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A note on the two-dipole problem
Solid State Communications, Vol. 9, pp. 1323—1328, 1971.
Pergainon Press.
Printed in Great Britain
A NOTE ON THE TWO-DIPOLE PROBLEM* M.J. Mandell Laboratory of Atomic and Solid State Physics Cornell University, Ithaca, New York 14850 (Received 27 May 1971 by N.B. Hannay)
The problem of two free-rotor quantum dipoles with fixed interparticle distance is considered. For various values of interaction strength, the correlation and the parallel and perpendicular susceptibilities are calculated as a function of temperature, and the zero-temperature polarization is calculated as a function of electric field. We discuss the nature of the elementary excitations of a lattice of electric dipoles. These results are applicable to understanding the observed susceptibility maximum in doped alkali halides, and the properties of such molecular crystals as the hydrogen halides.
THERE HAS been some recent interest in the properties of two nearby particles having a fixed interparticle distance but an orientation-dependeat interaction. The possibility of low-temperature ferroelectric-type phase transitions in electric dipole doped host crystals is provocative, and indeed low-temperature susceptibility maxima have been observed in alkali halides doped with relatively high concentrations of such dipolar impurities as 0H, CN, and Li+.l~ZInaddition, crystals of such dipolar molecules as the hydrogen halides are known to have ferroelectric phase transitions.3 This motivated the present study, which, as will be seen below, has been illuminating, although indicative of the necessity of more complicated models, An important example of a system with orientation-dependent potential is a pair of free rotors with electric dipole interaction, i.e., a pair of linear dipolar molecules. (Other examples would be six- or eight-well tunnelling systems, which have the disadvantage of depending on the orientation of the interparticle axis relative to *
.
the crystal axes.) The energy levels for this systern have been found by Jungst and v. Meyenn.’ In this paper we discuss the susceptibility, and speculate on applying the results of this problem to the susceptibilities of dipolar impurities in solids,2 and to ferroelectric molecular crystals such as the hydrogen halides. Our aim in this work is not to examine properties in the critical region, where long-range correlations are important and the precise nature of the interaction is irrelevant, but to restrict ourselves to regions of finite-range correlations where properties are determined by the details of the Hamiltonian. Problems in the lore of magnetism bearing some relation to this work are, for the lattice case, the Ising model with transverse field, ~ and a magnet with crystal-field splitting,6 and, for the disordered case, the work of Klein on Cu : Mn. .
This work has been supported by the U.S. Atomic Energy Commission under Contract No. AT (30—1)—3699, Technical Report No. NYO—3699 —55. 1323
The Hamiltonian for a pair of dipoles may be written as (1) 1’ 2 where L~is the square of the orbital angular
1324
A NOTE ON THE TWO-DIPOLE PROBLEM
momentum operator for particle i, having eigenvalues L(L + 1), Fr~)4,I is a parameter whose 2 is the moment of value inertia, of order pis the dipole(!p2/h moment, rthe interparticle distance, and
V(~Z~ 1l~) 2 cos 0~ cos sin O~ sin 0, cos (q~ ç~,). =
—
(2)
—
We have chosen the unit of ener~’ on the order of the libronic energy for an ordered array of dipoles.
Vol. 9, No. 15
states have m
1 = m2 = 0, and, in the basis IL ,L2> have the form [alOO> + increasing bill>] andF. [101> 10>], where b increases with The +latter state has a decrease in energy first order in the potential, and becomes nearly degenerate with the ground state as F becomes large. In an analogous way, a translationally invariant, infinite array of dipoles at zero temperature will, in absence of interaction, have excited states consisting of one dipole in the state L = 1, m = 0, ±1. For small F these states are more properly written
The potential can be written in operator notation by the substitutions cos 0
sin 0 ~
=
=
±a[
a4
+
a
(3)
~ L4~ 1
1
L±mj
± mj
±
(4) a~YL -~
L ±~Lm =
=
[L2 _m2l2 ________ j
L4L2
[(L ~ m)(L ±m
~
~Lm
1)~YLm +
~ y>
=
N~~ei-~!ac4
where ~Q> is the non-interacting ground state and c4 7~(k) is a linear combination of a~,
L~+a~, and L~_a~.These states form bands centered about E 2/F having bandwidth of order F. =
(5)
For some value~ofF of order unity the bottom of one or more of the bands becomes degenerate (or nearly degenerate) ofwith the ground state,The leading to the possibility a phase transition.
(6)
highly anharmonic nature of the model and the importance of the higher energy states make
As pointed out in reference 4, there are three good quantum numbers for the eignestates of this Hamiltonian: (1) the eigenfunctions will be symmetric (s) or antisymrnetric(a) under interchange of the two particles forming the pair. For a translationally invariant system, each state has a wavevector k.
quantitative calculations difficult for an infinite system, but it is straightforward to show that the lowest state of the band c~a= a1~is at k = 0 in the one-dimensional case, and k = (7T, 7T, 0) in the case of a simple cubic lattice. For the case of two dipoles we have calculated, as a function of inverse temperature, the quantities
(2) the quantity ~ L~will either be even (g) orodd(u). (3) the orbital angular momentum about the interparticle axis, ~ m~,is a good quanturn number.
-~_.
symmetries, and if, the properties of the those states having problem is reduced
to finding the eigenvalues and eigenvectors of relatively small matrices. The energy levels can be found in reference 4. It is worthwhile to point out the nature of the two lowest states. The main components of these
~
2= ~
~
<~ O,~, t> (7)
(cos01+ cosO2) and =
If we take advantage of these for the purpose of calculating low energy states, we neglect very high values of ~L L~,the
(k)j~>
a
.cç e~a /kT
FZL4
0,,~ j>/(E~
—
Es),
where 0,, =~cos 0,
O~= ~5ifl
O~C05
Z =~exp(—E~/kT).
(8)
Vol. 9, No. 15
A NOTE ON THE TWO-DIPOLE PROBLEM
1325
$0.0 F~3.2 F.ID
a
N
U
+ U, 0
F 1.6
~ 0.5 FIG. 1. The parallel correlation, [(cos
I-
2
0~+ cos 02)
4 —
8
$6
B/I’
2/3], as a function of inverse temperature.
The calculations were done using the exact eigenstates and eigenvectors for all states of energy less than 7/F in the restricted Hilbert space of £ L~~ 12, except that for the states
In an attempt to salvage a susceptibility peak, we have calculated the parallel polarization as a function of field at zero temperature. (Fig. 3). For small values of F the curve remains linear
[s, g, m
up to relatively high fields. For large F the curve
=
0] we considered ~ L~~ 14. For the
range of temperature and F considered the results seemed insensitive to these cut-offs; higher values of F would require enlargement of the Hi!bert space, and higher temperatures would require higher energy levels to be considered as well. In the scheme used the largest matrix to be diagonalized was 10 x 10, and most were 4 x 4 or 5x 5 The quantities (7) and (8) are plotted in Figs. 1 and 2. The behavior of the static susceptibilities is disappointingly dull. The parallel susceptibility rises with F, and the transverse susceptibility is correspondingly depressed. Neither shows a peak as a function of temperature. There is a slight peak in the correlation function, which would correspond to a peak in the high frequency susceptibility. This peak is due to the first excited state’s being more correlated than the ground state, at the expense of some kinetic energy.
bends over fairly soon. If we assume that a pair 2) is in a local field (at T = 0) given by (2lpE~ 00/h = F2, we find the differential susceptibility lowered by a factor of 0.7 for F = 0.8, and by a factor of 0.09 for F = 1.6. Such local fields are not all unreasonable in the light of 2measurements of remanent polarization by Fiory of several microcoulombs/m2.These value correspond to (2IpE/F12) 102. This average field will certainly be much enhanced in the immediate neighborhood of a strongly interacting pair.
Table 1 Distance Molecule or ion HF H~1 HI CO OH -
(A) 2.5 3 3 3 5
F2 (approximate) 60 10 5 1 3
1326
A NOTE ON THE TWO-DIPOLE PROBLEM
Vol. 9, No. 15
:.o
10.0
F=O.I
//
1.0
-
/
/
/
/
/
/
-‘-
/~..— -~7
/4(
I/I 1/
r=o.s
-~~J’
‘—‘-I
1/ 1/
I/I
/ L.
,-
/ I.
r~t.6
>< 0.1
/ /
_________.0I
FIG. 2. The parallel susceptibility (solid line; left scale) and perpendicular susceptibility (broken line; right scale) as defined in equation (8) as a function of inverse temperature. We next ask if, at the experimental concentrations, it is reasonable to expect large numbers of strongly interacting pairs. Table 1 shows that, for the case of OH in alkali halides, neglecting any lattice effects, only near-neighbor pairs would show significant effects. If we assume that individual impurities are randomly distributed, there are probably too few near-neighbor pairs to be significant. There are, however, two factors favoring the formation of such pairs. The first is that the lattice polarization enhances the interaction of more distant pairs. The second is that at high concentrations the assumption or random distribution of impurities is probably unrealistic,
The van der Waals forces between OW ions (pro2I) are comparable to those between portional to p HF molecules, and HF liquifies at about room temperature. Furthermore, the properties of crystals with the highest concentrations of Li~have been observed to change with time when stored at room temperature.8 This suggests that clustering effects may be highly significant. Finally, we would like to suggest that the detailed nature of the Hamiltonian (1) has bearing on the properties of molecular crystals such as those listed in Table 1. Among other crystals of possible interest are NO, SO, HD, deuterated
Vol. 9, No. 15
A NOTE ON THE TWO-DIPOLE PROBLEM
1327
FzI.6
1.2 .
P 1.0 -
0.8
-
0.6
-
0.4
-
r~o.e
0.2
0
I 0.1
I
I
I
I
I
0.2
0.3
0.4
0.5
0.6
0.7
I
I
I
0.8
0.9
1.0
E
FIG. 3. The polarization, (cos 0~+ cos 02), at T = 0°K as a function of electric field in units of h2/Fp. methanes, and alkali hydroxides. Dipolar and quadrupolar interactions [which can be cast in a form similar to (2)] are of roughly equal importance in these crystals. MC!, HBr, and HI are known to undergo phase transitions,3 largely due to these interactions, and CO is close to having a phase 9 transition at very low temperatures. We hope by further study of this Hamiltonian to gain further
insight into the dynamics and ferroelectric properties of these materials. Acknowledgements — The author would like to thank R.O. Pohi and co-workers, especially A. Fiory, for much encouragement and many helpful discussions, and J.A. Krumhansl for support, for many useful hints and references, and for many useful criticisms of the manuscript.
REFERENCES 1.
KANZIG W., HART H.R., Jr., and ROBERTS S., Phys. Rev. Lett. 13, 543 (1964).
2.
FIORY A.T., Cornell University Materials Science Center Report No. 1410. To be published in Phys.
Rev. 3.
A recent reference is E. Hanamura, Proc. 2nd Intern. Meeting Ferroelectricity, Kyoto, 1969, J. Phys. Soc. Japan 28 (1970) Suppi. p. 189.
4.
JUNGST K.L. and MEYENN K.v., Z. Phys. 242, 45 (1971).
5. ELLIOT R.J., PFEUTY P. and WOOD C., Phys. Rev. Leti. 25, 443 (1970). 6. WANG Y.L. and COOPER B.R., Phys. Rev. 185, 696 (1969). 7. KLEIN M.W., Phys. Rev. 188, 933 (1969). 8. 9.
FIORY A. and POHL R.O., private communication. CURL R.F., HOPKINS H.P. and PITZER K.S., J. Chem. Phys. 48, 4064 (1968).
1328
A NOTE ON THE TWO-DIPOLE PROBLEM Nous étudions le problème de deux dipoles, considérés comme rotateurs quantiques libres, ayant une distance interatomique constante. Nous calculons la susceptibilité parallèle et perpendiculaire ainsi que la correlation dipolaire en fonction de la temperature. La polarisation au zero absolu est également calculée en fonction du champs électrique. Nous discutons Ia nature des excitations élémeritaires d’un réseau de dipoles électriques. Ces résultats sont alors utilisés dans le but d’expliquer le maximum de la susceptibilité observée dans la halo-génure d’alcalins dopes, ainsi que les propriétés des cristaux moléculaires tels que les halogénures d’hydrogene.
Vol. 9, No. 15
Pergainon Press.
Printed in Great Britain
A NOTE ON THE TWO-DIPOLE PROBLEM* M.J. Mandell Laboratory of Atomic and Solid State Physics Cornell University, Ithaca, New York 14850 (Received 27 May 1971 by N.B. Hannay)
The problem of two free-rotor quantum dipoles with fixed interparticle distance is considered. For various values of interaction strength, the correlation and the parallel and perpendicular susceptibilities are calculated as a function of temperature, and the zero-temperature polarization is calculated as a function of electric field. We discuss the nature of the elementary excitations of a lattice of electric dipoles. These results are applicable to understanding the observed susceptibility maximum in doped alkali halides, and the properties of such molecular crystals as the hydrogen halides.
THERE HAS been some recent interest in the properties of two nearby particles having a fixed interparticle distance but an orientation-dependeat interaction. The possibility of low-temperature ferroelectric-type phase transitions in electric dipole doped host crystals is provocative, and indeed low-temperature susceptibility maxima have been observed in alkali halides doped with relatively high concentrations of such dipolar impurities as 0H, CN, and Li+.l~ZInaddition, crystals of such dipolar molecules as the hydrogen halides are known to have ferroelectric phase transitions.3 This motivated the present study, which, as will be seen below, has been illuminating, although indicative of the necessity of more complicated models, An important example of a system with orientation-dependent potential is a pair of free rotors with electric dipole interaction, i.e., a pair of linear dipolar molecules. (Other examples would be six- or eight-well tunnelling systems, which have the disadvantage of depending on the orientation of the interparticle axis relative to *
.
the crystal axes.) The energy levels for this systern have been found by Jungst and v. Meyenn.’ In this paper we discuss the susceptibility, and speculate on applying the results of this problem to the susceptibilities of dipolar impurities in solids,2 and to ferroelectric molecular crystals such as the hydrogen halides. Our aim in this work is not to examine properties in the critical region, where long-range correlations are important and the precise nature of the interaction is irrelevant, but to restrict ourselves to regions of finite-range correlations where properties are determined by the details of the Hamiltonian. Problems in the lore of magnetism bearing some relation to this work are, for the lattice case, the Ising model with transverse field, ~ and a magnet with crystal-field splitting,6 and, for the disordered case, the work of Klein on Cu : Mn. .
This work has been supported by the U.S. Atomic Energy Commission under Contract No. AT (30—1)—3699, Technical Report No. NYO—3699 —55. 1323
The Hamiltonian for a pair of dipoles may be written as (1) 1’ 2 where L~is the square of the orbital angular
1324
A NOTE ON THE TWO-DIPOLE PROBLEM
momentum operator for particle i, having eigenvalues L(L + 1), Fr~)4,I is a parameter whose 2 is the moment of value inertia, of order pis the dipole(!p2/h moment, rthe interparticle distance, and
V(~Z~ 1l~) 2 cos 0~ cos sin O~ sin 0, cos (q~ ç~,). =
—
(2)
—
We have chosen the unit of ener~’ on the order of the libronic energy for an ordered array of dipoles.
Vol. 9, No. 15
states have m
1 = m2 = 0, and, in the basis IL ,L2> have the form [alOO> + increasing bill>] andF. [101> 10>], where b increases with The +latter state has a decrease in energy first order in the potential, and becomes nearly degenerate with the ground state as F becomes large. In an analogous way, a translationally invariant, infinite array of dipoles at zero temperature will, in absence of interaction, have excited states consisting of one dipole in the state L = 1, m = 0, ±1. For small F these states are more properly written
The potential can be written in operator notation by the substitutions cos 0
sin 0 ~
=
=
±a[
a4
+
a
(3)
~ L4~ 1
1
L±mj
± mj
±
(4) a~YL -~
L ±~Lm =
=
[L2 _m2l2 ________ j
L4L2
[(L ~ m)(L ±m
~
~Lm
1)~YLm +
~ y>
=
N~~ei-~!ac4
where ~Q> is the non-interacting ground state and c4 7~(k) is a linear combination of a~,
L~+a~, and L~_a~.These states form bands centered about E 2/F having bandwidth of order F. =
(5)
For some value~ofF of order unity the bottom of one or more of the bands becomes degenerate (or nearly degenerate) ofwith the ground state,The leading to the possibility a phase transition.
(6)
highly anharmonic nature of the model and the importance of the higher energy states make
As pointed out in reference 4, there are three good quantum numbers for the eignestates of this Hamiltonian: (1) the eigenfunctions will be symmetric (s) or antisymrnetric(a) under interchange of the two particles forming the pair. For a translationally invariant system, each state has a wavevector k.
quantitative calculations difficult for an infinite system, but it is straightforward to show that the lowest state of the band c~a= a1~is at k = 0 in the one-dimensional case, and k = (7T, 7T, 0) in the case of a simple cubic lattice. For the case of two dipoles we have calculated, as a function of inverse temperature, the quantities
(2) the quantity ~ L~will either be even (g) orodd(u). (3) the orbital angular momentum about the interparticle axis, ~ m~,is a good quanturn number.
-~_.
symmetries, and if, the properties of the those states having problem is reduced
to finding the eigenvalues and eigenvectors of relatively small matrices. The energy levels can be found in reference 4. It is worthwhile to point out the nature of the two lowest states. The main components of these
~
2= ~
~
<~ O,~, t> (7)
(cos01+ cosO2) and =
If we take advantage of these for the purpose of calculating low energy states, we neglect very high values of ~L L~,the
(k)j~>
a
.cç e~a /kT
FZL4
0,,~ j>
—
Es),
where 0,, =~cos 0,
O~= ~5ifl
O~C05
Z =~exp(—E~/kT).
(8)
Vol. 9, No. 15
A NOTE ON THE TWO-DIPOLE PROBLEM
1325
$0.0 F~3.2 F.ID
a
N
U
+ U, 0
F 1.6
~ 0.5 FIG. 1. The parallel correlation, [(cos
I-
2
0~+ cos 02)
4 —
8
$6
B/I’
2/3], as a function of inverse temperature.
The calculations were done using the exact eigenstates and eigenvectors for all states of energy less than 7/F in the restricted Hilbert space of £ L~~ 12, except that for the states
In an attempt to salvage a susceptibility peak, we have calculated the parallel polarization as a function of field at zero temperature. (Fig. 3). For small values of F the curve remains linear
[s, g, m
up to relatively high fields. For large F the curve
=
0] we considered ~ L~~ 14. For the
range of temperature and F considered the results seemed insensitive to these cut-offs; higher values of F would require enlargement of the Hi!bert space, and higher temperatures would require higher energy levels to be considered as well. In the scheme used the largest matrix to be diagonalized was 10 x 10, and most were 4 x 4 or 5x 5 The quantities (7) and (8) are plotted in Figs. 1 and 2. The behavior of the static susceptibilities is disappointingly dull. The parallel susceptibility rises with F, and the transverse susceptibility is correspondingly depressed. Neither shows a peak as a function of temperature. There is a slight peak in the correlation function, which would correspond to a peak in the high frequency susceptibility. This peak is due to the first excited state’s being more correlated than the ground state, at the expense of some kinetic energy.
bends over fairly soon. If we assume that a pair 2) is in a local field (at T = 0) given by (2lpE~ 00/h = F2, we find the differential susceptibility lowered by a factor of 0.7 for F = 0.8, and by a factor of 0.09 for F = 1.6. Such local fields are not all unreasonable in the light of 2measurements of remanent polarization by Fiory of several microcoulombs/m2.These value correspond to (2IpE/F12) 102. This average field will certainly be much enhanced in the immediate neighborhood of a strongly interacting pair.
Table 1 Distance Molecule or ion HF H~1 HI CO OH -
(A) 2.5 3 3 3 5
F2 (approximate) 60 10 5 1 3
1326
A NOTE ON THE TWO-DIPOLE PROBLEM
Vol. 9, No. 15
:.o
10.0
F=O.I
//
1.0
-
/
/
/
/
/
/
-‘-
/~..— -~7
/4(
I/I 1/
r=o.s
-~~J’
‘—‘-I
1/ 1/
I/I
/ L.
,-
/ I.
r~t.6
>< 0.1
/ /
_________.0I
FIG. 2. The parallel susceptibility (solid line; left scale) and perpendicular susceptibility (broken line; right scale) as defined in equation (8) as a function of inverse temperature. We next ask if, at the experimental concentrations, it is reasonable to expect large numbers of strongly interacting pairs. Table 1 shows that, for the case of OH in alkali halides, neglecting any lattice effects, only near-neighbor pairs would show significant effects. If we assume that individual impurities are randomly distributed, there are probably too few near-neighbor pairs to be significant. There are, however, two factors favoring the formation of such pairs. The first is that the lattice polarization enhances the interaction of more distant pairs. The second is that at high concentrations the assumption or random distribution of impurities is probably unrealistic,
The van der Waals forces between OW ions (pro2I) are comparable to those between portional to p HF molecules, and HF liquifies at about room temperature. Furthermore, the properties of crystals with the highest concentrations of Li~have been observed to change with time when stored at room temperature.8 This suggests that clustering effects may be highly significant. Finally, we would like to suggest that the detailed nature of the Hamiltonian (1) has bearing on the properties of molecular crystals such as those listed in Table 1. Among other crystals of possible interest are NO, SO, HD, deuterated
Vol. 9, No. 15
A NOTE ON THE TWO-DIPOLE PROBLEM
1327
FzI.6
1.2 .
P 1.0 -
0.8
-
0.6
-
0.4
-
r~o.e
0.2
0
I 0.1
I
I
I
I
I
0.2
0.3
0.4
0.5
0.6
0.7
I
I
I
0.8
0.9
1.0
E
FIG. 3. The polarization, (cos 0~+ cos 02), at T = 0°K as a function of electric field in units of h2/Fp. methanes, and alkali hydroxides. Dipolar and quadrupolar interactions [which can be cast in a form similar to (2)] are of roughly equal importance in these crystals. MC!, HBr, and HI are known to undergo phase transitions,3 largely due to these interactions, and CO is close to having a phase 9 transition at very low temperatures. We hope by further study of this Hamiltonian to gain further
insight into the dynamics and ferroelectric properties of these materials. Acknowledgements — The author would like to thank R.O. Pohi and co-workers, especially A. Fiory, for much encouragement and many helpful discussions, and J.A. Krumhansl for support, for many useful hints and references, and for many useful criticisms of the manuscript.
REFERENCES 1.
KANZIG W., HART H.R., Jr., and ROBERTS S., Phys. Rev. Lett. 13, 543 (1964).
2.
FIORY A.T., Cornell University Materials Science Center Report No. 1410. To be published in Phys.
Rev. 3.
A recent reference is E. Hanamura, Proc. 2nd Intern. Meeting Ferroelectricity, Kyoto, 1969, J. Phys. Soc. Japan 28 (1970) Suppi. p. 189.
4.
JUNGST K.L. and MEYENN K.v., Z. Phys. 242, 45 (1971).
5. ELLIOT R.J., PFEUTY P. and WOOD C., Phys. Rev. Leti. 25, 443 (1970). 6. WANG Y.L. and COOPER B.R., Phys. Rev. 185, 696 (1969). 7. KLEIN M.W., Phys. Rev. 188, 933 (1969). 8. 9.
FIORY A. and POHL R.O., private communication. CURL R.F., HOPKINS H.P. and PITZER K.S., J. Chem. Phys. 48, 4064 (1968).
1328
A NOTE ON THE TWO-DIPOLE PROBLEM Nous étudions le problème de deux dipoles, considérés comme rotateurs quantiques libres, ayant une distance interatomique constante. Nous calculons la susceptibilité parallèle et perpendiculaire ainsi que la correlation dipolaire en fonction de la temperature. La polarisation au zero absolu est également calculée en fonction du champs électrique. Nous discutons Ia nature des excitations élémeritaires d’un réseau de dipoles électriques. Ces résultats sont alors utilisés dans le but d’expliquer le maximum de la susceptibilité observée dans la halo-génure d’alcalins dopes, ainsi que les propriétés des cristaux moléculaires tels que les halogénures d’hydrogene.
Vol. 9, No. 15