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Neutron scattering by magnetic ball solitons
ARTICLE IN PRESS
Journal of Magnetism and Magnetic Materials 266 (2003) 258–267
Neutron scattering by magnetic ball solitons V.V. Nietz* Frank Laboratory of Neutron Physics, Joint Institute for Nuclear Research, Dubna, Moscow region 141980, Russia
Abstract The character of neutrons scattering by ball solitons arising at the spin–flop transition in antiferromagnet is considered. Results of analysis of form-factor and dynamical structural factor for neutron scattering are presented. Possible arrangements for experiments on the observation of elastic and inelastic scattering by such solitons are discussed. r 2003 Published by Elsevier B.V. PACS: 75.25; 75.50.E; 29.30.H Keywords: Antiferromagnet; Spin–flop transition; Magnetic field; Ball soliton; Form-factor; Dynamic structural factor; Neutron scattering
1. Introduction Sole neutron research of magnetic solitons in crystals were carried out in a series of experiments with quasi-one-dimensional antiferromagnet CsNiF3 in 1978–1986 (see Refs. [1–11]). In the investigated compound the ferromagnetic exchange interaction between the neighbor magnetic moments of Ni2+ ions arranged in chains along the c-axis of the hexagonal structure exceeds the antiferromagnetic interaction between the ions in the neighboring chains by over an order of magnitude. As a result, analysis of the behaviour of this compound in an external magnetic field is reduced to a one-dimensional problem. Naturally, in the field perpendicular to the c-axis the magnetic moments are parallel to the field. However, there may exist so-called 2p-solitons in which the *Tel.: +7-96-21-65-552; fax: +7-96-21-65-882. E-mail address: [email protected] (V.V. Nietz). 0304-8853/$ - see front matter r 2003 Published by Elsevier B.V. doi:10.1016/S0304-8853(03)00244-0
magnetic moments turn by 360 in the plane perpendicular to the c-axis on some limited part of ( The number of such solitons, the chain (D100 A). whose formation is due to thermal fluctuations, obeys the Gibbs distribution. The publications [12–22] are devoted to theoretical analysis of 2psolitons and carry out a comparison of the experimental data with theory (see also an analysis of the data in review [23]). Another type of solitons with their energy comparable with the energy of thermal fluctuations in the temperature region corresponding to the existence of magnetic ordering are the solitons of a (quasi) spherical form that arise in the vicinity of metastability limits at magnetic phase transitions of the first order (see Refs. [24–28]). Let us call them the ball solitons (BS). In the present article, a scattering of neutrons by the BSs that arise at the spin–flop phase transitions in antiferromagnet with one-axis symmetry is analysed.
ARTICLE IN PRESS V.V. Nietz / Journal of Magnetism and Magnetic Materials 266 (2003) 258–267
QðrÞ is solution of equation
2. Magnetic solitons at spin–flop transition For the macroscopic energy of antiferromagnet Z ( B 2 C K1 W ¼ 2M0 jmj þ ðlmÞ2 þ jl> j2 2 2 2 " # K2 a ql 2 ql 2 þ jl> j4 mz Hz þ þ 2 qX qY 4 2 ) az ql þ dX dY dZ ð1Þ 2 qZ in external field H ¼ Hz ; at rather small values of jl> j2 ; the equation _ qMi * i þ G _ Mi qMi ¼ Mi H 2mB qt 2mB qt
ð2Þ
(i ¼ 1; 2; Mi are the magnetisation vectors of two * i is effective field acting on the sublattices, H corresponding sublattice) looks the form (see Ref. [26]) q2 l> ql> Dl> 2 2ih qt qt ¼ ð1 h2 Þ l> gjl> j2 l> þ g
ql> ; qt
ð3Þ
where l ¼ ð2M0 Þ1 ðM1 M2 Þ; m ¼ ð2M0 Þ1 ðM1 þ M2 Þ; l> ¼ ðlx þ ily Þ (x- and y-axes are in the basal plane of a crystal). In Eq. (2) h ¼ Hz B0:5 K10:5 ; g ¼ 0:5ð1 2K2 =K1 Þ; g ¼ GB0:5 K 0:5 ; the differentiation is with respect to the dimensionless time t ¼ 2mB ðK1 BÞ0:5 _1 t and dimensionless coordinates x ¼ K10:5 a0:5 X ; y ¼ K10:5 a0:5 Y ; z¼ 0:5 0:5 K1 az Z: The solution of Eq. (3) at g ¼ 0 corresponding to a BS is
d2 Q 2 dQ ¼ Qð1 Q2 Þ þ dr2 r dr
ð5Þ
(for the sake of simplicity we neglect d B=ðC þ BÞ ¼ wJ =w> and take a ¼ az ). The configuration of QðrÞ is shown in Fig. 1. The BS is defined by the precession frequency o and velocity v of translational moving, thus pffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðo þ hÞ 1 v2 ¼ 1 gQ2 ð6Þ 1 qs0 ; where qs0 jl> ðrs ¼ 0Þj is amplitude of BS, in the given case qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qs0 ¼ g1 ½1 ðo þ hÞ2 ð1 v2 Þ Q1 Q1 ¼ 4:33739: The energy of the soliton is (see Ref. [26]) pffiffiffiffiffi 8pM0 a az I1 Q1 Es ¼ pffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffi g gK1 1 v2 qs0 pffiffiffiffiffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g 1 h 1 v2 1 2 q2s0 Qn RN 2 2 Here, I1 ¼ 0 r Q ðrÞ dr ¼ 1:503796. The energy has the minimum pffiffiffiffiffi ffi 8pM0 a az I1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Es;min ¼ pffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 h2 ð1 v2 Þ K1 g 1 v2
4 3 Q
ð4Þ
where qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r ¼ 1 ðo þ hÞ2 ð1 v2 Þjrs j; pffiffiffiffiffiffiffiffiffiffiffiffiffi rs xs ¼ ðx vtÞ= 1 v2 ; ys ; zs
2 1 0 0
is a Cartesian coordinate fixed to the moving soliton, the xs -axis is directed along the x-axis,
ð7Þ
ð8Þ
when o ¼ 0 and the amplitude is qs0 ¼ Q1 g0:5 ½1 h2 ð1 v2 Þ 0:5 : The dependence of the energy Es on the amplitude for some fields at v ¼ 0 is illustrated in Fig. 2. Let a soliton be generated at time t=0 with a configuration described by Eq. (3) without the
ls ðrs ; tÞ ¼ g0:5 ½1 ðo þ hÞ2 ð1 v2 Þ 0:5 QðrÞ exp ½iðot ðo þ hÞ ðvrs ÞÞ ;
259
1
ρ
2
Fig. 1. The configuration of QðrÞ function.
ARTICLE IN PRESS V.V. Nietz / Journal of Magnetism and Magnetic Materials 266 (2003) 258–267
260 0.4
1
2
60
0.015
40
0.010
~
4 5
0.005
0
0.000
-20
-0.005
7
0.0
20
rs
6
0.0
ω
Es
0.020
rs
3
0.2
80
0.2
0.4
0.6
0.8
1.0
q0 Fig. 2. The dependencies of energy on the amplitude for the next h values: 1–0.98, 2–0.99, 3–0.993, 4–0.995, 5–0.997, * 6–1, pffiffiffiffiffi7–1.001. ffi ffi 1 in high field phase. Here Es ¼ pqffiffiffiffi¼ g K1 =8pM0 a az I1 Es : A change in the precession frequency of solitons is at h ¼ 0:99: The dashed line is the dependence of the effective radius on the amplitude.
additional term gðql=qtÞ: The change in configuration of the soliton in an approximation linear in t can be written [26]: 0 1 got B C ls ðrs ; tÞ ¼ ls ðrs ; t ¼ 0Þ@1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA eiot 2 2 2 1 gQ1 qs0
frequency o: Therefore, the interaction of neutrons with the BS is of inelastic character, which means that it is accompanied with exchange of energy with the internal degree of freedom. This is to some extent similar to what happens at neutron scattering on the spin wave, which is always inelastic. Of interest to us is the neighborhood of the reciprocal lattice centre that corresponds to the usual magnetic diffraction reflection. In the discussed model of spin–flop transition solitons are connected with changes of the components lx and ly of the antiferromagnetism vector. Therefore, for the cross section of magnetic scattering on BS we proceed from the expression: d2 s k2 ¼ ðr0 gn Þ2 P2 ðk2 k1 Þ Sðk; eÞ; dO de _k1 where the dynamical structural factor is
Sðk; eÞ ¼
Z 1 N ð1 þ eZ2 Þ dt eiet=_ 2p N 4 X ikRn ð0Þ ikRn0 ðtÞ e e
ð9Þ i.e. at o > 0 the solitons decrease and transform into initial (metastable) state of a crystal. The energy of BS in an approximation linear in t equals: pffiffiffiffiffi 8pM0 a az I1 Q1 Es ¼ pffiffiffiffiffiffi pffiffiffi K1 g gqs0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 1 h 1 gQ1 qs0 go t ð10Þ
3. Neutron scattering by BSs We analyse the neutron scattering by BSs that are the result of thermal fluctuations in quasiequilibrium state (i.e. at o > 0; in oo0 case the ls values increase and dissipation of energy transforms the soliton into the high field domain). As against the solitons in CsNiF3, the BSs have an internal degree of freedom, the precession
ð11Þ
n;n0 lnþ ð0Þ
ln0 ðtÞ þ ln ð0Þ lnþ0 ðtÞ
ð12Þ
(r0 ¼ 2:8 1013 cm is the classical radius of the electron; gn ¼ 1:913 mn is the magnetic moment of the neutron, eZ is the component of the scattering 7 x y vector). In Expression (12): n ðtÞ ¼ ln ðtÞ7iln ðtÞ; P lis j ðhklÞ pi ðtÞ where lnj ðtÞ ¼ 1=nM0 e M ðtÞ ðj ¼ ni i x; yÞ is the sum for the all magnetic ions in the elementary cell, Mnij ðtÞ is the magnetization component, n is the number of magnetic sublattices. Pðk2 k1 Þ is the magnetic form-factor of the ion, it is assumed that the crystal consists of magnetic ions of one kind); k ¼ k2 k1 2psðhklÞ ; where sðhklÞ is the vector of the reciprocal lattice corresponding to the investigated reciprocal lattice node. Since the size of the soliton is a few hundred Angstroms, neutron scattering on them concentrates in the region of jkj whose value does not exceed several hundredths of 2pjsðhklÞ j: Thus we can use Pð2psðhklÞ Þ instead of Pðk2 k1 Þ: Thus
ARTICLE IN PRESS V.V. Nietz / Journal of Magnetism and Magnetic Materials 266 (2003) 258–267
Z Fs2 ðk; o; vÞ ¼ drs QðrÞ 8" 9 # sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 = ffi < rffiffiffiffiffi a ðo þ hÞv zs exp i : v k pffiffiffiffiffiffiffiffiffiffiffiffiffi rs 1 : ; K1 rs 1 v2
from Eq. (4): qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi lnþ ðtÞ ¼ g1 ½1 ðo þ hÞ2 ð1 v2 Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 Q 1 ðo þ hÞ ð1 v Þ jrsn ðtÞj ei½otðoþhÞðvrn ðtÞÞ :
261
ð13Þ
ð17Þ
The following relation between the coordinates rn in the system of the crystal and the coordinates rsn in the system of the moving soliton holds
The following notations are used: Z1 ¼ RR dv do exp ½Es ðv; oÞ=kB T ; W ðo; vÞ ¼ ½1 ðo þ hÞð1p ffiffiffiffiffiffiffiffiffi vÞ2 ð1v2 Þ; R ¼ _a3 ð1 þ eZ2 Þ=4gK13 Vc2 ; U ¼ 2mB K1 B; Vc is the elementary cell volume in the coordinate system (X,Y,Z). To obtain Expression (16), in the beginning the dynamic structural factor of scattering on one soliton averaged over v and o had been calculated, and then the result for one soliton was multiplied by the average number of solitons in the sample, Ns ; which depends on relation of minimum energy of solitons to kB T value. This dependence can be obtained from thermodynamics of solitons in a fficrystal. pffiffiffiffiffiffiffiffiffiffi The energy shift by U a=K1 ðkvÞ ðo þ hÞv2 is due to Doppler displacement. In addition, neutron scattering on each precessing soliton is accompanied with a transfer of an energy quantum pffiffiffiffiffiffiffiffiffi equal to _O; where O ¼ 2mB _1 K1 B o: The energy of BS can be presented as an expansion in a power series in the velocity and frequency of BS restricting oneself to square-law members: " # 2 2 pffiffiffiffiffiffiffiffiffiffiffiffiffi o v Es ¼ K 1 h2 þ þ pffiffiffiffiffiffiffiffiffiffiffiffiffi ð18Þ 2ð1 h2 Þ1:5 2 1 h2
ðx vx tÞ ¼ cos yv cos jv xs sin jv ys pffiffiffiffiffiffiffiffiffiffiffiffiffi þ sin yv cos jv zs 1 v2 ; ðy vy tÞ ¼ cos yv sin jv xs cos jv ys pffiffiffiffiffiffiffiffiffiffiffiffiffi þ sin yv sin jv zs 1 v2 ; pffiffiffiffiffiffiffiffiffiffiffiffiffi ðz vz tÞ ¼ sin yv xs þ cos yv zs 1 v2 ;
ð14aÞ
ð14bÞ ð14cÞ
where yn is the angle between the z-axis and the vector v, jn is the angle between the x-axis and the plane formed by the z-axis and the vector v. We replace the summation over the cells of the crystal by the integration over the soliton volume, perform the integration over t and the averaging over the frequencies and velocities of the solitons. As a result, we obtain the following expression for the dynamical structural factor: Z Z RNs Sðk; eÞ ¼ do dv W ðo; vÞ eðEs ðo;vÞ=kB T Þ Z1 2 jFs1 ðk; o; vÞj d e þ Uo rffiffiffiffiffiffi a 2 U ðkvÞ ðo þ hÞv K1 2 þjFs2 ðk; o; vÞj d e Uo rffiffiffiffiffiffi a U ðkvÞ ðo þ hÞv2 ; ð15Þ K1 where the form-factors of the solitons are Z Fs1 ðk; o; vÞ ¼ drs QðrÞ 8" 9 # sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 = ffi < rffiffiffiffiffi a ðo þ hÞv zs ; exp i v k þ pffiffiffiffiffiffiffiffiffiffiffiffiffi rs 1 : ; 2 K1 rs 1v ð16Þ
pffiffiffiffiffiffi pffiffiffi (here K ¼ 8p M0 a aI1 =g K1 ). Using decomposition (18) the following expression is obtained: !2 kB T 2 2 13=4 pffiffiffiffiffiffiffiffiffiffiffiffiffi Z1 ¼ 2p ð1 h Þ K 1 h2 pffiffiffiffiffiffiffiffiffiffiffiffiffi exp ðK 1 h2 =kB TÞ: ð19Þ Let us estimate various factors that were taken into account as phase corrections of the formfactors of solitons. At ð1 hÞ ¼ 0:01: omax ¼ 0:01: In this case, in Expressions (15)–(17) we can neglect the o value in comparison with h within an accuracy of 1% ðhD1Þ: We introduce the notion v for the velocity value at which the BS formation
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262
probability decreases e times comparative to the formation probability of motionless solitons. At ð1 hÞ ¼ 0:01; T ¼ 200 K we have v ¼ 0:14 and the radius of the soliton Rs ¼ 1:22 106 cm: This means that basic scattering on solitons occurs within the limits Dk ¼ 72pR1 s ¼ 5:1 qffiffiffiffiffiffiffiffiffiffiffi 6 1 1 10 cm : The phase member aK1 ðkrs Þ becomes comparable with hðvrs Þ if v ¼ v only at pffiffiffiffiffiffiffiffiffiffiffiffiffi ko K a1 v ¼ 0:8 106 cm1 (at K ¼ 104 Oe; 1
1
a ¼ 3 1010 Oe cm2 ), i.e. at k not exceeding 16% of Dk: Actually, due to averaging over all the directions of soliton motion and exponential decrease of the number of solitons as their energy increases the role of the phase additive ðvrs Þ in the configuration of the form-factor only reveals itself at considerably smaller values of k. The phase correction in Eqs. (16) and (17) taking into account the deformation of solitons as they move is proportional to v2 and consequently, is not larger than 2% of the basic term qffiffiffiffiffiffiffiffiffiffiffi aK 1 ðkr Þ (at v ¼ v ). Taking into account s
1
the averaging over the values and directions of the velocity results in yet smaller changes of the expression for the form-factor. If the discussed corrections of the phase multipliers are neglected, in Eqs. (16) and (17) the integration is carried out over all the angles between the vectors k and rs ; we obtain the following expression for the dynamic structural factor within an accuracy of the terms proportional to v2 : Sðk; eÞ ¼
RNs
K
2p2 Uð1 h2 Þ21=4 Iðk; eÞ Fs2 ðkÞ;
pffiffiffiffiffiffiffiffiffiffiffiffiffi!2 1 h2 kB T ð20Þ
where Iðk; eÞ ¼
Z
Z do
" dv exp
( )# K o2 þ ð1 h2 Þv2
2ð1 h2 Þ3=2 kB T rffiffiffiffiffiffi e a þo ðkvÞ d U K1 rffiffiffiffiffiffi e a þd o ðkvÞ ð21Þ U K1
and form-factor of soliton is ! rffiffiffiffiffiffi a kr pffiffiffiffiffiffiffiffiffiffiffiffiffi sin Z N K 1 1 h2 2 rffiffiffiffiffiffi r QðrÞ dr Fs ðkÞ ¼ 4p a kr 0 pffiffiffiffiffiffiffiffiffiffiffiffiffi K 1 1 h2 ð22Þ (it is accepted that if hD1; o51 and it is believed qffiffiffiffiffiffiffiffiffiffiffi that v2 5 aK11 jðkvÞj). At small values of the transferred pulse it is necessary to take into account the dependence of the form-factor on the velocities. In the limiting qffiffiffiffiffiffiffiffiffiffiffi case aK11 k5v% this dependence takes form Z N sin ðvrð1 h2 Þ1 Þ Fs ðvÞD4p r2 QðrÞ dr: ð23Þ vrð1 h2 Þ1 0 Performing the integration in Eq. (21) over all the values of angles between the vectors k and v and next, over the values of v and o we obtain (in the approximation that the form-factor is independent of the velocity) pffiffiffiffiffiffiffiffiffiffiffiffiffi!1=2 pffiffiffi K 1 h2 2Fs2 ðkÞ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Sðk; eÞ ¼ kB T ak2 5=4 Uð1 h2 Þ 1þ 2 ð1 h ÞK1 1 0 pffiffiffiffiffiffiffiffiffiffiffiffiffi B K 1 h2 e2 C C B exp B 2 C: 2 A @ U ak 2kB Tð1 h2 Þ2 1 þ ð1 h2 ÞK1 RNs
ð24Þ Figs. 3 and 4 show the dependence curves of the function 1=2 ak2 f ðk; eÞ ¼ Fs2 ðkÞ 1 þ ð1 h2 ÞK1 " " pffiffiffiffiffiffiffiffiffiffiffiffiffi!# K 1 h2 exp kB T # 1 ak2 e2 1þ ð25Þ ð1 h2 ÞK1 2ð1 h2 Þ2 U 2 on the energy at specified values of the transferred momentum when ð1 hÞ ¼ 0:01; T ¼ 200 K; M0 ¼ 102 erg=ðOe cm3 Þ; K1 ¼ 104 Oe; a ¼ 3 1010 Oe cm2 ; gD0:5; and B ¼ 107 Oe:
ARTICLE IN PRESS V.V. Nietz / Journal of Magnetism and Magnetic Materials 266 (2003) 258–267
263
1000
k=0.1 x 10 6 cm-1 30 800
0.3 x 10 6 cm -1 600
Fs (k)
f (k,ε)
20
0.5 x 10 6 cm-1
400
0.7 x 10 6 cm-1
200
0
10
0 -0.2
-0.1
0.0 0.1 ε (meV)
0.2
0
Fig. 3. The dependence of the f ðk; eÞ function on the energy at rather small values of the transferred momentum (at 1 h ¼ 0:01). The broadening of the distribution is due to energy exchange with the precession degree of freedom of solitons. 1.0
0.8
f (k,ε)
0.6
0.4
3x10 6 cm -1 0.2
4x10 -0.4
-0.2
0.0
0.2
Fig. 5. The form-factor of BS 3 1010 Oe cm2 ; ð1 hÞ ¼ 0:01:
at
3
K1 ¼ 104 Oe;
a¼
(the width of the peak at half height). For the above outlined set of parameters at ð1 hÞ ¼ 0:01 it is 0:12 meV: At rather large k, Doppler-related broadening becomes dominating. For example, at k ¼ 4 106 cm1 : De ¼ DeD D0:62 meV:
cm-1
0.0 -0.6
2 k x 10 -6 cm -1
The configuration of the form-factor at ð1 hÞ ¼ 0:01 (from Formula (22)) is illustrated in Fig. 5. At small values of k the broadening is mainly associated with exchange of energy quanta _o: As a result sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi kB T 2 1=2 pffiffiffiffiffiffiffiffiffiffiffiffiffi De ¼ Dem ¼ 2 2ln 2Uð1 h Þ K 1 h2
6
k=2.5x10 cm -1
6
1
0.4
0.6
ε (meV)
Fig. 4. The dependence of the f ðk; eÞ function on the energy at rather large values of the transferred momentum (at 1 h ¼ 0:01). The distribution is mainly due to Doppler displacement caused by the motion of solitons.
Integrating Eq. (24) over the transferred energy we obtain the structural factor of scattering on solitons: Z N SðkÞ Sðk; eÞ de N pffiffiffiffiffiffiffiffiffiffiffiffiffi! 2RNs K 1 h2 exp ¼ pffiffiffi Fs2 ðkÞ: kB T pð1 h2 Þ11=4 ð26Þ
4. Neutron scattering methods of BS observation The greatest interest is in the BSs, that are, for the present, considered to be exotic in condensed matter physics. The actual problem is not only the experimental detection of such solitons, but also the elucidation of their role in the formation of domains of a new phase. At h values considerably less than 1; the BSs are, in principal, possible, but they have very large energy, so that it is difficult to observe them
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without a specific method for their generation. Therefore, in practice, such non-linear objects can appear in the region of phase spinodal, that is at hD1; where their energy decreases anomalously. When h exceeds the critical value of the field, the phase state becomes absolutely unstable and disintegrates. In such case the formation of solitons is not associated with the usual thermal fluctuations in the crystal (the energy of solitons becomes negative with respect to that of the initial metastable state) and consequently, the realization of experiments is possible at low temperatures. The phase reconstruction is accompanied with an intensive redistribution of energy and energy exchange with other degrees of freedom of the magnetic subsystem and with the fluctuations of the crystalline lattice. During the disintegration of the initial state the crystal turns into a mixture of regions with large anomalies in the distribution of l> whose values and the volume of occupied by the regions change fast. It is possible to assume that magnetic solitons play a special role in this chaos of large fluctuations and introduce an element of ordering in the process of phase reorganization. In the case when ho1 but close to the critical value 1; the energy of the BSs is positive but it decreases abnormally as the field approaches 1: If ð1 hÞ ¼ 0:01; for the typical values of the interaction constants: M0 ¼ 102 erg=ðOe cm3 Þ; a ¼ 3 1010 Oe cm2 ; K1 ¼ 104 Oe; gD0:5; the energy of the first order BS at v ¼ 0 (according to (8)) Es1 ¼ 34:5 meV: The effective radius of this soliton ( its amplitude qs0 ¼ 0:865 (maximum q Rs ¼ 122 A, equals 1). If ð1 hÞ ¼ 0:005; then correspondingly, ( qs0 ¼ 0:61: It is seen Es1 ¼ 24:5 meV; Rs ¼ 173 A, then that the energy of solitons is comparable with the energy of thermal fluctuations (at the temperature 200 K, kB TD17:25 meV) and hence, the probability of their spontaneous formation becomes essential. To detect solitons with the help of neutron scattering, it is necessary to transform the sample into the metastable state before phase reconstruction due to a motion of domain walls takes place. It is quite feasible technically to create a magnetic installation generating a rectangular pulse field with the amplitude up to 50–70 kOe, the duration
of the forward front about 1 ms and deviations from the constant value of the field on the top of the pulse equal or below 0.3%. In addition, the crystal must be as perfect as possible, which means that it must have minimum quantity of dislocations and admixtures as well as small mosaicity. It is important from the point of view of minimisation of the number of the centres of formation of ‘‘parasitic’’ domains of the new phase and reduction of the intensity of the normal diffraction peak. If at spin–flop transition the plane perpendicular to the initial direction of the antiferromagnetic vector is chosen as a reflecting plane, diffraction reflection will only appear in the final state. In another geometry, the registration of solitons will take place in the presence of varying intensity of the antiferromagnetic peak. At neutron diffraction on a single crystal neutrons over the wavelength range are scattered pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Dlm ¼ l0 =tg yB Z2 þ s2 for each fixed direction of incidence chosen from the primary beam, where yB is the Bragg angle determined by the condition l0 ¼ 2d sin yB ; d is the interplane distance, Z is the parameter of mosaicity, s ¼ 4d 2 tg yB jFt j=pVc ; jFt j is the module of the scattering amplitude on the elementary cell for the chosen diffraction reflection. Usually, s does not exceed several seconds of arc. The corresponding wavelength range for scattering on solitons Dls Dl0 d=ð2Rs Þ: If a perfect crystal, i.e. a crystal with weak mosaicity is used in the experiment, the value of Dls can be three or four orders of magnitude larger than Dlm : Actually in this case, basic diffraction scattering occurs in a thin surface layer of the crystal. The scattering intensity on solitons is proportional to the number of solitons, i.e. the volume of the sample. Therefore, it is possible to suppose that even the simplest diffraction geometry (see Fig. 6a), a rather thick single crystal in the Bragg position in the ‘‘white’’ beam of the pulsed source and the time-of-flight method of neutron analysis, may serve the purpose of first test measurements of solitonic formations. More essential information can be obtained from the measurement of the angular distribution of neutrons scattered around the central diffraction peak (see Fig. 6b). If ð1 hÞ ¼ 0:02 the central peak of scattering by solitons covers a diapason of
ARTICLE IN PRESS V.V. Nietz / Journal of Magnetism and Magnetic Materials 266 (2003) 258–267 D
D
D M2 L2
L2
L3
k2 L1
” te hi “w
(a)
“white”
M1
k1
S
S
265
“white”
S
(c)
(b)
Fig. 6. The geometry of measurements of neutron scattering on magnetic solitons: (a) measurement of integral scattering on solitons against the background of the diffraction peak, (b) measurement of the angular distribution of neutron scattering for the determination of the soliton form-factor, (c) measurement of inelastic coherent scattering to determine the dynamic structural factor.
( it means that about 8 106 cm1 : For l0 ¼ 4 A, the scattering is into the angular range 71.5 . It is much larger than the angular width of the normal diffraction peak, which is determined by collimation of incident neutrons and can be about 50 . The result of such Rmeasurements will be the structural factor SðkÞ ¼ de Sðk; eÞ: Synchronizing the registration of neutrons with the start of the rectangular pulse of the magnetic field, we obtain Sðk; tÞ; i.e. the information about the time change in neutron scattering on solitons. The time resolution of registration of this process is restricted by the average energy transfer De at interaction of neutrons with solitons: Dte D1:53L2 l30 De (here L2 is the sample to detector distance [m], l0 is ( De is measured in meV, Dte is expressed in A, expressed in ms). For example, for L2 ¼ 2 m; ( De ¼ 0:1 meV: Dte D1:25 ms: l0 ¼ 2 A, The character of the dynamic structural factor Sðk; eÞ is determined with the help of inelastic coherent scattering measurements. The geometry of such measurements is shown in Fig. 6c. The diagrams in Fig. 7, as an example of spin–flop transition in a Cr2O3 single crystal, illustrate the possibility of such measurements. The crystalline plane of magnetic reflection (2 2 4) near which it is possible to search the scattering on solitons is at an angle of about 38 to the plane (1 1 1). The magnetic field is horizontal and lies in the plane of neutron scattering (along the Z-axis, which is perpendicular to the (1 1 1) plane). The vector
k (224) k )
τ
2π
38 0
o
k1
24
(2
k2 φ
7
o
Fig. 7. Diagrams of inelastic scattering on spherical solitons at spin–flop transition in Cr2O3 in neighbourhood to node (2 2 4) of reciprocal lattice. Here k1 and k2 are the wave vectors for incident and scattering neutrons, correspondingly, k ¼ k2 k1 2psð224Þ : Incident and scattering beams are monochromatized. (The value of k vector on figure relatively to k1 and k2 is grossly exaggerated.)
2psð224Þ is perpendicular to the (2 2 4) plane and is equal to 2p=dð224Þ in magnitude, where dð224Þ is the interplanar space for (2 2 4). The primary neutron beam is directed at an angle of about 7 to the Zaxis. With this geometry, the scattering angle is close to 90 corresponding best to the use of pulsed magnets with a horizontal field direction. Let it be necessary to determine the dependence Sðe; tÞ at given k when the rectangular pulses of the field with an amplitude close to the critical value periodically affect the sample. The primary (with
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the wave vector k1) and scattered (with the vector k2) neutron beams are monochromatic, so that the following condition holds: k2 ¼ k þ k1 þ 2p& ð224Þ : The energy change at interaction of neutrons with solitons is equal to e ¼ ð_2 =2mn Þðk12 k22 Þ: As a result, the dependence of the intensity I on t is obtained at fixed values of k and e; where the time t is counted from the beginning of the rectangular pulse. For different e it is necessary to change k1 and k2 and, respectively, the angle f between them without changing k. In Fig. 7, three diagrams are for the energy transfer from neutron to soliton (vector k is directed right-downward) and the other three are for the case of energy transfer from soliton to neutron (vector k is directed left-upward). If one takes into consideration that k1 Dk2 and assumes that two monochromators have analogous characteristics, the time resolution of such inelastic scattering measurements is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Dt ¼ ct ðL2 þ L3 Þ2 Dl2 þ l22 ðDðL2 þ L3 ÞÞ2 ð27Þ (see notations in Fig. 7), where Dl ¼ ½ct L1 =t2s þ tg2 y1 =ðl1 Dy1 Þ2 0:5 is the degree of monochromatization of neutrons incident on the sample, DðL2 þ L3 Þ is the uncertainty of the flight distance past the sample. Similarly to the diffraction case, Dt for inelastic scattering may be about 2 3 ms:
5. Conclusion The uniaxial symmetry of the discussed model of phase transition in an antiferromagnet allows us to obtain expressions for the dynamic structural factor taking into account not only Doppler broadening at translational motion of solitons but also energy exchange between neutrons and the internal precession degree of freedom of solitons. In such model, neutron scattering is essentially inelastic (if oa0) in contrast to the case of 2p-solitons in which the width of the energy distribution is only associated with Doppler broadening. The obtained characteristics of the functions Fs ðkÞ and Fs ðk; eÞ evidence in favour of the possibility of using neutrons in the investigation
of ball magnetic solitons arising in metastabile phase at spin–flop transition. Some experimental setups of neutron scattering on solitons are suggested. The required angular and energy resolution as well as problems connected with locating of solitonic effects in the presence of diffraction, elastic incoherent or inelastic coherent scattering are similar to those in the study of 2psolitons in CsNiF3. BS studying experiments are more complicated because solitons arise and exist in dynamics, during the phase transition. The energy of solitons becomes abnormally small and, consequently, the probability of their origin increases only near the metastability limits of the phase state. However, in this region the phase reconstruction owing to processes that are not directly associated with the formation of solitons, takes place. It is necessary to locate solitonic effects against the background of effects connected, for example, with such processes as the motion of domain walls and growth of parasitic domains.
Acknowledgements Thanks are due to Tatiana Drozdova for the help in preparing this paper for publication.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
J.K. Kjems, M. Steiner, Phys. Rev. 41 (16) (1978) 1137. M.J. Steiner, Appl. Phys. 50 (11) (1979) 7395. M.J. Steiner, J. Magnetism, Magn. Mater. 14 (1979) 142. M. Steiner, K. Kakurai, W. Knop, R. Pynn, J.K. Kjems, Solid State Commun. 41 (4) (1982) 329. M. Steiner, K. Kakurai, J.K. Kjems, Zs. Phys. KI B Bd 53 (1983) 117. M. Steiner, K. Kakurai, J.K. Kjems, Phys. Ser B 120 (1983) 250. M. Steiner, J. Magn. Magn. Mater. 31–34 (1983) 1277. K. Kakurai, M. Steiner, J. Magn. Magn. Mater. 31–34 (1983) 1215. K. Kakurai, R. Pynn, B. Dorner, M. Steiner, J. Phys. Ser. C 17 (1984) L123. K. Kakurai, M. Steiner, R. Pynn, B. Dorner, J. Magn. Magn. Mater. 54–57 (1986) 835. K. Kakurai, M. Steiner, Phys. Ser. B 136 (1986) 349. H.J. Mikeska, J. Phys. C 11 (1978) L29.
ARTICLE IN PRESS V.V. Nietz / Journal of Magnetism and Magnetic Materials 266 (2003) 258–267 [13] K.M. Leung, D.L. Huber, Solid State Commun. 32 (1979) 127. [14] M.J. Steiner, Appl. Phys. 50 (11) (1979) 7395. [15] E. Allroth, H.J. Mikeska, J. Phys. C 13 (1980) L725. [16] D. Hone, K.M. Leung, Phys. Rev. B 22 (11) (1980) 5308. [17] G. Reiter, Phys. Rev. Lett. 46(3) (1981) 202; Erratum: Phys. Rev. Lett. 46 (1981) 518. [18] E. Allroth, H.J. Mikeska, Z. Phys. B-Condens. Matter 43 (1981) 209. [19] P. Kumar, V.K. Samalam, J. Appl. Phys. 53 (3) (1982) 1856. [20] P. Kumar, Physica D 5 (1982) 359. [21] H.J. Mikeska, Physica B 120 (1983) 235. [22] V.K. Fedyanin, J. Magn. Magn. Mater. 31–34 (1983) 1237.
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[23] Yu.A. Izyumov, Usp. Phizitcheskikh Nauk (in Russian) 155 (4) (1988) 553. [24] V.V. Nietz, Communication JINR No. P17-87-28 (Dubna: JINR, 1987). [25] V.Yu. Bezzabotnov, V.V. Nietz, S.A. Oleynik, Communication JINR No. P17-95-87 (Dubna: JINR, 1995). [26] V.V. Nietz, Journal of the Moscow Physical Society 9 (1) (1999) 63. [27] V.V. Nietz, S.A. Oleynik, Communication JINR No. P1795-88 (Dubna: JINR, 1995). [28] V.V. Nietz, Euro-Asian Symposium ‘‘Trends in Magnetism’’ Ekaterinburg, Russia, February 27–March 2 2001, see in The Physics of Metals and Metallography, Vol. 92(1), 2001, p. S243.
Journal of Magnetism and Magnetic Materials 266 (2003) 258–267
Neutron scattering by magnetic ball solitons V.V. Nietz* Frank Laboratory of Neutron Physics, Joint Institute for Nuclear Research, Dubna, Moscow region 141980, Russia
Abstract The character of neutrons scattering by ball solitons arising at the spin–flop transition in antiferromagnet is considered. Results of analysis of form-factor and dynamical structural factor for neutron scattering are presented. Possible arrangements for experiments on the observation of elastic and inelastic scattering by such solitons are discussed. r 2003 Published by Elsevier B.V. PACS: 75.25; 75.50.E; 29.30.H Keywords: Antiferromagnet; Spin–flop transition; Magnetic field; Ball soliton; Form-factor; Dynamic structural factor; Neutron scattering
1. Introduction Sole neutron research of magnetic solitons in crystals were carried out in a series of experiments with quasi-one-dimensional antiferromagnet CsNiF3 in 1978–1986 (see Refs. [1–11]). In the investigated compound the ferromagnetic exchange interaction between the neighbor magnetic moments of Ni2+ ions arranged in chains along the c-axis of the hexagonal structure exceeds the antiferromagnetic interaction between the ions in the neighboring chains by over an order of magnitude. As a result, analysis of the behaviour of this compound in an external magnetic field is reduced to a one-dimensional problem. Naturally, in the field perpendicular to the c-axis the magnetic moments are parallel to the field. However, there may exist so-called 2p-solitons in which the *Tel.: +7-96-21-65-552; fax: +7-96-21-65-882. E-mail address: [email protected] (V.V. Nietz). 0304-8853/$ - see front matter r 2003 Published by Elsevier B.V. doi:10.1016/S0304-8853(03)00244-0
magnetic moments turn by 360 in the plane perpendicular to the c-axis on some limited part of ( The number of such solitons, the chain (D100 A). whose formation is due to thermal fluctuations, obeys the Gibbs distribution. The publications [12–22] are devoted to theoretical analysis of 2psolitons and carry out a comparison of the experimental data with theory (see also an analysis of the data in review [23]). Another type of solitons with their energy comparable with the energy of thermal fluctuations in the temperature region corresponding to the existence of magnetic ordering are the solitons of a (quasi) spherical form that arise in the vicinity of metastability limits at magnetic phase transitions of the first order (see Refs. [24–28]). Let us call them the ball solitons (BS). In the present article, a scattering of neutrons by the BSs that arise at the spin–flop phase transitions in antiferromagnet with one-axis symmetry is analysed.
ARTICLE IN PRESS V.V. Nietz / Journal of Magnetism and Magnetic Materials 266 (2003) 258–267
QðrÞ is solution of equation
2. Magnetic solitons at spin–flop transition For the macroscopic energy of antiferromagnet Z ( B 2 C K1 W ¼ 2M0 jmj þ ðlmÞ2 þ jl> j2 2 2 2 " # K2 a ql 2 ql 2 þ jl> j4 mz Hz þ þ 2 qX qY 4 2 ) az ql þ dX dY dZ ð1Þ 2 qZ in external field H ¼ Hz ; at rather small values of jl> j2 ; the equation _ qMi * i þ G _ Mi qMi ¼ Mi H 2mB qt 2mB qt
ð2Þ
(i ¼ 1; 2; Mi are the magnetisation vectors of two * i is effective field acting on the sublattices, H corresponding sublattice) looks the form (see Ref. [26]) q2 l> ql> Dl> 2 2ih qt qt ¼ ð1 h2 Þ l> gjl> j2 l> þ g
ql> ; qt
ð3Þ
where l ¼ ð2M0 Þ1 ðM1 M2 Þ; m ¼ ð2M0 Þ1 ðM1 þ M2 Þ; l> ¼ ðlx þ ily Þ (x- and y-axes are in the basal plane of a crystal). In Eq. (2) h ¼ Hz B0:5 K10:5 ; g ¼ 0:5ð1 2K2 =K1 Þ; g ¼ GB0:5 K 0:5 ; the differentiation is with respect to the dimensionless time t ¼ 2mB ðK1 BÞ0:5 _1 t and dimensionless coordinates x ¼ K10:5 a0:5 X ; y ¼ K10:5 a0:5 Y ; z¼ 0:5 0:5 K1 az Z: The solution of Eq. (3) at g ¼ 0 corresponding to a BS is
d2 Q 2 dQ ¼ Qð1 Q2 Þ þ dr2 r dr
ð5Þ
(for the sake of simplicity we neglect d B=ðC þ BÞ ¼ wJ =w> and take a ¼ az ). The configuration of QðrÞ is shown in Fig. 1. The BS is defined by the precession frequency o and velocity v of translational moving, thus pffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðo þ hÞ 1 v2 ¼ 1 gQ2 ð6Þ 1 qs0 ; where qs0 jl> ðrs ¼ 0Þj is amplitude of BS, in the given case qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qs0 ¼ g1 ½1 ðo þ hÞ2 ð1 v2 Þ Q1 Q1 ¼ 4:33739: The energy of the soliton is (see Ref. [26]) pffiffiffiffiffi 8pM0 a az I1 Q1 Es ¼ pffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffi g gK1 1 v2 qs0 pffiffiffiffiffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g 1 h 1 v2 1 2 q2s0 Qn RN 2 2 Here, I1 ¼ 0 r Q ðrÞ dr ¼ 1:503796. The energy has the minimum pffiffiffiffiffi ffi 8pM0 a az I1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Es;min ¼ pffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 h2 ð1 v2 Þ K1 g 1 v2
4 3 Q
ð4Þ
where qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r ¼ 1 ðo þ hÞ2 ð1 v2 Þjrs j; pffiffiffiffiffiffiffiffiffiffiffiffiffi rs xs ¼ ðx vtÞ= 1 v2 ; ys ; zs
2 1 0 0
is a Cartesian coordinate fixed to the moving soliton, the xs -axis is directed along the x-axis,
ð7Þ
ð8Þ
when o ¼ 0 and the amplitude is qs0 ¼ Q1 g0:5 ½1 h2 ð1 v2 Þ 0:5 : The dependence of the energy Es on the amplitude for some fields at v ¼ 0 is illustrated in Fig. 2. Let a soliton be generated at time t=0 with a configuration described by Eq. (3) without the
ls ðrs ; tÞ ¼ g0:5 ½1 ðo þ hÞ2 ð1 v2 Þ 0:5 QðrÞ exp ½iðot ðo þ hÞ ðvrs ÞÞ ;
259
1
ρ
2
Fig. 1. The configuration of QðrÞ function.
ARTICLE IN PRESS V.V. Nietz / Journal of Magnetism and Magnetic Materials 266 (2003) 258–267
260 0.4
1
2
60
0.015
40
0.010
~
4 5
0.005
0
0.000
-20
-0.005
7
0.0
20
rs
6
0.0
ω
Es
0.020
rs
3
0.2
80
0.2
0.4
0.6
0.8
1.0
q0 Fig. 2. The dependencies of energy on the amplitude for the next h values: 1–0.98, 2–0.99, 3–0.993, 4–0.995, 5–0.997, * 6–1, pffiffiffiffiffi7–1.001. ffi ffi 1 in high field phase. Here Es ¼ pqffiffiffiffi¼ g K1 =8pM0 a az I1 Es : A change in the precession frequency of solitons is at h ¼ 0:99: The dashed line is the dependence of the effective radius on the amplitude.
additional term gðql=qtÞ: The change in configuration of the soliton in an approximation linear in t can be written [26]: 0 1 got B C ls ðrs ; tÞ ¼ ls ðrs ; t ¼ 0Þ@1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA eiot 2 2 2 1 gQ1 qs0
frequency o: Therefore, the interaction of neutrons with the BS is of inelastic character, which means that it is accompanied with exchange of energy with the internal degree of freedom. This is to some extent similar to what happens at neutron scattering on the spin wave, which is always inelastic. Of interest to us is the neighborhood of the reciprocal lattice centre that corresponds to the usual magnetic diffraction reflection. In the discussed model of spin–flop transition solitons are connected with changes of the components lx and ly of the antiferromagnetism vector. Therefore, for the cross section of magnetic scattering on BS we proceed from the expression: d2 s k2 ¼ ðr0 gn Þ2 P2 ðk2 k1 Þ Sðk; eÞ; dO de _k1 where the dynamical structural factor is
Sðk; eÞ ¼
Z 1 N ð1 þ eZ2 Þ dt eiet=_ 2p N 4 X ikRn ð0Þ ikRn0 ðtÞ e e
ð9Þ i.e. at o > 0 the solitons decrease and transform into initial (metastable) state of a crystal. The energy of BS in an approximation linear in t equals: pffiffiffiffiffi 8pM0 a az I1 Q1 Es ¼ pffiffiffiffiffiffi pffiffiffi K1 g gqs0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 1 h 1 gQ1 qs0 go t ð10Þ
3. Neutron scattering by BSs We analyse the neutron scattering by BSs that are the result of thermal fluctuations in quasiequilibrium state (i.e. at o > 0; in oo0 case the ls values increase and dissipation of energy transforms the soliton into the high field domain). As against the solitons in CsNiF3, the BSs have an internal degree of freedom, the precession
ð11Þ
n;n0 lnþ ð0Þ
ln0 ðtÞ þ ln ð0Þ lnþ0 ðtÞ
ð12Þ
(r0 ¼ 2:8 1013 cm is the classical radius of the electron; gn ¼ 1:913 mn is the magnetic moment of the neutron, eZ is the component of the scattering 7 x y vector). In Expression (12): n ðtÞ ¼ ln ðtÞ7iln ðtÞ; P lis j ðhklÞ pi ðtÞ where lnj ðtÞ ¼ 1=nM0 e M ðtÞ ðj ¼ ni i x; yÞ is the sum for the all magnetic ions in the elementary cell, Mnij ðtÞ is the magnetization component, n is the number of magnetic sublattices. Pðk2 k1 Þ is the magnetic form-factor of the ion, it is assumed that the crystal consists of magnetic ions of one kind); k ¼ k2 k1 2psðhklÞ ; where sðhklÞ is the vector of the reciprocal lattice corresponding to the investigated reciprocal lattice node. Since the size of the soliton is a few hundred Angstroms, neutron scattering on them concentrates in the region of jkj whose value does not exceed several hundredths of 2pjsðhklÞ j: Thus we can use Pð2psðhklÞ Þ instead of Pðk2 k1 Þ: Thus
ARTICLE IN PRESS V.V. Nietz / Journal of Magnetism and Magnetic Materials 266 (2003) 258–267
Z Fs2 ðk; o; vÞ ¼ drs QðrÞ 8" 9 # sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 = ffi < rffiffiffiffiffi a ðo þ hÞv zs exp i : v k pffiffiffiffiffiffiffiffiffiffiffiffiffi rs 1 : ; K1 rs 1 v2
from Eq. (4): qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi lnþ ðtÞ ¼ g1 ½1 ðo þ hÞ2 ð1 v2 Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 Q 1 ðo þ hÞ ð1 v Þ jrsn ðtÞj ei½otðoþhÞðvrn ðtÞÞ :
261
ð13Þ
ð17Þ
The following relation between the coordinates rn in the system of the crystal and the coordinates rsn in the system of the moving soliton holds
The following notations are used: Z1 ¼ RR dv do exp ½Es ðv; oÞ=kB T ; W ðo; vÞ ¼ ½1 ðo þ hÞð1p ffiffiffiffiffiffiffiffiffi vÞ2 ð1v2 Þ; R ¼ _a3 ð1 þ eZ2 Þ=4gK13 Vc2 ; U ¼ 2mB K1 B; Vc is the elementary cell volume in the coordinate system (X,Y,Z). To obtain Expression (16), in the beginning the dynamic structural factor of scattering on one soliton averaged over v and o had been calculated, and then the result for one soliton was multiplied by the average number of solitons in the sample, Ns ; which depends on relation of minimum energy of solitons to kB T value. This dependence can be obtained from thermodynamics of solitons in a fficrystal. pffiffiffiffiffiffiffiffiffiffi The energy shift by U a=K1 ðkvÞ ðo þ hÞv2 is due to Doppler displacement. In addition, neutron scattering on each precessing soliton is accompanied with a transfer of an energy quantum pffiffiffiffiffiffiffiffiffi equal to _O; where O ¼ 2mB _1 K1 B o: The energy of BS can be presented as an expansion in a power series in the velocity and frequency of BS restricting oneself to square-law members: " # 2 2 pffiffiffiffiffiffiffiffiffiffiffiffiffi o v Es ¼ K 1 h2 þ þ pffiffiffiffiffiffiffiffiffiffiffiffiffi ð18Þ 2ð1 h2 Þ1:5 2 1 h2
ðx vx tÞ ¼ cos yv cos jv xs sin jv ys pffiffiffiffiffiffiffiffiffiffiffiffiffi þ sin yv cos jv zs 1 v2 ; ðy vy tÞ ¼ cos yv sin jv xs cos jv ys pffiffiffiffiffiffiffiffiffiffiffiffiffi þ sin yv sin jv zs 1 v2 ; pffiffiffiffiffiffiffiffiffiffiffiffiffi ðz vz tÞ ¼ sin yv xs þ cos yv zs 1 v2 ;
ð14aÞ
ð14bÞ ð14cÞ
where yn is the angle between the z-axis and the vector v, jn is the angle between the x-axis and the plane formed by the z-axis and the vector v. We replace the summation over the cells of the crystal by the integration over the soliton volume, perform the integration over t and the averaging over the frequencies and velocities of the solitons. As a result, we obtain the following expression for the dynamical structural factor: Z Z RNs Sðk; eÞ ¼ do dv W ðo; vÞ eðEs ðo;vÞ=kB T Þ Z1 2 jFs1 ðk; o; vÞj d e þ Uo rffiffiffiffiffiffi a 2 U ðkvÞ ðo þ hÞv K1 2 þjFs2 ðk; o; vÞj d e Uo rffiffiffiffiffiffi a U ðkvÞ ðo þ hÞv2 ; ð15Þ K1 where the form-factors of the solitons are Z Fs1 ðk; o; vÞ ¼ drs QðrÞ 8" 9 # sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 = ffi < rffiffiffiffiffi a ðo þ hÞv zs ; exp i v k þ pffiffiffiffiffiffiffiffiffiffiffiffiffi rs 1 : ; 2 K1 rs 1v ð16Þ
pffiffiffiffiffiffi pffiffiffi (here K ¼ 8p M0 a aI1 =g K1 ). Using decomposition (18) the following expression is obtained: !2 kB T 2 2 13=4 pffiffiffiffiffiffiffiffiffiffiffiffiffi Z1 ¼ 2p ð1 h Þ K 1 h2 pffiffiffiffiffiffiffiffiffiffiffiffiffi exp ðK 1 h2 =kB TÞ: ð19Þ Let us estimate various factors that were taken into account as phase corrections of the formfactors of solitons. At ð1 hÞ ¼ 0:01: omax ¼ 0:01: In this case, in Expressions (15)–(17) we can neglect the o value in comparison with h within an accuracy of 1% ðhD1Þ: We introduce the notion v for the velocity value at which the BS formation
ARTICLE IN PRESS V.V. Nietz / Journal of Magnetism and Magnetic Materials 266 (2003) 258–267
262
probability decreases e times comparative to the formation probability of motionless solitons. At ð1 hÞ ¼ 0:01; T ¼ 200 K we have v ¼ 0:14 and the radius of the soliton Rs ¼ 1:22 106 cm: This means that basic scattering on solitons occurs within the limits Dk ¼ 72pR1 s ¼ 5:1 qffiffiffiffiffiffiffiffiffiffiffi 6 1 1 10 cm : The phase member aK1 ðkrs Þ becomes comparable with hðvrs Þ if v ¼ v only at pffiffiffiffiffiffiffiffiffiffiffiffiffi ko K a1 v ¼ 0:8 106 cm1 (at K ¼ 104 Oe; 1
1
a ¼ 3 1010 Oe cm2 ), i.e. at k not exceeding 16% of Dk: Actually, due to averaging over all the directions of soliton motion and exponential decrease of the number of solitons as their energy increases the role of the phase additive ðvrs Þ in the configuration of the form-factor only reveals itself at considerably smaller values of k. The phase correction in Eqs. (16) and (17) taking into account the deformation of solitons as they move is proportional to v2 and consequently, is not larger than 2% of the basic term qffiffiffiffiffiffiffiffiffiffiffi aK 1 ðkr Þ (at v ¼ v ). Taking into account s
1
the averaging over the values and directions of the velocity results in yet smaller changes of the expression for the form-factor. If the discussed corrections of the phase multipliers are neglected, in Eqs. (16) and (17) the integration is carried out over all the angles between the vectors k and rs ; we obtain the following expression for the dynamic structural factor within an accuracy of the terms proportional to v2 : Sðk; eÞ ¼
RNs
K
2p2 Uð1 h2 Þ21=4 Iðk; eÞ Fs2 ðkÞ;
pffiffiffiffiffiffiffiffiffiffiffiffiffi!2 1 h2 kB T ð20Þ
where Iðk; eÞ ¼
Z
Z do
" dv exp
( )# K o2 þ ð1 h2 Þv2
2ð1 h2 Þ3=2 kB T rffiffiffiffiffiffi e a þo ðkvÞ d U K1 rffiffiffiffiffiffi e a þd o ðkvÞ ð21Þ U K1
and form-factor of soliton is ! rffiffiffiffiffiffi a kr pffiffiffiffiffiffiffiffiffiffiffiffiffi sin Z N K 1 1 h2 2 rffiffiffiffiffiffi r QðrÞ dr Fs ðkÞ ¼ 4p a kr 0 pffiffiffiffiffiffiffiffiffiffiffiffiffi K 1 1 h2 ð22Þ (it is accepted that if hD1; o51 and it is believed qffiffiffiffiffiffiffiffiffiffiffi that v2 5 aK11 jðkvÞj). At small values of the transferred pulse it is necessary to take into account the dependence of the form-factor on the velocities. In the limiting qffiffiffiffiffiffiffiffiffiffiffi case aK11 k5v% this dependence takes form Z N sin ðvrð1 h2 Þ1 Þ Fs ðvÞD4p r2 QðrÞ dr: ð23Þ vrð1 h2 Þ1 0 Performing the integration in Eq. (21) over all the values of angles between the vectors k and v and next, over the values of v and o we obtain (in the approximation that the form-factor is independent of the velocity) pffiffiffiffiffiffiffiffiffiffiffiffiffi!1=2 pffiffiffi K 1 h2 2Fs2 ðkÞ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Sðk; eÞ ¼ kB T ak2 5=4 Uð1 h2 Þ 1þ 2 ð1 h ÞK1 1 0 pffiffiffiffiffiffiffiffiffiffiffiffiffi B K 1 h2 e2 C C B exp B 2 C: 2 A @ U ak 2kB Tð1 h2 Þ2 1 þ ð1 h2 ÞK1 RNs
ð24Þ Figs. 3 and 4 show the dependence curves of the function 1=2 ak2 f ðk; eÞ ¼ Fs2 ðkÞ 1 þ ð1 h2 ÞK1 " " pffiffiffiffiffiffiffiffiffiffiffiffiffi!# K 1 h2 exp kB T # 1 ak2 e2 1þ ð25Þ ð1 h2 ÞK1 2ð1 h2 Þ2 U 2 on the energy at specified values of the transferred momentum when ð1 hÞ ¼ 0:01; T ¼ 200 K; M0 ¼ 102 erg=ðOe cm3 Þ; K1 ¼ 104 Oe; a ¼ 3 1010 Oe cm2 ; gD0:5; and B ¼ 107 Oe:
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263
1000
k=0.1 x 10 6 cm-1 30 800
0.3 x 10 6 cm -1 600
Fs (k)
f (k,ε)
20
0.5 x 10 6 cm-1
400
0.7 x 10 6 cm-1
200
0
10
0 -0.2
-0.1
0.0 0.1 ε (meV)
0.2
0
Fig. 3. The dependence of the f ðk; eÞ function on the energy at rather small values of the transferred momentum (at 1 h ¼ 0:01). The broadening of the distribution is due to energy exchange with the precession degree of freedom of solitons. 1.0
0.8
f (k,ε)
0.6
0.4
3x10 6 cm -1 0.2
4x10 -0.4
-0.2
0.0
0.2
Fig. 5. The form-factor of BS 3 1010 Oe cm2 ; ð1 hÞ ¼ 0:01:
at
3
K1 ¼ 104 Oe;
a¼
(the width of the peak at half height). For the above outlined set of parameters at ð1 hÞ ¼ 0:01 it is 0:12 meV: At rather large k, Doppler-related broadening becomes dominating. For example, at k ¼ 4 106 cm1 : De ¼ DeD D0:62 meV:
cm-1
0.0 -0.6
2 k x 10 -6 cm -1
The configuration of the form-factor at ð1 hÞ ¼ 0:01 (from Formula (22)) is illustrated in Fig. 5. At small values of k the broadening is mainly associated with exchange of energy quanta _o: As a result sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi kB T 2 1=2 pffiffiffiffiffiffiffiffiffiffiffiffiffi De ¼ Dem ¼ 2 2ln 2Uð1 h Þ K 1 h2
6
k=2.5x10 cm -1
6
1
0.4
0.6
ε (meV)
Fig. 4. The dependence of the f ðk; eÞ function on the energy at rather large values of the transferred momentum (at 1 h ¼ 0:01). The distribution is mainly due to Doppler displacement caused by the motion of solitons.
Integrating Eq. (24) over the transferred energy we obtain the structural factor of scattering on solitons: Z N SðkÞ Sðk; eÞ de N pffiffiffiffiffiffiffiffiffiffiffiffiffi! 2RNs K 1 h2 exp ¼ pffiffiffi Fs2 ðkÞ: kB T pð1 h2 Þ11=4 ð26Þ
4. Neutron scattering methods of BS observation The greatest interest is in the BSs, that are, for the present, considered to be exotic in condensed matter physics. The actual problem is not only the experimental detection of such solitons, but also the elucidation of their role in the formation of domains of a new phase. At h values considerably less than 1; the BSs are, in principal, possible, but they have very large energy, so that it is difficult to observe them
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without a specific method for their generation. Therefore, in practice, such non-linear objects can appear in the region of phase spinodal, that is at hD1; where their energy decreases anomalously. When h exceeds the critical value of the field, the phase state becomes absolutely unstable and disintegrates. In such case the formation of solitons is not associated with the usual thermal fluctuations in the crystal (the energy of solitons becomes negative with respect to that of the initial metastable state) and consequently, the realization of experiments is possible at low temperatures. The phase reconstruction is accompanied with an intensive redistribution of energy and energy exchange with other degrees of freedom of the magnetic subsystem and with the fluctuations of the crystalline lattice. During the disintegration of the initial state the crystal turns into a mixture of regions with large anomalies in the distribution of l> whose values and the volume of occupied by the regions change fast. It is possible to assume that magnetic solitons play a special role in this chaos of large fluctuations and introduce an element of ordering in the process of phase reorganization. In the case when ho1 but close to the critical value 1; the energy of the BSs is positive but it decreases abnormally as the field approaches 1: If ð1 hÞ ¼ 0:01; for the typical values of the interaction constants: M0 ¼ 102 erg=ðOe cm3 Þ; a ¼ 3 1010 Oe cm2 ; K1 ¼ 104 Oe; gD0:5; the energy of the first order BS at v ¼ 0 (according to (8)) Es1 ¼ 34:5 meV: The effective radius of this soliton ( its amplitude qs0 ¼ 0:865 (maximum q Rs ¼ 122 A, equals 1). If ð1 hÞ ¼ 0:005; then correspondingly, ( qs0 ¼ 0:61: It is seen Es1 ¼ 24:5 meV; Rs ¼ 173 A, then that the energy of solitons is comparable with the energy of thermal fluctuations (at the temperature 200 K, kB TD17:25 meV) and hence, the probability of their spontaneous formation becomes essential. To detect solitons with the help of neutron scattering, it is necessary to transform the sample into the metastable state before phase reconstruction due to a motion of domain walls takes place. It is quite feasible technically to create a magnetic installation generating a rectangular pulse field with the amplitude up to 50–70 kOe, the duration
of the forward front about 1 ms and deviations from the constant value of the field on the top of the pulse equal or below 0.3%. In addition, the crystal must be as perfect as possible, which means that it must have minimum quantity of dislocations and admixtures as well as small mosaicity. It is important from the point of view of minimisation of the number of the centres of formation of ‘‘parasitic’’ domains of the new phase and reduction of the intensity of the normal diffraction peak. If at spin–flop transition the plane perpendicular to the initial direction of the antiferromagnetic vector is chosen as a reflecting plane, diffraction reflection will only appear in the final state. In another geometry, the registration of solitons will take place in the presence of varying intensity of the antiferromagnetic peak. At neutron diffraction on a single crystal neutrons over the wavelength range are scattered pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Dlm ¼ l0 =tg yB Z2 þ s2 for each fixed direction of incidence chosen from the primary beam, where yB is the Bragg angle determined by the condition l0 ¼ 2d sin yB ; d is the interplane distance, Z is the parameter of mosaicity, s ¼ 4d 2 tg yB jFt j=pVc ; jFt j is the module of the scattering amplitude on the elementary cell for the chosen diffraction reflection. Usually, s does not exceed several seconds of arc. The corresponding wavelength range for scattering on solitons Dls Dl0 d=ð2Rs Þ: If a perfect crystal, i.e. a crystal with weak mosaicity is used in the experiment, the value of Dls can be three or four orders of magnitude larger than Dlm : Actually in this case, basic diffraction scattering occurs in a thin surface layer of the crystal. The scattering intensity on solitons is proportional to the number of solitons, i.e. the volume of the sample. Therefore, it is possible to suppose that even the simplest diffraction geometry (see Fig. 6a), a rather thick single crystal in the Bragg position in the ‘‘white’’ beam of the pulsed source and the time-of-flight method of neutron analysis, may serve the purpose of first test measurements of solitonic formations. More essential information can be obtained from the measurement of the angular distribution of neutrons scattered around the central diffraction peak (see Fig. 6b). If ð1 hÞ ¼ 0:02 the central peak of scattering by solitons covers a diapason of
ARTICLE IN PRESS V.V. Nietz / Journal of Magnetism and Magnetic Materials 266 (2003) 258–267 D
D
D M2 L2
L2
L3
k2 L1
” te hi “w
(a)
“white”
M1
k1
S
S
265
“white”
S
(c)
(b)
Fig. 6. The geometry of measurements of neutron scattering on magnetic solitons: (a) measurement of integral scattering on solitons against the background of the diffraction peak, (b) measurement of the angular distribution of neutron scattering for the determination of the soliton form-factor, (c) measurement of inelastic coherent scattering to determine the dynamic structural factor.
( it means that about 8 106 cm1 : For l0 ¼ 4 A, the scattering is into the angular range 71.5 . It is much larger than the angular width of the normal diffraction peak, which is determined by collimation of incident neutrons and can be about 50 . The result of such Rmeasurements will be the structural factor SðkÞ ¼ de Sðk; eÞ: Synchronizing the registration of neutrons with the start of the rectangular pulse of the magnetic field, we obtain Sðk; tÞ; i.e. the information about the time change in neutron scattering on solitons. The time resolution of registration of this process is restricted by the average energy transfer De at interaction of neutrons with solitons: Dte D1:53L2 l30 De (here L2 is the sample to detector distance [m], l0 is ( De is measured in meV, Dte is expressed in A, expressed in ms). For example, for L2 ¼ 2 m; ( De ¼ 0:1 meV: Dte D1:25 ms: l0 ¼ 2 A, The character of the dynamic structural factor Sðk; eÞ is determined with the help of inelastic coherent scattering measurements. The geometry of such measurements is shown in Fig. 6c. The diagrams in Fig. 7, as an example of spin–flop transition in a Cr2O3 single crystal, illustrate the possibility of such measurements. The crystalline plane of magnetic reflection (2 2 4) near which it is possible to search the scattering on solitons is at an angle of about 38 to the plane (1 1 1). The magnetic field is horizontal and lies in the plane of neutron scattering (along the Z-axis, which is perpendicular to the (1 1 1) plane). The vector
k (224) k )
τ
2π
38 0
o
k1
24
(2
k2 φ
7
o
Fig. 7. Diagrams of inelastic scattering on spherical solitons at spin–flop transition in Cr2O3 in neighbourhood to node (2 2 4) of reciprocal lattice. Here k1 and k2 are the wave vectors for incident and scattering neutrons, correspondingly, k ¼ k2 k1 2psð224Þ : Incident and scattering beams are monochromatized. (The value of k vector on figure relatively to k1 and k2 is grossly exaggerated.)
2psð224Þ is perpendicular to the (2 2 4) plane and is equal to 2p=dð224Þ in magnitude, where dð224Þ is the interplanar space for (2 2 4). The primary neutron beam is directed at an angle of about 7 to the Zaxis. With this geometry, the scattering angle is close to 90 corresponding best to the use of pulsed magnets with a horizontal field direction. Let it be necessary to determine the dependence Sðe; tÞ at given k when the rectangular pulses of the field with an amplitude close to the critical value periodically affect the sample. The primary (with
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the wave vector k1) and scattered (with the vector k2) neutron beams are monochromatic, so that the following condition holds: k2 ¼ k þ k1 þ 2p& ð224Þ : The energy change at interaction of neutrons with solitons is equal to e ¼ ð_2 =2mn Þðk12 k22 Þ: As a result, the dependence of the intensity I on t is obtained at fixed values of k and e; where the time t is counted from the beginning of the rectangular pulse. For different e it is necessary to change k1 and k2 and, respectively, the angle f between them without changing k. In Fig. 7, three diagrams are for the energy transfer from neutron to soliton (vector k is directed right-downward) and the other three are for the case of energy transfer from soliton to neutron (vector k is directed left-upward). If one takes into consideration that k1 Dk2 and assumes that two monochromators have analogous characteristics, the time resolution of such inelastic scattering measurements is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Dt ¼ ct ðL2 þ L3 Þ2 Dl2 þ l22 ðDðL2 þ L3 ÞÞ2 ð27Þ (see notations in Fig. 7), where Dl ¼ ½ct L1 =t2s þ tg2 y1 =ðl1 Dy1 Þ2 0:5 is the degree of monochromatization of neutrons incident on the sample, DðL2 þ L3 Þ is the uncertainty of the flight distance past the sample. Similarly to the diffraction case, Dt for inelastic scattering may be about 2 3 ms:
5. Conclusion The uniaxial symmetry of the discussed model of phase transition in an antiferromagnet allows us to obtain expressions for the dynamic structural factor taking into account not only Doppler broadening at translational motion of solitons but also energy exchange between neutrons and the internal precession degree of freedom of solitons. In such model, neutron scattering is essentially inelastic (if oa0) in contrast to the case of 2p-solitons in which the width of the energy distribution is only associated with Doppler broadening. The obtained characteristics of the functions Fs ðkÞ and Fs ðk; eÞ evidence in favour of the possibility of using neutrons in the investigation
of ball magnetic solitons arising in metastabile phase at spin–flop transition. Some experimental setups of neutron scattering on solitons are suggested. The required angular and energy resolution as well as problems connected with locating of solitonic effects in the presence of diffraction, elastic incoherent or inelastic coherent scattering are similar to those in the study of 2psolitons in CsNiF3. BS studying experiments are more complicated because solitons arise and exist in dynamics, during the phase transition. The energy of solitons becomes abnormally small and, consequently, the probability of their origin increases only near the metastability limits of the phase state. However, in this region the phase reconstruction owing to processes that are not directly associated with the formation of solitons, takes place. It is necessary to locate solitonic effects against the background of effects connected, for example, with such processes as the motion of domain walls and growth of parasitic domains.
Acknowledgements Thanks are due to Tatiana Drozdova for the help in preparing this paper for publication.
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[23] Yu.A. Izyumov, Usp. Phizitcheskikh Nauk (in Russian) 155 (4) (1988) 553. [24] V.V. Nietz, Communication JINR No. P17-87-28 (Dubna: JINR, 1987). [25] V.Yu. Bezzabotnov, V.V. Nietz, S.A. Oleynik, Communication JINR No. P17-95-87 (Dubna: JINR, 1995). [26] V.V. Nietz, Journal of the Moscow Physical Society 9 (1) (1999) 63. [27] V.V. Nietz, S.A. Oleynik, Communication JINR No. P1795-88 (Dubna: JINR, 1995). [28] V.V. Nietz, Euro-Asian Symposium ‘‘Trends in Magnetism’’ Ekaterinburg, Russia, February 27–March 2 2001, see in The Physics of Metals and Metallography, Vol. 92(1), 2001, p. S243.