768
Journal of Magnetism and Magnetic Materials 90 & 91 (1990) 768-770 North-Holland
Thermal and magnetic properties of bee solid 3 He Tomio Iwasaki Institute of Physics, Unicersity of Tsukuba, Ibaraki 305, Japan
The mean-field properties for the high-field phase of the bee solid 3 He arc calculated including the two-, three-, Iour-, six-spin exchange processes. Using these exchange processes, we develop the spin wave theory for the U2Dl phase. We investigate the effects of the six-spin exchange process .
1. Introduction It is considered that the multiple-particle exchange processes, especially the pair, triple and quadruple spin exchange interactions play an essential role in the nuclear magnetic properties of the solid 3 He. These exchange processes cause the solid 3 He to possess the interesting magnetic structure for each phase.
2. Mean field theory for ordered phase The exchange Hamiltonian of the bee 3 He including the crown six-spin exchange process is given by [1-3] 3
H cx = -2
E EJn ( Sj ' SJ n=1
- S\
E
- 4 c
E
(Pjjklmn
ee
E
Ka[(Sj'SJ(Sk' S,)
+ (Sj'S')(Sj'Sd-(Sj'Sd(Sj'S,)]
P, F t-cj-ck «!
+ Pi}kl/mn -
5
2 X 2- ) ,
(1)
i
where Nn's are effective pair-exchange frequencies between the nth nearest neighbour sites which include three- and four-spin exchanges, and K's composed of K P and K F are the planar and folded four-spin exchange coupling constants, and S\ is the crown six-spin exchange coupling constant (3]. If the external magnetic field, H, is applied in the z direction, the total Hamiltonian is given by
(2)
H I OI = Hex - g/-lHESiz, i
where g and po are the g factor of 3 He and the nuclear magneton. First we investigate the properties of the high-field phase, which contains the canted normal antiferromagnetic phase and the pseudoferromagnetic phase. From eqs. (1) and (2), the mean-field energy is given by [3] E II F ( H, p, u) = - !N{ TJ1Fp 2 + 3( K
+ !S\)p4 + 24(K + SI)p4 U4 + 8[(J\ + ~S\) - 3(K + ~S\)p21 p2 U2
+ !S\p6(32u 6 - 4811 4 + I8u 2 - 1) + gu Hpu },
(3)
where TJ1F and K are defined by Tit = -4J\ + 3J2 + 6J) - 3/2S\ and K F + K p, and N is the total number of 3 He atoms. Here we define u = cos 8, 8 being the angle between the sublattice magnetization and the field H, and p is the sublattice polarization given by p = 2(Sz) . For spin 1/2, the entropy S(p) can be written as S(p) =kBN{ln 2-
H(1 + p) In(I + p) + (I-p) In(I-p)]},
(4)
as far as the mean-field theory is concerned. From eqs. (3) and (4), p and u are obtained by minimizing the free energy FllF(T, H, p, II) = EllF(H, p, u) - TS(p). In this way, the temperature dependence of cos 0 is shown in fig. 1. According to this result, two kinds of phase transitions seem to occur. One is the first-order transition between the 0304-8853/90/503.50 © 1990 - Elsevier Science Publishers B.V. (North-Holland) and Yamada Science Foundation
T. Iwasaki / Thermal and magnetic properties ofbee 3He
769
case 10
---------~~~:~------------] I
0.5
I I I
I
II I
II II I I
I I
a
1.0
I I
I I
1.441.56 T (mKI
Fig. 1. The value of cos 0 by the mean-field theory for the high-field phase is plotted versus T (JNN = -0.46 mK, J, K p = - 0.27 mK. K F = - 0.027 mK, S\ = - 0.035 mK, H = 0.5 1).
=
-0.19 mK,
pseudoferromagnetic phase and the normal anti ferromagnetic (naf) phase. The other is the second-order transition between the naf phase and the paramagnetic phase.
The uudd phase has four sublattices 1= 1, 2, 3 and 4. Introducing the spin-deviation operators, the Hamiltonian H o bilinear in magnon variable can be written Ho =
~ { w(k) t~l al(k)at(k) + (y(k)[ a\(k )a3( -k) + a2(~)a4( -k)] +v(k)[al(k)a4(-k)+aHk)aH-k)] +1J(k)[aT(k)a2(k)+aHk)a4(k)] +h.C}
(5)
Here w(k), y(k), v(k) and 1J(k) are defined by
w(k)
=
2S{2J2 - 4J3 - 2St - (48K p + 16K F-16S t)S2 - 2( J2 + 4KpS2)[cos{kxG) + cos(kyG)] - [4J3 + 4S t + (8K p + 24S t)S2] cos(kxG) cos(kyG)} ,
y(k)
=
4S cos(kzG){ -J2+ 4(Kp + 2K F)S2 + [-2J3 - St + 4(K p + St)S2] [cos(kxG) + cos(kyG)]},
v(k)
=
8S [ -Jt
-
~St - (12K p + 4K F
-
(6) (7)
2St)S2 + 4S tS 4 ] cos(kxG/2) cos( k yG/2) e i k,a/2
- SSt {(1- 8S 2 + 16S 4 ) cos{kxG/2) cos(kyG/2) e- 3ik ,a/ 2 + (1 + 24S 2 + 16S 4 ) [cos(3kxG/2) cos( k yG/2) + cos(kxG/2) cos(3kyG/2)] e i k,a/2},
(8)
and
1J(k) = 8S[ -Jt
-
~St + (12K p + 4K F + 2SdS 2 + 4S tS 4 ] cos(kxG/2) cos( k yG/2) ei k ,a/ 2
- SSt {(I - 8S 2 + 16S 4 ) cos(kxG/2) cos( k yG/2) e- 3 ik,a/2 + (1 - 8S 2 + 16S 4 ) [ cos(3k"G/2) cos( k yG/2) + cos(kxG/2) cos(3kyG/2)] e i k,a/2},
(9)
where G is a lattice constant and S is the magnitude of the spins, which is equal to 1/2 for 3 He. The Hamiltonian (5) is put into diagonal form [3]
Ho =
L {i\+(k)[ et(k )et(k) + el(k )e3(k)] + L(k) [el(k)e2(k) + e1(k )e4(k)] k
+[i\+(k) +L(k) - 2w(k)]},
'
(10)
770
T. Iwasaki / Thermal and magnetic properties of bee JH e
H
1.0
0.5L_----
o
0.01
0.02
T
IS\llmKI
Fig. 2. The value of CjNk B(k BT)3 by the spin wave theory for the U2D2 phase is plotted versus I SI I (iNN = -0.377 mK, J. = -0.155 mK, K p = -0.327 mK).
Fig. 3. The feature of the mean-field free energy in the magnetic phase diagram.
where the new operators e;(k) are boson operators, and A_(k) and A+(k) are acoustic and optical magnon frequencies. The specific heat is given by
c../v= n'll2k n(k nT)3( V;Vt) -t, where Vt and Vt are defined by L(k):; [v;(k; feature is shown in fig. 2..
(11)
+ k;) + v}k;f/2
at low k. The specific heat is sensitive to
I S, I. This
4. Discussion In·order to investigate the feature of the mean-field free energy (see fig. 3), we expand the free energy with respect to 8, F= (N/2) { [(8Jt
+ 9S I)p2 + (24K + 30SI)p4 + 9S Ip6 + gtt Hp/2] 8 2
- [(8JI + 9S I)p2/3
+ (32K + 34Sd p4 + 27S IP6 + gtt Hp/24] 8 4
+ [2(811 + 9S I)p2/45 + 8(32K/15 + 13SI/6)p4 -78SIP6/5 + gtt Hp/720] 0 6 - [(8J I + 9S I)p2/315 + (512K + 514SI)p4/105 -729SIP6/35 + gtt Hp/40320] 8 8 } .
(12)
Investigating this expansion, we find that the first- and second-order transition lines do not seem to touch. This feature seems to contradict the experimental results. However, these experimental results do not seem to be exact. Therefore, these mean-field properties may not be wrong. References [1] D.M. Ceperley and G. Jacucci, Phys. Rev. Lett. 58 (1987) 1648. [2] H. Godfrin and D.O. Osheroff, Phys. Rev. B 38 (1988) 4492. [3] M. Roger, J.H. Hetherington and J.M. Delrieu, Rev. Mod. Phys. 55 (1983) 1.