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Lattice vibrations in alkali azides
PHYSICS
Volume 28A, number 10
LI”““1”“‘-“J 48
40
32
24 0’ Fig. 2.
16
8
LETTERS
24 February 1969
the maximum is very diffuse, but its existence is also undoubtless, which means that any pattern on the photo emulsion can be regarded as three-dimensional structure. In fig. 2 the decrease of intensity of the transitting wave is shown, corresponding to the case of extinction, observed by a hologram, recorded in the film. It is interesting that to analyse it, it proved enough to carry out the measurements in the most transparent pieces of the same hologram for which the Borrmann effect has been marked. The last can be used to investigate experimentally the diffraction of electromagnetic waves in ideal three-dimensional structures. Realy, by means of directing both the thickness and darkening of photo emulsion it is possible to model different experimental situations by means of holograms, including those inaccessible for X-ray analysis. In particular, the hologram being analysed on photo emulsion can be regarded as an optical analogue of zone Huinier-Preston. It is pointed out by the experimentally detected considerable extent of knots of the reciprocal lattice being correlated with the hologram in the direction of the normal towards the surface of the film.
0
b) recording on an ordinary photo emulsion (thickness of photosensitive layer some microns). The dependence of the light intensity, having transitted the hologram, plotted against the angle of rotation is given in fig. 1, and fig. 2. The shape of the curves in fig. 1 is characteristic of the effect of abnormal transmission*. In fig. la the Borrmann effect is shown, corresponding to the hologram, recorded in the dyed crystal KBr. In case of an hologram on photo emulsion (fig. lb)
References V. V. Aristov, V. L. Broude, L. V. Kovalsky, V. K. Polijnsky, V. Sh. Shekhtman, V. B. Timofeev, Doklad. Akad. Nauk 177 (1967) 1, 65. 2. P. P. Ewald, Rev. Mod. Phys. 37 (1965) 46. 3. E. N. Leith, A. Kozma, J. Upatnieks, J. Marks and N. Massey, Appl. Opt. 8 (1966) 1303. 4. Edward J. Saccocio, Appl. Phys. 10 (1967) 3994. 1.
* The intensity increase of the “reading” beam having transitted the hologram on the condition that the angle of reading is equal to this of recording was discribed in ref. 3, where thick-layer emulsions (t N 40 pm) were utilized, In ref. 4 the relation of this phenomenon with the Borrmann effect is pointed out.
LATTICE
VIBRATIONS
J. GOVINDARAJAN Department
of Physics,
IN
ALKALI
AZIDES
and T. M. HARIDASAN
Indian Institute
of Science,
Bangalore-12,
India
Received 27 January 1969
The longwavelength lattice vibrations in potassium, rubidium and caesium azides have been using BornPs lattice dynamics.
In this note we report the lattice vibrations in KN3, RbN3 and CsN3 calculated on the basis of the rigid ion model, taking into account the long range electrostatic coupling coefficients com-
calculated
puted using the Ewald’s method and the nearest neighbour short range interaction with the Pauling potential [l] with n = 9. .The method of Gray et al. [2] has been followed in evaluating a general ra701
Volume
28A, number
PHYSICS
10
dius for the azide ion. These alkali azides crystallise in the body centered tetragonal system with four molecules in the unit cell. The longwavelength optical vibrations fall under the following irreducible representations of the point group D4h
The coordinates of the alkali ions 1 and 2 in the unit cell have been taken to be (OOi) and (00:) and those for the azide ions 3 and 4 as (060) and ($00) respectively. The symmetry relations between the coupling coefficients are given below:
for (Yf j3. The dynamical matrix (12 X 12) was diagonalised to obtain the eigen values which were assigned to the above symmetric species with the help of the eigenvectors. The results are presented in table 1 with our assignments and the experimenTable I Assignment
CsN3
KN3 Calc.
Expt . *
Calc.
Expt.*
Calc.
64
68(T;2g\
83
A%
133
A2u
206
228(Tj)
155
Blu
129
lZO(VL3)
114
74
65(Thg)
81 72
E
g
EU
136(T;)
RbN3
120(Ti2g)
* The assignments Bryant.
118(TA
2u
)
164
;;(T&,)
119
49
42(T;g)
47
101(vL2)
89
89(Tku)
62
81(vLl)
83
-
67
given in brackets
24 February
1969
tal values deduced by Bryant et al. [3,4] for KN3 and CsN3. There are no experimental results of RbN3. Even though our calculated values for the lattice vibrations of KN3 agree fairly with experimental values, our assignments do not correspond with those given by Bryant and may be due to the fact that his assignments are quite tentative and indirect as they have been inferred from the knowledge of the combination bands alone. The Raman bands in KN3 and CsN3 however agree both in frequency and in assignment. In CsN3, Bryant has tentatively assigned the weaker band at 38 cm-l and the stronger band at 89 cm-l as two E, modes and the broad band at 118 cm-l as A2u whereas we obtain an E, mode at 89 cm-1 and another at 83 cm-l and the A2u at 155 cm-l. We feel that the 118 cm-l assigned as A2u by him correspond to one of the E, modes. We also get for A2 a value of 64 cm-]- as against the value of 68 cm- gl estimated by him. Corresponding to the Blu mode we get a value of 114 cm-l and it may correspond to the 137 cm-l reported by Bryant, but assigned as a rotational band from a mere intensity consideration alone. The authors however feel that for a better understanding of the optical vibrations in azides, some work in the direction of laser Raman spectra and far infrared polarised spectra would be of much help. The authors are thankful to Prof. R. S. Krishnan for guidance and encouragement and to Dr. N. Krishnamurthy for useful discussions.
References L. Pauling, The nature of the chemical bond (Cornell University Press, third ed., 1960) p.523. P. Gray and T. C. Waddington, Proc. Roy. Sot. A235 (1956) 481. J. I. Bryant, J. Chem. Phys. 38 (1963) 2845: 45 (1966) 689. J. I. Bryant and R. L. Brooks, J. Chem. Phys. 43 (1965) 880.
are those by
*****
702
LETTERS
Volume 28A, number 10
LI”““1”“‘-“J 48
40
32
24 0’ Fig. 2.
16
8
LETTERS
24 February 1969
the maximum is very diffuse, but its existence is also undoubtless, which means that any pattern on the photo emulsion can be regarded as three-dimensional structure. In fig. 2 the decrease of intensity of the transitting wave is shown, corresponding to the case of extinction, observed by a hologram, recorded in the film. It is interesting that to analyse it, it proved enough to carry out the measurements in the most transparent pieces of the same hologram for which the Borrmann effect has been marked. The last can be used to investigate experimentally the diffraction of electromagnetic waves in ideal three-dimensional structures. Realy, by means of directing both the thickness and darkening of photo emulsion it is possible to model different experimental situations by means of holograms, including those inaccessible for X-ray analysis. In particular, the hologram being analysed on photo emulsion can be regarded as an optical analogue of zone Huinier-Preston. It is pointed out by the experimentally detected considerable extent of knots of the reciprocal lattice being correlated with the hologram in the direction of the normal towards the surface of the film.
0
b) recording on an ordinary photo emulsion (thickness of photosensitive layer some microns). The dependence of the light intensity, having transitted the hologram, plotted against the angle of rotation is given in fig. 1, and fig. 2. The shape of the curves in fig. 1 is characteristic of the effect of abnormal transmission*. In fig. la the Borrmann effect is shown, corresponding to the hologram, recorded in the dyed crystal KBr. In case of an hologram on photo emulsion (fig. lb)
References V. V. Aristov, V. L. Broude, L. V. Kovalsky, V. K. Polijnsky, V. Sh. Shekhtman, V. B. Timofeev, Doklad. Akad. Nauk 177 (1967) 1, 65. 2. P. P. Ewald, Rev. Mod. Phys. 37 (1965) 46. 3. E. N. Leith, A. Kozma, J. Upatnieks, J. Marks and N. Massey, Appl. Opt. 8 (1966) 1303. 4. Edward J. Saccocio, Appl. Phys. 10 (1967) 3994. 1.
* The intensity increase of the “reading” beam having transitted the hologram on the condition that the angle of reading is equal to this of recording was discribed in ref. 3, where thick-layer emulsions (t N 40 pm) were utilized, In ref. 4 the relation of this phenomenon with the Borrmann effect is pointed out.
LATTICE
VIBRATIONS
J. GOVINDARAJAN Department
of Physics,
IN
ALKALI
AZIDES
and T. M. HARIDASAN
Indian Institute
of Science,
Bangalore-12,
India
Received 27 January 1969
The longwavelength lattice vibrations in potassium, rubidium and caesium azides have been using BornPs lattice dynamics.
In this note we report the lattice vibrations in KN3, RbN3 and CsN3 calculated on the basis of the rigid ion model, taking into account the long range electrostatic coupling coefficients com-
calculated
puted using the Ewald’s method and the nearest neighbour short range interaction with the Pauling potential [l] with n = 9. .The method of Gray et al. [2] has been followed in evaluating a general ra701
Volume
28A, number
PHYSICS
10
dius for the azide ion. These alkali azides crystallise in the body centered tetragonal system with four molecules in the unit cell. The longwavelength optical vibrations fall under the following irreducible representations of the point group D4h
The coordinates of the alkali ions 1 and 2 in the unit cell have been taken to be (OOi) and (00:) and those for the azide ions 3 and 4 as (060) and ($00) respectively. The symmetry relations between the coupling coefficients are given below:
for (Yf j3. The dynamical matrix (12 X 12) was diagonalised to obtain the eigen values which were assigned to the above symmetric species with the help of the eigenvectors. The results are presented in table 1 with our assignments and the experimenTable I Assignment
CsN3
KN3 Calc.
Expt . *
Calc.
Expt.*
Calc.
64
68(T;2g\
83
A%
133
A2u
206
228(Tj)
155
Blu
129
lZO(VL3)
114
74
65(Thg)
81 72
E
g
EU
136(T;)
RbN3
120(Ti2g)
* The assignments Bryant.
118(TA
2u
)
164
;;(T&,)
119
49
42(T;g)
47
101(vL2)
89
89(Tku)
62
81(vLl)
83
-
67
given in brackets
24 February
1969
tal values deduced by Bryant et al. [3,4] for KN3 and CsN3. There are no experimental results of RbN3. Even though our calculated values for the lattice vibrations of KN3 agree fairly with experimental values, our assignments do not correspond with those given by Bryant and may be due to the fact that his assignments are quite tentative and indirect as they have been inferred from the knowledge of the combination bands alone. The Raman bands in KN3 and CsN3 however agree both in frequency and in assignment. In CsN3, Bryant has tentatively assigned the weaker band at 38 cm-l and the stronger band at 89 cm-l as two E, modes and the broad band at 118 cm-l as A2u whereas we obtain an E, mode at 89 cm-1 and another at 83 cm-l and the A2u at 155 cm-l. We feel that the 118 cm-l assigned as A2u by him correspond to one of the E, modes. We also get for A2 a value of 64 cm-]- as against the value of 68 cm- gl estimated by him. Corresponding to the Blu mode we get a value of 114 cm-l and it may correspond to the 137 cm-l reported by Bryant, but assigned as a rotational band from a mere intensity consideration alone. The authors however feel that for a better understanding of the optical vibrations in azides, some work in the direction of laser Raman spectra and far infrared polarised spectra would be of much help. The authors are thankful to Prof. R. S. Krishnan for guidance and encouragement and to Dr. N. Krishnamurthy for useful discussions.
References L. Pauling, The nature of the chemical bond (Cornell University Press, third ed., 1960) p.523. P. Gray and T. C. Waddington, Proc. Roy. Sot. A235 (1956) 481. J. I. Bryant, J. Chem. Phys. 38 (1963) 2845: 45 (1966) 689. J. I. Bryant and R. L. Brooks, J. Chem. Phys. 43 (1965) 880.
are those by
*****
702
LETTERS