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Lattice dynamics study of vibrational modes in gold Σ5[001] twist boundary
Acta metall, mater. Vol. 39, No. 1I, pp. 2489-2496, 1991 Printed in Great Britain. All rights reserved
0956-7151/91 $3.00 + 0.00 Copyright © 1991 Pergamon Press plc
LATTICE DYNAMICS STUDY OF VIBRATIONAL MODES IN GOLD E5[001] TWIST B O U N D A R Y S. AHARON and A. B R O K M A N Division of Applied Physics and Materials Science, Graduate School of Applied Science, The Hebrew University, Jerusalem 91904, Israel (Received 8 Not,ember 1990; in revised form 16 May 1991)
Aima'act--The dynamical matrix was constructed for the equilibrium structure precalculated by means of a molecular statics computer simulation for the case of a bi-crystal with a X5[001] twist boundary in gold. The eigenstates analysis indicates a pure boundary mode in a small volume of the Brillouin zone, while a resonance with the bulk states occurs in the complementary volume in momentum space. This degenerated state develops at a frequency equal approximately to 2 x 10t3 s -~. The eigenvectors are analyzed and it is shown that the local modes consist of a collective in-plane displacement of the four atoms around the O-lattice point. These modes represent an external libration mode and a coupled breathing mode. The total r.m.s, displacement in the boundary is close to the bulk one. The physical significance of the local mode together with possible implications and its effect on the scattered X-ray intensity are discussed. Ri~suntr--On construit la matrice dynamique pour une structure d'&luilibre pr~calcul~e au moyen d'une simulation mol~culaire statique par ordinateur dans le cas d'un bicristal d'or ayant un joint de grains de torsion E5[001]. L'analyse des valeurs propres r~v~le un mode de joint pur dans un petit volume de la zone de Brillouin, tandis qu' une rr'sonance avec les Stats massifs se produit dans le volume compl~mentaire dans respace des moments. L'(~tat d(~#m[r~ se d~veloppe ~i tree fr&luence ~gale environ ~i 2.10 ~3s -t. Les vecteurs propres sont analys~s et on montre que les modes locaux consistent en un d~placement collectif dans le plan des quatre atomes situt~s autour du point du r~seau 0. Ces modes repr~sentent un mode de libration externe et un mode de respiration couple. Le d~placement total r.m.s, dans le joint est voisin de celui de la masse. La signification physique du mode local, ainsi que ses implications possibles et son effet sur rintensit6 X diffus~e sont discut~s. Zuummenfassung--Fiir einen Gold-Bikristall mit einer X = 5 [001]-Zwillingsgrenze wird die dynamische Matrix der Gleichgewichtsstruktur, die vorher mit einer molekularstatistischen Computersimulation berechnet worden ist, konstruiert. Die Analyse der Eigenzus~nde weist darauf hin, dab in einem kleinen Volumen der Brillouin-Zone eine reine Korngrenzmode existiert, wohingegen im komplementiiren Volumen des Impulsraumes eine Resonanz mit den Volumenzust~den auftritt. Dieser entartete Zustand entwickelt sich bei einer Frequenz von etwa 2 x 10~3sec -t. Die Eigenvektoren werden analysiert und es wird gezeigt, dab die lokalen Moden aus einer kollektiven Verschiebung in der Ebene der vier Atome um den O-Gitter-Punkt bestehen. Diese Moden repr~isentieren eine externe Librationsmode und eine gekoppelte Atmungsmode. Die gesamte mittlere quadratische Verschiebung in der Korngrenze liegt nahe derjenigen im Volumen. Die physikalische Natur der lokalen Mode wird zusammen mit deren m6glicher Bedeutung und deren EinfluB auf die gestreute R6ntgenintensit~t diskutiert.
1. INTRODUCTION The technological importance of grain boundaries in polycrystalline materials has motivated long-standing efforts to establish and understand the physical properties of these defects. During the last two decades, work has been primarily devoted to the understanding of the core structure of high-angle grain boundaries. It is now established that the core of these boundaries is periodic, and its periodicity is associated with the Coincidence Site Lattice (CSL) [1]. The E5[001] twist boundary in gold is the simplest such structure investigated by X-ray diffraction [2-5]. As simple as is this structure, the interpretation of exper-
imental results was unsatisfactory until very recently and has been the subject of lengthy discussions. The c o m m o n approach to the analysis of the experimental results was based on a computer molecular statics simulation. This provides us with a significant amount of information about different types of possible equilibrium structures at 0 K [e.g. 6, 7]. A well known discrepancy between experimental observations and the computer simulations has shifted the focus of research towards structural studies of the Z5 twist boundary in gold, while many other properties of this simple grain boundary had been neglected for a long time. This unfortunate situation was changed last year when two independent sets of
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2490
AHARON and BROKMAN: VIBRATIONAL MODES IN GOLD TWIST BOUNDARY
experiments which considered the local displacement were reported. Taylor, Majid, Bristow and Ballufli restudied the absolute scattered X-ray intensity from Y-5 twist boundaries in gold rigorously and very accurately [8]. This experiment yields to a better agreement with the static computer simulation [9]. At the same time, Fitzsimmons, Burkel and Sass measured the local displacement in gold twist boundaries as a function of temperature [10]. It was argued that the boundary thermal displacement is significantly larger than in the bulk. At this early stage of the new experiments, there is considerable disagreement between the two groups [10, 12]. However, both studies demonstrate clearly that the experimental technique has reached the stage where local dynamics can be evaluated. Furthermore, it seems that a better understanding of these dynamics could be used in the interpretation of the new results. Therefore, it is the purpose of the present work to investigate the boundary vibrational modes. Beyond the direct implications to the above new data, the local mode is of major importance in the understanding of physical properties of the boundary. The dynamic study of other types of grain boundaries, by molecular dynamics techniques has been successfully carried out. For example, the study of migration of 125 two-dimensional tilt boundaries [13], the identification of diffusion mechanisms along the Y-5 tilt boundary in b.c.c, iron [14], the study of local premelting of the I25 tilt boundary in f.c.c. Lennard-Jones system [I 5]. Attempts have even been made to characterize the local modes in a Y.I 1 tilt boundary in gold by means of molecular dynamics simulation using the Morse potential [16]. Recently, Foiles [17] calculated the local r.m.s, displacement in 1213 twist boundary in gold, using an embedded atom-Monte Carlo simulation model. His results, while predicting the observed Debye-Waller factor [10], did not elaborate on the spectrum and character of the local vibrational modes. A different approach was adopted by Earmme et al. [18]. These authors studied the structure of the I25 tilt boundary (Morse potential) by means of lattice dynamics. This model can identify unstable structures when their configuration corresponds to negative eigenvalues of the dynamical matrix. This work is most significant since it demonstrates the possible application of the lattice dynamics techniques to the determination of grain boundary properties.
In the present work we used the latter approach to study the dynamics of the Y-5 tilt boundary in gold in the harmonic approximation. The advantage of this technique arises from its ability to characterize the local modes developed in the core of the grain boundary. In fact, the use of lattice dynamics approximation may be justified in view of the above mentioned X-ray studies. On one hand Majid, Bristow and Ballutti indicated that the molecular dynamics simulation is computationally too heavy, while on the other hand, Fitzsimmons, Burkel and Sass showed how the harmonic approximation is satisfactory for the interpretation of the local thermal displacive. Unlike Earmme et al., we started with a stable structure obtained by a molecular statics computer simulation. For this configuration we analyzed the local vibrational spectrum, and it was found that the local mode consists of the coupled collective in-plane libration and breathing displacements centered at the O-lattice points. This result tackles the long standing problem of the 125 twist boundary in gold from a yet-unexplored point of view, the application of which is discussed below. EQUILIBRIUM CONHGURATION
2. DETERMINATION OF THE
The long standing discrepancy between computer simulation and X-ray ray diffraction experiments led to the application of sophisticated simulation techniques, the latest being the embedded atom model. In the present work, we choose to use a simple bruteforce simulation for two reasons: (a) the free electron contribution to the dynamic properties is of a second order importance, and (b) the comparison between the structure calculated by the embedded atom and the brute-force technique show a very similar static structure. Having this comment as a background, we will further describe the simulation procedure. The first step in the analysis involved the determination of the atom positions in the vicinity of the grain boundary, by using molecular statics computer simulation. This technique is well documented [6], and here we briefly summarize our application. An eighteen (002)-plane 125[001] bi-crystal was generated with a periodic CSL boundary conditions in the plane of the boundary. The atoms interact in pairs via a Morse potential suggested by Cherns et al. [19] for gold-gold interaction
0+p{-077,( -07)t 2oxp{03,7(-07)t1 ~(r)[eV]
=
-06s 0.0
+ o o.s6
-07)
_
07)
r ~<0.70 a0 o.7o < _r ~ O.9747. ao r
-- > 0.9747 ao
(1)
AHARON and BROKMAN: VIBRATIONAL MODES IN GOLD TWIST BOUNDARY
o
A
[]
X ©
X
X ©
X X ]
o plone 1 plone 2
X
plone - I
/~
plone - 2
Fig. 1. The equilibrium structure calculated by molecular statics computer simulation. The atomic positions in the two planes adjacent to the boundary plane in both crystals are projected onto the boundary plane and numbered according to their distance from the boundary plane (positive and negative signs refer to the first and second crystals). Here a0 = 4.0786 ,~ is the f.c.c,gold lattice constant. This construction was then relaxed to a minimum potential energy by applying the conjugated gradient molecular static technique [20]. Different intergranular translation stages were used as initial configuration, and it was found that the CSL configuration (no translation in the plane--see Ref. [6]) is the most stable one. The relaxed structure is drawn in Fig. 1. The multiplicity of units suggested by Oh and Vitek [7] was not considered in the present work since the lattice dynamics technique requires a stable configuration (metastable structure complicates the analysis). Moreover, the strength of the multiplicity of units in explaining X-ray scattering observations is weakened by the new experimental results of Taylor et aL [8].
order to determine a representative smallest model size we relaxed statically a series of structures of different thicknesses, employing the periodic boundary conditions, and, we compared the obtained configuration with the equilibrium structure obtained in Section 2. It was found that the atomic configuration in the (002) planes included in the interaction range on both sides of the boundary plane is the equilibrium configuration--as long as the model size is kept larger than 5 times the lattice constant. It should be emphasized that the total potential energy was slightly changed by applying the periodic boundary condition as a result of the interaction between the virtual grain boundaries thus generated. This change in the potential energy reflects small displacements outside of the interaction range, and is thus of less importance for our lattice dynamics consideration-especially when dealing with specific vibrational normal modes localized in the core of the boundary. Therefore, it is concluded that for the sake of the dynamical matrix construction, the periodic structure well represents the equilibrium structure of the grain boundary. This periodic boundary condition in the Z-direction (perpendicular to the grain boundary) generates a superlattice with repeating virtual grain boundaries. We used this computational supercell for our lattice dynamics calculation. Let x (I) designate the vectors of the computational supercell, and x(x) the basis positions within the cell. Following the notation of Ref. [21], the force constants are then given by • ,~(l~; 1'~')
620
au~ (I, ~ )6u~ (1' ~')
$,a(l¢; l'¢') = - ~,~(/,;/'~'),
The large model size used for the configurational equilibrium procedure described above is impractical for the purpose of lattice dynamics calculation. In order to construct a dynamical matrix which can be diagonalized in a reasonable c.p.u, time, one must synthesize a thinner bi-crystal than the one described above. Thinning the bicrystal does not affect the force constant near the boundary as long as the interaction range is smaller than the size of the model. In principle, one could cut out, from the "thick" equilibrium structure considered above, a slab thicker than twice the interaction range which contains the grain boundary plane at its center, and carry the analysis within that slab. For practical reasons, we adopted an alternative strategy which incorporates a periodic boundary condition in the direction perpendicular to the grain boundary plane (in addition to the CSL periodic boundary condition in-plane). In
• ,~(/z; l'z') = ~, dp,#(lz; I'z')
39,'11--B
(2)
where • is the potential energy of the bicrystal, and u,(/r) is the a component of the displacement vector u(/z) of the atom positioned at x(IQ = x(l) + x(x). In the case of a central interaction # ( I x ( / z ) - x(l'x')l) between the lr and I'C atoms we have
3. THE DYNAMICAL MATRIX
AMM
2491
(1~)# (l'¢') (3)
/r'
where ~b~#is derived as follows
+ ~ $'(r). (4) r
In equation (4), r = I x ( h ) - x(l'T')l and the prime represents a radial derivative. The dynamical matrix is then defined 1
D,#(~z'Ik)= ~ ~ O,#(1~;I'T') x exp{ik.[x(1)-x(l')]} (5)
2492
AHARON and BROKMAN: VIBRATIONAL MODES IN GOLD TWIST BOUNDARY
G(LO) O.O8
G(LD) 0.08 ~0.06
0.06
~
0.04
0.04
0.02,
o.oog
0.02
/
i
0.000 3 4 COXlO~[sec' ' ] 2
2
3
4
x wo'3Esec-']
Fig. 2. The number of local modes G(w) as a function of frequency. (a) The modes counted in the histograms are the one with large eigenvector components associated with displacements in the first plane adjacent to the boundary pine of one crystal (the spectrum of the first plane in the second crystal is identical by symmetry). (b) The same spectrum obtained for a (002) plane in the bulk. Note that the histogram counts the number of states and not their density so the total area below the curve represents the number of modes with relative large local displacements (see text). where k is the wave vector, and M the gold atomic mass. The potential used in constructing the dynamical matrix was the same as the one used for the molecular static calculation [equation (I)]. Fifty particles located in 10 planes at positions calculated by the molecular static code were used for this calculation. The 150 x 150 dynamical matrix was constructed for various wave vectors (see below) in the Brillouin zone. The dynamical matrix [equation (5)] was diagonalized, as described in the next section, in order to identify the local vibrational modes. 4. DENSITY OF GRAIN BOUNDARY STATES A major difficulty in searching for vibrational modes which are localized at extended defects is the need for repeated diagonalization of a large matrix. This makes the detailed analysis of the density of state outside of our computation capability. Therefore, we adapted an iterative sampling strategy which reduces the number of wave vectors required for the construction of the local density of states: Accordingly, we calculated the density of states g(w) in the boundary for a variable net of wave vectors in the 2/16 irreducible volume of the Briliouin zone. At each iteration we tried to identify a local mode which is not in resonance with the bulk (i.e. a pure mode) by comparing our spectrum with the one obtained from the dynamical matrix constructed separately for the f.c.c, bulk under the same set of wave vectors. Once a pure mode was identified within a small k space volume, the state density was calculated by summing the eigenstates obtained on a finer wave vector grid. In order to illustrate this procedure we examine Fig. 2: here the spectrum confined to the grain boundary is calculated by counting the number of states of large eigenvector components associated with atoms in the two lattice planes forming the grain
boundary. For this purpose, the displacement amplitude of relevant components are defined larger than a quarter of the average displacement (taken over all 150 x 150 components). Figure 2(a), shows the number of states G(w) as calculated for a limited number of wave vectors (16 vectors in the irreducible volume of Brillouin zone). For the sake of comparison, Fig. 2(b) presents the equivalent calculation (same computation cell size and orientation) carried out on two bulk planes. As expected, the noise level in this spectrum is extremely high. However, the example seen in Fig. 2 demonstrates that the high noise level cannot screen the major differences between bulk and grain boundary spectrums, which are analysed next. Two difficulties arise when trying to analyze the localized modes seen in Fig. 2. Firstly, the noise level is too high for detailed statistical analysis, and secondly, these modes are developed resonantly with the bulk and the eigenvectors are therefore perturbed. Overcoming these difficulties is made possible due to the character of the local mode developed at specific wave vectors. Figure 3 shows examples of G(w [k). It is seen that when k equals (1, 0, 0) in the DSC lattice vector (the reciprocal CSL), the associated local mode is pure (out of resonance) and is developed at frequencies within the band gap of the bulk phonons. Therefore, we can investigate the local states by a dense sampling in a limited area of Brillouin zone close to that wave vector. Such analysis of these spectrums implies that a local mode is confined to the grain boundary and is found in the frequency range between 2 x 1013 and 2.2 x 1013s-l. Figure 4 shows the same spectrum calculated for successive planes parallel to the boundary (plane number 1 is closest to the boundary plane). It is noticed that the boundary mode is damped and mixed with the bulk modes at planes away from the boundary. These modes should identify the local vibrational modes which are the subject of the following chapter.
AHARON and BROKMAN: VIBRATIONAL MODES IN GOLD TWIST BOUNDARY
2493
G(~)
G (~)
2,0)
_
m
,I,0)
,0,0)
1,0)
3,OI
0,0) J
0
1
2 3 4 CO xlO t3 [sec"]
5
6
0
I
2 5 4 CO x 1013[sec-1]
5
6
Fig. 3. G(w) vs w (as calculated in Fig. 2) for different wave vectors (DSC) Lattice units, in the plane adjacent to the boundary (a); and in the (002) plane in the bulk (b). It is seen that a pure local mode develops at long wave length [e.g. k = (0, 0, 0), k = (I, 0, 0)] at frequency ~ 2 x I0 ~5s-~. G (~)
0
5. CHARACTERIZATION OF THE LOCAL DISPLACEMENT
1
2
3
4
5
6
× I01s[~c-'] Fig. 4. G(w) plotted for a series of successive planes out of the boundary. It is seen that the boundary mode is weakened and coupled to the other modes when approaching the bulk.
Having identified the local modes from the spectrum calculated above, we can now characterize their origin by considering the relevant eigenvectors. These describe the displacement pattern in the boundary. A closer examination of the numerical data indicates that the boundary mode is unique (we did not identify any other mode localized at the boundary) doubly degenerated according to the bicrystal symmetry, and therefore, two eigenvectors are associated with it. The components of the calculated eigenvector which are associated with atoms in the two planes across the boundary similarity transformation, are drawn in Fig. 5(a). The magnitude of the eigenvector component on the second and higher planes away from the boundary plane are negligible. Therefore, Fig. 5(a) represents the local mode at the boundary. It is seen that the CSL atom is not displaced from its equilibrium position, while the other atoms collectively vibrate in two groups of four atoms--each group of atoms belongs to a plane. In Fig. 5(a) we notice that the two eigenvectors are related by the bi-crystal symmetry group. Now, we linearly combine the two degenerated eigenvectors to obtain the new sets of displacements, seen in Fig. 5(b). Here the displacement of one mode represents a collective rotation of the four atom group around O-lattice
2494
AHARON and BROKMAN: VIBRATIONAL MODES IN GOLD TWIST BOUNDARY
b //
\\
/ ~NX
\
// ./ /
//
\\ \ \
/ ? ~
\\
x
\\
[~////
\ \\\\
\\
,
/ ////
\ca,.
,
\\
\
/// \x//
[
//.x\\ ~ \ / \
I
/
\.x,//
/
,x / /
//
/
/
\
[~
\\ \\\
~ \
l
/ / / ~ \ \ \ \
\\\ "'El / /3 /// //
x
(\\
\\
// \\ //
\
"
/ \\
CSL cell /
//(x
0- Loffice
Wigner- Seitz cell
x plane -I r-1 plane I
Fig. 5. The displacement pattern in the two planes forming the boundary. (a) The eigenvector components associated with the relevant atoms. This pattern was calculated by diagonalizing the dynamical matrix. The two patterns shown are associated with the same (degenerated) eigenstate and belong to the same bicrystal symmetry group representation. (b) The same eigenvector components after performing simple linear combination of the two vectors plotted in (a). In this representation the boundary mode consists of a coupled in-plane external libration (top figure) and internal breathing (bottom figure) displacements of the four-atoms around O-lattice point. The circle in (b) represents the magnitude of the r.m.s. displacement in the bundary (magnified x 10). point, and will be described as an external libration (a similar static relaxation was identified previously [22]). The rotation amplitude is approximately 0.9 ° . The displacement associated with the second mode represent the vibration of the four atoms towards the O-lattice point. The later will be described as an inplane breathing mode. it is interesting to notice that each vector remains invariant under the bi-crystal symmetry group. At the same time, however, no element of this group transforms one vector to the other, i.e. the similarity transformation from Fig. 5(a) to (b) reduces the representation of the dynamical matrix. Therefore, we conclude that the libration and breathing modes are accidentally (not by symmetry) degenerated. When considering the total boundary r.m.s, displacement, the bulk perturbations should also be taken into account. Therefore, while calculating the total displacement effect, we considered the cross correlation of two components of the eigenvectors associated with the atoms in question. This correlation is given in the classical limit (hco <
~(u,(lT)up(l'z')) ----~
1
~ k,:
e,(¢lkj)e~(z'lkj) W~('f)
(6)
w h e r e / / = I/KBT, % is the eigenfrequency of t h e j t h mode, e~(z Ikj) are the components of the associated eigenvector and N the number of supercells considered. The summation here is taken over k starvectors associated with the pure local states. The matrix defined by equation (6), contains all the information required, in order to analyze the effect of temperature on the X-ray scattered intensity [10]. In particular, one can calculate the r.m.s, displacement. The latter was found to be 0.022 A at room temperature, and is drawn as a circle in Fig. 5. It should be emphasized here that the displacements and frequencies so calculated are comparable to the bulk ones. Also, a high correlation (0.98 after normalization) was calculated for the in plane displacements which indicates that perturbation due to lattice coupling is small. A smaller value (0.87) was calculated for the case of intra-planes cross-correlation.
6. DISCUSSION We summarize our findings as follows: (a) A boundary mode was identified in gold ~E5 twist boundary. This mode is doubly degenerated and
AHARON and BROKMAN: VIBRATIONAL MODES IN GOLD TWIST BOUNDARY
-
-2o
G 2o 0 (degree)
40
Fig. 6. The potential well for external libration. This potential is calculated in the zeroth-order adiabatic approximation and neglects the coupling to the internal displacement.
confined to the boundary. It propagates in the plane and decays rapidly in the bulk lattice. (b) The boundary mode consists of external libration and internal breathing of the four atoms around the O-lattice point. (c) The local r.m.s, displacement is very close to the bulk one. The breathing mode is a consequence of the large near-neighbour force constant in the plane. The libration is a more complicated mode and is caused by the strong interaction between nearest neighbours across the boundary plane. It has been shown [23] that the nearest neighbours across the boundary plane are related via their position in the O-lattice. Therefore, the four-atom in the plane are displaced in the strong field of the near-neighbour across the plane, imposing the rotating force around an O-lattice point. In a previous work [22], we have recognized that static relaxation consists of rotational displacements in the boundary centered at O-lattice points. This mode of relaxation is motivated by the nearest-neighbour strong interactions across the boundary plane in a similar way as described above. The external libration may be discussed and understood in the frame of the local potential well for rotation. One way of constructing the libration potential well is by applying a zeroth order adiabatic approximation. According to this approximation we freeze all motions that do not participate directly in the libration of their equilibrium positions, and calculate the change in the system's internal energy as a function of the libration amplitude. Figure 6 plots the change in the potential energy due to a static rotation of the four-atom group. The equilibrum position is represented by zero rotation angle. This angle, for which the structure is fully relaxed by the molecular static technique, readily accounts for the static rotational relaxation angle (equals 7 °). In this calculation we considered a 3 x 3 CSL model with 12 (002) planes, where the four-atom-group in the central cell was rotated rigidly. The potential well so calculated is parabolic to a large extent and therefore, the harmonic approximation used in the present work is adequate for the description of local libration. It should be mentioned here that since the rotational
2495
constant (h2/2J, J being the momentum of inertia of the rotating group) is small compared with the potential well depth, the wave function of the libration is localized at the bottom of the potential well, and should be quantized with equidistant energy levels. At high temperature, a rotational jump between nearest minima may occur by thermal fluctuation. It should be emphasized that the zeroth order approximation neglects the coupling to the internal mode. The experimental confirmation of the boundary mode by scattering techniques is difficult since the number of scattering centres is very limited. However, the associated Debye--Waller effect may be recognized in the X-ray scattering from the periodic array of the boundary core. Our calculated r.m.s, displacement value is very close to the bulk value and is similar to the one assumed for ~5 boundary by Majid et al. in the interpretation of their results [9]. This value is smaller than the one measured by Fitzsimmons, Burkel and Sass for the case of ~ 13 boundary [10, 17]. However, Majid et al. [9] have argued that it is possible that the r.m.s, value varies with the degree of coincidence. The boundary mode may affect the physical properties of the bicrystal (or of polycrystal materials in general). Thermal properties (entropy, heat capacity etc.) are now being investigated in the frame suggested above; grain boundary migration may occur as a result of rotational jump of the four-atomgroup from one crystal orientation to the other. In the same manner, a roughening transition may occur at the plane of the boundary, which involves the O-lattice columns. These properties and others are the subject for future investigation. Acknowledgement--This work was supported by the U.S.-Israel Bi-national Science Foundation under Grant No. 00041/85. REFERENCES
I. See for review R. W. Balluffi, in lnterfackd Segregation (edited by W. C. Johnson and J. M. Blakely). Am. Soc. Metals, Metals Park, Ohio (1977). 2. W. Guadig, D. Y. Guan and S. L Sass, Phil. Mag. A34, 923 (1976). 3. D. Y. Guan and S. L. Sass, Phil. Mag. A39, 293 0979). 4. W. Guadig and S. L. Sass, Phil. Mag. A39, 725 (1979). 5. J. Budai, P. D. Bristowe and S. L. Sass, Acta metall. 31, 699 (1983). 6. P. D. Bristowe and A. G. Crocker, Phil. Mag. A3g, 487 (1978). 7. Y. Oh and V. Vitek, Acta metall. 34, 1941 (1986). 8. M. S. Taylor, F. Majid, P. D. Bristowe and R. W. Balluffi, Phys. Rev. B40, 2772 (1989). 9. I. Majid, P. D. Bristowe and R. W. BaUuffi, Phys. Rev. B40, 2779 (1989). 10. M. R. Fitzsimmons, E. Burkel and S. L. Sass, Phys. Rev. Lett. 61, 2237 (1988). I I. M. R. Fitzsimmons and S. L. Sass, Scripta metall. 23, 411 (1989). 12. I. Majid, P. D. Bristowe and R. W. Balluffi, Scripta metall. 23, 1639 (1989) and the following comment by S. L. Sass.
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AHARON and BROKMAN:
VIBRATIONAL MODES IN GOLD TWIST BOUNDARY
13. G. H. Bishop, R. J. Harrison, T. Kwok and S. Yip, Trans. Am. Nucl. Soc. 27, 323 (1977). 14. T. Kwok, P. S. Ho, S. Yip, R. W. Balluffi, P. D. Bristowe and A. Brokman, Phys. Rev. Lett. 47, 1148 (1981). 15. G. Ciccoti, M. Guillope and V. Pontikis, Phys. Rev. B27, 5576 (1983). 16. M. Hashimoto, H. lchinose and Y. Ishida, Japan J. appl. Phys. 19, 1045 (1980). 17. S. M. Foiles, Acta metall. 37, 2815 (1989). 18. Y. Y. Earmme, J. K. Lee, R. J. Harrison and G. H. Bishop, Surf Sci. 92, 174 (1980).
19. D. Cherns, M. W. Finnis, and M. D. Matthews, Phil. Mag. 5, 693 (1977). 20. M. J. Wein, Surf Sci. 31, 138 0972). 21. A.A. Maradudin, E. W. MontroU, G. H. Wi¢ss and F. P. Ipatova, in SolidState Physics (edited by H. Ehrenttich, F. Seitz and D. Turnbull), 2rid edn. Academic Press, New York (1977). 22. A. Brokman and R. W. Balluffi, Acta metall. 29, 1703 (1981). 23. W. Bollmann, Crystal Defects and Crystallyne Interfaces. Springer, Berlin (1970).
0956-7151/91 $3.00 + 0.00 Copyright © 1991 Pergamon Press plc
LATTICE DYNAMICS STUDY OF VIBRATIONAL MODES IN GOLD E5[001] TWIST B O U N D A R Y S. AHARON and A. B R O K M A N Division of Applied Physics and Materials Science, Graduate School of Applied Science, The Hebrew University, Jerusalem 91904, Israel (Received 8 Not,ember 1990; in revised form 16 May 1991)
Aima'act--The dynamical matrix was constructed for the equilibrium structure precalculated by means of a molecular statics computer simulation for the case of a bi-crystal with a X5[001] twist boundary in gold. The eigenstates analysis indicates a pure boundary mode in a small volume of the Brillouin zone, while a resonance with the bulk states occurs in the complementary volume in momentum space. This degenerated state develops at a frequency equal approximately to 2 x 10t3 s -~. The eigenvectors are analyzed and it is shown that the local modes consist of a collective in-plane displacement of the four atoms around the O-lattice point. These modes represent an external libration mode and a coupled breathing mode. The total r.m.s, displacement in the boundary is close to the bulk one. The physical significance of the local mode together with possible implications and its effect on the scattered X-ray intensity are discussed. Ri~suntr--On construit la matrice dynamique pour une structure d'&luilibre pr~calcul~e au moyen d'une simulation mol~culaire statique par ordinateur dans le cas d'un bicristal d'or ayant un joint de grains de torsion E5[001]. L'analyse des valeurs propres r~v~le un mode de joint pur dans un petit volume de la zone de Brillouin, tandis qu' une rr'sonance avec les Stats massifs se produit dans le volume compl~mentaire dans respace des moments. L'(~tat d(~#m[r~ se d~veloppe ~i tree fr&luence ~gale environ ~i 2.10 ~3s -t. Les vecteurs propres sont analys~s et on montre que les modes locaux consistent en un d~placement collectif dans le plan des quatre atomes situt~s autour du point du r~seau 0. Ces modes repr~sentent un mode de libration externe et un mode de respiration couple. Le d~placement total r.m.s, dans le joint est voisin de celui de la masse. La signification physique du mode local, ainsi que ses implications possibles et son effet sur rintensit6 X diffus~e sont discut~s. Zuummenfassung--Fiir einen Gold-Bikristall mit einer X = 5 [001]-Zwillingsgrenze wird die dynamische Matrix der Gleichgewichtsstruktur, die vorher mit einer molekularstatistischen Computersimulation berechnet worden ist, konstruiert. Die Analyse der Eigenzus~nde weist darauf hin, dab in einem kleinen Volumen der Brillouin-Zone eine reine Korngrenzmode existiert, wohingegen im komplementiiren Volumen des Impulsraumes eine Resonanz mit den Volumenzust~den auftritt. Dieser entartete Zustand entwickelt sich bei einer Frequenz von etwa 2 x 10~3sec -t. Die Eigenvektoren werden analysiert und es wird gezeigt, dab die lokalen Moden aus einer kollektiven Verschiebung in der Ebene der vier Atome um den O-Gitter-Punkt bestehen. Diese Moden repr~isentieren eine externe Librationsmode und eine gekoppelte Atmungsmode. Die gesamte mittlere quadratische Verschiebung in der Korngrenze liegt nahe derjenigen im Volumen. Die physikalische Natur der lokalen Mode wird zusammen mit deren m6glicher Bedeutung und deren EinfluB auf die gestreute R6ntgenintensit~t diskutiert.
1. INTRODUCTION The technological importance of grain boundaries in polycrystalline materials has motivated long-standing efforts to establish and understand the physical properties of these defects. During the last two decades, work has been primarily devoted to the understanding of the core structure of high-angle grain boundaries. It is now established that the core of these boundaries is periodic, and its periodicity is associated with the Coincidence Site Lattice (CSL) [1]. The E5[001] twist boundary in gold is the simplest such structure investigated by X-ray diffraction [2-5]. As simple as is this structure, the interpretation of exper-
imental results was unsatisfactory until very recently and has been the subject of lengthy discussions. The c o m m o n approach to the analysis of the experimental results was based on a computer molecular statics simulation. This provides us with a significant amount of information about different types of possible equilibrium structures at 0 K [e.g. 6, 7]. A well known discrepancy between experimental observations and the computer simulations has shifted the focus of research towards structural studies of the Z5 twist boundary in gold, while many other properties of this simple grain boundary had been neglected for a long time. This unfortunate situation was changed last year when two independent sets of
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AHARON and BROKMAN: VIBRATIONAL MODES IN GOLD TWIST BOUNDARY
experiments which considered the local displacement were reported. Taylor, Majid, Bristow and Ballufli restudied the absolute scattered X-ray intensity from Y-5 twist boundaries in gold rigorously and very accurately [8]. This experiment yields to a better agreement with the static computer simulation [9]. At the same time, Fitzsimmons, Burkel and Sass measured the local displacement in gold twist boundaries as a function of temperature [10]. It was argued that the boundary thermal displacement is significantly larger than in the bulk. At this early stage of the new experiments, there is considerable disagreement between the two groups [10, 12]. However, both studies demonstrate clearly that the experimental technique has reached the stage where local dynamics can be evaluated. Furthermore, it seems that a better understanding of these dynamics could be used in the interpretation of the new results. Therefore, it is the purpose of the present work to investigate the boundary vibrational modes. Beyond the direct implications to the above new data, the local mode is of major importance in the understanding of physical properties of the boundary. The dynamic study of other types of grain boundaries, by molecular dynamics techniques has been successfully carried out. For example, the study of migration of 125 two-dimensional tilt boundaries [13], the identification of diffusion mechanisms along the Y-5 tilt boundary in b.c.c, iron [14], the study of local premelting of the I25 tilt boundary in f.c.c. Lennard-Jones system [I 5]. Attempts have even been made to characterize the local modes in a Y.I 1 tilt boundary in gold by means of molecular dynamics simulation using the Morse potential [16]. Recently, Foiles [17] calculated the local r.m.s, displacement in 1213 twist boundary in gold, using an embedded atom-Monte Carlo simulation model. His results, while predicting the observed Debye-Waller factor [10], did not elaborate on the spectrum and character of the local vibrational modes. A different approach was adopted by Earmme et al. [18]. These authors studied the structure of the I25 tilt boundary (Morse potential) by means of lattice dynamics. This model can identify unstable structures when their configuration corresponds to negative eigenvalues of the dynamical matrix. This work is most significant since it demonstrates the possible application of the lattice dynamics techniques to the determination of grain boundary properties.
In the present work we used the latter approach to study the dynamics of the Y-5 tilt boundary in gold in the harmonic approximation. The advantage of this technique arises from its ability to characterize the local modes developed in the core of the grain boundary. In fact, the use of lattice dynamics approximation may be justified in view of the above mentioned X-ray studies. On one hand Majid, Bristow and Ballutti indicated that the molecular dynamics simulation is computationally too heavy, while on the other hand, Fitzsimmons, Burkel and Sass showed how the harmonic approximation is satisfactory for the interpretation of the local thermal displacive. Unlike Earmme et al., we started with a stable structure obtained by a molecular statics computer simulation. For this configuration we analyzed the local vibrational spectrum, and it was found that the local mode consists of the coupled collective in-plane libration and breathing displacements centered at the O-lattice points. This result tackles the long standing problem of the 125 twist boundary in gold from a yet-unexplored point of view, the application of which is discussed below. EQUILIBRIUM CONHGURATION
2. DETERMINATION OF THE
The long standing discrepancy between computer simulation and X-ray ray diffraction experiments led to the application of sophisticated simulation techniques, the latest being the embedded atom model. In the present work, we choose to use a simple bruteforce simulation for two reasons: (a) the free electron contribution to the dynamic properties is of a second order importance, and (b) the comparison between the structure calculated by the embedded atom and the brute-force technique show a very similar static structure. Having this comment as a background, we will further describe the simulation procedure. The first step in the analysis involved the determination of the atom positions in the vicinity of the grain boundary, by using molecular statics computer simulation. This technique is well documented [6], and here we briefly summarize our application. An eighteen (002)-plane 125[001] bi-crystal was generated with a periodic CSL boundary conditions in the plane of the boundary. The atoms interact in pairs via a Morse potential suggested by Cherns et al. [19] for gold-gold interaction
0+p{-077,( -07)t 2oxp{03,7(-07)t1 ~(r)[eV]
=
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-07)
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07)
r ~<0.70 a0 o.7o < _r ~ O.9747. ao r
-- > 0.9747 ao
(1)
AHARON and BROKMAN: VIBRATIONAL MODES IN GOLD TWIST BOUNDARY
o
A
[]
X ©
X
X ©
X X ]
o plone 1 plone 2
X
plone - I
/~
plone - 2
Fig. 1. The equilibrium structure calculated by molecular statics computer simulation. The atomic positions in the two planes adjacent to the boundary plane in both crystals are projected onto the boundary plane and numbered according to their distance from the boundary plane (positive and negative signs refer to the first and second crystals). Here a0 = 4.0786 ,~ is the f.c.c,gold lattice constant. This construction was then relaxed to a minimum potential energy by applying the conjugated gradient molecular static technique [20]. Different intergranular translation stages were used as initial configuration, and it was found that the CSL configuration (no translation in the plane--see Ref. [6]) is the most stable one. The relaxed structure is drawn in Fig. 1. The multiplicity of units suggested by Oh and Vitek [7] was not considered in the present work since the lattice dynamics technique requires a stable configuration (metastable structure complicates the analysis). Moreover, the strength of the multiplicity of units in explaining X-ray scattering observations is weakened by the new experimental results of Taylor et aL [8].
order to determine a representative smallest model size we relaxed statically a series of structures of different thicknesses, employing the periodic boundary conditions, and, we compared the obtained configuration with the equilibrium structure obtained in Section 2. It was found that the atomic configuration in the (002) planes included in the interaction range on both sides of the boundary plane is the equilibrium configuration--as long as the model size is kept larger than 5 times the lattice constant. It should be emphasized that the total potential energy was slightly changed by applying the periodic boundary condition as a result of the interaction between the virtual grain boundaries thus generated. This change in the potential energy reflects small displacements outside of the interaction range, and is thus of less importance for our lattice dynamics consideration-especially when dealing with specific vibrational normal modes localized in the core of the boundary. Therefore, it is concluded that for the sake of the dynamical matrix construction, the periodic structure well represents the equilibrium structure of the grain boundary. This periodic boundary condition in the Z-direction (perpendicular to the grain boundary) generates a superlattice with repeating virtual grain boundaries. We used this computational supercell for our lattice dynamics calculation. Let x (I) designate the vectors of the computational supercell, and x(x) the basis positions within the cell. Following the notation of Ref. [21], the force constants are then given by • ,~(l~; 1'~')
620
au~ (I, ~ )6u~ (1' ~')
$,a(l¢; l'¢') = - ~,~(/,;/'~'),
The large model size used for the configurational equilibrium procedure described above is impractical for the purpose of lattice dynamics calculation. In order to construct a dynamical matrix which can be diagonalized in a reasonable c.p.u, time, one must synthesize a thinner bi-crystal than the one described above. Thinning the bicrystal does not affect the force constant near the boundary as long as the interaction range is smaller than the size of the model. In principle, one could cut out, from the "thick" equilibrium structure considered above, a slab thicker than twice the interaction range which contains the grain boundary plane at its center, and carry the analysis within that slab. For practical reasons, we adopted an alternative strategy which incorporates a periodic boundary condition in the direction perpendicular to the grain boundary plane (in addition to the CSL periodic boundary condition in-plane). In
• ,~(/z; l'z') = ~, dp,#(lz; I'z')
39,'11--B
(2)
where • is the potential energy of the bicrystal, and u,(/r) is the a component of the displacement vector u(/z) of the atom positioned at x(IQ = x(l) + x(x). In the case of a central interaction # ( I x ( / z ) - x(l'x')l) between the lr and I'C atoms we have
3. THE DYNAMICAL MATRIX
AMM
2491
(1~)# (l'¢') (3)
/r'
where ~b~#is derived as follows
+ ~ $'(r). (4) r
In equation (4), r = I x ( h ) - x(l'T')l and the prime represents a radial derivative. The dynamical matrix is then defined 1
D,#(~z'Ik)= ~ ~ O,#(1~;I'T') x exp{ik.[x(1)-x(l')]} (5)
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AHARON and BROKMAN: VIBRATIONAL MODES IN GOLD TWIST BOUNDARY
G(LO) O.O8
G(LD) 0.08 ~0.06
0.06
~
0.04
0.04
0.02,
o.oog
0.02
/
i
0.000 3 4 COXlO~[sec' ' ] 2
2
3
4
x wo'3Esec-']
Fig. 2. The number of local modes G(w) as a function of frequency. (a) The modes counted in the histograms are the one with large eigenvector components associated with displacements in the first plane adjacent to the boundary pine of one crystal (the spectrum of the first plane in the second crystal is identical by symmetry). (b) The same spectrum obtained for a (002) plane in the bulk. Note that the histogram counts the number of states and not their density so the total area below the curve represents the number of modes with relative large local displacements (see text). where k is the wave vector, and M the gold atomic mass. The potential used in constructing the dynamical matrix was the same as the one used for the molecular static calculation [equation (I)]. Fifty particles located in 10 planes at positions calculated by the molecular static code were used for this calculation. The 150 x 150 dynamical matrix was constructed for various wave vectors (see below) in the Brillouin zone. The dynamical matrix [equation (5)] was diagonalized, as described in the next section, in order to identify the local vibrational modes. 4. DENSITY OF GRAIN BOUNDARY STATES A major difficulty in searching for vibrational modes which are localized at extended defects is the need for repeated diagonalization of a large matrix. This makes the detailed analysis of the density of state outside of our computation capability. Therefore, we adapted an iterative sampling strategy which reduces the number of wave vectors required for the construction of the local density of states: Accordingly, we calculated the density of states g(w) in the boundary for a variable net of wave vectors in the 2/16 irreducible volume of the Briliouin zone. At each iteration we tried to identify a local mode which is not in resonance with the bulk (i.e. a pure mode) by comparing our spectrum with the one obtained from the dynamical matrix constructed separately for the f.c.c, bulk under the same set of wave vectors. Once a pure mode was identified within a small k space volume, the state density was calculated by summing the eigenstates obtained on a finer wave vector grid. In order to illustrate this procedure we examine Fig. 2: here the spectrum confined to the grain boundary is calculated by counting the number of states of large eigenvector components associated with atoms in the two lattice planes forming the grain
boundary. For this purpose, the displacement amplitude of relevant components are defined larger than a quarter of the average displacement (taken over all 150 x 150 components). Figure 2(a), shows the number of states G(w) as calculated for a limited number of wave vectors (16 vectors in the irreducible volume of Brillouin zone). For the sake of comparison, Fig. 2(b) presents the equivalent calculation (same computation cell size and orientation) carried out on two bulk planes. As expected, the noise level in this spectrum is extremely high. However, the example seen in Fig. 2 demonstrates that the high noise level cannot screen the major differences between bulk and grain boundary spectrums, which are analysed next. Two difficulties arise when trying to analyze the localized modes seen in Fig. 2. Firstly, the noise level is too high for detailed statistical analysis, and secondly, these modes are developed resonantly with the bulk and the eigenvectors are therefore perturbed. Overcoming these difficulties is made possible due to the character of the local mode developed at specific wave vectors. Figure 3 shows examples of G(w [k). It is seen that when k equals (1, 0, 0) in the DSC lattice vector (the reciprocal CSL), the associated local mode is pure (out of resonance) and is developed at frequencies within the band gap of the bulk phonons. Therefore, we can investigate the local states by a dense sampling in a limited area of Brillouin zone close to that wave vector. Such analysis of these spectrums implies that a local mode is confined to the grain boundary and is found in the frequency range between 2 x 1013 and 2.2 x 1013s-l. Figure 4 shows the same spectrum calculated for successive planes parallel to the boundary (plane number 1 is closest to the boundary plane). It is noticed that the boundary mode is damped and mixed with the bulk modes at planes away from the boundary. These modes should identify the local vibrational modes which are the subject of the following chapter.
AHARON and BROKMAN: VIBRATIONAL MODES IN GOLD TWIST BOUNDARY
2493
G(~)
G (~)
2,0)
_
m
,I,0)
,0,0)
1,0)
3,OI
0,0) J
0
1
2 3 4 CO xlO t3 [sec"]
5
6
0
I
2 5 4 CO x 1013[sec-1]
5
6
Fig. 3. G(w) vs w (as calculated in Fig. 2) for different wave vectors (DSC) Lattice units, in the plane adjacent to the boundary (a); and in the (002) plane in the bulk (b). It is seen that a pure local mode develops at long wave length [e.g. k = (0, 0, 0), k = (I, 0, 0)] at frequency ~ 2 x I0 ~5s-~. G (~)
0
5. CHARACTERIZATION OF THE LOCAL DISPLACEMENT
1
2
3
4
5
6
× I01s[~c-'] Fig. 4. G(w) plotted for a series of successive planes out of the boundary. It is seen that the boundary mode is weakened and coupled to the other modes when approaching the bulk.
Having identified the local modes from the spectrum calculated above, we can now characterize their origin by considering the relevant eigenvectors. These describe the displacement pattern in the boundary. A closer examination of the numerical data indicates that the boundary mode is unique (we did not identify any other mode localized at the boundary) doubly degenerated according to the bicrystal symmetry, and therefore, two eigenvectors are associated with it. The components of the calculated eigenvector which are associated with atoms in the two planes across the boundary similarity transformation, are drawn in Fig. 5(a). The magnitude of the eigenvector component on the second and higher planes away from the boundary plane are negligible. Therefore, Fig. 5(a) represents the local mode at the boundary. It is seen that the CSL atom is not displaced from its equilibrium position, while the other atoms collectively vibrate in two groups of four atoms--each group of atoms belongs to a plane. In Fig. 5(a) we notice that the two eigenvectors are related by the bi-crystal symmetry group. Now, we linearly combine the two degenerated eigenvectors to obtain the new sets of displacements, seen in Fig. 5(b). Here the displacement of one mode represents a collective rotation of the four atom group around O-lattice
2494
AHARON and BROKMAN: VIBRATIONAL MODES IN GOLD TWIST BOUNDARY
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,
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x
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//(x
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Wigner- Seitz cell
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Fig. 5. The displacement pattern in the two planes forming the boundary. (a) The eigenvector components associated with the relevant atoms. This pattern was calculated by diagonalizing the dynamical matrix. The two patterns shown are associated with the same (degenerated) eigenstate and belong to the same bicrystal symmetry group representation. (b) The same eigenvector components after performing simple linear combination of the two vectors plotted in (a). In this representation the boundary mode consists of a coupled in-plane external libration (top figure) and internal breathing (bottom figure) displacements of the four-atoms around O-lattice point. The circle in (b) represents the magnitude of the r.m.s. displacement in the bundary (magnified x 10). point, and will be described as an external libration (a similar static relaxation was identified previously [22]). The rotation amplitude is approximately 0.9 ° . The displacement associated with the second mode represent the vibration of the four atoms towards the O-lattice point. The later will be described as an inplane breathing mode. it is interesting to notice that each vector remains invariant under the bi-crystal symmetry group. At the same time, however, no element of this group transforms one vector to the other, i.e. the similarity transformation from Fig. 5(a) to (b) reduces the representation of the dynamical matrix. Therefore, we conclude that the libration and breathing modes are accidentally (not by symmetry) degenerated. When considering the total boundary r.m.s, displacement, the bulk perturbations should also be taken into account. Therefore, while calculating the total displacement effect, we considered the cross correlation of two components of the eigenvectors associated with the atoms in question. This correlation is given in the classical limit (hco <
~(u,(lT)up(l'z')) ----~
1
~ k,:
e,(¢lkj)e~(z'lkj) W~('f)
(6)
w h e r e / / = I/KBT, % is the eigenfrequency of t h e j t h mode, e~(z Ikj) are the components of the associated eigenvector and N the number of supercells considered. The summation here is taken over k starvectors associated with the pure local states. The matrix defined by equation (6), contains all the information required, in order to analyze the effect of temperature on the X-ray scattered intensity [10]. In particular, one can calculate the r.m.s, displacement. The latter was found to be 0.022 A at room temperature, and is drawn as a circle in Fig. 5. It should be emphasized here that the displacements and frequencies so calculated are comparable to the bulk ones. Also, a high correlation (0.98 after normalization) was calculated for the in plane displacements which indicates that perturbation due to lattice coupling is small. A smaller value (0.87) was calculated for the case of intra-planes cross-correlation.
6. DISCUSSION We summarize our findings as follows: (a) A boundary mode was identified in gold ~E5 twist boundary. This mode is doubly degenerated and
AHARON and BROKMAN: VIBRATIONAL MODES IN GOLD TWIST BOUNDARY
-
-2o
G 2o 0 (degree)
40
Fig. 6. The potential well for external libration. This potential is calculated in the zeroth-order adiabatic approximation and neglects the coupling to the internal displacement.
confined to the boundary. It propagates in the plane and decays rapidly in the bulk lattice. (b) The boundary mode consists of external libration and internal breathing of the four atoms around the O-lattice point. (c) The local r.m.s, displacement is very close to the bulk one. The breathing mode is a consequence of the large near-neighbour force constant in the plane. The libration is a more complicated mode and is caused by the strong interaction between nearest neighbours across the boundary plane. It has been shown [23] that the nearest neighbours across the boundary plane are related via their position in the O-lattice. Therefore, the four-atom in the plane are displaced in the strong field of the near-neighbour across the plane, imposing the rotating force around an O-lattice point. In a previous work [22], we have recognized that static relaxation consists of rotational displacements in the boundary centered at O-lattice points. This mode of relaxation is motivated by the nearest-neighbour strong interactions across the boundary plane in a similar way as described above. The external libration may be discussed and understood in the frame of the local potential well for rotation. One way of constructing the libration potential well is by applying a zeroth order adiabatic approximation. According to this approximation we freeze all motions that do not participate directly in the libration of their equilibrium positions, and calculate the change in the system's internal energy as a function of the libration amplitude. Figure 6 plots the change in the potential energy due to a static rotation of the four-atom group. The equilibrum position is represented by zero rotation angle. This angle, for which the structure is fully relaxed by the molecular static technique, readily accounts for the static rotational relaxation angle (equals 7 °). In this calculation we considered a 3 x 3 CSL model with 12 (002) planes, where the four-atom-group in the central cell was rotated rigidly. The potential well so calculated is parabolic to a large extent and therefore, the harmonic approximation used in the present work is adequate for the description of local libration. It should be mentioned here that since the rotational
2495
constant (h2/2J, J being the momentum of inertia of the rotating group) is small compared with the potential well depth, the wave function of the libration is localized at the bottom of the potential well, and should be quantized with equidistant energy levels. At high temperature, a rotational jump between nearest minima may occur by thermal fluctuation. It should be emphasized that the zeroth order approximation neglects the coupling to the internal mode. The experimental confirmation of the boundary mode by scattering techniques is difficult since the number of scattering centres is very limited. However, the associated Debye--Waller effect may be recognized in the X-ray scattering from the periodic array of the boundary core. Our calculated r.m.s, displacement value is very close to the bulk value and is similar to the one assumed for ~5 boundary by Majid et al. in the interpretation of their results [9]. This value is smaller than the one measured by Fitzsimmons, Burkel and Sass for the case of ~ 13 boundary [10, 17]. However, Majid et al. [9] have argued that it is possible that the r.m.s, value varies with the degree of coincidence. The boundary mode may affect the physical properties of the bicrystal (or of polycrystal materials in general). Thermal properties (entropy, heat capacity etc.) are now being investigated in the frame suggested above; grain boundary migration may occur as a result of rotational jump of the four-atomgroup from one crystal orientation to the other. In the same manner, a roughening transition may occur at the plane of the boundary, which involves the O-lattice columns. These properties and others are the subject for future investigation. Acknowledgement--This work was supported by the U.S.-Israel Bi-national Science Foundation under Grant No. 00041/85. REFERENCES
I. See for review R. W. Balluffi, in lnterfackd Segregation (edited by W. C. Johnson and J. M. Blakely). Am. Soc. Metals, Metals Park, Ohio (1977). 2. W. Guadig, D. Y. Guan and S. L Sass, Phil. Mag. A34, 923 (1976). 3. D. Y. Guan and S. L. Sass, Phil. Mag. A39, 293 0979). 4. W. Guadig and S. L. Sass, Phil. Mag. A39, 725 (1979). 5. J. Budai, P. D. Bristowe and S. L. Sass, Acta metall. 31, 699 (1983). 6. P. D. Bristowe and A. G. Crocker, Phil. Mag. A3g, 487 (1978). 7. Y. Oh and V. Vitek, Acta metall. 34, 1941 (1986). 8. M. S. Taylor, F. Majid, P. D. Bristowe and R. W. Balluffi, Phys. Rev. B40, 2772 (1989). 9. I. Majid, P. D. Bristowe and R. W. BaUuffi, Phys. Rev. B40, 2779 (1989). 10. M. R. Fitzsimmons, E. Burkel and S. L. Sass, Phys. Rev. Lett. 61, 2237 (1988). I I. M. R. Fitzsimmons and S. L. Sass, Scripta metall. 23, 411 (1989). 12. I. Majid, P. D. Bristowe and R. W. Balluffi, Scripta metall. 23, 1639 (1989) and the following comment by S. L. Sass.
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AHARON and BROKMAN:
VIBRATIONAL MODES IN GOLD TWIST BOUNDARY
13. G. H. Bishop, R. J. Harrison, T. Kwok and S. Yip, Trans. Am. Nucl. Soc. 27, 323 (1977). 14. T. Kwok, P. S. Ho, S. Yip, R. W. Balluffi, P. D. Bristowe and A. Brokman, Phys. Rev. Lett. 47, 1148 (1981). 15. G. Ciccoti, M. Guillope and V. Pontikis, Phys. Rev. B27, 5576 (1983). 16. M. Hashimoto, H. lchinose and Y. Ishida, Japan J. appl. Phys. 19, 1045 (1980). 17. S. M. Foiles, Acta metall. 37, 2815 (1989). 18. Y. Y. Earmme, J. K. Lee, R. J. Harrison and G. H. Bishop, Surf Sci. 92, 174 (1980).
19. D. Cherns, M. W. Finnis, and M. D. Matthews, Phil. Mag. 5, 693 (1977). 20. M. J. Wein, Surf Sci. 31, 138 0972). 21. A.A. Maradudin, E. W. MontroU, G. H. Wi¢ss and F. P. Ipatova, in SolidState Physics (edited by H. Ehrenttich, F. Seitz and D. Turnbull), 2rid edn. Academic Press, New York (1977). 22. A. Brokman and R. W. Balluffi, Acta metall. 29, 1703 (1981). 23. W. Bollmann, Crystal Defects and Crystallyne Interfaces. Springer, Berlin (1970).