Quantum aspects of neutron spin behavior in homogeneous magnetic field

Physica B 304 (2001) 193–213

Quantum aspects of neutron spin behavior in homogeneous magnetic field N.K. Pleshanov* Petersburg Nuclear Physics Institute (PNPI), 188350, Gatchina, St. Petersburg, Russia Received 5 April 2000; received in revised form 21 November 2000

Abstract When speaking about quantum aspects of the spin behavior, one usually keeps in mind the peculiarities of the spin projections. Quantum aspects of different nature, related to the spin-dependent interaction of neutrons with magnetic fields, are considered. As the origin of the neutron spin precession is the birefregence of magnetic media, the precession phase planes are, generally, not perpendicular to the neutron momenta (non-frontal precession). The spin-dependent inelastic interaction of a particle with magnetic fields may result in the nutation of its spin in a homogeneous static field, the origin of such behavior of the spin being purely quantum mechanical. The conventional approach to the neutron polarization is shown to ensue from the exact solutions of the Schro. dinger equation. Quantum aspects in description of the neutron spin behavior play a role only in extreme cases. Quantum behavior not described in the Larmor precession picture may come into play for low energy (ultracold) neutrons in sufficiently strong fields. When not the total momentum but its component less by orders of magnitude is effective, as is the case in reflectometry, the quantum aspects may become essential even for thermal neutrons. They should be taken into account in the interpretation of some polarized neutron experiments. The superposition of two fermion states, the phase difference between which changes with a frequency o, is proved to be equivalent to a state with the spin rotating about an axis with a period 2p=o, this rotation being uniform only when the two states are orthogonal. # 2001 Elsevier Science B.V. All rights reserved. PACS: 03.75.Be; 03.75.Dg Keywords: Neutron spin; Precession; Nutation; Density waves

1. Introduction The behavior of the neutron polarization P is often analyzed with the Bloch equation dP ¼ g½PB: dt *Tel.: +7-81271-46973; fax: +7-81271-39053. E-mail address: [email protected] (N.K. Pleshanov).

According to this equation, the spin inclined to a homogeneous field B precesses about it with the classical Larmor frequency (mn is the neutron magnetic moment) oL ¼ jgBj ¼ 2jmn Bj=":

ð1Þ

ð2Þ

According to the exact QM approach to the neutron polarization (e.g., see Refs. [1–3]), the origin of the spin precessions is a change in the phase difference between the states with the

0921-4526/01/$ - see front matter # 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 1 ) 0 0 4 9 7 - 5

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spin up (+) and down () the field B. In an RF flipper the two states may acquire different energies. When the neutron kinetic energy considerably exceeds its magnetic Zeeman energy, the QM approach leads [4,5] to the classical Larmor precession picture. A more general case, when the neutron kinetic energy is arbitrary and two states with different energies are coherent states with the spins non-collinear not only to the static field direction but also to each other, is considered in the present paper. It will be shown that quantum aspects of the spin behavior not described by the Bloch equation may become essential for ultra-cold neutrons. Moreover, when not the total momentum but its component less by orders of magnitude is effective, as the case is in reflectometry, the quantum aspects may come into play even for thermal neutrons. The details of the neutron interaction with magnetic fields should exactly be followed in the interpretation of some polarized neutron experiments. Quantum peculiarities of the spin projections define only the character of observation of the spin behavior. The quantum aspects considered in this paper are related to the spin-dependent (elastic and inelastic) interaction of neutrons with magnetic fields. The consequences of the difference between ‘‘neutron polarization’’ and ‘‘neutron beam polarization’’ (which may be important when spin-dependent refraction is essential) having been analyzed earlier [6], we shall deal only with ‘‘neutron polarization’’. Neither shall we consider the consequences of averaging the neutron polarization over neutron wavelengths and trajectories. This averaging is conveniently performed by treating the propagation of an initial (at t0 ) neutron wave packet Z   ð3Þ c ¼ dkAðkÞexp ½i kr  EðkÞt0 ="  through the apparatus. Analyzing the propagation of a wave packet through the apparatus, we can study the neutron polarization behavior. Any wave packet can be represented as a superposition of plane waves. The superposition principle of quantum mechanics requires that the wave-packet and the plane-wave descriptions

be, in principle, equivalent. Therefore, we shall assume that the initial state of neutrons is a state with a sharp energy and momentum. It gets rid us of the necessity to take the shape of the neutron wave packet into account. Assuming, in addition, that the initial beam is completely polarized, we may equally speak about behavior of the neutron polarization and the neutron spin. The plane-wave description is widely used in papers dealing with neutron beams and usually simpler for analysis. The Bloch equation (1) is formulated in the neutron rest frame, i.e. also for neutrons in a sharp energy and momentum state. Apart from the peculiarities of the spin projections, the spin obeying the Bloch equation behaves quite classically. The spin inclined to a homogeneous magentic field B precesses with the Larmor frequency. In view of the correspondence principle, this result appears to be quite reliable and even obvious. Yet, one question still remains: does the same result ensue from the exact solutions of the Schro. dinger equation? The incident-plane-wave solutions C of the Schro. dinger equation may play a heuristic role. The assumptions about perfect monochromatization and polarization of the initial beam, perfect field homogeneity and perfect resonance parameters of the RF flipper may be relaxed without losing conclusions obtained from the analysis of these solutions. The quantity jCðr; tÞj2 is proportional to the probability to find a neutron in a volume dx dy dz at a point r in the time interval (t, t þ dt). The neutron spin orientation at this point and time is fully defined by Cðr; tÞ. The spin behavior in space and time is fully defined, if we know two laws (1) the law of the spin motion in a given point r0 (of course, when speaking about the spin motion in a point, we do not mean that the neutron rests in this point, we mean only that the spin orientation depends on the time when the neutron happened to be at this point); (2) the law of the spin evolution, when, at any given time, we start moving with a certain velocity V from this point r0 along any

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direction. As we shall use plane waves solutions, it will be sufficient to know the spin evolution when we move in the direction of the neutron wave vectors. Knowing the two laws, one may find the spin orientation at any position and time. Of course, it would be natural to move with the neutron velocity. However, the neutron state may be a superposition of two or more states with different velocities. Therefore, we shall not try to find a neutron rest frame, but look for a reference frame, in which the spin motion description is simpler. Of course, the picture as a whole does not depend on the choice of the reference frame, because in any frame the wavefunction in a given point and instant is the same. In this sense, all reference frames are equivalent. The existence of equivalent interpretations of the neutron spin evolution agrees with the probabilistic nature of measurements in quantum mechanics. In a homogeneous magnetic field the spin evolution for neutrons in a sharp energy and momentum state is usually described as the Larmor precession. It will be shown that this simple picture may fail when the neutron state is a superposition of two or more states with different and sufficiently low energies. Particularly, quantum nutations of the neutron spin come then into play for low energy neutrons (see also Ref. [7]). In textbooks on classical mechanics we find that nutation requires at least that the symmetry axis of a gyroscope be inclined to the axis of rotation of the gyroscope. For a 1/2 spin particle, such an asymmetry cannot exist, higher moments would be necessary. However, the possibility to prepare the spin particle in a state, which is a superposition of two or more states with different energies (in contrast to the classical particle the energy of which is definite in any time and position), yields a purely quantum condition for the nutation behavior of the spin in quantum mechanics. A detailed analysis is given below. Now we only mention that quantum nutations of the spin vanish in the classical limit (l ! 0), as one could expect.

Fig. 1. The neutron beam is polarized with the polarizer. The spins of neutrons leaving the precession coil are inclined to the guide field. After the RF flipper, the energies of neutrons in the states with the spin up and down the static field (Eþ and E ) are different. The behavior of the neutron spin in the homogeneous field (B) region is analyzed. The analyzer and the detector form the registering system.

2. Neutron spin precession For the setup given in Fig. 1 the solution in the region of a homogeneous field B is ! Aþ exp ðikþ r  itEþ ="Þ Cðr; tÞ ¼ ; ð4Þ A exp ðik r  itE ="Þ B     x d0 Aþ ¼ cos exp i ; 2 2     x d0 exp i ; ð5Þ A ¼ sin 2 2 where the subscript B shows that a representation with the quantization axis ZkB is used, the angles x and d0 (Fig. 2) define the neutron spin orientation in r ¼ 0 at t ¼ 0. The states with the spin parallel (+) and antiparallel () to B are the states with different neutron energies E and momenta rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2mn k ¼ ½E  ðUn jmn BjÞ ð6Þ "2 (Un is the nuclear potential). Consequently, the velocities of neutrons in these states are v ¼ ð"k =mn Þ:

ð7Þ

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[8,9]). The use of the velocity V ¼ vav extends the applicability of the Larmor precession picture to neutrons with low energies. Usually, the velocity V ¼ v0 , where (E0 is the neutron total energy before the RF flipper) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v0 ¼ 2E0 =mn ð13Þ

Fig. 2. The behavior of the neutron polarization P in a magnetic field B is described in a reference frame X; Y; Z with the quantization axis ZkB moving along the neutron beam with a velocity V. Generally, the quantization axis may be inclined to B (e.g., Z0 ).

The phase difference between the two spin states is dðr; tÞ ¼ ðkþ  k Þr  ðEþ  E Þt=" þ d0 :

ð8Þ

In a reference frame moving with a velocity V along the neutron path, we find

 dðtÞ ¼ ðkþ  k ÞV  ðEþ  E Þ=" t þ d0 ¼ o p t þ d0 ;

ð9Þ

where op is the frequency of the resultant spin precession about B. The choice of V determines the magnitude of op . Particularly, when Eþ ¼ E ¼ E, Eq. (4) may be written as !   Aþ exp ðitop =2Þ "vav E exp it V CðtÞ ¼ ; " mn A exp ðitop =2Þ B

ð10Þ where op ¼ ðV=vav Þ oL . Substituting V ¼ vav into Eq. (9), where vav ¼ ðvþ þ v Þ=2

ð11Þ

and, by definition, E ¼ mn v2 =2 þ Un jmn Bj, k ¼ mn v =", we find that dðtÞ ¼ 2jmn Bjt=" þ d0 ¼ oL t þ d0 ;

ð12Þ

i.e. op coincides with the classical Larmor frequency oL (the negative sign is due to the fact that mn 50). Therefore, we obtain the exact Larmor precession behavior of the spin for arbitrary values of Eþ and E (we do not consider here the case when at least one of the spin components propagates in regime of tunneling

is substituted (e.g., see Refs. [2–5,10–12]) into Eq. (9), rather than vav or, say, v . Designate the respective precession frequency as o0 . If neutrons are in magnetic vacuum (Un ¼ 0) and jmn Bj; jmn H0ðRFÞ j5E0 , the relative deviation of o0 from oL is very small !3 jmn jðB  H0ðRFÞ Þ oL  o0 ffi ð14Þ 16E0 oL (it was assumed that E ¼ E0 j mn H0ðRFÞ j, where H0ðRFÞ is the static field of the RF flipper). When Un 6¼ 0, the deviation is more conspicuous oL  o0 Un ffi : ð15Þ oL 2E0 It manifests itself as an additional precession in a non-magnetic matter (‘Neutron Optical Spin Rotation’ [10,11]) and was observed even with thermal neutrons traversing silicon plates [12]. The interpretation of the experiment [12] changes, if we take V ¼ vav . The precession frequency is always equal to oL (NOSR effect disappears), but it takes more time for neutrons to cross the silicon plate (Un > 0, vav 5v0 ) and the spins acquire an additional precession angle. This interpretation seems to be more appropriate. Indeed, the velocity v0 does not even lie in the interval between the values of vþ and v inside the silicon plate. When ignoring the refractive (mechanical) effect of the magnetic field, it would be more appropriate to use the velocity pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vn ¼ 2ðE0  Un Þ=mn ¼ v0 1  Un =E0 ð16Þ which, for the case under consideration, is undistinguishable from vav and, therefore, leads to the interpretation with op ¼ oL . The Bloch equation also yields precession with the Larmor precession frequency oL . It can be derived from the Schro. dinger equation on the assumption that, when the potential changes, the total energy of the neutron changes so that

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its velocity v0 remains constant (see further details in Ref. [13]). When the fields are static, these energy transfers contradict the nature of interaction and are not observed experimentally. Neither is the NOSR effect (V ¼ v0 ) taken into account by the Bloch equation. However, the deviation of the precession frequency from the Larmor frequency is usually so negligible that one may reliably use the Bloch equation. It should also be said clearly that the choice of V among vþ , v , vav , vn , v0 is then of purely academic interest. On the other hand, one might think that the differences between the velocities vþ , v , vav , vn , v0 are important only for neutrons with very low energies (ultracold neutrons). However, when neutrons traverse a boundary between media, only the velocity component normal to the boundary is effective. For very small glancing angles the differences between the normal components of the respective velocities (v?þ , v? , v?av , v?n , v?0 ) may be essential even for thermal neutrons. Indeed, consider the experiment [14] on measurement of the traversal time of neutrons through a magnetic film using the spin precession. The best fit for extra precessions in the film was obtained for v?n , whereas the predictions based on the use of v?0 , v?þ and v? led to noticeable deviations from the experiment. It was concluded that ‘‘the observed shifts of the NSE signals conform well to classical Larmor precession passing through a refracted path due only to the nuclear potential in a magnetic film’’ [14]. No explanation was given as to why the precession angle acquired in the film is determined by the spin-independent nuclear potential. Of course, as it should be, the precession angle is determined by the magnetic interaction. Due to different refraction in the magnetic film, the trajectories and the traversal times are different for the neutrons in states with the spins up and down the induction vector in the film. As it follows from the considerations given above, the precession angle the neutron spin acquires inside the film can be obtained on the assumption that the precession goes on with the classical Larmor frequency and the neutron moves across the film with the velocity vav? ¼ ðv?þ þ v? Þ=2, where v?þ and v? are the surface normal velocity components for the opposite spin states inside the

197

film (see Section 3). It may be concluded that the transverse NSE technique is sensitive to v?av . The NSE signal shift is related not to a traversal time, which is different for the spin-up (tþ ) and spindown (t ) neutrons, but to a quantity t ¼ ð2tþ t Þ=ðtþ þ t Þ. Of course, knowing t and the film parameters, one can find tþ and t . E.g., taking the parameters of the iron film given in Ref. ( neutrons [14] and the glancing angle 0.78, for 5.8 A we obtain that tþ =t ¼ 1:5, t =t ¼ 0:75. This means that the spin-down neutrons traverse the iron film two times quicker than the spin-up neutrons. The reason for the agreement of the experiment with the interpretation given in Ref. [14] is simple: v?n is very close to v?av . Yet, it should be mentioned that the use of v?n , instead of v?av , leads in the example given above to a 5% underestimation of t, since v?av =v?n ¼ 0:95. The discrepancy is essentially less for larger glancing angles, but the glancing angles smaller than 0.78 were also used in the experiment.

3. Non-frontal neutron spin precession Thus, the origin of the neutron spin precession is birefregence of magnetic media. In the reference frame moving with the velocity ð*þ þ * Þ=2 the spin was shown to precess exactly with the Larmor frequency. A neutron state which is a superposition of states with velocities differing not only in magnitude but also in direction may be prepared owing to the spin-dependent refraction at the boundaries of magnetic media (see below). When the velocities *þ and * differ in direction, the precession phase planes are no longer perpendicular to the neutron velocities. The precession front is then perpendicular to u ¼ ð*  *þ Þ=j*  *þ j. In the reference frame moving along u with the velocity V ¼ ð*þ þ * Þ=2u the spin precesses exactly with the Larmor frequency. It can be proved by substitution of r ¼ t ð*þ þ * Þ=2 into Eq. (8). E.g., a precession coil tilted with respect to the beam [15] induces such a nonfrontal precession (Fig. 3). The relation of this phenomenon to refraction at the boundaries between the magnetic media inside and outside the coil was pointed out in Ref. [16]. In addition,

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Fig. 3. The initial polarization P 0 is parallel to the guide field H. When the precession coil is tilted with respect to the mean neutron path, in the region behind the coil the beam cross sections with constant direction of the precessing polarization vector P are not perpendicular to neutron trajectories.

Fig. 4. Reflection and transmission of neutrons incident with the spin parallel (+) to an external field H. If the magnetic induction vector B film in the film is not collinear to H, there will be both ‘‘spin-plus’’ (spin up the field) and ‘‘spin-minus’’ (spin down the field) neutrons inside and outside the film. To simplify the scheme, the reflection from the second boundary of the film is assumed to be negligible.

non-frontal spin precession was shown [13] to be intrinsic to magnetic scattering due to refraction under SF scattering in a non-zero mean field (refraction under specular reflection [17,8] was recently demonstrated [18]). It has earlier been concluded [13] that observations of the angular splitting and the non-frontal precession related to that splitting could not be made simultaneously. The precession front may be almost parallel to the neutron velocities. To illustrate this, consider specular reflection of spin-up neutrons at a film magnetized in a direction inclined to the guide field H (Fig. 4). According to the optics laws, the parallel-to-the-surface components of the wave

vectors outside and inside the film are equal to the parallel-to-the-surface component of the incident wave vector. As the parallel-to-the-surface components of kþ and k in each medium are equal, only their perpendicular-to-the-surface components contribute into d (see Eq. (6)). As a consequence, the precession fronts inside the film as well as in the reflected and transmitted beams should be parallel to the sample surface. Note that, though the spin-dependent refraction is not described by the Bloch equation, it fails only when the refraction angle is comparable or even exceeds the beam divergence (e.g., see Ref. [18]), or when the lengths of the paths of neutrons in the opposite spin states noticeably differ (see discussion of Ref. [14] in Section 2). However, usually the angular splitting is very small and the correct description of the polarization vector behavior may be obtained by applying the Bloch equation to different neutron trajectories. Particularly, in Fig. 4 the spins of all incident neutrons are parallel to the guide field and the spins of neutrons (with a given wavelength) reflected at a certain angle have the same directions at the moment of reflection (the mirror is assumed to be homogeneous) and start precessing in the guide field. It is evident that, if the external field is homogeneous, the precession front in the reflected neutron beam will be parallel to the mirror surface. The precessing component of the polarization vector is reduced or even vanishes, when averaging is performed in a beam cross section not parallel to the sample surface (Fig. 4). A polarizer and an analyzer on the basis of remanent supermirrors have been used [19] to show that the precession front in the reflected beam for different wavelengths is parallel to the reflecting surface of the polarizer. The incident beam divergence was small enough and the surfaces of the polarizer and the analyzer were parallel to each other. It has been concluded that three components of the polarization vector of neutrons at the moment of scattering from a mirror can be measured in such a scheme. This possibility is of interest for the study of layered structures. Techniques such as neutron (3D) depolarization and neutron spin echo may, in principle, be formulated for beams polarized with non-frontal

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precession. Non-frontal precession may play an essential role in polarized neutron reflectometry with phase analysis [20] and for understanding details of magnetic scattering and depolarization mechanism (see Ref. [13]). It suffices to notice that the neutron scattering cross sections (specular reflectivities) may significantly differ (in principle, by orders of magnitude) in the cases when the incident neutron polarization precesses as usual,

Cðr; tÞ ¼

wavefunction (4) with B ¼ B1 and Un ¼ Un;1 . In order to find the wavefunction in medium 2, one should solve the problem of reflection and transmission of neutrons through the boundary of partition of magnetic media (see a detailed consideration in Refs. [8,9]). The superposition of the solutions for the spin components with energies Eþ and E yields the wavefunction in medium 2

þ þ þ Aþ " exp ðiK " r  itE" ="Þ þ A# exp ðiK # r  itE# ="Þ    A " exp ðiK " r  itE" ="Þ þ A# exp ðiK # r  itE# ="Þ

on the one hand, and when the precession front is parallel to the mirror sample surface, on the other hand; the reason being that the spins of the incident neutrons at the sample surface (at the moment of reflection) are oriented differently in the former case and identically in the latter case.

4. Neutron spin nutation In Sections 2 and 3 the two states with different energies were assumed to be orthogonal (the states with the spin parallel and antiparallel to the field), the neutron state being a superposition of such states. Generally, the states with different energies may be non-orthogonal (the states with the spins non-collinear to each other). Such a state can certainly be prepared in a neutron interferometer, but, what is more striking, the initially orthogonal states with different energies and antiparallel spins inclined to a homogeneous field may eventually become non-orthogonal, while the neutron is still in the homogeneous field region (see below). Assume that there is another homogeneous magnetic medium after the region with a field B (Fig. 1), the boundary between the two media is sharp and the two fields are not collinear. Designate the nuclear potentials and the fields in medium j as Un; j and B j ð j ¼ 1; 2Þ. The incident neutron state (in medium 1) is described by

! ð17Þ B2

(the subscript B2 shows that a representation with the quantization axis ZkB 2 is used), where rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2mn K"# ¼ ½E"#  ðUn;2 jmn B2 jÞ: ð18Þ "2 The neutron spin behavior is now defined by four velocities. The origin of the spatial coordinates is redefined so that r ¼ 0 at the boundary of the partition of two magnetic media. The superscripts ( ) designate the quantities related to the neutron states with the spin parallel to B 2 . The quantities related to the neutron states with different energies are designated by the arrows ( " ) and ( # ) (even when the respective spins are not antiparallel). Particularly, the energies Eþ and E in Eq. (4) are now designated, respectively, as E" and E# . The squared moduli of the amplitudes A "# yield the four probabilities to find the neutron in either (+) or () spin states with the energy either E" or E# . Designate the states with different energies as the ( " ) and ( # ) states. Certain spin orientations ( " spin and # -spin) correspond to these states at any instant and point. The neutron spin orientation is found from superposition of the ( " ) and ( # ) states, which are, generally, not orthogonal. Note that the neutron spin behavior is usually discussed in terms of the states with the spin up and down a quantization axis, as if they were individual spins (see Section 2). Further discussion of the neutron spin behavior in terms of the states with the spins ( " -spin and # -spin) which may be inclined to each other is based on the analysis in

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Appendix A of the superposition of two nonorthogonal fermion states. Such an approach is found to be extremely helpful and is well justified in the case under consideration. To describe the neutron spin behavior, rewrite Eq. (17) as ! þ Aþ " exp ðiK " r  itE" ="Þ Cðr; tÞ ¼  A " exp ðiK " r  itE" ="Þ B 2

Aþ # exp A # exp

þ

ðiK þ #r ðiK  #r

 itE# ="Þ  itE# ="Þ

! :

ð19Þ

B2

In the reference frame rotating about B2 with a frequency o" þ o# vp vav ¼  " # oL ; op ¼  2 vav vav where vav ¼ ðv"av þ v#av Þ=2

CðtÞ ¼

!

A " exp ðito" =2Þ



B2

Aþ # exp ðito# =2Þ

2

þ

A # exp ðito# =2Þ

 it  exp mn v"av V  E" "

! exp B2



 it  mn v#av V  E# ; "

where V v"av

oL ;

o# ¼

  þ  v"# ¼ v þ v av "# =2; "#

V v#av

oL ;

v "# ¼ ð"=mn ÞK"# ;

ð21Þ ð22Þ

It follows from Eq. (20) that the " -spin and the # -spin rotate about B 2 with the frequencies, respectively, o" and o# . If we choose the ‘‘pilot velocity’’ vp ¼ ðE"  E# Þ½mn ðv"av  v#av Þ1

þ

ð20Þ

o" ¼

ð26Þ

the spin motion is described by !  Aþ o"  o#  " exp it crot ðtÞ ¼ 4 A # B

For each of the spinors we may use the result formulated by Eq. (10). Then Aþ " exp ðito" =2Þ

ð25Þ

ð23Þ

for V, the spin behavior is determined by the function

Aþ # A "

! B2

 o o  " # exp it : 4

ð27Þ

One can see from Eq. (25) that the difference between |op | and oL (by definition, oL>0) is usually very small. Generally, both the magnitudes and the phases of A "# are arbitrary. ‘‘The first spin’’ and ‘‘the second spin’’ in Eq. (27) do not coincide with the " -spin and the # -spin. As it follows from Appendix A, Eq. (27) describes the neutron spin rotation about an axis (for reasons stated below, we shall call it ‘‘the nutation axis’’). In the rotating reference frame the orientation of the nutation axis with respect to B 2 is defined by the angles g (tilt) and r (rotation) that can be found from formulas (A.18) and (A.16), accordingly. The angle W between the nutation axis and the neutron spin may be found from (A.19). A geometric approach to finding the orientation of the nutation axis and calculating W is the use of formulas (A.24) and (A.25). The period of the neutron spin rotation about the nutation

0

0   o"  1 o#  1 þ exp it exp it Aþ A B " C B # C  o2 A þ@  o2 A CðtÞ ¼ @ " #   A" exp it A# exp it 2 2 B2 B2 0h   o  o i  o"  o#  o" þ o#  1 " # þ exp it exp it Aþ þ A exp it # B " C 4 4 4 ¼@ h  o o    o þo  A : o"  o# i " # " #   A" exp it þ A# exp it exp it 4 4 4 B2

ð24Þ

N.K. Pleshanov / Physica B 304 (2001) 193–213

axis is 2p=on , where 

on ¼ 

o"  o# 1 1 1 ¼  2 v"av v#av 2 1 E"  E# ¼ oL ; 2 mn v"av v#av

 v p oL ð28Þ

but the angular velocity of rotation changes during one period, when ‘‘the first spin’’ and ‘‘the second spin’’ are not antiparallel (see Appendix A). We may conclude that the neutron spin motion is a superposition of two rotations: the neutron spin rotates with a period 2p=on about the nutation axis, which, in its turn, rotates about B2 with a frequency op . As it follows from Eq. (28), the origin of quantum nutation is a difference in the frequencies of rotation of the " -spin (o " ) and the # -spin (o# ) about B 2 (Fig. 5). The mutual orientation of the " -spin and the # -spin changes with time. Consequently, the angle between the neutron spin and B 2 changes, leading to quantum nutation. The rotation about the nutation axis actually looks like ‘nutation’ known from Classical Mechanics. Of course, this analogy with CM cannot be stretched too far, more so as the nature of nutations is different. So the rotation of the neutron

201

spin about the nutation axis is more correctly to be called ‘quantum nutation’. The use of the word ‘nutation’ may be justified only by the apparent likelihood of the motion of the neutron spin (its expectation value) with that of a classical top when it precesses and nutates. It allows brief explanations and gives a helpful ‘visualization’ of the effect. It is noteworthy that precession is also treated differently in QM and in CM. Nevertheless, the notion ‘precession’ is not rejected in QM just because it has an exact definition in CM. Usually, the kinetic energy of neutrons is much greater than the neutron potential energy and E# ffi E" ffi E. Then the frequency of quantum nutations of the neutron spin is E"  E# oL : on ffi ð29Þ 4E If the difference in the energies E" and E# is acquired in an RF flipper, then on ffi 3:69107 H0ðRFÞ l2 oL ;

ð30Þ

( ) is the neutron where on (Hz), oL (Hz), l (A ðRFÞ wavelength, and H0 (T) is the static field of the RF flipper. Therefore, the frequency of the spin nutations is usually by orders of magnitude lower than the Larmor frequency. The nutation frequency is proportional to l2 , so the nutations in the classical limit (l ! 0) just vanish, as one could expect. The length at which the spin makes one nutation is   2p 4p 1 1 1 ¼ L n ¼ vp  : ð31Þ jon j oL v#av v"av In the approximation in which evaluation (30) for on was obtained we find that Ln ffi

369

;

ð32Þ

H0ðRFÞ B2 l3 ( ), H ðRFÞ (T), B2 (T), and Ln (m). E.g., if where l (A 0 ðRFÞ H0 =0.01 T, B2 ¼ 0:1 T, then Ln ¼ 369 km for

Fig. 5. In medium 1 the neutron spin precesses about B 1 . At the moment of entry into medium 2 (t ¼ 0) the " -spin and the # spin are antiparallel and the neutron spin is along P(0). Owing to a difference in the frequencies o" and o# , the angle between the " -spin and the # -spin changes, leading to a change in the angle between the neutron spin and B2 (and to quantum nutation).

( and Ln ¼ 0:369 mm for l ¼ 1000 A ( . In the l ¼ 1A overwhelming majority of cases the nutations may certainly be ignored. The strong dependence of Ln on the neutron wavelength l provides that the quantum nutations can be observed only for very large wavelengths. Of course, when not the total momentum but its component less by orders of

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magnitude is effective, as is the case in reflectometry, quantum aspects of the neutron spin behavior may come into play even for thermal neutrons. The length at which the spin makes one precession may also be found 2p v" v# 2p Lp ¼ vp   ¼ av av : vav oL op

ð33Þ

In the same approximations as above we find the evaluation 1:356104 ; ð34Þ B2 l ( ), B2 (T), and Lp (m). E.g., if B2 ¼ 0:1 T, where l (A ( then Lp ¼ 1:356 mm for l ¼ 1A and ( Lp ¼ 1:356 mm for l ¼ 1000 A. When reflection from the sharp boundary between media 1 and 2 is negligible, the neutron spin does not change its orientation when the neutron crosses the boundary and we find     w x d0  j Aþ ¼ cos cos exp i ; " 2 2 2     w x d0  j ¼ sin sin exp i ; Aþ # 2 2 2      w x d0 þ j ¼ sin cos exp i ; A " 2 2 2     w x d0 þ j ¼ cos sin exp i ; ð35Þ A # 2 2 2 Lp ffi

where the angles x (between B1 and the spin direction) and d0 define the spin orientation at the boundary (r ¼ 0) at t ¼ 0, w (between B 1 and B2 ) and j define the orientation of B1 (Fig. 6). Eq. (35) is obtained on the basis of transition from the quantization axis Z 0 kB 1 , X0 2(B1, B2) to the quantization axis ZkB2 by consecutive rotations about the Y 0 -axis by an angle w and about the Zaxis by an angle j. Comparing Eqs. (35) and (27) with Eq. (A.1), one may see that a1 a2 ¼ b1 b2 , b1  a1 ¼ j  d0 , b2  a2 ¼ j þ d0 . When the neutron spin at the moment of entry of the neutron into medium 2 lies in the plane (B 1 ; B2 ) (d0 ¼ 0 or d0 ¼ p) (see Fig. 6), ‘‘the first spin’’ and ‘‘the second spin’’ (see Eq. (27)) are antiparallel and collinear to the nutation axis. The nutation axis lies in the plane

Fig. 6. Transition from the quantization axis Z0 kB 1 , X 0 2 ðB 1 ; B 2 Þ to the quantization axis ZkB 2 can be made by consecutive rotations about the axis Y 0 ? ðB 1 ; B 2 Þ by an angle w and about the Z-axis by an angle j.

(B 1 ; B 2 ) (r ¼ j, when d0 ¼ 0, and r ¼ j þ p, when d0 ¼ p). The angle between the nutation axis and B 2 is equal to x (the angle between the neutron spin in medium 1 and B 1 ). The angle between the nutation axis and the neutron spin in medium 2 is equal to w (the angle between B 1 and B 2 ). The spin rotation corresponding to nutation is uniform (see Appendix A) with a frequency on . When the neutron spin at the moment of entry of the neutron into medium 2 does not lie in the plane (B 1 ; B 2 ), the nutation axis orientation is defined by formulas either (A.18) and (A.16) or (A.24), and the spin rotation about the nutation axis with a period 2p=on is not uniform. What is the resultant motion of the neutron polarization vector P in a cross section perpendicular to the beam (at L)? It can be found for each L by substitution of V ¼ 0 into Eqs. (9) or (20). First of all, note that, when E" ¼ E# , P does not depend on time. When E" 6¼ E# , P is timedependent in any beam cross section (the spin orientation will depend on the instant at which a neutron happened to be at the given point). It follows from Eq. (9) that P in medium 1 rotates about the field direction with a frequency oRF ¼ ðE"  E# Þ="

ð36Þ

defined by the RF flipper. In order to find the time dependence of P in medium 2 in a beam cross section at a distance

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L from the boundary, rewrite Eq. (19) as 80  1 þ > < Aþ exp iK L   " B " C   A exp itoRF =2 CðL; tÞ ¼ @ >  : A " exp iK" L B2 9 0  1 þ þ > A# exp iK# L  = B C   A exp itoRF =2 þ@ > ; A exp iK  L #

#

B2



itðE" þ E# Þ  exp  : 2"

ð37Þ

Note that at a point L the two terms define two spin orientations coinciding with the " -spin and the # -spin. Applying the analysis given in Appendix A, we find that the vector P rotates with a period tRF ¼ 2p=oRF around a fixed axis (‘local dynamic axis’), the orientation of which is defined by formulas either (A.18) and (A.16) or (A.24) and depends on the four phases K"# L as well as on the phases and moduli of the four amplitudes A "# . When the " -spin and the # -spin are not antiparallel, the rotation of P is not uniform (the angular velocity changes during one period tRF ). The change of the orientation of the local dynamic axis may be found by taking L as a variable and using the results of Appendix A. We find thereby that when we move along the beam, the local dynamic axis uniformly rotates about B2 with a period Lp , the angle between this axis and

203

# -spin (see the two terms in Eq. (37)) about B 2 due to the change in L are the same (K"þ  K" ffi K#þ  K# , if E# ffi E" much exceed the neutron potential), so their mutual orientation does not change (see also evaluation (32) for Ln ). Of special interest is the case when the angle between the " -spin and B 2 and the angle between the # -spin and B 2 are equal. Then the local dynamic axis will be collinear to the magnetic induction, even when the two spins are not antiparallel (see Appendix A). If the " -spin and the # -spin are antiparallel, it signifies a transition from solution of type (17) to a simpler solution of type (4). By adding another RF flipper, synchronized with the first, we may restore the initial beam polarization (as obtained after the polarizer), provided depolarization is negligible. When the two spins are not antiparallel, a physically new situation arises: (a) despite the fact that the spin rotates about the magnetic induction vector, this rotation is not uniform; (b) restoration of the initial polarization with another RF flipper is no longer possible; (c) at distances comparable with Ln the local dynamic axis becomes inclined to the magnetic induction vector. When the neutron spin does not change its orientation at the moment when the neutron crosses the boundary between media 1 and 2, we may substitute Eq. (35) into Eq. (37) and write the expression inside the braces as

0

 1  w d0  j þ   B cos   þ iK" L C exp i 2 2 x B itoRF C   cos exp  B C w A 2 @ 2 d0 þ j sin þ iK" L exp i 2 2 B2 0  1 w d0  j þ   B sin   þ iK# L C exp i 2 2 x B itoRF C   C exp þ sin : B  w A 2 @ 2 d0 þ j cos þ iK# L exp i 2 2 B2

B2 changes with a period Ln . However, the change in this angle is usually negligible, the reason being that the angles of rotation of the " -spin and the

ð38Þ

Comparing Eq. (38) with Eq. (A.1), we find that a1 a2 ¼ b1 b2 , b1  a1 ¼ j þ ðK"  K"þ ÞL,  b2  a2 ¼ j þ ðK#  K#þ ÞL. At the boundary

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(L ¼ 0) we obtain b1  a1 ¼ b2  a2 . It means (see Appendix A) that the vector P uniformly rotates with a frequency oRF about B1 (the local dynamic axis is parallel to B 1 ). As it should be expected, Eq. (4) yields the same behavior of P. As we move along the beam, the local dynamic axis behaves as discussed above. Note that the " -spin and the # -spin are antiparallel to each other at the moment when the neutron crosses the sharp boundary between two magnetic media. Usually the difference between o" and o# is negligible and these states remain to be orthogonal. Thus, on the one hand, we may usually describe the motion of P as the Larmor precession about the magnetic induction vector. On the other hand, we have concluded that, in a cross section perpendicular to the beam line, the vector P rotates about a local dynamic axis, which is, generally, inclined to the field. Yet, the latter does not contradict the Bloch equation. Solving the Bloch equation with the time of the neutron entry into the apparatus as a parameter, we obtain the same behavior of P in the lab frame. Note that the Bloch equation may fail even when quantum nutations are negligible. It is derived from the assumption that the neutron experiences the field along its trajectory point-bypoint. This condition is not satisfied when neutron scattering is essential. This condition is not satisfied in the interferometer where the existence of two paths for a neutron is essential. A state with arbitrary orientations of the " -spin and the # -spin may be prepared with the interferometer. When the " -spin and the # -spin are not antiparallel, the rotation of P about the local dynamic axis is not uniform and can by no means be derived from the Bloch equation. Of course, we may postulate the unusual behavior of P in a beam cross section and describe the further evolution of P by the Bloch equation. Another consequence of nonclassical preparation of the neutron beam, not described by the Bloch equation, is the neutron density variations (see Section 5). Of course, they were taken into account when we analyzed the neutron spin behavior on the basis of results from Appendix A. The possibility of numerous equivalent interpretations of the neutron spin behavior depending

on the choice of V is a consequence of the postulates of quantum mechanics according to which the state of each particle represents the properties of the beam, the result of each measurement is of a probabilistic nature. Direct calculations from Eq. (17) confirm all conclusions drawn in this section (e.g., see Fig. 7). It is to be emphasized that, when nutations exist, no reference frame can be found in which the spin

Fig. 7. The angle z between B 2 and the neutron spin is calculated from Eq. (17) as a function of time in the interval from 0 to 0.001 s for different velocities of the reference frame: þ þ # (a) vþ # =3.7922, (b) vav =3.8074, (c) v" =3.8225, (d) vav =3.9387, (e) vp ¼ 3:9479, (f ) vav ¼ 3:9533, (g) vav0 ¼ 3:9560, (h) v"av =   3.9679, (i) v # =4.0852, (j) vav =4.0993, (k) v" =4.1133 m/s (the velocities are defined in the text). It was assumed that ( , B2 ¼ 0:1 T, H ðRFÞ =0.01 T (oRF =2p ¼ 290 kHz), l ¼ 1000 A 0 and that the angles (see Fig. 6) giving the mutual orientation of B 1 and B2 and the initial (t ¼ 0) neutron spin orientation are w ¼ 458, j ¼ 0, x ¼ 308, d0 ¼ 0 (thus, z ¼ w þ x ¼ 758 at t ¼ 0). The complete description of the spin motion also includes its precession about B 2 with a frequency op . With the parameters used in the calculations, op exceeds the frequencies of variations of z by 3 orders of magnitude.

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behavior in space and time looks like a mere precession about the field direction. Neither does such a reference frame exist among the noninertial reference frames with V changing.

5. Neutron density waves Mezei [5] suggested producing ‘particle density waves on macroscopic scales’ by placing a static flipper into one arm and an RF flipper into the other arm of a Bonse–Hart neutron interferometer. The # -spin and the " -spin related to the states with different energies (E" and E# ) are then parallel to each other and antiparallel to the guide field. Consider now the general case when the # -spin and the " -spin are oriented arbitrarily. Then both the upper and lower spinor components of solution (17) contain two terms. Owing to their interference, both the spin-up (+) and spin-down () neutron densities (the squared moduli of the upper and lower spinor components) periodically change in time. It follows from Eq. (37) that at any point both the spin-up and spin-down neutron densities change in time with a frequency oRF . With such a change in time there corresponds a certain spatial periodicity (spin-up (+) and spin-down () neutron density waves). Fixing t in Eq. (37) and using the relations v "# ¼ ð"=mn ÞK"# and E"  2 2 E# ¼ mn ½ðv" Þ  ðv# Þ =2, we may find the lengths of the spin-up (+) and spin-down () density waves L ¼

v 2p 2p" " þ v# ¼ ¼ v av tRF ; E"  E# 2 K"  K#

ð39Þ where tRF ¼ 2p=oRF . The resultant spin-up (+) and spin-down () density waves propagate in medium 2 with different velocities v av

¼

v " þ v#

2

¼

E"  E# "ðK"  K# Þ

ð40Þ

(owing to the difference in refraction for the spin-up (+) and spin-down () neutrons). In the reference frame moving with the velocity V ¼  vþ av (V ¼ vav ), the spin-up (spin-down) neutron

205

density does not vary in time, its magnitude being dependent on the spin orientation when the neutron traverses the boundary. The neutron probability density jCj2 is the sum of the spin-up (+) and spin-down () neutron densities. In the reference frame moving with the velocity V ¼ vp , both the spin-up (+) and spindown () neutron densities change with a period tn ¼ 2p=on . As a consequence, the variations of jCj2 are with the same period tn and may usually be neglected. When ‘‘the first spin’’ and ‘‘the second spin’’ in Eq. (27) are antiparallel, the variations of the spin-up (+) and spin-down () neutron densities compensate each other, jCj2 is constant and the neutron spin uniformly rotates about the nutation axis. In a given beam cross section, i.e. in the lab reference frame (V ¼ 0), the neutron density is conserved in time only if the " -spin and the # spin are antiparallel. Otherwise, the neutron density varies in time with a period tRF (both the spin-up (+) and spin-down () neutron densities vary with this period). The mutual orientation of the " -spin and the # -spin changes along the neutron path with a period Ln (see Section 4). Therefore, the magnitude of the neutron density variations changes along the neutron path with a period Ln =2. It is to be noted that, owing to rotation of the neutron spin at the boundary of media 1 and 2 about B 1 with a frequency oRF , (see Section 4) the intensities of neutrons with the spin up (+) and down () the field B2 (Fig. 6) will vary in time. When the " -spin and the # -spin are not antiparallel (at a distance from the boundary comparable with Ln ), beats of the total intensity may be observed in the lab frame. The beats are reproduced in other reference frames (see Fig. 8). It is to be noted that jCj2 may exceed 1 within a period of time when the maxima of the spin-up (+) and spin-down () intensity modulations approach  each other (vþ av 6¼ vav ). Then the neutron density exceeds that of the initial beam. It is worth noting that the spin-up (+) and spin-down () neutron intensities change in time not only because the probabilities of the neutron spin projections on the directions up and down the field change (due to the neutron spin motion), but also because the total

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Fig. 9. 4D polarization analysis technique impose new requirements to the registering system.

Fig. 8. The neutron probability density jCj2 is calculated from Eq. (17) as a function of time in the interval from 0 to 0.001 s for different velocities of the reference frame. The velocities and the parameters are the same as used in the calculations of the curves in Fig. 7.

intensity (jCj2) varies in time. As pointed out above, the changes in the spin-up (+) and spindown () intensities may compensate each other, then jCj2 is constant (e.g., see Fig. 8e). In order that only neutrons with the same spin orientation be analyzed at a given instant, it is necessary that LA 5v av ð2p=oL;A Þ, where LA is the analyzer length (along the beam), and oL;A is the Larmor precession frequency in the field of the analyzer. Or else the neutron beam polarization with the precession front parallel to the analyzer surface should be used. In order that neutrons reflected from the analyzer at a given moment reach the detector simultaneously, it is also necessary that LA 5L (see Fig. 9a). Or else a neutron counter with a flat surface aligned parallel to that of the analyzer (Fig. 9b) should be used

(e.g., on the basis of 6Li glass with m  10 cm1 for thermal neutrons). When a transmission analyzer is used (Fig. 9c), the distance LSA between the analyzer and such a detector can be minimized to relax the restrictions due to a finite wavelength width (Dl=l5L =LSA ) and a finite beam divergence. In addition, it is necessary that D5L , where D is the mean neutron path in the detector counter. As the typical values for tRF are of the order of 105 (s), we obtain an estimation ( ). In order that the measureL (mm)40/l (A ments of the ( ) intensity modulations be possible, it is necessary that the detector be synchronized with the RF flippers. We may conclude that the ‘particle density waves on macroscopic scales’ produced with an interferometer fall into the category of experiments with amplitude modulators. The time-dependent phase difference between the neutron waves in the two arms of the interferometer (in the scheme described in Ref. [5]) leads to the amplitude modulation of the outgoing beams. The same physical consequences may be obtained with an absorbing modulator [21]. Other methods of the amplitude modulation have also been considered [22,23] and used [24,25]. Thus, the ‘particle density

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207

wave on macroscopic scales’ is equivalent to the intensity modulations propagating along the beam and can hardly be called a ‘‘new quantum phenomenon’’ [5], even when the beam preparation is fundamentally non-classical. The beats in the intensity resulting from summation of two (or more) density waves have counterparts in classical mechanics, too.

6. Discussion In the present paper the case when the neutron state is a superposition of two states with different energies and with the spins non-collinear not only to the static field direction but also to each other has been considered in detail. It has been shown that, usually, the Bloch equation describes the neutron spin behavior well. The spin evolution for neutrons in a sharp energy and momentum state is then described as the Larmor precession. In a beam cross section the neutron spin motion can be described as the rotation about a local dynamic axis. Such a local dynamic axis inclined to the field direction may be introduced by a static spin turner placed after an RF flipper (see Section 4). Another RF flipper will introduce a second dynamic axis inclined to the first, and the motion of the polarization vector P in a beam cross section will be a superposition of two rotations (P rotates about the second dynamic axis rotating about the first). The new local dynamic axis can be made noncollinear to the magnetic field by means of another static spin-turner. Adding pairs ‘‘static spin turner and RF flipper’’, any number of local dynamic axes can be introduced, the motion of P being more and more intricate. Of course, the RF flippers should be synchronized or else the neutron beam will be just depolarized. The resultant neutron state in a homogeneous field B will be a superposition of J states with different neutron energies ! Cþ ðr; tÞ Cðr; tÞ ¼ C ðr; tÞ B

PJ ¼

j¼1 PJ j¼1

þ Aþ j exp ðiK j r  itEj ="Þ  A j exp ðiK j r  itEj ="Þ

! B: ð41Þ

Fig. 10. The neutron spin orientation (P) at any instant is defined by superposition of the states jj> with different energies. To each such state there corresponds a certain spin orientation (see the dashed arrows).

Each state with a given energy is a superposition of two plane waves, in accordance with the number of eigenvalues of the wave vector operator K# j . As in the case of two energies, each state j ¼ 1; 2; . . . ; J with a certain energy Ej may formally be juxtaposed with a spin having a certain orientation at a given time and in a certain position. The neutron spin orientation (P) is defined by the superposition of such states (Fig. 10). In a reference frame moving with an ‘average’ velocity, the deviations of the neutron spin behavior from that predicted by the Bloch equation will be negligible, if the differences between the velocities are small. Indeed, this means that the frequencies of rotations of the J spins about the magnetic field are practically the same (oj ffi oL , j ¼ 1; 2; . . . ; J) and the mutual orientation of the spins does not change in time. The phase differences between these states do not change noticeably when the neutron crosses the apparatus. Consequently, the neutron spin precesses with the frequency oL. Particularly, in zero field the neutron spin precession is absent, so the notion ‘zero-field precession’ used in some papers seems to be rather confusing. The motion of P in a beam cross section will normally be a superposition of uniform rotations, each RF flipper producing coupled states (by

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splitting each energy level into two with the spin up and down the static field of the flipper). As a consequence, in a beam cross section after the Jth precession coil the vector P rotates about the Jth local dynamic axis, which rotates about the (J  1)th local dynamic axis, which rotates about the (J  2)th local dynamic axis, etc. For the states coupled in the manner described above all rotations are uniform (Ln much exceeds the length of the setup). Of course, such behavior is also obtained from the Bloch equation as a result of the transmission of neutrons through a sequence of regions with (static and time-dependent) magnetic fields. Generally, all the J spins are independent, the motion of P in a homogeneous field is not reduced to a superposition of rotations and cannot be deduced from the Bloch equation. The study of the motion of the polarization vector in a beam cross section may be complementary to 3D polarization analysis and informative of synchronization of processes in a sample under a periodic external force. To measure the neutron polarization as a function of time, it is necessary that the detector be synchronized w ith the external force. Additional requirements were mentioned earlier (see Fig. 9 and the text thereby). It is noteworthy that beams with unusual properties and with unusual polarization behavior may be produced with a neutron interferometer. The superposition of two orthogonal fermion states has been observed earlier in the fundamental works [1,26,27]. The superposition of two states with the spins non-collinear not only to the magnetic field direction, but also to each other may be studied experimentally, if a spin turner (a precession coil) is used before the interferometer to prepare a beam with the polarization vector inclined to the guide field. An arbitrary angle between the spins can be made, when an additional field collinear to the guide field is produced in one of the arms of the interferometer (e.g., see Ref. [28]). To imitate the time dependence of the phase (Appendix A), the phase difference between the two coherent neutron states (at the exit of the interferometer) may be changed with the conventional phase shifter. If the coil turns the spin to a direction perpendicular to the guide field, these

states can be made orthogonal. Unlike in experiments [1,26,27], the polarization vector component parallel to the guide field is informative about the result of the superposition, so no precession coil after the interferometer will be required to observe it. It is to be noted that the time dependence may also be obtained, if an RF flipper is placed after the precession coil (before the crystal plates of the interferometer). Of course, generally, the neutron spin behavior in a cross section of either O- or H-beam may be studied in the scheme with a precession coil, a p=2 spin turner and an analyzer. All the features of the superposition of non-orthogonal fermion states can be demonstrated. Particularly, one may show that the neutron spin motion resulting from superposition of the two states is always a rotation about an axis. Changing the phase difference linearly by means of the phase shifter, one may also demonstrate that this rotation is uniform only when the two spins are antiparallel. Introducing an absorber into one arm, we may attenuate the respective amplitude. The change of the phase shift will result in a neutron spin rotation about a different axis (see Appendix A). As before, this rotation is uniform when the two spins are antiparallel. In addition, mentioned here are two interesting situations, which can be realized in this experimental scheme (1) If the two spins are perpendicular to the guide field and are antiparallel, one may choose the phase shift so that the neutron spins will be along the guide field and then change their direction in the opposite). This spin-flip is produced by non-magnetic material of the phase shifter and seems to be quite exciting. (2) One can make the axis about which the neutron spin rotates collinear to the guide field even when the two spins are not antiparallel. A linear change in the phase difference produced by the phase shifter will rotate the neutron spin about the guide field direction as in common precession (the spin projection on the guide field direction remains unchanged), but such a rotation will not be uniform. The sense of this rotation can be made either positive or negative.

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These experiments would demonstrate general features of the neutron spin behavior not exploited so far in neutron beam techniques. An arbitrary mutual orientation of the " -spin and the # -spin (corresponding to states with different energies) may be obtained by placing the spin turner (precession coil) and the RF flipper into the opposite arms of the interferometer. Of interest might also be experiments with an RF flipper and a spin turner in each arm of the interferometer, with the use of different or the same frequencies. To emphasize the spinindependent nature of the interaction with the material of the phase shifter, it may be put, in succession, before and after the spin turners. The experiments can be treated on the basis of analysis from Appendix A. The use of an interferometer is not the only possibility to produce thermal and cold neutrons in a state which is a superposition of non-orthogonal ( " ) and ( # ) states (with different energies). Such states may be obtained if neutrons prepared in the scheme of Fig. 1 (in a state which is a superposition of the orthogonal ( " ) and ( # ) states) are reflected from a mirror made of layers with magnetizations non-collinear to the guide field and/or to each other. In conclusion, some general remarks. The considerations carried out show that the Bloch equation fails only in extreme cases. This is just for any number J of the coherent states with different energies. Therefore, the plane-wave description (the spin motion results from superposition of plane waves) and a semi-classical description (the spin motion coincides with that of a classical magnetic top) are usually equivalent. The wave-packet description is then equivalent to a semi-classical picture of particles moving through the instrument with different velocities along different trajectories, their spins experiencing the Larmor precession. One must bear this in mind when reading the papers in which the two approaches are opposed or the significance of one of the approaches is exaggerated at the expense of the other. Surely, the results obtained in one approach can also be obtained in the other approach (e.g., see Ref. [29]). The choice of an approach can be made only on the basis of its

convenience for theoretical analysis and interpretation of the experiment. Only when the Bloch equation fails, it is imperative to choose the planewave (wave-packet) description, which takes quantum aspects of the spin behavior into account.

Acknowledgements The work was supported by INTAS foundation (grant INTAS-97-11329).

Appendix A. Superposition of two non-orthogonal fermion states The following theorem will be proven: a superposition of two spin states, the phase difference between which changes with a frequency o, jCiZ ¼

a1 exp ðia1 Þ b1 exp ðib1 Þ þ

! Z

a2 exp ðia2 Þ b2 exp ðib2 Þ

 o exp it 2

! Z

 o exp it ; 2

ðA:1Þ

where the parameters aj , aj , bj , bj (j ¼ 1,2) are arbitrary real quantities, is equivalent to a state with the spin rotating with a period 2p=o about a certain axis Z 0 . This rotation is uniform during one period only if a2  a1 ¼ b2  b1 , a1 a2 ¼ b1 b2 , (or, equivalently, a2  a1 ¼ p þ b2  b1 , a1 =b1 ¼ b2 =a2 ), i.e., when the two spins are antiparallel (‘‘the first spin’’ and ‘‘the second spin’’ are orthogonal states). In Eq. (A.1) the states are given in a representation with an arbitrary quantization axis Z. Suppose now that the axis Z0 does exist and define it with respect to the Z-axis by the angles g (tilt) and r (rotation) (the same definitions as for the axis Z 0 in Fig. 2). Then transformation to the new coordinates (X 0 , Y 0 , Z 0 ), i.e., to the new quantization axis Z 0 , can be made by consecutive rotations of the old coordinate system (X, Y, Z) about Z-axis by an angle r and about Y 0 -axis by an angle g. The matrix of such a unitary

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transformation is 0  r g cos exp i B 2  2r Uðg; rÞ ¼@ g sin exp i 2 2

 r 1 g sin exp i C 2  2r A: g cos exp i 2 2 ðA:2Þ

Thus, in the representation with the quantization axis Z 0 , the superposition of the spinors (A.1) is transformed to the following form: ðCÞZ0 ¼ Uðg; rÞðCÞZ ¼

Cup Cdown

f ¼ 2f1 sin g þ f2 cos g; f1 ¼ a1 b2 cos ðot þ b2 1 rÞ

! ;

þ a2 b1 cos ðot þ a2  b1 þ rÞ

Z0

" # Cup ¼ a1 exp ðia1  ito=2Þ þ a2 exp ðia2 þ ito=2Þ " cos ðg=2Þexp ðir=2Þ þ b1 exp ðib1  ito=2Þ # þb2 exp ðib2 þ ito=2Þ sin ðg=2Þexp ðir=2Þ; " # Cdown ¼  a1 exp ðia1  ito=2Þ þ a2 exp ðia2 þ ito=2Þ " sin ðg=2Þexp ðir=2Þ þ b1 exp ðib1  ito=2Þ # þb2 exp ðib2 þ ito=2Þ cos ðg=2Þexp ðir=2Þ: ðA:3Þ

The projection of the neutron polarization vector onto the Z 0 -axis is   Cup 2 jCdown j2 f ðA:4Þ PZ0 ¼   :  Cup 2 þjCdown j2 g If the neutron spin rotates about the Z 0 -axis, this quantity does not depend on time t: f’g  f g’ d ¼ 0: ðA:5Þ P’ Z0  PZ0 ¼ g2 dt Note that the function g is the neutron probability density. It follows from Eq. (A.3) that g ¼ a21 þ 2a1 a2 cos ðot þ a21 Þ þ a22 þ b21

þ a2 b2 cos ða2  b2 þ rÞ;

ðA:8Þ

f2 ¼ a21 þ 2a1 a2 cos ðot þ a21 Þ þ a22  b21  2b1 b2 cos ðot þ b21 Þ  b22 : Consequently 2sin gð f’1 g  f1 g’ Þ þ cos gð f’2 g  f2 g’ Þ : P’ Z0 ¼ g2 It follows from P’ Z0 ¼ 0 that ! f’2 g  f2 g’ g ¼ arctan 2ð f’1 g  f1 g’ Þ  ¼ arctan

 h2 : 2h1

ðA:9Þ

ðA:10Þ

By simple transformations we find that f1 ¼ A cos ðotÞ  B sin ðotÞ þ C;

g ¼ G cos ðotÞ  H sin ðotÞ þ I;

þ b1 b2 sin ðot þ b21 Þ; ðA:6Þ where a21  a2  a1 ; b21  b2  b1 . Therefore, the neutron probability density is conserved in time only if the following two conditions are satisfied: ð2Þ a1 a2 ¼ b1 b2 :

þ a1 b1 cos ða1  b1 þ rÞ

f2 ¼ D cos ðotÞ  E sin ðotÞ þ F;

þ 2b1 b2 cos ðot þ b21 Þ þ b22 ; g’ ¼ 2o ½a1 a2 sin ðot þ a21 Þ

ð1Þ a21 ¼ b21 ;

(or, equivalently, a2  a1 ¼ p þ b2  b1 , a1 =b1 ¼ b2 =a2 ). The neutron probability density is not conserved in time for arbitrary parameters. This is taken into account in the further analysis (the physical content is discussed in Section 5). It follows from Eqs. (A.3) and (A.4) that

ðA:7Þ

h1 ¼ f’1 g  f1 g’ ¼ o½ðBI  CHÞcos ðotÞ þ ðAI  CGÞsin ðotÞ þ BG  AH; h2 ¼ f’2 g  f2 g’ ¼ o½ðEI  FHÞcos ðotÞ þ ðDI  FGÞsin ðotÞ þ EG  DH;

ðA:11Þ

N.K. Pleshanov / Physica B 304 (2001) 193–213

where A ¼ a1 b2 cos ðb2  a1  dÞ þ a2 b1 cos ða2  b1 þ rÞ; B ¼ a1 b2 sin ðb2  a1  dÞ þ a2 b1 sin ða2  b1 þ rÞ; C ¼ a1 b1 cos ða1  b1 þ dÞ þ a2 b2 cos ða2  b2 þ rÞ; D ¼ 2½a1 a2 cos ða21 Þ  b1 b2 cos ðb21 Þ; E ¼ 2½a1 a2 sin ða21 Þ  b1 b2 sin ðb21 Þ; F ¼ a21 þ a22  b21  b22 ; G ¼ 2½a1 a2 cosða21 Þ þ b1 b2 cosðb21 Þ; H ¼ 2½a1 a2 sinða21 Þ þ b1 b2 sinðb21 Þ; I ¼ a21 þ a22 þ b21 þ b22 :

Substitution of Eq. (A.12) leads to the following expression:   g ¼ arctan G1 =G2 ; "   G1 ¼ 2 b1 b2 a21 þ a22 sin ðb21 Þ   a1 a2 b21 þ b22 sin ða21 Þ # 2a1 a2 b1 b2 sin ða21  b21 Þ ; "   G2 ¼ 2 a2 b2 a21  b21 sin ðb2  a2  rÞ #   a1 b1 a22  b22 sin ðb1  a1  rÞ þ ða21 þ a22 þ b21 þ b22 Þ½a1 b2 sin ðb2  a1  rÞ

ðA:12Þ

In order that the angle between the neutron spin (polarization vector) and the Z0 -axis be unchanged in time, it is necessary that   d h2 h’2 h1  h’1 h2 ¼ 0: ðA:13Þ ¼ dt h1 h21 Since h’2 h1  h’1 h2 ¼ o3 ðH sin ðotÞ  G cos ðotÞ  IÞ ðCHD  HAF  BID þ AEI þ BGF  CGEÞ; ðA:14Þ we obtain the following equation for determination of the angle r:

þ a2 b1 sin ða2  b1 þ rÞ "  2 a1 b1 cos ðb1  a1  rÞ # þa2 b2 cos ðb2  a2  rÞ ½a1 a2 sin ða21 Þ þ b1 b2 sin ðb21 Þ:

Y1 ¼ a1 b2 cos ða1  b2 þ rÞ þ a2 b1 cos ða2  b1 þ rÞ þ a1 b1 cos ða1  b1 þ rÞ þ a2 b2 cos ða2  b2 þ rÞ; Y2 ¼ a21 þ 2a1 a2 cos ða21 Þ þ a22  b21

Substitution of Eq. (A.12) leads to the transcendent equation for r:

Y3 ¼ a21 þ 2a1 a2 cos ða21 Þ þ a22 þ b21

þ

a22 b21

þ

2a21 b21 Þsin

 2b1 b2 cos ðb21 Þ  b22 ;

þ 2b1 b2 cos ðb21 Þ þ b22 :

a1 b1 ða21 b22 þ a22 b21 þ 2a22 b22 Þsin ðb1  a1  rÞ ðb2  a2  rÞ

þ2a1 a2 b1 b2 sin ðb21  a21 Þ½a1 b1 cos ðb1  a1  rÞ þa2 b2 cos ðb2  a2  rÞ ¼ 0 ðA:16Þ which can be solved by standard numerical methods. For the obtained value of r the ratio h1 =h2 does not depend on time. Choosing now h1 and h2 at t ¼ 0, from (A.10) and (A.11) we find   EI  FH þ EG  DH g ¼ arctan : ðA:17Þ 2ðBI  CH þ BG  AHÞ

ðA:18Þ

The angles r and g obtained in this manner define the Z 0 -axis. Since the angle Z between the neutron spin direction and the Z 0 -axis does not change and, by definition, cos W ¼ PZ0 ¼ f =g, using the values of the functions f and g, say, at t ¼ 0; we find " # W ¼ arccos ð2Y1 sin g þ Y2 cos gÞ=Y3 ;

CHD  HAF  BID þ AEI þ BGF  CGE ¼ 0: ðA:15Þ

a2 b2 ða21 b22

211

ðA:19Þ

Let us note the following property of the solution. If the first spin and the second spin are turned by the same angle f about the Z-axis, the Z 0 -axis will rotate about the Z-axis by the same angle f. To prove this, it is sufficient to substitute a1 þ f and a2 þ f, instead of, respectively, a1 and a2 , into Eqs. (A.16)–(A.19). If a21 ¼ b21 , Eq. (A.16) with arbitrary aj , bj has a solution with b1  a1  r ¼ b2  a2  r ¼ 0. Therefore, using a21 ¼ b21 , we find r ¼ b1  a1 ¼ b2  a2 ¼ ðb1 þ b2 Þ=2  ða1 þ a2 Þ=2; ðA:20Þ

212

N.K. Pleshanov / Physica B 304 (2001) 193–213

and then g ¼ arctan



 2ða1 b1  a2 b2 Þ ; a21 þ b22  a22  b21

ðA:21Þ

g ¼ 2ða1 a2 þ b1 b2 Þcosðot þ b21 Þ þ a21 þ a22 þ b21 þ b22 :

ðA:22Þ

Of interest is that one can always choose a new quantization axis Z12 in the plane common for ‘‘the first spin’’ and ‘‘the second spin’’, such that the two spins lie in the same half of the plane divided by Z12 . In this representation the parameters aj , aj , bj , bj ( j ¼ 1,2) will be redefined and (from the choice of the quantization axis Z12 ) a21 ¼ b21 . As a consequence, the axis about which the neuron spin rotates (the Z 0 -axis) lies in this plane (as one could expect from symmetry considerations), the angle between the axes Z0 and Z12 being defined by substitution of the new parameters into Eq. (A.21). In terms of the angles xj between the Z12 -axis and the jth spin and the relative amplitude of ‘‘the second spin’’ state, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ða22 þ b22 Þ=ða21 þ b21 Þ, we find   sin x1  T 2 sin x2 g ¼ arctan : cos x1  T 2 cos x2

T¼

ðA:23Þ

Note that T does not depend on the choice of the quantization axis. By rotation of the Z12 -axis in the plane of the two spins, we can make x1 ¼ x, x2 ¼ p  x. The Z12 -axis chosen in the manner described1, one obtains from Eqs. (A.18) and (A.19) that   1  T2 tan x ; ðA:24Þ g ¼ arctan 1 þ T2 W ¼ arccos

sin g : sin x

ðA:25Þ

Thus, to find the orientation of the rotation axis and the angle between the neutron spin and this axis is now just a geometric problem. It follows from Eq. (A.24) that the rotation axis Z 0 lies between the Z12 -axis and the direction of either ‘‘the first spin’’ (T51) or ‘‘the second spin’’ (T > 1). When T ¼ 1, the axes Z 0 and Z12 coincide (g ¼ 0) and the neutron spin is perpendicular to 1

A.G. Wagh substantiated this choice in our discussion.

them (W ¼ p=2), its rotation being uniform only when ‘‘the first spin’’ and ‘‘the second spin’’ are antiparallel. If a21 ¼ b21 and a1 a2 ¼ b1 b2 (or, equivalently, a2  a1 ¼ p þ b2  b1 , a1 =b1 ¼ b2 =a2 ), i.e. ‘‘the first spin’’ and ‘‘the second spin’’ are antiparallel, we can find G and Y such that G Y a1 ¼ I 1=2 cos cos ; 2 2 G Y 1=2 a2 ¼ I sin sin ; 2 2 ðA:26Þ G Y 1=2 b1 ¼ I cos sin ; 2 2 G Y 1=2 b2 ¼ I sin cos : 2 2 Substitution of Eq. (A.26) into Eqs. (A.21) and (A.19) yields, respectively, b2 a2 g ¼ G ¼ 2 arctan ¼ 2 arctan ; ðA:27Þ a1 b1 W ¼ Y ¼ 2 arctan

b1 a2 ¼ 2 arctan : a1 b2

ðA:28Þ

Thus, the Z 0 -axis is collinear to the two spins. The same conclusion can be drawn from Eq. (A.24) (in the case under consideration x ¼ 0). On the other hand, when ‘‘the first spin’’ and ‘‘the second spin’’ are parallel (a21 ¼ b21 þ p), the neutron spin direction does not change in time and the rotation axis is not defined. The superposition of two spin states (A.1) may also be represented as a superposition of two states with the spins rotating in the opposite directions about the quantization axis Z with the frequency o: ! a1 exp ðia1  ito=2Þ jCiZ ¼ b2 exp ðib2 þ ito=2Þ Z ! a2 exp ðia2 þ ito=2Þ þ : ðA:29Þ b1 exp ðib1  ito=2Þ Z

As follows from the results obtained above, the superposition of such states with arbitrary real parameters aj , aj , bj , bj ( j ¼ 1, 2) is equivalent to a state with the spin rotating with a period 2p=o about the Z0 -axis. Calculations show that the rotation of the spin is uniform with the frequency o only when a1 a2 ¼ b1 b2 , a2  a1 ¼ b2  b1 (or,

N.K. Pleshanov / Physica B 304 (2001) 193–213

equivalently, a 2  a 2 ¼ p þ b 2  b2 , a1 =b2 ¼ b2 =a2 ). In conclusion, note that the angle between the resultant spin direction and the Z 0 -axis was proven to be constant in time, independently on the phase difference between the two spinors in (A.1). Consequently, a more general conclusion is that the superposition of two spinors ! a1 exp ðia1 Þ exp ½iW1 ðtÞ b1 exp ðib1 Þ Z ! a2 exp ðia2 Þ þ exp ½iW2 ðtÞ; ðA:30Þ b2 exp ðib2 Þ Z

the phase difference between which, W2 ðtÞ  W1 ðtÞ, is an arbitrary function of time, is equivalent to a state with the spin rotating about an axis. This axis is defined either by the angles g and r that can be found from Eqs. (A.19) and (A.16), accordingly, or, in the geometric approach, by the angle g as given by Eq. (A.24). References [1] G. Badurek, H. Rauch, J. Summhammer, Phys. Rev. Lett. 51 (1983) 1015. [2] R. Gaehler, R. Golub, Z. Phys. B 65 (1987) 269. [3] F. Mezei, Physica B 151 (1988) 74. [4] R. Golub, R. Gaehler, T. Keller, Am. J. Phys. 62 (1994) 779. [5] F. Mezei, J. Phys. Soc. Japan 65 (Suppl. A) (1996) 25. [6] N.K. Pleshanov, Phys. Lett. A 259 (1999) 29. [7] N.K. Pleshanov, Phys. Rev. B 62 (2000) 2994.

213

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