Molecular to atomic transformation in solid iodine under high pressure
TADEUSZLUTY
Institute of Organic and Physical Chernistn, Technical universih of Wrocla~v.Wroclaw, Poland
AND
J O H N C .RAICH
Department of Physics and Condensed Matter Sciences Laboratory, Colorado State University, Fort Collins, CO 80523, U.S.A.
Received October 16, 1987
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This paper is dedicated to Professor J. A. Morrison
TADEUSZ
LUTYand JOHNC. RAICH.
Can. J. Chem. 6 6 , 8 12 (1988).
Structural transformations in solid iodine under high pressure are analyzed from the point of view of symmetries of possible
charge density distortions coupled to phonons. The transition from the high pressure atomic phase to the low pressure molecular
phase is interpreted in terms of phonon softening and formation of molecular bonds and is seen as a condensation of a charge
density distortion wave coupled to a phonon mode.
TADEUSZ
LUTYet JOHNC. RAICH.
Can. J. Chem. 6 6 , 812 (1988).
Dans le but d'exarniner les symktries des distorsions possibles dans les densites de charges coupltes aux phonons, on a ttudie
les transformations structurales de l'iode solide, sous pression. On interprkte la transition de la phase atomique a haute
pression vers la phase moltculaire a basse pression en fonction d'un ramollissement du phonon et de la formation de liaisons
molCculaires et on la considkre comme une condensation d'une onde de distorsion de la densitt de charge qui est couplee avec
un mode phonon.
[Traduit par la revue]
1. Introduction
Solid iodine has been the subject of many experimental and
theoretical studies in part because of its interesting properties
under pressure. The dynamical properties of the crystal are of
particular interest because they appear to have characteristics
intermediate between those of molecular solids and of metals.
This intermediate character of solid iodine is evident from
experimental studies, which indicate an insulator-metal transition at about 16GPa (1). According to recent X-ray crystal
structure analysis, the insulator-metal transition is followed by
a structural phase transition associated with molecular dissociation at 21 GPa (2). It has been known for a long time that there is
a chemical bond formation involving 5 p electrons of nearest
iodine molecules in the crystal under atmospheric pressure (3).
A common opinion is that the intermolecular bond is of a charge
transfer nature, that is, a dative structure involving ionic and
covalent bonds between nearest atoms of different molecules
is formed. Theoretical studies of the lattice dynamics (4, 5 )
demonstrated the importance of charge distribution of the
intermolecular bond.
Structural studies of iodine under pressure (2) stimulated
this theoretical effort to construct a model for charge density
distortions and lattice dynamics which gives a phenomenological explanation for the observed structural changes.
The generally accepted viewpoint is that the application of
pressure changes the balance between intermolecular chargetransfer interactions and intramolecular covalent bonds, so
that the crystal becomes metallic and molecular dissociation
follows. Because of the relatively simple crystal structures, it
is believed that the pressure-induced transformations in solid
iodine can serve as a model to study charge density distortions
in molecular crystals caused by applied pressure. The current
studies elucidate the transition between the metallic bond and
the ionic-covalent bond, so far only weakly suggested by
McMillan (6) within the charge density wave concept. The
transformation of solid iodine under pressure can be viewed as a
solid state chemical reaction (2N IT-LNI,). Thus, it is believed
that concepts needed to understand the transformation in solid
iodine will also be useful for an understanding of solid state
reactions (dimerization, polymerization) in more complicated
systems.
Very recently the description of molecular dissociation of
iodine at p > 21 GPa became controversial. On the basis of
Mossbauer effect measurements (10) it has been argued that
"up to 30 GPa the I, molecules remain the building blocks of the
high pressure structural modifications". The need for assuming
a "zig-zag" structure is stressed in ref. 10. We hope that the
analysis presented in this paper will provide a compromise
(a polymeric structure) between the two interpretations of the
transition.
In the present paper we start with a brief description of the
iodine crystal structures and phase transitions under pressure.
In section 3 we formulate the 16-bond model and determine the
possible order parameters for the phase transformations. The
next section is devoted to demonstrating that the 16-bond model
is equivalent to a 2-bond model in reciprocal space. It is shown
that the order parameters can be identified with amplitudes
of charge density distortion waves. Finally, we consider the
coupling of the charge density distortion waves to phonons.
2. Structures of solid iodine
The low-pressure (LP), molecular phase of iodine is orthorhombic, space group Dl,,-Cmca, with two I, molecules per
primitive unit cell (7). The iodine molecules form layered
structure in the (bc) plane, as shown in Fig. 1. When pressure
is applied to the crystal the iodine molecules tilt around the
crystallographic n axis in such a way that the angle between the
two translationally nonequivalent molecules in the (Dc) plane
comes closer to ~ 1 2In. the range of stability of the molecular
phase ( p < 21 GPa) no change in the intramolecular bond length
has been observed (2) although an insulator-metal transition
occurs at about 16 GPa (1). This purely electronic transition is
813
.ND RAICH
45 GPa to a face-centred tetragonal structure (1 1) (space group
D4h-14/mmm) and the other at 55 GPa to a face-centred cubic
structure (12) (space group Oh-Fm3m). The formation of the
tetragonal structure (VHP phase) can be seen as decreasing the
shear strain defined in eq. [I], when the pressure increases
to approximately 45 GPa. In the present paper this very high
pressure tetragonal phase is labeled VHP. Electronic band
structure calculations have been performed for the tetragonal
structure (8).
A comparison of the atomic positions in the LP and HP phases
is the basis for a phenomenological model of the LP
HP
transition (2). The model uses atomic displacements as order
parameters without any reference to charge density distortions.
The present study also offers a simple phenomenological
model for the transition and also combines the charge density
distortions and phonon instabilities.
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-
FIG. 1. The (bc) planes of the molecular phase, stable at atmospheric pressure, of solid iodine.
3. 16-Bond model
In the considerations which follow we treat the tetragonal
VHP phase as the parent phase. Following arguments of
Takemura and co-workers (2) it will be assumed that the
dimerization of the iodine atoms occurs primarily in the (a, cH)
or (bLcL)plane as the applied pressure is decreased. Hence,
we believe that a two-dimensional model will be adequate to
describe the essential features of the phase transitions. The other
basic assumption of the postulated model is that we describe
the electronic bonds in solid iodine in a manner similar to the
description used by Blinc et al. (9) for hydrogen bonding.
Let us consider the VHP tetragonal phase. Each atom has four
nearest neighbors in the (a, cH)plane. In the plane the unit cell
is a square and electronic densities (or probabilities) in the four
bond directions are equal. This situation is shown schematically
in Fig. 3a. The HP phase results from the VHP phase by an
application of a shear strain resulting in c,
a, and electronic
densities which are different in the orthogonal directions. The
schematic of the electron densities for this case is given in
Fig. 3b. For the diagrams of Fig. 3 the lengths of the lines
indicate the strengths of the bonds and not the lengths of the
bonds. One would expect an inverse relationship between bond
strengths and bond lengths or interatomic distances.
The molecular LP phase has a larger unit cell than the VHP or
HP phases with four atoms in the (bc) plane in each unit cell.
The strengths of the bonds with nearest neighbors for a typical
atom in the molecular phase are shown schematically in Fig. 3c.
In order to describe all possible phases of iodine in terms
of Brillouin zone center instabilities we consider the unit cell
shown in Fig. 4. This unit cell contains eight atoms in the
crystal plane, hence is twice as large as needed to describe the
molecular phase. 'The two-dimensional basis is a square lattice.
From Fig. 4 it is clear that two choices for unit cells are possible.
In the following we shall consider the 16-bond model as
represented in Fig. 4. We assume that the electronic interaction
energy for two-particle interactions can be written in the form
+
FIG.2. The iodine atoms on the (ac) plane of the atomic phase at
30 GPa, according to ref. 2.
followed, according to ref. 2, by the first-order structural phase
transition at 21 GPa which is essentially molecular dissociation.
The high-pressure (HP), monoatomic phase forms a bodycentred orthorhombic lattice, space group D2h-lmmm,with one
iodine atom per primitive unit cell (2). The structure of the phase
at 30 GPa according to ref. 2 is shown in Fig. 2. In terms of the
basic vectors of the molecular (LP) phase the atomic phase
forms a monoclinic structure. If the larger unit cell of the LP
phase is considered, the HP monoatomic structure appears as
a shear strained LP molecular structure when projected on the
(bc) plane. The distortion angle a can be expressed in terms of
the strain, e,, = e4, and the basic vectors of the atomic phase,
a, and CH,
[l]
c i - a:{
e4 = ctg a = ~
~
H
C
H
Relationships between crystal axes of the LP phase ( a L , bl-,
and cL) and those of the HP phase (a,, bH, and cH) are:
Recent X-ray diffraction studies of solid iodine have revealed
that it undergoes two additional phase transitions, one at about
I
[31
1
U2-- 2 C
uapa p n a n p
where na is the occupation probability for the a t h bond in the
unit cell. Unit cell indices are neglected in eq. [3] because
the chosen cell is the largest one and amp is the interaction
parameter summed already over the unit cells. In eq. [3], a , P =
1, 2, ..., 16. The form of the interactions can be justified
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VHP
FIG. 3. A schematic representation of electronic densities in four directions around an iodine atom in the ( a ) atomic tetragonal VHP, ( b ) atomic
orthorhombic HP, and ( c ) molecular LP phases.
= &TIl6 =
[41
Sn
tin2 = Sn4 = Sn5 = Sn7 = 6 n l o = SizI2 = tinl3
= S n I 5= - S n
Here Sn denotes a change in the occupation probability n of a
particular bond. From the eigenvector [4] an order parameter
of Big symmetry can be constructed as follows:
[5]
-q = { n l
+ n 3 + n6 + n8 + n9 + rill + n 1 4 + n16
- n 2 - n 4 - n5 - n 7 - n l o -
FIG. 4. The 8-atom two-dimensional unit cell with 16 interatomic
bonds.
~112-
n13- n15}
Now, it is seen that a distortion in the bond probabilities which
is of B 1 , symmetry can be directly assigned to that observed in
the HP orthorhombic phase. The corresponding configuration
of bonds associated with a given iodine atom for that phase is
shown in Fig. 3b.
We expect that the LP molecular phase can be described by
changes in the bond occupation probabilities from those of the
parent phase which transform according to the E, representation. The eigenvector with this symmetry has the form:
on the basis of Coulomb interactions between bond charge
Snl = Sn7 = Sn9 = tinl5 = 81
distributions.
We postulate that structural phase transitions in solid iodine
are a consequence of changes in occupation probabilities of
interatomic bonds. The changes will correspond to eigenvectors
of the interaction matrix { a a p } . For the 16-bond model the
eigenvectors will transform according to following irreducible
It follows from the form of the interaction energy, given by
representations of the D4,, point group: 2A I,, 2A2,, 2 B l g ,
eq. [3], that the eigenvalue problem to be solved has the form:
2B2,, 4Eu. The eigenvectors with A l , symmetry correspond
to trivial solutions. Hence, all possible changes in the bond
occupation probabilities corresponding to the various phases of
solid iodine must be linear combinations of eigenvectors with
For example, if interactions between bond charges are limited
A2,, Big, B2,, and E, symmetries. Because we limit ourselves
to nearest neighbors, the eigenvalue problem with eigenvectors
to the two observed phases of solid iodine, an inspection of the
[6] can be simplified to:
schematic representation of HP and LP phases in Figs. 3b
and 3c allows us to reject eigenvectors with A2, and B?,
symmetries. None of the distributions shown in these figures is
symmetric with respect to the four-fold axis as AZgsymmetry
requires or with respect to diagonal symmetry planes (ad)
required by the B2, representation. Therefore, we can conclude
that the changes in bond occupation probabilities, which will
drive the structural changes to the HP and LP phases must
where a is the nearest neighbor interaction strength. The
transform according to B l , and E, representations.
solutions for the eigenvalues and eigenvectors for the nearest
An eigenvector corresponding to B l , symmetry has the form,
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LUTY AND RAICH
FIG.5. A representation of the four-component order parameter 5 representing four possible domains of the molecular phase.
neighbor case are:
Although the eigenvalues change when further neighbors are
considered, the symmetry of the eigenvectors given by eqs.
[gal, [ 8 b ]and [ 8 c ]does not.
The first solution to the eigenvalue problem, eq. [ g a l , must
be rejected because bond-charge conservation condition,
61 + 26m
+ 6n = 0
is not fulfilled. For the second solution, eq. [8b ], the eigenvector
has the form of eq. [6]with 61 = -6n, and 6m = 0 . For that
case the following order parameter of E, symmetry can be
constructed,
It is easy to verify that the order parameter 5 is orthogonal to
the order parameter q and satisfies the charge conservation
condition. As there are four possible choices of equivalent
eigenvectors of the form [ 6 ] ,one can construct three additional
equivalent order parameters,
[9dl t4= {nl + n3 + n l o + n12 - n2 - n ~ - n9 - rill)
The distortions in bond occupation probabilities corresponding
to E, symmetry and represented by the four-component order
parameter 5 correspond to the distortion shown in Fig. 3c. This
figure schematically represents the bonds around an iodine atom
in the molecular phase. Four components of the order parameter
lead to four equivalent structures which represent distortions
with E, symmetry from the assumed parent phase, the tetragonal VHP phase. The structures are represented in Fig. 5 .
It is seen that an increase in a bond occupation probability,
corresponding to a positive En, can be identified with a
formation of a chemical bond which will cause the dimerization
of the atomic phase and the formation of the molecular phase
of iodine.
The third solution [ 8 c ] of the eigenvalue problem [7] also
has E, symmetry. This distortion in the bond occupation
probabilities corresponds to the eigenvector of eq. [ 6 ] with
6m = - 61 = - 6n. The order parameter which corresponds to
this eigenvector is the four component order parameter 5 with
the components:
[IOU]
=
{ n l + n3 + n5 + n7 + n9 + n l 1+ n 1 3+ n 1 5
- n 2 - n4 - n6 - 118 - n l o - n 1 4- n 1 6 )
A graphical representation of the order parameter is shown in
Fig. 6 . It is seen that the charge density distortions characterized
by the order parameter 5 correspond to polymerization of
the iodine atoms leading to molecular chains (I),,.Thus, the
possibility for polymerization and the formation of a polymerized phase is permitted by the present model. It is quite
possible that this is the "zig-zag" chain phase postulated in
ref. 10 as an intermediate phase which exists in the pressure
range 16 < p < 21 GPa. One should notice that this postulated
"polymeric" phase has the same symmetry as the molecular
phase so that the metal-insulator transition at 16 GPa would
indeed be iso-structural.
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816
FIG. 7. The bond charge density distortion wave of B I , symmetry
in the tetragonal phase of solid iodine.
of the matrix + ( q ) ,
FIG. 6. A representation of the four-component order parameter
a hypothetical polymerized
phase of iodine.
5 representing four domains possible of
have to be positive definite. Eigenvectors which correspond to
the above eigenvalues are
4. Bond charge density distortion waves
The distortions in the bond occupation probabilities discussed
for the 16-bond model will now be reinterpreted as bond charge
density distortion waves. For this purpose we start with the
VHP tetragonal phase and choose the smallest unit cell as
the elementary unit. As before, we limit our discussion to
a two-dimensional square lattice. The unit cell contains one
iodine core and two electron charge densities, p a ( l ) . Here
a = x,y, and 1 denotes the unit cell vector. Charge densities
are assigned to the nearest neighbor interatomic bonds only.
Distortions in the charge densities at the wavevector q
Because of the charge conservation condition, the totally
symmetric eigenvector, S p A ( q ) is trivial, S p ( q ) = 0. Hence,
we are only interested in charge density distortions described
by the anti-symmetric eigenvector [16b]. The distortions
corresponding to the wavevector q will transform according
to one of the irreducible representation of the corresponding
wavevector group G ( q ) .
A. r-point, q = 0
At the center of the Brillouin zone distortions can be classified
according to the D4/,point group. The charge density distortion
described by the eigenvector [ 16b],
will cause a change in the electronic part of the crystal internal
energy
Ll21
su = ( 1 / 2 ) C C ~ ~ ~ ( q ) + ~ p ( q ) ~ p p ( - q )
aP
q
where
and x a p ( l , l r )is the charge density response in unit cell 1 to a
scalar potential in unit cell 1' and x is a simplified version of
an electronic susceptibility x ( r , r r ) .
For our two-dimensional square lattice with one atom in a
unit cell, the matrix + ( q ) has form
A condition for the crystal to be stable and to support the charge
density distortion described by eq. [ l 11 is that the eigenvalues
transforms according to the B1, representation and is shown
schematically on Fig. 7. It is clearly seen that the distortion
corresponds to changes in the bond occupation probabilities
of the same symmetry for the 16-bond model. Therefore, the
amplitude of the charge density distortion wave, SpB(q = O),
can be identified with the order parameter q, introduced in
section 3.
B. A-direction, q = ( r r / 2 a , 4 2 a , 0 )
The wavevector group for the A-direction is isomorphic with
the CZupoint group. Charge density distortions corresponding
to the eigenvector [16b] in this direction will transform
according to the B1 representation of the C2, group. 'The
changes in occupation probabilities described by the order
parameter 6 will correspond to the charge density distortion
wave of B1 symmetry for the wavevector qo = ( n / 2 a , n / 2 a ,
0). The "bond structures" shown in Fig. 5 and described by the
order parameters 6 can now be seen as patterns of the charge
density distortion waves for wavevector go. One of the latter
patterns is illustrated in Fig. 8. Thus, the amplitude of the
On introducing the renormalized coupling constants,
the expression for the renormalized phonon frequencies is
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As a consequence of the adiabatic approximation we obtain a
useful relation between the amplitudes of the charge density
distortion waves, Sp,(q), and the amplitude of the phonon
modes, Q ( q j ) ,
FIG. 8. The bond charge density distortion wave with wavevector
40 =
(7r/2a, 7r/2a, 0 ) in
the tetragonal phase of iodine.
charge density distortion wave in the A-direction can be
identified with the order parameter 6 , keeping in mind that
Sp(q,) transforms according to the B1 irreducible representation of the CZupoint group.
The distortions of the bond charge densities described here
are coupled to phonons. This coupling leads to indirect
interactions between iodine atoms. The indirect interaction
produces a softening of certain phonon modes of the tetragonal
phase, as described in the next section.
5. Dynamics of the tetragonal phase (two-dimensional model)
Propagating phonon modes are described by quantum numbers ( q j ) , representing the wavevector q and branch index j ,
with polarization vectors w(qj ) and frequency o ( q j ) . The
frequencies are given by the eigenvalues of the dynamical
matrix D(q), through the solution of the eigenvalue problem
1181
mo2(qj)wa(qj) = CDap(q)wp(qj)
P
In eq. [18], m is the atomic mass and the indices a , p represent
the Cartesian components. For the two-dimensional square
lattice with one atom per unit cell there are two branches ( j =
1, 2) for every wavevector.
The force constants which make up the dynamical matrix
can be viewed as coming from two separate contributions. The
first can be thought of as arising from the forces between
"bare" atoms and a second contribution comes from the indirect
interaction between the "bare" atoms mediated by charge
density distortions. It follows from considerations in the
previous section that the amplitudes Sp,(q), p = A , B, of the
bond charge density distortion waves can be chosen as the most
natural approximation for the electronic variables. Introducing
phenomenological coupling constants Aa,(q) between the
atomic displacement in the a direction and the charge density
distortion wave of type p with wavevector q , the following
expression for the dynamical matrix is obtained
[I91
1221 Sp,(q) = - + i l ( q ) A , ( q j ) Q ( q j )
Four acoustic modes the relation reads
Dap(9) = D$4(9) - c ~ ~ , ( q ) + , ~ ( q ) ~ ~ , ( - q )
P
where +,(q) are the eigenvalues of the inverse electronic
susceptibility, given by eq. [15]. Di%(q) is the part of the
dynamical matrix which comes from direct interactions between
"bare" atoms. The form of eq. [19] is a consequence of the
adiabatic approximation: whatever lattice displacement exists at
a given time, the charge density distortion amplitudes Sp,
instantly adjust themselves to minimize the electronic energy.
where the components of the strain tensor, eqp, conveniently
describe displacements of atoms due to acoustic phonons
[24]
cap
=
lim ( ~ m ) - " * i ~ ~ Q ( q ~ , a )
'1-0
In eq. [24], Q ( q p , a ) is the acoustic mode amplitude with
eigenvector polarized in the a direction and wavevector in the
p direction.
Relations [22] and [23] put restrictions on the coupling
constants A,(qj). It is clear that only phonon modes or strain
components and charge density distortions of the same symmetry can couple bilinearly. We now consider phonons in the
two-dimensional tetragonal lattice of solid' iodine in the longwave length limit q + 0 and in the A-direction of the Brillouin
zone.
A. q-+ 0 limit
For a lattice with one atom per unit cell, the dynamical
problem in the limit q + 0 reduces to the equations of motion of
elasticity theory. The dynamical matrix is expressed in terms of
the isothermal elastic constants cayps as
For the two-dimensional lattice, the eigenvalues of the elastic
constant matrix are
(1/2)(cl
+ c I 2 ) with eigenvector ( e l + e2) of
A,, symmetry,
(1/2)(cl - c12) with eigenvector ( e l
-
e2) of
BI, symmetry,
4c66 with eigenvector (e6) of B2, symmetry.
Here the Voigt notation for elastic constants and strain components is used.
In the previous section the charge density distortion wave at
q = 0 is shown to be of B,, symmetry. It follows from eqs. [I91
and [25] that the eigenvalue of the effective dynamical matrix
can be expressed as
Here As,, is the coupling constant between the charge density
distortion wave and the strain e7 = (e - e2). Note that e7 also
has B , , symmetry. cyl and cY2 are elastic constants of the
crystal due to "bare" interatomic interactions only. The relation
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CAN. J. CHEM. VOL. 66. 1988
wave with wave vector g o automatically produces simultaneously a longitudinal charge density wave with q = 290.
Moreover, the dimerization of the iodine atoms in our model is
due to the condensation of the phonon mode of B, symmetry
and, triggered by it, the condensation of the bond charge density
distortion wave of the same symmetry. This wave, together
with the corresponding atomic displacements, helps to form
the chemical bond between iodine atoms. This is visualized
schematically in Fig. 9.
The proposed phenomenological model correctly predicts
structural changes for transitions VHP + HP and VHP + LP.
The model also suggests a possible mechanism which causes
the instabilities of the (parent) tetragonal atomic VHP phase
leading to the HP and LP phases. The observed sequence of
the transformations VHPT - L H P T - ~ L Pwhich requires an
analysis of thermodynamical properties is not a topic for
discussion in this paper.
6. Discussion
FIG.9. 'The formation of the molecular phase of iodine as a result
of simultaneous condensation of the bond charge density distortion
wave (characterized by increases ( + ) and decreases ( - ) in the bond
charges) and the transverse phonon mode (characterized by atomic
displacements indicated by the arrows), both of BI symmetry.
indicates how the amplitude of the charge density distortion
wave adjusts to the crystal strain. It is clear from eq. [26] that for
sufficiently large coupling constant the eigenvalue ( l / 2 ) ( c l c12) of the elastic constant matrix vanishes. At that point the
crystal is unstable against a homogeneous deformation and will
distort continuously to a new structure with symmetry indicated
by the e7 strain. The new structure is orthorhombic and the
strain can be used as the primary order parameter to describe the
VHP + HP phase transition as the pressure is decreased.
Although the strain e, is a primary order parameter, the
transition is driven by distortions of the electronic charge
distributions.
B. A -direction
In this direction the two phonon branches of the square
lattice of the VHP phase of iodine are classified according to
symmetries Al and B1 of the CZu point group. The Al
phonon corresponds to a longitudinal mode and B I represents a
transverse mode. It follows from the previous discussion that
the frequency of the transverse mode depends on the coupling of
the B1 phonon to the charge density distortion wave of the
same B I symmetry,
On the basis of the present model the phase transformations in
solid iodine under high pressure can be explained as follows.
Atomic phase A molecular phase
dissociation
can be interpreted in terms of a coupling between a bond charge
density distortion wave and a transverse phonon. The driving
force for the transformation is the electron-phonon coupling
and the instability has a two-dimensional Peierls character. The
dimerization results from the condensation of the transverse
phonon mode which goes soft as a result of indirect interatomic
interactions mediated by charge density distortion waves. The
charge density fluctuation coupled to the transverse phonon
results in increased electronic charge densities between certain
atoms. The displacements corresponding to the soft phonons
bring these atoms closer together resulting in chemical bonds
between certain iodine atoms and the molecular phase appears.
1. R. W. LYNCH
and H. G. DRICKAMER.
J . Chem. Phys. 45, 1020
2.
3.
4.
Again, we expect that the phonon with the wavevector go =
(7r/2a, 7r/2a, 0) will show a tendency to go soft due to indirect
interactions mediated by charge density distortion waves of B1
symmetry. Therefore, we assume that the amplitude of the B l
phonon at go will be an order parameter for the phase transition
to the molecular phase of iodine (iodine dimerization). It has
to be stressed that the B1 phonon at go corresponds to the
transverse wave which is used in ref. 2 to explain the HP + LP
phase transition. There is, however, an important difference
between our approach and that of ref. 2. Here, no additional
longitudinal wave with wave vector q = 2qo is needed to
explain the structural changes during the iodine dimerization.
The reason is that the transverse bond charge density distortion
5.
6.
7.
8.
9.
10.
11.
12.
(1966); N. SAKAI,
K. TAKEMURA,
and K. TSUJI.J. Phys. Soc.
Jpn. 51, 1811 (1982).
0 . SHIMOMURA,
K. TAKEMURA,
Y. FUJII,S. MINOMURA,
M.
MORI,Y. NODA,and Y. YAMADA.
Phys. Rev. B18,715 (1978);
K. TAKEMURA,
Y. FUJII,S. MINOMURA,
and 0 . SHIMOMURA
Solid State Commun. 30, 137 (1979); K. TAKEMURA,
S.
MINOMURA,
0 . SHIMOMURA,
and Y. FUJII.Phys. Rev. Lett. 45,
1881 (1980); K. TAKEMURA,
S. MINOMURA,
0 . SHIMOMURA
Y. FUJU,and J. D. AXE.Phys. Rev. B26, 998 (1982).
C. H. TOWNES
and D. P. DAILEY.
J. Chem. Phys. 20,35 (1952).
A. PASTERNAK,
A. ANDERSON,
and J. W. LEECH.J. Phys. C10,
3261 (1977).
K. KOBASHI
and R. D. ETTERS.
J. Chem. Phys. 79,3018 (1983).
W. L. MCMILLAN.
Phys. Rev. B16, 643 (1977).
P. B. KOSTER,and T. MIGCHELSEN.
Acta
F. VAN BOLHUIS,
Crystallogr. 23, 90 (1967).
and T. SUZUKI.
Solid State Commun. 44, 1105
Y. NATSUME
(1982).
R. BLINC,B. ZEKS,and R. KIND.Phys. Rev. B17,3409 (1978).
M. PASTERNAK,
J. N. FARRELL,
and R. D. TAYLOR.
Phys Rev.
Lett. 58, 575 (1987).
and A. ONODERA
Y. FUJII,K. HASE,Y. OHISHI,N. HAMAYA,
Solid State Commun. 59, 85 (1986).
Y. FUJII,K. HASE,N. HAMAYA,
Y. OHISHI,A. ONODERA
0 . SHIMOMURA,
and K. TAKEMURA.
Phys Rev. Lett. 58, 796
(1987).