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Bound states dynamics in disordered N2O crystals
LUMINESCENCE
Journal of Luminescence 53(1992) 557—561
JOURNAL OF
Bound states dynamics in disordered N20 crystals J.-P. Lemaistre, B. Perrin, R. Ouillon and P. Ranson Déparlemeni de Recherches Physiques, Unirersitd P.&M. Curie, URA 71 du CNRS, Tour 22, 4, Place Jussieu, 75252 Paris Cédex 05, France
The nitrous oxide crystal is used as a prototype system to analyze the dynamics of bound states in disordered materials. Because of the intramolecular anharmonicity of the potential, a Fermi resonance interaction (F) takes place between the v1 and the 2v, vibron modes. Both modes are dispersed throughout the crystal by dipolar interactions (i). This generally leads, when F >~~i, to the appearance of bound states. In a N20 crystal a frozen-in disorder occurs because of the head—tail arrangements of the molecules. Although the treatment of the Fermi resonance in ordered crystals has been well understood the combination of Fermi resonance and of disorder remained a non-trivial problem. A theoretical analysis of the incoherent Raman response of the bound states is provided and compared to experimental data. Also provided are new results obtained by nanosecond CARS experiments in the frequency domain. A comparison between these two types of spectroscopies will be discussed by taking into account previous picosecond CARS data performed in the time domain.
1. Introduction Coherent anti-stokes Raman scattering (CARS) techniques in the time and frequency domains as well as high resolution conventional Raman scattering experiments allow us to get accurate measurements of the vibron states dynamics in molecular crystals. Molecular crystals made up of small molecules have a small number of vibron bands which are well separated from each other and from the lattice phonons. Such candidates are for example nitrogen (N2-a phase), carbon monoxide (CO-a phase), carbon dioxide (C02) or nitrous oxide (N2O). These four molecular crystals have the same FCC Bravais lattice structure and may serve as prototype systems to analyze their molecular properties through the IR and Raman responses of their vibron bands. The CO crystals exhibits, at all temperatures, a “frozen-in” head—tail orientational disorder. The internal stretching molecular vibrations (v1) are coupled through dipolar intermolecular interacCorrespondence to: Dr. J.-P. Lemaistre, Departement de Recherches Physiques, Université P.&M. Curie, URA 71 du CNRS, Tour 22,4, Place Jussieu. 75252 Cédex 05, France.
0022-2313/92/105.00 © 1992
—
tions (i). As a result the head—tail orientation disorder is of Ising type and can be easily worked out. In CO2 molecule the stretching mode v~,at frequency w1, and the overtone of the double 2W degenerate bending mode v2, at frequency 2, are in quasi-resonance and coupled by a cubic anharmonicity term (F). This phenomenon is called the molecular Fermi resonance. The anharmonic coupling mixes the wave functions associated to the v1 and v2 vibrations to an extent defined by F and the energy separation 2W 7 I. The problem of the Fermi resonance in molecular crystals was extensively studied by Agranovich et al. [1,21and by Bogani et al. [31.In CO2 ordered crystal v1 vibron mode is undispersed and the dispersion of the v2 vibron modes is smaller than the anharmonic coupling. In this strong Fermi resonance case two narrow bands made up of hybrid states (vt, v) emerge from each side of the free two~vibroncontinuum. N20 crystals exhibit, like CO a frozen-in head—tail orientational disorder and, like C02, a Fermi resonance interaction between2W the both quasi-degenerdispersed I 2 ate vibron modes at W and through dipolar interactions. Because of the large energy separation wi 2w 2 in N2O we already
Elsevier Science Publishers By. All rights reserved
—
—
558
.1. -F Lemaistre et al.
/ Bound .Oatcs dinamie,s
analyzed the effect of disorder in the upper v crystal eigenstates by neglecting the Fermi resonance interaction [41.Cardini et al. analyzed the Fermi resonance in N20 crystals by neglecting the orientational disorder [5]. More recently they performed molecular dynamics simulations of the disordered crystal taking into account orientational disorder as well as translational disorder hut neglecting the Fermi resonance [61.The comhination of the Fermi resonance and of the head—tail orientational disorder remained a nontrivial problem [7].
in disordered N,O crystals
the disorder is removed from the Hamiltonian I1~ showing that the eigencnergies of the disordered crystal are those of the ordered one but that the wavefunctions are modulated through the e,~. We assume that the Raman scattering originates mainly from the first order Raman tensor of the r’~ mode. In the oriented gas model the crystal polarizability is simply the sum of the polarizahilities, x~”.of all the molecules of the crystal. The Raman scattering intensity, in a classical right angle Raman geometry along the crystal axes (a, b). is given by
Lx~~( ~k)Ojk)
a ~
~(w
2. One-vibron crystal states with orientational disorder Let us calculate the one-vibron crystal states for the stretching mode v~ of N2O in the absence of the Fermi resonance. It was established by Zumofen [8], in the case of a-CO, that the head—tail orientational disorder does not change the energetics of the crystal but only modifies the phase of the wave functions when the vibron modes interact through long range dipole—dipole interactions. The disorder, which is of Ising type, can he introduced through pseudo-spins e~ ±I randomly distributed and defined relatively to a given ordered configuration. We shall use a cornposite index m which labels the lattice vector m, the molecule site s and the degeneracy x if any. The harmonic Hamiltonian for the i.’~mode can he written as =
H1
Ewja~a,~+
=
L~1~a~a,1.
=
=
~ (2) 1 means that the molecule m has flipped relatively to an ordered configuration for which E + 1 Setting E,,f,
=
E
=
—
=
a,,
(-~)
=
e,,a,,
.
with 05(q)
~em exp[—iq m]. (5) N k is the wave vector, ~ the branch index. N the number of unit cells. w(~k)the eigenmode frequency and sp~(~k) the amplitude of the mode on site .r. Since in Raman spectroscopy the light is scattered by a macroscopic system we calculate configuration-averaged properties. We assume that there are no correlations of orientations hetween the four sites of the unit cell and between the unit cells and setting p. the concentration of pseudo-spins which have not been flipped, we get =
(Ojq))
—
(3)
~exp[—iq
=
N (2p
mJKe~,~
I) ~(q). (6) 4p( 1 —p) Ko~(q)01( q’ )~ ~ N ~ q q’) 2p 1) i(q) ~.l(q’). (7) +( The first term describes the incoherent diffraction in a random substitutional mixture and equals zero for p 0 (or I) which corresponds to an =
In the above equation a~ and a~ are the crcation and annihilation operators for the v~ vibration at frequency w1 i~, (with S 0 if n m) describes theintermolecular interaction between molecules m and n. The potential, 5, of the disordered configuration is simply related to that, i, of the ordered one through the relation
~k)), (4)
(1)
,nn
~1f
w(
—
=
—
=
ordered configuration. The second term describes the coherent contribution and equals zero for p 0.5 which corresponds to a totally disordered configuration. In the latter case the 6~selection rule and the normalization condition reduces the expression of the Raman intensity to I x.~”=
f-P. Lemaistre et a!.
/ Bound states dynamics in
which does not depend on the k wave vector, Consequently the Raman intensity is proportional to the density of crystal states (DOS). 3. Fermi resonance with orientational disorder Nitrous oxide crystal exhibits a Fermi resonance interaction between the stretching mode v1, at frequency w1, and the first overtone of the doubly degenerate bending mode v2, at frequency 2w2. We add to the harmonic Hamiltonian H1 the Hamiltonian H2 associated to 2v2: H2 Lw2b,~7h,~ + ~~in,i(2)b,~bri. (8) =
disordered N
559
20 crystals
in which C is a product of the wave functions coefficients and the quantities 0, previously defined, give a spectral representation of the disorder. Generally, in the presence of disorder, the Fermi resonance term V mixes the polarizations ~ as well as the wave vectors k. Using a perturbation technique we define a configuration averaged Green’s function
Thus the intramolecular anharmonic interaction is given by ~
=
2
[a~(b1~)~
~
+
a,n(b,~)21.
a~ and b,~ are the annihilation (creation) operators of molecule m for v1 and v2 vibration modes respectively. The intermolecular potentials L1’(i) (i 1,2 for the v1 and v2 modes respectively) are modulated by e. Defining again primed operators to remove the disorder from H~ and =
—
w~(~k)— 2T2fdz
w —z The poles of the function g gives the bound states. This function is the quitepositions similar of to that derived for an ordered configuration but with p(z) being an averaged DOS which does not depend on the wave vector k. Thus the poles of g are not sensitive to the detailed structure of the free two-vibron continuum at 2w2 and we can ignore the dispersion of the v2 mode. The Raman intensity is reduced to
H2, /w~ \ In
,
In b’=mb~~,
a~=,,1a(1L (10) we note that such a transformation keeps a pseudo-spin in c,~in the anharmonic Hamiltonian. Consequently the eigenstates of the total Hamiltonian cannot be obtained exactly and we have to use a perturbation technique. The resolution of the Schrodinger equation, in the k wave vector representation, leads to a secular equation. The derivation of such an equation can be found elsewhere [71.We get [w
—
w~(~k)] A(~k) = ~
v(~~ )A(~’k’)~ (11)
where
co~(kk
2
~k
p
W
=
1
k2)01(k’
k1
k2)
—w2(~1k~) —w2(~,k2)
Jah(w)aP±~_____)
(15)
with r
~
~-Ii± L
d 2+212
w1
;
d
=
—
2w2 .
2
The densities p~ are the densities of the v1 modes shifted around the frequencies w ± of the free molecule.
4. Results and conclusions Lattice dynamics calculations were performed on a cubic lattice in the reduced Brillouin zone. The one-vibron DOS p~~~(w) was evaluated by means of the Ewald—Kornfeld method. The molecular parameters used for the calculations are w1 1287 cm’, w2= 1173 cm’, 1=20 cm~, (a~/8q1)=89 cm~, (~ji/dq2) 31 cm~ [5]. With these parameters d 57 cm~ and the =
k1k2
=
(12)
(16)
=
561)
.1. -P. Len,ai.stre c/ al.
N
/ Bound
stute.s dvna,nic,s in disordered N,() crystaLs
the v~.We display in fig. 1 the DOS, p(w), which is the self-convolution of the one-vibron density of states. The densities of the v and ~ modes were calculated with w1 (ak) running over the ii,
o 2 2V2
1
I
—10 5 0 I. Calculated DOS p1w)
5 10 ~ cmIhe self-convolution at
Fig. as the one-vibron states with the position and widlh of Ihe i’ bound states (dotted lines).
coefficients ~ involved in the Raman intensity are s~ 0.95 and ij 0.05. Thus we expect the Raman line shape of the v ~ modes to he slightly shifted from that of the v~ modes and described in the harmonic approximation. A more drastic effect is expected to occur for the a modes, Actually the width of the v~ line shape is re=
- =
hand. These densities of states are displayed in fig. 2 (vt) and in fig.~~ (v—). In both figures are plotted the experimental Raman line shapes recorded in a conventional VV geometry. The calculated DOS of the v + modes is identical to that of the i.~ modes meaning that the cubic anharmonicity plays a negligible role and the Raman line shape is well described by one-vibron states. The main difference between the v + Raman lineshape and the DOS of the v modes is a peak the with located calculated at w frequency 1291 cm of which the totally coincides symmetric mode (A) of a perfectly ordered crystal belonging to the space group T4. Moreover, re=
cent nanosecond CARS experiments performed in the frequency domain also revealed such a phenomenon. We formally reproduced this maximum by assuming a partial ordering in the N,O crystal. From the simulation of the Raman line shapes of the t’~ modes we deduced that all the modes have line widths smaller than 1 cm . This is in good agreement with picosecond CARS resuits (T 7 13 ps) obtained by Vallée et al. [9]. On the other hand the width of the a hand is reduced by one order of magnitude relative to that (11.4 cm I) of the t’ + hand as shown in fig. =
duced by the factor
i~
=
0.05 relatively to that of
______________
N20
...
.~- - -.
__________
Tr1OK
N2 0
1290A
~ 1295
1300
cm
110K
/
1
Fig. 2. C ~ilculated DOS (solid line) of the s hound states in N~O disordered crystals. The positions of the tour k = symmetry modes of a fictitious N 2O crystal are indicated. Observed Raman line shape (dotted line) recorded in a Conventional geometry and fitted by assuming a partially disordered crystal (p = )).47 instead of )).5 >.
1164
1165
1166
cm—i
Fig. 3. Calculated DOS (solid line) of the i’ hound states of lotally disordered N~Ocrystals. Observed Rarnan line shape recorded in ii conventional VV geomet~and fitted by assumlug a totally disordered crystal.
/ Bound States dynamics in
f-P. Lemaistre et al.
2. This result, which gives a value of ij 0.05 is in perfect agreement with that calculated. The observed shape of the v band as shown in fig. 3 is in agreement with the long exponential decay time of T2 15 ps measured by the picosecond
13] 14]
CARS technique [9].
15]
—
References [II 121
V.M. Agranovich and 1.1. Lalov, Soy. Phys. Solid State 13 (1971) 859. V.M. Agranovich, in: Spectroscopy and Excitation Dynam-
disordered N,O crystals
561
ics of Condensed Molecular Systems, eds. V.M. Agranovich and R.M. l-Iochstrasser (North-Holland, Amsterdam, 1983) p. 84. F. Bogani and P R. Salvi, J Chem. Phys 81 (1984) 4991. J.-P. Lemaistre, R. Ouillon and P. Ranson, J. Phys. Chem.
92 (1988) 1070. G. Cardini, P.R. Salvi and V. Sehettino, Chem. Phys. 119 (1988) 241. [61G. Cardini and V. Schettino, J. Chem. Phys. 94 (1991) 2502. [71B. Perrin, R. Ouillon. P. Ranson, J.-P. Lemaistre and A.A. Maradudin, Phys. Rev. B 43 (1991) 4451. 181 G. Zumofen, J. Chem. Phys. 68 (1978) 3747. 191 F. Vallee, G.M. Gale and C. Flytzanis. Chem. Phys. Let. 124 (1986) 216.
Journal of Luminescence 53(1992) 557—561
JOURNAL OF
Bound states dynamics in disordered N20 crystals J.-P. Lemaistre, B. Perrin, R. Ouillon and P. Ranson Déparlemeni de Recherches Physiques, Unirersitd P.&M. Curie, URA 71 du CNRS, Tour 22, 4, Place Jussieu, 75252 Paris Cédex 05, France
The nitrous oxide crystal is used as a prototype system to analyze the dynamics of bound states in disordered materials. Because of the intramolecular anharmonicity of the potential, a Fermi resonance interaction (F) takes place between the v1 and the 2v, vibron modes. Both modes are dispersed throughout the crystal by dipolar interactions (i). This generally leads, when F >~~i, to the appearance of bound states. In a N20 crystal a frozen-in disorder occurs because of the head—tail arrangements of the molecules. Although the treatment of the Fermi resonance in ordered crystals has been well understood the combination of Fermi resonance and of disorder remained a non-trivial problem. A theoretical analysis of the incoherent Raman response of the bound states is provided and compared to experimental data. Also provided are new results obtained by nanosecond CARS experiments in the frequency domain. A comparison between these two types of spectroscopies will be discussed by taking into account previous picosecond CARS data performed in the time domain.
1. Introduction Coherent anti-stokes Raman scattering (CARS) techniques in the time and frequency domains as well as high resolution conventional Raman scattering experiments allow us to get accurate measurements of the vibron states dynamics in molecular crystals. Molecular crystals made up of small molecules have a small number of vibron bands which are well separated from each other and from the lattice phonons. Such candidates are for example nitrogen (N2-a phase), carbon monoxide (CO-a phase), carbon dioxide (C02) or nitrous oxide (N2O). These four molecular crystals have the same FCC Bravais lattice structure and may serve as prototype systems to analyze their molecular properties through the IR and Raman responses of their vibron bands. The CO crystals exhibits, at all temperatures, a “frozen-in” head—tail orientational disorder. The internal stretching molecular vibrations (v1) are coupled through dipolar intermolecular interacCorrespondence to: Dr. J.-P. Lemaistre, Departement de Recherches Physiques, Université P.&M. Curie, URA 71 du CNRS, Tour 22,4, Place Jussieu. 75252 Cédex 05, France.
0022-2313/92/105.00 © 1992
—
tions (i). As a result the head—tail orientation disorder is of Ising type and can be easily worked out. In CO2 molecule the stretching mode v~,at frequency w1, and the overtone of the double 2W degenerate bending mode v2, at frequency 2, are in quasi-resonance and coupled by a cubic anharmonicity term (F). This phenomenon is called the molecular Fermi resonance. The anharmonic coupling mixes the wave functions associated to the v1 and v2 vibrations to an extent defined by F and the energy separation 2W 7 I. The problem of the Fermi resonance in molecular crystals was extensively studied by Agranovich et al. [1,21and by Bogani et al. [31.In CO2 ordered crystal v1 vibron mode is undispersed and the dispersion of the v2 vibron modes is smaller than the anharmonic coupling. In this strong Fermi resonance case two narrow bands made up of hybrid states (vt, v) emerge from each side of the free two~vibroncontinuum. N20 crystals exhibit, like CO a frozen-in head—tail orientational disorder and, like C02, a Fermi resonance interaction between2W the both quasi-degenerdispersed I 2 ate vibron modes at W and through dipolar interactions. Because of the large energy separation wi 2w 2 in N2O we already
Elsevier Science Publishers By. All rights reserved
—
—
558
.1. -F Lemaistre et al.
/ Bound .Oatcs dinamie,s
analyzed the effect of disorder in the upper v crystal eigenstates by neglecting the Fermi resonance interaction [41.Cardini et al. analyzed the Fermi resonance in N20 crystals by neglecting the orientational disorder [5]. More recently they performed molecular dynamics simulations of the disordered crystal taking into account orientational disorder as well as translational disorder hut neglecting the Fermi resonance [61.The comhination of the Fermi resonance and of the head—tail orientational disorder remained a nontrivial problem [7].
in disordered N,O crystals
the disorder is removed from the Hamiltonian I1~ showing that the eigencnergies of the disordered crystal are those of the ordered one but that the wavefunctions are modulated through the e,~. We assume that the Raman scattering originates mainly from the first order Raman tensor of the r’~ mode. In the oriented gas model the crystal polarizability is simply the sum of the polarizahilities, x~”.of all the molecules of the crystal. The Raman scattering intensity, in a classical right angle Raman geometry along the crystal axes (a, b). is given by
Lx~~( ~k)Ojk)
a ~
~(w
2. One-vibron crystal states with orientational disorder Let us calculate the one-vibron crystal states for the stretching mode v~ of N2O in the absence of the Fermi resonance. It was established by Zumofen [8], in the case of a-CO, that the head—tail orientational disorder does not change the energetics of the crystal but only modifies the phase of the wave functions when the vibron modes interact through long range dipole—dipole interactions. The disorder, which is of Ising type, can he introduced through pseudo-spins e~ ±I randomly distributed and defined relatively to a given ordered configuration. We shall use a cornposite index m which labels the lattice vector m, the molecule site s and the degeneracy x if any. The harmonic Hamiltonian for the i.’~mode can he written as =
H1
Ewja~a,~+
=
L~1~a~a,1.
=
=
~ (2) 1 means that the molecule m has flipped relatively to an ordered configuration for which E + 1 Setting E,,f,
=
E
=
—
=
a,,
(-~)
=
e,,a,,
.
with 05(q)
~em exp[—iq m]. (5) N k is the wave vector, ~ the branch index. N the number of unit cells. w(~k)the eigenmode frequency and sp~(~k) the amplitude of the mode on site .r. Since in Raman spectroscopy the light is scattered by a macroscopic system we calculate configuration-averaged properties. We assume that there are no correlations of orientations hetween the four sites of the unit cell and between the unit cells and setting p. the concentration of pseudo-spins which have not been flipped, we get =
(Ojq))
—
(3)
~exp[—iq
=
N (2p
mJKe~,~
I) ~(q). (6) 4p( 1 —p) Ko~(q)01( q’ )~ ~ N ~ q q’) 2p 1) i(q) ~.l(q’). (7) +( The first term describes the incoherent diffraction in a random substitutional mixture and equals zero for p 0 (or I) which corresponds to an =
In the above equation a~ and a~ are the crcation and annihilation operators for the v~ vibration at frequency w1 i~, (with S 0 if n m) describes theintermolecular interaction between molecules m and n. The potential, 5, of the disordered configuration is simply related to that, i, of the ordered one through the relation
~k)), (4)
(1)
,nn
~1f
w(
—
=
—
=
ordered configuration. The second term describes the coherent contribution and equals zero for p 0.5 which corresponds to a totally disordered configuration. In the latter case the 6~selection rule and the normalization condition reduces the expression of the Raman intensity to I x.~”=
f-P. Lemaistre et a!.
/ Bound states dynamics in
which does not depend on the k wave vector, Consequently the Raman intensity is proportional to the density of crystal states (DOS). 3. Fermi resonance with orientational disorder Nitrous oxide crystal exhibits a Fermi resonance interaction between the stretching mode v1, at frequency w1, and the first overtone of the doubly degenerate bending mode v2, at frequency 2w2. We add to the harmonic Hamiltonian H1 the Hamiltonian H2 associated to 2v2: H2 Lw2b,~7h,~ + ~~in,i(2)b,~bri. (8) =
disordered N
559
20 crystals
in which C is a product of the wave functions coefficients and the quantities 0, previously defined, give a spectral representation of the disorder. Generally, in the presence of disorder, the Fermi resonance term V mixes the polarizations ~ as well as the wave vectors k. Using a perturbation technique we define a configuration averaged Green’s function
Thus the intramolecular anharmonic interaction is given by ~
=
2
[a~(b1~)~
~
+
a,n(b,~)21.
a~ and b,~ are the annihilation (creation) operators of molecule m for v1 and v2 vibration modes respectively. The intermolecular potentials L1’(i) (i 1,2 for the v1 and v2 modes respectively) are modulated by e. Defining again primed operators to remove the disorder from H~ and =
—
w~(~k)— 2T2fdz
w —z The poles of the function g gives the bound states. This function is the quitepositions similar of to that derived for an ordered configuration but with p(z) being an averaged DOS which does not depend on the wave vector k. Thus the poles of g are not sensitive to the detailed structure of the free two-vibron continuum at 2w2 and we can ignore the dispersion of the v2 mode. The Raman intensity is reduced to
H2, /w~ \ In
,
In b’=mb~~,
a~=,,1a(1L (10) we note that such a transformation keeps a pseudo-spin in c,~in the anharmonic Hamiltonian. Consequently the eigenstates of the total Hamiltonian cannot be obtained exactly and we have to use a perturbation technique. The resolution of the Schrodinger equation, in the k wave vector representation, leads to a secular equation. The derivation of such an equation can be found elsewhere [71.We get [w
—
w~(~k)] A(~k) = ~
v(~~ )A(~’k’)~ (11)
where
co~(kk
2
~k
p
W
=
1
k2)01(k’
k1
k2)
—w2(~1k~) —w2(~,k2)
Jah(w)aP±~_____)
(15)
with r
~
~-Ii± L
d 2+212
w1
;
d
=
—
2w2 .
2
The densities p~ are the densities of the v1 modes shifted around the frequencies w ± of the free molecule.
4. Results and conclusions Lattice dynamics calculations were performed on a cubic lattice in the reduced Brillouin zone. The one-vibron DOS p~~~(w) was evaluated by means of the Ewald—Kornfeld method. The molecular parameters used for the calculations are w1 1287 cm’, w2= 1173 cm’, 1=20 cm~, (a~/8q1)=89 cm~, (~ji/dq2) 31 cm~ [5]. With these parameters d 57 cm~ and the =
k1k2
=
(12)
(16)
=
561)
.1. -P. Len,ai.stre c/ al.
N
/ Bound
stute.s dvna,nic,s in disordered N,() crystaLs
the v~.We display in fig. 1 the DOS, p(w), which is the self-convolution of the one-vibron density of states. The densities of the v and ~ modes were calculated with w1 (ak) running over the ii,
o 2 2V2
1
I
—10 5 0 I. Calculated DOS p1w)
5 10 ~ cmIhe self-convolution at
Fig. as the one-vibron states with the position and widlh of Ihe i’ bound states (dotted lines).
coefficients ~ involved in the Raman intensity are s~ 0.95 and ij 0.05. Thus we expect the Raman line shape of the v ~ modes to he slightly shifted from that of the v~ modes and described in the harmonic approximation. A more drastic effect is expected to occur for the a modes, Actually the width of the v~ line shape is re=
- =
hand. These densities of states are displayed in fig. 2 (vt) and in fig.~~ (v—). In both figures are plotted the experimental Raman line shapes recorded in a conventional VV geometry. The calculated DOS of the v + modes is identical to that of the i.~ modes meaning that the cubic anharmonicity plays a negligible role and the Raman line shape is well described by one-vibron states. The main difference between the v + Raman lineshape and the DOS of the v modes is a peak the with located calculated at w frequency 1291 cm of which the totally coincides symmetric mode (A) of a perfectly ordered crystal belonging to the space group T4. Moreover, re=
cent nanosecond CARS experiments performed in the frequency domain also revealed such a phenomenon. We formally reproduced this maximum by assuming a partial ordering in the N,O crystal. From the simulation of the Raman line shapes of the t’~ modes we deduced that all the modes have line widths smaller than 1 cm . This is in good agreement with picosecond CARS resuits (T 7 13 ps) obtained by Vallée et al. [9]. On the other hand the width of the a hand is reduced by one order of magnitude relative to that (11.4 cm I) of the t’ + hand as shown in fig. =
duced by the factor
i~
=
0.05 relatively to that of
______________
N20
...
.~- - -.
__________
Tr1OK
N2 0
1290A
~ 1295
1300
cm
110K
/
1
Fig. 2. C ~ilculated DOS (solid line) of the s hound states in N~O disordered crystals. The positions of the tour k = symmetry modes of a fictitious N 2O crystal are indicated. Observed Raman line shape (dotted line) recorded in a Conventional geometry and fitted by assuming a partially disordered crystal (p = )).47 instead of )).5 >.
1164
1165
1166
cm—i
Fig. 3. Calculated DOS (solid line) of the i’ hound states of lotally disordered N~Ocrystals. Observed Rarnan line shape recorded in ii conventional VV geomet~and fitted by assumlug a totally disordered crystal.
/ Bound States dynamics in
f-P. Lemaistre et al.
2. This result, which gives a value of ij 0.05 is in perfect agreement with that calculated. The observed shape of the v band as shown in fig. 3 is in agreement with the long exponential decay time of T2 15 ps measured by the picosecond
13] 14]
CARS technique [9].
15]
—
References [II 121
V.M. Agranovich and 1.1. Lalov, Soy. Phys. Solid State 13 (1971) 859. V.M. Agranovich, in: Spectroscopy and Excitation Dynam-
disordered N,O crystals
561
ics of Condensed Molecular Systems, eds. V.M. Agranovich and R.M. l-Iochstrasser (North-Holland, Amsterdam, 1983) p. 84. F. Bogani and P R. Salvi, J Chem. Phys 81 (1984) 4991. J.-P. Lemaistre, R. Ouillon and P. Ranson, J. Phys. Chem.
92 (1988) 1070. G. Cardini, P.R. Salvi and V. Sehettino, Chem. Phys. 119 (1988) 241. [61G. Cardini and V. Schettino, J. Chem. Phys. 94 (1991) 2502. [71B. Perrin, R. Ouillon. P. Ranson, J.-P. Lemaistre and A.A. Maradudin, Phys. Rev. B 43 (1991) 4451. 181 G. Zumofen, J. Chem. Phys. 68 (1978) 3747. 191 F. Vallee, G.M. Gale and C. Flytzanis. Chem. Phys. Let. 124 (1986) 216.