SURF ACE WAVES AND ROTATIONAL INVARIANCE IN LAmCE THEORY

Transcription

1 Institut für Reaktorwerkstoffe KERNFORSCHUNGSANLAGE JDLICH des Landes Nordrhein-Westfalen-e.V. SURF ACE WAVES AND ROTATIONAL INVARIANCE IN LAmCE THEORY by W.~dwig and B.J:!ngeler Jül-226-RW 1964

2 Berichte der Kernforschungsanlage Jülich Nr. 226 Institut für Reaktorwerkstoffe Jül RW Dok.: Crystol Lottices Surfoce Woves DK: 539.2/.31 : Zu beziehen durch: ZENTRALBIBLIOTHEK der Kernforschungsanlage Jülich, Jülich, Bundesrepublik Deutschland

3 Reprinted from SOLID STATE COMMUNICATIONS

4 Solid State Communications Vol. 2, pp 83-86, Pergamon Press, Inc. Printed in the United States SURFACE WAVES AND ROTATIONAL INVARIANCE IN LATTICE THEORY W. Ludwig and B. Lengeler Institut für Reaktorwerkstoffe der Kernforschungsanlage JÜiich, Jüiich;Lehrstuhl für physikalische Grundlagen der Reaktorwerkstoffe der Technischen Hochschule Aachen, Aachen, Germany (Received 2 February 1964 by G. Leibfried) A crystal lattice with a free surface may have localized modes of vibration at the surface (surface waves). In a consistent theory in the limit of long waves these modes have to agree with the elastic surface modes (Rayleigh waves). lt is shown, that it is necessary for this agreement to satisfy the condition of rotational invariance. IN recent years several attempts have been made to study surface waves in cryf ~ lattices with an infinitely extended surface. - In the limit of lang waves, these surface modes have to agree with the corresponding elastic waves, i. e. the famous surface modes of Rayleigh and their generalization. Also, the elastic himit has been discussed in different cases. 5- lt appeared, that the lattice surface waves in the limit of long waves in general da not agree with the elastic theory. This fact has been realized sometimes, in other cases it has not been spelled out. lt seems to us, that the reason of this disagreement has never been discussed. In the following we will show the necessary condition for the agreement between lattice theory in the long-wave-limit and the elastic theory. The conditions for rotational invariance have tobe satisfied for every atom, especially for those in the surface. As far as we can see, this condition is vlo~ted in all the recently discussed models. - Inl there is discussed a ( 001 )- surface in a simple cubic lattice with nearest-neighbor-interaction only. This gives no surface modes, in contrast to the elastic behavior, which allows for such modes. Takeno2 discusses the same model as in ref. 1, but by a different method. He states, that there are surface modes, which are in agreement with the elastic limit; but he di d not realize, that his surface inodes do not obey the dispersion law for elastic surface modes, i.e. w2 ~k2, but rathe{ w2= -A+Bk 2 with A>O; this means that w becomes negative with k... 0, which cannot be correct neither in lattice nor in elastic theory. In3 also a simple cubic lattice with ( 001)-, (Oll)- and ( 111 )- surface is discussed. There are surface modes in case of a ( 0 H )- and ( 111 ) surface. A comparison with the elastic limit has not been made, since the elastic theory in this case has not been discussed. The importance of rotational invariance has been mentioned, but it has not been taken into account because otherwise the calculations could not have been done in a simple way. Therefore these calculations cannot agree with the elastic limit, as we will show in the following. We begin with a few remarks an the elastic theory: The equation of motion is with psi (i=) = 0 ik 1 k summation convention (1) 0 ik=gik,jlejl = Gik,jlsjll (2) p is the mass density, s i ( r) the displacement field, oik and jl = 1/2 (Sj\l + Sllj) are stress and strain tensor. The elastic constants Cik, jl are symmetric toward interchange of i with k, j with 1 and ik with jl. If the elastic constants depend an position r, (2) and (1) gi.ve P 8 (r) = Gik 11ks 11 + cik 1s 1k1 1,J J,J J (3) In an (infinite) homogeneous medium the first term on the right side vanishes. If one is interested in the problem of a semi-infinite crystal with a free surface (assuming a homogeneous medium in the half-space with z >O e. g. ), one can easily show that solving equation 83

5 84 SURFACE WAVES AND ROTATIONAL INVARIANCE Vol. 2, No. 3 (3) ls equlvalent to the following problem: one can solve equatlon (3) without the first term on the rlght side, but with the boundary condition of vanishing forces in z = 0 on the surface. In thls case (4) m+h n+h cp i j and therefore mn cp i j k cprr r(x~- X~)=~ cp~ f X~= (11) Thls is the way in which surface states in elastic theory are investigated. lt implies the symmetry of the elastic constants, as the elastic constants enter the first term in (3) as well as the boundary conditions (4). Specially, the symmetry in j and 1 is conserved, as can be seen from the first term in (3). Lattice theory starts (in harmonic approximation, primitive lattices) with the equation of motion M.. m E. mn n 5 i = - nj cp i j 5 j Here the coupling parameters cp'«!? are symmetric in the pairs TP: 1 1 mn nm q>i j = cpji and they have to satisfy the conditions for translational invariance rcp1?-~=0 n i J and for rotational invariance mn( n m mn n m ~ to i j XI - X 1 ) = ~ cp i 1 (Xj - X j ) for rery atom m of the system. Equations (6-8 are valid for every system of mass-points without external forces, forming not necessarlly a lattlce To get the elastic llmlt, we introduce in the usual way a slowlyvarylng dlsplacement field s r = s r + s ~ 1 (X~ _ x~ ) + + 1/2 st 1 kl (~ - X~ ) (X~ - X~ ) which gives, using (7) and dividing by V z P s. (R m) 1 m n (Xn Xm) i - v ~j 1 l'.p i j 1-1 s j '. t z (5) (6) (7) (8) (9) (10) In infinite homogeneous lattices the first term on the right side vanishes, because mn -m -n cp i j = cp i j and The condition (8) has not been used here. The second term is Independent of m and gives, using the Born-Huang-relations, the usual elastic constants In inhomogeneous media, e. g. in a semi-infinite medium with a free surface, the relations (11) do not hold in a surface region, the dimensions depending on the range of the coupling-parameters. Therefore the first term in (10) does not vanish at the free surface; it corresponds exactly to the first term in (3), which is symmetric toward an interchange of j and 1, and therefore the first term in (10) must be symmetric in j and 1 too, to get the elastic limit from lattice theory. In order to have this symmetry guaranteed it is necessary, to have the condition of rotational invariance satisfied, as can be seen immediately from (8). The second term in (10) has to be identified with the elastic constants. This can be done in the usual way by using Born-Huangrelations But there is one restriction in inhomogeneous lattices: the dependence of the second term on m, i. e. the elastic constantsi depend on the position in the crystal, which ma' happen also in the elastic theory (equation 3). The first term in (10) of course, depends on m too. But if the inhomogeneities have a range which is small compared to the wave-length (elastic limit), this does not influence the frequencies. A crystal with a free surface is generall: described by assuming a homogeneous medium in z > 0, the elastic constants being described by a step-function (atz= O ). This means that the first term in (10) is different from zero only for valves of m at the surface, while the first term in (3) contains a factor ö (z). Consequently deviations from the conditions of rotational invariance (8) at the surface of a free crystal enter only through the boundary conditions. We will illustrate this with a simple example: A simple cubic lattice with nearestneighbor-interaction. The lattice sites are

6 Val. 2, No. 3 SURFACE WAVES AND ROTATIONAL INVARIANCE 85 given by... m... R =am; m 1, m 2 = 0, :1: 1, :1: 2, m 3 = 0, + 1, + 2, tor a semi-infinite lattice. Typical coupling matrices for surface atoms are (12) ~,nd g,_ ö (am ) = ~2 ö (m 3 ) a 3 a correspondingly oc441. = ß ö (m3) ; ioz a2 (16) 0 ±100 qi i j qi?o.01= -(~~~) 1 J (13) lt can be seen immediately that these matrices are compatible with the point group symmetry of the corresponding atoms. The coupling parameters in the interior of the crystal are assumed to be those of the homogeneous lattice. The rotation condition (8) requires ß = 2 ö A simple calculation of the first term in (10) gives for i = 1: ~ { ßsu ö s~ 1 } ö om 3 (14) (15) Therefore we have for the first term in (3), which is equivalent to the boundary condition (4) for ß i = 1 : ä2 { s + s } ö (m ) i = 2 : ~2 i = 3 : ~ { s2/3 + s3/2 } ö (m3) (l '/) l a. s3/3 - ß ( sl/1 + s2/2 J ö (m3) (15) and (17) do agree only, ü the condition (14) is satisfied. And it can be seen once more that violating the condition of rotational invariance corresponds to a change of the boundary conditions. Lattice surface waves in the limit of lang waves can a~ee with the elastic surface waves only, ü (8) is satisfied. Inl and 3 there has been put ö = 0, which means, that long-wave-lattice theory and elastic theory can not coincide. The elastic theory, assuming the ideal!lastic constants for z > 0, gives for the above nodel 01.1/ oz = c 11 ö (z) References 1. WALLIS R. F., Phys. Rev. 105, 540 (1557); lbid. 116, 302 (1959). 2. TAKENO SH., Progr. Theor. Phys. 30, 1 (1963). A calculation of surface waves which shows the influence of (8) in the dispersion curves directly as well as further examples and discussions will be given in. 4 We should like to thank Prof. G. Leibfried for some critical remarks. 3. LENGELER B. and LUDWIG W., J. Phys. Chem. Solids In press (:i,b64). 4. LENGELER B., Diplomarbeit Aachen (1964). 5. RAYLEIGH LORD, Proc. London Math. Soc. 17, 4 (1887). 6. STONELEY R., Proc. Roy. Soc. London A 232, 447 (1955). 7. SYNGE J. L., J. Math. Phys. 35, 323 (1957).

7 86 SURFACE WAVES AND ROTATIONAL INVARIANCE Vol. 2, No.~ 8. MUSGRAVE M. J. P., Rflp. Progr. Phys. 22, 74 (1959). 9. BUCHWALD V. T. and DAVIS A., Quart. J. Mech. Appl. Math. 16/3, 283 (1963). 10. LEIBFRIED G., Handb. d. Physik 7/1, 104 (1955). 11. LEIBFRIED G. and LUDWIG W., Solid State Phys. 12, 275 (1961). 12. HEDIN L. T. Arkiv f. Fysik 18, 369 (1960); LEIBFRIED G. and LUDWIG W., Z Physik 160, 80 (1960); LAX M., J. Phys. Chem. Solids In press (1964). Ein Kristallgitter mit einer freien Oberfläche kann an dieser Oberfläche lokalisierte Schwingungszustände besitzen (Oberflächenwellen). In einer konsistenten Theorie müssen diese Zustände im Grenzfall langer Wellen mit den elastischen Oberflächenzuständen (Rayleigh-Wellen) übereinstimmen. Es wird gezeigt, daß es für diese Uebereinstimmung notwendig ist, die Rotationsbedingung zu erfüllen. Die Rotationsbedingung folgt daraus, daß sich bei einer infinitesimalen Drehung des Kristalls seine potentielle Energie nicht ändern darf, wenn keine äu~ren Kräfte wirksam sind.