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p h y s ic a l r e v ie w
V O L U M E 46, N U M B E R 2
b
1 J U L Y 1992-11
T w o -v ib ro n e x c ita tio n s in th e fe rro e le c tric phase o f N a N 0 2
W. B. J. M. Janssen and A. van der Avoird
In stitu te o f Theoretical Chemistry, University o f Nijmegen, Toernooiveld, 6525 E D Nijmegen, The Netherlands
(Received 27 J a n u a ry 1992)
In relation with recently observed o v e rto n e sp ectra for the ferroelectric phase o f solid N a N O : , we
have p e rfo rm ed lattice-d y n am ics calculations.
F ro m a basis o f single-vibron functions c o m p u te d in a
previous p a p er and including both the in tra m o le c u la r and in te rm o le c u la r a n h a rm o n ic ity , we have c a lc u ­
lated the tw o -v ibro n states an d the R a m a n and lum inescence intensities. G o o d a g reem en t with the
different e x p e rim e n ta l sp ectra was o b ta in e d for an in tra m o le c u la r a n h a rm o n ic ity c o n stan t A = 1 . 3 cm
’.
We c o n clu d e th a t the line sh apes in the lum inescence and R a m a n sp e c tra are d e te rm in e d by the o c ­
c u rre n c e o f a q u a sib o u n d bivibron state w hich is w eakly coupled to the tw o-vibron states.
I. I N T R O D U C T I O N
Solid N a N O : is an interesting m olecular ionic crystal
which is extensively studied. Recently, K a t o et a l . ] m e a ­
sured R a m a n sca tterin g and singlet and triplet lum ines­
cence s p e c tra of the overtones of the v 2 vibration in the
ferroelectric phase of N a N O : . In these spectra they
found unexpected s h a rp peaks em b ed d ed in broad multivibron bands. T hey co n clu d ed that a q u a sib o u n d twovibron state was form ed with an a n h a r m o n ic ity p a r a m e ­
ter of 0 .8 ± 0 .5 cm 1 and that the higher vibron states are
truly bound. In this pap er we investigate by means of
lattice-dynamics calculations w h e th e r there is indeed a
quasibound tw'o-vibron state and we c o m p a r e o u r results
with the exp erim en tal spectra.
T he
vibration c o rr e s p o n d s to an internal bending
mode of the nitrite ions. If a vibration in the crystal is
purely h a rm o n ic , the excitation energy o f a doubly excit­
ed state will be exactly twice the excitation energy of the
fun dam en tal state. F o r vibrons this m ean s that there is
no difference between the excitation energy of two f u n d a ­
mental vibrons traveling independently th ro u g h the cry s­
tal (a two-vibron state) and a vibron in which each m ole­
cule is doubly excited (an overtone state). But if the
molecular vibration is a n h a r m o n ic and the energy of the
doubly excited vibration is lower th a n twice the f u n d a ­
mental excitation energy, there are different possibilities.
If the a n h a r m o n ic ity is large c o m p a re d with the vibron
ban d w id th , a bivibron state can be formed in which an
overtone vibration travels th ro u g h the crystal. This state
is som etim es called a bound state and has little coupling
to the states in the two-vibron band. If the a n h a r m o n i c i ­
ty is small, there is a stron g coupling between the o v e r­
tone and tw o-vibron states. T h e energy of the overtone
lies in the two-vibron band, and no bivibron is formed.
The theory of bivibron states was developed by A g r a n o ­
vich and L alo v ,2 B elousov/ and A g ran o v ic h , Dubovski,
and O r lo v 4 h and o t h e r s .7
Experim ental two-vibron
spectra are reported for a n u m b e r of crystals which c o n ­
tain small molecules such as N H 4 ‘ , C 0 32 - , N 0 3~, C 0 2,
N : 0 , and O C S . 10 13 O t h e r theoretical calculations on
two-vibron sp ectra have been perform ed by Dows and
S c h e t t i n o 10 for C O , and by Bogani^ for C O : , N : 0 , and
OCS.
46
In a previous p a p e r 14 we have calculated the f u n d a ­
mental p h o n o n and vibron states of the N a N O , crystal.
In these calculations the in term o lecu lar interactions w'ere
modeled by a sem iem pirical a to m -a to m potential s u p p le ­
m ented with point charges and the m olecular polarizabilities w'ere included by the shell model. T h e results w;ere
in fairly good agreem ent with the experim ental data. In
the calculations presented here, we used the same model
potential and calculated the single-vibron states as in the
previous paper. T he s tr u c tu r e o f the ferroelectric phase
o f N a N O : is also described in that paper.
II. T H E O R Y
A. Two-vibron states
F o r the derivation o f the expressions needed for the
calculation of the two-vibron energies in the crystal, we
used the same ansatz and n otatio n as A g r a n o v i c h , 15 but
follow' a different route. In second q u an tiz a tio n we write
the h a rm o n ic crystal H am ilto n ian for one internal vibra­
tion as
Ho
2 ' v~
nm b I b
n
(l)
n. m
w here 11 is the excitation energy o f the free m olecular vi­
bration, B n and B n are the excitation and deexcitation
o p e ra to rs for the m olecular vibration on the molecule
with position vector n, and V nm is the coupling between
the excitations on molecules n and m. Because of the
translational sy m m e try in the crystal, we can F o u rier
tra n sfo rm the m olecular excitation o p e ra to rs to crystal
excitation o p e ra to rs
1
B
VN
X R n exp( - / k n) ,
n
2
( )
B
t
1
v'N
n
where k is a vector in the first Brillouin zone and N is the
n u m b e r of unit cells in the crystal. T he crystal H a m il ­
tonian can now' be w'ritten as
(3)
831
© 1 9 9 2 T h e A m e ric a n Physical Society
832
W. B. J. M. J A N S S E N A N D A. van der A V O I R D
w it h
2
V(k)
L
(4)
e x p [ / k - ( m —n ) ] .
m ( * n)
T h e eigenfunctions of this H a m ilto n ia n are the vibrons
which can be labeled by k and can be w ritten as
1I- ) = ß LI0 ) ,
(5)
with energy
k
n + v( k).
( 6)
T h e a n h a r m o n ic ity in the crystal is intro d u ced in two
different ways. First, there is the in tra m o le c u la r a n h a rm onicity that is included by the o p e r a to r
(7)
HÜ
n
where A is defined as the difference between the f u n d a ­
mental m olecular excitation energy and half o f the e x c ita ­
tion energy of the doubly excited state. It can be show n
that if the in tra m o le c u la r a n h a rm o n ic itie s o f third and
fourth o rd e r are taken into a c co u n t by m eans of a con tact
tran sform ation", we obtain a H a m ilto n ia n of the form
H = H 0 + / / , . T h e in te rm o le c u la r a n h a r m o n ic ity is i n tr o ­
duced by
*)
T W „ m( B nVB m
// Î
each wave vector K = k + k'. T h e s t r u c t u r e o f the Ham il­
tonian is extremely simple; it can be considered as the
sum of a diagonal m a trix and a c o n sta n t m atrix.
T he formal solution for the eigenstates o f this Hamil­
tonian is given by A g r a n o v i c h 15 by the use of the
G r e e n ’s-function m e th o d . T h e tw o-vibron G r e e n 's func­
tion for a given K c o n ta in s an in tegratio n over the first
Brillouin zone. Since this integration over k can no t be
p e rfo rm ed analytically, it is replaced by a sum m ation
over a grid of k points. This is equivalent to the diagonalization o f the H a m ilto n ia n in Eq. (12), w here k runs
over this grid. F o r convergence, the n u m b e r o f points,
and thus the dim ension o f the m atrix, must becom e veryw
large. S ta n d a rd d iag on alization routines are too slow and
too storage intensive for solving this eigenvalue problem.
But because of the special s t r u c t u r e o f the H a m ilto n ia n , a
simple d iag on alization sch em e can be used. In matrix
form the H a m ilto n ia n can be w ritten as
H( K) = D( K) + C( K) ,
(13)
where D( K ) is the diagonal m a trix with elements
^qk^q^^K
(14a)
—
and O K ) is the c o n sta n t m atrix
2 A ± W( K )
^qk <K) = c
(14b)
N
( 8)
>
We omit the K d e p e n d e n c e of the m atrices and w rite C as
n, m
w here H ’nm describes the interaction between the doubly
excited slates on the molecules n and m. By the use of
Eq. (2), we can write Eqs. (7) and (8 ) as
A -jW
év ’Ik + k ')
+ //,
2
2
k
k'
2
w
k”
(9)
w 11 h
w here 1 is a c o lu m n vector o f length N with all elements
equal to i. F o r a certain eigenvector e \ the eigenvalue
problem looks like
(16)
We now define the scalar a as
a
2
(15)
C = c 111 ,
(£> + c 11 T)e' = (o'e‘
X B kB k B k >Bk + k< k- ,
W{ k )
46
H ’nm eXp[/ k -( m —n )] .
( 10)
c 1V
(17)
and write the eigenvalue problem o f Eq. (16) for row q as
m î f nî
(E q +
We want to obtain the tw o-vibron states and energies,
and so we have to diagonalize the total H a m ilto n ia n
/ / = H () + H u + ƒ-ƒ, in a basis o f h a rm o n ic two-vibron
states. T hese can be w ritten as
2 kk >= «k«k l0 ) •
(11)
Because these states co n tain two crystal excitations, they
can be labeled by two independent wave vectors. T he
m atrix elements o f the H a m ilto n ia n are easily calculated:
<2
Iƒ/ | 2 kk.
kk ) = 6 ( q + q' —k - k ' )
X 6 (q — k )(Ek + Ek<)
2 A - W( k + k' )
AT
( 12 )
F r o m the first 6 function, it is clear that only tw o-vibron
states with equal total wave vector are mixed. Therefore,
the H a m ilto n ia n (12) can be diagonalized separately for
£K
q K + a
If w ' ^ £ q - f £ K
is given by
L
o
' e
(18)
q •
1
, then the c o m p o n e n t q o f eigenvector e
a
(
o
‘
-
(
E
q
(19)
+
E
K
~
q
)
Inserting Eq. (19) into Eq. (17) gives
i=2
(
(0
(Eq + eK
20 )
)
T h e eigenvalues col can thus be found as the zeros of the
function
co
(
E
q
+ eK - q )
1
( 21)
E q uation (20) is the same as Eq. (20a) derived by A g r a n o ­
v i c h 15 by the use of G r e e n ’s functions. T h e function f U o )
is singular for co
K - q , i.e., when the state with en£
q
+
£
T W O - V I B R O N E X C I T A T I O N S IN T H E F E R R O E L E C T R I C
46
ergy co lies within the tw o-vibron band. In the G r e e n ’sfunction m e th o d , these poles are avoided by ad din g a
small im aginary c o m p o n e n t to co. H ere we deal with a
finite grid o f points q and we use an a lg o rith m that finds
l0 in a stable m a n n e r, even when it coincides with
f q + EK q or lies close to such a pole. It can be proved
(hat between every two values o f £q + £K -q there will be
exactly one co' which c a n n o t be equal to one o f the d ia g o ­
nal elements £q + £ K - q unless this element is d e g e n e r a t e . 10
Such a zero o f f ( c o ) can be found by the N e w to n Raphson procedure. If the eigenvalues co1 are k n o w n , the
corresponding eigenvectors can be found by the use of
Eq. (19), where a is simply the n o rm a liz a tio n c o n sta n t of
the eigenvector. N ext, we consider the case of degenerate
diagonal elements. If g is the degeneracy of a given d ia g ­
onal element, then there are g — 1 roots co1 which are ex­
actly equal to this element. In this case it follows from
Eq. (18) that a = 0. F r o m Eq. (17) and the fact that
0,
we obtain the following co n d itio n for the c o rre s p o n d in g
eigenvectors:
0
(
22 )
For all the c o m p o n e n ts <?q with vectors q for which
Eq + e K _ q^ c i / , we k n o w from Eq. ( 19) that e ‘q = 0 . This is
sufficient to d e te rm in e all the rem aining eigenvectors.
833
2k Vih>= 2 > g < K )/îX
- J 0V'b>
(27)
•
T h e electronic transition dipole m o m e n t is expand ed
up to second o rd e r in the norm al co o rd in a te s o f the vi­
bration,
Mn = d11'£?„ 4- d ' 2
2 4-
£
d ™ Q nQ m
m (
(28)
,
)
w it h
cl
d
( 11
0
9 <?n
^2
d
(
el
o Mn
2)
(29)
ÖQI
^2
o
el
2)
d nm
(
m
JO
We have o m itte d the label n for d 1 and d 2 because all
the nitrite ions in N a N O : are equivalent. W ith the aid of
the relation
2 <Ek r ' / 2 e x p ((k n ) ( ö k 4 - ß +_ k ) ,
(30)
V2N
we can calculate the transition dipole m om ent in Eq. (25)
elnvib
<od 2(.vlV l r k'ovlb>
B. L u m in escen ce and R a m a n -sc a tte rin g intensities
8( K _ k )
If we want to c o m p a r e the calculated two-vibron spec­
tra with laser lum inescence spectra, we have to calculate
the lum inescence intensities o f the tw o-vibron transitions.
The intensity is p ro p o rtio n a l to the oscillator strength of
the transition: \ ( f fi / ) | \ where / ) is the initial state,
ƒ ) the final state of the lum inescence process, and fj. is
the dipole m o m e n t o p e ra to r. In the lum inescence exp eri­
ments o f K a t o et u /.,1 the initial state is an electronically
excited state (singlet or triplet) with no vibrations excited:
/->
r k'ovib >
(23a)
.
rhe final state in the lum inescence process und er c o n ­
sideration is the electronic g ro u n d state with a twovibron state excited:
(23b)
ƒ > = i0el2k vib> •
For the electronically excited state, we can assum e the
exciton model
î
ik>
VN
V ?/v
n
H
!0^ ) .
(24)
1
< / | m|/) = _ ^ 2
e x p ( / k . n )<2{;v' V n l 0 vib> ,
(25)
n
where
M n ^ O n V J 'n >
q
q
x [ 2 d i2i+ i d q2,+ d K:Lq n
.
(31)
w it h
d
2
I
)
m
d „ m ex p [ / q - ( i n —n )]
(32)
nI
F ro m the 6 function follows the selection rule that the k
vector is conserved in the luminescence process.
In the calculation o f the luminescence intensities, we
assum ed that the transition dipole m o m e n t of the nitrite
ion is mainly dependent on the vibration of the molecule
itself an d is little influenced by the vibrations o f the o th e r
molecules: d > > d nm. This simplifies Eq. (31), and the
oscillator streng th now becomes
(oe'2iryib\u rJov,b>I2
(2 )
^
e q ( K ) ( £ q£ K _ q ) ~ 1/2 ’
(33)
m ( * n)
Fhe transition dipole m o m en t then becomes
X ^
q
q
d
Y e x p ( / k n ) r n'>
) 1/2
[g <( K ) ]*(e e
(26)
is the electronic transition dipole m o m e n t on molecule n.
The two-vibron states are solutions of Eq. ( 16):
T he R a m a n -sc a tte rin g intensity is p ro p o rtio n a l to
x j </1 a K\i l/#) l : » w here a /fl are the c o m p o n e n ts o f the
unit-cell polarizability tensor. In the scattering process
observed by K a t o et a /.,1 the initial state is the v ib ra tio n ­
al g ro u n d state and the final state is a two-vibron state
with K = 0. In o rd e r to calculate the scattering intensi­
ties, we can expand the polarizability in the same m a n n e r
as the transition dipole m o m e n t in Eq. (28). T h e final ex­
pression that we obtain for
a /fl\i ) 12 is identical
to the rig h t-h a n d side of Eq. (33) with K = 0. In this case
834
W. B. J. M. J A N S S E N A N D A. van der A V O I R D
w'here all the m o lecu lar degrees of freedom were taken
into a cco u n t sim ultaneously, it was clear that the cou­
pling between the v 2 vibron and o th e r vibrons and lattice
vibrations is very small and can be neglected. Therefore,
in these calculations, only the v 2 bending m od e o f the ni­
trite ion is considered.
T h e interaction between the doubly excited states on
the molecules n and m is given by
d : is replaced by the sum over the second derivatives of
the c o m p o n e n ts of the polarizability tensor with respect
to the n orm al c o o rd in a te o f the v 2 vibration and we have
again neglected the c o n trib u tio n s of the o th e r molecules.
III. C O M P U T A T I O N A L A S P E C T S
T h e vibron energies £q that form the diagonal elements
o f the tw o-vibron H a m ilto n ia n in Eq. (12) are calculated
as described in a previous p a p e r . 14 We have used the
a to m - a to m potential which was developed by LyndenBell, Impey, and Klein in m o le c u la r-d y n a m ic s c a lc u la ­
tions on solid N a N 0 2. ’ It has an exponential short
range and an r h dispersion c o n trib u tio n , and the elec­
tro static in teraction s are modeled by point charges. F o r
the in tra m o le c u la r potential of the nitrite ion, we used
the force field of W esto n and B r o d a s k y 18 developed for
the nitrite ion in crystalline N a N 0 2. In o rd e r to obtain
the correct L O -T O splittings of the fu n d a m e n ta l vibrons,
we adjusted the transition dipole m o m e n ts of the vibron
modes. T h e polarizability of the ions w'as taken into a c ­
count by m eans of the shell m o d e l . 19,20 Each ato m c o n ­
sists of a core that co n tain s the total mass and a massless
shell that follows the core m otions adiabatically. T h e in­
teraction between the core and shell is p a ra m e triz e d by a
force co n stan t which is related to the atom ic polarizabilities. F r o m o u r calculations in the previous p a p e r , 14
W nm
7
1
m
t
22 2
a ,ß A.,À'
46
W nm
<2 nO JH O n2 m> ,
(34)
w'here V is the crystal potential and l2n ) is the doubly ex­
cited m o lecu lar v 2 vibration on molecule n. If we make a
T a y lo r expansion o f the crystal potential up to fourth o r­
der in the n orm al c o o rd in a te s o f the m o le c u la r vibration
and apply the well-known rules for m a trix elements of
h a rm o n ic -o s c illa to r functions, we obtain
W nm
1
34K
8 il:
à Q l d Q lm
(35)
o
T h e ato m ic d isp lac em en ts are related to the n o rm a l co o r­
dinates by the eigenvectors / of the m olecular vibration,
u Kan
(36)
I'kciQ u »
w here a labels the a to m s in molecule n and À the c a rte ­
sian directions. If we consider the crystal potential as the
sum of a to m - a t o m potentials we obtain for Eq. (35)
a4Vnamß
du Aan du
an du nßm
(37)
n’ßva
o
where F n((m/j is the interaction between a to m a o f m ole­
cule n and atom ß o f molecule m. F o r the calculation of
W __,
nm we have used the sam e a to m - a t o m potential as in
the calculations o f the fu n d a m e n ta l excitations. A l ­
th o u g h the a to m - a to m potential co ntains C o u lo m b in­
teractions between point charges, we take the fourth
derivative o f this potential and there are no 0 —*2 tra n s i­
tion dipoles. So there will be no L O -T O splitting o f the
tw o-vibron levels. E xperim entally, this splitting has not
been observed e i t h e r . 1
F o r the calculation o f the tw o-vibron energies for
K = 0, w'e have used a basis o f 18 413 h a rm o n ic twovibron functions (q points). F o r the total density o f twovibron states and the lum inescence s p e c tru m induced by
b ro a d b a n d excitation, a fu rth e r integration over the Brillouin zone has to be perform ed. This was done by the a p ­
plication o f the q u a d ra tic integration schem e developed
by W iesenekker, te Velde, and B a e re n d s ,21,22 using 50 K
points. In this case the energies in every K point were
calculated with a basis o f 2411 h a rm o n ic two-vibron
functions.
IV. R E S U L T S A N D D I S C U S S I O N
First, w'e have calculated the tw o-vibron spectra for
K = 0. F ro m the diagonalization of the tw o-vibron H a m ­
iltonian results a c o n tin u o u s band of eigenvalues and no
bound bivibron state is split off. In Fig. 1 we have plot­
ted the n u m b e r o f eigenvalues as a function o f the energy
for K = 0. This K = 0 density o f states in Fig. 1 w'as cal­
culated using an a n h a r m o n i c i t y c o n sta n t A o f 0.8 cm
as estim ated by K a to et al. from luminescence s p e c t r a . 1
Frequency
( c m ’ 1)
F IG . 1. D ensity o f tw o-vibron states for K = 0 and A = 0.8
cm
T W O - V I B R O N E X C I T A T I O N S IN T H E F E R R O E L E C T R I C . .
46
Frequency
835
cence s p e c tru m at K = 0 calculated from Eq. (33) has the
same shape as the calculated R a m a n sp ec tru m ; they differ
only by a co n stan t factor. T h e results are plotted in Fig.
2. O ne observes that the line shape changes d ram a tically
if the a n h a r m o n ic ity co n sta n t is ch an g ed by only 1.0
cm
F o r A = 0 . 3 cm 1 the o v erto ne state is so s t r o n g ­
ly coupled to the two-vibron states that it is not visible in
the s p e c tru m and this s p e c tru m resembles the K = 0 d e n ­
sity of states. For A = 0 . 8 cm 1 we begin to observe a
q u a sib o u n d state. T h e peak resulting from this q u a si­
bo un d state appears on the low-energy side of the spec­
tru m . F o r , 4 = 1 . 3 cm 1 the quasib ou nd state becomes
m uch m ore p ro n o u n c e d and the s p e c tru m consists of a
s h a rp peak with a broad sh o u ld er on the high-energy
side. It is still q u a sib o u n d , however, because it lies within
the band o f two-vibron states.
We can c o m p a re the sh ape of the calculated K = 0
luminescence and R a m a n -s c a tte rin g sp ectru m with the
n a rro w -b a n d singlet luminescence a n d R a m a n -s c a tte rin g
spectra m easured by K a to et a l . ] In the luminescence ex­
perim ents the N a N O : crystal was excited by a laser with
a p h o to n energy of 17 cm 1 above the excitation energy
of the singlet exciton state. In this way, besides the sing­
let exciton, only p h o n o n s with energies below 17 cm 1
can be excited. These acoustic p h o n o n s have wave vec­
tors close to the zone center, and therefore, also, the exci­
tons will have small K | . Because of the low te m p e r a tu re
in the ex p erim en ts (2 K), the redistribution of the exciton
wave vectors over the Brillouin zone will be slow 23 24 and
the luminescence s p e c tru m is generated by states with
wave vectors close to zero. In Fig. 3 we have c o m p a re d
the 2 vt line shape m e asu red in these experim ents with the
luminescence and R a m a n -sc a tte rin g sp ectru m calculated
( c m ’ 1)
F IG . 2. L u m in escen ce intensities calcu lated for different
values o f the a n h a rm o n ic ity c o n stan t A.
From the eigenvalues and eigenvectors of the two-vibron
H am ilto n ian , it is clear that a q u asib o u n d bivibron state
is present on the lower-energy side o f the sp e c tru m , but is
not split off. It is not visible in the K = 0 density o f states
because it is only a single state a m o n g 18 412 two-vibron
states. C alcu lation s with o t h e r a n h a r m o n i c i ty c o n sta n ts
A show that the K = 0 density of state is not significantly
influenced by value o f A.
This is different for the luminescence and R a m a n in­
tensities o f the two-vibron line. Because o f the large u n ­
certainty in the a n h a r m o n ic ity c o n s ta n t given by K a to
et al . y , 4 = 0 . 8 ± 0 . 5 cm
, we have calculated these in­
tensities for , 4 = 0 . 3 , 0.8, and 1.3 cm '. T h e lum ines­
in
c
<u
1640
1630
1620
1640
1630
Frequency
1620
1640
1630
1620
( c m ')
F IG . 3. C o m p a riso n betw een the m easured R a m a n -sc a tte rin g line (left), n a rro w -b a n d singlet lum inescence line (middle), and c a l­
cu lated R a m a n and lum inescence lines at K = 0 (right). T he energy scale is reversed, and the ex p erim en tal peaks are shifted to the
position o f the calcu lated peak.
W. B. J. M. J A N S S E N A N D A. van der A V O I R D
836
1640
1635
Frequency
1625
1630
(cm
-1
)
1640
1635
Frequency
1630
46
1625
-1
(cm " )
F IG . 4. C o m p a riso n betw een the m easured (left) and calcu
lated (right) triplet lum inescence intensities.
for A = 1.3 cm ', which gives the best co rresp o n d en ce.
In o rd e r to c o m p a r e the calculation with the ex p eri­
ments, we have reversed the energy scale.
K a to et al. also perfo rm ed m e a s u r e m e n ts on the triplet
luminescence spectra o f the overtones of the v 2 vibration.
T he N a N O : crystal was excited by a m e rcu ry lamp,
which leads to a b ro a d b a n d excitation. Beside the triplet
exciton, lattice vibrations with wave vectors th ro u g h o u t
the Brillouin zone are generated. Triplet excitons with
wave vectors in the entire Brillouin zone give rise to
luminescence. We have integrated the lum inescence in­
tensities over the Brillouin zone, and under the a s s u m p ­
tion that the K distribution of triplet exciton states is u n i­
form, we can c o m p a re o u r result with the m easu red 2 v 2
line. T he c o rre s p o n d e n c e betw'een the m easured and c a l­
culated lines is good, as can be seen in Fig. 4, where we
have again used an a n h a r m o n ic ity c o n sta n t A = 1.3
cm '. T h e bivibron peak in the calculated s p e c tru m is
very n a rro w because the in te rm o le c u lar a n h a r m o n ic ity
B r( K ) that causes the dispersion o f the bivibron is very
small, typically 0.005 cm ', and the coupling to the twovibron states is weak.
Finally, we may c o m p a r e the calculated luminescence
intensities with the s p e c tru m m easured by b ro a d b a n d ex­
citation o f the singlet exciton state of the N a N O , cry s­
t a l .25 Since the dispersion of this singlet state is o f the
same o rd e r of m a g n itu d e as the dispersion of the twovibron band, we have to accou nt for the exciton d is p e r­
sion. In the calculation o f the luminescence s p e c tru m of
the single-vibron state in the previous paper, we have as­
sum ed a cosine-shape dispersion of the singlet exciton
band and a width of 5.0 cm ~ '. This yielded very re a s o n ­
able results for the line shape o f the v 2 lu m in escen ce , 14
and so we use the same model in the calculation o f the
line shape o f the 2 v : lum inescence following b ro a d b a n d
singlet excitation. F o r these calculations we again used
.4 = 1.3 cm 1 because in the previous calculations this
gave a good c o rre s p o n d e n c e with the ex perim ental re­
sults. In Fig. 5 we have c o m p a re d the calculated line
shape o f the b r o a d b a n d singlet luminescence peak with
the m easured one. We observe that the width of the mea-
Frequency
(cm
')
Frequency
(cm "’)
F IG . 5. C o m p a ris o n betw een the m easu red (left) a n d calcu
lated (right) b ro a d b a n d excited singlet lum inescence intensi
ties. T h e c a lc u la tio n s were p e rfo rm e d with A = 1.3 cm '.
sured band is about 3 cm 1 larger than the calculated
width. In the calculations in the previous p a p e r 14 on the
singlet lum inescence peak of the v 2 vibration, it was
show n th at the w idth of the calculated band was also
about 3 cm 1 too small. This leads to the conclusion that
the assum ed width o f the singlet exciton band is probably
slightly u n d e re stim a te d . T h e calculated band has a
s h a r p e r peak at the lower-energy side originating from
the q u asib o u n d states. In the m easured b and this peak is
not so p ro n o u n c e d . T h e 2 v : band was m easured at
r = 4.2 K. It was d e m o n s tr a te d that at such low t e m p e r ­
atures the singlet states are not uniformly distributed
th ro u g h o u t the Brillouin z o n e 23,24 and the m easured line
shape might be distorted because of this n o n u n ifo rm dis­
tribution.
In conclusion, we can say that the coupling between
the nitrite o vertone vibration and the two-vibron states in
solid N a N O : is weak enough to allow for a quasibound
bivibron to be formed, as was already concluded from the
luminescence ex p erim en ts by K a t o et a l . 1 If we assume
an a n h a r m o n ic ity c o n sta n t of 1.3 cm 1 for the v 2 bending
vibration of the nitrite ion, which is so m ew hat higher
th an the value of 0. 8 ± 0 . 5 cm 1 estim ated by K a to et al
we can very well explain the luminescence line shapes
m easured by n a rro w -b a n d excitation o f the singlet exci­
ton state and by b r o a d b a n d excitation o f the triplet exci­
ton state, as well as the R a m a n line shapes. Also the ca l­
culated luminescence line shape for b r o a d b a n d excitation
of the singlet exciton state is in reasonable agreem ent
46
T W O - V I B R O N E X C I T A T I O N S IN T H E F E R R O E L E C T R I C . . .
w ith the m easu red line shape, if we assum e a cosine­
s h a p e d dispersion for the singlet exciton band and a
w id th of s o m e w h a t m o re than 5 cm
1 for this band.
ACKNOW LEDGM ENTS
W e th a n k Jü rg en K ö h le r for stim ulating discussions on
837
viding the alg o rith m to find the eigenvalues and eigenvec­
tors. T h e investigations were s u p p o rte d in part by the
N e th e r la n d s F o u n d a tio n for C hem ical Research (SON)
with financial aid from the N e th e r la n d s O rg a n iz a tio n for
Scientific R esearch (NWO). Part of this work has been
p erfo rm ed as an IB M -A C IS project.
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