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Description of neutron beams in magnetic media
2 August 1999
Physics Letters A 259 Ž1999. 29–37 www.elsevier.nlrlocaterphysleta
Description of neutron beams in magnetic media N.K. Pleshanov Petersburg Nuclear Physics Institute, 188350, Gatchina, St. Petersburg, Russia 1 Received 25 November 1998; received in revised form 10 March 1999; accepted 24 May 1999 Communicated by P.R. Holland
Abstract The quantum aspects of the description of neutron beams in magnetic media are discussed. ‘Neutron polarization’ and ‘neutron beam polarization’ are not equivalent, as the former is related to the probability densities of neutron spin states whereas the latter is related to the respective current densities Žintensities.. The very presence of a magnetic field induces polarization of beam neutrons. As a consequence, neutron beams are polarized even by unpolarized nuclei. A possibility to produce the source of polarized ultracold neutrons is pointed out. q 1999 Elsevier Science B.V. All rights reserved. PACS: 61.12.y q; 61.12.Bt Keywords: Spin; Neutron polarization; Neutron optics; Theory of measurement
1. Introduction The interaction of a neutron with a magnetic field is described by the Schrodinger equation, the exact ¨ solution of which, quite often, cannot be found. Therefore, such standard approaches as the Born approximation and the Larmor precession approximation are used Že.g. w1–7x.. The neutron polarization is treated in the majority of works in the frame of these approaches. The polarization vector is a 3-dimensional vector classically projected onto any axis. So it is visual and convenient, unlike the ‘spin’. The measurement of its components is a purely technical problem solved in the classical 3D polar1
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ization analysis w8,9x. This approach to neutron polarization was enriched with important details in a number of papers w10–25x making use of that quite remarkable fact that the projections of the neutron spin onto the directions ‘up’ and ‘down’ the field are, actually, projections into two optically different media. Some revision of the conventional approach is given in the present paper. In what follows, to facilitate the analysis, we assume that the neutron beam polarization is formed by static fields only, though the arguments persist in a more general case. Consequently, the total energies of monochromatic neutrons in the opposite spin states are equal, whereas their kinetic energies are different. Indeed, when the refraction properties of the medium with an induc-
0375-9601r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 9 9 . 0 0 3 5 5 - 2
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tion B and a nuclear potential Vn are taken into account, the wave vectors for the neutron states with the spin ‘up’ Žq. and ‘down’ Žy. the field turn out to be different in length:
k "s
(
2m "2
E y Ž Vn " < mn B < . .
Ž 1.
Respectively, two velocities should be attributed to a neutron in magnetic field, zq and zy. This difference in velocities is lost in the Larmor precession approximation, as it means transition to the coordinate system moving with the neutron on assumption that zqs zy. This difference in velocities is lost also in the Born approximation Žthe standard Born approximation is meant here; in principle, one may use the Refracted Wave Born approximation, outlined in w25x and described in w26x, implying that different wave vectors for the neutron states with the spin ‘up’ and ‘down’ the field are used.. However, the difference between kq and ky may play an essential role and leads to a number of polarization effects in a homogeneous magnetic field w10–15,19,21x. When the difference in velocities of neutrons with the spin up and down the field is essential, the neutron motion and the spin motion are no longer related as usually: even if the expected spin orientation is known for a neutron at any given point and instant, the spin evolution in magnetic fields cannot be, without a loss in rigor, described by attaching the spin to a neutron moving with a certain velocity. Therefore, the usual description of the neutron motion and the related spin motion, including precession, is not exact, even though the spin behavior is fully defined. Only when this difference is very small, and usually it is, ‘precession in space’ Ža change of the spin orientation from point to point. is ‘precession in time’ for a neutron moving with a certain velocity along a classical trajectory. It is shown in several works Že.g. in Ref. w22x. that the two velocity picture leads then to the classical picture of Larmor precession Žto the first order in D k . whereas the quantum effects come in at higher orders. Still the quantum effects may become essential in polarized neutron experiments. Mention the latest
observations w26–29x in polarized neutron reflectometry in which subtle neutron optical effects of magnetic fields do play significant roles. In addition, even the minutest details of neutron magnetic interaction may be essential for ultra-cold neutrons. A number of quantum aspects of the difference between kq and ky have not been discussed in literature on neutron polarization and are considered below.
2. ‘Neutron polarization’ and ‘neutron beam polarization’ Introduce definitions of the neutron polarization vector Pn '
ž
N≠ y Nx
/ ž ž / eX q
N≠ q Nx
q
X
N≠ y Nx
N≠ y Nx N≠ q Nx
/
eY Y
eZ
N≠ q Nx
Ž 2.
Z
and the neutron beam polarization vector Pb '
ž
I≠ y Ix
/ ž ž /
I≠ q Ix
q
eX q
X
I≠ y Ix I≠ q Ix
eZ .
I≠ y Ix I≠ q Ix
/
eY Y
Ž 3.
Z
Here e X ,e Y ,e Z are three orths along which the vector components are defined, N≠ x and I ≠ x are, respectively, concentrations and intensities of neutrons with the spin ‘up’ Ž≠. and ‘down’ Žx. an axis shown as a subscript. Definitions Ž2. and Ž3. are assumed in literature to be equivalent Žsee e.g. w1–7x.. As we shall see, their difference is small in the majority of cases, and such an approach is very well justified. But not always. Of course, all one ever measures with beams is intensities. It would be of interest, therefore, to find the relation between Pn and Pb . To evaluate the difference between Pn and Pb , we could use the
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well-known solution of the Schrodinger equation for ¨ a neutron in a homogeneous magnetic medium, which is a superposition of two plane waves with amplitudes A ".
Cq Cs Cy
Aq exp Ž ikqr .
ž / ž s
/
Ay exp Ž iky r .
B
, B
< Aq < 2 q < Ay < 2 s 1
Ž 4.
Žhere the quantization axis Z is parallel to B .. It is to be noticed from the very beginning that the presence of precession Ž Pn is inclined to B . or its absence Ž Pn is collinear to B . is not essential for further considerations. However, in the case of precessing polarization beams, one should take into account that the orientations of neutron spins change along the beam line, i.e. along k " Žwe assume that kqI ky .. A fixed beam cross section is assumed to be perpendicular to the precessing polarization beam direction Ž k ". , so that the spins of all neutrons passing through this cross section are oriented in one direction. We do not analyze here the trivial consequences of neutron spin precession when either the orientation or the position of the beam cross section is changed. Consider the beam cross section in which the vectors B, Pn and a new quantization axis C lie in one plane, as shown in Fig. 1 Žthe general case adds nothing but complications in mathematics.. This cross section is given by the condition Ž kqy ky . r s 2p m, m s 0," 1," 2, . . . . In a new basis with the quantization axis C, which subtends an angle g with Z, the wave function is
CC s
Cx
C
g s
s
ž
Aq cos
2
exp Ž ikq r . q Ay sin
g y Aq sin
2
g 2
exp Ž iky r .
exp Ž ikq r . q Ay cos
A ≠ exp Ž i w ≠ . A x exp Ž i w x .
/
, C
where A ≠ s cosŽ br2., A x s sinŽ br2.. By definition, the positive direction for g and b coincides with that from the Z-axis to the X-axis, i.e. the clockwise direction in Fig. 1; b is the angle counted from the C-axis to Pn and g is the angle counted from the Z-axis to the C-axis. Thus, b q g is the angle counted from the Z-axis to Pn and
bqg Aqs cos
2
bqg ,
Ays sin
2
.
Ž 6.
It is to be noted that the four quantities A ≠ , A x , Aq, Ay are real Žpositive or negative. numbers. Once the solution is known, the probability of the spin projection onto C can be found: PnŽC . s
ž
N≠ y Nx N≠ q Nx
/ ž s
C
A2≠ y A2x A2≠ q A2x
/
s cos b , C
Ž 7. and one may define the three components of Pn . Relation Ž7. confirms the known fact that the projection of Pn onto any arbitrary axis is the vector projection. Next we may find the total current density
C≠
ž /
Fig. 1. A beam cross section is chosen, in which the magnetic field B, the neutron polarization vector Pn and an axis C lie in one plane; b and g are the angles between the axis C and, respectively, Pn and B. The intensity Žflux. is defined unambiguously only for neutron states with the spin collinear to B.
g 2
exp Ž iky r .
0
ž ž
j tot ' Re C † P C
Ž 5.
s Re CC† P
" im n " im n
"
/
=C s
=CC
/
mn
Ž Aq2 kqq Ay2 ky . Ž 8.
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and for the selected beam cross section the current densities for neutrons in the states with the spin ‘up’ Ž≠. and ‘down’ Žx. the axis C: "
ž
j ≠ Ž x . ' Re C ≠ Ž x . P " j≠ s
2 Aq cos 2
mn "
im n
g
ž / 2
g
A ≠ Aq cos
ž /
mn
1 q
2
2
g
"
Ž 9.
Aq AyP sin Ž g . P kq
2
ž /
mn s
1 q
2 Ay sin2
q
/
=C ≠ Ž x . ,
Aq AyP sin Ž g . P ky
P kqq Ay sin
2
g
ž / 2
P ky ,
Ž 10 . " jx s
2 Aq sin2
mn " q
mn
g
ž / 2
2 Ay cos 2
mn
2
Aq Ay sin Ž g . P kq
g
ž /
1 y
2
g
" s
1 y
A x yAq sin
ž / 2
2
Aq AyP sin Ž g . P ky
P kqq Ay cos
g
ž / 2
P ky ,
Ž 11 . Accordingly,
P bŽC . s
ž
j≠ y jx j≠ q jx
q cos 2
ž
b 2
/ ž
s cos 2
C
y cos 2
g 2
/
b 2
y sin2
P ky jtot .
g 2
/
P kq
Ž 12 .
Calculations for b s ypr7 and g s pr2 Žthe vector Pn is directed as shown in Fig. 1, whereas the C-axis coincides with the X-axis. yield the X-projection of Pn , PnŽ X . ( 0.90, and for neutrons with wavelengths 0.1 nm, 1 nm, 10 nm and 100 nm in a magnetic field 1T the difference P bŽ X . y PnŽ X . s 1.4 = 10y7 , 1.4 = 10y5 , 1.4 = 10y3 , 0.211, respectively. Note that formula Ž12. for neutrons of wave-
length 100 nm gives P bŽ X . ( 1.11, i.e. the X-projection of Pb exceeds 1. This is possible only if j x - 0. For b s ypr7 and g s pr2 the quantities A ≠ , Aq, Ay) 0 and A x - 0. Therefore, j x - 0 only g g if Ay cos Ž 2 . P ky) Aq sin Ž 2 . P kq. Owing to birefrigence of the magnetic medium for neutrons, k ) kq Žfor neutrons with very small kinetic energies, ky may exceed kq in any number of times. and j x may become negative when refraction is strong enough, as in the case of neutrons with the wavelength 100 nm in the field 1T. However, the reverse flux Ž j x - 0. in a homogeneous medium makes no sense. Moreover, it follows from jtot s j ≠ q j x that j ≠ ) jtot which certainly makes no sense. To understand why the formulas fail when kq/ k I, more detailed analysis should be given Žif kqs kys k, then j ≠ s m"n w Aq cosŽgr2. q Ay sinŽgr2.x 2 P k G 0, j x s m"n w Aq sinŽgr2. y Ay cosŽgr2.x 2 P k G 0, and Pb s Pn .. To begin with, notice that the use of formulas Ž9–12. assumes the current densities Žintensities. of neutrons with the spin both up and down any arbitrary quantization axis to be well-defined quantities given by Eq. Ž9.. However, when the quantization axis is inclined to the field direction, such quantities are not defined, the reason being that the current density operator jˆ and the operator of the spin projection onto a direction inclined to the magnetic field do not commute, i.e. do not have a common set of eigenfunctions. Indeed, note that the stationary Hamiltonian for the neutron of energy E in magnetic media is p2 Hˆ s 2 ˆm n q Vn y mn sˆ B s E Ž pˆ is the neutron mop2 mentum operator.. It follows that 2 ˆm n s E y Vn qmn sˆ B and pˆ s "2 Ž kqq ky . q "2 Ž kqy ky .Ž sˆ Br B .. Consequently, jˆs prm ˆ n and sˆ e j Žthe operator of the spin projection onto a direction e j . commute only if either e j is collinear to B or kqs ky. Conclusion: the possibility of representation of any spin state as a superposition of two orthogonal states with the spin up or down an arbitrary axis does not imply the possibility of unambiguous redistribution of the total flux Žintensity. between these orthogonal states. Thus, even though the portion of neutrons with the spin along an arbitrary axis is known Žfor a given beam cross section, if we speak of a precessing polarization beam., the corresponding flux is not
N.K. PleshanoÕr Physics Letters A 259 (1999) 29–37
always defined. Quantum Mechanics gives no definite answer about the flux Žintensity. of neutrons with the spin inclined to the magnetic field direction. Consequently, formula Ž9. just cannot be used. E.g., the state of neutrons polarized along the field Ž Pn s 1, Pn I B . can be represented as a superposition of states with the opposite spins perpendicular to B, but one cannot give formulas for the corresponding fluxes Žthose obtained from Ž9. do not correspond to any measurement and cannot be used to find the redistribution of fluxes between the states Ž≠. and Žx... The situation can be understood only if we suggest that any attempt to sever neutrons with the spin Ž≠. inclined to the field direction from those with the opposite spin Žx. should introduce perturbations into each of the orthogonal states, i.e., lead to spinflip Žwith respect to the quantization axis inclined to the field direction. processes, Ž≠x. and Žx≠., that do not balance each other. This makes measurements of the respective intensities impossible. Inevitably, neutrons with the right spin will be partly lost and those with the wrong spin will be partly transmitted by the analyzer. The magnitude of the perturbations depends on the method of measurement, but they can never be completely eliminated. Of course, knowing the method of measurement Žthe ‘construction’ of the analyzer., one may theoretically calculate from the measured intensities the probability of the spin projection onto any direction. Yet, rigorously speaking, no measured intensity could be interpreted as the intensity related to a spin eigenstate in representation with the quantization axis inclined to the magnetic field. When all neutron spins in a certain beam cross section are along one direction inclined to the field direction, there is a temptation to relate a priori the total flux to the respective spin eigenstate. However, no experiment would confirm this assumption Žmeasurement of the total intensity without analysis of polarization proves nothing.. Within reasonable assumptions about the measurement conditions Žthe magnitude of the fields, etc.., the perturbations may usually be assumed negligible Ži.e., they can be made very low by the proper choice of the analyzer, though they cannot be made infinitely small.. Only in this case the notion ‘beam polarization’ makes sense. On the other hand, for very cold neutrons in sufficiently strong fields, when the kinetic energy of a neutron is comparable with its
33
Zeeman energy in magnetic field, the perturbations may be always significant. Thus, application of the notion ‘beam polarization’ is restricted. Of course, the perturbations can be eliminated in measurements of intensities related to the states with the spin parallel and antiparallel to the field direction Žin practice, the efficiency of the analyzer never reaches unity, however, this is a purely technical problem. and one may speak about the projection of Pb onto B: PbŽ B . s
ž
Iqy Iy Iqq Iy
/ ž s
B
kq< Aq < 2 y ky < Ay < 2 2
kq < Aq < q ky < Ay <
2
/
B
Ž 13 . Žthe intensities Iq and Iy are for the same beam cross section, so in the right part of the equation they are substituted by the corresponding flux densities.. The other components of Pb are undefined, even when all components of Pn are known. Thus, the absolute quantities in magnetic field are the spin and its average Ž Pn ., the total intensity Žthe total flux density kq< Aq < 2 q ky < Ay < 2 ., the ‘spin-plus’ and ‘spin-minus’ intensities Žthe flux densities kq< Aq < 2 and ky< Ay < 2 , respectively.. It is not the beam polarization, but the spin state of neutrons in the beam Ži.e. the neutron polarization Pn s 2² sˆn : s ² sˆ :. that defines the features of the spin-dependent neutron interaction. Note that the density operator rˆ s 12 Ž1 q Pn P sˆ . is related to the probability densities, whereas the expected intensities Žcross sections, etc.. are related to the current densities. If refraction in magnetic medium cannot be neglected, the density operator apparatus should be reformulated w30x. The nature of the distinction between ‘neutron polarization’ and ‘neutron beam polarization’ is illustrated in Fig. 2. The fact that neutrons are in a beam is of decisive importance. Comparing formulas Ž7. and Ž13., one might conclude that the difference between ‘neutron polarization’ and ‘neutron beam polarization’ may be essential only for very cold and ultracold neutrons. However, the refractive indices may be related not to the total momentum, but to its component, less by orders of magnitude, as the case is in reflectometry, and the difference may become quite significant even for thermal neutrons.
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N.K. PleshanoÕr Physics Letters A 259 (1999) 29–37
Fig. 2. In a volume V with a magnetic field B the number of neutrons with the spin along the field differs from the number of neutrons with the opposite spin Ž Nq / Ny ., because of the difference in their velocities Ž kq / ky ., though the source emits in one second the same number of neutrons with the opposite spins Ž Iq s Iy ..
It is to be emphasized that the analysis given above is not reduced to the well-known statement: no measurement in quantum mechanics can be thought of without specifying the instrument. Even if we can make the instrument perfect and specify it completely, the spin in magnetic field is ‘hidden’ from direct observation in beam experiments, because all one ever measures with beams is a current through the polarizer, rather than probability densities. As a matter of fact, introduction of a perfect instrument is a common method used, often tacitly, in theoretical analysis. Say, the expected intensity for neutrons with the spin up Ždown. the field is defined unambiguously, for one may always propose at least a hypothetical construction of the analyzer that transmits only spin-up Žspin-down. neutrons. The construction of such an analyzer should provide matching of the refractive index of medium in which neutrons move with that of materials of the analyzer. On the other hand, the refractive index of a magnetic medium for neutrons with the spin inclined to the field direction is unambiguous Žbifurcate.. That is one of the reasons why the analyzer transmitting all neutrons in such a spin state and not transmitting neutrons in the opposite spin state is impossible. Since the analyzer transmitting only neutrons with the spin parallel to the field can be produced, we can measure the probability of the respective spin projection. By means of a combination of sudden and slow Žadiabatic. changes in field magnitudes and directions we can turn the spin from any direction in the field B to any direction in the guide field BA near the analyzer. It is usually believed, therefore, that, turning the spins by pr2, one can always determine
the intensities corresponding to the components of Pn perpendicular to B. This suggestion, however, requires more detailed analysis. Assume first that all spins are parallel to B, so that all neutrons reach the detector ŽFig. 3a., the respective intensity being Itot . To measure the perpendicular-to-B component of Pn Žit is equal to 0., turn the spins by pr2 ŽFig. 3b.. Then the spins are perpendicular to BA , and the probability densities of the states with the spins up and down BA are equal, i.e., < Aq < s < Ay <. Consequently, if BA / 0, then kq- ky and jqs kq < Aq < 2 - jys ky< Ay < 2 , i.e., the detected neutron intensity will be lower than Itotr2, in apparent contradiction with the fact that the probability of the spin projection onto BA is 1r2. Even more surprisingly, the detected intensity depends on the magnitude of the guide field. Therefore, the intensities of neutrons with the spin inclined to B are not defined unambiguously, in full agreement with the conclusion drawn above. Only when BA s 0 the intensity is equal to Itotr2 and the use of Eq. Ž3. correctly defines Pn in the field B. Of course, we might suggest that the ‘correct’ measurements are those with zero field in the analyzer region. Yet, this does not generally solve the problem. Indeed, assume that the neutron spins are perpendicular to B. The flux of
Fig. 3. The experimental schemes to measure components of the neutron polarization vector Pn in the field B. The schemes include an analyzer that transmits only neutrons with the spin parallel to the guide field BA in the analyzer region and a spin-turner that may turn the spin from any direction in the field B to any direction in the field BA . See the text for further details.
N.K. PleshanoÕr Physics Letters A 259 (1999) 29–37
neutrons with the spin parallel to B is a well-defined quantity. Using the arguments just given above, we infer that jq- jtotr2. On the other hand, if the spin-turner conserves the spin orientation, i.e., it is the same at its entrance and in the zero field region ŽFig. 3c., the detected intensity will be Itotr2. Therefore, the flux of neutrons with the spin along B I BA ™ 0 is different in the field B and in the region of the analyzer, i.e., the spin-turner causes some spin flipping and perturbations of the respective spin states. Consequently, the ‘correct’ measurement with zero field in the analyzer region fails. Using the spin-turner, one may change the angle j between Pn and BA ŽFig. 3b.. However, contrary to what is usually expected, the detected intensity is not proportional to cos 2 Ž jr2., since Iqs
kqcos 2 Ž jr2 . q
2
y
2
k cos Ž jr2 . q k sin Ž jr2 .
Itot .
Ž 14 .
Of course, this deviation is important only in extreme cases when kq and ky differ noticeably.
elsewhere w31x. Now mention only that the angle j decreases when the magnitude of the field increases, and for small tilts the change of the angle j is
Dj opt (
ž(
nq ny
/
y 1
Ž 16 . where j 0 is the angle between Pn and the field of vanishing magnitude and Pn,opt is the magnitude of the optically induced neutron polarization defined below. More exact formulas may be obtained on the basis of the ‘refracted plane wave’ solution w31x of the Schrodinger equation for slowly changing fields: ¨ Ž 0. Aq
C Ž l. s
ž
Cq Ž l . Cy Ž l .
/
s
(n
q
Ž l.
Ž 0. Ay
(n
y
Ž l.
exp ikq Ž l . l
exp iky Ž l . l
0
,
Ž 17 .
where the constants AŽ0. " are the amplitudes of the plane waves beyond the action of the potential, and l is the coordinate along the neutron trajectory.
3. Optically induced neutron spin tilt Note that it was tacitly assumed that the field between the regions of homogeneous fields B and BA in Fig. 3a changes slowly and neutron spins follow the field direction adiabatically. Such a scheme excludes spin flipping Žalso when BA ™ 0. and is not equivalent to that in Fig. 3c. In contrast to the scheme in Fig. 3c, the flux Žintensity. of neutrons with the spin collinear to the field direction is not changed along the path. If the refractive index changes gradually along the neutron trajectory from n1 to n 2 , then, according to the flux conservation law, the amplitude of a plane wave is changed from A1 to A 2 so that k 1 < A1 < 2 s k 2 < A 2 < 2 , or n1 < A1 < 2 s n 2 < A 2 < 2 .
35
Ž 15 .
Therefore, when the magnitude of the magnetic field slowly changes, the refractive indices Ž n " . and the amplitudes Ž A " . of the ‘spin-plus’ and ‘spinminus’ components change differently and there will be a change in the angle j between Pn and the magnetic field direction Žoptically induced neutron spin tilt.. A detailed analysis of this effect is given
4. Optically induced neutron polarization An unpolarized beam is usually supposed to be such a mixture of states with opposite spins that, independently of the choice of the quantization axis, the intensities Žfluxes. of neutrons with the opposite spins are equal. As it follows from above considerations, only the equality of the intensities of neutrons with the spins up and down the field Ž Iqs Iy . can be postulated for an unpolarized beam in magnetic field. The intensities of neutrons with other spin projections are just not defined. The condition Iqs Iy signifies that the following relation exists between the moduli of the amplitudes of the Žincoherent. ‘spin-plus’ and ‘spin-minus’ states: kq< Aq < 2 s ky< Ay < 2 . As kq/ ky in magnetic field, hence < Aq < / < Ay <. It follows that the unpolarized beam in magnetic field consists of partly polarized Žalong the field. neutrons Žoptically induced neutron polarization.. This paradox is easily understood from Fig. 2: in spite of the equality of Iq and Iy, the concentra-
N.K. PleshanoÕr Physics Letters A 259 (1999) 29–37
36
tion of the ‘spin-plus’ neutrons exceeds that of the ‘spin-minus’ neutrons. If the total energy E of a neutron much exceeds both its nuclear and magnetic potentials, Pn ,opt ( c opt Bl2 ,
It is to be noted that optically induced spin asymmetry of absorption of beam neutrons by unpolarized nuclei in magnetic field can be generalized for arbitrary spin particles in a spin dependent potential.
Ž 18 .
˚ T . If E is compawhere c opt s 0.37 P 10 wA rable with the potentials, the exact formulas should ˚ Vn s 0 and B s 1 T, be used. E.g. for l s 1000 A, one obtains Pn,opt s 0.44. y6
y2
y1 x
5. Optically induced spin asymmetry of absorption It is to be noticed that optically induced neutron polarization cannot be observed directly. This is just an establishment, put in other words, of the difference between the probability densities of the states ‘spin-plus’ Žspin up the field. and ‘spin-minus’ Žspin down the field.. This difference may reveal itself in interaction with scatterers in magnetic medium. For example, neutrons in the state with a greater probability density Žspin up the field. are stronger absorbed by unpolarized nuclei Žoptically induced spin asymmetry of absorption.. A formal explanation is that, though the imaginary part of the optic potential of unpolarized nuclei is independent of the neutron spin state, the imaginary parts of kq and ky are different in the presence of a field, and the transmitted beam is polarized ‘down’ the field. More simple explanation is that it takes more time for slower ‘spin-plus’ neutrons to traverse the target, so they are captured by the nuclei of the target with a greater probability. If E is sufficiently large, the effect is P b ,abs ( ca log 10 Ž f a . Bl2 , y6
Ž 19 . y2 y1 ˚ wA T x, fa is the beam
where ca s 0.84 = 10 attenuation factor, l is the neutron wavelength. If E is comparable with the potentials, the exact formula should be used. For the same beam attenuation, strong and weak absorbers yields effects of the same order of magnitude, e.g. the exact calculation for l s 100 nm and for a beam attenuation in 10 times Ž f a s 0.1. yields, when B s 0.1 T, the beam polarization Pabs s y0.1 ŽTi., y0.43 ŽAl., y0.43 ŽSi., y0.44 ŽCd. and y0.12 ŽGd., and when B s 0.01 T, respectively, Pabs s 0.01 ŽTi., y0.044 ŽAl., y0.044 ŽSi., y0.046 ŽCd. and y0.012 ŽGd..
6. Summary Two basic conclusions: Ž1. ‘neutron polarization’ and ‘neutron beam polarization’ are not equivalent, Ž2. ‘neutron polarization’ is a well-defined quantity, whereas the application of the notion ‘neutron beam polarization’ is restricted. Even if the portion of neutrons with the spin inclined to the magnetic field direction is known, quantum mechanics gives no definite answer about the respective intensity Žintensities measured without analysis of polarization cannot be attributed to certain spin states and are not considered.. It has been concluded that the very presence of magnetic field induces polarization of beam neutrons. As a consequence, neutron beams in magnetic media are polarized even by unpolarized nuclei Žoptically induced spin asymmetry of neutron absorption.. It is to be emphasized that the conclusions drawn and new details in description of neutron beams in magnetic media play role only in extreme cases. The formulas to evaluate the differences between the conventional and introduced approaches have been given. Usually, the differences are not measurable. This should be said clearly not to make the impression that until now everything about polarized neutrons was wrong. The effect of magnetic field on the neutron spin state depends on whether neutrons are in a beam or in a closed volume. Say, there will be no optically induced tilt of the spin of neutrons in a closed volume, because the probability densities of the Žq. and Žy. states do not depend on the magnitude of the field. On the other hand, confined neutrons will be polarized when a weakly-reflecting absorber is introduced into the volume: the frequency of collisions in the presence of magnetic field is different for neutrons in the Žq. and Žy. states, because of the difference in velocities Žif the volume is pumped out, thermalization of neutrons may be neglected.. Also we note the following. If a UCN source Žmoderator. is placed into a magnetic field, the
N.K. PleshanoÕr Physics Letters A 259 (1999) 29–37
kinetic energy of moderated neutrons in the Žq. and Žy. states will be the same Žit depends on the motion of atoms of the moderator.. If the magnetic field is very strong Žsay, 1–2 T., the magnitude of the magnetic potential in the region of the neutron source may exceed the kinetic energy of neutrons and the total energy of the ‘spin-minus’ neutrons is then negative. It means that the ‘spin-minus’ neutrons will be trapped in the strong field region, and the out-coming beam of ultracold neutrons will be polarized. Of course, magnetic fields in the UCN channel should provide adiabatic passage of the ‘spin-plus’ neutrons from the moderator to the region of relatively weak guide fields. Acknowledgements The work was supported by INTAS foundation ŽGrant No. INTAS-97-11329.. Also I would like to thank my colleagues, esp. Roland Gahler, Theo ¨ Rekveldt and Aleksandr Petukhov, for encouraging discussions. References w1x O. Halpern, T. Holstein, Phys. Rev. 59 Ž1941. 960. w2x I.I. Gurevich, L.V. Tarasov, Physics of Low Energy Neutrons, Nauka, Moscow, 1965. w3x F. Mezei, Z. Phys. 255 Ž1972. 146. w4x M.Th. Rekveldt, Z. Phys. 259 Ž1973. 391. w5x S.W. Lovesey, Theory of Neutron Scattering from Condensed Matter, Vols. 1r2, Clarendon Press, Oxford, 1984. w6x W.G. Williams, Polarized Neutrons, Clarendon Press, Oxford, 1988.
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w7x S.V. Maleyev, Preprint PNPI-2110, 1996. w8x A.I. Okorokov, V.V. Runov, V.I. Volkov, A.G. Gukasov, Zh. ETP 69 Ž1975. 590. w9x M.Th. Rekveldt, J. Van Woesik, J. Meijer, Phys. Rev. 181 Ž1977. 920. w10x V.G. Baryshevskii, Pis’ma Zh. Eksp. Teor. Fiz. 33 Ž1981. 78. w11x V.G. Baryshevskii, S.V. Cherepitza, Phys. Lett. A 90 Ž1982. 267. w12x G. Badurek, H. Rauch, J. Summhammer, Phys. Rev. Lett. 51 Ž1983. 1015. w13x M. Buettiker, Phys. Rev. B 27 Ž1983. 6178. w14x A.I. Frank, Voprosy Atomnoy Nauki i Tekhniki, ser.: Obschaya i Yadernaya Fiz. 3 Ž1986. 69; Nucl Instr. Meth. A 284 Ž1989. 161. w15x R. Gahler, R. Golub, Z. Phys. B 65 Ž1987. 269. ¨ w16x F. Mezei, Physica B 151 Ž1988. 74. w17x V.G. Baryshevskii, S.V. Cherepitza, A.I. Frank, Phys. Lett. A 153 Ž1991. 299. w18x R. Golub, R. Gahler, T. Keller, Am. J. Phys. 62 Ž1994. 779. ¨ w19x V. Nunez, N.F. Berk, C.F. Majkrzak, Bull. Am. Phys. Soc. 39 Ž1994. 313. w20x N.K. Pleshanov, Z. Phys. B 94 Ž1994. 233. w21x M. Hino, N. Achiva, S. Tasaki, T. Ebisawa, T. Akiyoshi, Physica B 213r214 Ž1995. 842. w22x F. Mezei, J. Phys. Soc. Jpn. ŽSuppl. A. 65 Ž1996. 25. w23x W. Besenboeck, P. Hank, M. Koeppe, R. Gahler, T. Keller, ¨ R. Golub, J. Phys. Soc. Jpn. ŽSuppl. A. 65 Ž1996. 215. w24x T. Keller, R. Golub, F. Mezei, R. Gahler, Physica B 234–236 ¨ Ž1997. 1126. w25x N.K. Pleshanov, Physica B 234–236 Ž1997. 516. w26x G.P. Felcher, S. Adenwalla, V.O. De Haan, A.A. Van Well, Nature 377 Ž1995. 409. w27x C. Fermon, Physica B 213r214 Ž1995. 910. w28x G.P. Felcher, S. Adenwalla, R.J. Goyette, to be published in J. Appl. Cryst. w29x F. Ott, C. Fermon, Physica B 234–236 Ž1996. 522. w30x N.K. Pleshanov, to be published in Phys. Rev. B. w31x N.K. Pleshanov, submitted to European Phys. J. B ŽRef. No. B8221..
Physics Letters A 259 Ž1999. 29–37 www.elsevier.nlrlocaterphysleta
Description of neutron beams in magnetic media N.K. Pleshanov Petersburg Nuclear Physics Institute, 188350, Gatchina, St. Petersburg, Russia 1 Received 25 November 1998; received in revised form 10 March 1999; accepted 24 May 1999 Communicated by P.R. Holland
Abstract The quantum aspects of the description of neutron beams in magnetic media are discussed. ‘Neutron polarization’ and ‘neutron beam polarization’ are not equivalent, as the former is related to the probability densities of neutron spin states whereas the latter is related to the respective current densities Žintensities.. The very presence of a magnetic field induces polarization of beam neutrons. As a consequence, neutron beams are polarized even by unpolarized nuclei. A possibility to produce the source of polarized ultracold neutrons is pointed out. q 1999 Elsevier Science B.V. All rights reserved. PACS: 61.12.y q; 61.12.Bt Keywords: Spin; Neutron polarization; Neutron optics; Theory of measurement
1. Introduction The interaction of a neutron with a magnetic field is described by the Schrodinger equation, the exact ¨ solution of which, quite often, cannot be found. Therefore, such standard approaches as the Born approximation and the Larmor precession approximation are used Že.g. w1–7x.. The neutron polarization is treated in the majority of works in the frame of these approaches. The polarization vector is a 3-dimensional vector classically projected onto any axis. So it is visual and convenient, unlike the ‘spin’. The measurement of its components is a purely technical problem solved in the classical 3D polar1
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ization analysis w8,9x. This approach to neutron polarization was enriched with important details in a number of papers w10–25x making use of that quite remarkable fact that the projections of the neutron spin onto the directions ‘up’ and ‘down’ the field are, actually, projections into two optically different media. Some revision of the conventional approach is given in the present paper. In what follows, to facilitate the analysis, we assume that the neutron beam polarization is formed by static fields only, though the arguments persist in a more general case. Consequently, the total energies of monochromatic neutrons in the opposite spin states are equal, whereas their kinetic energies are different. Indeed, when the refraction properties of the medium with an induc-
0375-9601r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 9 9 . 0 0 3 5 5 - 2
N.K. PleshanoÕr Physics Letters A 259 (1999) 29–37
30
tion B and a nuclear potential Vn are taken into account, the wave vectors for the neutron states with the spin ‘up’ Žq. and ‘down’ Žy. the field turn out to be different in length:
k "s
(
2m "2
E y Ž Vn " < mn B < . .
Ž 1.
Respectively, two velocities should be attributed to a neutron in magnetic field, zq and zy. This difference in velocities is lost in the Larmor precession approximation, as it means transition to the coordinate system moving with the neutron on assumption that zqs zy. This difference in velocities is lost also in the Born approximation Žthe standard Born approximation is meant here; in principle, one may use the Refracted Wave Born approximation, outlined in w25x and described in w26x, implying that different wave vectors for the neutron states with the spin ‘up’ and ‘down’ the field are used.. However, the difference between kq and ky may play an essential role and leads to a number of polarization effects in a homogeneous magnetic field w10–15,19,21x. When the difference in velocities of neutrons with the spin up and down the field is essential, the neutron motion and the spin motion are no longer related as usually: even if the expected spin orientation is known for a neutron at any given point and instant, the spin evolution in magnetic fields cannot be, without a loss in rigor, described by attaching the spin to a neutron moving with a certain velocity. Therefore, the usual description of the neutron motion and the related spin motion, including precession, is not exact, even though the spin behavior is fully defined. Only when this difference is very small, and usually it is, ‘precession in space’ Ža change of the spin orientation from point to point. is ‘precession in time’ for a neutron moving with a certain velocity along a classical trajectory. It is shown in several works Že.g. in Ref. w22x. that the two velocity picture leads then to the classical picture of Larmor precession Žto the first order in D k . whereas the quantum effects come in at higher orders. Still the quantum effects may become essential in polarized neutron experiments. Mention the latest
observations w26–29x in polarized neutron reflectometry in which subtle neutron optical effects of magnetic fields do play significant roles. In addition, even the minutest details of neutron magnetic interaction may be essential for ultra-cold neutrons. A number of quantum aspects of the difference between kq and ky have not been discussed in literature on neutron polarization and are considered below.
2. ‘Neutron polarization’ and ‘neutron beam polarization’ Introduce definitions of the neutron polarization vector Pn '
ž
N≠ y Nx
/ ž ž / eX q
N≠ q Nx
q
X
N≠ y Nx
N≠ y Nx N≠ q Nx
/
eY Y
eZ
N≠ q Nx
Ž 2.
Z
and the neutron beam polarization vector Pb '
ž
I≠ y Ix
/ ž ž /
I≠ q Ix
q
eX q
X
I≠ y Ix I≠ q Ix
eZ .
I≠ y Ix I≠ q Ix
/
eY Y
Ž 3.
Z
Here e X ,e Y ,e Z are three orths along which the vector components are defined, N≠ x and I ≠ x are, respectively, concentrations and intensities of neutrons with the spin ‘up’ Ž≠. and ‘down’ Žx. an axis shown as a subscript. Definitions Ž2. and Ž3. are assumed in literature to be equivalent Žsee e.g. w1–7x.. As we shall see, their difference is small in the majority of cases, and such an approach is very well justified. But not always. Of course, all one ever measures with beams is intensities. It would be of interest, therefore, to find the relation between Pn and Pb . To evaluate the difference between Pn and Pb , we could use the
N.K. PleshanoÕr Physics Letters A 259 (1999) 29–37
31
well-known solution of the Schrodinger equation for ¨ a neutron in a homogeneous magnetic medium, which is a superposition of two plane waves with amplitudes A ".
Cq Cs Cy
Aq exp Ž ikqr .
ž / ž s
/
Ay exp Ž iky r .
B
, B
< Aq < 2 q < Ay < 2 s 1
Ž 4.
Žhere the quantization axis Z is parallel to B .. It is to be noticed from the very beginning that the presence of precession Ž Pn is inclined to B . or its absence Ž Pn is collinear to B . is not essential for further considerations. However, in the case of precessing polarization beams, one should take into account that the orientations of neutron spins change along the beam line, i.e. along k " Žwe assume that kqI ky .. A fixed beam cross section is assumed to be perpendicular to the precessing polarization beam direction Ž k ". , so that the spins of all neutrons passing through this cross section are oriented in one direction. We do not analyze here the trivial consequences of neutron spin precession when either the orientation or the position of the beam cross section is changed. Consider the beam cross section in which the vectors B, Pn and a new quantization axis C lie in one plane, as shown in Fig. 1 Žthe general case adds nothing but complications in mathematics.. This cross section is given by the condition Ž kqy ky . r s 2p m, m s 0," 1," 2, . . . . In a new basis with the quantization axis C, which subtends an angle g with Z, the wave function is
CC s
Cx
C
g s
s
ž
Aq cos
2
exp Ž ikq r . q Ay sin
g y Aq sin
2
g 2
exp Ž iky r .
exp Ž ikq r . q Ay cos
A ≠ exp Ž i w ≠ . A x exp Ž i w x .
/
, C
where A ≠ s cosŽ br2., A x s sinŽ br2.. By definition, the positive direction for g and b coincides with that from the Z-axis to the X-axis, i.e. the clockwise direction in Fig. 1; b is the angle counted from the C-axis to Pn and g is the angle counted from the Z-axis to the C-axis. Thus, b q g is the angle counted from the Z-axis to Pn and
bqg Aqs cos
2
bqg ,
Ays sin
2
.
Ž 6.
It is to be noted that the four quantities A ≠ , A x , Aq, Ay are real Žpositive or negative. numbers. Once the solution is known, the probability of the spin projection onto C can be found: PnŽC . s
ž
N≠ y Nx N≠ q Nx
/ ž s
C
A2≠ y A2x A2≠ q A2x
/
s cos b , C
Ž 7. and one may define the three components of Pn . Relation Ž7. confirms the known fact that the projection of Pn onto any arbitrary axis is the vector projection. Next we may find the total current density
C≠
ž /
Fig. 1. A beam cross section is chosen, in which the magnetic field B, the neutron polarization vector Pn and an axis C lie in one plane; b and g are the angles between the axis C and, respectively, Pn and B. The intensity Žflux. is defined unambiguously only for neutron states with the spin collinear to B.
g 2
exp Ž iky r .
0
ž ž
j tot ' Re C † P C
Ž 5.
s Re CC† P
" im n " im n
"
/
=C s
=CC
/
mn
Ž Aq2 kqq Ay2 ky . Ž 8.
N.K. PleshanoÕr Physics Letters A 259 (1999) 29–37
32
and for the selected beam cross section the current densities for neutrons in the states with the spin ‘up’ Ž≠. and ‘down’ Žx. the axis C: "
ž
j ≠ Ž x . ' Re C ≠ Ž x . P " j≠ s
2 Aq cos 2
mn "
im n
g
ž / 2
g
A ≠ Aq cos
ž /
mn
1 q
2
2
g
"
Ž 9.
Aq AyP sin Ž g . P kq
2
ž /
mn s
1 q
2 Ay sin2
q
/
=C ≠ Ž x . ,
Aq AyP sin Ž g . P ky
P kqq Ay sin
2
g
ž / 2
P ky ,
Ž 10 . " jx s
2 Aq sin2
mn " q
mn
g
ž / 2
2 Ay cos 2
mn
2
Aq Ay sin Ž g . P kq
g
ž /
1 y
2
g
" s
1 y
A x yAq sin
ž / 2
2
Aq AyP sin Ž g . P ky
P kqq Ay cos
g
ž / 2
P ky ,
Ž 11 . Accordingly,
P bŽC . s
ž
j≠ y jx j≠ q jx
q cos 2
ž
b 2
/ ž
s cos 2
C
y cos 2
g 2
/
b 2
y sin2
P ky jtot .
g 2
/
P kq
Ž 12 .
Calculations for b s ypr7 and g s pr2 Žthe vector Pn is directed as shown in Fig. 1, whereas the C-axis coincides with the X-axis. yield the X-projection of Pn , PnŽ X . ( 0.90, and for neutrons with wavelengths 0.1 nm, 1 nm, 10 nm and 100 nm in a magnetic field 1T the difference P bŽ X . y PnŽ X . s 1.4 = 10y7 , 1.4 = 10y5 , 1.4 = 10y3 , 0.211, respectively. Note that formula Ž12. for neutrons of wave-
length 100 nm gives P bŽ X . ( 1.11, i.e. the X-projection of Pb exceeds 1. This is possible only if j x - 0. For b s ypr7 and g s pr2 the quantities A ≠ , Aq, Ay) 0 and A x - 0. Therefore, j x - 0 only g g if Ay cos Ž 2 . P ky) Aq sin Ž 2 . P kq. Owing to birefrigence of the magnetic medium for neutrons, k ) kq Žfor neutrons with very small kinetic energies, ky may exceed kq in any number of times. and j x may become negative when refraction is strong enough, as in the case of neutrons with the wavelength 100 nm in the field 1T. However, the reverse flux Ž j x - 0. in a homogeneous medium makes no sense. Moreover, it follows from jtot s j ≠ q j x that j ≠ ) jtot which certainly makes no sense. To understand why the formulas fail when kq/ k I, more detailed analysis should be given Žif kqs kys k, then j ≠ s m"n w Aq cosŽgr2. q Ay sinŽgr2.x 2 P k G 0, j x s m"n w Aq sinŽgr2. y Ay cosŽgr2.x 2 P k G 0, and Pb s Pn .. To begin with, notice that the use of formulas Ž9–12. assumes the current densities Žintensities. of neutrons with the spin both up and down any arbitrary quantization axis to be well-defined quantities given by Eq. Ž9.. However, when the quantization axis is inclined to the field direction, such quantities are not defined, the reason being that the current density operator jˆ and the operator of the spin projection onto a direction inclined to the magnetic field do not commute, i.e. do not have a common set of eigenfunctions. Indeed, note that the stationary Hamiltonian for the neutron of energy E in magnetic media is p2 Hˆ s 2 ˆm n q Vn y mn sˆ B s E Ž pˆ is the neutron mop2 mentum operator.. It follows that 2 ˆm n s E y Vn qmn sˆ B and pˆ s "2 Ž kqq ky . q "2 Ž kqy ky .Ž sˆ Br B .. Consequently, jˆs prm ˆ n and sˆ e j Žthe operator of the spin projection onto a direction e j . commute only if either e j is collinear to B or kqs ky. Conclusion: the possibility of representation of any spin state as a superposition of two orthogonal states with the spin up or down an arbitrary axis does not imply the possibility of unambiguous redistribution of the total flux Žintensity. between these orthogonal states. Thus, even though the portion of neutrons with the spin along an arbitrary axis is known Žfor a given beam cross section, if we speak of a precessing polarization beam., the corresponding flux is not
N.K. PleshanoÕr Physics Letters A 259 (1999) 29–37
always defined. Quantum Mechanics gives no definite answer about the flux Žintensity. of neutrons with the spin inclined to the magnetic field direction. Consequently, formula Ž9. just cannot be used. E.g., the state of neutrons polarized along the field Ž Pn s 1, Pn I B . can be represented as a superposition of states with the opposite spins perpendicular to B, but one cannot give formulas for the corresponding fluxes Žthose obtained from Ž9. do not correspond to any measurement and cannot be used to find the redistribution of fluxes between the states Ž≠. and Žx... The situation can be understood only if we suggest that any attempt to sever neutrons with the spin Ž≠. inclined to the field direction from those with the opposite spin Žx. should introduce perturbations into each of the orthogonal states, i.e., lead to spinflip Žwith respect to the quantization axis inclined to the field direction. processes, Ž≠x. and Žx≠., that do not balance each other. This makes measurements of the respective intensities impossible. Inevitably, neutrons with the right spin will be partly lost and those with the wrong spin will be partly transmitted by the analyzer. The magnitude of the perturbations depends on the method of measurement, but they can never be completely eliminated. Of course, knowing the method of measurement Žthe ‘construction’ of the analyzer., one may theoretically calculate from the measured intensities the probability of the spin projection onto any direction. Yet, rigorously speaking, no measured intensity could be interpreted as the intensity related to a spin eigenstate in representation with the quantization axis inclined to the magnetic field. When all neutron spins in a certain beam cross section are along one direction inclined to the field direction, there is a temptation to relate a priori the total flux to the respective spin eigenstate. However, no experiment would confirm this assumption Žmeasurement of the total intensity without analysis of polarization proves nothing.. Within reasonable assumptions about the measurement conditions Žthe magnitude of the fields, etc.., the perturbations may usually be assumed negligible Ži.e., they can be made very low by the proper choice of the analyzer, though they cannot be made infinitely small.. Only in this case the notion ‘beam polarization’ makes sense. On the other hand, for very cold neutrons in sufficiently strong fields, when the kinetic energy of a neutron is comparable with its
33
Zeeman energy in magnetic field, the perturbations may be always significant. Thus, application of the notion ‘beam polarization’ is restricted. Of course, the perturbations can be eliminated in measurements of intensities related to the states with the spin parallel and antiparallel to the field direction Žin practice, the efficiency of the analyzer never reaches unity, however, this is a purely technical problem. and one may speak about the projection of Pb onto B: PbŽ B . s
ž
Iqy Iy Iqq Iy
/ ž s
B
kq< Aq < 2 y ky < Ay < 2 2
kq < Aq < q ky < Ay <
2
/
B
Ž 13 . Žthe intensities Iq and Iy are for the same beam cross section, so in the right part of the equation they are substituted by the corresponding flux densities.. The other components of Pb are undefined, even when all components of Pn are known. Thus, the absolute quantities in magnetic field are the spin and its average Ž Pn ., the total intensity Žthe total flux density kq< Aq < 2 q ky < Ay < 2 ., the ‘spin-plus’ and ‘spin-minus’ intensities Žthe flux densities kq< Aq < 2 and ky< Ay < 2 , respectively.. It is not the beam polarization, but the spin state of neutrons in the beam Ži.e. the neutron polarization Pn s 2² sˆn : s ² sˆ :. that defines the features of the spin-dependent neutron interaction. Note that the density operator rˆ s 12 Ž1 q Pn P sˆ . is related to the probability densities, whereas the expected intensities Žcross sections, etc.. are related to the current densities. If refraction in magnetic medium cannot be neglected, the density operator apparatus should be reformulated w30x. The nature of the distinction between ‘neutron polarization’ and ‘neutron beam polarization’ is illustrated in Fig. 2. The fact that neutrons are in a beam is of decisive importance. Comparing formulas Ž7. and Ž13., one might conclude that the difference between ‘neutron polarization’ and ‘neutron beam polarization’ may be essential only for very cold and ultracold neutrons. However, the refractive indices may be related not to the total momentum, but to its component, less by orders of magnitude, as the case is in reflectometry, and the difference may become quite significant even for thermal neutrons.
34
N.K. PleshanoÕr Physics Letters A 259 (1999) 29–37
Fig. 2. In a volume V with a magnetic field B the number of neutrons with the spin along the field differs from the number of neutrons with the opposite spin Ž Nq / Ny ., because of the difference in their velocities Ž kq / ky ., though the source emits in one second the same number of neutrons with the opposite spins Ž Iq s Iy ..
It is to be emphasized that the analysis given above is not reduced to the well-known statement: no measurement in quantum mechanics can be thought of without specifying the instrument. Even if we can make the instrument perfect and specify it completely, the spin in magnetic field is ‘hidden’ from direct observation in beam experiments, because all one ever measures with beams is a current through the polarizer, rather than probability densities. As a matter of fact, introduction of a perfect instrument is a common method used, often tacitly, in theoretical analysis. Say, the expected intensity for neutrons with the spin up Ždown. the field is defined unambiguously, for one may always propose at least a hypothetical construction of the analyzer that transmits only spin-up Žspin-down. neutrons. The construction of such an analyzer should provide matching of the refractive index of medium in which neutrons move with that of materials of the analyzer. On the other hand, the refractive index of a magnetic medium for neutrons with the spin inclined to the field direction is unambiguous Žbifurcate.. That is one of the reasons why the analyzer transmitting all neutrons in such a spin state and not transmitting neutrons in the opposite spin state is impossible. Since the analyzer transmitting only neutrons with the spin parallel to the field can be produced, we can measure the probability of the respective spin projection. By means of a combination of sudden and slow Žadiabatic. changes in field magnitudes and directions we can turn the spin from any direction in the field B to any direction in the guide field BA near the analyzer. It is usually believed, therefore, that, turning the spins by pr2, one can always determine
the intensities corresponding to the components of Pn perpendicular to B. This suggestion, however, requires more detailed analysis. Assume first that all spins are parallel to B, so that all neutrons reach the detector ŽFig. 3a., the respective intensity being Itot . To measure the perpendicular-to-B component of Pn Žit is equal to 0., turn the spins by pr2 ŽFig. 3b.. Then the spins are perpendicular to BA , and the probability densities of the states with the spins up and down BA are equal, i.e., < Aq < s < Ay <. Consequently, if BA / 0, then kq- ky and jqs kq < Aq < 2 - jys ky< Ay < 2 , i.e., the detected neutron intensity will be lower than Itotr2, in apparent contradiction with the fact that the probability of the spin projection onto BA is 1r2. Even more surprisingly, the detected intensity depends on the magnitude of the guide field. Therefore, the intensities of neutrons with the spin inclined to B are not defined unambiguously, in full agreement with the conclusion drawn above. Only when BA s 0 the intensity is equal to Itotr2 and the use of Eq. Ž3. correctly defines Pn in the field B. Of course, we might suggest that the ‘correct’ measurements are those with zero field in the analyzer region. Yet, this does not generally solve the problem. Indeed, assume that the neutron spins are perpendicular to B. The flux of
Fig. 3. The experimental schemes to measure components of the neutron polarization vector Pn in the field B. The schemes include an analyzer that transmits only neutrons with the spin parallel to the guide field BA in the analyzer region and a spin-turner that may turn the spin from any direction in the field B to any direction in the field BA . See the text for further details.
N.K. PleshanoÕr Physics Letters A 259 (1999) 29–37
neutrons with the spin parallel to B is a well-defined quantity. Using the arguments just given above, we infer that jq- jtotr2. On the other hand, if the spin-turner conserves the spin orientation, i.e., it is the same at its entrance and in the zero field region ŽFig. 3c., the detected intensity will be Itotr2. Therefore, the flux of neutrons with the spin along B I BA ™ 0 is different in the field B and in the region of the analyzer, i.e., the spin-turner causes some spin flipping and perturbations of the respective spin states. Consequently, the ‘correct’ measurement with zero field in the analyzer region fails. Using the spin-turner, one may change the angle j between Pn and BA ŽFig. 3b.. However, contrary to what is usually expected, the detected intensity is not proportional to cos 2 Ž jr2., since Iqs
kqcos 2 Ž jr2 . q
2
y
2
k cos Ž jr2 . q k sin Ž jr2 .
Itot .
Ž 14 .
Of course, this deviation is important only in extreme cases when kq and ky differ noticeably.
elsewhere w31x. Now mention only that the angle j decreases when the magnitude of the field increases, and for small tilts the change of the angle j is
Dj opt (
ž(
nq ny
/
y 1
Ž 16 . where j 0 is the angle between Pn and the field of vanishing magnitude and Pn,opt is the magnitude of the optically induced neutron polarization defined below. More exact formulas may be obtained on the basis of the ‘refracted plane wave’ solution w31x of the Schrodinger equation for slowly changing fields: ¨ Ž 0. Aq
C Ž l. s
ž
Cq Ž l . Cy Ž l .
/
s
(n
q
Ž l.
Ž 0. Ay
(n
y
Ž l.
exp ikq Ž l . l
exp iky Ž l . l
0
,
Ž 17 .
where the constants AŽ0. " are the amplitudes of the plane waves beyond the action of the potential, and l is the coordinate along the neutron trajectory.
3. Optically induced neutron spin tilt Note that it was tacitly assumed that the field between the regions of homogeneous fields B and BA in Fig. 3a changes slowly and neutron spins follow the field direction adiabatically. Such a scheme excludes spin flipping Žalso when BA ™ 0. and is not equivalent to that in Fig. 3c. In contrast to the scheme in Fig. 3c, the flux Žintensity. of neutrons with the spin collinear to the field direction is not changed along the path. If the refractive index changes gradually along the neutron trajectory from n1 to n 2 , then, according to the flux conservation law, the amplitude of a plane wave is changed from A1 to A 2 so that k 1 < A1 < 2 s k 2 < A 2 < 2 , or n1 < A1 < 2 s n 2 < A 2 < 2 .
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Ž 15 .
Therefore, when the magnitude of the magnetic field slowly changes, the refractive indices Ž n " . and the amplitudes Ž A " . of the ‘spin-plus’ and ‘spinminus’ components change differently and there will be a change in the angle j between Pn and the magnetic field direction Žoptically induced neutron spin tilt.. A detailed analysis of this effect is given
4. Optically induced neutron polarization An unpolarized beam is usually supposed to be such a mixture of states with opposite spins that, independently of the choice of the quantization axis, the intensities Žfluxes. of neutrons with the opposite spins are equal. As it follows from above considerations, only the equality of the intensities of neutrons with the spins up and down the field Ž Iqs Iy . can be postulated for an unpolarized beam in magnetic field. The intensities of neutrons with other spin projections are just not defined. The condition Iqs Iy signifies that the following relation exists between the moduli of the amplitudes of the Žincoherent. ‘spin-plus’ and ‘spin-minus’ states: kq< Aq < 2 s ky< Ay < 2 . As kq/ ky in magnetic field, hence < Aq < / < Ay <. It follows that the unpolarized beam in magnetic field consists of partly polarized Žalong the field. neutrons Žoptically induced neutron polarization.. This paradox is easily understood from Fig. 2: in spite of the equality of Iq and Iy, the concentra-
N.K. PleshanoÕr Physics Letters A 259 (1999) 29–37
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tion of the ‘spin-plus’ neutrons exceeds that of the ‘spin-minus’ neutrons. If the total energy E of a neutron much exceeds both its nuclear and magnetic potentials, Pn ,opt ( c opt Bl2 ,
It is to be noted that optically induced spin asymmetry of absorption of beam neutrons by unpolarized nuclei in magnetic field can be generalized for arbitrary spin particles in a spin dependent potential.
Ž 18 .
˚ T . If E is compawhere c opt s 0.37 P 10 wA rable with the potentials, the exact formulas should ˚ Vn s 0 and B s 1 T, be used. E.g. for l s 1000 A, one obtains Pn,opt s 0.44. y6
y2
y1 x
5. Optically induced spin asymmetry of absorption It is to be noticed that optically induced neutron polarization cannot be observed directly. This is just an establishment, put in other words, of the difference between the probability densities of the states ‘spin-plus’ Žspin up the field. and ‘spin-minus’ Žspin down the field.. This difference may reveal itself in interaction with scatterers in magnetic medium. For example, neutrons in the state with a greater probability density Žspin up the field. are stronger absorbed by unpolarized nuclei Žoptically induced spin asymmetry of absorption.. A formal explanation is that, though the imaginary part of the optic potential of unpolarized nuclei is independent of the neutron spin state, the imaginary parts of kq and ky are different in the presence of a field, and the transmitted beam is polarized ‘down’ the field. More simple explanation is that it takes more time for slower ‘spin-plus’ neutrons to traverse the target, so they are captured by the nuclei of the target with a greater probability. If E is sufficiently large, the effect is P b ,abs ( ca log 10 Ž f a . Bl2 , y6
Ž 19 . y2 y1 ˚ wA T x, fa is the beam
where ca s 0.84 = 10 attenuation factor, l is the neutron wavelength. If E is comparable with the potentials, the exact formula should be used. For the same beam attenuation, strong and weak absorbers yields effects of the same order of magnitude, e.g. the exact calculation for l s 100 nm and for a beam attenuation in 10 times Ž f a s 0.1. yields, when B s 0.1 T, the beam polarization Pabs s y0.1 ŽTi., y0.43 ŽAl., y0.43 ŽSi., y0.44 ŽCd. and y0.12 ŽGd., and when B s 0.01 T, respectively, Pabs s 0.01 ŽTi., y0.044 ŽAl., y0.044 ŽSi., y0.046 ŽCd. and y0.012 ŽGd..
6. Summary Two basic conclusions: Ž1. ‘neutron polarization’ and ‘neutron beam polarization’ are not equivalent, Ž2. ‘neutron polarization’ is a well-defined quantity, whereas the application of the notion ‘neutron beam polarization’ is restricted. Even if the portion of neutrons with the spin inclined to the magnetic field direction is known, quantum mechanics gives no definite answer about the respective intensity Žintensities measured without analysis of polarization cannot be attributed to certain spin states and are not considered.. It has been concluded that the very presence of magnetic field induces polarization of beam neutrons. As a consequence, neutron beams in magnetic media are polarized even by unpolarized nuclei Žoptically induced spin asymmetry of neutron absorption.. It is to be emphasized that the conclusions drawn and new details in description of neutron beams in magnetic media play role only in extreme cases. The formulas to evaluate the differences between the conventional and introduced approaches have been given. Usually, the differences are not measurable. This should be said clearly not to make the impression that until now everything about polarized neutrons was wrong. The effect of magnetic field on the neutron spin state depends on whether neutrons are in a beam or in a closed volume. Say, there will be no optically induced tilt of the spin of neutrons in a closed volume, because the probability densities of the Žq. and Žy. states do not depend on the magnitude of the field. On the other hand, confined neutrons will be polarized when a weakly-reflecting absorber is introduced into the volume: the frequency of collisions in the presence of magnetic field is different for neutrons in the Žq. and Žy. states, because of the difference in velocities Žif the volume is pumped out, thermalization of neutrons may be neglected.. Also we note the following. If a UCN source Žmoderator. is placed into a magnetic field, the
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kinetic energy of moderated neutrons in the Žq. and Žy. states will be the same Žit depends on the motion of atoms of the moderator.. If the magnetic field is very strong Žsay, 1–2 T., the magnitude of the magnetic potential in the region of the neutron source may exceed the kinetic energy of neutrons and the total energy of the ‘spin-minus’ neutrons is then negative. It means that the ‘spin-minus’ neutrons will be trapped in the strong field region, and the out-coming beam of ultracold neutrons will be polarized. Of course, magnetic fields in the UCN channel should provide adiabatic passage of the ‘spin-plus’ neutrons from the moderator to the region of relatively weak guide fields. Acknowledgements The work was supported by INTAS foundation ŽGrant No. INTAS-97-11329.. Also I would like to thank my colleagues, esp. Roland Gahler, Theo ¨ Rekveldt and Aleksandr Petukhov, for encouraging discussions. References w1x O. Halpern, T. Holstein, Phys. Rev. 59 Ž1941. 960. w2x I.I. Gurevich, L.V. Tarasov, Physics of Low Energy Neutrons, Nauka, Moscow, 1965. w3x F. Mezei, Z. Phys. 255 Ž1972. 146. w4x M.Th. Rekveldt, Z. Phys. 259 Ž1973. 391. w5x S.W. Lovesey, Theory of Neutron Scattering from Condensed Matter, Vols. 1r2, Clarendon Press, Oxford, 1984. w6x W.G. Williams, Polarized Neutrons, Clarendon Press, Oxford, 1988.
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