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A spin-echo like effect in polarized neutron scattering from magnetized media
Physica B 156 & 157 (1989) 663-665 North-Holland, Amsterdam
A SPIN-ECHO LIKE EFFECT IN POLARIZED NEUTRON SCATTERING FROM MAGNETIZED MEDIA B.P. TOPERVERG’ ‘Leningrad Nuclear Physics ‘Sektion Physik, Technische
and J. WENIGER Institute, 188350 Leningrad, USSR Universitiit Magdeburg, PSF 124, 3010 Magdeburg,
German
Dem. Rep.
An effect resembling the neutron spin echo is predicted for the scattering of polarized neutrons from magnetized media. It arises from the Larmor precession of the polarization vector in the internal sample field.
1. Introduction In most cases polarization analysis experiments on magnetized samples are performed choosing the incident polarization P, and the direction of analysis of the scattered beam polarization P parallel or antiparallel to the internal sample field B. Otherwise the components of PO and P perpendicular to B carry out precessions with the Larmor frequency w,_and because of the wavelength spread of the beam are depolarized. This fact prevents the application of the Neutron Spin Echo (NSE) method [l] to study the low energy excitations in magnets and had caused the modification of the original NSE arrangement [2]. Here the depolarization was primarily due to the nonuniform field in the sample (domain structure). Since experiments involving the analysis of neutron wavelengths allow, in principle, to measure also the components of P perpendicular to B [3] we consider here the general case of noncollinear orientation of P,, P and B. If P,, has components perpendicular to B and also the perpendicular components of P are analysed an effect resembling the NSE should be observed, which could be used to measure small energy transfers. Furthermore, such a scheme enables to extract the contribution arising from the scattering by the longitudinal spin correlations. For simplicity, only the case of small-angle scattering is considered.
2. Precession of the polarization in the sample field As it was mentioned, in the case of arbitrary orientation of P,,, P and B the motion of the spin of both the incident and the scattered neutron wave in the internal sample field has to be taken into account in the formalism of scattering theory. Thus, according to [4] the polarization of the scattered beam can be written
where the neutron scattering amplitude f,,+,is the sum of nuclear and magnetic one. The effective spin operator a,,, given by a,,,(k’,
d - x, b) = b(b o) + [a - b(b u)] cos 4(k’,
d - x)
+ b x u sin 4(k’, d-x)
, (2)
b = BB-‘,
c$(k’,x)=o,_m/k’,
Osxsd, (3)
describes the precession of the scattered beam polarization in the field of the sample whose thickness is d. The angular brackets in (1) denote the averaging over the coordinates of the scattering event x and over the incident neutron spin states. The latter has to be carried out by
0921-4526/89/$03.50 CQ Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
B. P. Toperverg
664
means of the effective Peff
=
Po[Qeff(k~
x>
-b)l
and J. Weniger I Spin-echo effect in scattering from magnets
density matrix operator (4)
7
p” = l/2(1 + P”0) )
(5)
which contains the precession of the incident polarization in the sample field. The cross section is given by an expression analogous to (1) (cf. [4]). The polarization resulting from (1) can be divided into two parts, dependent on and independent of the incident polarization. The polarization-independent term describes the polarization created in the scattering and is determined by the scattering from the asymmetric perpendicular spin correlations and by the nuclear-magnetic interference scattering. The polarization-dependent term arises from nuclear scattering and from magnetic scattering by the longitudinal and symmetric perpendicular spin correlations (cf. [5]). Both the cross section and the polarization after scattering contain terms being oscillating functions of the magnetic field, the neutron wavelength and the thickness of the sample. One of these terms in the polarization has an oscillation frequency which is proportional to the energy transfer w. This term together with the non-oscillating terms gives the main contribution to the observed polarization and will be analysed in the next section.
3. NSE-like effect We assume that the incident polarization is aligned in the plane perpendicular to b. Then in the case w 4 E we get from (1) for the polarization component parallel to P, [5] &
P = z
sin(ot/2)
x cos(w,t, - wt/2)S,.,
,
(6)
where t, = dmlk and t = t,o,l(2E). The function S, o contains the nuclear scattering cross section ‘and the longitudinal and symmetric perpendicular spin pair correlation functions g” and g ‘, respectively:
d2Pc’ s 9.w =--d0dE’ -
(r,y)*(l -
e2,)[e2,gL + (1 - e’b)g”l , (7)
where eb = bqq-‘. Additionally we have supposed that wL1, B 1, as usually realized in the NSE method. In the case of quasielastic scattering we obtain for the intensity measured without energy analysis (cf. [5])
= P,, cos(w,t,,)
I
dw S,,w sin(wt)/(wt)
,
(8)
from which follows Z’(t) = $ rZ(t) = POcos(w,t,)
I
dw cos(wt) S,,, ,
(9) which is proportional to the Fourier transform of s . Apart from the oscillating factor this result is”iompletely analogous to the standard NSE result. The dependence on the “time parameter” t can be obtained by performing the measurement in several magnetic fields. If the subject of investigation is the magnetic subsystem one has to keep in mind, however, that the spectrum of the magnetic excitations can be strongly affected by changing the magnetic field. Then it should be more convenient to perform the measurements without varying the field (also to avoid the beam depolarization) and to change either the magnetic field region traversed by the neutrons or the neutron wavelength. The latter possibility becomes very attractive if the neutron wavelength can be measured, e.g. by the time-offlight technique. In contrast to the standard NSE method one has to bear in mind that by varying the wavelength A not only the parameter t changes but also q - A -‘O, where 0 is the scattering angle. A constant-q time scan can be achieved by varying simultaneously with t the scattering angle according to 0 - A [6]. We note that because of the rapidly oscillating factor in (8) a sufficiently high wavelength resolution is necessary to observe the NSE effect. If the condition Bh /A < (CC+_&-’is violated, then the effect connected with Z(t) becomes rather small.
B. P. Toperverg
and J. Weniger
I Spin-echo effect in scattering from magnets
If the requirement of quasielasticity w 4 2EO is satisifed then the contribution of nuclear scattering is measured by aligning the field in the scattering plane perpendicular to the incident beam. By repeating the measurement with the field perpendicular to the scattering plane the contribution of the longitudinal spin correlations can be determined. If these results are completed by measurements with other angles between q and b the perpendicular spin correlations can be extracted. We mention here that the situation is much more complicated if the condition of quasielas-
ticity is violated. of q cannot be perimental data comes extremely
665
In this case the o-dependence neglected. An analysis of exwithout model assumptions bedifficult.
References [l] F. Mezei, J. Phys. (Paris) 43 (1982) C7-9. [2] B. Farago and F. Mezei, Physica B 136 (1986) 100. [3] Y. Endoh, Y. Sasaki, H. Ono, S. Mitsuda and S. Ikeda, Physica B 120 (1983) 45. [4] B. Toperverg and J. Weniger, submitted to 2. Phys. B. [5] B. Toperverg and J. Weniger, Z. Phys. B, in press. [6] F. Mezei, Nucl. Instr. Meth. 164 (1979) 153.
A SPIN-ECHO LIKE EFFECT IN POLARIZED NEUTRON SCATTERING FROM MAGNETIZED MEDIA B.P. TOPERVERG’ ‘Leningrad Nuclear Physics ‘Sektion Physik, Technische
and J. WENIGER Institute, 188350 Leningrad, USSR Universitiit Magdeburg, PSF 124, 3010 Magdeburg,
German
Dem. Rep.
An effect resembling the neutron spin echo is predicted for the scattering of polarized neutrons from magnetized media. It arises from the Larmor precession of the polarization vector in the internal sample field.
1. Introduction In most cases polarization analysis experiments on magnetized samples are performed choosing the incident polarization P, and the direction of analysis of the scattered beam polarization P parallel or antiparallel to the internal sample field B. Otherwise the components of PO and P perpendicular to B carry out precessions with the Larmor frequency w,_and because of the wavelength spread of the beam are depolarized. This fact prevents the application of the Neutron Spin Echo (NSE) method [l] to study the low energy excitations in magnets and had caused the modification of the original NSE arrangement [2]. Here the depolarization was primarily due to the nonuniform field in the sample (domain structure). Since experiments involving the analysis of neutron wavelengths allow, in principle, to measure also the components of P perpendicular to B [3] we consider here the general case of noncollinear orientation of P,, P and B. If P,, has components perpendicular to B and also the perpendicular components of P are analysed an effect resembling the NSE should be observed, which could be used to measure small energy transfers. Furthermore, such a scheme enables to extract the contribution arising from the scattering by the longitudinal spin correlations. For simplicity, only the case of small-angle scattering is considered.
2. Precession of the polarization in the sample field As it was mentioned, in the case of arbitrary orientation of P,,, P and B the motion of the spin of both the incident and the scattered neutron wave in the internal sample field has to be taken into account in the formalism of scattering theory. Thus, according to [4] the polarization of the scattered beam can be written
where the neutron scattering amplitude f,,+,is the sum of nuclear and magnetic one. The effective spin operator a,,, given by a,,,(k’,
d - x, b) = b(b o) + [a - b(b u)] cos 4(k’,
d - x)
+ b x u sin 4(k’, d-x)
, (2)
b = BB-‘,
c$(k’,x)=o,_m/k’,
Osxsd, (3)
describes the precession of the scattered beam polarization in the field of the sample whose thickness is d. The angular brackets in (1) denote the averaging over the coordinates of the scattering event x and over the incident neutron spin states. The latter has to be carried out by
0921-4526/89/$03.50 CQ Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
B. P. Toperverg
664
means of the effective Peff
=
Po[Qeff(k~
x>
-b)l
and J. Weniger I Spin-echo effect in scattering from magnets
density matrix operator (4)
7
p” = l/2(1 + P”0) )
(5)
which contains the precession of the incident polarization in the sample field. The cross section is given by an expression analogous to (1) (cf. [4]). The polarization resulting from (1) can be divided into two parts, dependent on and independent of the incident polarization. The polarization-independent term describes the polarization created in the scattering and is determined by the scattering from the asymmetric perpendicular spin correlations and by the nuclear-magnetic interference scattering. The polarization-dependent term arises from nuclear scattering and from magnetic scattering by the longitudinal and symmetric perpendicular spin correlations (cf. [5]). Both the cross section and the polarization after scattering contain terms being oscillating functions of the magnetic field, the neutron wavelength and the thickness of the sample. One of these terms in the polarization has an oscillation frequency which is proportional to the energy transfer w. This term together with the non-oscillating terms gives the main contribution to the observed polarization and will be analysed in the next section.
3. NSE-like effect We assume that the incident polarization is aligned in the plane perpendicular to b. Then in the case w 4 E we get from (1) for the polarization component parallel to P, [5] &
P = z
sin(ot/2)
x cos(w,t, - wt/2)S,.,
,
(6)
where t, = dmlk and t = t,o,l(2E). The function S, o contains the nuclear scattering cross section ‘and the longitudinal and symmetric perpendicular spin pair correlation functions g” and g ‘, respectively:
d2Pc’ s 9.w =--d0dE’ -
(r,y)*(l -
e2,)[e2,gL + (1 - e’b)g”l , (7)
where eb = bqq-‘. Additionally we have supposed that wL1, B 1, as usually realized in the NSE method. In the case of quasielastic scattering we obtain for the intensity measured without energy analysis (cf. [5])
= P,, cos(w,t,,)
I
dw S,,w sin(wt)/(wt)
,
(8)
from which follows Z’(t) = $ rZ(t) = POcos(w,t,)
I
dw cos(wt) S,,, ,
(9) which is proportional to the Fourier transform of s . Apart from the oscillating factor this result is”iompletely analogous to the standard NSE result. The dependence on the “time parameter” t can be obtained by performing the measurement in several magnetic fields. If the subject of investigation is the magnetic subsystem one has to keep in mind, however, that the spectrum of the magnetic excitations can be strongly affected by changing the magnetic field. Then it should be more convenient to perform the measurements without varying the field (also to avoid the beam depolarization) and to change either the magnetic field region traversed by the neutrons or the neutron wavelength. The latter possibility becomes very attractive if the neutron wavelength can be measured, e.g. by the time-offlight technique. In contrast to the standard NSE method one has to bear in mind that by varying the wavelength A not only the parameter t changes but also q - A -‘O, where 0 is the scattering angle. A constant-q time scan can be achieved by varying simultaneously with t the scattering angle according to 0 - A [6]. We note that because of the rapidly oscillating factor in (8) a sufficiently high wavelength resolution is necessary to observe the NSE effect. If the condition Bh /A < (CC+_&-’is violated, then the effect connected with Z(t) becomes rather small.
B. P. Toperverg
and J. Weniger
I Spin-echo effect in scattering from magnets
If the requirement of quasielasticity w 4 2EO is satisifed then the contribution of nuclear scattering is measured by aligning the field in the scattering plane perpendicular to the incident beam. By repeating the measurement with the field perpendicular to the scattering plane the contribution of the longitudinal spin correlations can be determined. If these results are completed by measurements with other angles between q and b the perpendicular spin correlations can be extracted. We mention here that the situation is much more complicated if the condition of quasielas-
ticity is violated. of q cannot be perimental data comes extremely
665
In this case the o-dependence neglected. An analysis of exwithout model assumptions bedifficult.
References [l] F. Mezei, J. Phys. (Paris) 43 (1982) C7-9. [2] B. Farago and F. Mezei, Physica B 136 (1986) 100. [3] Y. Endoh, Y. Sasaki, H. Ono, S. Mitsuda and S. Ikeda, Physica B 120 (1983) 45. [4] B. Toperverg and J. Weniger, submitted to 2. Phys. B. [5] B. Toperverg and J. Weniger, Z. Phys. B, in press. [6] F. Mezei, Nucl. Instr. Meth. 164 (1979) 153.