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Some Magnetic and Spin Effects in Neutron Interferometry* A. Zeilinger

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Some Magnetic and Spin Effects in Neutron Interferometry* A. Zeilinger i r Some Magnetic and Spin Effects in Neutron Interferometry* li A. Zeilinger Massachusetts Institute of Technology Cambridge, Mass., 02139, U.S.A. Abstract We propose and discuss several experiments where the neutron spin and the neutron magnetic moment are utilized in interferoir.etry. One of these is concerned with an explicit testing of the quantum mechanical laws of spin superposition. Additionally ferromag- nets as phase shifters will produce some qualitatively new polari- zation effects, and magnetic poase contrast topography can be ex- pected to provide images of magnetic domains and domain walls in coherent neutron transmission. Work supported through Wissenschaftsstipendium des Kulturamtes der Stadt Wien. Permanent Address: Atominstitut der Osterreichischen Universitaten, r Wien, Austria with -1- 1. Introduction K-" A whole field of new experiments in neutron jnterferometry can be expected as a consequence of the fact, that the neutron carries spin and magnetic moment. The first experiments in this field were the experimental demonstra- tion of the kn rotational symmetry of soinors (Rauch et al. 1975i Vterner et al. 1975) and of the capability of the neutron interferometer to act I: as a neutron polarizer (Badurek et al. 1?7f). First theoretical work was devoted to the study of effects due to ferronagnets and helical magnetic fields (Eder and Zeilinger 197^) and to a general analysis of spin rotations and spin-state superposition (Zeilinper 1976). In the present paner v/e will first deal with a more detailed discussion of spin-state stiDprnosition secondly v/e v/il 1 present explicit results for ferronacinets as phase shifters and finally we will discuss some possible magnetic phase contrast experirents. In most cases we shall present only the results. Details of the calculations can be found in Eder and Zeilinner (l?7ft) and Zeilinqer (1?7$). As the neutron is a spin-^ particle its wave function can be written as th.. spinor with a, b, ?, $ real quantities which are in general functions of space snd -2- time. It is usual to choose the coordinate system in such a way that (Q) and m are the eiaenstates of az with the Paul! spin vector /a = (ox,a ,a2) defined as I o' i 0 oj In Equ.(la) 5 denotes the spatial phase and (J the internal phase of the I: spinor. The intensity of a spinor as described by this equation is I = a2 + bz and its polarization vector is la.Q Other ways of representing a spinor are for example (It and, if the spinor is normalized It,9,,9, 11 -3- ft 1 m j ?:\ where, e.g. 11^/' is the.unit spinor with spin up in the z-direction, 8 is the angle of the neutron polarization vector with the z-axis and as above $ Is the angle between its projection onto the x-y-plane and the x-axis. Spin rotations are described by the unitary operator 11 ~ e Cl Here a is the angle of rotation and "of =~a'/a is a unit vector parallel to the axis of rotation. It is of particular interest for the interpretation of the experiments discussed in the next paragraph to study the unitary operator describing a 180° rotation iZ) Thus the rotation of a spin pointing initially in the +z direction around an axis normal to it can be written as f f\ Here, x is the angle betv/sen the rotation axis and the x-axis. Thus, after a spin flip process a neutron wave contains a phase factor which depends upon the axis around which the rotation was performed. •fr- -k- To avoid an unnecessary complication by a detailed discussion of dynamical diffraction effects and to provide a general analysis applicable to many Interferometer types we assume - with one exception - an idealized inter- ferometer experiment using 'half-silvered' mirrors. This idealized inter- ferometer provides two final beams as coherent superpositions of equal amplitudes from both beam paths. These contributions are in phase for the so-called forward beam and out of phase for the deviated heam because of a phase change by ir/2 upon each refrection. 2. Spin-State Superposition A neutron interferotneter with polarized incident neutrons can be used to demonstrate explicitly some aspects of spin-state superposition. For the experiments proposed in this paragraph we assume that the incoming neutrons are in the spin-up state with respect to the z-direction. To obtain a coherent amplitude in the spin-down state we turn the neutron spin in one beam path by 180° around the y-direction. The forward and the deviated beam are then both coherent superpositions of spin-up and spin-down states with equal amplitudes. These two states are in phase with each other according to Equ.(7). Thus the neutron spin in the forward beam is in the positive x-direction, the. wave function being L o) ¥' -5- Similarly the spin in the deviated beam is in the negative x-direction with the wave function I tl Probably the best way to measure, these spins is to use d.c. spin turners of the type first introduced by Rekveldt (1973). This experiment would be the realization of a Gedankenexperir.ent proposed by Uioner (12^3) where he imagines a double Stern-Gerlach arrangement to separate and recomblne coherently the z-spin components of a x-spin particle. This Gedanken- experiment and its realization as proposed above demonstrate explicitly the coherent spin superposition law of Equs.(l). The outcome of these experiments could never be interpreted as an incoherent mixture of the superposed spin states. The possibilities for direct experimental verifications of Equs.(l) Include additional detailed demonstrations of the spin superposition lav;. So the internal spin phsse can be changed directly by shifting the phase of the spin-down beam in Fig.I by x using for example a nuclear phase shifter. Thus we obtain for the forward bean This means that the spin of this beam can be rotated by a scalar inter- it-', «-! -6- action. Thlsfeerns to be paradoxical but the complete wave function includes the deviated bean and its spin is obtained as UO) Thus the spins of the forward and the deviated beam are always opposite to each other. This implies, that the scalar phase shift x does n°t rotate the spin of the complete wave function. If finally an absorber is brought into one of the beam paths, the final neutron spins can be tipped out of the xy-plane because absorption changes the amplitude of one of the states of Equs.(l). If the intensity of the -a neutrons passing through the absorber is reduced by e we have to reduce the amplitude of its wave function by the factor e"a/^2. As a particular example we attenuate the down spin intensity by e" = 3" 2)? or its a 2 amplitude by e " ' - 1. Thus we obtain for the forward beam with fl being the normalization factor. Thus in this example the spin points in a direction 45° up from the x-direction. """ ! -7- If performed, the class of experiments proposed in this paragraph would constitute the first direct test of the first principles of quantum mechanics of spin. Thus these experiments should be performed despite the fact that the principles involved are implicitly tested in existing experiments. I! 3. Magnetic phase shifters Other applications of the neutron spin in interferonntry can be expected when ferromagnets are used as phase shifters. Their effect on a trans- mitted neutroni bean can he described as a combination of a scalar phase shift and a spin rotation angle. To demonstrate the kind of experimental outcomes we assume that a single dorain magnet, i.e. a magnetically saturated sample is placed into one beam of the interferometer. In that case the spin rotation angle can be expressed in terns of the forward magnetic scattering length po which contains the magnetization as ill) where H is the atomic density and d is the thickness of the sample. This is an analog to the familiar expression of the nuclear phase shift x as a function of the nuclear scattering length bc -8- CO) Thus, both the nuclear phase shift and the spin rotation anqle vary linear with sample thickness keeping their ratio constant. If we assume that the initial neutron spin points in the x-direction and the sample is magnetized in the 2-direction we obtain for the intensity of the forward beam 1=1. Z where IQ is its intensity without a sample. The polarization of the forward bean is Figures 2, 3, and k show the intensity and the z-component of the polarization for Fe, Co and Mi as a function of sample thickness. It has to be noted, that these particular curves also hold for unpolarized incident neutrons. The complete behaviour of the polarization vector (Equ.15) is shown in Fig.5 as the trace of the unit spin vector on a unit sphere again for neutrons ^—! V- -9- Initially polarized in x-direction and a single domain sample magnetized In z-direction. The polarization vector starts at the point where the x-axis meets the equator. Y/ith increasing sample thickness the polari- zation vector starts to follow loops, their particular form being related to the ratio bc/po. The connection with absolute thickness values can be obtained by comparison of the P -components with those of Figs. 2-k. We point out that, as the motion of the neutron spin in the samples is simply the LariTior precession around the magnetization direction, the tip of the neutron spin would follow the equator in our representation with no difference between the different ferromagnnts.
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