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Vibrational and optical properties of the mixed crystal NaCℓ1−xBrx
967—972.
Solid State Coninunications, Vol.32,in pp. Pergansrn Press Ltd. 1979. Printed Great Britain.
*
VIBRATIONAL AND OPTICAL PROPERTIES OF THE MIXED CRYSTAL NaCL l—xBr x K. H. Wanser and R. F. Wallis
Department of Physics, University of California, Irvine, Irvine CA 92717 (Received 30 July 1979 by A. A. Maradudin)
A one—dimensional model is developed for the [111] transverse modes of the mixed crystal NaCi Br • The frequencies and eigenvectors of the normal modes of ~i~~~i 0A are calculated for ten random configurations at each of a number of compositions. The imaginary part of the dielectric function is calculated for the average configuration at each composition. The behavior of the system cannot be classified in terms of the usual one—mode or two—mode categories. Representative eigenvectors are displayed, and the random element Isodisplacement model is discussed in the light of our results.
In the past severa~,years, many studies have been carried out~’ on the vibrational properties of mixed crystals of the type AB l-xCx. A number of methods have been employed in an attempt to understand the properties of these crystals. A summary of both the experimental and theoretical situations has been presented in a review article by Barker and Sievers. A simple model which has had some success in correlating the optical properties of mixed 12 random In thiselement model,isodisplacement the individual crystals is the atomic species are assumed to form sublattices (REt) model. which move rigidly with respect to one another in the modes important for optical absorption. Although the EEl model has been applied to many cases, there has been relatively little work done on assessing the validity of the model on a microscopic level. An investigation which bears on this point, however, is that of Haag et al. who studied the optical spectra of the CdSl—x Sex and Cd lx — Zn S systems using a one— x dimensional model. The normal mode displace— ment patterns which they found are not in ac— cord with the REI model, In the present paper we consider mixed crystals of NaCL and NaBr, NaCL Br , a l—x x system which apparently has not been previously investigated. As in the work of Hass et al. we use a finite one—dimensional model, bu~i~lude both first and second neighbor Interactions and not just first neighbor interactions. Also, we adopt cyclic boundary conditions rather than the problem of harmonics encountered by avoid Hass fixed—end boundary conditions and thus ~
where m~is the mass and u~is the displacement of atom L, w is the frequency ~ and kf,f+l are nearest neighbor force constants, and kL,f_2 and ki f+2 are second neighbor force con— stants. If the halogen of a nearest neighbor pair is CL or Br, the nearest neighbor force constant is taken to be k12 or k13, respective— ly, where k12 and k13 are the nearest neighbor force constants of pure NaCL and NaBr, respec— tively. If the ions of a second neighbor pair are both CL or both Br, the second neighbor force constant is k or k 33, respectively, where k 2 and k a~ the second neighbor force constan~sfor h~ogensin pure NaCL and NaBr, respectively. If one CL ion and one Br ion make up a second neighbor pair, the second neighbor force constant is taken to be k23 (1 — x)k22 + x k33 . (2a) If two Na ions make up a second neighbor pair, the second neighbor force constant is taken to be k11 — (1 — x) ~ + x ~ (2b) where k and ~ are the second neighbor force constants for Na in pure NaCL and NaBr, respectively. The force constants for the pure crystals are 8’9determined for the TA by and fitting TO phonon the experi— frequen— des at data k — 0 and at the zone boundary in the mental [111] direction. Having determined the force constants, we are now able to solve the equations of motion for a given composition of the mixed crystal.
~
The equations of motion can be written in the form m~w2u 1— ~
_ui_1)+ki,t+1(ut_ui+1) +
+ k&,~...2(ui_ui...2) + k~L+2(uf _uL+2)
This yields the normal mode frequencies u and the eigenvector components ~ of the dynLuical matrix. The eigenvector comp&~ents satisfy the orthonormality condition
(1)
*
This research was supported In part by NSF Grant No. DNR—78O943O. 967
968
VIBRATIONAL AND OPTICAL PROPERTIES OF NAC&
Er £ “is is — ~as and the closure condition
~ ~
~i’s~ii’
‘(3)
If one now considers the equations for the in—
the normal modes which give large contributions to the dielectric constant. Let us first con— sider chains with composition Na 36CL9Br27. Here
teracting ions driven by an electromagnetic field of frequency u, one cam calculate the di— electric constant E(w). In the presence of damping characterized by damping constants r5, the result is
and in the following we designate the composi— tion by specifying the number of atoms of each species in the 72 atom chain. In Fig, 1 we plot the imaginary part of the dielectric con—
E(w)
•
1 + 4wn
—
(4)
Br Vol. 32, No. II 1—x x 0 although the damping is known to vary as a function of concentration,6’11’ NaC&,~’ One of our principal objectives has been to investigate the form of the eigenvectors for
~ s
(w
2 2 p5 2
—
iwr )
(5)
—
8
where p
5
5 is given by *
p
E Q~F;~fm~ , £
—
~
(6)
* is the number of chains per unit volume. We is the— effective charge of the ith ion and n take ~r — “~Na — _Q* and focus attention
on the imagi~rypart of E(~)which is propor— tional to nQ . The accuracy of the computer calculations was checked with the aid of two sum rules. The first is configuration dependent and is based on the invariance of the trace of the dynamical matrix, N
(k&,i+
1+k&,i,,,1+ki.&_2)
L—l
m~
(7) where N is the number of ions in the chain. The second is configuration independent and is based on the sum of the squares of the quantities p5, 15N + 1— mCL~ +—!—\ 1~rI . — ~ “~2._ (8) s s 2 I~ In discuasin~ ~ intensity of individual modes; it is convenient to define a mode strength S ~
by
2 4irp 5— 8Nu2
We take
( ,
4ir
~2
*
QLF;Ls)
(l)L+1 and measure
.
(9)
s and m£ in the numerical units of ~0(NaCL) evaluation and mCL of , S~. respectively, in Q~ =
ü
Calculations were made of the eigenvalues and eigenvectors of the dynamical. matrix, the mode strength of each mode, and the imaginary part of the dielectric constant for a number of compositions of the mixed crystal NaCL 1_ xx Br , Chains containing a total of 72 atoms were ~ ployed. For the intermediate compositions, ten centration and the results forused the imaginary different configurations were for each con— part of the dielectric constant were averaged over the configurations of that concentration. Equations (7) and (8) were verified for each case. The damping constants were chosen to be 0.04 c,i which is a realistic value for S
vectors can be associated with the various peaks. In FLg. 2 we show four such eigenvectors associated, with the four peaks identified above. 1 peak is essen— tially a gap mode for an isolated chlorine atom The eigenvector for the 85 cm” immersed in a background of bromine atoms. The eigenvector for the 135 cm”1’ peak is associated with groups of consecutive bromine atoms moving with the same phase but different amplitudes and the interspersed sodium atoms moving with op— posite phase to the bromine atoms. The mode giving rise to the peak at 160 cni” is basic— ally a local mode of an isolated chlorine atom. The eigenvector for the 170 cm”1 peak is associ—
4~&,&+ 24
E ss
stant, E2~5), as a function of frequency for an average of ten random There are two principal peaksconfigurations. at -‘135 cm~’~’and 1 — -‘160 cm 1 and two subsidiary peaks at —85 cm” and 170 cm’~. Particular normal mode eigen—
ated with groups of consecutive chlorine atoms moving with the same phase and the interspersed sodium atoms moving with opposite phase to the chlorine atoms. In each of the eigenvectors discussed above, the motion is characterized by relatively small groups of atoms whose amplitudes of displacement are significant. Such groups of atoms may occur vidual establishes freseveral small times group in thethat chain, but it isthethe mdi— quency to a good approximation. The importance of small groups of atoms has also been empha— sized b 1 Hass et al. and by Rosenstock and McGill. 2 —— Let us now turn to chains with composition Na36C&18Br18. In Fig. 3 is plotted E2(w) versus frequency for an average of ten random configurations. Just as with the previous composition, there are p~aksin the vicinity of 85, l35,l16O~ 1’ and 170 cm” ; however the peak near 170 cm is considerably considerably weaker stronger than and the corresponding that near 135 cm” peaks for the previous composition. Typical eigemvectors associated with these four peaks are shown in Fig. 4. The similarity of the displacement patterns to those for the Na36CL9Br27 composition is evident. In addition to these four peaks, however, the composition 1’. exhibits Eigenvectors Na36CL18Br18 also smallassociated peaks nearwith the latter 105 and 150two cm’ peaks are also shown in Fig, 4. In both cases the eigenvectors are character— ized by a group of three consecutive CL atoms which have relatively large amplitudes, but do not have the sane phase. The Na atoms adjacent to these CL atoms also have relatively large
Vol. 32, No. 11
VIBRATIONAL AND OPTICAL PROPERTIES OF NACLI_X Br~ 4.5
u
969
u
4.0 3.5
-
3.0 .~
2.5-
~
hull
20
40 Fig. 1’.
60
Imaginary part of the dielectric constant as a function of frequency for an average of ten random configurations of Na 36CL9Br27.
(a)
(b) w~l34.5Cm” 5-2.50
w.82.4Cm” S~0.075
5
II II
~~
.:...~.i.lIIlllIl1IlIi.......:.i.i..:
,.
~
Cc)
(d) II II II
I
w.159.7Cm”
II II II
~l~l ~]l
°
liii
11111 :~,.
ii
z
200 220
80 100 120 140 160 180 FREQUENCY w (cm~)
S-2.44 ~I
U-
iii
167.5cm”
5-1.13
III
~l~I
—
tftftt$t1fi~ttttfIff$ttIftt~fttttIli LATTICE SITE Pig. 2.
$Ittitit$ttftt~~$Utttl$ LATTICE SITE
Eigenvectore of four major infrared— active modes of Na36Ct9Br27.
970
VIBRATIONAL AND OPTICAL PROPERTIES OF NACL
1—x
Br
Vol. 32, No. 11
x
2.5
2.0
1.5 C”
w
1.0
0.5
-
0.0 20
-
40
60
80
100
120
FREQUENCY
Fig. 3.
140
160
180 200 220
w (cm’)
Imaginary part of the dielectric constant as a function of frequency for an average of ten random configurations of Na 36CL18Br18.
displacements The eigenve~torscorresponding but not the same phase,to the 135, 160, and 170 cm’ peaks are qualitatively similar to those discussed by Mass et al, for the Cd0 75Zn0 25S system. Eigenvectors analogous
3 served in the mixed experimentally crystal NaCL by MacDonald et al)’ 090Br010. The normal modes responsible for this peak have relatively large displacements of the Naneighbors atoms which the nearest and next—nearest of are a Br
to those corresponding to the relatively 1 are not dis— weak cussednear by Bass peaks 85, et 105,al.and 150 cm” The sensitivity of the imaginary part of the dielectric constant to the number of con— figurations averaged was studied for Na 36CL18Br18 by averaging over twenty configur—
atom. The REI model. for the system under consid— eration predicts two-mode behBytor. The f~e— qu~nciesof the two modes are plotted as func— tions of composition in Fig. 5. Also plotted are the frequencies of the principal peaks in
ations instead of ten. The peak positions were indistinguishable in the two cases, and the peak intensities differed by at most a few percent. The composition Na36CL27Br9 has also been investi~ated. It shows a single major peak near 170 cm”’’ with a shoulder at 160 ctC . Eigen— vectors associated with these peaks are quali— tatively very similar to those associated with the corresponding peaks at the other compositions previously discussed. ‘ The final composition which we discuss’ is Na36CL32Br4. In addition to the strong peak near 170 cm’~’, corresponds there is a small peak 1’ which closely to near that ob—
mode of the REt model might be said to represent a crude average of the positions of the nearby modes in our model—namely, the modes which be— come the CL local mode in the limit x +1 and the fundamental optical mode of NaCL in the limit x ‘*0. The lower mode of tbe.REI model is also seen to represent a crude average of the nearby modes of the linear chain model. The eigenvectors of the EEl model are not in accord. with those which are responsible for the princ~pa1peaks of the linear chain model. In the REt model, the atoms of a given species are assumed to form a sublattice which moves rigidly in the is important infrared—active modes. Such behavior not at all that exhibited in
140 cm”
our linear chain model.
We see that the upper
Vol. 32, No. II
VIBRATIONAL AND OPTICAL PROPERTIES OF NACLI_X Br
(I)u—
(a)
w - 83.8 crir’
gI.~
w -104.1 S0.15
‘
8-0.19 II ~ii~
S
9fl
cm”
II
I
~ Htt1ttt1$tt11ittttt1ttt~ttttt1ttt11 tt~11ttiitt1f1ttftft11tt1tttt1t111ti ~ 5
Cc) w S
Cd) -
134.4
•
2.57
cm-’ iI~
w
•
148.8
S
•
0.42
cm’
I I II ii *11111u1~ II
III ,iIlII
I-
I
U” (-)
tU~111t1ttf11t11~tIt~d$$t$ff11t111t1$t11fi1ttt,1~tt,tt1tf1$ti~tt111t1 Ce)
~ Z
(f)
w-159.8cm’ S-I.82 ii II till lIlt
::, Iii
liii 111111
1:;
liii
w-171.Scm” S-4.20
liii 11111*
I~l~I~if I ..IIi~I{[~i
f~t1itttt1f11i11~tttttittttttTttt1t tttt11t1tttt1111tt$titttttt~titt111 LATTICE SITE
Fig. 4.
LATTICE SITE
Eigenvectors of six infrared—active modes of Na36Ct18Br18.
Figs. 2 and 4 for the linear chain. In fact, the EEl behavior is not found in any of the con— figurations of any of the compositions which we have studied. Although the EEl model gives a simple means of correlating the infrared spectra of mixed crystals, we conclude that it does not give a reliable microscopic picture of the in frared—active modes. amountOne ofmight disorder a system, widths of thinkin that as one the increases the the absoprtion lines should increase. This notion is clearly borne out by the dependence of the wi4th I’ of the line at -~l35 cm1 on compos— ition as given in Table I. At x — 1. this node
is the fundamental optical mode of NaBr and has a relatively small width. As x decreases and the disorder increases, the width increases, eventually reaching a value which is a factor of two larger then the x — 1 value. -_______________________________________________ 1. composition Table I. Linewidth as a function of for the line at ‘ 135 cm” x 0.39 0.50 0.61 0.75 0.92 1.0 F/u 0.080 0.063 0.050 0.047 0.043 0.039 ____________________________________________
972
VIBRATIONAL AND OPTICAL PROPERTIES OF NACL
>,
1—x
Br
Vol. 32, No. 11
x
155
C’,
z
w150’ 0
0
0
o
0
I~O11Ol1u111l
0.0
0.1
Fig. 5.
0.2
0.3
0.4 0.5 0.6 cOMPOSITION X
0.7
0.8
0.9
1.0
Frequencies of infrared—active modes as functions of composition for the REI model (solid lines) and the linear chain model (open circles). A given line does not appear for certain compositions because the line is too weak or merges with at~otherline.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.
CHEN, Y. S., SHOCKLEY, W. and PEARSON, G. L., Phys. Rev. l5~648 (1966). CHAR, I. R. and MITRA, S. S., Phys. Rev. 172, 924 (1968). GENZEL, L., MARTIN, T. P. and PERRY, C. B., Phys. Status Solidi (b) 62, 83 (1974). BASS, H., ROSENSTOCK, H. B. and MCGILL, R. E., Solid State Commun. 7, 1 (1969). BARKER, A. S. and SIEVERS, A. J., Rev, of Modern Physics 47, Suppl. No. 2, Sl (1975). FERTEL, 3. B. and PERRY, C. B., Phys. Rev. 184, 874 (1969). TAYLOR, D. W., Solid State Common. 13, 117 (1973). REID, J. S., SMITh, T. and BUYERS, W. J. L., Phys. Rev. B1, 1833 (1970). RAUNIO, G., ALZ4QVIST, L. and STEDMAN, R., Phys. Rev. 178, 1496 (1969). BASS, N., Phys. Rev. 117, 1497 (1960). CHANG, R. K., LACINA, B. and PERSUAN, P. S., Phys. Rev. Letters 17, 755 (1966). ROSENSTOCK, H. B. and W~GILL, R. E., Phys. Rev. 176, 1004 (1968). MACDONALD, B. F., KLEIN, N. V. and MARTIN, T. P., Phys. Rev. 177, 1292 (1969).
Solid State Coninunications, Vol.32,in pp. Pergansrn Press Ltd. 1979. Printed Great Britain.
*
VIBRATIONAL AND OPTICAL PROPERTIES OF THE MIXED CRYSTAL NaCL l—xBr x K. H. Wanser and R. F. Wallis
Department of Physics, University of California, Irvine, Irvine CA 92717 (Received 30 July 1979 by A. A. Maradudin)
A one—dimensional model is developed for the [111] transverse modes of the mixed crystal NaCi Br • The frequencies and eigenvectors of the normal modes of ~i~~~i 0A are calculated for ten random configurations at each of a number of compositions. The imaginary part of the dielectric function is calculated for the average configuration at each composition. The behavior of the system cannot be classified in terms of the usual one—mode or two—mode categories. Representative eigenvectors are displayed, and the random element Isodisplacement model is discussed in the light of our results.
In the past severa~,years, many studies have been carried out~’ on the vibrational properties of mixed crystals of the type AB l-xCx. A number of methods have been employed in an attempt to understand the properties of these crystals. A summary of both the experimental and theoretical situations has been presented in a review article by Barker and Sievers. A simple model which has had some success in correlating the optical properties of mixed 12 random In thiselement model,isodisplacement the individual crystals is the atomic species are assumed to form sublattices (REt) model. which move rigidly with respect to one another in the modes important for optical absorption. Although the EEl model has been applied to many cases, there has been relatively little work done on assessing the validity of the model on a microscopic level. An investigation which bears on this point, however, is that of Haag et al. who studied the optical spectra of the CdSl—x Sex and Cd lx — Zn S systems using a one— x dimensional model. The normal mode displace— ment patterns which they found are not in ac— cord with the REI model, In the present paper we consider mixed crystals of NaCL and NaBr, NaCL Br , a l—x x system which apparently has not been previously investigated. As in the work of Hass et al. we use a finite one—dimensional model, bu~i~lude both first and second neighbor Interactions and not just first neighbor interactions. Also, we adopt cyclic boundary conditions rather than the problem of harmonics encountered by avoid Hass fixed—end boundary conditions and thus ~
where m~is the mass and u~is the displacement of atom L, w is the frequency ~ and kf,f+l are nearest neighbor force constants, and kL,f_2 and ki f+2 are second neighbor force con— stants. If the halogen of a nearest neighbor pair is CL or Br, the nearest neighbor force constant is taken to be k12 or k13, respective— ly, where k12 and k13 are the nearest neighbor force constants of pure NaCL and NaBr, respec— tively. If the ions of a second neighbor pair are both CL or both Br, the second neighbor force constant is k or k 33, respectively, where k 2 and k a~ the second neighbor force constan~sfor h~ogensin pure NaCL and NaBr, respectively. If one CL ion and one Br ion make up a second neighbor pair, the second neighbor force constant is taken to be k23 (1 — x)k22 + x k33 . (2a) If two Na ions make up a second neighbor pair, the second neighbor force constant is taken to be k11 — (1 — x) ~ + x ~ (2b) where k and ~ are the second neighbor force constants for Na in pure NaCL and NaBr, respectively. The force constants for the pure crystals are 8’9determined for the TA by and fitting TO phonon the experi— frequen— des at data k — 0 and at the zone boundary in the mental [111] direction. Having determined the force constants, we are now able to solve the equations of motion for a given composition of the mixed crystal.
~
The equations of motion can be written in the form m~w2u 1— ~
_ui_1)+ki,t+1(ut_ui+1) +
+ k&,~...2(ui_ui...2) + k~L+2(uf _uL+2)
This yields the normal mode frequencies u and the eigenvector components ~ of the dynLuical matrix. The eigenvector comp&~ents satisfy the orthonormality condition
(1)
*
This research was supported In part by NSF Grant No. DNR—78O943O. 967
968
VIBRATIONAL AND OPTICAL PROPERTIES OF NAC&
Er £ “is is — ~as and the closure condition
~ ~
~i’s~ii’
‘(3)
If one now considers the equations for the in—
the normal modes which give large contributions to the dielectric constant. Let us first con— sider chains with composition Na 36CL9Br27. Here
teracting ions driven by an electromagnetic field of frequency u, one cam calculate the di— electric constant E(w). In the presence of damping characterized by damping constants r5, the result is
and in the following we designate the composi— tion by specifying the number of atoms of each species in the 72 atom chain. In Fig, 1 we plot the imaginary part of the dielectric con—
E(w)
•
1 + 4wn
—
(4)
Br Vol. 32, No. II 1—x x 0 although the damping is known to vary as a function of concentration,6’11’ NaC&,~’ One of our principal objectives has been to investigate the form of the eigenvectors for
~ s
(w
2 2 p5 2
—
iwr )
(5)
—
8
where p
5
5 is given by *
p
E Q~F;~fm~ , £
—
~
(6)
* is the number of chains per unit volume. We is the— effective charge of the ith ion and n take ~r — “~Na — _Q* and focus attention
on the imagi~rypart of E(~)which is propor— tional to nQ . The accuracy of the computer calculations was checked with the aid of two sum rules. The first is configuration dependent and is based on the invariance of the trace of the dynamical matrix, N
(k&,i+
1+k&,i,,,1+ki.&_2)
L—l
m~
(7) where N is the number of ions in the chain. The second is configuration independent and is based on the sum of the squares of the quantities p5, 15N + 1— mCL~ +—!—\ 1~rI . — ~ “~2._ (8) s s 2 I~ In discuasin~ ~ intensity of individual modes; it is convenient to define a mode strength S ~
by
2 4irp 5— 8Nu2
We take
( ,
4ir
~2
*
QLF;Ls)
(l)L+1 and measure
.
(9)
s and m£ in the numerical units of ~0(NaCL) evaluation and mCL of , S~. respectively, in Q~ =
ü
Calculations were made of the eigenvalues and eigenvectors of the dynamical. matrix, the mode strength of each mode, and the imaginary part of the dielectric constant for a number of compositions of the mixed crystal NaCL 1_ xx Br , Chains containing a total of 72 atoms were ~ ployed. For the intermediate compositions, ten centration and the results forused the imaginary different configurations were for each con— part of the dielectric constant were averaged over the configurations of that concentration. Equations (7) and (8) were verified for each case. The damping constants were chosen to be 0.04 c,i which is a realistic value for S
vectors can be associated with the various peaks. In FLg. 2 we show four such eigenvectors associated, with the four peaks identified above. 1 peak is essen— tially a gap mode for an isolated chlorine atom The eigenvector for the 85 cm” immersed in a background of bromine atoms. The eigenvector for the 135 cm”1’ peak is associated with groups of consecutive bromine atoms moving with the same phase but different amplitudes and the interspersed sodium atoms moving with op— posite phase to the bromine atoms. The mode giving rise to the peak at 160 cni” is basic— ally a local mode of an isolated chlorine atom. The eigenvector for the 170 cm”1 peak is associ—
4~&,&+ 24
E ss
stant, E2~5), as a function of frequency for an average of ten random There are two principal peaksconfigurations. at -‘135 cm~’~’and 1 — -‘160 cm 1 and two subsidiary peaks at —85 cm” and 170 cm’~. Particular normal mode eigen—
ated with groups of consecutive chlorine atoms moving with the same phase and the interspersed sodium atoms moving with opposite phase to the chlorine atoms. In each of the eigenvectors discussed above, the motion is characterized by relatively small groups of atoms whose amplitudes of displacement are significant. Such groups of atoms may occur vidual establishes freseveral small times group in thethat chain, but it isthethe mdi— quency to a good approximation. The importance of small groups of atoms has also been empha— sized b 1 Hass et al. and by Rosenstock and McGill. 2 —— Let us now turn to chains with composition Na36C&18Br18. In Fig. 3 is plotted E2(w) versus frequency for an average of ten random configurations. Just as with the previous composition, there are p~aksin the vicinity of 85, l35,l16O~ 1’ and 170 cm” ; however the peak near 170 cm is considerably considerably weaker stronger than and the corresponding that near 135 cm” peaks for the previous composition. Typical eigemvectors associated with these four peaks are shown in Fig. 4. The similarity of the displacement patterns to those for the Na36CL9Br27 composition is evident. In addition to these four peaks, however, the composition 1’. exhibits Eigenvectors Na36CL18Br18 also smallassociated peaks nearwith the latter 105 and 150two cm’ peaks are also shown in Fig, 4. In both cases the eigenvectors are character— ized by a group of three consecutive CL atoms which have relatively large amplitudes, but do not have the sane phase. The Na atoms adjacent to these CL atoms also have relatively large
Vol. 32, No. 11
VIBRATIONAL AND OPTICAL PROPERTIES OF NACLI_X Br~ 4.5
u
969
u
4.0 3.5
-
3.0 .~
2.5-
~
hull
20
40 Fig. 1’.
60
Imaginary part of the dielectric constant as a function of frequency for an average of ten random configurations of Na 36CL9Br27.
(a)
(b) w~l34.5Cm” 5-2.50
w.82.4Cm” S~0.075
5
II II
~~
.:...~.i.lIIlllIl1IlIi.......:.i.i..:
,.
~
Cc)
(d) II II II
I
w.159.7Cm”
II II II
~l~l ~]l
°
liii
11111 :~,.
ii
z
200 220
80 100 120 140 160 180 FREQUENCY w (cm~)
S-2.44 ~I
U-
iii
167.5cm”
5-1.13
III
~l~I
—
tftftt$t1fi~ttttfIff$ttIftt~fttttIli LATTICE SITE Pig. 2.
$Ittitit$ttftt~~$Utttl$ LATTICE SITE
Eigenvectore of four major infrared— active modes of Na36Ct9Br27.
970
VIBRATIONAL AND OPTICAL PROPERTIES OF NACL
1—x
Br
Vol. 32, No. 11
x
2.5
2.0
1.5 C”
w
1.0
0.5
-
0.0 20
-
40
60
80
100
120
FREQUENCY
Fig. 3.
140
160
180 200 220
w (cm’)
Imaginary part of the dielectric constant as a function of frequency for an average of ten random configurations of Na 36CL18Br18.
displacements The eigenve~torscorresponding but not the same phase,to the 135, 160, and 170 cm’ peaks are qualitatively similar to those discussed by Mass et al, for the Cd0 75Zn0 25S system. Eigenvectors analogous
3 served in the mixed experimentally crystal NaCL by MacDonald et al)’ 090Br010. The normal modes responsible for this peak have relatively large displacements of the Naneighbors atoms which the nearest and next—nearest of are a Br
to those corresponding to the relatively 1 are not dis— weak cussednear by Bass peaks 85, et 105,al.and 150 cm” The sensitivity of the imaginary part of the dielectric constant to the number of con— figurations averaged was studied for Na 36CL18Br18 by averaging over twenty configur—
atom. The REI model. for the system under consid— eration predicts two-mode behBytor. The f~e— qu~nciesof the two modes are plotted as func— tions of composition in Fig. 5. Also plotted are the frequencies of the principal peaks in
ations instead of ten. The peak positions were indistinguishable in the two cases, and the peak intensities differed by at most a few percent. The composition Na36CL27Br9 has also been investi~ated. It shows a single major peak near 170 cm”’’ with a shoulder at 160 ctC . Eigen— vectors associated with these peaks are quali— tatively very similar to those associated with the corresponding peaks at the other compositions previously discussed. ‘ The final composition which we discuss’ is Na36CL32Br4. In addition to the strong peak near 170 cm’~’, corresponds there is a small peak 1’ which closely to near that ob—
mode of the REt model might be said to represent a crude average of the positions of the nearby modes in our model—namely, the modes which be— come the CL local mode in the limit x +1 and the fundamental optical mode of NaCL in the limit x ‘*0. The lower mode of tbe.REI model is also seen to represent a crude average of the nearby modes of the linear chain model. The eigenvectors of the EEl model are not in accord. with those which are responsible for the princ~pa1peaks of the linear chain model. In the REt model, the atoms of a given species are assumed to form a sublattice which moves rigidly in the is important infrared—active modes. Such behavior not at all that exhibited in
140 cm”
our linear chain model.
We see that the upper
Vol. 32, No. II
VIBRATIONAL AND OPTICAL PROPERTIES OF NACLI_X Br
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LATTICE SITE
Eigenvectors of six infrared—active modes of Na36Ct18Br18.
Figs. 2 and 4 for the linear chain. In fact, the EEl behavior is not found in any of the con— figurations of any of the compositions which we have studied. Although the EEl model gives a simple means of correlating the infrared spectra of mixed crystals, we conclude that it does not give a reliable microscopic picture of the in frared—active modes. amountOne ofmight disorder a system, widths of thinkin that as one the increases the the absoprtion lines should increase. This notion is clearly borne out by the dependence of the wi4th I’ of the line at -~l35 cm1 on compos— ition as given in Table I. At x — 1. this node
is the fundamental optical mode of NaBr and has a relatively small width. As x decreases and the disorder increases, the width increases, eventually reaching a value which is a factor of two larger then the x — 1 value. -_______________________________________________ 1. composition Table I. Linewidth as a function of for the line at ‘ 135 cm” x 0.39 0.50 0.61 0.75 0.92 1.0 F/u 0.080 0.063 0.050 0.047 0.043 0.039 ____________________________________________
972
VIBRATIONAL AND OPTICAL PROPERTIES OF NACL
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Vol. 32, No. 11
x
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0
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0.2
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Frequencies of infrared—active modes as functions of composition for the REI model (solid lines) and the linear chain model (open circles). A given line does not appear for certain compositions because the line is too weak or merges with at~otherline.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.
CHEN, Y. S., SHOCKLEY, W. and PEARSON, G. L., Phys. Rev. l5~648 (1966). CHAR, I. R. and MITRA, S. S., Phys. Rev. 172, 924 (1968). GENZEL, L., MARTIN, T. P. and PERRY, C. B., Phys. Status Solidi (b) 62, 83 (1974). BASS, H., ROSENSTOCK, H. B. and MCGILL, R. E., Solid State Commun. 7, 1 (1969). BARKER, A. S. and SIEVERS, A. J., Rev, of Modern Physics 47, Suppl. No. 2, Sl (1975). FERTEL, 3. B. and PERRY, C. B., Phys. Rev. 184, 874 (1969). TAYLOR, D. W., Solid State Common. 13, 117 (1973). REID, J. S., SMITh, T. and BUYERS, W. J. L., Phys. Rev. B1, 1833 (1970). RAUNIO, G., ALZ4QVIST, L. and STEDMAN, R., Phys. Rev. 178, 1496 (1969). BASS, N., Phys. Rev. 117, 1497 (1960). CHANG, R. K., LACINA, B. and PERSUAN, P. S., Phys. Rev. Letters 17, 755 (1966). ROSENSTOCK, H. B. and W~GILL, R. E., Phys. Rev. 176, 1004 (1968). MACDONALD, B. F., KLEIN, N. V. and MARTIN, T. P., Phys. Rev. 177, 1292 (1969).