research papers
Larmor neutron diffraction with one precession arm
^{a}Department of Radiation Science and Technology, Faculty of Applied Sciences, Delft University of Technology, 2629JB Delft, The Netherlands
^{*}Correspondence email: m.t.rekveldt@tudelft.nl
A new variant of Larmor neutron diffraction, applying only a single precession arm in the initial beam, is proposed. The single arm, together with the polarizer and analyser and possible rotators, are mounted in front of the sample. The great advantage with respect to the case with another precession arm in the reflected beam is that magnetic samples can now be investigated, because depolarization and the Larmor phase change of the beam polarization in the sample after the analyser is no longer of importance for
The application has lower resolution than the doublearm precession geometry, but is still better than conventional diffraction instruments. The differences will be discussed.1. Introduction
In Larmor diffraction, introduced as Larmor precession in highresolution neutron diffraction in 1999 (Rekveldt & Kraan, 1999; Rekveldt et al., 2001), the wavelength of the diffracted neutrons is determined by the precession angle in a field region of known dimensions and known field strength. The inclinations of the front and end faces of the precession region are chosen such that, after Bragg reflection of the neutron beam, the spread in precession angles due to the spread in wavelengths is greatly reduced, which means that the precession can be followed to very large precession angles. The precession angle that can be followed from a reflection on a specific lattice plane distance d is just the inverse of the resolution Δd/d of that lattice spacing. In the original method, a doublearm precession geometry (DAG) is used, which means that the incident and diffracted beams both pass a precession device, as sketched in Fig. 1. In doing so, as explained below in more detail, the Larmor precession of a polarized neutron beam encodes both the wavelength and the incident neutron direction in such a way as to cancel the angular and wavelength dependence of the Larmor precession of the polarization of the diffracted neutrons. In this way the measured Larmor precession, which can be measured to very high values, directly yields the lattice spacing with extremely high precision (Δd/d ≃ 10^{−6}).
The method has been applied successfully in the determination of latticespacing variations in Al and Si wafers (Rekveldt et al., 2002), the determination of the absolute lattice spacing of Si powders and the study of the superalloy Inconel (Repper et al., 2010), the study of very small latticeconstant variations in MnSi at low temperatures (Pfleiderer et al., 2007), the study of the distribution of lattice constants and the antiferromagnetic moment in highpurity URu_{2}Si_{2} (Niklowitz et al., 2010), the investigation of the temperature dependence and distribution of lattice constants in CePt_{3}Si (Ritz et al., 2010; Sokolov et al., 2011), and most recently the investigation of the link between structural and magnetic ordering in the BaM_{2}(XO_{4})_{2} family (M = Co, Ni; X = As, P) (Martin et al., 2012) and of the magnetic ordering in metallic Ba(Fe_{1−x}Mn_{x})_{2}As_{2} (Innosov, 2013).
With somewhat less accuracy in latticespacing determination, Larmor diffraction can also be used in a much simpler setup that is discussed in this article. We will describe and discuss a singlearm variant (singlearm precession geometry, SAG), which has different properties but which is simpler to use and also has important advantages in diffraction studies of magnetized samples.
2. Instrument description
Although the DAG was historically first, from an instrumental point of view we prefer to describe here the SAG first and compare its properties with those of the DAG.
The precession arm consists of a parallelogramshaped precession area with induction B perpendicular to that area and an inclination angle θ_{0} of the front and end faces of the parallelogram with the main beam direction, and a diffracting sample with diffraction planes making a well defined angle ∊ with the inclined faces of the precession area, as sketched in Fig. 2. To enable precession around B, the polarization vector created by the polarizer P in general has to be rotated first over π/2 by a rotator to a direction perpendicular to B in the precession region and then rotated back before the analyser A by a second rotator. For simplicity in describing these elements, the polarizer, analyser and rotators have been drawn as two units, P and A.
Consider a neutron with wavelength λ, entering a single or polycrystalline sample at an angle θ with respect to the main axis, that reflects to an angular position x on the detector. When diffraction takes place from a crystal plane with lattice spacing d and an orientation θ_{0} + ∊ (where ∊ = ∊_{c} + ∊_{v}, with ∊_{c} a constant and ∊_{v} the variable part, varying within Δ∊), then the following parameters are of relevance for the measured polarization at x:
The angle θ represents the divergence of the incident beam, which we assume to be Gaussian with a width Δθ. In the case of a single crystal, the variable part ∊_{v} is simply the mosaic spread, and in a polycrystalline material it is the average deviation around the constant ∊_{c}. The variable ∊_{v} is determined by both the incident angle θ and the detection angle x′. The latter has an uncertainty Δx determined by the detector uncertainty and the finite size of the sample.
The polarization of the diffracted beam can be derived from the cosine of the precession angle averaged over θ and λ in the SAG
with
c = γh/m = 4.6368 × 10^{14} T^{−1} m^{−2}, where h is the Planck constant, γ the gyromagnetic ratio and m the mass of a neutron. The precession angle φ_{s}(θ, ∊, d, B) for some lattice spacing d and magnetic field B is no longer dependent on λ but only on θ and ∊.
We first discuss the situation for a pointlike detector with Δx = 0. Then it appears that x′ = 2∊_{v} + θ must be constant and ∊_{v} and θ are necessarily coupled variables. We can split the precession angle φ_{s}(θ, ∊, d, B) into a constant part, φ_{sc}(∊_{c}, x′, d, B), and a variable part, φ_{sv}(∊_{v}, ∊_{c}, d, B). The former determines the oscillation frequency in P as a function of B, and the latter expresses the damping
where we have used the firstorder expansion of cot(θ_{0} + θ) = cotθ_{0} − θ/sin^{2}θ_{0} for small θ. The variable part φ_{sv} can be made zero at a value of ∊_{c} = ∊_{c0} given by
However, the constant part φ_{sc}(φ_{c}, x′, d, B) now varies linearly in x′.
To compare these results with the case of the DAG, consider the sketch of the DAG in Fig. 1. We find here for the precession angle the sum of the precessions through two regions:
We now also split φ_{d}(θ, ∊, d, B) into a constant part, φ_{dc}(∊_{c}, x′, d, B), and a variable part, φ_{dv}(∊_{v}, ∊_{c}, d, B), again using 2∊_{v} − x′ = −θ, to yield
and here it appears that the variable part can be made zero at ∊_{c} = 0, while at this value the constant φ_{dc}(∊_{c}, x′, d, B) also does not change with varying x′.
The ratio φ_{dv}(∊_{v}, ∊_{c}, d, B):φ_{dc}(∊_{c}, x′, d, B) for the SAG and DAG determines the damping of the measured polarization as a function of B. This ratio equals the achievable resolution Δd/d of the setups. Fig. 3 shows the results for the SAG and DAG at x′ = 0 and θ_{0} = π/4. The figure shows that, outside the points ∊_{c0} where φ_{(s,d)v}(∊_{v}, ∊_{c}, d, B) is minimal, the resolution is roughly proportional to the variable part ∊_{v} with a proportionality factor about 0.01 times the deviation of ∊_{c} from ∊_{c0}.
The offset angle ∊_{c0}, where precession damping is minimal, is dependent on the inclination angle θ_{0}, as found from equation (6). This dependence is shown in Fig. 4, where the θ_{B0} = θ_{0} + ∊_{c0}, with minimal damping, is plotted as a function of the inclination angle θ_{0}, together with its values when the Bragg reflection is out of the drawing plane (see Fig. 2) at the two angles given in the figure. The offset angle ∊_{c0} can be chosen as required for an experiment.
It appears that the focusing condition derived in the
is present over an angular range greater than 5° out of the plane. This angle should not be allowed to become too large, in order to avoid the outofplane nonfocused contribution to the becoming significant. This relatively large outofplane range in the detection area may be useful in studying polycrystalline materials.Fig. 3 shows that the resolution of the SAG at the proper ∊_{c} value is about the same as for the DAG.
Now we will discuss the influence of the uncertainty Δx in the scattering angle x′ on the resolution of both the DAG and the SAG.
For the DAG, the resolution remains high over a relatively large angular range x′, because φ_{dc}(∊_{c}, x′, d, B) is firstorder independent of x′. This is not the case for the SAG, where φ_{sc}(∊_{c}, x′, d, B) changes linearly with x′ within the uncertainty Δx, so equations (4) and (5) change to
The inclusion of the detector uncertainty in φ_{sv}(∊_{v}, ∊_{c}, x′, d, B), which is not present in the DAG case, equations (8) and (9), is the main advantage of the DAG over the SAG.
For the SAG, at each position x_{0} = 2(θ_{0} + ∊_{c}), the damping of the precession due to a single Bragg reflection can now be written as
with
and
where we have assumed a Gaussian distribution for the initial divergence and α(∊_{c}, ∊_{v}, Δx) is the halfwidth of the Gaussian exp[−(φ_{v} − φ_{c}/αφ_{c})^{2}]. The maximum resolution expressed by α(∊_{c}, Δ∊, Δx) is just Δx if Δx < Δ∊. A resolution of Δx = 2 × 10^{−3} may be achievable.
Fig. 5 shows the resolution function α(∊_{c}, Δx) as a function of ∊_{c} for two different values of Δx and ∊_{v} in three different combinations, together with the resolution expected using conventional diffraction with the same angular parameters. The figure shows that, at the focusing condition ∊_{c0} = −0.32, the resolution is fully determined by Δx and, at ∊_{c} = 0, it is fully determined by ∊_{v}. Depending on the value of ∊_{v}, we find a range around ∊_{c0} of − 0.33 < ∊_{c} < −0.25 where the resolution is more or less constant and only weakly dependent on the initial divergence Δθ, which may be much larger [see equations (10)–(14)]. Knowing the instrumental resolution at a certain detector position, the extra damping caused by a diffracting sample as a function of the precession angle directly delivers the Fourier transform of the line shape of that particular reflection.
3. Simulations with SAG
3.1. Polarization for a single crystal
Simulations have been performed in the singlecrystal case at different constant angles ∊_{c} around ∊_{c0}, with a mosaic spread Δ∊ = 0.01 rad, and with Δx = 10^{−3} around x′ = 0, on a highresolution positionsensitive detector. Fig. 6(a) shows the amplitude of A(Δx, Δ∊, ∊_{c}, φ_{sc}) as a function of the precession angle φ_{sc} at three different ∊_{c} values around ∊_{c0}. Fig. 6(b) shows the precession at ∊_{c} = ∊_{c0} at three values of x′ over a small phase interval, indicating the shift in phase with changing x′. At ∊_{c} = ∊_{c0}, the precession phase can be followed to about 150 revolutions, and the lattice spacing and changes therein can be determined with an accuracy Δd/d of about 10^{−5}. Thus, the phase of the damped precession and changes therein can still be determined with reasonable precision. For that purpose one does not need to measure the whole B range, only one period of the precession angle. With a phase accuracy of 10 mrad the lattice spacing can be determined with a relative accuracy of Δd/d = 10^{−5}.
3.2. Polarization for polycrystalline samples
In this section we wish to show that the SAG can be used as an inversegeometry diffractometer. We will use a white beam as the initial beam, with a normalized wavelength spectrum given by
Here λ_{T} = h/(mv_{T}), v_{T} = (2kT/m)^{1/2}, T = 300 K, k is the λ is the neutron wavelength and m is the neutron mass. We measure the polarization of the scattered beam at various scattering angles. At each scattering angle, the diffracted intensity of the polycrystal can be considered as coming from a single crystal with a constant average value ∊_{c} = (x_{0}/2) − θ_{0} and a variable part ∊_{v} with a distribution width given by
We use in our simulations a facecentred cubic (f.c.c.) polycrystalline sample as an example, with the following specifications: lattice planes (111), (200), (220), (311), (222) and (400), with a lattice spacing of 0.3 nm.
Because in a polycrystal at each angle x there are lattice planes in reflection, we find for the average polarization by all precession angles φ_{sc}(∊_{c}, d, B) from equations (10)–(14) using ∊_{c} = (x_{0}/2) − θ_{0} the average of the N_{d} weighted contributions from the different lattice planes:
with
Here, λ_{d} = 2dsin(x_{0}/2) and N_{d} is the multiplicity factor of plane d in an isotropic distribution in 4π, for which we use N_{111} = N_{222} = 8, N_{220} = 12, N_{200} = N_{400} = 6 and N_{311} = 24. Note that the damping as a function of B will be different for each Bragg reflection, meaning that for larger B values the oscillations from the smaller d values are predominantly present. The measured polarized intensity in the detector range Δx is
with
and
Here, V(x_{0}, t) contains the initial intensity of the white beam I_{w0} with a wavelength spectrum given by equation (15) using λ_{T} = 0.2 nm, losses at the polarizers A_{p}, the beam height b_{h}, the beam width b_{w}, the wavelength region dλ/λ = cot(x_{0}/2)Δθ contributing to each plane reflection, the reflectivity R per plane, the measuring time t per field point B in seconds, and the quantity dΩ determining the fraction of the total scattered neutrons captured by the detector of height h_{d} and width equal to the beam width at a distance L_{d} from the sample. In our simulation we took the parameters as given in Table 1. This results in
Using these numbers, Fig. 7 shows the simulated intensity as a function of the precession magnetic field B (in mT) at various values of x_{0} around 2(θ_{0} + ∊_{c0}). For the distance L between the inclined precession faces, we used L = 1 m. In the intensity I(x, B, t) we included a random number with amplitude ΔI = I(x, B, t)^{1/2} to account for the expected statistical spread at each point.

From Fig. 7 one sees that, for B > 20 mT, the damping becomes considerable for detector angles deviating more than 0.1 rad from the optimal x = 2(θ_{0} + ∊_{c0}) = 0.93 rad. The number of visible Larmor precessions is inversely proportional to the observable line width. By Fourier transforming the spectra of Fig. 7, we find the wavelengthdependent intensities shown in Fig. 8.
In the simulation, the instrumental parameters of Table 1 are used with a datacollection time of 10 000 s for 50 similar spectra simultaneously over a total detector angular range of 6°, 50 times the detector width used in the calculations. When the spectra are plotted against λ/sin(x_{0}/2) and averaged over the whole angular range, the shown spectrum represents the average measured in only 200 s. It has already been shown in §2 that, if a detector height ten times larger is used, the focusing condition in ∊_{c0} changes by a maximum of the order of 1% (see Fig. 4), which means that in an optimal detector design the shown spectrum can be obtained in about 20 s measuring time with a resolution much higher than in conventional diffraction.
4. Experiments on single crystals
Experiments have been carried out on an improvised Larmor diffractometer at the reactor institute in Delft, as sketched in Fig. 9. The setup makes use of the resonance technique, where the precession angle is determined by the resonance fields B in two coils a distance L_{c} apart, similar to what was described by Dalgliesh et al. (2011). The precession angle φ_{s}(θ, λ, B) = 2cBL_{c}dcosθ_{B} is varied in large steps by the resonance field B and in smaller steps by the distance L_{c} between the magnets by shifting M_{1} in the direction of M_{2}.
Because we could not vary the resonance frequency automatically, we varied the distance L_{c} only between 80 and 40 cm at various constant fields B and corresponding resonance frequencies to obtain a large enough range of precession angles. At the sample position we placed successively an Si crystal and an Fe_{3}Si single crystal, at the same scattering angle θ = 2θ_{0} = 84° and with different mosaic spreads. Fig. 10 shows the results for the Si crystal. The Si crystal consisted of a stack of thin wafers with nearly the same orientation, bent around an axis in the to increase the diffracted intensity by deformation. For the damping constant α of the amplitude of the precession defined in equation (13), we found α(...) = 4.7 × 10^{−3} using an initial polarization of 0.7, shown also in the figure. In the figure the measurements at different fields overlap, which means that in the top part of the figure the measurement at 5 mT is fully hidden by the neighbouring fields.
This α(...) value corresponds to an uncertainty in the wafer orientation of 0.28 mm over the diameter of 50 mm, which is a mosaic spread that may be expected for a stack of separately polished wafers.
Next, we investigated how the amplitude of the precession changes with deviation of the ). With a fixed open detector over about 10°, we measured the reflected intensity and polarization as a function of the change in Δθ_{B0}. At the minimum and maximum values of Δθ_{B0}, where the intensity drops, the polarization amplitude is increased by the effectively smaller angle range contributing to polarization in the detector. In the range where the intensity is roughly constant, the polarization amplitude changes from about 0.1 to 0.05, which corresponds to a change in α(...) of 4.7 to 5.4 × 10^{−3}.
from the inclination angle (Fig. 11Next, we investigated the reflected intensity and polarization of an Fe_{3}Si single crystal at the same of 42°. Before measuring the fielddependent polarization, we adjusted the crystal by hand to optimize the detector intensity. Because here we have contributions of different order reflections with different wavelengths, Fig. 12 shows the measured polarization as a function of f_{d} = 2cLBsinθ_{B0}, together with the calculated polarization of the three order reflections of Bragg planes (200), (300) and (400) of Fe_{3}Si in a field of 2.5 mT. The quantity f_{d} is related to the Larmor phase φ_{d} of each reflection from the lattice plane d_{hkl} by φ_{d} = f_{d}d. The polarization of the sum of the three orders, P(f_{d}), fits perfectly with the measured data, with a fitted intensity ratio for the orders I_{200}:I_{300}:I_{400} = 30:15:13, a lattice constant d_{100} = 0.565 nm and polarization amplitude damping according to
Here, α(...) is again the uncertainty in the being the mosaic spread of the crystal. In the fit we have used α(...) = 4.4 × 10^{−3}, close to the damping factor found in the Si diffraction in Fig. 10. For the measurement accuracy of the found lattice spacing, which is only dependent on the phase of the measured polarization, we estimate from this preliminary experiment with three contributing reflections Δd_{100} = 5 × 10^{−4} nm.
5. Discussion
A comparison of the singlearm Larmor diffractometer with the doublearm geometry shows that the DAG is applicable with very little damping at large precession angles and relatively large deviation angles ∊_{c} around ∊_{c0} = 0, while the SAG is applicable with somewhat more damping around ∊_{c0} ≠ 0, albeit with the precession angle linearly dependent on x, making this mode strongly dependent on a highresolution positionsensitive detector and small angular dimensions of the sample. Note that the resolutions of both modes are, to the first order, independent of the incoming beam divergence Δθ. The value of ∊_{c0} can be chosen by adjusting the inclination angle θ_{0}. Both the DAG and the SAG may be candidates for an inversegeometry diffraction instrument with an initial white beam, with excellent resolution for the DAG and a still high resolution for the SAG, but the latter can have much better counting statistics because it has a much larger angular region x around ∊_{c0} and a high angular region perpendicular to the plane determined by the angle x. The latter may be problematic using the DAG because a polarization analyser should also be applied over the large angular region. The large detector solid angle may make the SAG an excellent candidate for an inversegeometry diffraction instrument with an initial white beam for studying polycrystalline samples, because its resolution and intensity promise to be better than those of conventional diffraction instruments. In comparison with the DAG, the SAG has an advantage for the field arrangement around the precession regions, because it needs only one precession region in front of the sample, and no field arrangements around the sample and detector positions that are fully outside the precession region. As a consequence, the SAG can also be used for magnetic samples without difficult modifications of the setup. The SAG is especially applicable in magnetic and nonmagnetic samples. We estimate that sufficient accuracy can be obtained to study relative lattice plane changes larger than 10^{−5}. The SAG is more flexible than the DAG and can be built as a special tool to be implemented in the primary beam of a diffractometer. To compare the DAG and SAG with normal diffraction, both can work with a white beam, thus much higher intensity. Using a large detection angle, in principle all wavelengths of the initial white beam are used in the data collection. Using the latter option, one may concentrate on the variation of different lattice spacings simultaneously by measuring the change in the precession phases of the different lattice spacings in a small field range at high field, thus saving much measuring time in comparison with standard diffraction.
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