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Submitted on 1 Jan 1981
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MEANS SQUARE DISPLACEMENTS OF A SUBSTITUTIONAL DEFECT AND ITS NEAREST
NEIGHBOURS IN KI
S. Duraiswamy, T. Haridasan
To cite this version:
S. Duraiswamy, T. Haridasan. MEANS SQUARE DISPLACEMENTS OF A SUBSTITUTIONAL DEFECT AND ITS NEAREST NEIGHBOURS IN KI. Journal de Physique Colloques, 1981, 42 (C6), pp.C6-899-C6-901. �10.1051/jphyscol:19816267�. �jpa-00221358�
JOURNAL DE PHYSIQUE
Co Lloque C6, supp Lgrnent- au n o 12, Tome 42, d6cembre 1981 page C6-899
MEANS SQUARE DISPLACEMENTS OF A SUBSTITUTIONAL DEFECT AND I T S NEAREST NEIGHBOURS I N K I
S. Duraiswarny and T.M. Haridasan
SchooL o f P h y s i c s , Madurai Kamaraj U n i v e r s i t y , Madurai 625021, I n d i a
Abstract.- The means square displacements of both the cation and anion substitutional defects and those of their nearest neighbours in KI are calculated in the frame work of a scattering matrix formalism. Our results .how that $he means square amplitudes of the defect atom such as the Rb in Kf is much larger than the value of ~ b * in RbI as also of K in KI. The means square displacements of the nearest neigh- bours in the defect environment are not that significanltly altered from their values in the host crystal environment.The other salient features observed from our calculations are also discussed.
1.Introduction.- hwenthough many aspects associted with point defects in alkalihalides are well investigated both theoretically and experimentally in the past, the means square displacements of point defects and their immediate neighbours are not explored that extensively so far. In this paper we report the results of our investigations on the means square displacements of ~ a + , ~ b + , ~ l - and Br- impurities and those of their nearest neighbours in KI using the lattice Green's functions in the scattering matrix formalism.
2.Method of Calculation.- It is well known that the impurity modes associated with a point defect in as alkalihalide is given by the solutions of the determinantal equation 1
-
) ( uJ )\
-0for various d values. However fo1;owing t e same theory one can formulate a scattering matrix and show that the matrix associated with the scattered amplitude U, of the defect and its neighbours can be written corresponding to the incident plane wave as 1
a, = r l t g ~ ~ ~ - 4 ~ j - 1 3 - -
U; ( 1 )Here g is the Green's function matrix of the host lattice in the
-
defect space constituted by the impurity and its first six neigh- bours.
'5
is the matrix desribing the perturbation in the same space due to the presence of the defect. So knowing&,
g and the plane0 N
wave amplitude u(q,j) for the incident mode(q,j) one can compute ( K , d ) I
I and then obtain the means square amplitudes and the Debye Waller factors as
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19816267
C6-900 JOURNAL DE PHYSIQUE
The elements of @ matrix in the (21x21) defect space are available from the early work on impurity modes2 .The elements of $1 matrix are also available likewiee. The relevant elements of g matrix are evaluated ueing the phonon frequencies and ei envectors from a shell model fitted to nuetron scattering data
$ .
The $1 matrix involve only one unknown parameterdA,the change of force conetant between the defect and its first neighbour. These are fitted to the experimental gap modes4. The values o f A A obtained are given in Table 1. The u(q,j) are again fed from the shell model used in the estimation of g. Then using Eqs 2 and 3V
<
u:(K)> and B ( K ~for the impurity ion and their first neighboure are computed.
3.Results and discussion.- The results of our calculations are given in Tables 2 and3. The main conclusions are the following.
a) The B factor for the impurity is in general larger than the corresponding B factor of the host atoms.For example B factor vf Rb in KI is more than that of K in KI .Also it is larger than that of Rb in RbI.
b) The E factor of Na+ in KI is greater than the B factor of ~ b + in KI whereas for the negntive ion impurities the trend is reversed.
c)The B factor of the first neighbour of the defect is different from the B factor of the same ion in the host crystal environment.
However the difference in these two B factors is not that prominent.
d) The occurances of possible resonances in ( I Z ~ )-I may be reeponsible for getting larger B factors for the defectd
In calculations such as diffusion where the jump frequeny depende on the jumping atom(which is an impurity) and its neigh- bours one should employ these mean8 square amplitudes. In the absence of experimental results we hope that these estimations would be of use to scattering experimentalists to see how the scattered intensities would be influenced by such defects.
Table 1. Change in force conetant in KX The defect Change in
4
inunitsof 10 dyntsfcm
Table 2. B factors of various defects and nearest neighboure in KI at 300 K in units of
Defect B(Defect )
nearest
ne ighbour )
Table 3. B facotrs in A * for ~ b + in K I at different temperaturee.
Temperature (B+
IRb
inRbI (B ) (B )+Fib in KI I' inK1:Rb (*-)I in KI
References
.-
l.Maradudin.A.A.,Montroll.E.W.,Welss.G.H,and Ipatova.l.P,
Theory of lattice dynamics in the harmonic approximation,2nd Ed, Academic Press, NewYork (1971)
3.Dolling.G.,Cowley.H.A.,Schittenhelm.C. and Thorson.1.M. Phys.
Rev
147,
577 (1966)4.Kalyani.S. and Haridasan.T.M, J.Phys.Chem,Solids 38,735(1977).
