We have carried out an ab initio theoretical study of the potential energy and molecular dynamics of Rb2ZnCl4. These calculations demonstrate that the incommensurate transition in this compound is caused by the relaxation of imperfect-hexagonal spirals of highly unstable ZnCl42- ions. This leads to angstrom-size displacements with essentially zero energy change. We argue that such  latent  (or imperfect) symmetry could be the general cause of incommensurate transitions in insulators

Boyer, L. L.

Edwardson, P. J.

Hardy, John R.

Katkanant, V.

English

Figures 2(a) and 2(b) of this paper have been inadvertently interchanged; thus, the caption for Fig. 2(a) relates to Fig. 2(b) and the caption for Fig. 2(b) relates to Fig. 2(a)

Katanant, V.

Edwardson, P.J.

Hardy, John

DigitalCommons@University of Nebraska

University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln John R. Hardy Papers Research Papers in Physics and Astronomy 12-1-1986 Errata: First-Principles Theoretical Explanation of Incommensurate Behavior in Rb2ZnC4 V. Katanant P.J. Edwardson John Hardy University of Nebraska - Lincoln L. L. Boyer Follow this and additional works at: https://digitalcommons.unl.edu/physicshardy  Part of the Physics Commons Katanant, V.; Edwardson, P.J.; Hardy, John; and Boyer, L. L., "Errata: First-Principles Theoretical Explanation of Incommensurate Behavior in Rb2ZnC4" (1986). John R. Hardy Papers. 59. https://digitalcommons.unl.edu/physicshardy/59 This Article is brought to you for free and open access by the Research Papers in Physics and Astronomy at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in John R. Hardy Papers by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. VOLUME 57, NUMBER 22 PHYSICAL REVIEW LETTERS 1 DECEMBER 1986 ERRATA First-Principles Theoretical Explanation of Incommensu- rate Behavior in Rb$3nC4. V. KATKANANT, P. J. EDWARDSON, J. R. HARDY, and L. L. BOYER [Phys. Rev. Lett. 57, 2033 (1986)l. Figures 2(a) and 2(b) of this paper have been inadver- tently interchanged; thus, the caption for Fig. 2(a) re- lates to Fig. 2(b) and the caption for Fig. 2(b) relates to Fig. 2(a). 

Errata: First-Principles Theoretical Explanation of Incommensurate Behavior in Rb\u3csub\u3e2\u3c/sub\u3eZnC\u3csub\u3e4\u3c/sub\u3e

University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln John R. Hardy Papers Research Papers in Physics and Astronomy 10-20-1986 First-Principles Theoretical Explanation of Incommensurate Behavior in RbZnC4 V. Katkanant University of Nebraska - Lincoln P. J. Edwardson University of Nebraska - Lincoln John R. Hardy University of Nebraska - Lincoln L. L. Boyer U. S. Naval Research Laboratory, Washington, D. C. Follow this and additional works at: https://digitalcommons.unl.edu/physicshardy  Part of the Physics Commons Katkanant, V.; Edwardson, P. J.; Hardy, John R.; and Boyer, L. L., "First-Principles Theoretical Explanation of Incommensurate Behavior in RbZnC4" (1986). John R. Hardy Papers. 58. https://digitalcommons.unl.edu/physicshardy/58 This Article is brought to you for free and open access by the Research Papers in Physics and Astronomy at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in John R. Hardy Papers by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. VOLUME 57, NUMBER 16 P H Y S I C A L  R E V I E W  L E T T E R S  20 OCTOBER 1986 First-Principles Theoretical Explanation of Incommensurate Behavior in RbZnC4 V. Katkanant, P. J. Edwardson, and J. R. Hardy Behlen Laboratory of Physics, University of Nebraska- Lincoln, Lincoln, Nebraska 68588 and L. L. Boyer U. S. Naval Research Laboratory, Washington, D. C. 20375 (Received 20 December 1985) We have carried out an ab initio theoretical study of the potential energy and molecular dynamics of Rb2ZnC14. These calculations demonstrate that the incommensurate transition in this compound is caused by the relaxation of imperfect-hexagonal spirals of highly unstable ZnCb2- ions. This leads to angstrom-size displacements with essentially zero energy change. We argue that such "latent" (or imperfect) symmetry could be the general cause of incommensurate transitions in in- sulators. PACS numbers: 64.70.Rh, 63.70.+h, 63.75.+z, 77.80.Bh In 1977 Iizumi et al.' demonstrated conclusively that the 129-K phase transition in K2Se04 was associated with the freezing in of a soft 2 2  phonon with a wave vector of $a*( 1 - 6 1, where 6 << 1. The resultant displacement pattern is thus incommensurate and remains so until 6 vanishes at 93 K, the commensurate transition temperature. This immediately presented the problem of identify- ing the basic driving mechanism, since Fermi-surface instabilities2 (the only then-known mechanism) could hardly be invoked for a large-band-gap insulator. This problem became even more perplexing as similar in- commensurate phases were discovered in K2Se04 iso- morphs (e.g., ~ b ~ ~ n ~ r ~ , ~  and ~ b ~ z n ~ b ~ )  without however, any associated soft modes. Indeed, the at- tempt to find one in Rb2ZnBr4' gave results which, while not fully conclusive, indicated strongly that no such mode exists. In this communication we wish to report the identi- fication of the basic mechanism driving incommensu- rate transitions in Rb2ZnC4, probably all the K2Se04 isomorphs, and quite possibly all such transitions in insulators. Moreover, the mechanism appears suffi- ciently novel for these transitions to constitute a new and more general class, possibly embracing conven- tional "soft mode" transitions as a subclass. Specifically, we find the transition to be driven by a relaxation of the quasihexagonal spirals formed by the znCb2- ions when viewed along the a axis. This sym- metry would be exact if all zncb2-  groups were equivalent. Associated with this distortion is an ex- tremely flat area of the crystal potential surface which we demonstrate to exist by showing that during the reconstruction process drastic structural rearrangement occurs, which involves angstrom-size displacements, at virtually no cost in energy. We were led to these findings by an attempt to apply to these systems an approach that we have previously found to be highly effective in explaining melting,6 su- perioni~ity,~ and, most recently, zone-boundary insta- bilities in halide perovskites.8~9 The approach used the   or don-~im" technique to derive accurate potential surfaces for the systems under study. Since this ap- pears to work well for RbCl and KCI, there appears to be no reason why these same potentials should not be equally good in the chlorozincates. The only new po- tential required is the zinc-chlorine interaction (all other short-range interactions involving zn2+ are negligible). For this ion pair, the Gordon-Kim poten- tial was manifestly too hard (e.g., the zinc-chlorine bond length was - 10% too large). However, when the prefactor in the Gordon-Kim potential was re- duced, to minimize the unbalanced forces on the zinc and chlorine ions at the observed lattice positions, these forces decreased by 90% and the free-ion bond lengths were in good agreement with experiment. Subsequently, the lattice relaxed smoothly and quickly to the equilibrium Pnam ( D I ~ )  structure with the lengths of the cell sides constrained to the experimen- tal values. The resultant structure is shown in Fig. l(a). Apart from the one adjustment of the in- tramolecular potential to ensure that the free ion has the correct size, all interionic potentials are parameter- free. Consequently, we regard this as an a priori calcu- lation as far as the crystal is concerned. We then proceeded to investigate the lattice dynam- ics of this structure. This revealed a very unstable sys- tem with 7 or 8 completely unstable branches in the disperison curves. Evidently, given the results for other systems which indicate that our potentials are reliable, we must be dealing with a novel situation. In particular, it would appear unlikely that conventional soft-mode theory can be applied. Consequently, we then proceeded to examine the energy of the low-temperature phase in which the unit cell is tripled along the a axis. We do @ 1986 The American Physical Society 2033 [ Includes Erratum from December 1986. ]VOLUME 57, NUMBER 16 P H Y S I C A L  R E V I E W  L E T T E R S  20 OCTOBER 1986 (b) FIG. 1. (a) ac and bc cross sections of the relaxed static-Pnam structure of Rb2ZnCI4. The two triangles join the triplets of zincs of width - a17 along a (in this and sub- sequent figures all ZnC142- tetrahedra are shown but not all Rb+ ions). Note: a = 9.257 A, b = 12.726 A, and c = 7.282 A. (b) ac and bc cross sections of the relaxed Pna2, struc- ture of RbzZnC4-note the relative torsional movement of the ZnCh2- triplets in the other two cells with respect to the two triangles of triplets shown for the first of the three cells. not address the incommensurate phase directly, since we need a periodic structure for our computations. As we shall see, this is unimportant. Once again the system relaxed smoothly and quickly to the final configuration shown in Fig. 1 (b). In this new structure approximately half the un- stable normal modes for the Pnam phase had stabi- lized, but the most striking feature was that the energy per molecule had changed by only 0.0037 eV. This result is particularly striking when one compares the ionic positions in the Pnam and Pna2, ( Cg ) struc- tures, as shown in Figs. 1 (a) and 1 (b). It can be seen that the ionic positions are very different and, in par- ticular, the znc14*- tetrahedra have displaced and ro- tated through angstrom -size distances. Evidently, this implies that the potential energy sur- face in the hyperspace of ionic coordinates is extreme- ly flat, so much so that the two configurations are ef- fectively degenerate with one another, and with the large volume of hyperspace between them. Examination of Figs. 1 (a) and 1 (b) provided a quali- tative explanation of this most unusual behavior. It would appear that the apparently hexagonal array of znC14?- ions can rotate synchronously over large an- gles with little appreciable change in energy. However, the apparent near-hexagonal symmetry in projection is revealed in three dimensions as an approximate siwfold screw axis along a. This imperfect symmetry, which we choose to call "latent symmetry," is the origin of in- commensurate behavior. The true situation is that there are two trigonal ar- rays of tetrahedra, as indicated in Fig, l (a) ,  rotated with respect to one another by .n about a. However, while the two layers within each array are separated by - a17 along the a axis, the two arrays are a12 apart. Thus, what has taken place in the transformation between the two ideal lattices, shown in Fig. 1, can be decomposed into two components: (a) synchronous rotations within the trigonal layers, which cost virtually no energy until the rotational displacements are a sig- nificant fraction of the distance between first-neighbor tetrahedra; and (b) correlated motion between adja- cent trigonal layers, again at a low cost in energy. In addition to these primary motions, there are secondary (but still angstrom-size) displacements, transverse to the a axis, of the zn2+  and Rb' ions. This type of motion provides a very efficient means of relaxing any significant accumulation of energy due to the rotations; it does so because the ~ n C 1 4 ~ -  radicals, as we shall demonstrate, are very loosely attached to the orthorhombic "backbone" of the lattice. For the energy to remain nearly stationary as both types of dis- placements build up, the secondary displacements must be periodic around the hexagonal spiral. Hence, we obtain a tripled cell. However, if we decompose these displacements into plane waves, only two basic modulations are present, since all component wave vectors Q, must satisfy the requirements, Q, II a and Thus, when n = 3 (or some multiple of 31, we have a q, = O  modulation (where q, is restricted to the first zone); otherwise we have a q, =a*/3 component, i.e., a threefold modulation. However, Eq. (1) can be expressed in another form: (where xk and xk, are the a components of the basis vectors of the nearest-neighbor ion pairs around the hexagonal spirals). In this form, it is apparent that this condition, be- cause of the latency of the symmetry, cannot be identical- ly  satisfied for all pairs of ions. As a consequence, the near-sixfold helices tend to pucker and rotate. This implies intrahelical torques which would, were q, free to vary, favor a slight unwinding. The result would be an incommensurately modulated structure. The proviso that the unwinding be small in this case arises from the strongly commensurate nature of the rotational motion-indeed, as the amplitude of the modulation increases, this forces the commensurate lockin. Thus far, our discussion has been conducted in VOLUME 57, NUMBER 16 PHYSICAL REVIEW LETTERS 20 OCTOBER 1986 terms of the ideal lattices and static energies. Howev- er, we know that both lattices have strong instabilities, and we have already p~s tu l a t ed"~ '~  that the tetrahedra are disordered in some measure, in order to explain anomalies in the Raman spectra of the internal modes. We thus need to establish the basic validity of the foregoing arguments when the system is disordered and at finite temperature. We therefore carried out a molecular dynamics simulation which will be fully re- ported elsewhere.13 The results essential to the present discussion are shown in Figs. 2(a) and 2(b). In Fig. 2(a), we show the system at 335 K, above, but close to the transition. It can be seen that the tri- gonal layers are becoming correlated in precisely the manner we have predicted for the static structure. In Fig. 2(b), we show the same system at 255 K: The cell-tripling transition has clearly taken place since the correlation within the trigonal layers has been joined by the modulated interlayer correlation, again as predicted for the static system. The large C1- ellip- soids indicate rotational disorder of the zncld2- units. This partially decouples the spirals from the "back- bone" of Rb' ions. We have emphasized the structural aspect of latent symmetry since it seems likely that it can account for many, if not all, examples of incommensurate behavior in insulators. A completely different exam- ple is ThBr4, l4 which does not contain ill-fitting molec- ular groups, but has eight bromide ions clustered above each thorium ion. In this case, from Fig. 1 of Ref. 14, the latent sym- metry along the z (c )  axis appears to be that of an ap- proximate fourfold screw axis. Thus, for this case, we have, for the ionic z coordinates zk, where q, = (0, O,qz). Since if n = 1, it follows that qs = (O,O,q,) - c*/3 (the observed value). Again, the symmetry is not exact, since there are other basis-vector differences; hence an incommensu- rate phase appears. Its wide range of stability may be due to the absence of associated large molecular motions which favor commensurateness. Thus we feel that we have isolated the mechanism which produces incommensurate behavior in Rb2ZnCI4 and its isomorphs. Briefly: Incommensurate behavior arises j o m  the presence in the parent (high-temperature) structure of imperfect and mechanically unstable helices of atoms or groups. The imperfection causes intrahelical torques and an incommensurate structure results, when the helices open to relieve these particular stresses during the transition. (a) The work at the University of Nebraska was sup- a s ported by the U. S. Army Research Office. We should also like to acknowledge very early discussions wlth the late I. Lefkowitz in which he expressed the strong intuitive belief that inccommensurate behavior had some simple physical origin. FIG. 2. (a) ac and bc cross sections of the molecular- dynamics simulation of RbzZnC4 at 335 K. The ellipsoids are drawn about the average positions of the ions, and have principal axes equal to the appropriate root-mean-square de- viations. Note the correlations parallel to c. The arrows in- dicate the senses of rotation about a of their associated tetrahedra. These clearly show mixed periodicity, i.e., disor- der, along a. (b) a c  and bc cross sections of the molecular-dynamics simulation of Rb2ZnC4 at 255 K. Again the arrows show the senses of the tetrahedra rota- tions. This now has a single periodicity: Three positive fol- lowed by three negative, i.e., the unit cell has tripled. (Note that in both figures half the rubidium ions in the ac projec- tion are almost concealed by zinc ions lying above them.) lM. Iizumi, J. D.  Axe, G. Shirane, and K. Shimaoka, Phys. Rev. B 15, 4392 (1977). 2J. A. Wilson, F. J. DiSalvo, and S. Mahajan, Adv. Phys. 24, 117 (19751, and Phys. Rev. Lett. 32, 882 (1974). 3K. Gesi and M. Iizumi, J. Phys. Soc. Jpn. 45, 1777 (1978). 4K. Gesi and M. Iizumi, J .  Phys. Soc. Jpn. 4 6 , 6 9 7  (1979). 5C. J. de Pater and C. van Dijk, Phys. Rev. B 18, 1281 (1978). 6L. L. Boyer, Phys. Rev. Lett. 42, 584 (1979). A fuller ac- count is given in Phys. Rev. B 23, 3673 (1981). 7L. L. Boyer, Phys. Rev. Lett. 45, 1858 (19801, and 46, 1172 (1981). *L. L. Boyer and J. R. Hardy, Phys. Rev. B 24, 2527 (1981). VOLUME 57, NUMBER 16 PHYSICAL REVIEW LETTERS 20 OCTOBER 1986 9L. L. Boyer, J. Phys. C 17, 1825 (1984). Appl. Phys., Suppl. Pt. 2 24, 790 (1985). loR. G .  Gordon and Y. S. Kim, J. Chem. Phys. 56, 3122 13P. J .  Edwardson, V. Katkanant, J. R. Hardy, and L. L. (1971). Boyer, to be published. 11N. E. Massa, F. G. Ullman, and J .  R. Hardy, Phys. Rev. l4L. Bernard, R. Currat, P. Delamoye, C. M. E. Zeyen, B 27, 1523 (1980). S. Hubert, and R.  De Kouchovsky, J. Phys. C 16, 433 12V. Katkanant, F. G .  Ullman, and J .  R.  Hardy, Jpn. J .  (1983). VOLUME 57, NUMBER 22 PHYSICAL REVIEW LETTERS 1 DECEMBER 1986 ERRATA First-Principles Theoretical Explanation of Incommensu- rate Behavior in Rb$3nC4. V. KATKANANT, P. J. EDWARDSON, J. R. HARDY, and L. L. BOYER [Phys. Rev. Lett. 57, 2033 (1986)l. Figures 2(a) and 2(b) of this paper have been inadver- tently interchanged; thus, the caption for Fig. 2(a) re- lates to Fig. 2(b) and the caption for Fig. 2(b) relates to Fig. 2(a). 

First-Principles Theoretical Explanation of Incommensurate Behavior in RbZnC\u3csub\u3e4\u3c/sub\u3e

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University of Nebraska - Lincoln 
DigitalCommons@University of Nebraska - Lincoln 
John R. Hardy Papers Research Papers in Physics and Astronomy 
12-1-1986 
Errata: First-Principles Theoretical Explanation of 
Incommensurate Behavior in Rb2ZnC4 
V. Katanant 
P.J. Edwardson 
John Hardy 
University of Nebraska - Lincoln 
L. L. Boyer 
Follow this and additional works at: https://digitalcommons.unl.edu/physicshardy 
 Part of the Physics Commons 
Katanant, V.; Edwardson, P.J.; Hardy, John; and Boyer, L. L., "Errata: First-Principles Theoretical Explanation 
of Incommensurate Behavior in Rb2ZnC4" (1986). John R. Hardy Papers. 59. 
https://digitalcommons.unl.edu/physicshardy/59 
This Article is brought to you for free and open access by the Research Papers in Physics and Astronomy at 
DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in John R. Hardy Papers by an 
authorized administrator of DigitalCommons@University of Nebraska - Lincoln. 
VOLUME 57, NUMBER 22 PHYSICAL REVIEW LETTERS 1 DECEMBER 1986 
ERRATA 
First-Principles Theoretical Explanation of Incommensu- 
rate Behavior in Rb$3nC4. V. KATKANANT, P. J. 
EDWARDSON, J. R. HARDY, and L. L. BOYER [Phys. 
Rev. Lett. 57, 2033 (1986)l. 
Figures 2(a) and 2(b) of this paper have been inadver- 
tently interchanged; thus, the caption for Fig. 2(a) re- 
lates to Fig. 2(b) and the caption for Fig. 2(b) relates to 
Fig. 2(a). 


University of Nebraska - Lincoln 
DigitalCommons@University of Nebraska - Lincoln 
John R. Hardy Papers Research Papers in Physics and Astronomy 
10-20-1986 
First-Principles Theoretical Explanation of Incommensurate 
Behavior in RbZnC4 
V. Katkanant 
University of Nebraska - Lincoln 
P. J. Edwardson 
University of Nebraska - Lincoln 
John R. Hardy 
University of Nebraska - Lincoln 
L. L. Boyer 
U. S. Naval Research Laboratory, Washington, D. C. 
Follow this and additional works at: https://digitalcommons.unl.edu/physicshardy 
 Part of the Physics Commons 
Katkanant, V.; Edwardson, P. J.; Hardy, John R.; and Boyer, L. L., "First-Principles Theoretical Explanation of 
Incommensurate Behavior in RbZnC4" (1986). John R. Hardy Papers. 58. 
https://digitalcommons.unl.edu/physicshardy/58 
This Article is brought to you for free and open access by the Research Papers in Physics and Astronomy at 
DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in John R. Hardy Papers by an 
authorized administrator of DigitalCommons@University of Nebraska - Lincoln. 
VOLUME 57, NUMBER 16 P H Y S I C A L  R E V I E W  L E T T E R S  20 OCTOBER 1986 
First-Principles Theoretical Explanation of Incommensurate Behavior in RbZnC4 
V. Katkanant, P. J. Edwardson, and J. R. Hardy 
Behlen Laboratory of Physics, University of Nebraska- Lincoln, Lincoln, Nebraska 68588 
and 
L. L. Boyer 
U. S. Naval Research Laboratory, Washington, D. C. 20375 
(Received 20 December 1985) 
We have carried out an ab initio theoretical study of the potential energy and molecular dynamics 
of Rb2ZnC14. These calculations demonstrate that the incommensurate transition in this compound 
is caused by the relaxation of imperfect-hexagonal spirals of highly unstable ZnCb2- ions. This 
leads to angstrom-size displacements with essentially zero energy change. We argue that such 
"latent" (or imperfect) symmetry could be the general cause of incommensurate transitions in in- 
sulators. 
PACS numbers: 64.70.Rh, 63.70.+h, 63.75.+z, 77.80.Bh 
In 1977 Iizumi et al.' demonstrated conclusively that 
the 129-K phase transition in K2Se04 was associated 
with the freezing in of a soft 2 2  phonon with a wave 
vector of $a*( 1 - 6 1, where 6 << 1. The resultant 
displacement pattern is thus incommensurate and 
remains so until 6 vanishes at 93 K, the commensurate 
transition temperature. 
This immediately presented the problem of identify- 
ing the basic driving mechanism, since Fermi-surface 
instabilities2 (the only then-known mechanism) could 
hardly be invoked for a large-band-gap insulator. This 
problem became even more perplexing as similar in- 
commensurate phases were discovered in K2Se04 iso- 
morphs (e.g., ~ b ~ ~ n ~ r ~ , ~  and ~ b ~ z n ~ b ~ )  without 
however, any associated soft modes. Indeed, the at- 
tempt to find one in Rb2ZnBr4' gave results which, 
while not fully conclusive, indicated strongly that no 
such mode exists. 
In this communication we wish to report the identi- 
fication of the basic mechanism driving incommensu- 
rate transitions in Rb2ZnC4, probably all the K2Se04 
isomorphs, and quite possibly all such transitions in 
insulators. Moreover, the mechanism appears suffi- 
ciently novel for these transitions to constitute a new 
and more general class, possibly embracing conven- 
tional "soft mode" transitions as a subclass. 
Specifically, we find the transition to be driven by a 
relaxation of the quasihexagonal spirals formed by the 
znCb2- ions when viewed along the a axis. This sym- 
metry would be exact if all zncb2-  groups were 
equivalent. Associated with this distortion is an ex- 
tremely flat area of the crystal potential surface which 
we demonstrate to exist by showing that during the 
reconstruction process drastic structural rearrangement 
occurs, which involves angstrom-size displacements, at 
virtually no cost in energy. 
We were led to these findings by an attempt to apply 
to these systems an approach that we have previously 
found to be highly effective in explaining melting,6 su- 
perioni~ity,~ and, most recently, zone-boundary insta- 
bilities in halide perovskites.8~9 The approach used the 
  or don-~im" technique to derive accurate potential 
surfaces for the systems under study. Since this ap- 
pears to work well for RbCl and KCI, there appears to 
be no reason why these same potentials should not be 
equally good in the chlorozincates. The only new po- 
tential required is the zinc-chlorine interaction (all 
other short-range interactions involving zn2+ are 
negligible). For this ion pair, the Gordon-Kim poten- 
tial was manifestly too hard (e.g., the zinc-chlorine 
bond length was - 10% too large). However, when 
the prefactor in the Gordon-Kim potential was re- 
duced, to minimize the unbalanced forces on the zinc 
and chlorine ions at the observed lattice positions, 
these forces decreased by 90% and the free-ion bond 
lengths were in good agreement with experiment. 
Subsequently, the lattice relaxed smoothly and quickly 
to the equilibrium Pnam ( D I ~ )  structure with the 
lengths of the cell sides constrained to the experimen- 
tal values. The resultant structure is shown in Fig. 
l(a). Apart from the one adjustment of the in- 
tramolecular potential to ensure that the free ion has 
the correct size, all interionic potentials are parameter- 
free. Consequently, we regard this as an a priori calcu- 
lation as far as the crystal is concerned. 
We then proceeded to investigate the lattice dynam- 
ics of this structure. This revealed a very unstable sys- 
tem with 7 or 8 completely unstable branches in the 
disperison curves. 
Evidently, given the results for other systems which 
indicate that our potentials are reliable, we must be 
dealing with a novel situation. In particular, it would 
appear unlikely that conventional soft-mode theory 
can be applied. Consequently, we then proceeded to 
examine the energy of the low-temperature phase in 
which the unit cell is tripled along the a axis. We do 
@ 1986 The American Physical Society 2033 
[ Includes Erratum from December 1986. ]
VOLUME 57, NUMBER 16 P H Y S I C A L  R E V I E W  L E T T E R S  20 OCTOBER 1986 
(b) 
FIG. 1. (a) ac and bc cross sections of the relaxed 
static-Pnam structure of Rb2ZnCI4. The two triangles join 
the triplets of zincs of width - a17 along a (in this and sub- 
sequent figures all ZnC142- tetrahedra are shown but not all 
Rb+ ions). Note: a = 9.257 A, b = 12.726 A, and c = 7.282 
A. (b) ac and bc cross sections of the relaxed Pna2, struc- 
ture of RbzZnC4-note the relative torsional movement of 
the ZnCh2- triplets in the other two cells with respect to the 
two triangles of triplets shown for the first of the three cells. 
not address the incommensurate phase directly, since 
we need a periodic structure for our computations. As 
we shall see, this is unimportant. 
Once again the system relaxed smoothly and quickly 
to the final configuration shown in Fig. 1 (b). 
In this new structure approximately half the un- 
stable normal modes for the Pnam phase had stabi- 
lized, but the most striking feature was that the energy 
per molecule had changed by only 0.0037 eV. This 
result is particularly striking when one compares the 
ionic positions in the Pnam and Pna2, ( Cg ) struc- 
tures, as shown in Figs. 1 (a) and 1 (b). It can be seen 
that the ionic positions are very different and, in par- 
ticular, the znc14*- tetrahedra have displaced and ro- 
tated through angstrom -size distances. 
Evidently, this implies that the potential energy sur- 
face in the hyperspace of ionic coordinates is extreme- 
ly flat, so much so that the two configurations are ef- 
fectively degenerate with one another, and with the 
large volume of hyperspace between them. 
Examination of Figs. 1 (a) and 1 (b) provided a quali- 
tative explanation of this most unusual behavior. It 
would appear that the apparently hexagonal array of 
znC14?- ions can rotate synchronously over large an- 
gles with little appreciable change in energy. However, 
the apparent near-hexagonal symmetry in projection is 
revealed in three dimensions as an approximate siwfold 
screw axis along a. This imperfect symmetry, which we 
choose to call "latent symmetry," is the origin of in- 
commensurate behavior. 
The true situation is that there are two trigonal ar- 
rays of tetrahedra, as indicated in Fig, l (a) ,  rotated 
with respect to one another by .n about a. However, 
while the two layers within each array are separated by - a17 along the a axis, the two arrays are a12 apart. 
Thus, what has taken place in the transformation 
between the two ideal lattices, shown in Fig. 1, can be 
decomposed into two components: (a) synchronous 
rotations within the trigonal layers, which cost virtually 
no energy until the rotational displacements are a sig- 
nificant fraction of the distance between first-neighbor 
tetrahedra; and (b) correlated motion between adja- 
cent trigonal layers, again at a low cost in energy. 
In addition to these primary motions, there are 
secondary (but still angstrom-size) displacements, 
transverse to the a axis, of the zn2+  and Rb' ions. 
This type of motion provides a very efficient means of 
relaxing any significant accumulation of energy due to 
the rotations; it does so because the ~ n C 1 4 ~ -  radicals, 
as we shall demonstrate, are very loosely attached to 
the orthorhombic "backbone" of the lattice. For the 
energy to remain nearly stationary as both types of dis- 
placements build up, the secondary displacements 
must be periodic around the hexagonal spiral. Hence, 
we obtain a tripled cell. However, if we decompose 
these displacements into plane waves, only two basic 
modulations are present, since all component wave 
vectors Q, must satisfy the requirements, Q, II a and 
Thus, when n = 3 (or some multiple of 31, we have 
a q, = O  modulation (where q, is restricted to the first 
zone); otherwise we have a q, =a*/3 component, i.e., 
a threefold modulation. 
However, Eq. (1) can be expressed in another form: 
(where xk and xk, are the a components of the basis 
vectors of the nearest-neighbor ion pairs around the 
hexagonal spirals). 
In this form, it is apparent that this condition, be- 
cause of the latency of the symmetry, cannot be identical- 
ly  satisfied for all pairs of ions. As a consequence, the 
near-sixfold helices tend to pucker and rotate. This 
implies intrahelical torques which would, were q, free 
to vary, favor a slight unwinding. The result would be 
an incommensurately modulated structure. The proviso 
that the unwinding be small in this case arises from the 
strongly commensurate nature of the rotational 
motion-indeed, as the amplitude of the modulation 
increases, this forces the commensurate lockin. 
Thus far, our discussion has been conducted in 
VOLUME 57, NUMBER 16 PHYSICAL REVIEW LETTERS 20 OCTOBER 1986 
terms of the ideal lattices and static energies. Howev- 
er, we know that both lattices have strong instabilities, 
and we have already p~s tu l a t ed"~ '~  that the tetrahedra 
are disordered in some measure, in order to explain 
anomalies in the Raman spectra of the internal modes. 
We thus need to establish the basic validity of the 
foregoing arguments when the system is disordered 
and at finite temperature. We therefore carried out a 
molecular dynamics simulation which will be fully re- 
ported elsewhere.13 The results essential to the 
present discussion are shown in Figs. 2(a) and 2(b). 
In Fig. 2(a), we show the system at 335 K, above, 
but close to the transition. It can be seen that the tri- 
gonal layers are becoming correlated in precisely the 
manner we have predicted for the static structure. In 
Fig. 2(b), we show the same system at 255 K: The 
cell-tripling transition has clearly taken place since the 
correlation within the trigonal layers has been joined 
by the modulated interlayer correlation, again as 
predicted for the static system. The large C1- ellip- 
soids indicate rotational disorder of the zncld2- units. 
This partially decouples the spirals from the "back- 
bone" of Rb' ions. 
We have emphasized the structural aspect of latent 
symmetry since it seems likely that it can account for 
many, if not all, examples of incommensurate 
behavior in insulators. A completely different exam- 
ple is ThBr4, l4 which does not contain ill-fitting molec- 
ular groups, but has eight bromide ions clustered 
above each thorium ion. 
In this case, from Fig. 1 of Ref. 14, the latent sym- 
metry along the z (c )  axis appears to be that of an ap- 
proximate fourfold screw axis. Thus, for this case, we 
have, for the ionic z coordinates zk, 
where q, = (0, O,qz). Since 
if n = 1, it follows that 
qs = (O,O,q,) - c*/3 (the observed value). 
Again, the symmetry is not exact, since there are 
other basis-vector differences; hence an incommensu- 
rate phase appears. Its wide range of stability may be 
due to the absence of associated large molecular 
motions which favor commensurateness. 
Thus we feel that we have isolated the mechanism 
which produces incommensurate behavior in Rb2ZnCI4 
and its isomorphs. Briefly: Incommensurate behavior 
arises j o m  the presence in the parent (high-temperature) 
structure of imperfect and mechanically unstable helices of 
atoms or groups. The imperfection causes intrahelical 
torques and an incommensurate structure results, when 
the helices open to relieve these particular stresses during 
the transition. 
(a) The work at the University of Nebraska was sup- 
a s ported by the U. S. Army Research Office. We should 
also like to acknowledge very early discussions wlth 
the late I. Lefkowitz in which he expressed the strong 
intuitive belief that inccommensurate behavior had 
some simple physical origin. 
FIG. 2. (a) ac and bc cross sections of the molecular- 
dynamics simulation of RbzZnC4 at 335 K. The ellipsoids 
are drawn about the average positions of the ions, and have 
principal axes equal to the appropriate root-mean-square de- 
viations. Note the correlations parallel to c. The arrows in- 
dicate the senses of rotation about a of their associated 
tetrahedra. These clearly show mixed periodicity, i.e., disor- 
der, along a. (b) a c  and bc cross sections of the 
molecular-dynamics simulation of Rb2ZnC4 at 255 K. 
Again the arrows show the senses of the tetrahedra rota- 
tions. This now has a single periodicity: Three positive fol- 
lowed by three negative, i.e., the unit cell has tripled. (Note 
that in both figures half the rubidium ions in the ac projec- 
tion are almost concealed by zinc ions lying above them.) 
lM. Iizumi, J. D.  Axe, G. Shirane, and K. Shimaoka, 
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(1981). 
VOLUME 57, NUMBER 16 PHYSICAL REVIEW LETTERS 20 OCTOBER 1986 
9L. L. Boyer, J. Phys. C 17, 1825 (1984). Appl. Phys., Suppl. Pt. 2 24, 790 (1985). 
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VOLUME 57, NUMBER 22 PHYSICAL REVIEW LETTERS 1 DECEMBER 1986 
ERRATA 
First-Principles Theoretical Explanation of Incommensu- 
rate Behavior in Rb$3nC4. V. KATKANANT, P. J. 
EDWARDSON, J. R. HARDY, and L. L. BOYER [Phys. 
Rev. Lett. 57, 2033 (1986)l. 
Figures 2(a) and 2(b) of this paper have been inadver- 
tently interchanged; thus, the caption for Fig. 2(a) re- 
lates to Fig. 2(b) and the caption for Fig. 2(b) relates to 
Fig. 2(a). 


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First-Principles Theoretical Explanation of Incommensurate Behavior in RbZnC\u3csub\u3e4\u3c/sub\u3e

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