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A high-frequency magnetic chopper for polarized neutrons
NUCLEAR INSTRUMENTS
AND METHODS
i4o (I977) 269-274; © N O R T H - H O L L A N D P U B L I S H I N G CO.
A H I G H - F R E Q U E N C Y MAGNETIC C H O P P E R FOR P O L A R I Z E D NEUTRONS KLAUS WEISE, ALBRECHT MEHL and WINFRIED WEIRAUCH
Physikalisch-Technische Bundesanstalt, Postfach 3345, D-3300 Braunschweig, IV. Germany Received 22 September 1976 The described high-frequency chopper for polarized neutrons works with a periodically operating magnetic spin-flip arrangement. It consists of one coil of rectangular cross-section which is supplied with a sinusoidally alternating current. The chopper was investigated theoretically and tested experimentally at a frequency of 95 kHz.
1. Introduction
Experiments with thermal neutrons often demand high resolution time-of-flight measurement. For this purpose a neutron current has to be modulated with a high frequency, preferably of the order 100 kHz. Heinonen et al. 1) achieved this by means of refined mechanical choppers. In addition, different types of magnetic high-frequency choppers have been constructed. Mook et al. 2-4) developed a magnetic chopper which modulates the neutron current by Bragg :reflection in a 7Li-ferrite single crystal. The reflectivity of this crystal depends on its magnetization and can therefore be varied with time. In another group of magnetic choppers a beam of polarized neutrons is used. The beam can be produced, for instance, by Bragg reflection in a magnetized C o - F e single crystal. The polarized neutrons pass through a magnetic device which varies the direction of polarization with time. The neutrons finally strike an analyzer which is only permeable for neutrons with one direction of spin. In consequence the neutron beam leaving it is modulated and polarized. Usually the direction of polarization is altered by means of a resonance spin-flip coil. Here the spins are turned by a magnetic field which oscillates with the Larmor precession frequency of the neutrons in a guiding fieldS). To alter the direction of polarization as a function of time, the system has to be switched into and out of resonance6-16). Mezei 17) used static magnetic fields to change the spin direction. The fields were produced by coils of rectangular cross-section oriented perpendicularly both to a guiding field and to the direction of neutron flight. Badurek et al. 1s, 19) used this method for the construction o f a chopper. Their arrangement consisted of two coils, the fields of which were switched on and
off. It operated quite well up to repetition rates of 100 kHz. In this paper it will be shown that a coil of rectangular cross-section too can be used to chop a polarized neutron beam in a more dynamic manner than by Badurek et al. The change of spin direction in a periodically alternating magnetic field was theoretically investigated in a rather general way. The results of the calculations were examined experimentally for the simplest case. Here the neutrons traverse the coil whose axis is perpendicular to the direction of the initial polarization, as well as to the beam direction (see fig. 1). A configuration with two coils will also be discussed. The mode of operation to be described is only applicable to periodic beam modulation. However, it may be advantageous because of its particular simplicity. 2. Theoretical considerations 2.1. PRINCIPLE OF THE MAGNETIC NEUTRON BEAM MODULATION
A beam of monoenergetic and polarized neutrons 2
3
~
5
Fig. 1. Set-up for polarizing and modulating a neutron beam. (1) Neutron beam entering the apparatus, (2) polarizing and monochromizing Co-Fe single crystal, (3) modulating coil with periodic induction B(t), (4) diaphragm, (5) analyzing Co-Fe single crystal, (6) detector, Bc crystal magnetizing induction, Po and p(t) polarization vectors of the neutron beam in front of and behind the coil, l(t) neutron current of the modulated beam,
270
K L A U S W E I S E et al.
with neutron current Io and polarization vector P0 may be produced, for example, from a primary neutron beam by Bragg reflection in a magnetized C o - F e single crystal. Fig. 1 shows the arrangement suitable for high-frequency magnetic modulation. During the time interval of length z, from t' = t - r to t ' = t, the neutrons pass through a homogeneous magnetic field, the induction B(t') of which varies periodically with time. Due to B(t') the spin state [Z) of a neutron changes according to the Schr6dinger equation
ihdlz)/dt' = - B(t') lalZ) .
(1)
(p =/~na vector operator of the magnetic moment /~, of the neutron, a spin vector operator, the components of which can be represented by the Pauli spin matrices.) The polarization vector
p(t') -- (zl~rlz>,
(2)
too, varies from P0 to p(t) during the passage through the magnetic field. The average in eq. (2) has to be taken over the initial distribution of the neutrons in the beam among the two spin eigenstates. The projection of p in any direction is equal to the difference of the probabilities of finding a neutron with its spin parallel or antiparallel to that direction. From eqs. (1) and (2) the differential equation
dp(t')/dt' = (2/~./h) p(t') × B(t')
(3)
can be derived; this has to be integrated from t' = t - r to t' = t, with p ( t - r ) = P 0 , to get p(t) for the neutrons leaving the magnetic field. With the magnetic induction B(t), the polarization vector p(t) is a periodic function of time with angular frequency ~o. The absolute value Po of p is constant. Hence the tip of the vector p(t) moves along a closed path on a spherical surface. After leaving the magnetic field, the periodically polarized neutron beam is analyzed e.g. by means of Bragg reflection in a second magnetized C o - F e single crystal, which would produce the polarization vector pl (absolute value p~) for a hitherto unpolarized beam. This causes the modulation of the neutron current l(t) of the beam, which, disregarding the reflectivity, has the form
l(t) = ½10[1 + p ( t ) P l ] .
(4)
To verify this equation, let I0~ and I ± be the currents of the neutrons entering and leaving the magnetic field respectively, with their spins parallel ( + ) or antiparallel ( - ) to Pl, Io = I ~ - + I o ; 1 = I + + / ' - , and further w ± = ½(1 ___p~) the probabilities of these neutrons to remain in the beam, so that 1 -+ = w ± l ~ .
With the projection p ' = ( 1 ~ - I o ) / I o o f p in the direction ofp~, the current I = w +I o+ + w - I o =½Io(1 +p'p~) becomes identical to I(t) from eq. (4). The neutron current l(t) is also a periodic function of time with angular frequency 09. 2.2.
REQUIREMENTS FOR EXPERIMENTAL APPLICATIONS
To use magnetic neutron beam modulation in experiments the periodic neutron current I(t) has to fulfil certain requirements: l) To obtain maximum modulation of the neutron current I(t) it is necessary that at certain times p(t) is directed parallel and antiparallel to Pl, so that p(t)pl = +POP1 respectively. Then the modulation M has the value
M = (]-l)/(i+l)
= POP,.
(5)
[ ] , / a r e maximum and minimum values of I(t).] 2) Preferably l(t) should be of simple functional form with symmetrical, congruent, and equally spaced extrema, and should not depend strongly on the physical parameters and conditions, particularly those of the magnetic field. 3) Moreover, only simple and realizable magnetic field configurations can be taken into account. For high-frequency modulation with o3 ~ 105-106 Hz these may be superpositions of fields with constant, sinusoidal or perhaps trapezoidal shape in time, built up easily using long coils with plain parallel surfaces perpendicular to the neutron beam, as shown in fig. 1. This achieves homogeneity, prevents stray fields, and ensures equal transit time r through the magnetic field for all neutrons. Multiple magnetic field configurations are possible. 2.3. SOLUTIONS OF THE MODULATION PROBLEM The differential eq. (3) can be solved easily for a magnetic induction with constant direction. With (21~n/h) B ( t ) = b f ( t ) [b constant unit vector, f ( t ) periodic] the solution is
P(O = b(bpo) + Pox b sin
f(t') dt'+ --r
+ b x (P0 x b) cos
f ( t ' ) dt'.
(6)
-r
For maximum modulation the radius of the circle on which the tip of the vector p(t) moves around b must have its greatest possible value Po, and once during the period p(t) has to point in the direction of p~. It follows that Po and Pl should be orthogonal to b.
HIGH-FREQUENCY
MAGNETIC
Hence
(10)
f(t')
cos
(7)
271
CHOPPER
and
merely having to write
(11),
fll = 4(y.B1/h(0) s(½(0z)
(12)
and to substitute cos ¢ by with sin~ = - ( P o x b) th/PoP~ ; c¢= 4z (Po, Pl). A t certain times the cosine function in eq. (7) has to take the values + 1 and - 1 . Superposition o f a constant magnetic induction Bob and a sinusoidal one, B~b cos((0t+~,0) (~o phase constant), means
j'(t) = (2/tnih) [ B o + B 1 cos((0t+cp)],
(8)
and supplies the result
l(t) = ½Io[I +PoP~ cos ~(t)] ; ~ ( t ) = 3 o + 3 1 cos 0 ( 0 ;
flo
O(t) = (0t+q)-½(0r ;
= ~ + 2 y , rBo/h ; fll = 4(p. Bflh(0) sin(½ (or). (9)
W e a k dependency o f the extremal values o f I(t) on the induction parameter /~1 requires that dcos~b/dB~ = 0, together with dcos~b/dt = 0, for nontrivial fl~ ~ 0, yielding the condition sin~b = 0 at all o f the existing relative extrema, particularly at those two during a period for which d~b/dt = 0. C o n g r u e n c y in the n e i g h b o u r h o o d o f the maxima (minima) demands in addition equal values o f Id 2 cosq~/dt21 =(d~b/dt) 2 ,'at all o f them. T w o solutions exist: 1) If d4)/dt ~ 0 for both cos ~b = + 1 and - 1, then there are no further extrema during a period and it has 'to be flo = ½ ( 2 k + l ) ~ , fll = +½~ for some integer k, hence
I(t) = I ~ ( 0 ) = ½Io[l +PoP~ sin(½n c o s ~ ) ] ;
c(O,(0r) = [ s ( O + ½ ( 0 r ) - s ( ¢ - ½ ( 0 r ) ] / 2 s ( ½ ( 0 r ) , with the triangular function s(0) =
sgn cos 0 ' d 0 ' .
(14)
c(0, cot) depends explicitly on (0z and is a trapezoidal function o f ¢ with the same amplitude and zeros as cos 0. The neutron current functions I~, 4 (0) correspond to the I ~ , 2 ( 0 ) of eqs. (10) and (11). I ~ ( ¢ ) and I 2 ( 0 ) are shown in fig. 2. F o r the special case ( 0 r = ( 2 n + l ) r c (n integer), there are additional solutions because c[0, ( 2 n + 1)r~] is triangular and hence IdZcosd~/dt21-~(d(o/dt) 2 = const, and all o f the minima or maxima are congruent. With the condition sin q5 = 0 at the extrema and sin {½(21+ l) nc[O, ( 2 n + l ) n]} = ( - 1 ) / x × cos [(21 + 1) ¢ ] ;
cos{lnc[O, (2n + 1) hi} = ( - 1 ) t cos(210) ,
(15)
the solutions J,± o f the neutron current are (k, l integer)
I(t) = J~(O) = ½Io (1 +POP1 cos 10) (flo = ½ k n ;
fll = ½In;
k+ieven).
(16)
(10)
2) If, on the other hand, d ~ b / d t = 0 only for cos q5 = + 1 or - 1 two additional extrema exist during a period and it must be flo = k n , fl~ = +_n, yielding
....
I
/ l(t) = I ~ ( 0 ) = ½10[1 ±POP1 cos(n c o s ¢ ) ] .
(13)
~---- x ---.~ .
.
IX
7, MX.', \ \'/, \
(11)
The functions I + ( ~ ) , I + ( ¢ ) , and I~(~k) o f eqs. (10) and (11) are shown in fig. 2. The signs of the second terms depend on k and on the sign o f ill. I ~ ( t ) has angular frequency 2(o. The parameters c~, z, Bo and B~ must be chosen so that with eqs. (9) the conditions for //o and fix in one o f the two cases are met. F o r an abruptly alternating second magnetic induction term in eq. (8) o f the form B~ sgn [cos ((0t +~o)] a similar but more complicated calculation can be carried out. The results are analogous to eqs. (9),
•
~
\.
I a/Z
Fig. 2. The periodic neutron current I as a function of the phase qt for some optimal cases, corresponding to I~+ (i = 1.... ,4), 1~', J~ and J~-, described in section 2.3. x = Is(½cov)l, sou) triangular function defined in eq. (14), r~v phase angle referring to the time the neutrons remain in the periodic magnetic field, and f minimum and maximum values of I respectively.
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K L A U S W E I S E et al.
J~- (~b) and J 2 (~b) are shown in fig. 2. For coz --- (2n + 1) rc they are identical with 13+ (¢) and I~-(if) respectively. l f f o r the time intervals zl and z2 the neutrons remain in two homogeneous magnetic fields with equally and constantly directed inductions (h/21~n)bfl,z(t) [see above eq. (6), f l , 2 ( t ) periodic], and enter the second field at time z o after leaving the first, then to get the solution of the differential eq. (3) the integral in eq. (6) has to be replaced as follows:
f ( t ' ) dt' ---, --~
f~(t') dt' + --~0--~1
+
--r2
f2(t') dt'.
(17)
-- ~2
Here the right and the left sides are identical if the relations T 1 "~-T5 2 =
T, ",
f l ( t - - z o ) = fz(t) = f ( t )
(18)
are fulfilled. This means that the same modulated neutron current I(t) can be obtained by passage of the neutron beam through two or possibly more parallel magnetic fields instead of through a single one, but additional conditions must be satisfied. To find more solutions for the neutron current I(t) suitable for experimental applications computer calculations have been carried out for a series of practicable magnetic field configurations, including crossed constant and sinusoidal fields, and with various sets of parameters. Unfortunately all the resulting functions I(t) have more or less complicated shapes and do not fulfill the requirements of section 2.2. The method proposed by Badurek et al. la' 19) for magnetically chopping a polarized neutron beam, in contrast to the one described here, is as follows: The polarization vector Po of the neutrons orthogonal to the beam direction is reversed by two successive magnetic fields perpendicular to each other and to the beam direction and inclined by _+45° to the polarization vector Po. The fields have equal strengths and are such that VLZ = ½ (rE Larmor precession frequency, z time the neutrons spend in each field). They are built up by superposition of a guiding field parallel to Po and the fields of two coils which are perpendicularly oriented both to the guiding field and to the direction of neutron flight. The coil fields are abruptly switched on and off in order to chop the neutron current. This method is quasistatic; the number of neutrons being within the coils during the field alter-
nation must be negligible, requiring that the switching frequency is not too large.
3. Experimental verification of the modulation principle 3.1. THE APPARATUS The modulation principle discussed in section 2.3 was experimentally tested at the Research and Measuring Reactor, Braunschweig (FMRB). The set-up used is schematically shown in fig. 1. Before entering this arrangement, the neutrons were reflected out of the primary neutron beam by a Cu single crystal. Polarization of the neutrons was achieved by Bragg reflection at the (111) planes of a C o - F e single crystal magnetized to saturation. In order to change the direction of polarization only one coil was used, perpendicularly oriented to the magnetization of the C o - F e crystal and to the direction of neutron flight. The coil had a rectangular cross-section of 20 m m x 60 m m and a length of 200 mm. The latter dimension was sufficiently large to avoid disturbance by the stray field of the coil. The coil bobbin was made of plexiglass and was provided with a window for the neutron beam. The coil consisted of only one layer with 190 windings of copper wire, the inductance being 250/~H. Immediately behind the coil the diameter of the neutron beam was restricted to l0 m m by a diaphragm. The polarization of the neutrons leaving the latter was analyzed by Bragg reflection in a second C o - F e crystal magnetized in a direction antiparallel to the magnetic field of the first one. The neutrons were finally detected by a LiI scintillation crystal which was 2 mm thick. For all measurements neutrons with a wavelength of 0.21 nm were used. The type of modulation characterized by eq. (11) should be tested. Here, inside the modulating coil no magnetic field directed perpendicularly to the coil axis must be present. To meet this condition the two C o - F e crystals were magnetized in opposite directions. Consequently the stray fields of the electromagnets cancelled each other in the space around the coil. This arrangement had the disadvantage that there was no guiding field for the polarized neutrons in the vicinity of the coil. However, the experiments showed that the polarization was not affected too much by this (see fig. 3). 3.2.
MEASUREMENT WITH A CONSTANT MAGNETIC FIELD
In a preliminary experiment the coil was supplied with direct current. The neutron current / was measur-
HIGH-FREQUENCY MAGNETIC CHOPPER ed as a function o f the magnetic induction B within the coil. The result is plotted in fig. 3. The neutron current was corrected for background. F o r this measurement the diaphragm shown in fig. 1 was removed in order to increase the neutron current. Since the two C o - F e crystals were magnetized in opposite directions, I was m i n i m u m for B = 0, whilst it was m a x i m u m for the induction such that the neutron spins were turned by rc on passing t h r o u g h the coil. F r o m the velocity o f the neutrons and the thickness o f the coil it was calculated that the magnetic induction B = 1.62 m T should produce m a x i m u m neutron current. According to fig. 3, the result o f the measurement is compatible with this value. The ratio o f maximum to m i n i m u m n e u t r o n current was Ref = 4.6.
273
The neutron current was measured as a function o f the time difference At between the m o m e n t each neutron was detected and a time corresponding to a fixed phase o f the coil current. F o r this purpose the electric pulses coming f r o m the detector were led to a time-to-pulse height converter which was connected to a multichannel analyzer. The converter was started by the detector pulses and stopped by pulses related to the fixed phase o f the coil current. Fig. 4 shows the measured time dependent neutron current, corrected for background. It was almost sinusoidally modulated with a frequency of 94.6 kHz. This was double the frequency o f the alternating current, as expected f r o m eq. (11). The ratio o f maxim u m to m i n i m u m neutron current was Raf---2.1, corresponding to a modulation M = 0.35.
3.3. MEASUREMENTWITH AN ALTERNATINGMAGNETIC FIELD
3.4. DISCUSSION OF THE EXPERIMENTALRESULTS
In the main experiment the coil was supplied with alternating current. The parameters co, Bo and B 1 were chosen in such a way that eq. (l l) was valid. F r o m the eq. (9) for fll and the condition fll = +re, it follows that the required induction B 1 is m i n i m u m for a given co, if cox = ( 2 n + l ) r c , n being an integer. The case n = 0 only is o f experimental interest because in all others some neutron spins are rotated by an angle greater than re. To satisfy this condition, the frequency v = c o / 2 r c = 4 7 . 3 k H z was used. The corresponding amplitude o f the induction was B~ = 2 . 5 5 m T . There was no constant induction: Bo = 0. This agrees with the eq. (9) for flo and the condition flo = kTr. Because ct = ~ (antiparallel magnetization o f the two C o - F e crystals) this case corresponds to k = 1.
The constant field experiment proved that the coil operated quite well and the polarization o f the neutrons was not affected too much by magnetic stray fields. In the alternating field measurement the ratio Raf o f m a x i m u m to m i n i m u m neutron current was much smaller than the corresponding value Rcf in the constant field measurement. F o r theoretical reasons the two should be equal. This discrepancy is largely due to the low time resolution o f the apparatus, which had not been designed for time-of-flight measurement. There are two reasons: firstly, the C o - F e crystals were used in the transmission mode. Hence, there were flight paths o f different lengths between the modulating coil and the detector. In order to diminish the differences, the diameter of the neutron beam was limited by the diaphragm (see fig. 1). Nevertheless, the O.B S-I
---....IK
0.6
/
x.
\,, /
0.4
2
D
/
0.2
h
0.5
I
I
1.0
1.5
mT
2.0
I
I
5
18
I
ps
15
At
B
Fig. 3. Result of the constant field measurement: Neutron current I as a function of the induction B within the spin rotating coil.
Fig. 4. Result of the alternating field measurement: Neutron current 1 as a function of the time difference At between the moment each neutron was detected and a time corresponding to a fixed phase of the coil current.
274
K L A U S W E I S E et al.
measurement was noticeably affected by this. Secondly, the neutrons had somewhat different velocities owing to the beam divergence and to the rather large mosaic spread of the C o - F e crystals (about 0.3 °) and the Cu crystal, by which the neutrons were reflected out of the primary neutron beam. In the experiment the two C o - F e crystals were magnetized in opposite directions. It was therefore expected that the neutron current should vary according to the function I~- drawn in fig. 2. However, the measured time function (fig. 4) is almost sine shaped. This is also a consequence of the low time resolution of the apparatus. The time resolution was calculated. Using this and the measured ratio Rcf = 4.6, the ratio R'af, which would be in agreement with R~f, was derived. The result is: R'af = 2.9. The measured ratio Raf is even considerably smaller than R'af. This can be attributed to neutrons which were subjected to higher order reflections in the single crystals. Both the constant and the alternating field measurement were affected by this, but as a consequence of the different kind of spin rotation in the two cases the effect was larger in the latter case. In order to estimate this effect the reflectivities of the single crystals for the different orders were calculated using the model of the mosaic crystal. It followed from these and from the energy spectrum of the neutrons in the primary beam, that the measurement was considerably affected by second order reflections, while even higher order reflections were negligible. Consequently the difference between Rat and R'af is presumably due to second order reflections. The applicability of the proposed method has been proved by the described experiments. When a chopper is designed it should be possible to avoid the discussed disturbances. To prevent flight paths of different lengths between the modulating coil and the detector the analyzer crystal should be used in the reflection mode. If this is not possible, the same effect can be attained by obliquely orienting the axis of the coil or the entrance surface of the detector to the direction of the neutron beam. 4. Conclusion
A new method of magnetically chopping a polarized
neutron beam by a coil of rectangular cross-section has been described and theoretically examined. The applicability of the method was tested experimentally at a chopping frequency of 94.6 kHz, using a sinusoidally alternating magnetic field. The frequency is limited by the induction which can be produced in the coil. The authors are greatly indebted to Prof. Dr. H. Maier-Leibnitz for stimulating this investigation. Thanks are due to Mr. S. Fischer for his assistance in performing the numerical calculations and to Mr. A. Lescow, who helped with the measurements. References ~) R. Heinonen, P. Hiism/iki, A. Piirto, H. P6yry a n d A. Tiitta, Proc. Conf. on New methods and techniques in neutron d~ffraction, Report R C N 234 (Petten, 1975) p. 347. 2) H. A. M o n k a n d M. K. Wilkinson, J. Appl. Phys. 39 (1968) 447. 3) H. A. Monk and M. K. Wilkinson, Proc. Conf. on Instrumentation for neutron inelastic scattering research (IAEA, Vienna, 1970) p. 173. 4) H . A . M o n k , F . W . Snodgrass and D . D . Bates, Nucl. Instr. and Moth. 116 (1974) 205. 5) L. W. Alvarez and F. Bloch, Phys. Rev. 57 (1940) 11 I. 6) H. Rauch, J. H a r m s and H. Moldaschl, Proc. Conf. on Neutron inelastic scattering, vol. 2 ([AEA, Vienna, 1968) p. 387. 7) H. Rauch, J. H a r m s and H. Moldaschl, Nucl. Instr. and Meth. 58 (1968) 261. 8) O. Steinsvoll and A. Virjo, Proc. Conf. on Neutron inelastic scattering, vol. 2 (1AEA, Vienna, 1968) p. 395. 9) O. Steinsvoll and A. Virjo, Phys. Lett. 26A (1968) 469. ~o) L. Pill, N. Kroo, F. Szlilvik a n d I. Vizi, Proc. Conf. on Neutron inelastic scattering, vol. 2 ([AEA, Vienna, 1968) p. 407. ~ ) J. G o r d o n , N. Kro6, G. Orbfin, L. Pal, P. Pellionisz, F. Szlilvik and I. Vizi, Phys. Lett. 26A (1968) 122. 12) H. Kendrick, J. S. King, S. A. Werner and A. Arrott, Nucl. Instr. and Meth. 79 (1970) 82. ~3) H. Rauch, Proc. Conf. on Instrumentation Jor neutron inelastic scattering research (IAEA, Vienna, 1970) p. 181. 14) R. L. Forgacs, Rev. Sci. Instr. 42 (1971) 1007. ~s) R. Papp a n d H. Rauch, Nucl. Instr. and Meth. 96 (1971) 29. ~6) H. Freisleben and H. Rauch, Nucl. Instr. and Meth. 98 (1972) 61. ~7) F. Mezei, Z. Physik 255 (1972) 146. is) G. Badurek, G. P. Westphal and P. Ziegler, Nucl. Instr. and Meth. 120 (1974) 351. ~9) G. Badurek and G. P. Westphal, Nucl. Instr. and Meth. 128 (1975) 315.
AND METHODS
i4o (I977) 269-274; © N O R T H - H O L L A N D P U B L I S H I N G CO.
A H I G H - F R E Q U E N C Y MAGNETIC C H O P P E R FOR P O L A R I Z E D NEUTRONS KLAUS WEISE, ALBRECHT MEHL and WINFRIED WEIRAUCH
Physikalisch-Technische Bundesanstalt, Postfach 3345, D-3300 Braunschweig, IV. Germany Received 22 September 1976 The described high-frequency chopper for polarized neutrons works with a periodically operating magnetic spin-flip arrangement. It consists of one coil of rectangular cross-section which is supplied with a sinusoidally alternating current. The chopper was investigated theoretically and tested experimentally at a frequency of 95 kHz.
1. Introduction
Experiments with thermal neutrons often demand high resolution time-of-flight measurement. For this purpose a neutron current has to be modulated with a high frequency, preferably of the order 100 kHz. Heinonen et al. 1) achieved this by means of refined mechanical choppers. In addition, different types of magnetic high-frequency choppers have been constructed. Mook et al. 2-4) developed a magnetic chopper which modulates the neutron current by Bragg :reflection in a 7Li-ferrite single crystal. The reflectivity of this crystal depends on its magnetization and can therefore be varied with time. In another group of magnetic choppers a beam of polarized neutrons is used. The beam can be produced, for instance, by Bragg reflection in a magnetized C o - F e single crystal. The polarized neutrons pass through a magnetic device which varies the direction of polarization with time. The neutrons finally strike an analyzer which is only permeable for neutrons with one direction of spin. In consequence the neutron beam leaving it is modulated and polarized. Usually the direction of polarization is altered by means of a resonance spin-flip coil. Here the spins are turned by a magnetic field which oscillates with the Larmor precession frequency of the neutrons in a guiding fieldS). To alter the direction of polarization as a function of time, the system has to be switched into and out of resonance6-16). Mezei 17) used static magnetic fields to change the spin direction. The fields were produced by coils of rectangular cross-section oriented perpendicularly both to a guiding field and to the direction of neutron flight. Badurek et al. 1s, 19) used this method for the construction o f a chopper. Their arrangement consisted of two coils, the fields of which were switched on and
off. It operated quite well up to repetition rates of 100 kHz. In this paper it will be shown that a coil of rectangular cross-section too can be used to chop a polarized neutron beam in a more dynamic manner than by Badurek et al. The change of spin direction in a periodically alternating magnetic field was theoretically investigated in a rather general way. The results of the calculations were examined experimentally for the simplest case. Here the neutrons traverse the coil whose axis is perpendicular to the direction of the initial polarization, as well as to the beam direction (see fig. 1). A configuration with two coils will also be discussed. The mode of operation to be described is only applicable to periodic beam modulation. However, it may be advantageous because of its particular simplicity. 2. Theoretical considerations 2.1. PRINCIPLE OF THE MAGNETIC NEUTRON BEAM MODULATION
A beam of monoenergetic and polarized neutrons 2
3
~
5
Fig. 1. Set-up for polarizing and modulating a neutron beam. (1) Neutron beam entering the apparatus, (2) polarizing and monochromizing Co-Fe single crystal, (3) modulating coil with periodic induction B(t), (4) diaphragm, (5) analyzing Co-Fe single crystal, (6) detector, Bc crystal magnetizing induction, Po and p(t) polarization vectors of the neutron beam in front of and behind the coil, l(t) neutron current of the modulated beam,
270
K L A U S W E I S E et al.
with neutron current Io and polarization vector P0 may be produced, for example, from a primary neutron beam by Bragg reflection in a magnetized C o - F e single crystal. Fig. 1 shows the arrangement suitable for high-frequency magnetic modulation. During the time interval of length z, from t' = t - r to t ' = t, the neutrons pass through a homogeneous magnetic field, the induction B(t') of which varies periodically with time. Due to B(t') the spin state [Z) of a neutron changes according to the Schr6dinger equation
ihdlz)/dt' = - B(t') lalZ) .
(1)
(p =/~na vector operator of the magnetic moment /~, of the neutron, a spin vector operator, the components of which can be represented by the Pauli spin matrices.) The polarization vector
p(t') -- (zl~rlz>,
(2)
too, varies from P0 to p(t) during the passage through the magnetic field. The average in eq. (2) has to be taken over the initial distribution of the neutrons in the beam among the two spin eigenstates. The projection of p in any direction is equal to the difference of the probabilities of finding a neutron with its spin parallel or antiparallel to that direction. From eqs. (1) and (2) the differential equation
dp(t')/dt' = (2/~./h) p(t') × B(t')
(3)
can be derived; this has to be integrated from t' = t - r to t' = t, with p ( t - r ) = P 0 , to get p(t) for the neutrons leaving the magnetic field. With the magnetic induction B(t), the polarization vector p(t) is a periodic function of time with angular frequency ~o. The absolute value Po of p is constant. Hence the tip of the vector p(t) moves along a closed path on a spherical surface. After leaving the magnetic field, the periodically polarized neutron beam is analyzed e.g. by means of Bragg reflection in a second magnetized C o - F e single crystal, which would produce the polarization vector pl (absolute value p~) for a hitherto unpolarized beam. This causes the modulation of the neutron current l(t) of the beam, which, disregarding the reflectivity, has the form
l(t) = ½10[1 + p ( t ) P l ] .
(4)
To verify this equation, let I0~ and I ± be the currents of the neutrons entering and leaving the magnetic field respectively, with their spins parallel ( + ) or antiparallel ( - ) to Pl, Io = I ~ - + I o ; 1 = I + + / ' - , and further w ± = ½(1 ___p~) the probabilities of these neutrons to remain in the beam, so that 1 -+ = w ± l ~ .
With the projection p ' = ( 1 ~ - I o ) / I o o f p in the direction ofp~, the current I = w +I o+ + w - I o =½Io(1 +p'p~) becomes identical to I(t) from eq. (4). The neutron current l(t) is also a periodic function of time with angular frequency 09. 2.2.
REQUIREMENTS FOR EXPERIMENTAL APPLICATIONS
To use magnetic neutron beam modulation in experiments the periodic neutron current I(t) has to fulfil certain requirements: l) To obtain maximum modulation of the neutron current I(t) it is necessary that at certain times p(t) is directed parallel and antiparallel to Pl, so that p(t)pl = +POP1 respectively. Then the modulation M has the value
M = (]-l)/(i+l)
= POP,.
(5)
[ ] , / a r e maximum and minimum values of I(t).] 2) Preferably l(t) should be of simple functional form with symmetrical, congruent, and equally spaced extrema, and should not depend strongly on the physical parameters and conditions, particularly those of the magnetic field. 3) Moreover, only simple and realizable magnetic field configurations can be taken into account. For high-frequency modulation with o3 ~ 105-106 Hz these may be superpositions of fields with constant, sinusoidal or perhaps trapezoidal shape in time, built up easily using long coils with plain parallel surfaces perpendicular to the neutron beam, as shown in fig. 1. This achieves homogeneity, prevents stray fields, and ensures equal transit time r through the magnetic field for all neutrons. Multiple magnetic field configurations are possible. 2.3. SOLUTIONS OF THE MODULATION PROBLEM The differential eq. (3) can be solved easily for a magnetic induction with constant direction. With (21~n/h) B ( t ) = b f ( t ) [b constant unit vector, f ( t ) periodic] the solution is
P(O = b(bpo) + Pox b sin
f(t') dt'+ --r
+ b x (P0 x b) cos
f ( t ' ) dt'.
(6)
-r
For maximum modulation the radius of the circle on which the tip of the vector p(t) moves around b must have its greatest possible value Po, and once during the period p(t) has to point in the direction of p~. It follows that Po and Pl should be orthogonal to b.
HIGH-FREQUENCY
MAGNETIC
Hence
(10)
f(t')
cos
(7)
271
CHOPPER
and
merely having to write
(11),
fll = 4(y.B1/h(0) s(½(0z)
(12)
and to substitute cos ¢ by with sin~ = - ( P o x b) th/PoP~ ; c¢= 4z (Po, Pl). A t certain times the cosine function in eq. (7) has to take the values + 1 and - 1 . Superposition o f a constant magnetic induction Bob and a sinusoidal one, B~b cos((0t+~,0) (~o phase constant), means
j'(t) = (2/tnih) [ B o + B 1 cos((0t+cp)],
(8)
and supplies the result
l(t) = ½Io[I +PoP~ cos ~(t)] ; ~ ( t ) = 3 o + 3 1 cos 0 ( 0 ;
flo
O(t) = (0t+q)-½(0r ;
= ~ + 2 y , rBo/h ; fll = 4(p. Bflh(0) sin(½ (or). (9)
W e a k dependency o f the extremal values o f I(t) on the induction parameter /~1 requires that dcos~b/dB~ = 0, together with dcos~b/dt = 0, for nontrivial fl~ ~ 0, yielding the condition sin~b = 0 at all o f the existing relative extrema, particularly at those two during a period for which d~b/dt = 0. C o n g r u e n c y in the n e i g h b o u r h o o d o f the maxima (minima) demands in addition equal values o f Id 2 cosq~/dt21 =(d~b/dt) 2 ,'at all o f them. T w o solutions exist: 1) If d4)/dt ~ 0 for both cos ~b = + 1 and - 1, then there are no further extrema during a period and it has 'to be flo = ½ ( 2 k + l ) ~ , fll = +½~ for some integer k, hence
I(t) = I ~ ( 0 ) = ½Io[l +PoP~ sin(½n c o s ~ ) ] ;
c(O,(0r) = [ s ( O + ½ ( 0 r ) - s ( ¢ - ½ ( 0 r ) ] / 2 s ( ½ ( 0 r ) , with the triangular function s(0) =
sgn cos 0 ' d 0 ' .
(14)
c(0, cot) depends explicitly on (0z and is a trapezoidal function o f ¢ with the same amplitude and zeros as cos 0. The neutron current functions I~, 4 (0) correspond to the I ~ , 2 ( 0 ) of eqs. (10) and (11). I ~ ( ¢ ) and I 2 ( 0 ) are shown in fig. 2. F o r the special case ( 0 r = ( 2 n + l ) r c (n integer), there are additional solutions because c[0, ( 2 n + 1)r~] is triangular and hence IdZcosd~/dt21-~(d(o/dt) 2 = const, and all o f the minima or maxima are congruent. With the condition sin q5 = 0 at the extrema and sin {½(21+ l) nc[O, ( 2 n + l ) n]} = ( - 1 ) / x × cos [(21 + 1) ¢ ] ;
cos{lnc[O, (2n + 1) hi} = ( - 1 ) t cos(210) ,
(15)
the solutions J,± o f the neutron current are (k, l integer)
I(t) = J~(O) = ½Io (1 +POP1 cos 10) (flo = ½ k n ;
fll = ½In;
k+ieven).
(16)
(10)
2) If, on the other hand, d ~ b / d t = 0 only for cos q5 = + 1 or - 1 two additional extrema exist during a period and it must be flo = k n , fl~ = +_n, yielding
....
I
/ l(t) = I ~ ( 0 ) = ½10[1 ±POP1 cos(n c o s ¢ ) ] .
(13)
~---- x ---.~ .
.
IX
7, MX.', \ \'/, \
(11)
The functions I + ( ~ ) , I + ( ¢ ) , and I~(~k) o f eqs. (10) and (11) are shown in fig. 2. The signs of the second terms depend on k and on the sign o f ill. I ~ ( t ) has angular frequency 2(o. The parameters c~, z, Bo and B~ must be chosen so that with eqs. (9) the conditions for //o and fix in one o f the two cases are met. F o r an abruptly alternating second magnetic induction term in eq. (8) o f the form B~ sgn [cos ((0t +~o)] a similar but more complicated calculation can be carried out. The results are analogous to eqs. (9),
•
~
\.
I a/Z
Fig. 2. The periodic neutron current I as a function of the phase qt for some optimal cases, corresponding to I~+ (i = 1.... ,4), 1~', J~ and J~-, described in section 2.3. x = Is(½cov)l, sou) triangular function defined in eq. (14), r~v phase angle referring to the time the neutrons remain in the periodic magnetic field, and f minimum and maximum values of I respectively.
272
K L A U S W E I S E et al.
J~- (~b) and J 2 (~b) are shown in fig. 2. For coz --- (2n + 1) rc they are identical with 13+ (¢) and I~-(if) respectively. l f f o r the time intervals zl and z2 the neutrons remain in two homogeneous magnetic fields with equally and constantly directed inductions (h/21~n)bfl,z(t) [see above eq. (6), f l , 2 ( t ) periodic], and enter the second field at time z o after leaving the first, then to get the solution of the differential eq. (3) the integral in eq. (6) has to be replaced as follows:
f ( t ' ) dt' ---, --~
f~(t') dt' + --~0--~1
+
--r2
f2(t') dt'.
(17)
-- ~2
Here the right and the left sides are identical if the relations T 1 "~-T5 2 =
T, ",
f l ( t - - z o ) = fz(t) = f ( t )
(18)
are fulfilled. This means that the same modulated neutron current I(t) can be obtained by passage of the neutron beam through two or possibly more parallel magnetic fields instead of through a single one, but additional conditions must be satisfied. To find more solutions for the neutron current I(t) suitable for experimental applications computer calculations have been carried out for a series of practicable magnetic field configurations, including crossed constant and sinusoidal fields, and with various sets of parameters. Unfortunately all the resulting functions I(t) have more or less complicated shapes and do not fulfill the requirements of section 2.2. The method proposed by Badurek et al. la' 19) for magnetically chopping a polarized neutron beam, in contrast to the one described here, is as follows: The polarization vector Po of the neutrons orthogonal to the beam direction is reversed by two successive magnetic fields perpendicular to each other and to the beam direction and inclined by _+45° to the polarization vector Po. The fields have equal strengths and are such that VLZ = ½ (rE Larmor precession frequency, z time the neutrons spend in each field). They are built up by superposition of a guiding field parallel to Po and the fields of two coils which are perpendicularly oriented both to the guiding field and to the direction of neutron flight. The coil fields are abruptly switched on and off in order to chop the neutron current. This method is quasistatic; the number of neutrons being within the coils during the field alter-
nation must be negligible, requiring that the switching frequency is not too large.
3. Experimental verification of the modulation principle 3.1. THE APPARATUS The modulation principle discussed in section 2.3 was experimentally tested at the Research and Measuring Reactor, Braunschweig (FMRB). The set-up used is schematically shown in fig. 1. Before entering this arrangement, the neutrons were reflected out of the primary neutron beam by a Cu single crystal. Polarization of the neutrons was achieved by Bragg reflection at the (111) planes of a C o - F e single crystal magnetized to saturation. In order to change the direction of polarization only one coil was used, perpendicularly oriented to the magnetization of the C o - F e crystal and to the direction of neutron flight. The coil had a rectangular cross-section of 20 m m x 60 m m and a length of 200 mm. The latter dimension was sufficiently large to avoid disturbance by the stray field of the coil. The coil bobbin was made of plexiglass and was provided with a window for the neutron beam. The coil consisted of only one layer with 190 windings of copper wire, the inductance being 250/~H. Immediately behind the coil the diameter of the neutron beam was restricted to l0 m m by a diaphragm. The polarization of the neutrons leaving the latter was analyzed by Bragg reflection in a second C o - F e crystal magnetized in a direction antiparallel to the magnetic field of the first one. The neutrons were finally detected by a LiI scintillation crystal which was 2 mm thick. For all measurements neutrons with a wavelength of 0.21 nm were used. The type of modulation characterized by eq. (11) should be tested. Here, inside the modulating coil no magnetic field directed perpendicularly to the coil axis must be present. To meet this condition the two C o - F e crystals were magnetized in opposite directions. Consequently the stray fields of the electromagnets cancelled each other in the space around the coil. This arrangement had the disadvantage that there was no guiding field for the polarized neutrons in the vicinity of the coil. However, the experiments showed that the polarization was not affected too much by this (see fig. 3). 3.2.
MEASUREMENT WITH A CONSTANT MAGNETIC FIELD
In a preliminary experiment the coil was supplied with direct current. The neutron current / was measur-
HIGH-FREQUENCY MAGNETIC CHOPPER ed as a function o f the magnetic induction B within the coil. The result is plotted in fig. 3. The neutron current was corrected for background. F o r this measurement the diaphragm shown in fig. 1 was removed in order to increase the neutron current. Since the two C o - F e crystals were magnetized in opposite directions, I was m i n i m u m for B = 0, whilst it was m a x i m u m for the induction such that the neutron spins were turned by rc on passing t h r o u g h the coil. F r o m the velocity o f the neutrons and the thickness o f the coil it was calculated that the magnetic induction B = 1.62 m T should produce m a x i m u m neutron current. According to fig. 3, the result o f the measurement is compatible with this value. The ratio o f maximum to m i n i m u m n e u t r o n current was Ref = 4.6.
273
The neutron current was measured as a function o f the time difference At between the m o m e n t each neutron was detected and a time corresponding to a fixed phase o f the coil current. F o r this purpose the electric pulses coming f r o m the detector were led to a time-to-pulse height converter which was connected to a multichannel analyzer. The converter was started by the detector pulses and stopped by pulses related to the fixed phase o f the coil current. Fig. 4 shows the measured time dependent neutron current, corrected for background. It was almost sinusoidally modulated with a frequency of 94.6 kHz. This was double the frequency o f the alternating current, as expected f r o m eq. (11). The ratio o f maxim u m to m i n i m u m neutron current was Raf---2.1, corresponding to a modulation M = 0.35.
3.3. MEASUREMENTWITH AN ALTERNATINGMAGNETIC FIELD
3.4. DISCUSSION OF THE EXPERIMENTALRESULTS
In the main experiment the coil was supplied with alternating current. The parameters co, Bo and B 1 were chosen in such a way that eq. (l l) was valid. F r o m the eq. (9) for fll and the condition fll = +re, it follows that the required induction B 1 is m i n i m u m for a given co, if cox = ( 2 n + l ) r c , n being an integer. The case n = 0 only is o f experimental interest because in all others some neutron spins are rotated by an angle greater than re. To satisfy this condition, the frequency v = c o / 2 r c = 4 7 . 3 k H z was used. The corresponding amplitude o f the induction was B~ = 2 . 5 5 m T . There was no constant induction: Bo = 0. This agrees with the eq. (9) for flo and the condition flo = kTr. Because ct = ~ (antiparallel magnetization o f the two C o - F e crystals) this case corresponds to k = 1.
The constant field experiment proved that the coil operated quite well and the polarization o f the neutrons was not affected too much by magnetic stray fields. In the alternating field measurement the ratio Raf o f m a x i m u m to m i n i m u m neutron current was much smaller than the corresponding value Rcf in the constant field measurement. F o r theoretical reasons the two should be equal. This discrepancy is largely due to the low time resolution o f the apparatus, which had not been designed for time-of-flight measurement. There are two reasons: firstly, the C o - F e crystals were used in the transmission mode. Hence, there were flight paths o f different lengths between the modulating coil and the detector. In order to diminish the differences, the diameter of the neutron beam was limited by the diaphragm (see fig. 1). Nevertheless, the O.B S-I
---....IK
0.6
/
x.
\,, /
0.4
2
D
/
0.2
h
0.5
I
I
1.0
1.5
mT
2.0
I
I
5
18
I
ps
15
At
B
Fig. 3. Result of the constant field measurement: Neutron current I as a function of the induction B within the spin rotating coil.
Fig. 4. Result of the alternating field measurement: Neutron current 1 as a function of the time difference At between the moment each neutron was detected and a time corresponding to a fixed phase of the coil current.
274
K L A U S W E I S E et al.
measurement was noticeably affected by this. Secondly, the neutrons had somewhat different velocities owing to the beam divergence and to the rather large mosaic spread of the C o - F e crystals (about 0.3 °) and the Cu crystal, by which the neutrons were reflected out of the primary neutron beam. In the experiment the two C o - F e crystals were magnetized in opposite directions. It was therefore expected that the neutron current should vary according to the function I~- drawn in fig. 2. However, the measured time function (fig. 4) is almost sine shaped. This is also a consequence of the low time resolution of the apparatus. The time resolution was calculated. Using this and the measured ratio Rcf = 4.6, the ratio R'af, which would be in agreement with R~f, was derived. The result is: R'af = 2.9. The measured ratio Raf is even considerably smaller than R'af. This can be attributed to neutrons which were subjected to higher order reflections in the single crystals. Both the constant and the alternating field measurement were affected by this, but as a consequence of the different kind of spin rotation in the two cases the effect was larger in the latter case. In order to estimate this effect the reflectivities of the single crystals for the different orders were calculated using the model of the mosaic crystal. It followed from these and from the energy spectrum of the neutrons in the primary beam, that the measurement was considerably affected by second order reflections, while even higher order reflections were negligible. Consequently the difference between Rat and R'af is presumably due to second order reflections. The applicability of the proposed method has been proved by the described experiments. When a chopper is designed it should be possible to avoid the discussed disturbances. To prevent flight paths of different lengths between the modulating coil and the detector the analyzer crystal should be used in the reflection mode. If this is not possible, the same effect can be attained by obliquely orienting the axis of the coil or the entrance surface of the detector to the direction of the neutron beam. 4. Conclusion
A new method of magnetically chopping a polarized
neutron beam by a coil of rectangular cross-section has been described and theoretically examined. The applicability of the method was tested experimentally at a chopping frequency of 94.6 kHz, using a sinusoidally alternating magnetic field. The frequency is limited by the induction which can be produced in the coil. The authors are greatly indebted to Prof. Dr. H. Maier-Leibnitz for stimulating this investigation. Thanks are due to Mr. S. Fischer for his assistance in performing the numerical calculations and to Mr. A. Lescow, who helped with the measurements. References ~) R. Heinonen, P. Hiism/iki, A. Piirto, H. P6yry a n d A. Tiitta, Proc. Conf. on New methods and techniques in neutron d~ffraction, Report R C N 234 (Petten, 1975) p. 347. 2) H. A. M o n k a n d M. K. Wilkinson, J. Appl. Phys. 39 (1968) 447. 3) H. A. Monk and M. K. Wilkinson, Proc. Conf. on Instrumentation for neutron inelastic scattering research (IAEA, Vienna, 1970) p. 173. 4) H . A . M o n k , F . W . Snodgrass and D . D . Bates, Nucl. Instr. and Moth. 116 (1974) 205. 5) L. W. Alvarez and F. Bloch, Phys. Rev. 57 (1940) 11 I. 6) H. Rauch, J. H a r m s and H. Moldaschl, Proc. Conf. on Neutron inelastic scattering, vol. 2 ([AEA, Vienna, 1968) p. 387. 7) H. Rauch, J. H a r m s and H. Moldaschl, Nucl. Instr. and Meth. 58 (1968) 261. 8) O. Steinsvoll and A. Virjo, Proc. Conf. on Neutron inelastic scattering, vol. 2 (1AEA, Vienna, 1968) p. 395. 9) O. Steinsvoll and A. Virjo, Phys. Lett. 26A (1968) 469. ~o) L. Pill, N. Kroo, F. Szlilvik a n d I. Vizi, Proc. Conf. on Neutron inelastic scattering, vol. 2 ([AEA, Vienna, 1968) p. 407. ~ ) J. G o r d o n , N. Kro6, G. Orbfin, L. Pal, P. Pellionisz, F. Szlilvik and I. Vizi, Phys. Lett. 26A (1968) 122. 12) H. Kendrick, J. S. King, S. A. Werner and A. Arrott, Nucl. Instr. and Meth. 79 (1970) 82. ~3) H. Rauch, Proc. Conf. on Instrumentation Jor neutron inelastic scattering research (IAEA, Vienna, 1970) p. 181. 14) R. L. Forgacs, Rev. Sci. Instr. 42 (1971) 1007. ~s) R. Papp a n d H. Rauch, Nucl. Instr. and Meth. 96 (1971) 29. ~6) H. Freisleben and H. Rauch, Nucl. Instr. and Meth. 98 (1972) 61. ~7) F. Mezei, Z. Physik 255 (1972) 146. is) G. Badurek, G. P. Westphal and P. Ziegler, Nucl. Instr. and Meth. 120 (1974) 351. ~9) G. Badurek and G. P. Westphal, Nucl. Instr. and Meth. 128 (1975) 315.