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Polarised neutrons and axial-vector interactions in magnetic materials
Physica B 297 (2001) 67}74
Polarised neutrons and axial-vector interactions in magnetic materials S.V. Maleyev* Theory Department, Petersburg Nuclear Physics Institute, 188350 St. Petersburg, Gatchina, Russia
Abstract Inelastic polarised neutron scattering is analysed when the magnetic interaction includes a part characterised by an axial vector. In this case the dynamical chiral (DC) scattering and inelastic nuclear-magnetic interference (INMI) may appear. As examples we consider magnetic "eld and antiferromagnets with k "0. Possibilities of other axial-vector $ interactions such as the Dzialoshinskii}Moriya interaction and elastic torsion are mentioned too. The DC scattering in magnetic "eld is determined by the three-spin correlations. Their experimental investigation in iron above ¹ allows to con"rm predictions of the Polyakov}Kadano!}Wilson operator algebra and demonstrate crossover to the dipolar critical dynamics. The INMI in the antiferromagnets with k "0 is considered theoretically and appearance of the $ infra-red divergence in the case of quasi-2D compound BaNi (PO ) was demonstrated. 2001 Elsevier Science B.V. All rights reserved. PACS: 75.25.#z; 75.30.GW; 75.40.Gb; 75.50.Ee Keywords: Polarised neutrons; Chiral scattering; Nuclear-magnetic interference
1. Introduction Increasing a role of polarised neutrons (PN) in investigation of magnetic materials demands an additional theoretical analysis. It was done partly during PNCMI'98 [1}3]. In this paper we continue these studies. Our main idea is the following: as the cross section and initial neutron polarisation P are scalar and an axial vector, respectively, the cross section may depend on P if the system is charac terised by an axial vector. The same holds for the polarisation of the scattered neutrons too: any
* Fax: #7-812-713-1963. E-mail address: [email protected] (S.V. Maleyev).
additional change of this axial vector comparing to its rotation due to conventional magnetic scattering may appear in the presence of an axial-vector interaction only. As examples of the axial-vector interactions one can point out the interaction with external magnetic "eld, the Dzialoshinskii}Moriya interaction [3}6], interaction of the spin chirality with elastic torsion [7], spin interaction with the atomic helix structures, etc. Antiferromagnets with k "0 pro $ vide an additional example of such systems where the staggered magnetisation leads to the rotation of the neutron polarisation due to nuclear-magnetic interference. In this paper we present general expressions for polarised neutron scattering in magnetic materials
0921-4526/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 0 ) 0 0 8 3 8 - 3
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S.V. Maleyev / Physica B 297 (2001) 67}74
(Section 2). Then in Section 3 we consider chiral scattering mediated by external magnetic "eld. Section 4 is devoted to antiferromagnets with k "0. $ In Section 5 we summarise the main results of the paper.
2. General expressions for the PN scattering For the following we should present general expressions for the PN scattering. Neglecting the interaction between nuclei and neutron spins for the scattering amplitude we have a well-known expression
1 FQ "! b e\ QRL !r f (Q)e QRL (S, ) L K K N L K "N #M, , / /
(1)
where R and b are the position and scattering L L length of the nth nucleus, r"0.54 fm, f (Q), S and K K R are the magnetic form-factor, spin and position K of the mth magnetic ion in the lattice, respectively. Here and below `Na labels the perpendicular to the scattering vector Q part of the corresponding vectors. We also omit the orbital part of the amplitude which may be easily taken into account and does not change our results appreciably. For the elastic scattering, the cross section and the "nal polarisation are given by [8}10] "N Q NQ #M, Q M,Q #iM, Q ;M,Q \ \ \ #N Q M,Q #M, Q NQ P , \ \
(2)
(P) "N Q NQ P \ #M, Q (M,Q P )#(P M, Q )M,Q \ \ !P M, Q M,Q !iM, Q ;M,Q \ \ #N Q M,Q #M, Q NQ \ \ #i[N Q M,Q !M, Q NQ ;P ], \ \
(3)
where 2 means both the thermal and disorder average. The expressions for the inelastic scattering follow from Eqs. (2) and (3) if one replaces AB by
(k /k )S (), where S () is a Van Hove function: D G
1 S ()" dt exp(it)A(t)B(0) 2 \
"A, B 1!exp ! S ¹
\
(4)
and A, B is the absorptive part of the generalisS ed susceptibility A, B determined by S A, B "i S
dt exp(it)[A(t), B(0)].
(5)
\
For A"B> we have A, B "ImA, B S [1}3,11]. From conventional representation of A, B as a sum over intermediate states we have S [3] A, B
S> B
"B, A
\S\ B
,
A, B "!B, A . S \S
(6)
The symmetry under time re#ection tP!t demands [12]: A, B
S H
"$BA
S \H
,
(7)
where H is a magnetic "eld or spontaneous magnetisation of the sample. Signs `plusa and `minusa correspond to equal and di!erent time oddness of the operators A and B, respectively. The "rst and second terms in Eqs. (2) and (3) describe nuclear and conventional magnetic scattering, respectively. The third terms are connected to the chiral scattering and the last terms describe the nuclear magnetic interference. One can show that the conventional and chiral inelastic magnetic scattering are proportional to Im 1(Q,) and Im[i (Q,)], respectively, where ?@ ?@ 1 "( $ ) are symmetric and antisym?@ ?@ @? metric parts of the magnetic susceptibility [13,14]. ?@ The nuclear-magnetic interference is well-known in the case of elastic scattering in ferromagnets. Recently, this interference appears very useful for determination of complex antiferromagnetic structures (see Refs. [15}17] and references therein). Due to recent development of the PN scattering tech-
S.V. Maleyev / Physica B 297 (2001) 67}74
nique and particularly CRYOPAD II [15] a possibility appears to study inelastic nuclear-magnetic interference (INMI) which provides direct information about spin-lattice interaction [18,19]. General theory of the INMI and its analysis for the spin Peierls compound are presented in Ref. [3]. Theoretical calculation of this interference for ferromagnets is given in Ref. [20].
3. Dynamical chirality Elastic chiral scattering takes place in the case of helix magnetic structure. It was studied experimentally in many cases (see for example Refs. [4,21]). Inelastic chiral scattering or dynamical chirality (DC) was discussed theoretically in Refs. [13,22,23]. These studies initiated experimental investigation of the DC in ferromagnets [24}28] and triangular lattice antiferromagnets [14,29}31]. As was pointed above the DC is determined by antisymmetric part of the magnetic susceptibility (Q,). According to Eq. (7) we have ?@ (Q,,H )" (!Q,,!H ), and the antisym?@ @? metric part of may appear in magnetic "eld or in the case of an axial-vector interaction, if the Fourier transform of corresponding axial vector is an odd function of Q. The Dzialoshinskii}Moriya interaction (DM) is a corresponding example [4}6]. It was shown that the DC scattering is determined by Im i (Q,) [13,14]. The -parity of this ?@ function is opposite to the t-parity of corresponding axial-vector. Particularly, it is -even in the case of magnetic "eld and -odd for the DM interaction [5,6]. Any antisymmetric tensor is related to an axial vector. In the case of uniaxial crystal one has [13] (Q,)"!i ?@ ?@A [hK S (q,)#c( (c( ) hK )S (Q,)], A & A 67
(8)
where hK and c( are unit vectors along the axial vector and preferred crystal axis, respectively. For isotropic Heisenberg interaction one has S ,0 67 and S ,0 for the X> model. &
69
The chiral contribution to the cross section has the form
k 2 (Q,)" [rf (Q)] D (1!e\S 2)\ k G [(P QK )(QK hK ) Im S (Q,) & #(P Q)(QK c( )(c( hK )Im S (Q,)]. 67
(9)
Below we limit ourselves with the case of magnetic "eld. The DM interaction is considered in Refs. [4}6]. As stated above Im S are even functions & 67 of due to t-oddness of H. As a result () changes sign with . It was con"rmed experimentally for critical scattering in iron [24] and triangular-lattice antiferromagnets CsMnBr and CsNiCl [29}31]. If for given Q characteristic en ergy ;¹, the static chiral scattering is zero:
¹ (Q)"
\
d (Q,)"0.
(10)
The -integrated chiral cross section does not vanish in quantum limit only, when one cannot replace the Bose-factor in Eq. (9) by (¹/). In weak magnetic "eld (Q,) is linear in H and connected to the three-spin correlation function. It is a new object for experimental studies and may provide an additional information which is not accessible to conventional neutron scattering dealing with the two-spin correlations. This general statement is illustrated below by DC critical scattering experiments in iron [24}28]. Recently, DC critical scattering in triangular lattice antiferromagnets CsMnBr and CsNiCl al lowed to determine the chiral critical exponent which could not be measured by other methods [29,31]. 3.1. Polyakov}Kadanow}Wilson (PKW) algebra and critical factorisation of the tree-spin correlations in iron [25] Conventional neutron magnetic scattering is determined by two-spin correlation function and depends on one momentum Q. However, in the strong correlated spin systems there are n-spin nontrivial
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S.V. Maleyev / Physica B 297 (2001) 67}74
correlations where n'2, which depend on n!1 momenta q . The PKW algebra deals with n-spins G correlations near the second order phase transition. It predicts that if one momentum, say q , is much larger than others q and the inverse correlation G$ length " \, then the dependence on q appears as a factor q\V, where x"5! !1/ , where and
are the Fisher's and the correlation length exponents, respectively [32}35]. This phenomenon may be called as critical factorisation. In the case of chiral scattering along with the momentum transfer Q we have the momentum of the uniform magnetic "eld which is zero. The "eld H is conjugated to the total magnetisation which is a relevant variable for ferromagnets. As a result on the base of static and dynamic scaling theories for cubic ferromagnets where S "0 we have [23,36] 67
gH . Im S (Q,)" & ¹(Qa)V( a)W ¹ (Qa)X
(11)
Here H is the internal magnetic "eld, a is of the order of interatomic distance, y"(1/ )!(1# )/2,
"a\ T, "(¹!¹ )/¹ and z is the dynamical critical exponent. From this expression we see that if Q< , the chiral scattering is proportional to \>J>E K \ , where in the right-hand side we put "0 and vK. In triangular lattice antifer romagnets the magnetisation is not a relevant variable and in Eq. (11) we have x"3!( #1)/ , where is the chiral crossover exponent [31,32,36]. This nontrivial theoretical prediction for ferromagnets has been con"rmed by smallangle scattering in iron [26]. This, as well as some other results were obtained using original method for the investigation of small-angle chiral scattering [23,24]. According to Eq. (10) for given Q integrated DC is zero. However, one can avoid this disappointing result changing weights of positive and negative contributions to the integral. For the small-angle scattering it may be done if the magnetic "eld is inclined to the direction of the incident beam. Indeed, if the "eld lies in the scattering plane at angle to the beam direction and P is along the "eld, the kin ematic factor in Eq. (9) before S is equal to & (2EP sin 2)/[#(2E)], where is the scattering angle, E is the neutron energy and
Fig. 1. Temperature dependence of the integrated DC scattering in iron [26]. (1) "553; (2) "683; and are de"ned in the text.
;E. As a result instead of Eq. (10) we get 4E¹P sin 2 ()"[rf(Q)]
Im S (k ,, H) & G d . #(2E)
(12)
Obviously, this contribution may be easily extracted experimentally due to its dependence on P and . Corresponding results for k < are shown in G Fig. 1 [25]. They are "tted by \JW, where
y"0.67$0.07 in an agreement with the PKW algebra predictions. To the best of my knowledge this is the only experimental con"rmation of this algebra. 3.2. Crossover to dipolar spin dynamics in the chiral channel It is well known that the exchange approximation is not applicable for critical dynamics of ferromagnets near ¹ in the region determined by the conditions [37,35] 4<1; Q(q "[4(g )a\] , (13) where is the static magnetic susceptibility and q is the dipolar wave vector. In this region the magnetic dipolar interaction plays a crucial role due to demagnetisation of the critical #uctuations.
S.V. Maleyev / Physica B 297 (2001) 67}74
71
correlated spin system except the Ising magnets. Corresponding experimental studies would be very fruitful in any soft magnetic systems. As an example we may point out KagomeH magnets. The study of the DC in the spin-Peierls compound CuGeO would be very desirable too. In this case ¹;, where is the spin gap and one can study quantum three-spin correlations in the copper chains.
Fig. 2. Crossover to dipolar critical dynamics in the chiral channel; , q and q are de"ned in the text. Dashed line shows the prediction in the exchange approximation.
As a result there are two characteristic energies of the critical #uctuations: the exchange one (Q)"¹ (Qa)X , where z "(5! )/2K2.5 and the dipolar one "¹ (q a)X \X (Qa)X with new dipolar critical exponent z . It was an unsuccessful attempt to observe the crossover to dipolar dynamics by using neutron scattering in iron [38]. This result was explained in Ref. [39]. In Fig. 2 results are shown for the chiral scattering in iron near ¹ , where P " ()/() and q "a\[2E/(¹ k a)] [28]. Qualitative exG G planation of these results is as follows [22]. For k 'q the scattering is inelastic and P &(k )\. G G G There is a crossover to quasielastic scattering at k &q and for (k (q one has G G G G P &(k )/ (k ), (14) G G where (k ) is a characteristic energy of the critical G #uctuations. Hence, in the exchange approximation P & (dashed line in Fig. 2). However, there is the second maximum approximately at q . This max imum may be explained if z (. According to Refs. [35,37] one has z "z !1/ K1. Hence, we have qualitative experimental con"rmation of this theoretical prediction. It should be pointed out that the value of z was calculated using the PKW algebra. According to [37] the crossover to dynamics with z K1 may depend on the spin value S and for large S it takes place for k ;q . In this respect G it would be very desired to do chiral scattering experiments in ferromagnets EuO and EuS with S". Concluding this section we have to note that the DC scattering should take place in any strongly
4. Antiferromagnets with k"0 and the inelastic nuclear-magnetic interference Many complex magnetic structures were deciphered with the help of the elastic nuclear-magnetic interference (see Refs. [16,17] and references therein). The INMI "rst time has been studied in Refs. [18,19] and reveals unexpected features of the spin-lattice interactions. Observed INMI in spinPeierls compound CuGeO was explained quali tatively by the Dzialoshinskii}Moriya interaction between adjacent spin chains modulated by the lattice vibrations [3]. However, quantitative evaluation of this interference in this quasi-one-dimensional compound is hardly possible. Meanwhile in ordered magnets corresponding calculations may be performed using the spin-wave theory. For ferromagnets it was done in Ref. [20]. Here, we present the results for the quasi-2D antiferromagnet BaNi (PO ) [40]. In this compound spin-wave spectrum was determined in Ref. [41]. The "rst observation of the INMI was reported in Ref. [19]. The compound BaNi (PO ) belongs to a wide class of antiferromagnets with k "0, where there $ are even number of magnetic ions in the centrosymmetric chemical unit cell. These ions arrange antiferromagnetically below ¹ and the whole , magnetic structure remains unchanged after the time re#ection R and the inversion I. As a result these antiferromagnets are magnetoelectrics and a single-domain state may be achieved by cooling in parallel magnetic and electric "elds [16,17]. This state is characterised by an axial vector of the staggered magnetisation S !S , where S and S are average spins of the antiparallel magnetic sublattices. As a result there is elastic nuclear magnetic interference described by the last term in Eq. (3). Corresponding experimental studies allow
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S.V. Maleyev / Physica B 297 (2001) 67}74
to determine magnetic structure of several antiferromagnets with k "0 [16,17]. We discuss now $ general properties of the inelastic PN scattering which follow from the above-mentioned IR invariance. Namely, instead of Eq. (7) we have now (15) A Q BQ "$B Q AQ , S \ S \ where QP!Q due to the inversion. As a result the antisymmetric part of the magnetic susceptibility is zero and we have no DC scattering. Meanwhile the susceptibility attains nondiagonal part. If the z-axis is along the sublattice magnetisation there is VW(Q,)O0, and using Eq. (6) we get Im VW(Q,)"!Im VW(!Q,!). The spin-wave theory for BaNi (PO ) predicts that Im VW(Q,)" !Im VW(!Q,), i.e. Im VW is an even function of [40]. As a result if the spin-wave energy ;¹, / the Q-odd contribution to the -integrated cross section disappears (cf. Eq. (10)). There are two contributions to the INMI, which are determined by (Q,)"N Q MQ $ S ! \ M Q NQ and using Eq. (15) we get (Q,)"0. \ S > Hence, according to Eqs. (2) and (3) the cross section does not depend on P and if P "0 scat tered neutrons remain unpolarised. At the same time using Eq. (6) we obtain (Q,)" \ (!Q,!). \ Let us consider now the BaNi (PO ) . This compound was studied in Ref. [41]. It consists of antiferromagnetic layers with two Ni> ions in the 2D chemical unit cells and S"1. We neglect below very weak interlayer coupling. There are three superexchange intraplane interactions shown in Fig. 3 and according to Ref. [41] we have J "2.2 K, J "0.3 and J "8.8 K. Besides there is single-ion easy plane anisotropy D"7.3 K. Corresponding Fourier transforms of these interactions have the form IQ " [J exp(iQ )#J exp(iQr )], H H H JQ "2J cos(Qr), where "(a #a )/3, "(a !2a )/3; " (2a !a )/3, r "!2 # # "0 and H H r"a !a . The INMI appears due to the spin-lattice interaction [3]. The strength of the exchange interaction
Fig. 3. Ni> plane in BaNi (PO ) . Arrows are Ni spins. Squares, full and open circles are spins connected to the crossed one by the exchange interactions J , J and J , respectively.
depends on the positions of surrounding ions. For simplicity, we assume that J depends on the disL tance R between corresponding Ni> ions only. JYJ As a result we get J (R )"J#(u !u ) L JYJ L JY J J (R ), where J is the exchange interaction L JYJ L between ions in the equilibrium positions and u is J the deviation from this position due to lattice vibrations. As a result the spin-lattice interaction is described by 1 (16) < " (u !u )J (R ). JY J L JYJ 1* 2 L J JY Using this interaction as perturbation one can evaluate the INMI. Omitting corresponding rather tedious calculations we may express this interference in the following way for Q"Q!;a\ [42]:
k [P(Q,)] " D (1!e\S 2)\ ' k G i[ ;P ]G(Q,), (17) where "!2irN(Q)z( sin Q is the expression , for in the case of elastic scattering, N(Q) is the nuclear structure factor and G(Q,)"Q D (q,)(tq) (q,), (18) ? ?@ @ where D (q,) is the phonon Green's function, tq is ?@ the "rst term of the expansion of (J)Q at small
S.V. Maleyev / Physica B 297 (2001) 67}74
q;a\ and (q,) is proportional to Im (Q,) XX at Q"#q, where is the longitudinal part of XX the spin susceptibility. Eq. (18) describes transformation of the phonon to the longitudinal spin #uctuations due to spinlattice interaction. Similar origin of the INMI appears in ferromagnets [20]. For BNiPO we have s;c, where s and c are the spin-wave and average sound velocity, respectively. As a result, for the phonon Green's function we have D & ?@ !(Mcq)\, where M is the mass of the unit cell. Hence, the INMI increases as qP0. However, it cannot be larger than the sum of nuclear and magnetic cross sections but an analysis of corresponding conditions is out of scope of this paper. The factor in Eq. (18) is a sum of terms containing even numbers of the spin-waves in intermediate states. Evaluated in one-loop (two spin wave) approximation it reveals the infra-red divergence as the staggered longitudinal susceptibility in conventional antiferromagnets with k O0 [42]. In 2D $ case, taking into account the gapless spin-wave mode only at sq; we have
S(I #D)v cth , (q,)"! 4¹ 16S
(19)
where v is the 2D unit cell volume. This expression is proportional to \ if <¹ and (¹/)sgn for ;¹. Similar expressions take place if sq< [40,42]. Obviously, such strong singularity is an unphysical and should be screened by multi-spinwave terms. Corresponding theoretical analysis is a very nontrivial problem. Its solution may be stimulated by experimental studies of the INMI as well as the longitudinal spin susceptibility.
5. Summary General expressions for the cross section and the "nal polarisation of the inelastically scattered neutrons are analysed. It is shown that dynamical chiral scattering and inelastic nuclear-magnetic interference may appear if the sample as a whole possesses any axial-vector interaction. This general statement is illustrated by examples of the chiral scattering in magnetic "eld and the INMI in anti-
73
ferromagnets with k "0. The case of the $ Dzialoshinskii}Moriya interaction in paramagnetic state is mentioned too (see Refs. [4}6]). The chiral critical exponent was measured "rst time using the chiral scattering in the triangular lattice antiferromagnets CeMnBr and CeNiCl [33]. It was suggested that DC scattering in any soft magnetic system would provide important information concerning the three-spin dynamical #uctuations. The INMI in antiferromagnets with k "0 is $ considered. The example of the quasi-2D compound BaNi (PO ) is analysed theoretically. It is shown that in this case the INMI reveals the infrared divergence which experimental study would provide important information concerning interaction between low-energy spin waves.
Acknowledgements The author thanks A.I. Okorokov, D.N. Aristov, A.V. Syromyatnikov, V.P. Plakhty, R.V Pisarev, and L.P. Regnault for very useful discussions. This work was supported by the Russian State Program for Statistical Physics (Grant No. VIII-2), RFBR (Grants No. 00-02-16873 and 00-15-96814) and Russian program `Neutron studies of condensed mattera.
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S.V. Maleyev / Physica B 297 (2001) 67}74 P.J. Brown et al., Physica B 267}268 (1999) 215. P.J. Brown, Physica B, these Proceedings. L.P. Regnault et al., Physica B 267}268 (1999) 236. L.P. Regnault et al., Physica B, these proceedings. A.V. Syromyatnikov, S.V. Maleyev, Physica B, these Proceedings. K. Siratori et al., Phys. Soc. Japan 48 (1980) 1111. A.V. Lazuta et al., Sov. Phys. JETP 48 (1981) 782. A.V. Lazuta et al., Sov. Phys. JETP 54 (1978) 386. A.I. Okorokov et al., Sov. Phys. JETP 51 (1981) 775. A.I. Okorokov et al., Sov. Phys. JETP Lett. 37 (1983) 319. A.I. Okorokov et al., Sov. Phys. JETP Lett. 37 (1983) 513. A.I. Okorokov, Physica B 276}278 (2000) 2004. B.P. Toperverg et al., Physica B 183 (1993) 326. V.P. Plakhty et al., Europhys. Lett. 48 (1999) 215. V.P. Plakhty et al., Physica B 267}268 (1999) 259. V.P. Plakhty et al., Physica B, these Proceedings.
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Polarised neutrons and axial-vector interactions in magnetic materials S.V. Maleyev* Theory Department, Petersburg Nuclear Physics Institute, 188350 St. Petersburg, Gatchina, Russia
Abstract Inelastic polarised neutron scattering is analysed when the magnetic interaction includes a part characterised by an axial vector. In this case the dynamical chiral (DC) scattering and inelastic nuclear-magnetic interference (INMI) may appear. As examples we consider magnetic "eld and antiferromagnets with k "0. Possibilities of other axial-vector $ interactions such as the Dzialoshinskii}Moriya interaction and elastic torsion are mentioned too. The DC scattering in magnetic "eld is determined by the three-spin correlations. Their experimental investigation in iron above ¹ allows to con"rm predictions of the Polyakov}Kadano!}Wilson operator algebra and demonstrate crossover to the dipolar critical dynamics. The INMI in the antiferromagnets with k "0 is considered theoretically and appearance of the $ infra-red divergence in the case of quasi-2D compound BaNi (PO ) was demonstrated. 2001 Elsevier Science B.V. All rights reserved. PACS: 75.25.#z; 75.30.GW; 75.40.Gb; 75.50.Ee Keywords: Polarised neutrons; Chiral scattering; Nuclear-magnetic interference
1. Introduction Increasing a role of polarised neutrons (PN) in investigation of magnetic materials demands an additional theoretical analysis. It was done partly during PNCMI'98 [1}3]. In this paper we continue these studies. Our main idea is the following: as the cross section and initial neutron polarisation P are scalar and an axial vector, respectively, the cross section may depend on P if the system is charac terised by an axial vector. The same holds for the polarisation of the scattered neutrons too: any
* Fax: #7-812-713-1963. E-mail address: [email protected] (S.V. Maleyev).
additional change of this axial vector comparing to its rotation due to conventional magnetic scattering may appear in the presence of an axial-vector interaction only. As examples of the axial-vector interactions one can point out the interaction with external magnetic "eld, the Dzialoshinskii}Moriya interaction [3}6], interaction of the spin chirality with elastic torsion [7], spin interaction with the atomic helix structures, etc. Antiferromagnets with k "0 pro $ vide an additional example of such systems where the staggered magnetisation leads to the rotation of the neutron polarisation due to nuclear-magnetic interference. In this paper we present general expressions for polarised neutron scattering in magnetic materials
0921-4526/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 0 ) 0 0 8 3 8 - 3
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(Section 2). Then in Section 3 we consider chiral scattering mediated by external magnetic "eld. Section 4 is devoted to antiferromagnets with k "0. $ In Section 5 we summarise the main results of the paper.
2. General expressions for the PN scattering For the following we should present general expressions for the PN scattering. Neglecting the interaction between nuclei and neutron spins for the scattering amplitude we have a well-known expression
1 FQ "! b e\ QRL !r f (Q)e QRL (S, ) L K K N L K "N #M, , / /
(1)
where R and b are the position and scattering L L length of the nth nucleus, r"0.54 fm, f (Q), S and K K R are the magnetic form-factor, spin and position K of the mth magnetic ion in the lattice, respectively. Here and below `Na labels the perpendicular to the scattering vector Q part of the corresponding vectors. We also omit the orbital part of the amplitude which may be easily taken into account and does not change our results appreciably. For the elastic scattering, the cross section and the "nal polarisation are given by [8}10] "N Q NQ #M, Q M,Q #iM, Q ;M,Q \ \ \ #N Q M,Q #M, Q NQ P , \ \
(2)
(P) "N Q NQ P \ #M, Q (M,Q P )#(P M, Q )M,Q \ \ !P M, Q M,Q !iM, Q ;M,Q \ \ #N Q M,Q #M, Q NQ \ \ #i[N Q M,Q !M, Q NQ ;P ], \ \
(3)
where 2 means both the thermal and disorder average. The expressions for the inelastic scattering follow from Eqs. (2) and (3) if one replaces AB by
(k /k )S (), where S () is a Van Hove function: D G
1 S ()" dt exp(it)A(t)B(0) 2 \
"A, B 1!exp ! S ¹
\
(4)
and A, B is the absorptive part of the generalisS ed susceptibility A, B determined by S A, B "i S
dt exp(it)[A(t), B(0)].
(5)
\
For A"B> we have A, B "ImA, B S [1}3,11]. From conventional representation of A, B as a sum over intermediate states we have S [3] A, B
S> B
"B, A
\S\ B
,
A, B "!B, A . S \S
(6)
The symmetry under time re#ection tP!t demands [12]: A, B
S H
"$BA
S \H
,
(7)
where H is a magnetic "eld or spontaneous magnetisation of the sample. Signs `plusa and `minusa correspond to equal and di!erent time oddness of the operators A and B, respectively. The "rst and second terms in Eqs. (2) and (3) describe nuclear and conventional magnetic scattering, respectively. The third terms are connected to the chiral scattering and the last terms describe the nuclear magnetic interference. One can show that the conventional and chiral inelastic magnetic scattering are proportional to Im 1(Q,) and Im[i (Q,)], respectively, where ?@ ?@ 1 "( $ ) are symmetric and antisym?@ ?@ @? metric parts of the magnetic susceptibility [13,14]. ?@ The nuclear-magnetic interference is well-known in the case of elastic scattering in ferromagnets. Recently, this interference appears very useful for determination of complex antiferromagnetic structures (see Refs. [15}17] and references therein). Due to recent development of the PN scattering tech-
S.V. Maleyev / Physica B 297 (2001) 67}74
nique and particularly CRYOPAD II [15] a possibility appears to study inelastic nuclear-magnetic interference (INMI) which provides direct information about spin-lattice interaction [18,19]. General theory of the INMI and its analysis for the spin Peierls compound are presented in Ref. [3]. Theoretical calculation of this interference for ferromagnets is given in Ref. [20].
3. Dynamical chirality Elastic chiral scattering takes place in the case of helix magnetic structure. It was studied experimentally in many cases (see for example Refs. [4,21]). Inelastic chiral scattering or dynamical chirality (DC) was discussed theoretically in Refs. [13,22,23]. These studies initiated experimental investigation of the DC in ferromagnets [24}28] and triangular lattice antiferromagnets [14,29}31]. As was pointed above the DC is determined by antisymmetric part of the magnetic susceptibility (Q,). According to Eq. (7) we have ?@ (Q,,H )" (!Q,,!H ), and the antisym?@ @? metric part of may appear in magnetic "eld or in the case of an axial-vector interaction, if the Fourier transform of corresponding axial vector is an odd function of Q. The Dzialoshinskii}Moriya interaction (DM) is a corresponding example [4}6]. It was shown that the DC scattering is determined by Im i (Q,) [13,14]. The -parity of this ?@ function is opposite to the t-parity of corresponding axial-vector. Particularly, it is -even in the case of magnetic "eld and -odd for the DM interaction [5,6]. Any antisymmetric tensor is related to an axial vector. In the case of uniaxial crystal one has [13] (Q,)"!i ?@ ?@A [hK S (q,)#c( (c( ) hK )S (Q,)], A & A 67
(8)
where hK and c( are unit vectors along the axial vector and preferred crystal axis, respectively. For isotropic Heisenberg interaction one has S ,0 67 and S ,0 for the X> model. &
69
The chiral contribution to the cross section has the form
k 2 (Q,)" [rf (Q)] D (1!e\S 2)\ k G [(P QK )(QK hK ) Im S (Q,) & #(P Q)(QK c( )(c( hK )Im S (Q,)]. 67
(9)
Below we limit ourselves with the case of magnetic "eld. The DM interaction is considered in Refs. [4}6]. As stated above Im S are even functions & 67 of due to t-oddness of H. As a result () changes sign with . It was con"rmed experimentally for critical scattering in iron [24] and triangular-lattice antiferromagnets CsMnBr and CsNiCl [29}31]. If for given Q characteristic en ergy ;¹, the static chiral scattering is zero:
¹ (Q)"
\
d (Q,)"0.
(10)
The -integrated chiral cross section does not vanish in quantum limit only, when one cannot replace the Bose-factor in Eq. (9) by (¹/). In weak magnetic "eld (Q,) is linear in H and connected to the three-spin correlation function. It is a new object for experimental studies and may provide an additional information which is not accessible to conventional neutron scattering dealing with the two-spin correlations. This general statement is illustrated below by DC critical scattering experiments in iron [24}28]. Recently, DC critical scattering in triangular lattice antiferromagnets CsMnBr and CsNiCl al lowed to determine the chiral critical exponent which could not be measured by other methods [29,31]. 3.1. Polyakov}Kadanow}Wilson (PKW) algebra and critical factorisation of the tree-spin correlations in iron [25] Conventional neutron magnetic scattering is determined by two-spin correlation function and depends on one momentum Q. However, in the strong correlated spin systems there are n-spin nontrivial
70
S.V. Maleyev / Physica B 297 (2001) 67}74
correlations where n'2, which depend on n!1 momenta q . The PKW algebra deals with n-spins G correlations near the second order phase transition. It predicts that if one momentum, say q , is much larger than others q and the inverse correlation G$ length " \, then the dependence on q appears as a factor q\V, where x"5! !1/ , where and
are the Fisher's and the correlation length exponents, respectively [32}35]. This phenomenon may be called as critical factorisation. In the case of chiral scattering along with the momentum transfer Q we have the momentum of the uniform magnetic "eld which is zero. The "eld H is conjugated to the total magnetisation which is a relevant variable for ferromagnets. As a result on the base of static and dynamic scaling theories for cubic ferromagnets where S "0 we have [23,36] 67
gH . Im S (Q,)" & ¹(Qa)V( a)W ¹ (Qa)X
(11)
Here H is the internal magnetic "eld, a is of the order of interatomic distance, y"(1/ )!(1# )/2,
"a\ T, "(¹!¹ )/¹ and z is the dynamical critical exponent. From this expression we see that if Q< , the chiral scattering is proportional to \>J>E K \ , where in the right-hand side we put "0 and vK. In triangular lattice antifer romagnets the magnetisation is not a relevant variable and in Eq. (11) we have x"3!( #1)/ , where is the chiral crossover exponent [31,32,36]. This nontrivial theoretical prediction for ferromagnets has been con"rmed by smallangle scattering in iron [26]. This, as well as some other results were obtained using original method for the investigation of small-angle chiral scattering [23,24]. According to Eq. (10) for given Q integrated DC is zero. However, one can avoid this disappointing result changing weights of positive and negative contributions to the integral. For the small-angle scattering it may be done if the magnetic "eld is inclined to the direction of the incident beam. Indeed, if the "eld lies in the scattering plane at angle to the beam direction and P is along the "eld, the kin ematic factor in Eq. (9) before S is equal to & (2EP sin 2)/[#(2E)], where is the scattering angle, E is the neutron energy and
Fig. 1. Temperature dependence of the integrated DC scattering in iron [26]. (1) "553; (2) "683; and are de"ned in the text.
;E. As a result instead of Eq. (10) we get 4E¹P sin 2 ()"[rf(Q)]
Im S (k ,, H) & G d . #(2E)
(12)
Obviously, this contribution may be easily extracted experimentally due to its dependence on P and . Corresponding results for k < are shown in G Fig. 1 [25]. They are "tted by \JW, where
y"0.67$0.07 in an agreement with the PKW algebra predictions. To the best of my knowledge this is the only experimental con"rmation of this algebra. 3.2. Crossover to dipolar spin dynamics in the chiral channel It is well known that the exchange approximation is not applicable for critical dynamics of ferromagnets near ¹ in the region determined by the conditions [37,35] 4<1; Q(q "[4(g )a\] , (13) where is the static magnetic susceptibility and q is the dipolar wave vector. In this region the magnetic dipolar interaction plays a crucial role due to demagnetisation of the critical #uctuations.
S.V. Maleyev / Physica B 297 (2001) 67}74
71
correlated spin system except the Ising magnets. Corresponding experimental studies would be very fruitful in any soft magnetic systems. As an example we may point out KagomeH magnets. The study of the DC in the spin-Peierls compound CuGeO would be very desirable too. In this case ¹;, where is the spin gap and one can study quantum three-spin correlations in the copper chains.
Fig. 2. Crossover to dipolar critical dynamics in the chiral channel; , q and q are de"ned in the text. Dashed line shows the prediction in the exchange approximation.
As a result there are two characteristic energies of the critical #uctuations: the exchange one (Q)"¹ (Qa)X , where z "(5! )/2K2.5 and the dipolar one "¹ (q a)X \X (Qa)X with new dipolar critical exponent z . It was an unsuccessful attempt to observe the crossover to dipolar dynamics by using neutron scattering in iron [38]. This result was explained in Ref. [39]. In Fig. 2 results are shown for the chiral scattering in iron near ¹ , where P " ()/() and q "a\[2E/(¹ k a)] [28]. Qualitative exG G planation of these results is as follows [22]. For k 'q the scattering is inelastic and P &(k )\. G G G There is a crossover to quasielastic scattering at k &q and for (k (q one has G G G G P &(k )/ (k ), (14) G G where (k ) is a characteristic energy of the critical G #uctuations. Hence, in the exchange approximation P & (dashed line in Fig. 2). However, there is the second maximum approximately at q . This max imum may be explained if z (. According to Refs. [35,37] one has z "z !1/ K1. Hence, we have qualitative experimental con"rmation of this theoretical prediction. It should be pointed out that the value of z was calculated using the PKW algebra. According to [37] the crossover to dynamics with z K1 may depend on the spin value S and for large S it takes place for k ;q . In this respect G it would be very desired to do chiral scattering experiments in ferromagnets EuO and EuS with S". Concluding this section we have to note that the DC scattering should take place in any strongly
4. Antiferromagnets with k"0 and the inelastic nuclear-magnetic interference Many complex magnetic structures were deciphered with the help of the elastic nuclear-magnetic interference (see Refs. [16,17] and references therein). The INMI "rst time has been studied in Refs. [18,19] and reveals unexpected features of the spin-lattice interactions. Observed INMI in spinPeierls compound CuGeO was explained quali tatively by the Dzialoshinskii}Moriya interaction between adjacent spin chains modulated by the lattice vibrations [3]. However, quantitative evaluation of this interference in this quasi-one-dimensional compound is hardly possible. Meanwhile in ordered magnets corresponding calculations may be performed using the spin-wave theory. For ferromagnets it was done in Ref. [20]. Here, we present the results for the quasi-2D antiferromagnet BaNi (PO ) [40]. In this compound spin-wave spectrum was determined in Ref. [41]. The "rst observation of the INMI was reported in Ref. [19]. The compound BaNi (PO ) belongs to a wide class of antiferromagnets with k "0, where there $ are even number of magnetic ions in the centrosymmetric chemical unit cell. These ions arrange antiferromagnetically below ¹ and the whole , magnetic structure remains unchanged after the time re#ection R and the inversion I. As a result these antiferromagnets are magnetoelectrics and a single-domain state may be achieved by cooling in parallel magnetic and electric "elds [16,17]. This state is characterised by an axial vector of the staggered magnetisation S !S , where S and S are average spins of the antiparallel magnetic sublattices. As a result there is elastic nuclear magnetic interference described by the last term in Eq. (3). Corresponding experimental studies allow
72
S.V. Maleyev / Physica B 297 (2001) 67}74
to determine magnetic structure of several antiferromagnets with k "0 [16,17]. We discuss now $ general properties of the inelastic PN scattering which follow from the above-mentioned IR invariance. Namely, instead of Eq. (7) we have now (15) A Q BQ "$B Q AQ , S \ S \ where QP!Q due to the inversion. As a result the antisymmetric part of the magnetic susceptibility is zero and we have no DC scattering. Meanwhile the susceptibility attains nondiagonal part. If the z-axis is along the sublattice magnetisation there is VW(Q,)O0, and using Eq. (6) we get Im VW(Q,)"!Im VW(!Q,!). The spin-wave theory for BaNi (PO ) predicts that Im VW(Q,)" !Im VW(!Q,), i.e. Im VW is an even function of [40]. As a result if the spin-wave energy ;¹, / the Q-odd contribution to the -integrated cross section disappears (cf. Eq. (10)). There are two contributions to the INMI, which are determined by (Q,)"N Q MQ $ S ! \ M Q NQ and using Eq. (15) we get (Q,)"0. \ S > Hence, according to Eqs. (2) and (3) the cross section does not depend on P and if P "0 scat tered neutrons remain unpolarised. At the same time using Eq. (6) we obtain (Q,)" \ (!Q,!). \ Let us consider now the BaNi (PO ) . This compound was studied in Ref. [41]. It consists of antiferromagnetic layers with two Ni> ions in the 2D chemical unit cells and S"1. We neglect below very weak interlayer coupling. There are three superexchange intraplane interactions shown in Fig. 3 and according to Ref. [41] we have J "2.2 K, J "0.3 and J "8.8 K. Besides there is single-ion easy plane anisotropy D"7.3 K. Corresponding Fourier transforms of these interactions have the form IQ " [J exp(iQ )#J exp(iQr )], H H H JQ "2J cos(Qr), where "(a #a )/3, "(a !2a )/3; " (2a !a )/3, r "!2 # # "0 and H H r"a !a . The INMI appears due to the spin-lattice interaction [3]. The strength of the exchange interaction
Fig. 3. Ni> plane in BaNi (PO ) . Arrows are Ni spins. Squares, full and open circles are spins connected to the crossed one by the exchange interactions J , J and J , respectively.
depends on the positions of surrounding ions. For simplicity, we assume that J depends on the disL tance R between corresponding Ni> ions only. JYJ As a result we get J (R )"J#(u !u ) L JYJ L JY J J (R ), where J is the exchange interaction L JYJ L between ions in the equilibrium positions and u is J the deviation from this position due to lattice vibrations. As a result the spin-lattice interaction is described by 1 (16) < " (u !u )J (R ). JY J L JYJ 1* 2 L J JY Using this interaction as perturbation one can evaluate the INMI. Omitting corresponding rather tedious calculations we may express this interference in the following way for Q"Q!;a\ [42]:
k [P(Q,)] " D (1!e\S 2)\ ' k G i[ ;P ]G(Q,), (17) where "!2irN(Q)z( sin Q is the expression , for in the case of elastic scattering, N(Q) is the nuclear structure factor and G(Q,)"Q D (q,)(tq) (q,), (18) ? ?@ @ where D (q,) is the phonon Green's function, tq is ?@ the "rst term of the expansion of (J)Q at small
S.V. Maleyev / Physica B 297 (2001) 67}74
q;a\ and (q,) is proportional to Im (Q,) XX at Q"#q, where is the longitudinal part of XX the spin susceptibility. Eq. (18) describes transformation of the phonon to the longitudinal spin #uctuations due to spinlattice interaction. Similar origin of the INMI appears in ferromagnets [20]. For BNiPO we have s;c, where s and c are the spin-wave and average sound velocity, respectively. As a result, for the phonon Green's function we have D & ?@ !(Mcq)\, where M is the mass of the unit cell. Hence, the INMI increases as qP0. However, it cannot be larger than the sum of nuclear and magnetic cross sections but an analysis of corresponding conditions is out of scope of this paper. The factor in Eq. (18) is a sum of terms containing even numbers of the spin-waves in intermediate states. Evaluated in one-loop (two spin wave) approximation it reveals the infra-red divergence as the staggered longitudinal susceptibility in conventional antiferromagnets with k O0 [42]. In 2D $ case, taking into account the gapless spin-wave mode only at sq; we have
S(I #D)v cth , (q,)"! 4¹ 16S
(19)
where v is the 2D unit cell volume. This expression is proportional to \ if <¹ and (¹/)sgn for ;¹. Similar expressions take place if sq< [40,42]. Obviously, such strong singularity is an unphysical and should be screened by multi-spinwave terms. Corresponding theoretical analysis is a very nontrivial problem. Its solution may be stimulated by experimental studies of the INMI as well as the longitudinal spin susceptibility.
5. Summary General expressions for the cross section and the "nal polarisation of the inelastically scattered neutrons are analysed. It is shown that dynamical chiral scattering and inelastic nuclear-magnetic interference may appear if the sample as a whole possesses any axial-vector interaction. This general statement is illustrated by examples of the chiral scattering in magnetic "eld and the INMI in anti-
73
ferromagnets with k "0. The case of the $ Dzialoshinskii}Moriya interaction in paramagnetic state is mentioned too (see Refs. [4}6]). The chiral critical exponent was measured "rst time using the chiral scattering in the triangular lattice antiferromagnets CeMnBr and CeNiCl [33]. It was suggested that DC scattering in any soft magnetic system would provide important information concerning the three-spin dynamical #uctuations. The INMI in antiferromagnets with k "0 is $ considered. The example of the quasi-2D compound BaNi (PO ) is analysed theoretically. It is shown that in this case the INMI reveals the infrared divergence which experimental study would provide important information concerning interaction between low-energy spin waves.
Acknowledgements The author thanks A.I. Okorokov, D.N. Aristov, A.V. Syromyatnikov, V.P. Plakhty, R.V Pisarev, and L.P. Regnault for very useful discussions. This work was supported by the Russian State Program for Statistical Physics (Grant No. VIII-2), RFBR (Grants No. 00-02-16873 and 00-15-96814) and Russian program `Neutron studies of condensed mattera.
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