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Neutron Charge Radius Deduced from Interferometric Bragg Reflection Technique

The possibility of the determination of the neutron mean square charge radius from high-precision thermal-neutron measurements of the nuclear scattering length and of the scattering amplitudes of Bragg reflections is considered. Making use of the same interferometric technique as Shull in 1968, the scattering amplitudes of about eight higher-order Bragg reflections in silicon could be measured without contamination problem. This would provide a value of the neutron charge radius as precise as the disagreeing Argonne-Garching and Dubna values, as well as a Debye-Waller factor of silicon ten times more precise than presently available.

The so-called mean square charge radius of the neutron r 2 n is an important structure constant which reflects the internal charge structure of the neutron. In terms of the generally used form factors, r 2 n is related to the derivative of the Sachs form factor G n E at vanishing transferred momentum Q, Low-energy neutron-atom scattering experiments have been found useful [1,2] to determine r 2 n via high-precision measurements of the neutron-electron scattering length b ne , where m n is the mass of the neutron. Because of the fundamental importance of these quantities for our understanding of the nucleon many ambitious experiments have been performed in the last decades (see e.g. [3] and references therein).
Despite these efforts the determination of b ne is still unsatisfactory because there exist two sets of results which differ more than three standard deviations from each other. In this paper we propose an independent method based on high-precision measurements of neutron Bragg reflections on silicon. Assuming for all higher-order reflections a precision similar to that achieved by Shull [4] one can solve the discrepancy in the values of b ne .
In a first theoretical estimate, b ne is expected to be given by the Foldy scattering length corresponding to the well-known value of the anomalous magnetic moment of the neutron [6]. The actual b ne value can be deduced from low-energy neutron-atom scattering, for which the "scattering length" reads [1] b where b nuclear is the nuclear-interaction scattering length, b nucleus is the neutron-nucleus scattering length (nuclear and electrostatic interactions), Z is the atomic number and f (Q) is the atomic form factor normalized to the forward direction: f (0) = 1. In Eq. (4), the Q-dependent electrostatic term is typically three orders of magnitude smaller than the constant nuclear term, hence it is reasonable to speak of "scattering length" despite the Q dependence. To get a sufficiently high sensitivity to this term scattering processes at Q = 0 (hence 0 < f (Q) < 1) and high Z should be chosen.
The best experimental values of b ne (see e.g. Table I in [7]) can be grouped into two sets. Those deduced from thermal-neutron angular scattering on noble gases [8] and from thermal-and epithermal-neutron transmission on liquid lead and bismuth [3,9]  obtained from the Bi-transmission experiments [3,10] is well understood [7,9,12], the Bragg-reflection method of Ref. [11] has never been revisited. In the latter experiment, the where (hkl) are the Miller indices, a 0 is the side of the conventional cubic unit cell, θ hkl is the Bragg angle and λ is the neutron wavelength.
For two different crystals, two very different values of b ne were obtained in Ref. [11]: i.e., b ne = −1.06 and −2.2 × 10 −3 fm, which led the authors to postulate the existence of an additional scattering process. We rather consider that this discrepancy reveals the systematic uncertainty of the experiment, where b(Q hkl ) is deduced from an integral-intensity measurement of the Bragg peak. This systematic uncertainty has probably been underestimated in Ref. [11] for different reasons: (i) extinction is neglected, (ii) the uncertainty on the temperature factor (which is shown below to be of crucial importance) is neglected.
Consequently, the accuracies obtained for b(Q hkl ) (typically 0.01 fm) seem unrealistic as compared with accuracies generally obtained by this kind of measurement (typically 0.1 fm) [13]. A hint to our conjecture is also given by the strong disagreement between other results [14] obtained inter alia with an integral-intensity measurement on monocrystals, and the Christiansen-filter results of Ref. [15] for W isotopes.
Let us now consider the method proposed by Shull in Ref. [4], which also aims at measuring b(Q hkl ) for Bragg reflections on monocrystals. We refer to Ref. [4] for details about the experimental setup. A collimated full-spectrum incident beam is Bragg reflected on a monocrystal blade (typical thickness t = 1 cm) in Laue transmission geometry. The wavelength selection is made through the Bragg condition (6), the crystal and the detector being moved simultaneously to form angles θ and 2θ respectively with the incident-beam direction.
As shown in the dynamical theory of Bragg reflection for neutrons [6,13], the reflected beam inside the crystal consists of two coherent waves, the so-called Pendellösung (pendulum solutions), interfering with each other and generating Pendellösung oscillations. An entrance and a scanning gadolinium slits (width 0.13 mm) are placed on both faces of the crystal to extract these fringes. Making use of the symmetry of the interference pattern a very precise alignment of the scanning slit with respect to the entrance slit is possible. The intensity at the center of the interference pattern is then measured as a function of the wavelength and compared with the theoretical expression. This expression may be obtained either by the Kato spherical-wave dynamical theory [16] or by solving the Takagi-Taupin equations [6].
For a perfectly-flat crystal and with infinitely-narrow slits, this intensity reads where I 0 is the incident intensity, J 0 is a Bessel function and |F hkl | is the unit-cell structure factor of the crystal. This expression has to be slightly modified to take into account the finite curvature of the crystal, which adds the curvature radius R (typically 20 km) as a parameter to be fitted to the data, and to take into account the finite opening of the slits [16]. Typically, forty fringes are observed for neutrons ranging the full spectrum of a thermal reactor. The structure factor can then be deduced with high precision from the period of these fringes. Shull's method is thus in essence an interferometric method, which explains why it provides very high accuracies (contrary to the integral-intensity method).
For a diamond-structure crystal like silicon, the unit cell consists of four elementary cells with two atoms. The corresponding structure factor reads [13] where b meas (Q) is related to b(Q) through the Debye-Waller temperature factor [13] b To get a precise b(Q hkl ) value, as needed to estimate b ne , one needs precise values of both the structure factor F hkl and the temperature parameter B. For instance, for the (111) reflection on Si (Z = 14, a 0 = 5.43072Å), Shull's method provides [16] b meas (Q 111 ) = 4.1053(8) fm.
Using the Debye-Waller factor obtained by measuring X-ray Pendellösung fringes at room where three additional units in the error are due to the error on B. Equation (9) shows that this additional uncertainty increases exponentially with Q 2 (see Fig. 1).
Following the approach of Ref. [16], we can deduce b nuclear from value (11) with Eq.
(4), using the Argonne-Garching b ne and the atomic form factor [18]: f (Q 111 ) = 0.7526 (a precise value of f (Q) is actually not required [1]). This yields b nuclear = 4.1495(11) fm (which slightly differs from the value of Ref. [16] because of the more recent temperature factor), in agreement with the more precise value obtained recently by non-dispersive-sweep neutron interferometry [19] b nuclear = 4.1507(2) fm.
Conversely, b ne can be deduced from values (11) and (12) with Eq. (4), which provides (see This value agrees with the Argonne-Garching value but does not exclude the Dubna value (see Fig. 2). The same method applied to germanium (diamond structure with a 0 = 5.6575 A) using b nuclear = 8.1929 (17) fm [20], b meas (Q 111 ) = 8.0829(15) fm [21], B = 0.57(1)Å 2 [22] and f (Q 111 ) = 0.8542 [18], leads to the value b Ge ne = 0.28(83) × 10 −3 fm. Although measurements on Ge (Z=32) could in principle lead to a value of b ne two times more precise than on Si (Z = 14), this advantage is compensated by the two-times-larger scattering length, which implies a two-times-larger absolute error on b(Q hkl ), the relative accuracy of Shull's method being 0.02% in both cases. Moreover, no b nuclear value for Ge of comparable precision to (12) is available at present. Hence we only consider Si in the following.
In Fig. 1, values (11) and (12)  Let us now determine the possibly-observable reflections. A diamond-structure crystal has a face-centered cubic lattice, which implies that all Miller indices have to be either even or odd [13]. Moreover, for a diamond structure, the structure factor (8) leads to three types of reflections: (i) forbidden when h + k + l = 2 + 4n, where n is natural, (ii) weak when h + k + l is odd, (iii) strong when h + k + l = 4 + 4n. A thermal-neutron beam has typically a maximum flux for λ = 1.2Å and the accessible angular range of the setup is assumed to be 0 ≤ 2θ ≤ 110 • . These conditions combined with Eq. (6) imply that sixteen reflections between (111) and (642) could be measured with reasonable intensities for Si. However, if the incident beam has a typical full spectrum 0.8 ≤ λ ≤ 2.5Å, contamination has to be taken into account. For instance, while the (111) reflection is pure for 15 ≤ 2θ ≤ 45 • thanks to the absence of the (222) reflection which is forbidden (hence the result of Refs. [4,16]), (333) and (444) would be measured on 45 ≤ 2θ ≤ 110 • and 61 ≤ 2θ ≤ 110 • respectively. This mixing implies that the reflected intensity would not be described by Eq. (7) and that no structure factor could easily be extracted. Among the sixteen considered reflections, seven could be contaminated. The remaining nine reflections are listed in Table I Table I); the intensities range between 0.7 and 1.7 times the intensity of the already-measured (111) weak reflection.
Finally, Eq. (7) allows to estimate the number of fringes which would be measured for each of them. For instance, 44 fringes would be observed for the (711) reflection if t = 1 cm.
These considerations show that these higher reflections should not be more complicated to measure than the (111) reflection already measured in Refs. [4,16]. Hence, we can reasonably assume that the precision on b meas would be equal to that of Ref. [16]: 0.0008Å.
With this assumption, the attainable precision for B and b ne can be estimated. For B, we apply the approximate linear relation [Eqs. (4) and (9)] while for b ne we use Eq. (4). The uncertainty of a linear-fit slope can be calculated from the abscissas of the experimental points and from their errors [23]. Measuring the three strong reflections would give while with the eight new reflections of Table I, one gets σ B = 0.00027Å 2 , σ bne = 0.06 × 10 −3 fm.
The temperature factor would be ten times more precise than the most precise value (10) and the b ne value would be nearly as precise as the Argonne-Garching and Dubna values (see Fig. 2). Since the present method is similar to the Dubna method on W, such a new result would certainly be of great help to decide the b ne discrepancy.
In conclusion, the measurement of Pendellösung fringes in Bragg reflections is a very powerful technique. This is known for X-rays [17,24]   The simulated points correspond to the reflections of Table I; they are calculated with the Argonne-Garching b ne value, with error bars only due to the uncertainty on the temperature factor (10).   [18], λ the neutron-wavelength interval, 2θ the corresponding angles between the incident and reflected beams. |F hkl | 2 is the squared structure factor which provides the reflection intensity through Eq. (8).