The Kru ¨ ger problem and neutron spin games Vladimir K. Ignatovich Joint Institute for Nuclear Research, Dubna, Russia 141980 Filipp V. Ignatovich a) Institute of Optics, University of Rochester, Rochester, New York14627 Received 3 September 2002; accepted 28 March 2003 We present a compact solution to the one-dimensional problem of neutron scattering from two mutually perpendicular magnetic fields: permanent and rotating ones. Both fields are confined inside a layer of matter or a layer of a free space. The applications of this solution to the interpretation of some experiments with polarized neutrons are considered. © 2003 American Association of Physics Teachers. DOI: 10.1119/1.1575763 I. INTRODUCTION The combination of two perpendicular magnetic fields, one permanent dc and one rotating radio-frequency magnetic field, occurs in almost every experiment with polarized neu- trons. In particular, such a combination is used in resonance spin flippers or spin rotators see, for example, Ref. 1. The goal of these rotators is to change the direction of the neu- tron’s spin. This change is accompanied by the absorption or emission of a rf quantum of energy and therefore by the change of the neutron’s energy. Usually the change in energy is small in comparison to the initial neutron’s energy and can be neglected. However such a change, regardless of how small it is, plays a crucial role in some experiments. In 1980 Kru ¨ ger 2 proposed accelerating neutrons with a combination of dc and rf magnetic fields. He considered a simple geometry of the fields confined in free space see Fig. 1, and demonstrated that a neutron, which is initially polar- ized opposite to the dc field B 0 , after interacting with the combination of fields may flip its polarization and acquire additional energy 2, where 2 is the frequency of the rf field. To find the efficiency of the acceleration, he calculated the probability of the spin flip for arbitrary polarization of the incident neutron, which was equivalent to solution of a system of 16 linear algebraic equations 2 with 16 unknowns. We call it ‘‘the Kru ¨ ger problem.’’ Kru ¨ ger solved it only approximately. We will show how to solve it precisely and analytically. It is understandable why the solution of the Kru ¨ ger prob- lem requires so many equations. The wave function of the neutron after scattering from the magnetic fields consists of elastically and inelastically reflected and transmitted waves. Because each part of the wave function describes two spin components, four complex coefficients are required to de- scribe the reflected part of the wave function, and four com- plex coefficients are required to describe the transmitted part of the wave function. Matching the wave function outside the area of the magnetic fields to the wave function inside this area gives equations for eight coefficients. The wave function inside the region of the magnetic fields consists of four waves with four unknown coefficients traveling in one direction and the same number of waves traveling in the opposite direction for a total of eight coefficients. Thus matching the total wave function and its derivative at the two interfaces gives 16 equations with 16 unknown coefficients. An attempt to solve this system of 16 equations was un- dertaken in Ref. 3. However, the authors approximated the final result, as was done in Ref. 2. Later a slightly different problem was considered. 4 In Refs. 2 and 3 the magnetic fields were confined in a vacuum, and in Ref. 4 the magnetic fields were in a semi-infinite medium where only one inter- face was present. For simplicity, the authors of Ref. 4 con- sidered neutrons initially polarized only along the magnetic field. They presented the final result, but did not discuss their method of solution. Here we discuss a compact solution that is physically transparent and does not require solving 16 linear algebraic equations. The compact formulas that we will derive allow us, with the help of the algebra of Pauli matrices, to find the coefficients for every component of the wave function ex- actly. Moreover, we generalize the original Kru ¨ ger problem by including a medium. We assume that all the fields are confined within a layer of matter 0 x D , and have no other dependence on the spatial coordinates. The interaction of neutrons with matter is characterized by optical potential (  2 /2m ) u , where u =4  N 0 b , N 0 is the atomic density and b is the coherent scattering amplitude. The quantity (  2 /2m ) u is called the optical potential, because it deter- mines the reflection and refraction of the neutron waves similarly to the reflection and refraction of the light waves in optics. The total field consists of a dc magnetic field along the z axis, B 0 =(0,0,B 0 ), and a rf magnetic field rotating in the x – y plane with frequency 2, B 1 =B 1 (cos 2t,sin 2t,0). The use of the frequency 2 instead of  is convenient be- cause it eliminates the factor 1/2 in some formulas; however, it is necessary to remember that the quantum of this rf field is 2 and not . The incident neutron is described by a plane wave  =exp(ik 0 x-iE 0 t )  0 , where k 0 =m v 0 /  , E 0 =E 0 /  , and  0 is an arbitrary spinor related to the polarization of the neutron. We wish to find the transmission and reflec- tion matrix amplitudes,  ˆ and  ˆ , which contain information about the polarization and energy of the neutron after the interaction. This article is constructed as follows. In Sec. II we review a common solution, which does not take into account the full dynamics of the neutron in the fields. In Sec. III we show our compact solution to the Kru ¨ ger problem. This solution yields the reflection and transmission matrices, which are derived for ideal interfaces in Sec. IV. With the help of these matrices it is possible to calculate the matrix elements for the elastic 1013 1013 Am. J. Phys. 71 10, October 2003 http://aapt.org/ajp © 2003 American Association of Physics Teachers