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ISSN: 1600-5767

Larmor neutron diffraction with one precession arm

aDepartment of Radiation Science and Technology, Faculty of Applied Sciences, Delft University of Technology, 2629JB Delft, The Netherlands
*Correspondence e-mail:

(Received 4 July 2013; accepted 13 December 2013; online 30 January 2014)

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 diffraction analysis. The application has lower resolution than the double-arm 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 high-resolution neutron diffraction in 1999 (Rekveldt & Kraan, 1999[Rekveldt, M. T. & Kraan, W. H. (1999). J. Neutron Res. 8, 52-70.]; Rekveldt et al., 2001[Rekveldt, M. T., Keller, T. & Golub, R. (2001). Europhys. Lett. 54, 342-346.]), 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 double-arm precession geometry (DAG) is used, which means that the incident and diffracted beams both pass a precession device, as sketched in Fig. 1[link]. 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).

[Figure 1]
Figure 1
A sketch of the DAG. P and A are, respectively, the polarizer and analyser producing and analysing the polarization component perpendicular to the precession field B. The precession regions PA have faces inclined by θ0 to the main axis and approximately parallel to the diffraction plane of the sample S, which makes an angle with θ0.

The method has been applied successfully in the determination of lattice-spacing variations in Al and Si wafers (Rekveldt et al., 2002[Rekveldt, M. T., Kraan, W. & Keller, T. (2002). J. Appl. Cryst. 35, 28-33.]), the determination of the absolute lattice spacing of Si powders and the study of the superalloy Inconel (Repper et al., 2010[Repper, J., Keller, T., Hofmann, M., Krempaszky, C., Petry, W. & Werner, E. (2010). Acta Mater. 58, 3459-3467.]), the study of very small lattice-constant variations in MnSi at low temperatures (Pfleiderer et al., 2007[Pfleiderer, C., Böni, P., Keller, T., Rössler, U. K. & Rosch, A. (2007). Science, 316, 1871-1874.]), the study of the distribution of lattice constants and the antiferromagnetic moment in high-purity URu2Si2 (Niklowitz et al., 2010[Niklowitz, P. G., Pfleiderer, C., Keller, T., Vojta, M., Huang, Y. K. & Mydosh, J. A. (2010). Phys. Rev. Lett. 104, 106406.]), the investigation of the temperature dependence and distribution of lattice constants in CePt3Si (Ritz et al., 2010[Ritz, R., Mühlbauer, S., Pfeiderer, C., Keller, T., White, J., Laver, M., Forgan, E. M., Cubitt, R., Dewhurst, C., Niklowitz, P. G., Prokofiev, A. & Bauer, E. (2010). J. Phys. Conf. Ser. 200, 012165.]; Sokolov et al., 2011[Sokolov, D. A., Ritz, R., Pfeiderer, C., Keller, T. & Huxley, A. D. (2011). J. Phys. Conf. Ser. 273, 012085.]), and most recently the investigation of the link between structural and magnetic ordering in the BaM2(XO4)2 family (M = Co, Ni; X = As, P) (Martin et al., 2012[Martin, N., Regnault, L.-P. & Klimko, S. (2012). J. Phys. Conf. Ser. 340, 012012.]) and of the magnetic ordering in metallic Ba(Fe1−xMnx)2As2 (Innosov, 2013[Innosov, D. (2013). Phys. Rev. B, 87, 224425.]).

With somewhat less accuracy in lattice-spacing determination, Larmor diffraction can also be used in a much simpler setup that is discussed in this article. We will describe and discuss a single-arm variant (single-arm 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 parallelogram-shaped precession area with induction B perpendicular to that area and an inclination angle θ0 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[link]. 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.

[Figure 2]
Figure 2
A sketch of the SAG. P and A are, respectively, the polarizer and analyser setting and analysing the polarization component (in the plane of the figure) perpendicular to the precession field B (perpendicular to the plane of the figure). The precession region has faces inclined by θ0 to the main axis and a distance L between the faces. Intensity is measured on the detector (Det) at an angle x. The angle is the deviation angle between the diffracting plane and the inclination angle θ0.

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 θ0 +  (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:

[\eqalign{ & x = {x_0} + x^{\prime} = 2{\theta _{\rm B}} - \theta = 2 \left ({\theta _0} + {\varepsilon _{\rm c}} + {\varepsilon _{\rm v}} \right) + \theta, \cr & {x_0} = 2{\theta _{\rm B0}} \equiv 2 \left ({\theta _0} + {\varepsilon _{\rm c}} \right), \cr & {\rm{thus }} \,\, x^{\prime} = \theta + 2{\varepsilon _{\rm v}}. } \eqno(1)]

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

[P = \left \langle {\cos {\varphi _{\rm s}} \left (\theta, \varepsilon, d, B \right)} \right \rangle _{\rm av}, \eqno(2)]


[\eqalignno{{\varphi _{\rm s}} \left (\theta, \varepsilon, d, B \right) & = \, c \lambda \left (\theta, \varepsilon \right) B {L \over {\sin \left ({{\theta _0} + \theta } \right)}} \cr & \equiv \, 2 c B L d {{\sin \left ({\theta _0} + \theta + \varepsilon \right)} \over {\sin \left ({\theta _0} + \theta \right)}} \cr & = \, 2 c B L d \left [{\cos \varepsilon + \sin \varepsilon \cot \left ({\theta _0} + \theta \right)} \right]. & (3)}]

c = γh/m = 4.6368 × 1014 T−1 m−2, where h is the Planck constant, γ the gyromagnetic ratio and m the mass of a neutron. The precession angle φs(θdB) 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 point-like detector with Δx = 0. Then it appears that x′ = 2v + θ must be constant and v and θ are necessarily coupled variables. We can split the precession angle φs(θ, , d, B) into a constant part, φsc(c, x′, d, B), and a variable part, φsv(v, c, d, B). The former determines the oscillation frequency in P as a function of B, and the latter expresses the damping

[\varphi _{\rm sc} (\varepsilon _{\rm c}, x^{\prime}, d, B) = 2 c B L d \left ({\cos {\varepsilon _{\rm c}} + \sin {\varepsilon _{\rm c}} \cot {\theta _0} - {{ x^{\prime} \sin {\varepsilon _{\rm c}}} \over {{{\sin }^2} {\theta _0} }}} \right), \eqno (4)]

[\eqalignno{ & {\varphi _{\rm sv}} ({\varepsilon _{\rm v}}, {\varepsilon _{\rm c}}, d, B) \cr & \, = 2 c B L d \left [{{\varepsilon _{\rm v}} \left ({ - \sin {\varepsilon _{\rm c}} + \cos {\varepsilon _{\rm c}} \cot {\theta _0} + {{2\sin {\varepsilon _{\rm c}}} \over {{{\sin }^2} {\theta _0} }}} \right)} \right] , &(5)}]

where we have used the first-order expansion of cot(θ0 + θ) = cotθ0θ/sin2θ0 for small θ. The variable part φsv can be made zero at a value of c = c0 given by

[\tan {\varepsilon _{\rm c0}} = {{\cos {\theta _0} \sin {\theta _0}} \over {{{\sin }^2}{\theta _0} - 2}} \,\, {\rm or } \,\, 2\tan \left ({{{x_0}} \over 2} \right) = \tan {\theta _0}. \eqno(6)]

However, the constant part φsc(φc, 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[link]. We find here for the precession angle the sum of the precessions through two regions:

[\eqalignno{ {\varphi _{\rm d}} (\theta, \varepsilon, d, B) = & \, c \lambda (\theta, \varepsilon) B \left [{{L \over {\sin \left ({{\theta _0} + \theta } \right)}} + {L \over {\sin \left ({{\theta _0} + \theta + 2\varepsilon} \right)}}} \right] \cr = & \, 2 c B L d \left [{{{\sin ({\theta _0} + \theta + \varepsilon)} \over {\sin ({\theta _0} + \theta)}} + {{\sin ({\theta _0} + \theta + \varepsilon)} \over {\sin ({\theta _0} + \theta + 2\varepsilon)}}} \right] \cr = & \, 2 c B L d \bigl \{ {2\cos \varepsilon} \cr & \, {+ \sin \varepsilon \left [{\cot ({\theta _0} + \theta) - \cot ({\theta _0} + \theta + 2\varepsilon)} \right] } \bigr \}. &(7)}]

We now also split φd(θ, , d, B) into a constant part, φdc(c, x′, d, B), and a variable part, φdv(v, c, d, B), again using 2v − x′ = −θ, to yield

[\eqalignno{& {\varphi _{\rm dc}} ({\varepsilon _{\rm c}}, x^{\prime}, d, B) \cr & \, = 2 c B L d \biggl \{ {2\cos {\varepsilon _{\rm c}} + \sin {\varepsilon _{\rm c}}} \left [\cot {\theta _0} - \cot ({\theta _0} + 2{\varepsilon _{\rm c}}) \right] \cr & \quad - x^{\prime} \sin {\varepsilon _{\rm c}} \left [{{1 \over {{{\sin }^2}{\theta _0}}} - {1 \over {{{\sin }^2}({\theta _0} + 2{\varepsilon _{\rm c}})}}} \right] \biggr \}, & (8)}]

[\eqalignno{& {\varphi _{{\rm dv}}} ({\varepsilon _{\rm v}}, {\varepsilon _{\rm c}}, d, B) \ = 2 c B L d \biggl \{ - 2{\varepsilon _{\rm v}} \sin {\varepsilon _{\rm c}} \cr & + {\varepsilon _{\rm v}} \cos {\varepsilon _{\rm c}} \left [{\cot {\theta _0} - \cot ({\theta _0} + 2{\varepsilon _{\rm c}}) } \right] + \sin {\varepsilon _{\rm c}} \left ({{{2{\varepsilon _{\rm v}}} \over {{{\sin }^2}{\theta _0}}}} \right) \biggr \}, & (9)}]

and here it appears that the variable part can be made zero at c = 0, while at this value the constant φdc(c, x′, d, B) also does not change with varying x′.

The ratio φdv(v, c, d, B):φdc(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[link] shows the results for the SAG and DAG at x′ = 0 and θ0 = 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.

[Figure 3]
Figure 3
The resolution plotted as a function of c at θ0 = π/4, constant position x′, and maxima of v = 0.01 and 0.005 for SAG and DAG, respectively.

The offset angle c0, where precession damping is minimal, is dependent on the inclination angle θ0, as found from equation (6)[link]. This dependence is shown in Fig. 4[link], where the Bragg angle θB0 = θ0 + c0, with minimal damping, is plotted as a function of the inclination angle θ0, together with its values when the Bragg reflection is out of the drawing plane (see Fig. 2[link]) at the two angles given in the figure. The offset angle c0 can be chosen as required for an experiment.

[Figure 4]
Figure 4
The average Bragg angle θB0 = θ0 + c0 where minimal damping is present, plotted as a function of the inclination angle θ0, in the drawing plane of the figure (red), 0.3 rad out of the plane (blue) and 0.5 rad out of the plane (green).

It appears that the focusing condition derived in the scattering plane 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 out-of-plane non-focused contribution to the Bragg angle becoming significant. This relatively large out-of-plane range in the detection area may be useful in studying polycrystalline materials.

Fig. 3[link] 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 φdc(c, x′, d, B) is first-order independent of x′. This is not the case for the SAG, where φsc(c, x′, d, B) changes linearly with x′ within the uncertainty Δx, so equations (4)[link] and (5)[link] change to

[{\varphi _{\rm sc}} ({\varepsilon _{\rm c}}, d, B) = 2 c B L d \left ({\cos {\varepsilon _{\rm c}} + \sin {\varepsilon _{\rm c}} \cot {\theta _0}} \right), \eqno(10)]

[\eqalignno{& {\varphi _{\rm sv}} ({\varepsilon _{\rm v}}, {\varepsilon _{\rm c}}, x^{\prime}, d, B) \cr & \, = 2 c B L d \left [{{\varepsilon _{\rm v}} \left ({ - \sin {\varepsilon _{\rm c}} + \cos {\varepsilon _{\rm c}} \cot {\theta _0} + {{2\sin {\varepsilon _{\rm c}}} \over {{{\sin }^2}{\theta _0}}}} \right) - {{x^{\prime} \sin {\varepsilon _{\rm c}}} \over {{{\sin }^2} {\theta _0} }}} \right]. \cr &&(11)}]

The inclusion of the detector uncertainty in φsv(v, c, x′, d, B), which is not present in the DAG case, equations (8)[link] and (9)[link], is the main advantage of the DAG over the SAG.

For the SAG, at each position x0 = 2(θ0 + c), the damping of the precession due to a single Bragg reflection can now be written as

[P ({\varepsilon _{\rm c}}, {\varepsilon _{\rm v}}, d, B) = A \left [{\Delta x}, {\Delta \varepsilon }, {\varphi _{\rm c}} ({\varepsilon _{\rm c}}, d, B) \right] \cos {\varphi _{\rm c}} ({\varepsilon _{\rm c}}, d, B), \eqno(12)]


[\eqalignno{ A &\left [{\Delta x}, {\Delta \varepsilon }, {\varepsilon _{\rm c}}, {\varphi _{\rm c}} ({\varepsilon _{\rm c}}, d, B) \right] = {\left \langle {\cos {\varphi _{\rm v}} ({\varepsilon _{\rm c}}, {\varepsilon _{\rm v}}, d, B)} \right \rangle _{x^{\prime}, {\varepsilon _{\rm v}}}} \cr &\simeq \, \exp \left \{ { - {{\left [{{{\alpha ({\varepsilon _{\rm c}}, {\Delta \varepsilon }, {\Delta x}) {\varphi _{\rm c}} ({\varepsilon _{\rm c}}, d, B)} \over 2}} \right] }^2}} \right \}, & (13)}]


[\alpha ({\varepsilon _{\rm c}}, {\Delta \varepsilon }, {\Delta x}) \simeq {\left \langle {{{{\varphi _{\rm v}} ({\varepsilon _{\rm c}}, {\varepsilon _{\rm v}}, x^{\prime}, d, B)} \over {{\varphi _{\rm c}} ({\varepsilon _{\rm c}}, d, B)}}} \right \rangle _{x^{\prime}, {\varepsilon _{\rm v}}}}, \eqno(14)]

where we have assumed a Gaussian distribution for the initial divergence and α(c, v, Δx) is the half-width of the Gaussian exp[−(φv − φc/αφc)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[link] 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)[link]–(14)[link][link][link][link]]. 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.

[Figure 5]
Figure 5
The resolution function α(c, Δx) at θ0 = π/4 as a function of c for two values of Δx and v in three different combinations, labelled (1), (2) and (3). For comparison, the resolution that would be achieved in conventional diffraction for the same angular parameters is shown by the purple line, labelled (4). Here, the incoming divergence was taken as Δθ = 0.01.

3. Simulations with SAG

3.1. Polarization for a single crystal

Simulations have been performed in the single-crystal 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 high-resolution position-sensitive detector. Fig. 6[link](a) shows the amplitude of A(Δx, Δ, c, φsc) as a function of the precession angle φsc at three different c values around c0. Fig. 6[link](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.

[Figure 6]
Figure 6
(a) The amplitude A = A(Δx, Δ, c, φsc) plotted as a function of the precession phase in the SAG with θ0 = π/4, Δx = 2 × 10−3 and Δ = 0.01 for three different c values as indicated in the key. (b) The polarization plotted as a function of the precession phase in a small phase interval with Δφsc = φsc − 1000 rad at x′ = 0 (red line), x′ = 0.0005 (green line) and x′ = −0.0005 (blue line). Here, c = c0 = −0.32.

3.2. Polarization for polycrystalline samples

In this section we wish to show that the SAG can be used as an inverse-geometry diffractometer. We will use a white beam as the initial beam, with a normalized wavelength spectrum given by

[I_0 (\lambda) = {4 \over { { {\pi }^{1/2} } }} \left ({{{\lambda _T^3} \over {{\lambda ^4}}}} \right) \exp \left ({{{\lambda _T^2} \over {{\lambda ^2}}}} \right). \eqno(15)]

Here λT = h/(mvT), vT = (2kT/m)1/2, T = 300 K, k is the Boltzmann constant, 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 = (x0/2) − θ0 and a variable part v with a distribution width given by

[(\Delta \varepsilon) ^2 = {\left ({{{\Delta \theta } \over 2}} \right) ^2} + {\left ({{{\Delta x} \over 2}} \right)^2}. \eqno(16)]

We use in our simulations a face-centred 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 φsc(c, d, B) from equations (10)[link]–(14)[link][link][link][link] using c = (x0/2) − θ0 the average of the Nd weighted contributions from the different lattice planes:

[\eqalignno { P({x_0},B,{\Delta x}, {\Delta \varepsilon }) = & \, \Bigl \{ \sum\limits_d {{N_d}} I({\lambda _d}) \cos {\varphi _{\rm sc}} \left ({{{x_0}} \over 2} - {\theta _0},d,B \right) \cr & \, \times A \left [{\Delta x}, {\Delta \varepsilon }, {{{x_0}} \over 2} - {\theta _0}, {\varphi _{\rm sc}} \left ({{{x_0}} \over 2} - {\theta _0}, d, B \right) \right] \Bigr \} \cr & \, \bigg / {\sum\limits_d {{N_d}} I ({\lambda _d}) }, &(17)}]


[\eqalignno{& A \left ({\Delta x}, {\Delta \varepsilon }, {{{x_0}} \over 2} - {\theta _0}, {\varphi _{\rm sc}} \right) \cr &\quad= \exp \left \{ { - {{ \left [{{{\alpha \left ({{{x_0}} / 2} - {\theta _0}, {\Delta x}, {\Delta \varepsilon } \right) {\varphi _{\rm sc}}} \over 2}} \right] }^2}} \right \}. &(18)}]

Here, λd = 2dsin(x0/2) and Nd is the multiplicity factor of plane d in an isotropic distribution in 4π, for which we use N111 = N222 = 8, N220 = 12, N200 = N400 = 6 and N311 = 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

[I({x_0},B,t) = V({x_0},t) \left [{{{1 + P({x_0},B,{\Delta x}, {\Delta \varepsilon })} \over 2}} \right] , \eqno(19)]


[V({x_0},t) = {I_{\rm w0}} {A_{\rm p}} {b_{\rm h}} {b_{\rm w}} {\rm d}\Omega ({x_0}) \cot \left ({{{x_0}} \over 2} \right) {\Delta \theta } Rt \sum\limits_d {{N_d}} \eqno(20)]


[{\rm d} \Omega ({x_0}) = {{h_d} \over {{L_d} \sin {x_0}}}. \eqno(21)]

Here, V(x0, t) contains the initial intensity of the white beam Iw0 with a wavelength spectrum given by equation (15)[link] using λT = 0.2 nm, losses at the polarizers Ap, the beam height bh, the beam width bw, the wavelength region dλ/λ = cot(x0/2)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 hd and width equal to the beam width at a distance Ld from the sample. In our simulation we took the parameters as given in Table 1[link]. This results in

[V({x_0},t) = 0.032 t \textstyle\sum\limits_d {N_d} = 20.4t. \eqno(22)]

Using these numbers, Fig. 7[link] shows the simulated intensity as a function of the precession magnetic field B (in mT) at various values of x0 around 2(θ0 + c0). For the distance L between the inclined precession faces, we used L = 1 m. In the intensity I(xBt) we included a random number with amplitude ΔI = I(xBt)1/2 to account for the expected statistical spread at each point.

Table 1
Instrumental parameters used in the simulations

Iw0 Ap bh bw hd Ld dΩ Δx θ0 Δθ R
106 cm−2 s−1 0.25 2 cm 2 mm 2 cm 1 m 0.003 0.002 π/4 0.01 0.1
[Figure 7]
Figure 7
The simulated intensity, in counts per 10 000 s, collected in one detector pixel of Δx = 2 mrad, plotted as a function of B at various detector angles. 150 has been added to each spectrum to make them more readily distinguishable from each other. Note the different damping of the spectra at the various x values indicated in the figure.

From Fig. 7[link] 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(θ0 + 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[link], we find the wavelength-dependent intensities shown in Fig. 8[link].

[Figure 8]
Figure 8
The simulated intensity of an f.c.c. polycrystal plotted as a function of the wavelength λ at three detector positions around the optimal x = 2(θ0 + c0) = 0.93 rad. The spectra were obtained by Fourier transformation of the simulated intensities of Fig. 7[link]. The dash–dotted line shows the wavelength spectrum used in the simulations.

In the simulation, the instrumental parameters of Table 1[link] are used with a data-collection 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(x0/2) 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[link] 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[link]), 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[link]. The setup makes use of the resonance technique, where the precession angle is determined by the resonance fields B in two coils a distance Lc apart, similar to what was described by Dalgliesh et al. (2011[Dalgliesh, R. M., Langridge, S., Plomp, J., de Haan, V. O. & van Well, A. A. (2011). Physica B, 406, 2346-2349.]). The precession angle φs(θ, λ, B) = 2cBLcdcosθB is varied in large steps by the resonance field B and in smaller steps by the distance Lc between the magnets by shifting M1 in the direction of M2.

[Figure 9]
Figure 9
A sketch of the improvised Larmor diffractometer, where P and A are the polarizer and analyser consisting of two polarizing mirror systems, R1 and R2 rotate the polarization direction from the initial polarization along z to the xy precession plane and back, M1 and M2 are magnets generating the resonance fields B in the resonance coils C1 and C2, and S is the sample single crystal to investigate. The precession angle is varied by changing the length Lc by shifting the position of magnet M1.

Because we could not vary the resonance frequency automatically, we varied the distance Lc 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 Fe3Si single crystal, at the same scattering angle θ = 2θ0 = 84° and with different mosaic spreads. Fig. 10[link] 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 scattering plane to increase the diffracted intensity by deformation. For the damping constant α of the amplitude of the precession defined in equation (13)[link], 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.

[Figure 10]
Figure 10
Measured polarization of the reflected beam and amplitude fit on a stack of Si(100) wafers of dimension 50 × 0.3 mm (diameter × thickness). The scattering angle of 84° was twice the inclination angle (42°) of the precession region. The colours and insets refer to the different B fields and resonance frequencies used in the precession region. The lower two figures show P(φ) in two small φ ranges at different B values in mT according to the key shown in the inset.

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 Bragg angle from the inclination angle (Fig. 11[link]). With a fixed open detector over about 10°, we measured the reflected intensity and polarization as a function of the change in Bragg angle, ΔθB0. At the minimum and maximum values of ΔθB0, 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.

[Figure 11]
Figure 11
(a) Measured intensity as a function of the change in Bragg angle ΔθB0 = . (b) Polarization derived from (a) at a fixed field of 5 mT and distance L between the resonance coils of 80 cm, corresponding to a precession angle of 600 rad at ΔθB0 = 0. (c) The change in precession angle derived from (b).

Next, we investigated the reflected intensity and polarization of an Fe3Si single crystal at the same Bragg angle of 42°. Before measuring the field-dependent 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[link] shows the measured polarization as a function of fd = 2cLBsinθB0, together with the calculated polarization of the three order reflections of Bragg planes (200), (300) and (400) of Fe3Si in a field of 2.5 mT. The quantity fd is related to the Larmor phase φd of each reflection from the lattice plane dhkl by φd = fdd. The polarization of the sum of the three orders, P(fd), fits perfectly with the measured data, with a fitted intensity ratio for the orders I200:I300:I400 = 30:15:13, a lattice constant d100 = 0.565 nm and polarization amplitude damping according to

[P({f_d}) = \sum\limits_{hkl} {\cos ({d_{hkl}} \, {f_d}) \exp \left \{ { - {{\left [{{{\alpha (...) {d_{hkl}} {f_d}} \over 2}} \right] }^2}} \right \} }. \eqno(23)]

Here, α(...) is again the uncertainty in the Bragg angle, 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[link]. 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 Δd100 = 5 × 10−4 nm.

[Figure 12]
Figure 12
Measured polarization as a function of fd of the sum of the three order reflections (200), (300) and (400) of an Fe3Si single crystal (red solid line), together with the calculated polarization (blue dotted line).

5. Discussion

A comparison of the single-arm Larmor diffractometer with the double-arm 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 high-resolution position-sensitive 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 θ0. Both the DAG and the SAG may be candidates for an inverse-geometry 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 inverse-geometry 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 non-magnetic 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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