J. Phys. Chem. 1983, 87, 4609-4613
4809
SUPERSATURATION Figure 3. Plot of size (Le., total number of monomer units independent of whether they are in polymer OT not) of the nucleus against monomer supersaturation for the case of vinyl acetate at 280 K.
lifetime as short as 6 X lo4 s. This last result is a formal mathematical one. The collision frequency of monomers with the nucleus is not high enough to provide such a short lifetime. For supersaturations lower than S = 1.2, there is no free energy barrier to nucleation. Thus, every polymer of any size will eventually produce a drop. Therefore, the table does not exhibit data for supersaturations lower than S = 1.2. The results of this investigation therefore indicate that experiments with vinyl acetate, emphasizing the unary process, are feasible. As we have indicated, we report the results of such experiments in a later paper. ~ Figure 2 is a plot of the nominal values of x which at each value of S will lead, according to Table 11, to immediate nucleation, i.e., to 1 ps. Polymers in a narrow group of sizes smaller than this size will have nucleating ability. Figure 3 is a plot of the corresponding size (in total monomer units) of the nucleus for each of these cases.
cause the immediate formation of a drop. In this case very large polymers can be sustained in the vapor phase. For example, a polymer of size x = 2000 has a computed lifetime of s. In contrast, one of size x = 3000 has a
Acknowledgment. This work was supported by NSF grant no. CHE82-07432. Registry No. Vinyl acetate, 108-05-4.
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MONOMER
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Thermodynamics of Deformable Spin-l/2 Heisenberg Antiferromagnetic Chains S. Jagannathan and 2. G. Soos' Department of Chemistry, Princeton Universlty, Princeton, New Jersey 08544 (Received: April 14, 1983)
Isotropic Heisenberg exchange Hamiltonians conserve total S for arbitrary spin-phonon coupling, J ( R ) ,and require a separate Born-Oppenheimer approximation in each S manifold. We examine the fully relaxed limit of slow scattering when the geometry of each spin state IS,r) in chains of 6,8, or 10 sites is optimized for linear or Gaussian J(R)and a harmonic lattice with force constant k. On extrapolating to infinite chains, the resulting thermodynamics and in particular the magnetic susceptibility, x( T),resemble a dimerized alternating chain for kT J(Ro),the average exchange. The reduced susceptibilityfrom the regular-chain value depends on c 2 / k ,where c = J'(Ro)/J(Ro)is the linear spin-phonon coupling. Soft lattices with c 2 / k = 1have significantly lower x ( T ) well above the mean-field spin Peierls transition temperature, T,. The reduction then depends on the functional form of J(R). Organic ion-radical chains with intermediate J(Ro)are good candidates, since competing mechanisms for reducing x( T) are then minimized.
Introduction Bray et al.' have critically reviewed spin-Peierls transitions in several ion-radical organic solids. The simplest exchange Hamiltonian for a deformable regular array of s = 112 sites is
where J > 0 is the unit of energy, y n = u , + ~- u, is the distortion of successive sites in Figure la, and r,(O) = 1 holds for the regular array. The harmonic lattice has force constant k in units of J. The kinetic energy and a three-dimensional lattice are usually added to (1). Pytte2 proved that Ising chains coupled to a three-dimensional lattice have a finite spin-Peierls transition temperature, (1)J. W.Bray, L. V. Interrante, J. S. Jacobs, and J. C. Bonner in "ExtendedLinear Chain Compounds",Vol. 111, J. S. Miller, Ed., Plenum, New York, 1983,p 353. (2) E. Pytte, Phys. Rev. B , 10,4637 (1974). 0022-365418312087-4609.$0 1.5010
T,.This rationalizes a finite T,for Heisenberg chains and a mean field description, which always leads to finite T,. The Taylor expansion through linear terms gives xn(yn)
= 1 - cyn
(2)
where c = -x'(O) > 0. The dimerized ground state in Figure l b has y n = (-1)V. The order parameter 6 ( T ) vanishes at T,, is finite for T < T,, and closely parallels the BCS energy gap in the mean field analysis of (1). The similarity can be understood through the Jordan-Wigner transformation of (1)into interacting fermions.' Mean field analysis now introduces anis~tropies,~ since the xy components of (1) are exactly soluble while the z components are quartic in the fermion operators. Thus, XY or Heisenberg models become similar in mean field and the S quantum number of (1) is lost. Since S is conserved for any x,(y,) in (l), we rigorously have noninteracting subspaces with potentially different order parameters in each S subspace. The (3)A. J. Silverstein and 2. G . Soos, J . Chem. Phys., 53,326 (1970).
0 1983 American Chemical Society
4810
(a) R e g u l a r
(b) A l t e r n a t i n g S=O
(c)
Jagannathan and Soos
The Journal of Physlcal Chemistfy, Vol. 87, No. 23, 1983
s=l
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.
2 . .
I
2
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.
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Spin correlations in state r are defined as
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and resemble bond orders in being partial derivatives of (1) with respect to x,. A purely singlet coupling between n and n 1 results in p n = 2, while triplet coupling yields p n = 0. Intermediate correlations in the chain yield 0 I pn 5 2 . The expectation value of (1) with respect to Ir) may consequently be written as
+
* . 2n
t
E,((Y))= C(-xn(yn)Pn n
2
Figure 1. (a) Deformable regular chain with distortions up at site p. (b) Dhnerired chain with alternation 6. (e) Lowest triplet state and bond orders for a 10-spin chain with linear spin-phonon coupling and c 2 / k = 1.
choice 6 ( T ) is appropriate for the S = 0 ground states; S = 1 yields quite different, soliton-like distortions; and the ferromagnetic S = N / 2 subspace remains undistorted for linear distortions in (2) because the total exchange then remains (N - 1) for fixed overall length. It is instructive to restate the problem in terms of the Born-Oppenheimer approximation. Sufficiently rapid electron-electron or spin-spin scattering leads to a fixed equilibrium geometry for the lattice, with at most a single order parameter like 6( T ) to describe dimerization below T, and a regular lattice above T,. Such a picture is persuasive for metallic electrons, where k-state scattering rates easily exceed phonon frequencies, but not for spin systems with long spin-lattice relaxation times and exchanges small or comparable to phonon energies. For isotropic exchange, the Born-Oppenheimer approximation must in principle be made for each S. Of course, corrections to (1) involving anisotropic exchange contributions due to spin-orbit interactions or to dipolar and hyperfine interactions may justify a single order parameter in particular systems for which (1) is inadequate in this sense. We postulate here the other extreme, a fully relaxed lattice in which individual spin states I S j ) have distortions ly,) minimizing (1).Spin-spin scattering is assumed to be sufficiently slow for the lattice to readjust to excited-state geometries (y,) relative to the regular lattice. Such local geometrical changes correspond to topological solitons for the lowest excited state^.^ We study such distortions in all excited state for finite segments of up to 10 spins5 and find that only low-lying states are significantly distorted from a regular array. The thermodynamics and in particular the magnetic susceptibility x ( T ) are obtained and extrapolated to the infiite chain. Such deviations to lower x ( T ) from the rigid lattice are often observed in organic solids that may be “generalized”spin-Peierls candidates.@
Fully Relaxed Lattice We seek solutions of (1) in which the distortions ly,) have been chosen to minimize the energy in a given spin state r. The overall length of the lattice is fixed, so that cy, = 0 n
(4)
(3)
is a constraint introduced via a Lagrange multiplier A. (4) T. Nakano and H. Fukuyama,J.Phys. SOC. Jpn., 49, 1679 (1980); W.P.Su,J. R. Schrieffer, and A. J. Heeger, Phys. Reu. Lett., 42, 1698
(1979);Phys. Rev. B , 22, 2099 (1980). (5) S. R. Bondeson and Z. G. Soos, Phys. Reu. B, 22, 1793 (1980). (6) J. B. Torrance, Ann. N . Y. Acad. Sci. 313, 210 (1978). (7)Y.Lepine, A. Caill6, and V. Larochelle, Phys. Reu. B , 18,35 85 (1978). (8) L. J De Jongh, H. tJ. M. de Groot, and J. Reedijk, preprint, 1982.
+ (1/2)kyn2 - xyn)
(5)
where we have omitted the superscript for the bond orders in (4). Minimization with respect to yn gives aE,/ay, = o = ky, - x - pnxf(yn)
(6)
It follows that (7) Iterative solutions of (6) are straightforward, since X in (7) and p,x’(yn) can be evaluated for any initial choice to give ky, and a new value for y,, x ( y , ) , and x’b,). The linear dependence of x ( y ) in (2) results in a constant x’(y) = -c and
As expected, the distortions are reduced for a stiffer lattice (large k) and increased for a stronger spin-phonon coupling c = -x’(O). The first-order corrections to a rigid chain with x,(O) = 1 involve distortions yn(l)evaluated at yn = 0 in either (8) or ( 6 ) . Then ( N - l ) X in (7) reduces to cEk(0) and we have Er(bn(’’])
= Ek(O) - (1/2)kE(y,“’)* n
(9)
The first-order correction is always negative, as expected, and is simply proportional to c2/k as can be verified by inserting (8) into (9) and recalling that y,,(I)contains regular-lattice bond orders. Furthermore, y; is large when p n deviates from the average bond order in state k. The dimerized ground state in Figure l b is lowered the most, since p n is alternately larger and smaller than the average bond order. The ferromagnetic state with S = N / 2 has vanishing bond orders and distortions. The thermodynamic^^*^ of fully relaxed Heisenberg chains simply require E,(ly))from (5) instead of E,(O). The first-order result in (9) no longer holds for large distortions, when the bond orders reflect the relaxed geometry. Shifts closely proportional to c2/k are nevertheless found for linear x ( y ) in (2). The c2/k = 1 levels in Figure 2b, for example, are indistinguishable for I18% distortions for c = 5 and -80% distortions for c = 1, although small differences would show up on a finer scale. Selected bond orders are listed in Table I for c2/k = 1 and linear or Gaussian exchanges. All states of 6,8, and 10 spin chains were relaxed. The lowest states in Figure 2 often required 5-10 iterations for c2/k = 1,while most of the upper states had small or negligible shifts.
Magnetic Susceptibility of Relaxed Heisenberg Chains The thermodynamics of rigid regular Heisenberg have been extensively10applied to both inorganic and organic (9) J. C. Bonner and M.
E. Fisher, Phys. Reu. A , 135, 640 (1964).
The Journal of Physical Chemistry, Vol. 87, No. 23, 1983 4611
Thermodynamics of Deformable Heisenberg Chains
TABLE I: Bond Orders (4)for Fully Relaxed Chains of 6,8, and 10 Spins in the Lowest S = 0 and S = 1 States for c'/k = 1 and Linear Exchange ( 2 ) or Gaussian Exchange (13)
(b)
s=o
S=l
~
bond orders
linear, Gaussian, N
6 8 10 6 8 10 6 8 10 8 10 10
p1 pz p3
p4
p5
~
eq 2
eq 13
linear
Gaussian
2.000 1.999 1.999 0.529 0.540 0.546 1.999 1.998 1.997 0.540 0.546 1.997
2.000 2.000 2.000 0.523 0.526 0.528 1.999 2.000 1.999 0.526 0.528 1.999
0.822 1.944 1.958 1.937 0.812 0.769 0.556 0.600 0.751 1.980 1.960 0.552
0.617 1.994 1.995 1.990 0.601 0.592 0.546 0.559 0.592 1.993 1.993 0.542
c2/k = 0
Flgure 2. Low-lying singlets (S), triplets (T), and quintets (Q) of 10-spin chains: (a) regular chain: (b) linear exchange, eq 2, and c 2 / k = 1; (c) Gaussian exchange, eq 13 and c 2 / k = 1.
insulators. The reduced temperature t = k T / J suffices for yn = 0 and x ( 0 ) = 1in (1). The magnetic susceptibility is denoted as xo(t),while that of the relaxed lattice is X(t,c2/k)for force constant k in (1)and linear coupling c in (2). Both the t 0 and t m limits may be obtained in general, without recourse to finite N computations. Thermally decoupled spin for t m reduce p>) in (4) to 1/2 for all bonds n or states r. Linear spin-phonon coupling and fixed overall length then makes the average spin energy independent of yn. I t follows that the total energy (5) is minimized by yn = 0 for all n, since the regular chain has the lowest lattice energy by hypothesis. Linear spin-phonon coupling and fixed length thus guarantees that rigid, regular results are regained as t m. The t 0 limit, on the other hand, resembles the alternating (6 > 0) rigid lattice, which corresponds to the S = 0 ground state of the relaxed chain. The regular (6 = 0) chain" has vanishing magnetic gap and finite ~ ~ (as0 ) Flgure 3. Susceptlbiltiyratios, eq 11, of deformable and regular chains t 0. Rigid alternating chains have a finite singlet-triplet of 6, 8 and 10 spins for liner spin-phonon coupling and fully relaxed 0 energy gap,12J3h E T ( 6 ) > 0, that dominates as t geometries. The dashed lines are least-squares extrapolations to N w. xo(T,6) T1exp[-hEd6)/kTl (10) The lowest S = 1excitation in deformable chains is lower, hibitive to relax all excited states and obtain the therbut still finite, on relaxing the lattice as sketched in Figure modynamics. It is consequently natural to define a reduction factor IC and given in Table I for c 2 / k = 1. The relaxed geometry resembles a pair of domain walls separating regions with f ( t , C 2 / k )= X(t,C2/k)/Xo(t) (11) reversed alternation4 The width 4 of such spin solitons describe the equilibrium bond orders and lattice distortions with t = k T / J to relate relaxed and rigid regular Heisen< A&(@ occurs in given in Table I. Thus 0 < A&(&[) berg chains. The t limit has f 1,while t 0 gives (10) for t 0, without changing the qualitative features. f 0 due to the finite magnetic gap. Similar ratios may Longer chains with N I20 spins13give more stringent N be defined for any other thermodynamic quantity. As m extrapolations in support of A&(&[) > 0 for 6 > 0. shown in Figure 3, f 1for c 2 / k = 0: the lattice is either The far larger (2N) number of spin states makes it prorigid ( k m) or there is no electronic gain for deformations (c = 0). The N = 6, 8, 10 curves for f(N,t,c2/k)in Figure (10) w. E. Hatfield, w . E. Estes, w . E. Marsh, M. w. Pickens, L. w. 3 are based on E,(M) from (5) and fully relaxed chains with ter Haar,and R. R. Weller in "ExtendedLinear Chain Compounds",Vol. c 2 / k = 1 / 2 and 1. Even for the softer lattice, many of the 111, J. S. Miller, Ed., Plenum, New York, 1983, pp 43-133; R. D. Willett, high-energy states are negligibly relaxed. We estimate R. M. Gaura, and C. P.Landee, ibid., pp 143-91; Z.G. Soos and S. R. 10% uncertainty for the extrapolated f ( t , c 2 / k )curves in Bondeson, ibid., pp 193-252. (11) R. B. Griffiths, Phys. Reu. A, 133, 768 (1964). 3. To better 0.10 over the entire range, we may use Figure (12) J. N. Fields, H. W. J. Blote, and J. C. Bonner, J . Appl. Phys., 60, the two-parameter fit 1807 (1907); M. C. Cross and D. S. Fisher, chys. Reu. E,19,402 (1979). (13) S. Ramasesha and Z. G. Soos, Solrd State Commun., 46, 509 2 f ( t ) = 1 - tanh A ( t - to)-' (12) (1983).
- -
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-
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4812
Jagannathan and Soos
The Journal of Physical Chemistry, Vol. 87, No. 23, 1983 \
\
0 08
1
I30
0.05.
/
il
Gaussian
I -.5
/goussian
N - 8
-C * _
\ \
0
k
5
0.01.
Figure 4. Linear and Gaussian exchanges, eq 2 and 13, respectively, with x ( 0 ) = 1 and -x'(O) = 1 and c = 5. Equilibrium positions have -0.20 < y < 0.20 and similar exchanges.
with A = 1.70, to = 0.74 for c 2 / k = 1 and A = 0.40, to = 0.24 for c 2 / k = 1/2. The simple form (12) connects the known T = 0 and T limits, but is not as accurate as (10) in the low-temperature regime of kT ll2,as seen from Figure 3. Such T > T , fluctuations correspond to the pseudogap regime in conventional Peierls systems,16where the conductivity rather than susceptibility is lowered. While the identification of spin-Peierls transitions T , remains open except for a few systems, T,< J / k may safely be anticipated. Downward deviations of x ( r ) around T J / k should best signal deformation effects in systems with minimal gtensor deviations and intermediate exchanges.
-
Acknowledgment. We gratefully acknowledge NSF support for this work through DMR-77-27418A01, (16) H. Z. Zeller in "Low-Dimensional Cooperative Phenomena", NATO AS1 Series B7,H.J. Keller, Ed., Plenum, New York, 1975,p 215.
