Neutron Spin Pendellӧsung Resonance

Abstract

The Spin–Orbit (SO) interaction is known to produce a very weak contribution to scattering of thermal neutrons. Schwinger proposed this type of interaction in the case of high energy neutron-nuclear scattering. He suggested the neutron would sense the atomic electric field by SO interaction between the neutron magnetic moment and the magnetic field due to its motion. In this chapter a diffraction theory is used to predict, and experimental evidence confirms a novel enhancement of SO scattering studied by diffraction. The experiment, using unpolarized thermal neutrons, demonstrates that SO and nuclear scattering interactions can be resonantly coupled in the pendellӧsung phenomenon of dynamical (perfect crystal) diffraction. On the theoretical side the Schwinger scattering contribution is included in describing the neutron-crystal potential used in equations of dynamical diffraction. Solutions of these equations predict a dramatic enhancement of the effects of SO scattering when a static magnetic field causes precession of the propagating neutron moment at a spatial period matched to the pendellӧsung (characteristic) length. We demonstrate this enhancement by performing a resonance experiment in which a large change in pendellӧsung oscillation intensity is observed over a small change in magnetic field strength. From these measurements we obtain a more precise value for the strength of SO scattering than has been previously possible.

Appendices

Appendix

The 17.5% discrepancy pointed to in the Analysis Section above can be accounted for using general wavefield solutions of the NSPR problem. These planewave solutions are used to match boundary conditions at the crystal entrance surface, when the incident beam does not precisely satisfy Bragg’s Law. Equation (A1) lists X-, Z-dependent solutions of Eq. (4.2) derived by Herb Bernstein. Conventions follow Theoretical Considerations Section.

$$ \begin{aligned} & \alpha_{+}^{\prime \prime }{{ = {\text{ e}}^{{{\text{i}}\left( {{\text{PX }} + { }\sqrt + Z} \right)}} }} \left( {\begin{array}{*{20}c} 1 \\ {RB_{ - } } \\ R \\ { - B_{ - } } \\ \end{array} } \right) \\ & \alpha_{-}^{\prime \prime }{{ = {\text{ e}}^{{{\text{i}}\left( {{\text{PX }} - { }\sqrt - Z} \right)}} }} \left( {\begin{array}{*{20}c} {RA_{ - } } \\ 1 \\ { - A_{ - } } \\ R \\ \end{array} } \right) \\ & \beta_{+}^{\prime \prime }{{ = {\text{ e }}^{{{\text{i}}\left( {{\text{PX }} + { }\sqrt - Z} \right)}} }} \left( {\begin{array}{*{20}c} {RA_{ + } } \\ 1 \\ { - A_{ + } } \\ R \\ \end{array} } \right) \\ & \beta_{-}^{\prime \prime }{{ = {\text{ e}}^{{{\text{ i}}\left( {{\text{PX }} - { }\sqrt + Z} \right)}} }} \left( {\begin{array}{*{20}c} 1 \\ {RB_{ + } } \\ R \\ { - B_{ + } } \\ \end{array} } \right) \\ \end{aligned} $$

(A1)

$B_{ - } = \left[ {\gamma + \delta - \sqrt + } \right]/K{\text{Sin}} \theta_{SO}$, $B_{ + } = \left[ {\gamma + \delta + \sqrt + } \right]/K{\text{Sin}} \theta_{SO}$,

$A_{ - } = \left[ { - \gamma + \delta - \sqrt - } \right]/K{\text{Sin}} \theta_{SO}$, $A_{ + } = \left[ { - \gamma + \delta + \sqrt - } \right]/K{\text{Sin}} \theta_{SO}$,

Also recommend using notation:

$B_{ \pm } = \left[ {\gamma + \delta \pm \sqrt + } \right]/K{\text{Sin}} \theta_{SO}$, $A_{ \pm } = \left[ {-\gamma + \delta \pm \sqrt - } \right]/K{\text{Sin}} \theta_{SO}$

$\sqrt \pm = \surd \left[ { \left( {\delta \pm \gamma } \right)^{2} + \left( {K {\text{Sin}} \theta_{SO} } \right)^{2} } \right]$, $\gamma = \surd \left[ { \left( {K{\text{Cos}} \theta_{SO} } \right)^{2} + \left( {P {\text{Tan}} \theta_{B} } \right)^{2} } \right]$,

$R = Y - \surd \left( {1 + Y^{2} } \right)$, $Y = - \left( {P {\text{Tan}} \theta_{B} } \right)/K {\text{Cos}} \theta_{SO}$.

P is error, from perfect Bragg incidence, of wave vector (X) component parallel to entrance face. All other parameters are defined in the Theoretical Considerations Section.

Inner wavefields ($\alpha_{ - }^{\prime \prime }$, $\beta_{ + }^{\prime \prime }$) display resonant behavior when $\delta \sim \gamma$, the outer pair do not. Wavefields of Eq. (A1) apply when $\delta \ge \gamma$, i.e. when magnetic field is greater or equal to resonance for perfect Bragg incidence (P = 0). The sign of $\sqrt {-}$ is opposite when $\delta < \gamma$.

Field parameter $\delta$ at resonance shifts with $\gamma$ which in turn depends on P, so off-Bragg incidence radiation excites wavefields satisfying the resonance condition at higher field. The crystal-slit system (Sample Preparation Section) limits range of P contributing signal exiting the crystal, and weights each ray according to propagation angle inside the crystal as illustrated in Fig. 4.6.

Fig. 4.6

Illustrates beam defining slits of sample crystal, and associated range of ray angles and weight (number of rays) contributing at each internal propagation angle. Incident beam |F > is assumed to be an incoherent sum of plane waves spanning angle range large compared to Darwin width of (111) reflection

Fig. 4.7

a Calculated intensity (vertical axis) exiting crystal in the Bragg diffracted direction, over a range of transverse incident wave vector P/K (horizontal axis). Signal exhibits spatial Pendellösung behavior [5]. Calculation corresponds to Pendellösung minimum T/${\Delta }_{0}$ = 25, when P/K = 0. δ⁄K = 0.05 i.e. magnetic field 5% of resonance. The region close to central minimum corresponds to range of P/K in our experiments. Signal from adjacent maxima and steeply rising regions are excluded by slits on front and back side of crystal. b Calculated NSPR intensity, with increased P/K wave vector resolution, near resonance (δ⁄K = 1). Signal (points on the U-shape curves) corresponds to δ⁄K = 0.95 (blue), 0.98 (red), 1.0 (black), 1.02 (green) with Bragg angle 8.7 degrees and $\theta_{SO} \approx \frac{1}{102}$

Measured strength of SO scattering, larger than first principles calculation, can be understood qualitatively by comparing, near resonance in Fig. 4.7, signal versus P/K to that at P/K equals zero. Here we explore the case corresponding to Fig. 4.4c where T/${\Delta }_{0}$ = 25, a zero-field pendellosung intensity minimum. Signal rises away from P/K = 0. This spatial pendellösung effect increases signal size measured through finite slit openings. Our data analysis used pendellosung peak-to-valley intensity ratio determined by scanning neutron wavelength. NSPR signals are assumed to scale with this difference, but intensity measured at each magnetic field is slightly increased by acceptance of signals away from the P/K = 0 minimum. The net result, increased resonant intensity at pendellösung minimum, increased resonant dip at pendellösung maximum (T/${\Delta }_{0}$ = 24.5) are consistent with larger $\theta_{SO}$ in our data reduction. The 17.5% discrepancy can be quantitatively accounted for by integrating signal over |P/K| ≤ 0.71, a range consistent with crystal slit width, yields the signal expected for Eq. (4.5) when $\theta_{SO} \simeq \frac{1}{102}$. Going further with this analyses is beyond the scope of this chapter.

Personal Reflection

The work reported here is a small example of synergy at Cliff Shull’s MIT neutron diffraction lab. Mike Horne was a most welcome weekly visitor, who inspired and mentored grad students, as he participated (often motivated) animated discussions, on every conceivable subject (including physics) with visiting scientists, students, and anyone who stopped by the office at NW13. This work evolved from Cliff’s interest in pushing measurement sensitivity to explore unexpected physics. It is my understanding that he came up with the NSPR concept. Mike worked out the dynamical diffraction description and found the “on-Bragg wavefields” using symmetry and energy conservation. Herb Bernstein used mathematical wizardry to derive general solutions. Meantime, Anton Zeilinger was my early mentor on experimental neutron diffraction methods. Tony Klein, a visitor from Australia, dispensed sage wisdom on the MIT qualifying exams.

Every Tuesday Mike would arrive by 10 AM, inject a shot of insulin, take out a notebook and pen, put up his feet, and get to work. Notebook entries always seemed to contain finished ideas; I cannot remember him using a scratch pad. Mike was a master teacher, patient, able to reduce problems to whatever level the student needed. He excelled at finding interesting explanations, and enjoyed recalling how he used them in lectures at Stonehill College. With everything he accomplished, social distance vanished in Mike’s relationships.

Mike taught me most of what I know about dynamical diffraction. Visualization was the key to understanding what went on inside the crystal and what the equations said. He was the most approachable person I met at MIT (after my wife). His wisdom extended way beyond physics. He had a refined palette that analyzed the ingredients in chili I’d bring back for him from MIT pushcart vendors. Mike loved music; he introduced me to SunRa and Robert Cray, played a very cool double bass, and he and Carole made a tape entitled “Our love is here to stay” that accompanied us on our honeymoon…

Editors’ note: Mike remained an active member of the Shimony Group at Boston University throughout his career and encouraged the further exploration of his neutron physics work. One of the results of this was the derivation from first principles, in 1989, of assumptions made in Horne et al. (1988), later published as: Jaeger, Shimony (1999) ‘An extremum principle for a neutron diffraction experiment’ Found Phys 29: 435.