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The Néel temperature of bcc 3He
Volume 41A, number 5
PHYSICS LETTERS
23 October 1972
THE NkEL TEMPERATURE OF bee 3He L.I. ZANE* Department
of Physics, Colorado State University, Fort Collins, Colorado 80521,
USA
Received 21 August 1972 The N&e1temperature is calculated for bee 3He, including the effect of triple exchange, using the 2-particle clutter approximation of Strieb, Callen and Horwitz. The inclusion of triple exchange raises the expected Niel temperature to between 2.5 and 2.9 mK.
Triple exchange [ 1, 21 has been postulated as an added degree of freedom in bee 3He which makes recent anomalous experiments [3-S] less mysterious. Now the effect of triple exchange on the NCel temperature is calculated using the results of Callen and Callen [6]. Callen and Callen applied the 2-particle cluster approximation of Strieb, Callen and Horwitz [7] to systems with antiferromagnetic ordering of the first kind, ordering in which all the spins are either up or down. The result of this application is that the expected Ne’el temperature of solid 3He near the melting curve is raised to between 2.5 and 2.9 mK. The new Nkel temperature agrees with the temperature of a transition observed by Osheroff, Richardson, and Lee [8] . The exact nature of their transition is still not clear but Horner and Nosanow [9] have suggested that it is indeed the expected second order antiferromagnetic transition taking place at an elevated temperature due to triple exchange. This letter lends credence to the above suggestion by demonstrating that the increase in the Ndel temperature due to triple exchange is of the desired magnitude. The Hamiltonian for bee 3He can be written as [2],
H,,(bcc)
=
-2hJ(nn)
i
@i’Uj -2Wnnn)
1:
Ui’Qj 9 0)
where J(nn) and J(nnn) are respectively the total exchange rates between nearest and next nearest neigh* Work supported by NSF Grant No. GP-22553.
bors (i.e. they include contributions from pairs, triples, etc.), Ui is the nuclear spin of the ith atom and the nn and nnn above the summations restrict them to nearest and next nearest neighbors respectively. Eq. (1) may be rewritten as,
Hex@4 = -2Wnn)
,nn
nnn
(,gai*Qj+A
,g
ei*aj) , (2)
where A = J(nnn)/J(nn)
.
(3)
The magnetic pressure experiment of Kirk and Adams [4] was analyzed in terms of the Hamiltonian in eq. (2) and a value of -0.22 was obtained for the parameter A. Eq. (37a) of ref. [2] is used to find J(nn) in terms of known quantities: J(nn) = .$(I’)/( 1+$A 2B)” , where J;(P) is the two particle nearest neighbor exchange rate found by Panczyk and Adams [lo] . The factor ( 1+$A2B) = 1.03, where B is the ratio of the Griineisen constants for next nearest neighbor to nearest neighbor exchange and is equal to 1.9. Therefore we can equate J(nn) to .I@‘) without incurring much error. At a molar volume of 24 cm3 J(nn) is equal to -0.7 mK. The ferromagnetic ordering of the next nearest neighbors which is due to the positive sign of J(nnn), tends to help the crystal as a whole align antiferromagnetically. This tendency explains why an increase in the Nkel temperature is expected when triple exchange is taken into account. There are different types of ordering possible in a bee lattice with nearest and next nearest neighbor exchange. The ratio A de421
Volume
41A, number
5
PHYSICS
LETTERS
23 October
1972
Callen has been found reliable for ferromagnets with nearest and next nearest exchange [ 131. If we assume that the theory of Baker et al. scales in a manner similar to the theory of Callen and Callen we obtain a value of about 2.5 mK for the high temperature expansion theory Niel temperature. Due to the limited information available on the reliability of either theory when applied to antiferromagnets we state with qualified confidence that the N&e1temperature for solid 3He near melting should fall between 2.5 mK and 2.9 mK.
Fig. I. Upper Niel temperature as a function of the ratio of s,uchange constants, J(nnn)/lJ(nn)l, for a bee lattice with antiferromagnetic ordering of the fist kind.
termines which type of order dominates. For the value ofA given above bee 3He is expected to have an antiferromagnetic transition of the first kind [I 11 . For this type of ordering in a bee lattice all the nearest neighbors of an atom align antiparallel to the central atom while all the next nearest neighbors align parallel to the central atom. Fig. 2 of ref. [6] is reproduced, in part, as our fig. 1. This figure is used in conjunction with the parameter A to find the Nkel temperature of bee 3He. The result is a transition temperature of approximately 2.9 mK. For the case when A = 0, i.e. only nearest neighbor exchange, the high temperature expansion theory of Baker et al. [ 121 gives T = 2.9 J(nn) = 7.0 mK while the theory of Callen and Callen gives T = 3.3 J(nn) = 2.3 mK_ The theory of Baker et al. is thought to be reliable and the theory of Callen and
422
[I] L.1. Zanc, Phys. Rev. Lett. 28 (1972) 420. [ 21 L.I. Zane, J. Low Temp. Phys., to be published. [3] E.B. Osgood and M. Garber, Phys. Rev. Lett. 26 (1971) 353. [ 4 1 W.P. Kirk and R.1). Adams, Phys. Rev. Lett. 27 (1971) 392. 151 R.T. Johnson, R.1:. Rapp, and J.(‘. Wheatley, J. Low Temp. Phys. 6 (1972) 445. f1.B. Callen and F.. Callen, J. Phys. Sot. Japan 20 (1965) 1980. 1~. Strieb, 11.13. Callen and G. Horwitz. Phys. Rev. 130 (1963) 1798. D.D. Osheroff, RX’. Richardson, and D.M. Lee, Phys. Rev. l.ett. 28 (1972) 885. tl. Horner and L.11. Nosanow, Phys. Rev. Lett. 29 (1972) 88. M.F. l’anczyk and 1.1.1~.Adams, Phys. Rev. 187 (1969) 321. J. Samuel Smart, Effcctivt field theories of magnetism (W.B. Saunders Co.. Philadelphia and London, 1966), Chap. 7. GA. Baker Jr. et al., Phys. Rev. 164 (1967) 800. H.B. Callen and 1:. Callen. Phys. Rev. 136 (1964) 1675A.
PHYSICS LETTERS
23 October 1972
THE NkEL TEMPERATURE OF bee 3He L.I. ZANE* Department
of Physics, Colorado State University, Fort Collins, Colorado 80521,
USA
Received 21 August 1972 The N&e1temperature is calculated for bee 3He, including the effect of triple exchange, using the 2-particle clutter approximation of Strieb, Callen and Horwitz. The inclusion of triple exchange raises the expected Niel temperature to between 2.5 and 2.9 mK.
Triple exchange [ 1, 21 has been postulated as an added degree of freedom in bee 3He which makes recent anomalous experiments [3-S] less mysterious. Now the effect of triple exchange on the NCel temperature is calculated using the results of Callen and Callen [6]. Callen and Callen applied the 2-particle cluster approximation of Strieb, Callen and Horwitz [7] to systems with antiferromagnetic ordering of the first kind, ordering in which all the spins are either up or down. The result of this application is that the expected Ne’el temperature of solid 3He near the melting curve is raised to between 2.5 and 2.9 mK. The new Nkel temperature agrees with the temperature of a transition observed by Osheroff, Richardson, and Lee [8] . The exact nature of their transition is still not clear but Horner and Nosanow [9] have suggested that it is indeed the expected second order antiferromagnetic transition taking place at an elevated temperature due to triple exchange. This letter lends credence to the above suggestion by demonstrating that the increase in the Ndel temperature due to triple exchange is of the desired magnitude. The Hamiltonian for bee 3He can be written as [2],
H,,(bcc)
=
-2hJ(nn)
i
@i’Uj -2Wnnn)
1:
Ui’Qj 9 0)
where J(nn) and J(nnn) are respectively the total exchange rates between nearest and next nearest neigh* Work supported by NSF Grant No. GP-22553.
bors (i.e. they include contributions from pairs, triples, etc.), Ui is the nuclear spin of the ith atom and the nn and nnn above the summations restrict them to nearest and next nearest neighbors respectively. Eq. (1) may be rewritten as,
Hex@4 = -2Wnn)
,nn
nnn
(,gai*Qj+A
,g
ei*aj) , (2)
where A = J(nnn)/J(nn)
.
(3)
The magnetic pressure experiment of Kirk and Adams [4] was analyzed in terms of the Hamiltonian in eq. (2) and a value of -0.22 was obtained for the parameter A. Eq. (37a) of ref. [2] is used to find J(nn) in terms of known quantities: J(nn) = .$(I’)/( 1+$A 2B)” , where J;(P) is the two particle nearest neighbor exchange rate found by Panczyk and Adams [lo] . The factor ( 1+$A2B) = 1.03, where B is the ratio of the Griineisen constants for next nearest neighbor to nearest neighbor exchange and is equal to 1.9. Therefore we can equate J(nn) to .I@‘) without incurring much error. At a molar volume of 24 cm3 J(nn) is equal to -0.7 mK. The ferromagnetic ordering of the next nearest neighbors which is due to the positive sign of J(nnn), tends to help the crystal as a whole align antiferromagnetically. This tendency explains why an increase in the Nkel temperature is expected when triple exchange is taken into account. There are different types of ordering possible in a bee lattice with nearest and next nearest neighbor exchange. The ratio A de421
Volume
41A, number
5
PHYSICS
LETTERS
23 October
1972
Callen has been found reliable for ferromagnets with nearest and next nearest exchange [ 131. If we assume that the theory of Baker et al. scales in a manner similar to the theory of Callen and Callen we obtain a value of about 2.5 mK for the high temperature expansion theory Niel temperature. Due to the limited information available on the reliability of either theory when applied to antiferromagnets we state with qualified confidence that the N&e1temperature for solid 3He near melting should fall between 2.5 mK and 2.9 mK.
Fig. I. Upper Niel temperature as a function of the ratio of s,uchange constants, J(nnn)/lJ(nn)l, for a bee lattice with antiferromagnetic ordering of the fist kind.
termines which type of order dominates. For the value ofA given above bee 3He is expected to have an antiferromagnetic transition of the first kind [I 11 . For this type of ordering in a bee lattice all the nearest neighbors of an atom align antiparallel to the central atom while all the next nearest neighbors align parallel to the central atom. Fig. 2 of ref. [6] is reproduced, in part, as our fig. 1. This figure is used in conjunction with the parameter A to find the Nkel temperature of bee 3He. The result is a transition temperature of approximately 2.9 mK. For the case when A = 0, i.e. only nearest neighbor exchange, the high temperature expansion theory of Baker et al. [ 121 gives T = 2.9 J(nn) = 7.0 mK while the theory of Callen and Callen gives T = 3.3 J(nn) = 2.3 mK_ The theory of Baker et al. is thought to be reliable and the theory of Callen and
422
[I] L.1. Zanc, Phys. Rev. Lett. 28 (1972) 420. [ 21 L.I. Zane, J. Low Temp. Phys., to be published. [3] E.B. Osgood and M. Garber, Phys. Rev. Lett. 26 (1971) 353. [ 4 1 W.P. Kirk and R.1). Adams, Phys. Rev. Lett. 27 (1971) 392. 151 R.T. Johnson, R.1:. Rapp, and J.(‘. Wheatley, J. Low Temp. Phys. 6 (1972) 445. f1.B. Callen and F.. Callen, J. Phys. Sot. Japan 20 (1965) 1980. 1~. Strieb, 11.13. Callen and G. Horwitz. Phys. Rev. 130 (1963) 1798. D.D. Osheroff, RX’. Richardson, and D.M. Lee, Phys. Rev. l.ett. 28 (1972) 885. tl. Horner and L.11. Nosanow, Phys. Rev. Lett. 29 (1972) 88. M.F. l’anczyk and 1.1.1~.Adams, Phys. Rev. 187 (1969) 321. J. Samuel Smart, Effcctivt field theories of magnetism (W.B. Saunders Co.. Philadelphia and London, 1966), Chap. 7. GA. Baker Jr. et al., Phys. Rev. 164 (1967) 800. H.B. Callen and 1:. Callen. Phys. Rev. 136 (1964) 1675A.