(Nm = fl,).

Corollary 8.3. The only case where negative and positive lattices coincide is for m = 1 and n = 3, i.e., Ni = Ma.

natural

We shall also mention two other properties that give a geometrical interpretation of the least positive solution of Pell’s equation. We shall omit the proof as it can be derived straightforwardly. Proposition 9. Suppose dE.Z a given positive discriminant (d E 2) and (ni, zdi) the least positive solution of the corresponding Pell (plus) equation. The positive natural lattice M,, is a sublattice of index ~1 in the principal lattice Pd. Proposition 10. Suppose d a given positive discriminant appearing in a Pell (minus) equation, which has non trivial solutions, and let (ml, vi) be the least positive one. The negative natural lattice Mm, is a sublattice of index zli in the principal lattice Pd. Quite generally, as all primitive lattices of same discriminant have equal surface of their elementary cell (d = -4 det g(B)), there is an obvious generalization of the last two propositions.

54

A. JANNER

AND

E. ASCHER

5. Some properties of ambiguous lattices. The language of quadratic forms is a very convenient one for the matter discussed in this section and for referring to the literature. Only primitive quadratic forms are considered here, and in this section by equivalency (2.8). We recall that a form (a, b, c) is ambiguous if a divides b. In this case the proper arithmetic class coincides with the arithmetic one (see ref. 17, p. 7 1) : the lattice represented by the forms of the class is ambiguous and vice versa. An ambiguous form is called rectangular if b/a is even and rhombic if b/a is odd. This nomenclature is understandable on the basis of the following proposition. Proposition 11. A rectangular form is equivalent a rhombic form to one with b = a. More explicitly: (a, 2ka, c) N (a, 0, c (a, (2k + 1) a, c) -

k2a)

to one with b = 0;

VkEZ,

(a, a, c -

k(k +

(5.1)

1) a).

Proof.

2k;

1 a

‘“;:

“)(:

-:)=i,

c_k(ki)J’

It is in general not easy to decide whether given a quadratic form (a, b, c) is equivalent to an ambiguous one or not, the number of equivalent forms being infinite. The standard approach is to go over to equivalent reduced forms (see ref. 18, p. 100). The conditions for a form (a, b, c) to be reduced are : 0 < Jd

-

b < 2 Ial -=c Jd

+ b

and

O
(5.2)

and imply also 0 < Jd

-

b < 2 ICI < ,/d + b.

(5.3)

The number of equivalent reduced forms is finite (ref. 17 theorem 79, p. 103). These can be ordered into a chain of equivalent reduced forms. Prop o sit i o n 12. A rectangular form is equivalent to a reduced rectangular form. The same is true if one replaces rectangular by rhombic. Proof. We show first that any primitive (A, 0, C) is equivalent to a reduced form (a, 2ka, c). Suppose IA / < (Cl and take for k the largest rational integer satisfying the inequality : 0 < J ]C j - lkl J IA / (d is not a

TWO-DIMENSIONAL

RELATIVISTIC

LATTICES

55

square). The form (A, 0, C) is equivalent to the reduced one (A, 2KA, C + @A), If /A1 > ICI the same can be proved by considering the (equivalent) form (C, 0, A), called the associate of (A, 0, C). In the case of a rhombic form, one verifies that any form (a, (a(, c) with -UC > 0 is reduced. Now a rhombic form can always be given as (&A, A, &C) with A, C > 0. The only non-reduced cases are (A, A, C) and (--A, A, -C). The first form is equivalent to (4C - A, A - 4C, C), the second one to (A - 4C, A - 4C, C) which are both reduced rhombic forms. We may distinguish between three sorts of ambiguous lattices: (i) A lattice is rectangular (R) if any ambiguous form representing it is rectangular. (ii) A lattice is rhombic (D) if any ambiguous form representing it is rhombic. (iii) A lattice is mixed (RD) if it is represented by rhombic as well as by rectangular forms. Corollary 12.1. In a chain of reduced equivalent forms representing an ambiguous lattice, there are always two ambiguous forms. If both of these are rectangular (or both rhombic) the lattice is rectangular (or rhombic). If one form is rectangular and the other is rhombic, the lattice is mixed (RD). Proof. See ref. 17, p. 116 and use proposition 12. Let us remark that this classification of ambiguous lattices has a deeper meaning than one might be tempted to think initially. One can e.g. show that a RD-lattice cannot be of type I. This result will be proved and discussed in a subsequent paper 21). 6. Isotropic lattices. A lattice is called isotropic if it contains isotropic lattice vectors. One may then always choose a lattice basis B’ = (ei, eh) with metric tensor g(B’) = (0, b, c) h aving a square discriminant d = b2 (6 E R). In particular if the lattice is primitive, then its discriminant is a squared integer. This property is characteristic for isotropic primitive lattices. Proposition 13. A primitive lattice is isotropic if and only if its discriminant is a squared integer. Proof. We may suppose the lattice given by a primitive g(B) = (a, b, c). The lattice is isotropic if and only if the corresponding quadratic form: f(x, y) = ax2 + bxy + cyz,

Vx,yeZ

(6.1)

represents zero over 2. The necessary and sufficient condition for (a, b, c) to represent zero over the rationals Q (i.e. for x, y E Q) is that the discriminant d be a squared integer (see ref. 20, p. 395, theorem 10) ; conversely, if d is a squared integer, than (a, b, c) also represents zero over 2.

A. JANNER

56

AND

E. ASCHER

Proposition 14. The only proper automorphs of an isotropic are E and -E. Proof. Consider the metric tensor g(B) = (0, b, c) of an isotropic and its automorph SE

a

B 6

( > Y

E .x(2,

lattice

Z).

Then (for b # 0, because

d > 0) : (YB= 1.

and

a!=6

p=y=o, Proposition

lattice

15. A primitive

isotropic

lattice

(6.2) given by g(B) = (0, b, c)

is: ambiguous

if b divides cs -

1,

(ii)

stable

if b divides ca +

1,

(iii)

negative

if b divides 2.

(i)

imply the third one.

Two of the three conditions Proof. Let

s= and g(B) = (0, b, c) with c # 0. If S is an improper then : ab + yc = 0,

a =

&j -

and

-8

automorph

of (0, b, c)

By = --I.

This means

s-i: y.

If S is a proper negautomorph, 01=

ab + yc = 0,

(6.3) then : -6

and

016 -

By=

016

-1,

giving

%*(-, ‘“j. If now S is an improper

y = 0,

negautomorph,

,9b = (a -

6) c

then and

:

1,

TWO-DIMENSIONAL

RELATIVISTIC

LATTICES

57

(6.5) As b and c are relatively prime, it follows that b divides 2. Clearly two of the three relations imply the third one. If now c = 0, we have for the primitive lattice g(B) = (0, 1, 0), and the isotropic

lattice is ambiguous,

stable and negative.

In fact we have in this

case : improper

automorphs :

proper negautomorphs

improper

:

negautomorphs

(6.6) :

As in table (4.3) one may distinguish between different types of primitive isotropic lattices. Note, however, that type IV lattice (enantiomorphic, unstable and negative) does not occur, because if b = 2, then c is odd and b divides c2 * 1; the same is trivially true for b = 1. The general situation is illustrated in the following table. TABLE (6.7) Classification of isotropic primitive lattices with metric tensor g(B) = (0, b, c) Lattice type

Characterization of the lattice A

Conditions on the metric tensor

ambiguous stable negative

A = A’ A=li A = /i’

b=l b=2 andc

II

ambiguous unstable positive

A = A’ A#/r A#z

b/,2-1 bfc”+ b # 192

III

enantiomorphic stable positive

A # A’ A=/i A #6’

bfc”b/cs+l b # 1,2

V

enantiomorphic unstable positive

A # A’ A#jl A #A’

bfc2bfc2+ b # 1,2

I

or -

1 (mod2)

1

Examples of g(B) (0, 1, 0) (0,2, -1)

and

2 (l,O, -1)

(0, 3, -4

1 (0, 5, -2)

1 1

(0, 7, -2)

The conditions for an ambiguous primitive isotropic lattice to be rhombic or rectangular are expressed by the following proposition.

A. JANNER

58

AND

E. ASCHER

Proposition 16. An ambiguous primitive isotropic lattice with metric tensor g(B) = (0, b, c) is rectangular if b is even and rhombic if b is odd. Proof. Let us first remark that an ambiguous isotropic lattice is either rhombic or rectangular, because it admits only two mirrors (&&I), which then belong to the same arithmetic class. Suppose the lattice rectangular;

(0, b, c) 2 (A, 0, C)

then:

by

giving the conditions : 2A6 = yb, 2(Sc -

-2Cy &)

= Sb,

2(yc + ,&A) = --LYb,

= -_Bb,

(6.8)

which implies b -O(mod2). Suppose the lattice rhombic, then by S as above (0, b, c) & (A, A, C), giving olb = A(2S however,

y),

db = A(6 -

2~);

(6.9)

B or y is odd, thus :

b = 1 (mod2). 7. Bravais classes. One has to distinguish between Bravais classes and relativistic Bravais classes. We have already defined the first concept by condition (2.10). The second arises only if one considers the identification of isomorphic space-time groups as discussed in ref. 3. In this section we consider only the subdivision of relativistic lattices into usual Bravais classes. Having these, it is very easy to go over to relativistic Bravais classes. A first and important distinction can be made between relativistic lattices with holohedry of finite order (the ametric case) and lattices with holohedry infinite order (the metric case). (a) Lattices with holohedries of finite order. In section 2 and 6 we have seen that lattices with metric tensors proportional to the integral ones, with discriminant not a squared integer, always have an infinite holohedry. Therefore only real incommensurable (a, b, c) and/or isotropic (0, b, c) give rise to lattices with finite holohedries. Proposition 17. The holohedry of a lattice is of finite order if and only if the lattice is isotropic and/or has an incommensurable metric tensor

TWO-DIMENSIONAL

RELATIVISTIC

LATTICES

59

g(B) = (a, b, c). Th is means that for no real positiv k are Ku, kb and kc rational integers. Any lattice is left invariant by the identity E and by the total inversion -E. Lattices that are invariant only with respect to these two elements have holohedries that belong to a single arithmetic class, which defines the oblique Bravais class. The corresponding holohedry is No = {-E} z Cz. In the relativistic case, mirrors M are the only other lattice symmetries of finite order. We therefore consider ambiguous lattices. The holohedry of isotropic and/or incommensurable ambiguous lattices is : H = {-E,

M} g Ds.

(7.1)

In fact, two mirrors Ml and Ms, where Mr f &Ma generate a proper Lorentz transformation of infinite order (cf. 3.4). We have now to find out the arithmetic classes of (7.1). These are the same as the arithmetic classes of crystallographic Lorentz mirrors. Proposition 18. There are two arithmetic classes of crystallographic class (MR} and the improper Lorentz transformations : the rectangular rhombic class {MD} where

,,=(A _i).

MR=(:, _y> Proof.

Any Lorentz

M& = f

mirror is of the form:

cash $

sinh C$

- sinh C$

M+ leaves invariant

the space-like

,u = k(cosh $4 ei with k real positive;

M*=fp

sinh #

C$ >* direction

es)

and M_ leaves invariant

p = k(sinh &rj ei with k real positive,

-cash

the time-like

direction

cash $4 es),

where as usual g(B) = (1, 0, - 1). One has and

M+ being crystallographic,

Mkp = +

(7.2)

there is a S E GL(2, R) and a basis

BS = B’ = (ei, eh), such that: M;

= S-lM*

S E GL(2,Z)

(7.3)

and there are lattice vectors in the ,u and p directions (for more details, see refs. 21 and 22). According to the results of section 2 of this paper, there is, in the relativistic geometric class of lattices having the same g(B’), a lattice (1 left invariant by M; which lattice has p as first basis vector (,u = e;). This

A. JANNER

60

AND

E. ASCHER

means that there is a basis B” = (e;, e; =) B’S’ of n such that: S’-rM;S’=MI;

=(i

_:>

with

7~2.

(7.4)

The same can be said for M_ taking el = p. By:

(:,

-:)(6

-:>(:,

Y) = (:

y ‘:“)j

\dLwE.z

(7.5)

one sees that: (;

_;)“(6

Furthermore MD. Finally

(7.6)

y-y).

one verifies that 44~ is not in the same arithmetic

MR,Z--_fi

and

MO & -

MD.

class as

(7.7)

To prove this, it is sufficient to interchange the role of p and ,u in the demonstration above. According to proposition 18, holohedries (7.1) belong to two arithmetic classes. Lattices having holohedries equivalent to the rectangular class belong to the rectangular Bravais class; those with holohedries equivalent to the rhombic class belong to the rhombic Bravais class. Note that invariance with respect to MR implies g(B) = (a, 0, c) ; invariance with respect to MD gives g(B) = (a, a, c). This explains the name of the class. To conclude, there are the following three Bravais classes of lattices in the ametric and isotropic cases. (i) The rectangular Bravais class. The lattices AR belonging to this class have either an incommensurable or an isotropic metric tensor g(B’) = (a’, b’, c’ ) such that there are real u, c with: a > 0.

(a’, b’, c’) 2L (a, 0, c), Their holohedry HR = {-E,

(7.8)

(with respect to a rectangular

basis) is:

MR} g Dz.

(7.9)

(ii) The rhombic class. The lattices AD belonging to this class have either an incommensurable or an isotropic metric tensor g(B’) = (a’, b’, c’) such that there are real a, c with

Their holohedry HD = {--E,

(with respect to a rhombic n/rD>s

(7.10)

a > 0.

(a’, b’, c’) r% (a, a, c),

D2.

basis) is: (7.11)

TWO-DIMENSIONAL

RELATIVISTIC

LATTICES

61

All incommensurable or isotropic (iii) The oblique Bravais class. lattices that are neither in the rectangular nor in the rhombic Bravais class belong to the oblique Ho = {-E} (b) Lattices

s

Bravais

class. Their holohedry

is: (7.12)

c2.

with

holohedries

of

infinite

order.

As we have

already seen in section 2, we may restrict our attention to primitive lattices whose discriminant is not a squared integer. In what follows by primitive lattices we mean non-isotropic primitive lattices. According to proposition 3 any proper automorph of a primitive lattice is given by (3.14) and is therefore generated by the total inversion -E and the fundamental Lorentz transformation LO, which leave the lattice invariant. The holohedry of an enantiomorphic primitive lattice is thus given by: H, = {-E,

Lo} E C,

x Cs.

(7.13)

Any improper automorph M of an ambiguous primitive lattice is the product of a particular mirror MO leaving the lattice invariant and a certain proper automorph. It follows that the holohedry of a primitive ambiguous lattice is : H, = {-E,

Lo, MO} cg D,

x Cz.

(7.14)

We do not have to investigate the arithmetic classes of (7.13) and of (7.14), for we know already the result: The proper arithmetic classes of primitive indefinite quadratic forms are in one-to-one correspondence with the Bravais classes of metric lattices. In each Bravais class we have singled out by (2.13) a primitive lattice. A list of chains of equivalent reduced quadratic forms corresponding to primitive lattices is given in the appendix for a number of discriminants. The table needs some comments. Usually reduced quadratic forms are tabulated according to the increasing value of the discriminant. It is well known that the number of different arithmetic classes for a given discriminant is finite (see ref. 17, theorem 79, p. 103). The least positive solutions of Pell’s equation are determined by a given discriminant. In our classification, however, we have adopted another ordering criterion that is based on the kinematical interpretation of the proper Lorentz transformations leaving a relativistic lattice invariant. To a proper Lorentz transformation of infinite order there corresponds the transformation from an inertial frame to one moving with relative velocity z, in the x direction (the only spatial dimension in two-dimensional Minkowskian space). The absolute value of z, is determined by the trace G E 2 of the Lorentz transformation

62

A. JANNER

considered.

AND

E. ASCHER

One has :

n=2cosh+=

(7.15) 2

giving :

(7.16)

We indicate below the correspondence

for low values of n.

TABLE (7.17) Kinematical interpretation of n n

-

v?z

3

4

5

6

7

8

Js 3

J3 2

J21 5

242 3

345 7

p_ 4

0.745

0.866

0.916

0.942

0.958

c 0.968

From the physical point of view we expect that the lowest values of n are the most interesting ones. Therefore we have ordered the Bravais classes of primitive lattice according to increasing value of 12. For a given n it is easy (by looking at the factor decomposition of 12s - 4) to determine all possible integral values of u and d in the Pell equation. Knowing the discriminant d, the corresponding arithmetic classes are derived by standard methods (see e.g. ref. 17). 8. Concluding remarks. By means of this paper we have laid down the foundation of a two-dimensional relativistic crystallography. The next step consists in determining of the arithmetic classes of relativistic point groups. These can then be used for the derivation of all two-dimensional space-time groups. Acknowledgements. The financial contribution accorded by the Curatoren of the Katholieke Universiteit Nijmegen is gratefully acknowledged. The authors wish to thank Professor W. Opechowski for critical remarks.

TWO-DIMENSIONAL

RELATIVISTIC

63

LATTICES

APPENDIX Bravais 1221

5 6

7

5 1 1

9 10

21 32

2

8

3 8

12

1

1

1

1 1

12

45 5 60

77 96

Type

Lattices Mg N1

[l, 2, --2x-2, 2, 11 [2, 2, - I][- 1, 2, 21

M4 -

r1, 3, -3x-3, [3, 3, -lx_-

3, 11 1, 3, 31

IID

[l> 4, -41[-4, [4, 4, -111-

4, 11 1, 4, 41

IIRD

MB M6

[l, 2, -1x-

1, 2, 11

IR

NB

[l, 5, -511-5, 5, I] [5, 5%- llC- 1, 5,51

IID

M, -

[l, 1, -ll[-1,

ID

N1

[l, 6, -6][-6, 6, 11 [6, 6, - I][-- 1, 6, 63

IIRD

M, Mg

[2, 6, -a][-3, [3, 6, -2][-2,

IIRD

M4

M7

1, 11

6, 21 6, 31

IID

[l, 8, -8][-8, 8, 1] [8, 8, - l][-- 1, 8, 81

IIRD

4, 4][4, 4, -5][-5, 6, 5][5, 4, -4][-4,

Mg M3

[l, 7, -71[-7, 7, 11 c7, 7, - lx,- 1, 7, 71

Mg Ms

6, 31 4, 5]

Mlg Mlo

IIRD

2

24

[l, 4, -2x-2, [2, 4, - ll[-

4, 11 1, 4, 21

IIR

1

117

[l, 9, -9][-9, [9, 9, - ll[-

9, I] 1, 9, 91

IID

Ml1 Ml1

3

13

Cl, 3, -1x-1,

3, 11

ID

N3

1

140

IIRD

Ml2 -

[l, 10, -lO][-10, [lo, 10, - 1x-1,

1

165

[l, 11, -ll][-11, [II, 11, -l][-1, [3. 9, -7x-7. [7? 9, -3x-3,

14

lattice (for n 5 25)

ID

1, 11

[2, 10, -5x-5, [S, 10, -2][-5, 13

relativistic

IIRD

[l, 1, -1x-1.

[3, 6, -5][-5, [5, 6, -3][--3,

11

(non-isotropic)

Chains of equivalent primitive reduced forms

d

31 4

classes of primitive

1

192

10, I] 10, lo]

M12

IIRD

10, 21 10, 21

IID

11, 1] 11, 111 5, 51[5, 5, --77[-7, 9, 7x7, 5, -5x-5,

p24 p24

M13 M13

9, 31 9, -31

IID

[1, 12, -12][-12, 12, 1] [- 1, 12, 12][12, 12, - 1]

IIRD

[3, 12, -4][-4, 12, 31 [-3, 12, 4![4, 12, -33

IIRD

Ml4 M14

A.

64

(A$$endix nu 14

AND

E.

ASCHER

continued) d

2

JANNER

Chains of equivalent

48

[l, 6, -3][-3,

primitive

reduced

forms

6, l]

Type

Lattice

IIR

P48

I-- 1, 6, 3][3, 6, -11 4

12

r1, 2, -2x-2, L--l,

15

1

221

2, 2x2,

[I, 13, c-1,

2, l] 2, -

13][-

[5> 11, -51r-5, 16

1

3 17

1

252

28 285

1

320

9, -7l[-7,

IIRD

14, 21 14, -21

80

19

20

20

8

5

IID

[3, 15, -5x-5,

15, 31

IID

1

357

1

3

396

44

M17 M17

IIRD

16, l] 16, -11 10,5][5,

10, -ll][-11,

12,4]

10, ll][ll,

12, -43

IIRD IIR

8, l]

1, 8, 4][4, 8, -11

[l, 4, -1x--1, l, -1x-I,

IR

N4

1, 11

ID

N1

IID

-ll][-11,

L-3,

7, 11][11,

7, -7][-7,

15,3]

18, 11 18, -11

IIRD IIRD

[5, 16, -7][-7, L-5, 16, 7][7,

12, 9][9, 6, -lo][-10, 12, -9][-9. 6, lO][lO,

14, 51 14, -51

L-7, 16, 5][5, [7, 16, -5][-5,

14, -lO][-10, 6, 9][9, 14, lO][lO, 6, -9][-9,

12, -71 12, 7J

6, l]

Mzo M20

18, 21 18, -21

1, 6, 2X2, 6, -11

IID

15, -31

[2, 18, -9][-9, [--2, 18, 9][9,

[1, 6, -2][-2,

Mrs =19

[3, 15, -1ll[-11.7,71[7,7, [1, 18, -18][-18, [-1, 18, 18][18,

PSO PSO

4, l]

[l, 17, -17l[-17, 17, l] [-- 1, 17, 17J[17, 17, -11

15, ll][ll,

Ml8 El8

12, 111[11, 10, -5][-5,

[l, 8, -4][-4,

[-

Pz8

15, 51[5, 15, -31

[1, 16, -16][-16, [-1, 16, 16][16,

[l,

Ml6 Ml6

p28

15, l] 15, -11

[4

IIRD

2, -3x-3, 4, 11 2, 3][3, 4, - 11

[l, 15, -15][-15, [--1, 15, 15][15,

[4, 12, -lll[-11, 2

III

9, 51

5, 71[7, 9, -51

[2, 14, -7x-7, L-2, 14, 7][7,

2, 2x2, 2, -2][-2,

M15 JQ15

IIRD

L-4, 18

IID

13, -11

[l, 14, - 14][- 14, 14, l] [- 1, 14, 14][14, 14, - 11

[l, 4, -31[-3, [- 1. 4, 3x3,

Me M4

9, 7][7, 5, -7][-7,

11, 5x5,

L-3, 18

11

13, 13, I]

13, 13x13,

c-5,

P48

IIRD

V

IIRD

P44 P44

TWO-DIMENSIONAL

RELATIVISTIC

65

LATTICES

(Appendix continzced) nu 21

Chains of equivalent

d 1

437

[l,

19, -19][-19,

[-1, 22

1

480

16, 7][7,

[3, 18, C-3, 2

120

[l,

12, 12][12,

1

525

8, 8][8, 8, -13][-13,

5 24

1

21 572

3, 11 -

1, 3, 3x3,3,

1

621

[l, 22, -22][-22,

69

15, -

21, 5H5, 19, -13x-13,

7, -13][-13, 11, 7, 13][13,

1l][-

7, ll][ll,

19, 13][13,

7, -ll][-11.

3, 3][3, 3, -5][-5,

1, 7, 51[5, 3, -31[-3,

M25 B25

15, ll][ll,

L-9,

[-

IID

23, l] 23, -11

21, 9][9,

[l, 7, -51[-5,

IIRD

22, 21

L-5,

[9, 21, -51[--5, 3

M24 M24

22, -21

1, 23, 23][23,

[5, 21, -9][-9,

IIRD

22, -11

22, ll][ll,

p21 61

22, l]

1, 22, 22][22,

[l, 23, -23][-23, [-

IID

11

[2, 22, -ll][-11, 25

M23 M23

IID

21, 31

[l, 3, -31[-3,

[-2,

IID

21, -31

[[-

IIR

8, 21

21, -11

c3, 21, -7x-7,

p120 p120

6, 71[7, 8, -21

21, l]

1, 21, 21][21,

21, 7x7,

IIR

6, 3X3, 6, -7][-7,

8, 7][7, 6, -3][-3,

r-3,

IIRD

18, 31

8, 13][13, 8, -31

10, l]

[l, 21, -21-j-21, [-

IIRD

16, 81 16, -81

10, --I]

[2, 8, --71[-7, 23

12, -7][-7, 12, 7][7,

8, -8][-8,

10, -5][-5,

C-2,

Mz2 M22

12, -12][-12,

13][-13,

10, 5][5,

M21 M21

IIRD

20, 41

18, 13][13,

c-1.

IIRD

11

Lattices

20, -43

[8, 16, --71[-7, [-8,

IID

20, l] 20, -

[4, 20, -5][-5, 20, 5x5,

Type

forms

19, -11

1, 20, 20][20,

[-4,

reduced

19, l]

19, 19x19,

[l, 20, -2O][-20, [-

primitive

19, 51 V

19, 51 15, -91 15, 91

7, l]

IID

3, 5][5, 7, -11

P69 p69

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