Calculations which demonstrate a physically important four-atom exchange process in bodycentered cubic He crystal and thus an important four-spin term in the exchange Hamiltonian are discussed. A simple, mean-field analysis of this Hamiltonian appears to account for a number of the perplexing properties of bodycentered cubic He crystal. It is suggested that an understanding of other properties may require treatment of the exact four-spin term

Mcmahan, A. K.

English

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nIndex No. 67.80 Jd
Four Particle Exchange in Solid 3lie
A. K. McMahan* and J. W. Wilkinst
Laboratory of Atomic and Solid State Physics
Cornell University
Ithaca, New York 14853
Abstract
We report calculations which suggest that there is a physically-important
four-atom exchange process in boo 3 H and thus an important four-spin term in
the exchange Hamiltonian. A simple, mean-field analysis of this Hamiltonian
appears to account for a number of the perplexing properties of bee 3He. An
understanding of other properties may require treatment of the exact four-spin
term. It is our hope to stimulate such effort by this Letter.
I
a^
(NASA-CR-142801) FOUR PARTICLE EXCHANGE IN
SOLID He-3 Interim Report (Cornell Univ.)
10 p HC $3.25
	 CSCL 20L
N75-24503
Unclas
G3/76 21799
- 2 -
We report the results of calculations which suggest there is a physically
important four-atom exchange process in bee solid 3lie. The process gives rise
to n four-spin term in the effective spin or exchange Hamiltonian with an
exchange energy comparable to the nearest-neighbor two-spin term. A simple-
minded mean-field treatment suggests this four-spin term could lead to a tcLp
erature-dependent exchange frequency which offers partial insight to the several
perplexing properties of bee solid 3Ile.1
To facilitate discussion we define the exchange Hamiltonian including
2
pair, triple, and the important cyclic quadruple exchange:
H = -2 CJl - 6J
	
+ 3J
ex	 112	 1111,23) i<j li
	
13
(2)	 ..,
-2 (J2 - 4J112 + J1111,231 i<j li • 1^
(1)
-4J1111,23	 E	 [(Ii Ii )(Ik IA) + (I'^k)( It1<^Qc<,(
(li lk)a	 Y1.
The first two-spin term involves nearest-neighbor spins (the "(1)" over the sum),
while the second involves next-nearest neighbor spins (the "(2)"). Finally the
one four-s pin term involves four atoms located at the corners of the rhombus,
lying in the (110) plane, whose sides 1j, jk, kj„ ki are first-neighbor dis-
tances (the subscript "1111" in J1111 23) and whose diagonals ik, J.2 are second-
..	 ,
and third-nearest neighbors (the 23 after the comma).
Later in this letter we will present a summary of the calculations for
the various exchange constants. The relative large size of J 	 appears to1:111,23
arise from the generally larger amplitude of atomic vibrations associated with
transverse phonons. Moreover the ratio of J1111,23/1 is-  further enhanced by
making the atomic wavef unction less localized than is indite€,tad by most calculations.
I
ti
I3 _
of the ground state energy, which are relntively insensitive to the degree of
localization.
First we want to briefly discuss the anomalies in 3lie solid within the con-
text of a mean-field approximation for (1) in which, for simplicity, we ignore
the next-nearest-neighbor exchange terms except to indicate those situations
where they may play a noticeable role. Then (1) reduces to
where
(1)
H = - 2J1 (T)	 E Ii	Ij	(2)
i<j
11(T) = J  - 6J112 + 3J1111,23(1 + 4 (I 	 t/ 1 ).	 (3)
In (3) the term (I	 I/ 1 corresponds to the thermal expectation value of the
scalar product of nearest-neighbor spins. Since we expect 3 H solid to become
anti-ferromagnetic at low temperatures, then (I • I) 1 will change from zero at
high temperature to something like -(3/4) in the ordered phase. Taking a simple
analytic form for this switching behavior, and using numbers we have calculated,
we might approximate (3) above the transition.by
11(T) p - 0.65 + 0.4 tanh (2/T)	 (4)
where JI and T are measured in mK. (The factor 2 in the tanh (2/T) is an esti-
mate of the transition temperature in the absence of J1111,23*)
There are several puzzling anomalies in the 3 H solid data which we discuss
in terms of (2) and (4). we restrict ourselves to a single density, corresponding
to a molar volume of 24.0 cc/mole.
High temperature results (T > 20 mK): In this regime, JI is a temperature-
independent constant. High temperature susceptibility measurements  indicate
it is negative (anti-ferromagnetic), with its best value of -0.65 mK coming
from the pressure measurements. 4 When the pressure measurements are extended
to finite magnetic fields, 5 it appears necessary to assume an additional
ferromagnetic next-nearest neighbor interaction of strength , /J1 -0.2. 6'7
- 4 -
I
All of the high temperature data can be reconciled within such a two-parameter
model, in which J1 is still approximately -0.65 mK, and J2 is ferromagnetic,G,7
Ratios in the vicinity of -0.2 have been theoretically calculated. 6'8'1 In the
present work we can only specify +0.03 >J 2 rJ I  > -0.2 because of 10% or so
uncertainties in the various exchange frequencies, and large cancellations in
J2 - 4J112 + J1111,23 (the high temperature value of J2(T)).
Low temperature results (T <20 mK):
(i) Transition temperature. A constant first-neighbor exchange frequency of
-0.65 mK corresponds (via TN = 2.75IJI) to an anti-ferromagnetic transition
temperature of 1.8 mK (a bit higher if J2 > 0), whereas the first observed
transition in the solid is I.1 mK. 9 Clearly (4) would produce a lower transi-
tion temperature.
(ii) Specific heat. A high temperature expansion for the specific heat in
powers of T-1  using an anti-ferromagnetic Heisenberg Hamiltonian results in a
T-3  term with a positive coefficient, (even larger if J2 > 0); whereas the
experiments 9210 indicate a negative one. A temperature-dependent exchange such
as (4) can result in a negative coefficient, although our crude agrument appears
to give too small a coefficient.
Near the transition temperature of the solid the specific heat has a pro-
pounced bump 9,10 near 2 mK prior to the entropy anomaly at 1.1 mK. Our mean
field model cannot explain this structure. On the other hand, the original
llwiii.ltonian (1) we propose may be able to explain it. Specifically we propose
3.r; the phase transition theory community the following question: does HeX (1)
with its four-spin tern have the same behavior as a Heisenberg Hamiltonian in
the vicinity of the (depressed) transition temperature? Or can it exhibit
structure consistent with that observed in bee 3lie?
(iii) Spin Diffusion. - As additional evidence that something extraordinary is
occuring at about 2 mK in the solid, there is the fact that the spin diffusion
coefficient D drops sharply by over 30% for decreasing temperature about 2 mK,11
leading once again to the idea that the effective J (o: D) is strongly temperature
dependent.
(iv) II-T phase boundary. In terms of a mean-field model, an external magnetic
field would iiavor n positive (I • 1) l in (3) and hence increase V11. Pre-
sumably this effect would increase the magnetic field at which the spin-flop
to paramagnetic-solid transition occurs over that predicted by a Heisenberg
Ilamiltonian. Only preliminary experimental data 12 exist for the H-T phase
boundary.
Finally we turn to the actual calculation of 31111,23 and the other exchange
frequencies. There is an extensive literature :3 which underpins our calculation.
The exchange frequencies can be written in terms of integrals involving the
Jastrow-Gaussian ansatz for the many-body wave function in solid 3Ile:
TT e JA( ri - R i )	 it f(rij),
	
(5)
1	 i<3
where r  and R  specify the atom and lattice site positions, respectively.
The scalar coefficient A is proportional to the (Einstein) frequency
describing the harmonic motion of an atom about its lattice site. In our calcu-
lation we generalize A to a matrix 14 in order to take into account a more
realistic description of the atomic motion based on a self-consistent calculation is
of the phonon frequencies. Accordingly the Aeff listed in Table I for the four
largest exchange frequencies reflects phonon frequencies which predominate in
that exchange process.
The Jastrow function f(r ij ) is roughly a step function used to describe
the short-range correlations, by prev:nting two atoms from approaching closer
than their hard-core diameter. The energy of solid He can be written in terms
of integrals involving *. Various calculations 16 which in effect minimize the
=
energy with respect to functional form and parameters of (5), yield very similar
Y- G -
Jastrow functions. On the oti t hand the energy is not very sensitive to A. Never-
theless, it now appears on the basis of recent Monte Carlo calculations, 8 that
A may be 300 smaller than the range of values previously accepted. 	 We exploit
this possibility in our work.
In terms of the wave function (5) the exchange frequency for a given ex-
change p of atoms is
Jp . -ap
 e A dp2/4 I
p(A, tt])
	
(G)
where d  is a distance in configuration space between the original and exchanged
atoms. Values of d  in units of the nearest neighbor distance H are given for
various exchanges in Table I. The attempt frequency for the exchange process,
within the Einstein approximation for the atomic motion, is given by
0 = (h M) d  (A3/Tf) i .
The exchange frequencies are dominated by the other two factors in (G).
Gaussian overlap. The exponential factor is a measure of the overlap of the
wavefunctions of the original and exchanged atoms. Note that it is a very
sensitive function of the frequencies of the exchanging atoms. In Table I Aeff
for next-nearest-neighbor exchange is smaller than for nearest-neighbor which
is a measure of the relative ease of distortions for exchange in (100) as
opposed to (111) directions.
Hard-core effects. The final factor I corrects for the fact that the Gaussian
p
overlap overestimates the probability for atoms to exchange by not excluding
routes of exchange forbidden by the hard-core repulsion of the atoms. This
effect is most important for pair exchange where straight-line exchange would
have the atoms sitting on top of each other. In this case I 1 includes the
dominant factor exp (- 2 a2) 25 where Q is the hard-core radius. This strong
0u
	
- 7 -
6-	 vc+
dependence on A does not occur in I  for three- and four-atom exchange, where
the short-range correlation effects are largely geometric in nature. 1 The
values for I  in Table I are extrapolated from the results of Ref. 1 based on
Monte Carlo integrations (for the Einstein case), with the exception of the
last (and crucial) entry which has been estimated on the basis of the experience
gained from three-atom exchange.
In the next to the last column of Table I are the results for J 	 It is
P
clear that J1111 23 is too small but so is J 1 by at least a factor of 20, ii' it
is to agree with the high temperature measurements. At this stage we take
advantage of the fact that the more accurate Monte Carlo work  suggests that
the true A values may be 30%, or so smaller than those consistent with Ref. 15
on which our work is based. Accordingly we decrease A, or more accurately,
rescnle downwar the phonon frequencies until the value of (J l
 = 6J112 + 3J1111 23 )r
(see (1)] agrees with the exchange frequency deduced from high temperature
measurements. i7 The resulting numbers are shown in the last column. We note
that 9IJ1111,23I is _.4 mK, the factor used in (1). As a result of these cal-
culations we argue that this four-atom exchange (the next largest one is an
order of magnitude smaller) must be taken into account in any treatment of the
phase transition of bee 3He.
We gratefully acknowledge the construdtive comments and continuous encourage-
ment of R. C. Richardson. In addition the assistance of W. P. Ralperin, D. R.
Nelson, and M. E. Fisher is much appreciated.
k^
,TABLE I
Pair	 (1)
(2)
Triple "(112)
Quadruple 1111,23
scaled
(dP/21t)2 Aeff (A 2)
n e AeffdP /4	 1 JP(mK) JP(nix)P
1/2 1.350 2.00 .014 -.02B -1.02
2/3 1.129 .602 .0067 -.0040 -	 .25
5/6 1.295 .0104 .086 -.00089 - .085
1 1.136 .0042 .09 -.00038 - .045
Table I. Exchange frequencies and related intermediate numbers for bee 3Ile
(24.0 cc/mole).
- 8 -
References
* Supported in part by Lite National Aeronautics and Space Administration
through grant number NGII-33-010-188. Permawent address: L504, Lawrence
Livermore Laboratory, Livermore, California, 94550.
} Supported in part by the National Science Foundation through grant number
DMR74-23494.
1. R. A. Guyer, Phys. Rev. A 9, 1452 (1974) first suggested that a four-spin
exchange might explain the anamolous data for bee 3Ile. We offer no advance
over his menn-field investigation of the effects but do suggest why the
exchange frequency could be large.
2. See A. K. McMahan and R. A. Guyer, Phys. Rev. A 7, 1105 (1973) for a dis-
cussion of such Ilamiltonians and methods for calculating the exchange constants.
3. See review article, S. B. Trickey, W. P. Kirk, and E. D. Adams, Rev. Mod.
Phys. 44, 668 (1972), and references therein.
i
4. M. F. Panczyk and E. D. Adams, Phys. Rev. 187, 321 (1969); A 1, 1356 (1970)
5. W. P. Kirk and E. D. Adams, Phys. Rev. Lett. 27, 392 (1971).
6. L. I. Zane, Phys. Rev. Lett. 28, 420 (1972); J. Low Temp. Phys. 9, 219 (1972).
7. L. Goldstein, Phys. Rev. A 8, 2160 (1973).
8. W. L. Lim and L. H. Nosanow (to be published); W. L. Lim, Doctoral Thesis,
University of Minnesota (1973); no. 73-29, 416 University Microfilms, Ann
Arbor, Michigan.
9. W. P. Halperin, C. N. Archie, F. B. Rasmussen, R. A. Buhrman, and R. C.
Richardson, Phys. Rev. Lett. 32, 927 (1974) plus private communication.
10. J. M. Bandon and J. M. Goodkind, Phys. Rev. Lett. 32, 1343 (1974).
11. R. T. Johnson, 0. G. Symko, and J. C. Wheatley, Phys. Letters 39A, 173 (1972).
12. R. B. Kummer, E. D. Adams, W. P. Kirk, A. S. Greenberg, R. M. Mueller, C. V.
Britton and D. M. Lee, Phys. Rev. Lett. 34, 517 (1975).
- 9 -
6- 10 -
0
13. See Refs. 1 and 6 for extensive bibleogrnphy.
14. k similar calculation for the no arcs t-neighbor-pair exchange frequency hsts
been reported by L. 11. Nosanow and C. M. Varma, Phys. Rev. 187, 660 (1969).
15. The self-consistent phonon frequencies that went into our calculation were
very kindly provided by Dr. 11. R. Glyde. See 11. R. Glyde, Can. J. Phys.
49, 761 (1971).
16. It. A. Guyer and L. I. Zane, Phys. Rev. 188, 445 (1969); R. I[orner, Phys.
Rev. A 1, 1732 (1970); C. Ebner and C. C. Sung, Phys. Rev. A 4, 3.099 (1971).
17. Ref. 8 also reported Einstein exchange calculations of J1 - 61112 in better
than factor of two agreement with the high temperature experiments. Con-
versely, our procedure of actually fitting the data results in resealed
values of Al (Aeff for first-neighbor pair exchange) in fairly good agree-
ment with the Einstein A values reported in Ref. 8 at both densities considered
there. The agreement becomes excellent when differences in short-range
correlation functions are taken into account.
w


Four particle exchange in solid He-3

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