High angular resolution neutron interferometry.

The currently largest perfect-crystal neutron interferometer with six beam splitters and two interference loops offers novel applications in neutron interferometry. The two additional lamellas can be used for quantitative measurements of a phase shift due to crystal diffraction in the vicinity of a Bragg condition. The arising phase, referred to as "Laue phase," reveals an extreme angular sensitivity, which allows the detection of beam deflections of the order of 10(-6) s of arc. Furthermore, a precise measurement of the Laue phase at different reflections might constitute an interesting opportunity for the extraction of fundamental quantities like the neutron-electron scattering length, gravitational short-range interactions in the sub-micron range and the Debye Waller factor. For that purpose several harmonics can be utilized at the interferometer instrument ILL-S18.

Introduction

Perfect crystal X-ray and neutron interferometry has a long history [1-3], but an interesting feature of crystal interferometers, namely their extreme angular sensitivity, has been exploited only by intensity measurements [4,5] but not yet by phase measurements. Only recently, the machining of large-scale crystal interferometers, specially designed [6], opened novel applications in neutron optics and fundamental physics [7]. The interferometer of Mach–Zehnder type has been cut with two additional lamellas L1 and L2 in the middle (Fig. 1), which can be used for the measurement of crystal phases due to dynamical diffraction. We refer to this phase as “Laue phase” [8] because the underlying diffraction process is known as Laue transmission in crystallography. Due to the large interferometer dimensions, different prism configurations can be employed for coherent beam deflection with smoothly tunable deflection angles [7]. It turns out that the Laue phase is extremely sensitive to beam deflections in front of the crystal lamellas (Section 2), which become detectable with the new interferometer setup (Section 3). After the first successful tests and phase measurements new applications can be envisaged, like the nice idea presented by Ioffe et al. [9] for measuring the neutron–electron scattering length, or a proposal made by Greene et al. [10] for testing a hypothetical short-range interaction (Section 4).

Fig. 1

The currently largest perfect crystal neutron interferometer.

Laue phase and angular sensitivity

A distinct phase shift is generated when the neutron wave passes a perfect crystal plate close to the Bragg condition [11,8]. In the following, we are considering only the diffracted wave in forward direction. The Laue phase is then defined as the argument of the complex transmission amplitude (t) as shown in Fig. 2.

Fig. 2

Comparison of the phase shifter case (left) where only refraction occurs, and the Laue transmission (right) where dynamical diffraction causes strong phase variations.

The parameters y and δθ=θ−θ describe the beam deviation from the Bragg angle θ, E the neutron energy and |V| the crystal potential [12]. The Laue phase in an analytical form reads

At certain positions of δθ, the arctan term induces a distinct fine-structure to the phase which is extremely sensitive to the Pendellösung length Δ. The Pendellösung length is an important parameter in crystal optics because it contains important quantities like the atomic scattering length (batom) and the Debye Waller factor (W)batom and W depend on the momentum transfer (q=2πn/d); N denotes the atomic density, d the lattices spacing and n the order of reflection. It turns out that the slope of the Laue phase, and hence the angular resolution, reaches its maximum in the vicinity of the Bragg condition δθ=θ−θ→0. There ϕLaue increases linearly with δθ and the angular resolution is determined by the geometric ratio of thickness (D) and lattice spacing (d)or alternatively it can be written as a function of momentum transfer

Higher order reflections (nh,nk,nl), or equivalently larger momentum transfers, are therefore more sensitive to beam deflections (Fig. 3).

Fig. 3

Laue phase calculated for collimated beams in the vicinity of the Bragg condition. Higher reflections orders (shorter wavelengths) yield higher angular sensitivity (D=15 mm).

Angular resolution in the perfect crystal interferometer

First measurements of the Laue phase have been performed at an instrument ILL-S18 with the configuration shown in Fig. 4a. The fine-tuning of beam deflection δθ was achieved with a simultaneous rotation of four aluminium prisms about the axis α (Fig. 4b). The averaging of the Laue phase over the angular distribution slightly reduces the instrumental resolution yielding an angular sensitivity ofnear the Bragg condition [13]. In fact, the current interferometer geometry in Fig. 4b is a compromise because further experiments are intended, in which the two-loop feature becomes essential [7]. The angular sensitivity would further be enhanced by machining a new interferometer with thicker lamellas, for example L1=L2=15 mm as sketched in Fig. 4c, and by generating two opposite beam deflections. Assuming a realistic phase resolution of 0.1°1 and considering all possible improvements an angular resolution of 10−6 s of arc (5×10−12 rad) seems feasible.

Fig. 4

Setup for detecting small beam deflections δθ in the six plate interferometer: (a) Present setup with four identical prisms and 3 mm thick lamellas. The prism in front of L2 creates a beam deflection and thereby a phase difference between L2 and L1. The other prisms are necessary to avoid dephasing and defocusing; (b) Experimental realization at ILL-S18; (c) Proposal for a new design to enhance phase sensitivity and angular resolution.

Another method for achieving high angular sensitivity can be realized with the same interferometer crystal in the non-interfering arrangement [4]. Then the angular sensitivity depends again on the geometric ratio d/D, but it can be expected that the high phase resolution makes the interference method favorable over the analysis of rocking curve widths.

Applications

As primary goal, the accuracy of Laue phase measurements has to be improved, in particular at angular positions, where deviations from the linear behavior are expected (see the Pendellösung-structures in the phase plot in Fig. 5). Such nonlinear structures reveal a pronounced sensitivity to the Pendellösung length and the parameters therein, namely the atomic scattering length and the Debye Waller factor. According to a proposal by Greene et al. [10], the atomic scattering length comprises three contributions, the dominant nuclear scattering length (b), the neutron–electron scattering length (b) [16], and a hypothetical short-range interaction (b)

Fig. 5

Top: effect of the Coriolis force on the neutron trajectories in large interferometers. Bottom: calculated and measured phase shifts at varying deflections δθ using the prism configuration in Fig. 4b. The mid-point between the two plateaus would be expected at δθ=0; however, it is slightly shifted by Coriolis deflection (λ=2.72 Å).

The b and b terms become distinguishable by their different q-dependence of the form factors f, f. A precise measurement of batom at several reflections would allow separating the different contributions. A short-range interaction would modify the silicon scattering length, with the effective range parameter (λ) as dominant parameter (). Recent years have witnessed renewed interest in departures from Newtonian gravity [14,15]. Corrections to Newton’s law of gravity are mainly expected at submillimeter, even at nuclear length scales. The most common assumption consists in a modified potential according towhere G is the gravitational constant, m and M are the interacting masses and α characterizes the strength of the correction. One now has to find constraints for α over a wide range of λ (e.g. α=1022, λ=1 Å→f220b=1.4×10−5 ▒fm). Especially in the range λ<100▒ nm, the constraints for α are still very weak, as most methods are not sensitive to this region.

Interferometers with long beam paths become increasingly sensitive to the Coriolis force due to Earth’s rotation. This causes an additional, wavelength-dependent, beam deflection and an offset to the Laue phase as shown in Fig. 5. Experimentally a phase offset of approximately 50° has been obtained for the (2 2 0) reflection, and accordingly 25° offset for the (4 4 0) reflection. The Coriolis deflection can be compensated by inserting suitable wedges in both interfering beams, whereby the visibility maximum is restored. Another, much larger Coriolis phase arises due to the interferometer’s large enclosed beam area (=area vector perpendicular to the enclosed beam, see Table 1). This so-called Sagnac phase [17] depends on the product (=Earth’s rotation vector); it is wavelength-independent and becomes larger than 900° considering the present beam orientation at instrument S18. The Sagnac phase becomes crucial in gravitation experiments [18], where it has to be eliminated experimentally by an interleaved scanning procedure [19].

Table 1

Key features of the new interferometer setup.

Interferometer length/path length	23.5/25 cm
Maximum enclosed beam area	A=100 cm2
Lamella thickness (current)/optimized	D=0.3 cm/≥1.5 cm for L1,2
Available reflections (S18)	(2 2 0), (4 4 0), (6 6 0)
Wavelengths	2.72, 1.36, 0.91 Å
Achieved interference contrast	60% (2 2 0), 73% (4 4 0), 85% (6 6 0)
Attainable angular resolution:
present geometry	6×10−6 s of arc (3×10−11 rad)
Improved geometry	10−6 s of arc (5×10−12 rad)
Momentum resolution	Δq≥2×10−10 nm−1

Summary

A new interferometer setup has been developed for the sensitive detection of beam deflections. It turns out that perfect crystal interferometers are extremely sensitive to beam deviations close to the Bragg condition. By further evolving the interferometer design an angular resolution of the order of 10−6 s of arc and momentum resolution Δq≥2×10−10 nm−1 becomes attainable. Large crystal interferometers consisting of six or more beam splitters open new application fields like the precise measurement of the Laue phase and associated fundamental quantities like the neutron–electron scattering length or the recently derived short-range interaction scattering length. Improving the angular sensitivity and phase precision is the basis for these measurements.