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Coupling of defect-clusters in a linear chain-identical defects
Physica 96B (1979) 116 - 121 ©North-Holland Publishing Company
COUPLING OF DEFECT-CLUSTERS IN A LINEAR CHAIN-IDENTICAL DEFECTS K. P. SRIVASTAVA Department o f Ceramic Engineering
and S. P. OJHA Applied Physics Section, Institute o f Technology, Banaras Hindu University, Varanasi-221005, India
Received 6 February 1978 Revised 9 August 1978
A numerical study on the coupling of two identical defect-clusters of light atoms embedded in a linear chain of heavy atoms taking second neighbour interaction into account has been made. The efficacy of the rule established by Rosenstock and McGiUfor the determination of defect frequencies has been discussed. On the basis of the present study a method has been suggested for computing defect frequencies very accurately. 1. Introduction
[4]. Very interesting results were obtained by including second neighbour interaction [3]. The positions as well as intensities of peaks are greatly affected by varying the strength of second neighbour interaction. Tfiese findings were attributed to the fact that inclusion of second neighbour interaction couples various light atom clusters and the number of possible configurations of defect-clusters increases. No study has been carried out on the coupling of defect-clusters separated by heavy atoms embedded in a linear chain in next-nearest neighbour interaction. In this paper a numerical study on the coupling of two identical defect-clusters of light atoms is presented. Moreover, the dependence of their frequencies on the strength of second neighbour force is also studied. Using the rule established by Rosenstock and McGill [1] a method has been suggested for computing the defect-frequencies very accurately.
Rosenstock and McGill [1] established an approximate rule, while computing the defect-frequencies, that a group of light atoms surrounded by heavy atoms will produce localized modes if and only if the cluster is bounded by two or more heavy atoms. The atoms outside the defect-region can be ignored and might be considered as rigid boundaries separating the defectregion. Thus, the localized vibrations, if they exist in large lattices, can be completely analyzed by the study of a much smaller lattice containing the defectcluster under fixed boundary conditions. By including second neighbour interaction in the calculation of defect-frequencies, Sah and Srivastava [2] found that four heavy atoms at each side of the defect-cluster are sufficient to give localized modes and, thus, their accurate determination is possible. Inaccuracy involved in the determination of lower modes can be reduced by increasing the number of heavy atoms surrounding the defect. The rule has been employed recently by Srivastava and Ojha [3] for dealing with the dependence of defect-frequencies over strength of second neighbour force. The defect-frequencies have been used for construc. tion of the frequency spectra of disordered lattices
2. Theory and procedure A linear chain of N atoms having the defect-cluster in the middle has been considered. I f m i (= m L or mH) denotes the mass and u i the displacement of the ith atom from equilibrium position, the equations of 116
K. P. Srivastava and S. P. Ojha/Coupling o f identical defect-clusters
motion for the system are mifti = X(ui- 1 - 2ui + ui + 1)
+/a(ui_ 2 - 2 u i + u i + 2 ) ( i =
1,2,3 . . . . . At),
(1)
where X and ta are the nearest and next-nearest neighbour force constants, respectively. It is assumed that these are related by the relation 3, + 4/a = constant, so the static elastic constant of the chain is the same for every value of c (=/a/X). The relation (2)
(t i = - 6 0 2 u i
and the transformation (3)
U i = m~i2ui,
gives the set of equations 7i Ui - 2 + [Ji Ui - 1 + (°~i - 602) Ui + ~i + 1 Ui + 1 +Ti+2Ui+2=O(fori=l,2,3
(4)
..... N)
with ~1 = ")'1 = ")'2 = 0 (fixed boundary condition). 602 being the squared angular frequency (= 4rr2v2) of the system including the defect-region and cti = 2(X + t2)/mi, lJi = - X / ( m i mi _ 1)1/2, 7i = - u / ( m i m i - 2) 1/2.
(5)
The condition for the set of equations (4) to have a non-trivial solution is [~2 - 602~ = 0,
(6)
where the eigenvalues of the dynamical matrix
fi=
/32 °t2 /33 3'4 132 73 r3 a3 ~4 ")'5
OC~N¢t (7)
N-1
N-1
will give frequencies of the whole system.
117
This is a (N × N) symmetric matrix of band width five. The eigenvalues of these matrices were computed on IBM-7044 computer. The programme first calculates the coefficients of the characteristic polynomial by calculating N powers of the matrices and their traces. The roots of the polynomial are calculated by NewtonRaphson method. The programme is in double precision and takes nearly 13 seconds for calculating the eigenvalues of a matrix of order (15 × 15). The defect-clusters - L - , - L L - , -LHJL - and - L L H / - L L - (for/" = 1, 2 and 3) have been investigated. The calculations were performed at a finite mass ratio ( m H / m L = 3) by keeping X + 4/a equal to constant (= 3). The maximum mass and nearestneighbour force constant are so chosen that the maximum frequency [602max(H)] of the chain of heavy atoms becomes 4.0. Only those frequencies are considered which lie above the maximum frequency corresponding to the chain of heavy atoms. The localized frequencies of a defect-cluster c~ are denoted by co2(H- 9) for k = 1,2 . . . . ,1 where 1 stands for the number of light atoms forming the defect.
3. Results and discussion
The localized frequencies of the clusters - L - and - L H / L - (j = 1,2 and 3) for different values of c are presented in the table I. The former cluster gives only one localized frequency whereas the later clusters give two frequencies for every value of/'. The frequencies o f - L H J L - are designated by co21(H-LH/L) and 602(H-LH/L). These two frequencies lie close to each other and their differences go on decreasing with increasing j. For j equal to three, the difference between co2(H-LHiL) and 602(H-LH/L) becomes very small and their average value, denoted by 602 (H - LH/L), approaches 602(H-L). The frequen~es co2 v (H - LH/L) are compared with co2(H-L) for all the three values off' and are listed in table I. The dependence of 602(H-LH/L) and co2(HLH/L) over] gives an idea about the coupling between two single - L - clusters present in the defect -LHJL - . For lower values of] (j = 1 and 2) a marked difference in co2(H-LH/L) and co2(H-LHJL) is observed, which suggests an existence of coupling between the two single - L - clusters appeared in -LHJL - . For/' equal to 3, no substantial difference in their values is
118
K. P. Srivastava and S. P. O]ha/Coupling o f identical defect-clusters
I
;>
3
I
3
I
I
> 3
I
3
I
I
3 *I ~J
I
i
3
I I
I
I
g., 3 0 o d o d d N N d N
.
observed. Thus, an obvious conclusion from this observation is that the coupling between two single - L - clusters ceases if they are separated by three or more heavy atoms and hence they behave independently. The duster - L L - is known to give two localized frequencies if placed in a chain of heavy atoms. If two - L L - clusters, separated by Hj (j = 1,2 and 3) heavy atoms, are placed in the some chain, four localized frequencies are obtained. These frequencies, denoted by ¢o2(H-~), 6o2(H-~]), w 2 ( H - ~ j ) a n d ¢o2(H-~;), are shown in tables IIA and liB. It can also be seen from these two tables that the two localized frequencies ¢o2(H-LL) and ¢o2(H-LL) of the - L L - cluster lie close to 6o2(H-~i) and ¢o2(H-~,.), respectively. Here the defect-cluster ~ j denotes the cluster - L L H / L L- . The frequencies ¢o~(H-~j) and w2(H-~;)~_ - - - are the average values of w~(H-~j) and w2(H-~j), and w2(H - ~j) and w2(H -,~,~!, respectively. The difference in values between ¢Ol(H-~/) and 6o2(H- ~/) decreases with increasing the number of heavy atoms. Table IIA shows the variation of co2(H-~j) and ~2(H-~/.) withj values. Similarly, 2 the variation of w 23 ( H - g / ) and 6o4(H-~/) for the different values o f j is presented in the table IIB. There is a significant difference between the results of tables IIA and IIB, although they correspond to the localized frequencies of the same cluster - L L - . The frequencies w2(H-~i) come closer to ~o2(H-LL) asj increases and atj equal to 3, the difference between these two frequencies becomes almost zero for nearly all values of c (table IIA). These frequencies correspond to the highest localized mode of the cluster - L L - . The frequency corresponding to the lower localized mode of the cluster - L L - shows somewhat different behaviour. In this case, a marked difference is observed in the values of w 2 ( n - L L ) and w2(H~/) even if the clusters are separated by three heavy atoms, i.e. forj equal to three. This suggests that the lower frequencies can not be determined accurately, if the clusters are separated by only three heavy atoms. In order to maintain the accuracy of the result for lower localized modes one has to increase the number of heavy atoms. With an increase in the number of heavy atoms separating the clusters, the coupling between clusters will decrease and the clusters would be less affected due to each other; it seems as if these identical clusters detach from their original configuration and, thus, their presence does not affect each other.
9.4124 8.0504 7.3721 6.6966 6.1586 5.6237 5.0928 4.5675 4.0499
0 0.050 0.083 0.125 0.167 0.219 0.286 0.375 0.500
9.7775 8.2427 7.5254 6.8022 6.1590 5.7108 5.1842 4.7318 4.2990
j= 1
9.0423 7.8573 7.2168 6.5890 6.1570 5.5350 5.0018 4.4016 3.8012
4.4994 3.9502 3.6861 3.4306 3.2328 3.0415 2.8573 2.6808 2.5121
0 0.050 0.083 0.125 0.167 0.218 0.286 0.375 0.500
* Denoted by ~j.
~I(H-LL)
c
9.4099 8.0500 7.3711 6.6956 6.1580 5.6229 5.0930 4.5667 4.0501
4.9026 4.1680 3.8780 3.6069 3.2858 3.0778 2.8593 2.7135 2.7844
3.9708 3.6248 3.3742 3.2047 3.1764 3.0020 2.8519 2.4871 2.2130
4.4367 3.8964 3.6261 3.4058 3.2311 3.0399 2.8556 2.6003 2.4977
J=1 o-)] ( H - ~ j ) t o ] ( H - ~ j ) t o ~ i ( H - ~
Table IIB Localized frequencies of clusters - L L - and - LLHJLL - *
* Denoted by ~/.
~(H-LL)
c
Table IIA Localized frequencies of clusters - L L - and - L L H I L L - *
j) 4.6162 4.0341 3.7484 3.4672 3.2462 3.0532 2.8934 2.7379 2.5869
J=2 t~(H-~j)
9.4385 8.0543 7.3782 6.7115 6.1795 5.6492 5.2111 4.5960 4.0750
j: 2
9.4123 8.0504 7.3722 6.6966 6.1587 5.6238 5.0931 4.5679 4.0508
4.2980 3.8276 3.6033 3.3862 3.2175 3.0297 2.8187 2.6147 2.4198
4.4571 3.9309 3.6759 3.4277 3.2319 3.0415 2.8561 2.6763 2.5034
to~(H-~j)to2i(H-
9.3860 8.0464 7.3661 6.6817 6.1378 5.5984 5.0650 4.5398 4.0265
~ j)
9.4088 8.0503 7.3710 6.6951 6.1575 5.6237 5.0908 4.5621 4.0396
4.5609 3.9846 3.7058 3.4358 3.2379 3.0546 2.8752 2.6998 2.5297
4.4129 3.9084 3.6640 3.4250 3.2277 3.0281 2.8386 2.6604 2.4945
4.4869 3.9465 3.6849 3.4304 3.2328 3.0414 2.8569 2.6801 2.5116
9.4124 8.0504 7.3721 6.6966 6.1588 5.6238 5.0929 4.5675 4.0449
f=3 eo~(H-~j)~(H-~j)w21(H-
9.4160 8.0504 7.3732 6.6980 6.1600 5.6238 5.0949 4.5728 4.0602
j= 3
6~flj )
120
K. P. Srivastava and S. P. O/ha/Coupling o f identical defect-clusters
~12 (H-LL HILL) =2 CH'LLHJLL )
j.I
j-2
j =3 - -
4 ( H-I.L )
~
t,a2 (. H-L )
/
~12 (H-LHJL) =2 CH'LHJL .)
f
~
G~ (H-LLHJLL)
~.(H-LL ) ~2 (H_LLHJLL)
_ / f - -
Fig. 1. Splitting of levels for defect-clusters- L - and - L L - with varying/. The coupling between two identical clusters is depicted in fig. 1. The level schemes o f - L - , - L L - , - L H / L - and - LLH/LL- for all values of/(equal to 1, 2 and 3) are shown. The upper and lower levels correspond to frequencies of - L L - and - L L H / L L whereas the middle levels to that o f - L - - and - L H f L clusters. As can be seen from fig. 1, the levels corresponding either to - L - or - L L - come close to each other as] increases and, finally, for j equal to three, they merge into each other. At this stage, no appreciable difference is observed between the levels o f - L and - L H / L - , and also between - L L - and - L L H / L L - . Moreover, the splitting of levels for lower modes is comparatively larger than that of higher localized modes. However, with increasing j the interval between levels for a particular frequency decreases and, finally, the levels merge into each other and their average values become nearly equal to that of the original level. From tables I and II, it can be seen that there is a lowering in values of defect-frequencies with increasing c, a fact which has already been reported in our earlier paper [3]. The splitting of levels due to the occurrence of two identical clusters continues to appear and the trend remains unchanged even if the strength of second neighbour force is varied. This may be attributed to the fact that the amount of second neighbour force does not alter the configuration of the defect-region and hence the coupling between two identical light atom clusters remains unaffected. Besides the results discussed in the preceding
sections, this study also gives a method for calculating defect-frequencies very accurately. Rosenstock and McGill [1] calculated the frequencies by placing the cluster in a heavy chain having at least three heavy atoms at each side of the defect. It is evident from the tables I and II, and also from fig. 1, that the levels of -LHJL - and - L L H / L L - split in such a way that their average values he near to the frequencies of - L - and - L L - clusters, respectively. Thus, if correct values of defect-frequencies are required for a cluster - 9 - , it would be worthwhile to consider the cluster - g H / ~ instead of treating the cluster -~--separately. The average values of frequencies corresponding to the duster - ~ - - Hi~ - will be more accurate (even for lower values off) as compared to the frequencies of the cluster - 9 - placed alone in the chain of heavy atoms. Moreover, the clusters separated by only one heavy atom will yield an average value accurate enough for the construction of frequency spectrum of disordered chains. The difference observed in frequencies, thus calculated, for different ] is always less than 0.01, less than the mesh size used in plotting the frequency spectrum and the difference involved does not affect the spectrum at all. Extension of this work would be useful in many directions. Numerical work should be performed for higher dimensions and with more realistic models. Particularly, the study of unlike defects should also be made, as the occurrence of unlike defects in a disordered two-component chain is equally probable.
K. P. Srivastava and S. P. Ojha/Coupling o f Men tical defect-clusters
Acknowledgements
References
The authors wish to express their sincere thanks to Professor P. Nath for encouragement and keen interest. One of us (KPS) is grateful to the Council of Scientific and Industrial Research (India) for financial assistance.
[1] H.B. Rosenstock and R. E. McGill,Phys. Rev. 176 (1968) 1004. [2] P. Sah and K. P. Srivastava, Physica 48 (1970) 146. [3] K.P. Srivastavaand S. P. Ojha, Canadian J. Phys. 56 (1978) 763. [4] F. Brouers and J. Deltour, Physica 37 (1967) 139. [5] J. L. Martin, Proc. Roy. Soc. A260 (1961) 139. F. Brouers and J. Deltour, Physica 37 (1967) 139. P. Sah and K. P. Srivastava,Physica 45 (1970) 537.
121
COUPLING OF DEFECT-CLUSTERS IN A LINEAR CHAIN-IDENTICAL DEFECTS K. P. SRIVASTAVA Department o f Ceramic Engineering
and S. P. OJHA Applied Physics Section, Institute o f Technology, Banaras Hindu University, Varanasi-221005, India
Received 6 February 1978 Revised 9 August 1978
A numerical study on the coupling of two identical defect-clusters of light atoms embedded in a linear chain of heavy atoms taking second neighbour interaction into account has been made. The efficacy of the rule established by Rosenstock and McGiUfor the determination of defect frequencies has been discussed. On the basis of the present study a method has been suggested for computing defect frequencies very accurately. 1. Introduction
[4]. Very interesting results were obtained by including second neighbour interaction [3]. The positions as well as intensities of peaks are greatly affected by varying the strength of second neighbour interaction. Tfiese findings were attributed to the fact that inclusion of second neighbour interaction couples various light atom clusters and the number of possible configurations of defect-clusters increases. No study has been carried out on the coupling of defect-clusters separated by heavy atoms embedded in a linear chain in next-nearest neighbour interaction. In this paper a numerical study on the coupling of two identical defect-clusters of light atoms is presented. Moreover, the dependence of their frequencies on the strength of second neighbour force is also studied. Using the rule established by Rosenstock and McGill [1] a method has been suggested for computing the defect-frequencies very accurately.
Rosenstock and McGill [1] established an approximate rule, while computing the defect-frequencies, that a group of light atoms surrounded by heavy atoms will produce localized modes if and only if the cluster is bounded by two or more heavy atoms. The atoms outside the defect-region can be ignored and might be considered as rigid boundaries separating the defectregion. Thus, the localized vibrations, if they exist in large lattices, can be completely analyzed by the study of a much smaller lattice containing the defectcluster under fixed boundary conditions. By including second neighbour interaction in the calculation of defect-frequencies, Sah and Srivastava [2] found that four heavy atoms at each side of the defect-cluster are sufficient to give localized modes and, thus, their accurate determination is possible. Inaccuracy involved in the determination of lower modes can be reduced by increasing the number of heavy atoms surrounding the defect. The rule has been employed recently by Srivastava and Ojha [3] for dealing with the dependence of defect-frequencies over strength of second neighbour force. The defect-frequencies have been used for construc. tion of the frequency spectra of disordered lattices
2. Theory and procedure A linear chain of N atoms having the defect-cluster in the middle has been considered. I f m i (= m L or mH) denotes the mass and u i the displacement of the ith atom from equilibrium position, the equations of 116
K. P. Srivastava and S. P. Ojha/Coupling o f identical defect-clusters
motion for the system are mifti = X(ui- 1 - 2ui + ui + 1)
+/a(ui_ 2 - 2 u i + u i + 2 ) ( i =
1,2,3 . . . . . At),
(1)
where X and ta are the nearest and next-nearest neighbour force constants, respectively. It is assumed that these are related by the relation 3, + 4/a = constant, so the static elastic constant of the chain is the same for every value of c (=/a/X). The relation (2)
(t i = - 6 0 2 u i
and the transformation (3)
U i = m~i2ui,
gives the set of equations 7i Ui - 2 + [Ji Ui - 1 + (°~i - 602) Ui + ~i + 1 Ui + 1 +Ti+2Ui+2=O(fori=l,2,3
(4)
..... N)
with ~1 = ")'1 = ")'2 = 0 (fixed boundary condition). 602 being the squared angular frequency (= 4rr2v2) of the system including the defect-region and cti = 2(X + t2)/mi, lJi = - X / ( m i mi _ 1)1/2, 7i = - u / ( m i m i - 2) 1/2.
(5)
The condition for the set of equations (4) to have a non-trivial solution is [~2 - 602~ = 0,
(6)
where the eigenvalues of the dynamical matrix
fi=
/32 °t2 /33 3'4 132 73 r3 a3 ~4 ")'5
OC~N¢t (7)
N-1
N-1
will give frequencies of the whole system.
117
This is a (N × N) symmetric matrix of band width five. The eigenvalues of these matrices were computed on IBM-7044 computer. The programme first calculates the coefficients of the characteristic polynomial by calculating N powers of the matrices and their traces. The roots of the polynomial are calculated by NewtonRaphson method. The programme is in double precision and takes nearly 13 seconds for calculating the eigenvalues of a matrix of order (15 × 15). The defect-clusters - L - , - L L - , -LHJL - and - L L H / - L L - (for/" = 1, 2 and 3) have been investigated. The calculations were performed at a finite mass ratio ( m H / m L = 3) by keeping X + 4/a equal to constant (= 3). The maximum mass and nearestneighbour force constant are so chosen that the maximum frequency [602max(H)] of the chain of heavy atoms becomes 4.0. Only those frequencies are considered which lie above the maximum frequency corresponding to the chain of heavy atoms. The localized frequencies of a defect-cluster c~ are denoted by co2(H- 9) for k = 1,2 . . . . ,1 where 1 stands for the number of light atoms forming the defect.
3. Results and discussion
The localized frequencies of the clusters - L - and - L H / L - (j = 1,2 and 3) for different values of c are presented in the table I. The former cluster gives only one localized frequency whereas the later clusters give two frequencies for every value of/'. The frequencies o f - L H J L - are designated by co21(H-LH/L) and 602(H-LH/L). These two frequencies lie close to each other and their differences go on decreasing with increasing j. For j equal to three, the difference between co2(H-LHiL) and 602(H-LH/L) becomes very small and their average value, denoted by 602 (H - LH/L), approaches 602(H-L). The frequen~es co2 v (H - LH/L) are compared with co2(H-L) for all the three values off' and are listed in table I. The dependence of 602(H-LH/L) and co2(HLH/L) over] gives an idea about the coupling between two single - L - clusters present in the defect -LHJL - . For lower values of] (j = 1 and 2) a marked difference in co2(H-LH/L) and co2(H-LHJL) is observed, which suggests an existence of coupling between the two single - L - clusters appeared in -LHJL - . For/' equal to 3, no substantial difference in their values is
118
K. P. Srivastava and S. P. O]ha/Coupling o f identical defect-clusters
I
;>
3
I
3
I
I
> 3
I
3
I
I
3 *I ~J
I
i
3
I I
I
I
g., 3 0 o d o d d N N d N
.
observed. Thus, an obvious conclusion from this observation is that the coupling between two single - L - clusters ceases if they are separated by three or more heavy atoms and hence they behave independently. The duster - L L - is known to give two localized frequencies if placed in a chain of heavy atoms. If two - L L - clusters, separated by Hj (j = 1,2 and 3) heavy atoms, are placed in the some chain, four localized frequencies are obtained. These frequencies, denoted by ¢o2(H-~), 6o2(H-~]), w 2 ( H - ~ j ) a n d ¢o2(H-~;), are shown in tables IIA and liB. It can also be seen from these two tables that the two localized frequencies ¢o2(H-LL) and ¢o2(H-LL) of the - L L - cluster lie close to 6o2(H-~i) and ¢o2(H-~,.), respectively. Here the defect-cluster ~ j denotes the cluster - L L H / L L- . The frequencies ¢o~(H-~j) and w2(H-~;)~_ - - - are the average values of w~(H-~j) and w2(H-~j), and w2(H - ~j) and w2(H -,~,~!, respectively. The difference in values between ¢Ol(H-~/) and 6o2(H- ~/) decreases with increasing the number of heavy atoms. Table IIA shows the variation of co2(H-~j) and ~2(H-~/.) withj values. Similarly, 2 the variation of w 23 ( H - g / ) and 6o4(H-~/) for the different values o f j is presented in the table IIB. There is a significant difference between the results of tables IIA and IIB, although they correspond to the localized frequencies of the same cluster - L L - . The frequencies w2(H-~i) come closer to ~o2(H-LL) asj increases and atj equal to 3, the difference between these two frequencies becomes almost zero for nearly all values of c (table IIA). These frequencies correspond to the highest localized mode of the cluster - L L - . The frequency corresponding to the lower localized mode of the cluster - L L - shows somewhat different behaviour. In this case, a marked difference is observed in the values of w 2 ( n - L L ) and w2(H~/) even if the clusters are separated by three heavy atoms, i.e. forj equal to three. This suggests that the lower frequencies can not be determined accurately, if the clusters are separated by only three heavy atoms. In order to maintain the accuracy of the result for lower localized modes one has to increase the number of heavy atoms. With an increase in the number of heavy atoms separating the clusters, the coupling between clusters will decrease and the clusters would be less affected due to each other; it seems as if these identical clusters detach from their original configuration and, thus, their presence does not affect each other.
9.4124 8.0504 7.3721 6.6966 6.1586 5.6237 5.0928 4.5675 4.0499
0 0.050 0.083 0.125 0.167 0.219 0.286 0.375 0.500
9.7775 8.2427 7.5254 6.8022 6.1590 5.7108 5.1842 4.7318 4.2990
j= 1
9.0423 7.8573 7.2168 6.5890 6.1570 5.5350 5.0018 4.4016 3.8012
4.4994 3.9502 3.6861 3.4306 3.2328 3.0415 2.8573 2.6808 2.5121
0 0.050 0.083 0.125 0.167 0.218 0.286 0.375 0.500
* Denoted by ~j.
~I(H-LL)
c
9.4099 8.0500 7.3711 6.6956 6.1580 5.6229 5.0930 4.5667 4.0501
4.9026 4.1680 3.8780 3.6069 3.2858 3.0778 2.8593 2.7135 2.7844
3.9708 3.6248 3.3742 3.2047 3.1764 3.0020 2.8519 2.4871 2.2130
4.4367 3.8964 3.6261 3.4058 3.2311 3.0399 2.8556 2.6003 2.4977
J=1 o-)] ( H - ~ j ) t o ] ( H - ~ j ) t o ~ i ( H - ~
Table IIB Localized frequencies of clusters - L L - and - LLHJLL - *
* Denoted by ~/.
~(H-LL)
c
Table IIA Localized frequencies of clusters - L L - and - L L H I L L - *
j) 4.6162 4.0341 3.7484 3.4672 3.2462 3.0532 2.8934 2.7379 2.5869
J=2 t~(H-~j)
9.4385 8.0543 7.3782 6.7115 6.1795 5.6492 5.2111 4.5960 4.0750
j: 2
9.4123 8.0504 7.3722 6.6966 6.1587 5.6238 5.0931 4.5679 4.0508
4.2980 3.8276 3.6033 3.3862 3.2175 3.0297 2.8187 2.6147 2.4198
4.4571 3.9309 3.6759 3.4277 3.2319 3.0415 2.8561 2.6763 2.5034
to~(H-~j)to2i(H-
9.3860 8.0464 7.3661 6.6817 6.1378 5.5984 5.0650 4.5398 4.0265
~ j)
9.4088 8.0503 7.3710 6.6951 6.1575 5.6237 5.0908 4.5621 4.0396
4.5609 3.9846 3.7058 3.4358 3.2379 3.0546 2.8752 2.6998 2.5297
4.4129 3.9084 3.6640 3.4250 3.2277 3.0281 2.8386 2.6604 2.4945
4.4869 3.9465 3.6849 3.4304 3.2328 3.0414 2.8569 2.6801 2.5116
9.4124 8.0504 7.3721 6.6966 6.1588 5.6238 5.0929 4.5675 4.0449
f=3 eo~(H-~j)~(H-~j)w21(H-
9.4160 8.0504 7.3732 6.6980 6.1600 5.6238 5.0949 4.5728 4.0602
j= 3
6~flj )
120
K. P. Srivastava and S. P. O/ha/Coupling o f identical defect-clusters
~12 (H-LL HILL) =2 CH'LLHJLL )
j.I
j-2
j =3 - -
4 ( H-I.L )
~
t,a2 (. H-L )
/
~12 (H-LHJL) =2 CH'LHJL .)
f
~
G~ (H-LLHJLL)
~.(H-LL ) ~2 (H_LLHJLL)
_ / f - -
Fig. 1. Splitting of levels for defect-clusters- L - and - L L - with varying/. The coupling between two identical clusters is depicted in fig. 1. The level schemes o f - L - , - L L - , - L H / L - and - LLH/LL- for all values of/(equal to 1, 2 and 3) are shown. The upper and lower levels correspond to frequencies of - L L - and - L L H / L L whereas the middle levels to that o f - L - - and - L H f L clusters. As can be seen from fig. 1, the levels corresponding either to - L - or - L L - come close to each other as] increases and, finally, for j equal to three, they merge into each other. At this stage, no appreciable difference is observed between the levels o f - L and - L H / L - , and also between - L L - and - L L H / L L - . Moreover, the splitting of levels for lower modes is comparatively larger than that of higher localized modes. However, with increasing j the interval between levels for a particular frequency decreases and, finally, the levels merge into each other and their average values become nearly equal to that of the original level. From tables I and II, it can be seen that there is a lowering in values of defect-frequencies with increasing c, a fact which has already been reported in our earlier paper [3]. The splitting of levels due to the occurrence of two identical clusters continues to appear and the trend remains unchanged even if the strength of second neighbour force is varied. This may be attributed to the fact that the amount of second neighbour force does not alter the configuration of the defect-region and hence the coupling between two identical light atom clusters remains unaffected. Besides the results discussed in the preceding
sections, this study also gives a method for calculating defect-frequencies very accurately. Rosenstock and McGill [1] calculated the frequencies by placing the cluster in a heavy chain having at least three heavy atoms at each side of the defect. It is evident from the tables I and II, and also from fig. 1, that the levels of -LHJL - and - L L H / L L - split in such a way that their average values he near to the frequencies of - L - and - L L - clusters, respectively. Thus, if correct values of defect-frequencies are required for a cluster - 9 - , it would be worthwhile to consider the cluster - g H / ~ instead of treating the cluster -~--separately. The average values of frequencies corresponding to the duster - ~ - - Hi~ - will be more accurate (even for lower values off) as compared to the frequencies of the cluster - 9 - placed alone in the chain of heavy atoms. Moreover, the clusters separated by only one heavy atom will yield an average value accurate enough for the construction of frequency spectrum of disordered chains. The difference observed in frequencies, thus calculated, for different ] is always less than 0.01, less than the mesh size used in plotting the frequency spectrum and the difference involved does not affect the spectrum at all. Extension of this work would be useful in many directions. Numerical work should be performed for higher dimensions and with more realistic models. Particularly, the study of unlike defects should also be made, as the occurrence of unlike defects in a disordered two-component chain is equally probable.
K. P. Srivastava and S. P. Ojha/Coupling o f Men tical defect-clusters
Acknowledgements
References
The authors wish to express their sincere thanks to Professor P. Nath for encouragement and keen interest. One of us (KPS) is grateful to the Council of Scientific and Industrial Research (India) for financial assistance.
[1] H.B. Rosenstock and R. E. McGill,Phys. Rev. 176 (1968) 1004. [2] P. Sah and K. P. Srivastava, Physica 48 (1970) 146. [3] K.P. Srivastavaand S. P. Ojha, Canadian J. Phys. 56 (1978) 763. [4] F. Brouers and J. Deltour, Physica 37 (1967) 139. [5] J. L. Martin, Proc. Roy. Soc. A260 (1961) 139. F. Brouers and J. Deltour, Physica 37 (1967) 139. P. Sah and K. P. Srivastava,Physica 45 (1970) 537.
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