Measurement of the coherent neutron scattering length of [.sup.3]He

Journal of Research of the National Institute of Standards and Technology, May-June, 2005 by W. Ketter, W. Heil, G. Badurek, M. Baron, R. Loidl, H. Rauch

By means of neutron interferometry the s-wave neutron scattering length of the [.sup.3]He nucleus was re-measured at the Institut Laue-Langevin (ILL). Using a skew symmetrical perfect crystal Si-interferometer and a linear twin chamber cell, false phase shifts due to sample misalignment were reduced to a negligible level. Simulation calculations suggest an asymmetrically alternating measuring sequence in order to compensate for systematic errors caused by thermal phase drifts. There is evidence in the experiment's data that this procedure is indeed effective. The neutron refractive index in terms of Sears' exact expression for the scattering amplitude has been analyzed in order to evaluate the measured phase shifts. The result of our measurement, [b'.sub.c] = (6,000 [+ or -] 0.009) fm, shows a deviation towards a greater value compared to the presently accepted value of [b'.sub.c] = (5.74 [+ or -] 0.07) fm, confirming the observation of the partner experiment at NIST. On the other hand, the results of both precision measurements at NIST and ILL exhibit a serious 12[sigma] (12 standard uncertainties) deviation, the reason for which is not clear yet.

Key words: elastic neutron scattering; few body systems; neutron interferometry; properties of the [.sup.3]He nucleus; real part of the coherent scattering length.

1. Introduction

The scattering length a is defined as the zero-energy limit of the scattering amplitude f. Owing to the spin dependence of the underlying strong interaction, the neutron scattering length must in general also be spin dependent. The spin independent part of the quantity is referred to as the coherent scattering length [a.sub.c]. In order to expand the scattering formalism to absorption, the scattering length is made complex, a = a' - ia". For a homogeneous, isotropic, monatomic and dilute gas sample of particle number density [rho], neutron scattering can be described by the so-called coherent wave that scatters off a macroscopic optical potential. Since this effective potential is weak in general, one can define a neutron refractive index and treat the scattering problem similarly to geometrical light optics. When describing the scattering off a nucleus of mass number A in the laboratory system, one often includes the kinematical transformation from the centre of mass system by defining the bound scattering length b = [[A + 1]/A]a. Sears [1] has given an exact expression of the neutron refractive index in terms of the scattering amplitude. Taking into account that the scattering amplitude f and the scattering length a are proportional only in the zero-energy limit, but that in general higher-order terms in the neutron wave number k contribute, one arrives at the following expression for the real (nonabsorptive) part of the refractive index in terms of the bound coherent scattering length [b.sub.c]:

n' = 1 - [[2[pi]]/[k.sup.2]][rho][b'.sub.c][1 - 2k[b".sub.c] + O([k.sup.2])]. (1)

Since scattering lengths are of the order of a few femtometers, the second term in the bracket is of the order of [10.sup.-4] for thermal neutrons and the term in [k.sup.2] can safely be ignored for our purposes.

In the following section, we describe an interferometric experiment that measures the phase shift

[DELTA][phi] = (n' - 1)kd (2)

that the partial wave function of a neutron of kinetic energy [E.sub.k] = [h.sup.2][k.sup.2]/2m experiences when passing through a sample of thickness d.

2. Experiment

The experiment was performed in May 2002 at the S18 CRG facility at ILL using the skew symmetrically shaped perfect crystal interferometer IFM4. The setup is essentially the same as described in [2], the most important changes being a massive water-cooled Helmholtz coil of 50 cm diameter with the interferometer at its centre and two layers of Mylar (1) foil between the coils' body and the interferometer. The coil was initially installed for polarized interferometry but turned out to stabilize the instrument's temperature considerably when operated.

[FIGURE 1 OMITTED]

On their optical path through the interferometer, the partial beams gather a phase difference [DELTA][phi] that causes modulations of the detectable intensity behind the instrument:

[I.sub.0,H] [proportional] 1 + cos[DELTA][phi]. (3)

By rotating of a plane parallel Al-plate through the divergent partial neutron beams after the dividing plate of the interferometer, an additional relative phase [DELTA][[phi].sub.A1] between the partial wave functions can be superimposed. The total phase shift is then the sum of this phase shifter contribution, a sample contribution [DELTA][[phi].sub.sample] and an additional, time-dependent instrumental intrinsic phase [DELTA][[phi].sub.int]:

[DELTA][phi] = [DELTA][[phi].sub.A1] + [DELTA][[phi].sub.sample] + [DELTA][[phi].sub.int]. (4)

Modulating the total phase by means of the phase shifter enables one to extract both modulus and sign of [DELTA][[phi].sub.sample] + [DELTA][[phi].sub.int]. The intrinsic phase shift of the instrument can be monitored when the phase shift through an identical but evacuated sample cell is measured periodically. The skew symmetry of the instrument has the consequence that the partial beams are almost parallel in the volume where the sample is placed, see Fig. 1. Therefore, the aluminum windows of the gas container do not act as phase shifter and any misalignment of the container is compensated. Further, misalignment of the cell to better than 2.5[degrees] (which is easily achieved using a theodolite) results in an increase of the effective sample thickness of less than [10.sup.-4].