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An extension of the Saxon-Hutner theorem in the relativistic domain
Volume 137, number 7,8
PHYSICS LETTERS A
29 May 1989
AN EXTENSION OF THE SAXON-HUTNER THEOREM IN THE RELATIVISTIC DOMAIN Ivailo M. MLADENOV’ CentralLaboratory ofBiophysics, Bulgarian Academy ofSciences, Acad. G. Bonchev Sir., Bl. 21, 1113 Sofia, Bulgaria Received 27 January 1989; accepted for publication 31 March 1989 Communicatedby J.P. Vigier
It is shown that to any real localized potential ofthe one-dimensional Dirac equation one mayjuxtapose a vector of special type in the three-dimensional complex vector space. A sufficient condition for the validity of the Saxon—Hutner theorem is given in terms of these vectors. Subramanian and Bhagwat’s relativistic generalization of Luttinger’s proof is confirmed and the “converse” theorem is briefly discussed.
1. Introduction In the treatment of the energy bands of heavy atom crystals the relativistic effects are too large to be neglected. At the same time the complexity of the problems forces us to work with different kinds of approximations whose accuracy is difficult to evaluate. The attainment of exact results is rare and the one-dimensional models are usually the laboratories through which the real world can be viewed. The motion of the electron in one-dimensional ordered or disordered lattices is the simplest one. The regularly ordered scatters divide the continuous spectrum of the electron energy into allowed and forbidden bands the so-called electron bands and energy gaps. The properties of conducting or semiconducting devices depend strongly on their presence and distribution. The way the energy gaps arise, become narrow and gradually disappear with the growth of the energy is usually demonstrated through the Kronig—Penney model [1—51.The relativistic analysis of the Kronig—Penney model [1] shows that the relativistic effects reduce the electron bands, while the band gaps tend to remain constant. Concerning the energy gaps predicted by the non-relativistic Schrodinger equation for one-dimensional binary alloys, Saxon and Hutner [6] have conjectured the following: —
Theorem. Forbidden energies that are common to the pure A crystal and the pure B crystal (with the same lattice constant) will always be forbidden in any arrangement of A and B atoms in a substitutionalsolid solution. Luttinger [7] was the first who provided the proof of such a theorem for the special case of ö-potentials of different strengths situated symmetrically in a lattice of a fixed constant. Curiously enough he was able to show that the Saxon—Hutner conjecture has an analogy in the case of wavetransmission down a line loaded with two terminals. The further development in the non-relativistic field of the problem may be found in ref. [8]. On the other hand, according to our knowledge there is only one paper dealing with the relativistic case, namely, the successful extension of Luttinger’s proof by Subramanian and Bhagwat [9]. Without any doubt it is interesting to know to what extent this spectral theorem remains valid for other potentials in the relativistic situation. In this paper group theory is used for discussion of the energy bands in the spectrum of the one-dimensional Partially supported by contracts 330/89 and 911/89 ofthe Ministry of Culture, Science and Education.
0375-9601/89/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
313
Volume 137, number 7,8
PHYSICS LETTERS A
29 May 1989
Dirac equation. Our treatment of the relativistic Saxon—Hutner theorem is based on the transfer-matrix approach invented by Kramers [101 for the Schrodinger equation. Just as in the non-relativistic case we can replace the real localised potential with a transfer matrix belonging to a concrete Lie group and what remains after the realization of this strategy is only to use the universal applicability of the representation theory to abstractly formulated problems. The paper is organized as follows: in section 2 the general transfer matrix for the one-dimensional Dirac equation is recognized as an element of the group SL(2, P). Section 3 contains the precise mathematical formulation of the Saxon—Hutner theorem and description of Fedorov’s parametrization of the group SL (2, P), which turns out to be extremely appropriate for the problem we are dealing with. In section 4 we find the sufficient conditions for the validity of the Saxon—Hutner theorem in terms of a special type of vectors in the three-dimensional complex space. The “converse” theorem is briefly discussed in section 5.
2. A transfer matrix for the one-dimensional Dirac equation The one-dimensional time-independent Dirac equation for an electron moving in a potential V(x) is ~
(1)
where W is a two-component spinor, and o~,a~are the Pauli matrices: /0
0~r=(\l
l’\ o)’
/1
7Z1\0
0 —l
Integrating (1) around the origin with the potential of the form V(x) = ~ô(x),~>0, Subramanian and Bhagwat [91 have found (in the real basis) the transfer matrix for a unit cell of length a: M— cos 0 cos ka )~sin 0 sin ka cos 0 sin ka +)~sin 0 cos ka (2) —
(
—
—
—
cos 0 sin ka
—
-
sin 0 cos ka
cos 0 cos ka
—
-
sin 0 sin ka
where E—mc2 E2—m2c4 k2= h2c2 hck 0=2arctg~-. (3) The conserved quantity for the system (1) is the total current density (see ref. [11] for an alternative spatial constant) ‘
j=cw+o.vw and consequently the transfer matrices are the matrices from SL(2, C) satisfying: (4) Firstly, we would like to point out the group structure of the matrices which fulfill the conditions (4). Secondly, the subgroup of SL (2, C) selected by defining relations (4) is a three-dimensional one, (a ~
ib’\ d)’
detu=ad+bc=l.
(5)
Finally we would like to recall the result from the inverse scattering theory (see ref. [12]), extremely important for our further considerations: the real localized potentials and the transfer matrices are in bijection. This allows for the replacement of the potentials with the corresponding transfer matrices and motivates the detailed 314
Volume 137, number 7,8
PHYSICS LETTERS A
29 May 1989
study of the latter which follows. We shall begin by establishing an isomorphism (realization) between the matrices in (5) and the group of real unimodular 2 x 2 matrices SL( 2, R). Explicitly r is given by
~) ~),
(°~= (a
~)4M=
u=(2
detM=1.
(6)
It is easy to see that this is a group homomorphism which preserves the trace. It may be represented in matrix form: (a
ib~ (a1
ib1’\ ~ic d) ~ic1 d1
(
a ks—c
I
b’\( a1 b1’\ d,/~—c1 d11
(
aa1—bc~ i(ab1+bd1) ~i(ca1+dc1) dd1—cb1
(
aa1—bc1 ab1+bd1 k~—(ca1+dc1) dd1—cb1
or in abstract form: r(p2)=r(j1)=r(~t)r(~u1)=MM1=M2, tr(u)=tr(r(~u))=tr(M). We would like to lay a particular stress on the fact that the transfer matrix M of Subramanian and Bhagwat has exactly the real form described above. In comparison with the nonrelativistic case, the general transfer matrix for the one-dimensional Schrodinger equation is an element of the group SU (1, 1) (see e.g. ref. [8]). Accidentally (or not) the last group is isomorphic to the group SL(2, IP).
3. MathematicaL formulation of the Saxon—Hutner theorem Let us consider a one-dimensional lattice composed by two types of atoms A and B, each having r•, d1eZ~ atoms in the ith period, 5 ...ArkBSk. (7) A~B The group nature of the individual transfer matrices MA and MB representing A and B atoms permits one to define the total transfer matrix of an arbitrary sequence (7) as the product 5k
MB
~k
AJsI
A”~
B
rl A~
The forbidden energies for an electron propagating in the periodic lattice are given by the condition HtrMI>l, where M is a transfer matrix for a unit cell. Using the transfer-matrix approach, the theorem may be expressed as follows: Is it true that for any arrangements A~B51...Ark BSk of A and B atoms we have
,
~ (9) provided that ~Jtr(MA)I, ~Itr(MB)I>l? The mathematical difficulties for proving this comes from the fact that each transfer matrix is described by four real parameters (a, fi, y, 5) under a constraint (aö— Oy= 1) instead of three independent ones. Fortu315
Volume 137, number 7,8
PHYSICS LETTERS A
29 May 1989
nately, we may use the complex vector parametrization developed by Fedorov [131. Any group element from SL (2, P) having a nonzero trace may be written in the form A(c)=
I+iCaaa
~
ceC
3
2
l+c >0.
,
In this way we put in correspondence a complex three-dimensional vector c to any transfer matrix. Using coordinates this reads M= (a
(ô—a)i).
~
(10)
Let us remark that the vectors in (10) span a linear subspace V c characteristic features of the parameterization (10) are: (i) The simple composition law
C ~, which
is stable under cross product. The
A(c)A(c’)=A(c”) c”
=
>
=
(c+c’ +cxc’
)/( 1—cc’)
(11)
.
(ii) Linearity under conjugations A(c)A(c’ )A ‘(c) =4(ê) where ê= O(c)c’, 0(c) means rotation around vector c in V. (iii) Naturality, which amounts to —
A(0)=I
and A(—c)=A~(c)
Obviously, the parameters in (5) do not possess any of these nice properties. We shall close this section introducing some useful notations: ~ tr(MA)=
=t(A),
~tr(MAMB)= ~
CB>~ =t(AB),
(12)
and so on.
4. Proof of the Saxon—Hutner theorem By definition we have band gaps if I 1 F of V: F={ceVI—l
/~/i~~ I > 1, or in other words if the vector c lies in the subregion (13)
Let us consider two such vectors cA and c~belonging to F. The question arises when their composition will lie also in F? According to (11) and (13) we must have
2 — c~+c~+2cAcB+(cAxcB)2 (I ~2 1,1—cA CB)
<0.
Evidently for this to happen it is sufficient that our vectors cA and cB satisfy (14)
cAcB~(0
and (cA xcR)
316
<0.
(15)
Volume 137, number 7,8
PHYSICS LETTERS A
29 May 1989
On the other hand we have the general identity t(AB) =t(A)t(B)(l
(16)
—CA~CB) ,
which shows that having (14), the condition (15) is superfluous. The AB lattice may be enlarged in two ways, i.e., adding an atom A or B. We shall have CABA =
CAB +CA +CAB XCA ILABACA
I
+ VABACB
+COABACA XCB
or CAB +CB +CAB ~ 1—CAB C
CABB=
XCB UABBCA+VABBCB+WABBCAXCB,
8
as well as t(ABA)l =It(AB)t(A)(l—cABcA)I >1 or It(ABB) I = It(AB)t(B) (1 —CABCB) 1>1 because of (14), (16) and the positivity of the coefficients eUABA, /~ABB,VABA, ~ABB•This step may be repeated as many times as we wish to simulate an arbitrary arrangement of A and B atoms. The full induction ensures the desired Proposition. A sufficient condition for the validity of the Saxon—Hutner theorem is
CACB<
0.
In order to extend this statement for arbitrary number of potentials A, B,..., X, Y, Z we need both conditions (14) and (15) to be statisfied by any pair of vectors CA, CB,..., c,~,~ c~.The invariance of (14) and (15) with respect to rotations in V reflects the invariance ofthe trace map under conjugation. The fact that for the binary alloys we need only (14) follows directly from this invariance. Indeed rotating CA and CB, we may send them simultaneously in the pseudo-Euclidean plane of index one where (15) is always at hand. The same refers also to Luttinger’s case treated by Subramanian and Bhagwat, since for ö-potentials we have 1 1 + CA
=It(A)t(B)sec(9A—OB)I>1 CB
So there have not been any obstacles for a successful relativistic generalization of Luttinger’s proof (cf. also ref. [141). The second part of Luttinger’s paper [7] deserves our attention as well, at least because of the form of the transfer matrix. For a section of a transmission line of length 1 Luttinger wrote the following transfer matrix: A— —
(
—
cos /31— (~/2k)sin /31 (i/k) [sin /31— (~/2k)(1 —cos /31)]
—ik[sin /31+ (,~/2k)(1 +cos $1)] cos /31— (~/2k)sin /31
It is exactly of the form (5) predicted by the Dirac equation. The factthat the common stop bands are present also in any mixture of reactances x and x’ spaced in the same way follows from the simple observation 1
=
It(A)t(A’)I >1
l+CACA’
and our proposition. Because of the group isomorphism SL( 2, P)
SU (1, 1) the statement of the proposition covers the non317
Volume 137, number 7,8
PHYSICS LETTERS A
29 May 1989
relativistic domain as well. A detailed comparison ofthe different criteria which determine the validity of the Saxon—Hutner conjecture, has been made by Hermann [151.
5. Remarks
(1) In the process of the proof use has not been made of the original assumption of Saxon and Hutner for the equality of cell lengths. So our proof includes the spatial as well as the shape disorder. (2) We shall close the paper with a brief discussion of what Tong and Tong [16] refer to as the “converse” theorem. “A level which is allowed both in A and B crystal is allowed in any substitutional alloy of A, B.” That such a theorem is difficult to prove can be seen after a closer examination of the regular lattice AB. Allowed energies are defined by cj, c~>0. In this situation it seems that the rotationally invariant assumption CA CB >0
is sufficiently appropriate for the converse theorem to be true. But even the above condition does not guarantee that It(AB) I <1 as long as 1(A) I, It(B)I <1. This explains why at the very end oftheir paper Tong and Tong [161 on the basis of computer analysis, have written: “It is probably safe to say that the forbidden gaps are remembered better than the allowed levels.” The last statement has found further support in ref. [171, where it is converted into a nonrelativistic inverse Saxon—Hutner theorem. Definitely the situation in the relativistic case is the same.
References [1] M.L. Glasser and S.G. Davison, Int. J. Quantum Chem. Symp. 3 (1970) 867. [2] C. Lange and T. Janssen, Physica A 127 (1984) 125. [3] J. Kollár and A. SOtO, Phys. Lett. A 117 (1986) 203. [4] F. Dominguez-Adame, Am. J. Phys. 55 (1987) 1003. [5] D. WOrtz, M. Soerensen and T. Schneider, Helv. Phys. Ada 61(1988) 345. [6] D.S. Saxon and R.A. Hutner, Philips Res. Rep. 4 (1949) 81. [7] J.M. Luttinger, Philips Res. Rep. 6 (1951) 303. [8] P. Erdds and R.C. Herndon, Adv. Phys. 31(1982) 65. [9] R. Subramanian and K.V. Bhagwat, Phys. Stat. Sol. (b) 48 (1971) 399. [10] H.A. Kramers, Physica 2 (1935)483. [Ii] C.L. Roy, Phys. Lett. A 130(1988)203. [121 S.P. Novikov, ed., Soliton theory (Nauka, Moscow, 1980) [in Russian]. [13] F.I. Fedorov, The Lorentz group (Nauka, Moscow, 1979) [in Russian]. [14] F. Gesztesy and P. Sëba, Lett. Math. Phys. 13 (1987) 345. [15] M. Hermann, Lett. Math. Phys., to be published. [16] B.Y. Tong and S.Y. Tong, Phys. Rev. 180 (1969) 739. [17]H.English,Phys.Stat.Sol. (b) 118 (1983) Kl7.
318
PHYSICS LETTERS A
29 May 1989
AN EXTENSION OF THE SAXON-HUTNER THEOREM IN THE RELATIVISTIC DOMAIN Ivailo M. MLADENOV’ CentralLaboratory ofBiophysics, Bulgarian Academy ofSciences, Acad. G. Bonchev Sir., Bl. 21, 1113 Sofia, Bulgaria Received 27 January 1989; accepted for publication 31 March 1989 Communicatedby J.P. Vigier
It is shown that to any real localized potential ofthe one-dimensional Dirac equation one mayjuxtapose a vector of special type in the three-dimensional complex vector space. A sufficient condition for the validity of the Saxon—Hutner theorem is given in terms of these vectors. Subramanian and Bhagwat’s relativistic generalization of Luttinger’s proof is confirmed and the “converse” theorem is briefly discussed.
1. Introduction In the treatment of the energy bands of heavy atom crystals the relativistic effects are too large to be neglected. At the same time the complexity of the problems forces us to work with different kinds of approximations whose accuracy is difficult to evaluate. The attainment of exact results is rare and the one-dimensional models are usually the laboratories through which the real world can be viewed. The motion of the electron in one-dimensional ordered or disordered lattices is the simplest one. The regularly ordered scatters divide the continuous spectrum of the electron energy into allowed and forbidden bands the so-called electron bands and energy gaps. The properties of conducting or semiconducting devices depend strongly on their presence and distribution. The way the energy gaps arise, become narrow and gradually disappear with the growth of the energy is usually demonstrated through the Kronig—Penney model [1—51.The relativistic analysis of the Kronig—Penney model [1] shows that the relativistic effects reduce the electron bands, while the band gaps tend to remain constant. Concerning the energy gaps predicted by the non-relativistic Schrodinger equation for one-dimensional binary alloys, Saxon and Hutner [6] have conjectured the following: —
Theorem. Forbidden energies that are common to the pure A crystal and the pure B crystal (with the same lattice constant) will always be forbidden in any arrangement of A and B atoms in a substitutionalsolid solution. Luttinger [7] was the first who provided the proof of such a theorem for the special case of ö-potentials of different strengths situated symmetrically in a lattice of a fixed constant. Curiously enough he was able to show that the Saxon—Hutner conjecture has an analogy in the case of wavetransmission down a line loaded with two terminals. The further development in the non-relativistic field of the problem may be found in ref. [8]. On the other hand, according to our knowledge there is only one paper dealing with the relativistic case, namely, the successful extension of Luttinger’s proof by Subramanian and Bhagwat [9]. Without any doubt it is interesting to know to what extent this spectral theorem remains valid for other potentials in the relativistic situation. In this paper group theory is used for discussion of the energy bands in the spectrum of the one-dimensional Partially supported by contracts 330/89 and 911/89 ofthe Ministry of Culture, Science and Education.
0375-9601/89/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
313
Volume 137, number 7,8
PHYSICS LETTERS A
29 May 1989
Dirac equation. Our treatment of the relativistic Saxon—Hutner theorem is based on the transfer-matrix approach invented by Kramers [101 for the Schrodinger equation. Just as in the non-relativistic case we can replace the real localised potential with a transfer matrix belonging to a concrete Lie group and what remains after the realization of this strategy is only to use the universal applicability of the representation theory to abstractly formulated problems. The paper is organized as follows: in section 2 the general transfer matrix for the one-dimensional Dirac equation is recognized as an element of the group SL(2, P). Section 3 contains the precise mathematical formulation of the Saxon—Hutner theorem and description of Fedorov’s parametrization of the group SL (2, P), which turns out to be extremely appropriate for the problem we are dealing with. In section 4 we find the sufficient conditions for the validity of the Saxon—Hutner theorem in terms of a special type of vectors in the three-dimensional complex space. The “converse” theorem is briefly discussed in section 5.
2. A transfer matrix for the one-dimensional Dirac equation The one-dimensional time-independent Dirac equation for an electron moving in a potential V(x) is ~
(1)
where W is a two-component spinor, and o~,a~are the Pauli matrices: /0
0~r=(\l
l’\ o)’
/1
7Z1\0
0 —l
Integrating (1) around the origin with the potential of the form V(x) = ~ô(x),~>0, Subramanian and Bhagwat [91 have found (in the real basis) the transfer matrix for a unit cell of length a: M— cos 0 cos ka )~sin 0 sin ka cos 0 sin ka +)~sin 0 cos ka (2) —
(
—
—
—
cos 0 sin ka
—
-
sin 0 cos ka
cos 0 cos ka
—
-
sin 0 sin ka
where E—mc2 E2—m2c4 k2= h2c2 hck 0=2arctg~-. (3) The conserved quantity for the system (1) is the total current density (see ref. [11] for an alternative spatial constant) ‘
j=cw+o.vw and consequently the transfer matrices are the matrices from SL(2, C) satisfying: (4) Firstly, we would like to point out the group structure of the matrices which fulfill the conditions (4). Secondly, the subgroup of SL (2, C) selected by defining relations (4) is a three-dimensional one, (a ~
ib’\ d)’
detu=ad+bc=l.
(5)
Finally we would like to recall the result from the inverse scattering theory (see ref. [12]), extremely important for our further considerations: the real localized potentials and the transfer matrices are in bijection. This allows for the replacement of the potentials with the corresponding transfer matrices and motivates the detailed 314
Volume 137, number 7,8
PHYSICS LETTERS A
29 May 1989
study of the latter which follows. We shall begin by establishing an isomorphism (realization) between the matrices in (5) and the group of real unimodular 2 x 2 matrices SL( 2, R). Explicitly r is given by
~) ~),
(°~= (a
~)4M=
u=(2
detM=1.
(6)
It is easy to see that this is a group homomorphism which preserves the trace. It may be represented in matrix form: (a
ib~ (a1
ib1’\ ~ic d) ~ic1 d1
(
a ks—c
I
b’\( a1 b1’\ d,/~—c1 d11
(
aa1—bc~ i(ab1+bd1) ~i(ca1+dc1) dd1—cb1
(
aa1—bc1 ab1+bd1 k~—(ca1+dc1) dd1—cb1
or in abstract form: r(p2)=r(j1)=r(~t)r(~u1)=MM1=M2, tr(u)=tr(r(~u))=tr(M). We would like to lay a particular stress on the fact that the transfer matrix M of Subramanian and Bhagwat has exactly the real form described above. In comparison with the nonrelativistic case, the general transfer matrix for the one-dimensional Schrodinger equation is an element of the group SU (1, 1) (see e.g. ref. [8]). Accidentally (or not) the last group is isomorphic to the group SL(2, IP).
3. MathematicaL formulation of the Saxon—Hutner theorem Let us consider a one-dimensional lattice composed by two types of atoms A and B, each having r•, d1eZ~ atoms in the ith period, 5 ...ArkBSk. (7) A~B The group nature of the individual transfer matrices MA and MB representing A and B atoms permits one to define the total transfer matrix of an arbitrary sequence (7) as the product 5k
MB
~k
AJsI
A”~
B
rl A~
The forbidden energies for an electron propagating in the periodic lattice are given by the condition HtrMI>l, where M is a transfer matrix for a unit cell. Using the transfer-matrix approach, the theorem may be expressed as follows: Is it true that for any arrangements A~B51...Ark BSk of A and B atoms we have
,
~ (9) provided that ~Jtr(MA)I, ~Itr(MB)I>l? The mathematical difficulties for proving this comes from the fact that each transfer matrix is described by four real parameters (a, fi, y, 5) under a constraint (aö— Oy= 1) instead of three independent ones. Fortu315
Volume 137, number 7,8
PHYSICS LETTERS A
29 May 1989
nately, we may use the complex vector parametrization developed by Fedorov [131. Any group element from SL (2, P) having a nonzero trace may be written in the form A(c)=
I+iCaaa
~
ceC
3
2
l+c >0.
,
In this way we put in correspondence a complex three-dimensional vector c to any transfer matrix. Using coordinates this reads M= (a
(ô—a)i).
~
(10)
Let us remark that the vectors in (10) span a linear subspace V c characteristic features of the parameterization (10) are: (i) The simple composition law
C ~, which
is stable under cross product. The
A(c)A(c’)=A(c”) c”
=
>
=
(c+c’ +cxc’
)/( 1—cc’)
(11)
.
(ii) Linearity under conjugations A(c)A(c’ )A ‘(c) =4(ê) where ê= O(c)c’, 0(c) means rotation around vector c in V. (iii) Naturality, which amounts to —
A(0)=I
and A(—c)=A~(c)
Obviously, the parameters in (5) do not possess any of these nice properties. We shall close this section introducing some useful notations: ~ tr(MA)=
=t(A),
~tr(MAMB)= ~
CB>~ =t(AB),
(12)
and so on.
4. Proof of the Saxon—Hutner theorem By definition we have band gaps if I 1 F of V: F={ceVI—l
/~/i~~ I > 1, or in other words if the vector c lies in the subregion (13)
Let us consider two such vectors cA and c~belonging to F. The question arises when their composition
2 — c~+c~+2cAcB+(cAxcB)2 (I ~2 1,1—cA CB)
<0.
Evidently for this to happen it is sufficient that our vectors cA and cB satisfy (14)
cAcB~(0
and (cA xcR)
316
<0.
(15)
Volume 137, number 7,8
PHYSICS LETTERS A
29 May 1989
On the other hand we have the general identity t(AB) =t(A)t(B)(l
(16)
—CA~CB) ,
which shows that having (14), the condition (15) is superfluous. The AB lattice may be enlarged in two ways, i.e., adding an atom A or B. We shall have CABA =
CAB +CA +CAB XCA ILABACA
I
+ VABACB
+COABACA XCB
or CAB +CB +CAB ~ 1—CAB C
CABB=
XCB UABBCA+VABBCB+WABBCAXCB,
8
as well as t(ABA)l =It(AB)t(A)(l—cABcA)I >1 or It(ABB) I = It(AB)t(B) (1 —CABCB) 1>1 because of (14), (16) and the positivity of the coefficients eUABA, /~ABB,VABA, ~ABB•This step may be repeated as many times as we wish to simulate an arbitrary arrangement of A and B atoms. The full induction ensures the desired Proposition. A sufficient condition for the validity of the Saxon—Hutner theorem is
CACB<
0.
In order to extend this statement for arbitrary number of potentials A, B,..., X, Y, Z we need both conditions (14) and (15) to be statisfied by any pair of vectors CA, CB,..., c,~,~ c~.The invariance of (14) and (15) with respect to rotations in V reflects the invariance ofthe trace map under conjugation. The fact that for the binary alloys we need only (14) follows directly from this invariance. Indeed rotating CA and CB, we may send them simultaneously in the pseudo-Euclidean plane of index one where (15) is always at hand. The same refers also to Luttinger’s case treated by Subramanian and Bhagwat, since for ö-potentials we have 1 1 + CA
=It(A)t(B)sec(9A—OB)I>1 CB
So there have not been any obstacles for a successful relativistic generalization of Luttinger’s proof (cf. also ref. [141). The second part of Luttinger’s paper [7] deserves our attention as well, at least because of the form of the transfer matrix. For a section of a transmission line of length 1 Luttinger wrote the following transfer matrix: A— —
(
—
cos /31— (~/2k)sin /31 (i/k) [sin /31— (~/2k)(1 —cos /31)]
—ik[sin /31+ (,~/2k)(1 +cos $1)] cos /31— (~/2k)sin /31
It is exactly of the form (5) predicted by the Dirac equation. The factthat the common stop bands are present also in any mixture of reactances x and x’ spaced in the same way follows from the simple observation 1
=
It(A)t(A’)I >1
l+CACA’
and our proposition. Because of the group isomorphism SL( 2, P)
SU (1, 1) the statement of the proposition covers the non317
Volume 137, number 7,8
PHYSICS LETTERS A
29 May 1989
relativistic domain as well. A detailed comparison ofthe different criteria which determine the validity of the Saxon—Hutner conjecture, has been made by Hermann [151.
5. Remarks
(1) In the process of the proof use has not been made of the original assumption of Saxon and Hutner for the equality of cell lengths. So our proof includes the spatial as well as the shape disorder. (2) We shall close the paper with a brief discussion of what Tong and Tong [16] refer to as the “converse” theorem. “A level which is allowed both in A and B crystal is allowed in any substitutional alloy of A, B.” That such a theorem is difficult to prove can be seen after a closer examination of the regular lattice AB. Allowed energies are defined by cj, c~>0. In this situation it seems that the rotationally invariant assumption CA CB >0
is sufficiently appropriate for the converse theorem to be true. But even the above condition does not guarantee that It(AB) I <1 as long as 1(A) I, It(B)I <1. This explains why at the very end oftheir paper Tong and Tong [161 on the basis of computer analysis, have written: “It is probably safe to say that the forbidden gaps are remembered better than the allowed levels.” The last statement has found further support in ref. [171, where it is converted into a nonrelativistic inverse Saxon—Hutner theorem. Definitely the situation in the relativistic case is the same.
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