(IUCr) Real-space interpretation of spin-echo small-angle neutron scattering

Volume 36
Part 1
Pages 117-124
February 2003

Received 3 June 2002
Accepted 7 November 2002

Real-space interpretation of spin-echo small-angle neutron scattering

Timofei Krouglov,^a ^* Ignatz M. de Schepper,^a Wim G. Bouwman ^a and M. Theo Rekveldt ^a

^aInterfaculty Reactor Institute, Delft University of Technology, Mekelweg 15, Delft, The Netherlands
Correspondence e-mail: krouglov@iri.tudelft.nl

Spin-echo small-angle neutron scattering (SESANS) is a novel real-space scattering technique. SESANS measures a correlation-like function G(Z), the meaning of which was unknown until now. Here a direct real-space interpretation of G(Z) through the particle scattering density and pair correlation function is given. One-dimensional and two-dimensional SESANS are compared. The case of non-interacting particles is considered in detail with an explicit geometrical interpretation. General methods for the calculation of structural parameters, such as the total scattering length and the radius of gyration, are developed. Analytical expressions of G(Z) for non-interacting solid spheres, hollow spheres and Gaussian coils are derived. The case of solid spheres is compared with experimental data.

Keywords: spin-echo small-angle neutron scattering; correlation function.

1. Introduction

The spin-echo small-angle neutron scattering (SESANS) technique (Keller et al., 1995; Gähler et al., 1996; Rekveldt, 1996; Bouwman et al., 2000; Rekveldt et al., 2002) is a novel real-space method to determine the structure of materials using small-angle neutron scattering (SANS). The idea to use spin echo for elastic scattering was initially proposed by Pynn (1978). The method is based on the Larmor precession of polarized neutrons in a magnetic field, which encodes the transmission angle of the neutron through a precession device. Advantages of this technique are the high intensity of the beam and the large length scales that can be reached, up to the micrometre range. A recent study (Rekveldt et al., 2002) showed that multiple scattering can easily be taken into account in SESANS. Another significant difference of SESANS compared with conventional SANS is that it measures a real-space function. This function G(Z) depends on the spin-echo length, Z. Previous numerical calculations showed that the SESANS correlation function G(Z) is highly sensitive to the structure and size of the particles (Bouwman & Rekveldt, 2000; Uca et al., 2003). Until now, however, the meaning of this function has not been understood.

Gähler et al. (1998) presented a real-space approach to SESANS. They showed that the SESANS correlation function is equal to a density-density autocorrelation function. This result included integration over all three components of reciprocal space and assumed that the spin-echo length is independent of the scattering vector. Here we discuss a SESANS technique that works with a single wavelength and does not include integration over wavelength.

There has been an attempt to justify alternative geometries for SESANS (Zhao, 2001) with pinholes in front of the sample, which produce simply interpretable real-space functions. But the presence of pinholes gives a considerable loss of intensity, which reduces the practicality of such a setup.

This paper establishes explicit relations between G(Z) and real-space properties of a scattering medium for practical setups, which uses a divergent neutron beam with a large cross section and thus high intensity.

2. SESANS concept

Let us consider a sample with cross section $[{\rm d}\Sigma_S({\bf Q})/{\rm d}\Omega]$ illuminated by the neutron beam of cross section S. Before and after scattering by a sample, neutrons undergo Larmor precession in magnetic fields (Rekveldt et al., 2002). The precession length is different for neutrons scattered by different angles (see Fig. 1). As a result of this precession, the initially polarized neutron beam with polarization P_n0 becomes depolarized. The total detected polarization P_n of the neutron beam is a function of the spin-echo length Z:

$[{{P_{n}(Z)}\over{P_{n0}}} = \exp[\tilde{G}(Z)-\tilde{G}(0)] = \exp\{\tilde{G}(0)[G_{0}(Z)-1]\}, \eqno (1)]$

where the SESANS correlation function is

$[\tilde{G}(Z) = {{1}\over{k_{0}^{2}}}\int\limits_{-\infty}^{+\infty} {\rm d}Q_{y}\int\limits_{-\infty}^{+\infty} {\rm d}Q_{z}{{{\rm d}\Sigma_{S} ({\bf Q})}\over{S{\rm d} \Omega}}\cos(Q_{z}Z) \eqno (2)]$

where Q = (0, Q_y, Q_z),

$[Z = {{c \lambda ^{2}BL\cot \theta_{0}}/{2\pi}} \eqno (3)]$

and G₀(Z) is the normalized function

$[G_{0}(Z) = {{\tilde{G}(Z)}/{\tilde{G}(0)}}. \eqno (4)]$

We are considering here elastic small-angle scattering at fixed wavelength. Equation (1) is valid for single scattering as well as for multiple scattering. For single scattering, expression (1) can be linearized:

$[{{P_{n}(Z)}/{P_{n0}}} = 1-\tilde{G}(0)-\tilde{G}(0)G_{0}(Z).\eqno (5)]$

In practice, angular integration in equation (2) is performed within the detectable angles and has limited range. Here in (2) and in the following expressions we will consider infinite integration limits for momentum transfer components, since for sufficiently large particles (R > 10 nm) scattering angles are very small and almost all scattered neutrons reach the detector (Rekveldt, 1996).

Figure 1
Schematic side-view diagram of a SESANS setup built with magnetized foils. The spin-echo length Z can be varied by changing the sample (denoted by S) position L or by varying the magnetic field B. The gray areas indicate the precession regions. The definition of the X, Y and Z directions is given in the diagram.

Equation (2) gives the total scattering probability of a sample at Z = 0:

$[\tilde{G}(0) = {{\Sigma_{S}}/{S}} = \Sigma t \eqno (6)]$

and

$[\exp (- \Sigma_{S} / S ) = T]$

is the sample transmission.

3. SESANS: from reciprocal to real space

Let us assume that we have a sample with thickness t, illuminated by a neutron beam of cross section S. The volume of the sample illuminated by the neutron beam is then V_S = tS. The sample has scattering length density $[\rho]$ (r). The Patterson function of such a sample is

$[P({\bf R}) = \int\limits_{V_{S}}{\rm d}{\bf r}\,\rho ({\bf r })\rho ({\bf r}+{\bf R}) \eqno (7)]$

and the differential scattering cross section measured by means of conventional SANS is

$[{{{\rm d}\Sigma_{S} ({\bf Q})}\over{{\rm d} \Omega}} = \int\limits_{V_{S}}{\rm d}{\bf R}\exp(-i{\bf QR})\,P({\bf R}).\eqno (8)]$

The essential point of this article is to eliminate reciprocal space from the definition of the SESANS correlation function given by equation (2). To do so it is more convenient to use the reduced SESANS correlation function

$[G(Z) = ({{S}/{\lambda^{2}}})\,\tilde{G}(Z).\eqno (9)]$

We are considering here elastic scattering measured at fixed wavelength. Then

$[G(Z) = {{1}\over{4\pi^{2}}}\int\limits_{-\infty}^{+\infty} {\rm d}Q_{y}\int\limits_{-\infty}^{+\infty} {\rm d}Q_{z}{{{\rm d}\Sigma_{S} ({\bf Q})}\over{{\rm d} \Omega}}\cos(Q_{z}Z).\eqno (10)]$

Substituting (8) in (10) we obtain

$[G(Z') = {{1}\over{4\pi^{2}}}\int\limits_{V}\!{\rm d}{\bf R }\int\limits_{-\infty}^{+\infty} \!\!{\rm d}Q_{y}\int\limits_{-\infty}^{+\infty} \!\!{\rm d}Q_{z}\exp[-i({\bf QR}-Q_{z}Z')]\,P({\bf R}).\eqno (11)]$

Here the cosine is substituted by the complex exponent since we have bilateral integration and the complex part, represented by the sine, cancels.

After subsequent integration over Q_z, Z, Q_y and Y, and putting Q_x = 0, we arrive at the following expression:

$[G(Z) = \int\limits_{-\infty}^{+\infty}{\rm d}XP(X,0,Z).\eqno (12)]$

This expression is the key which allows a direct real-space interpretation of G(Z) in terms of sample spatial correlation functions. Here we are considering so-called one-dimensional SESANS, which performs Fourier transform over one component only, namely Z according to our setup.

So far it has been assumed that we are dealing with a scattering length density that is fixed in time and space. In practice, we measure the thermodynamic average, which includes orientational averaging:

$[G(Z) = \int\limits_{-\infty}^{+\infty}{\rm d}X\langle P(X,0,Z)\rangle_{\boldOmega}.\eqno (13)]$

For isotropic particles we have

$[P(R) = N\left[\gamma(R)+n\int\limits_{V_{S}}{\rm d}^{3}r\,\gamma(R+r)h(r)\right]\eqno (14)]$

where n = N/V_S is a number density of the particles, $[\gamma]$ (R) is the particle density autocorrelation function and h(r) = g(r) - 1 is a pair correlation function (see, for example, McQuarrie, 1976).

3.1. Two-dimensional SESANS and anisotropy

Let us consider two-dimensional SESANS, which performs a two-dimensional Fourier transform along the Z and Y directions independently. For such two-dimensional SESANS, the correlation function is [cf. equation (10)]

$[G(Y,Z) = {{1}\over{4\pi^{2}}}\int\limits_{-\infty}^{+\infty} {\rm d}Q_{y}\int\limits_{-\infty}^{+\infty} dQ_{z}{{{\rm d}\Sigma_{S} ({\bf Q})}\over{{\rm d} \Omega}}\cos(Q_{z}Z)\cos(Q_{y}Y).\eqno (15)]$

After elimination of reciprocal space and orientational averaging for an isotropic sample, we have

$[G(Y,Z) = \int\limits_{-\infty}^{+\infty}{\rm d}X\,\langle P(X,Y,Z)\rangle_{\boldOmega} = G( Y^{2}+Z^{2} )^{1/2} .\eqno (16)]$

Apparently, for an isotropically scattering sample, two-dimensional SESANS has no advantages compared with the one-dimensional case. If the sample is anisotropic and its correlation function along Y and Z can be decoupled, then measurement along two perpendicular axes of the sample can be made to restore the two-dimensional correlation function. If the correlation function of the anisotropic sample cannot be decoupled in Cartesian coordinates, then the two-dimensional space can be covered by measuring G(Z) at several orientations of the sample with respect to the Z and Y axes:

$[G(Z)_{\varphi} = \int\limits_{-\infty}^{+\infty}{\rm d}X\, P(X,Z\sin\varphi,Z\cos\varphi) = G(Z\sin\varphi,Z\cos\varphi) ,\eqno (17)]$

where $[\varphi]$ is the angle between the sample axis and the Z axis.

Therefore, two-dimensional SESANS does not give any advantage over one-dimensional SESANS, not only for isotropic samples but also for anisotropic samples.

4. Isotropic media formed by non-interacting particles

The most important and simple case is the case of non-interacting particles, when h(r) = 0. Then the equation (13) can be rewritten as

$[G(Z) = NG_{p}(Z), \eqno (18)]$

where G_p(Z) is the contribution of one particle:

$[G_{p}(Z) = \int\limits_{-\infty}^{+\infty}{\rm d}X\,\gamma( X^{2}+0^{2}+Z^{2} )^{1/2}.\eqno (19)]$

Switching from X, Z to R, Z coordinates for this expression gives

$[G_{p}(Z) = 2\int\limits_{Z}^{+\infty}{\rm d}R\,{{\gamma(R)R}\over{(R^{2}-Z^{2})^{1/3}}}.\eqno (20)]$

This formula allows us to calculate G(Z) using known $[\gamma]$ (R). Most of the particles have some maximal diameter, D, at which autocorrelations disappear and $[\gamma]$ (D) = 0 as well as G(D) = 0. For solid and hollow spheres it is just its outer diameter. In order to use the correlation function with a dimensionless argument, we can reduce all spatial variables by R_max = D/2 and rewrite equation (20) with dimensionless variables:

$[G_{p}(\xi) = 2R_{\rm max}\int\limits_{\xi}^{2}{\rm d}\chi\,{{\gamma(\chi)\chi}\over{(\chi^{2}-\xi^{2})^{1/2}}},\eqno (21)]$

where $[\chi]$ = R/R_max and $[\xi]$ = Z/R_max. Both variables $[\chi]$ and $[\xi]$ may vary from 0 to 2, which limits all correlations by 2 maximal radii.

4.1. Total scattering length

The scattering length of one particle is

$[b = \int\limits_{V}{\rm d}{\bf r}\rho ({\bf r}).]$

Now we would like to express the total scattering length in terms of G(Z). The right-hand side of equation (19) represents integration along the X axis. In order to obtain the integral over a three-dimensional volume, we have to complement the integration along the X axis by the integration over a cylindrical layer $[2\pi Z\,{\rm d}Z]$ in both sides of equation (19):

$[2\pi\int\limits_{0}^{\infty}G_{p}(Z)Z\,{\rm d}Z = \int\limits_{-\infty}^{+\infty} {\rm d}X\int\limits_{0}^{\infty}2\pi Z\,{\rm d}Z\,\gamma( X^{2}+0^{2}+Z^{2} )^{1/2}\eqno (22)]$

$[2\pi\int\limits_{0}^{\infty}G(Z)Z\,{\rm d}Z = N\int\limits_{V}\gamma(R)\,{\rm d}{\bf R} = Nb^{2}.\eqno (23)]$

4.2. Guinier radius

The Guinier radius can be expressed via the distance distribution function (see, for example, Glatter & Kratky, 1982):

$[R_{g}^{2} = {{\int\limits_{0}^{\infty}p(R)R^{2}\,{\rm d}R }\bigg/{2\int\limits_{0}^{\infty}p(R)\,{\rm d} R}}, \eqno (24)]$

where

$[p(R) = \gamma(R)R^{2}.]$

First we recalculate the denominator in terms of G(Z). Using

$[\int\limits_{V}\gamma(R)\,{\rm d}{\bf R} = \int\limits_{V}\gamma(R) 4 \pi R^{2}\,{\rm d}R = 4\pi\int\limits_{V}p(R)\,{\rm d}R,]$

equation (22) can be rewritten as

$[\int\limits_{0}^{\infty}G_{p}(Z)Z\,{\rm d}Z = 2\int\limits_{0}^{\infty}p(R)\,{\rm d}R, ]$

which is a normalization factor for the distance distribution function. We also need to recalculate the numerator of (24) in terms of G(Z) (see Appendix A):

$[\int\limits_{0}^{\infty}p(R)R^{2}\,{\rm d}R = {{3}\over{4}}\int\limits_{0}^{\infty}G(Z)Z^{3}\,{\rm d}Z.]$

After normalization we have the Guinier radius expressed directly through G(Z):

$[R_{g}^{2} = {{3}\over{4}}{{\int\limits_{0}^{\infty}G(Z)Z^{3}\,{\rm d}Z}\bigg/{\int\limits_{0}^{\infty}G(Z)Z\,{\rm d}Z}}.]$

4.3. Normalization

As mentioned above, $[\tilde{G}(0)]$ is an important invariant and it gives the total scattering probability of a sample. Using (9) and (18) we have

$[\tilde{G}(0) = ({{\lambda^{2}}/{S}})NG_{p}(0).\eqno (25)]$

It follows from (19) that

$[G_{p}(0) = 2\int\limits_{0}^{\infty} {\rm d}R\,\gamma(R)\eqno (26)]$

and

$[\tilde{G}(0) = 2{{\lambda^{2}}\over{S}}N\int\limits_{0}^{\infty}{\rm d}R\,\gamma(R) .\eqno (27)]$

This value allows us to calculate the intensity of the SESANS signal.

4.4. Eigenfunction and Gaussian coil

With respect to $[\gamma]$ (R), equation (20) is an Abel integral equation and its solution is

$[\gamma(R) = {{1}\over{\pi}}{{1}\over{R}}{{\rm d}\over{{\rm d}R}}\int\limits_{R}^{+\infty}{\rm d}Z\,{{G(Z)Z}\over{(Z^{2}-R^{2})^{1/2}}}.\eqno (28)]$

This permits us to recalculate measured G(Z) into $[\gamma]$ (R), but this approach to the treatment of experimental data is not very useful since this transformation includes divergent integration and differentiation. It is more convenient to fit experimental data by an analytically known G(Z). An analytical representation of G(Z) for most conventional isotropic particles is discussed below.

The eigenfunction of equation (20) is Gaussian:

$[G_{0}(Z) = \exp(- Z^{2}/6\langle R_{g}^{2}\rangle).]$

This correlation function corresponds to the unperturbed Gaussian coil model for polymers.

4.5. Sphere

For uniform homogeneous spherical particles the correlation function is known analytically (see, for example, Glatter & Kratky, 1982). If a sphere has radius R [volume V = (4/3) $[\pi]$ R³] and scattering length density $[\rho]$ , then

$[\gamma(\chi) = \rho^{2}V\gamma_{0}(\chi) ,\eqno (29)]$

where

$[\gamma_{0}(\chi) = 1-{{3}\over{4}}\chi+{{1}\over{16}}\chi^{3} \eqno (30)]$

and $[\chi]$ = r/R for 0 [less-than or equal to] $[\chi]$ 2. Let the volume fraction of spheres in a dilute solution be [varphi] _V = NV/V_S. Then we integrate (29) using (21) and express the result according to (9) and (4):

$[\tilde{G}(Z) = \tilde{G}(0)G_{0}(Z) \eqno (31)]$

where

$[\eqalignno{G_{0}(\xi) = \hskip.2em&\left[1-\left({{\xi}\over{2}}\right)^{2}\right]^{1/2}\left(1+{{1}\over{8}}\xi^{2}\right)\cr &\!+ {{1}\over{2}}\xi^{2}\left[1-\left({{\xi}\over{4}}\right)^{2}\right]\ln\left[{{\xi}\over{2+(4-\xi^{2})^{1/2}}}\right], &\hfill\llap{(32)}}]$

in which $[\xi]$ = Z/R for 0 [less-than or equal to] $[\xi]$ 2. The total scattering probability is [see equation (27)]

$[\tilde{G}(0) = {{2N\rho^{2}V{{3}\over{4}}R\lambda^{2}}\over{S}} = {{3}\over{2}}\varphi_{V}\rho^{2}\lambda^{2}tR. \eqno (33)]$

The rightmost expression has been obtained earlier (see Bouwman et al., 2002) by evaluating (6).

For homogeneous particles, $[\gamma]$ (R) has a clear physical meaning. It is the shared volume of two phantom particles shifted with respect to each other by the distance R:

$[\gamma(R) = \int\limits_{V}\rho(r)\rho(r+R)\,{\rm d}{\bf r} = \rho^{2}V_{\rm shared}.]$

Integration of this quantity along the X axis at fixed Z gives an average shared volume of two phantom spheres, sliding with respect to each other along the X axis with Z shift (see Fig. 2 ).

Figure 2
Geometrical interpretation of G(Z) for homogeneous spheres.

Figure 3
Density autocorrelation function (dotted line) and SESANS correlation function (solid line) for homogeneous spheres.

SESANS measurements on a dilute solution of uniform polystyrene spheres of 100 nm radius in water (Bouwman et al., 2002) can be satisfactorily fitted by the analytical function [equation (1)] using $[\tilde{G}(Z)]$ defined by (31) and (33). A least-squares fit yields R = 99.3 nm (Fig. 4), which is in excellent agreement with the information provided by the manufacturer.

Figure 4
Circles: measured SESANS signal for homogeneous latex spheres of 100 nm radius. Solid line: fit with R = 99.3 nm

4.6. Hollow sphere

The hollow sphere is an important practical model describing a wide range of colloids. Let us consider the dimensionless case and characterize our hollow sphere by reduced parameters. The hollow sphere has an outer radius, R_out, and inner radius, R_in. Then the reduced radius of a hollow sphere is

$[\chi = {{r}/{R_{\rm out}}}]$

and the reduced thickness is

$[\Delta = ( R_{\rm out}-R_{\rm in} )/ R_{\rm out} .]$

The reduced inner radius is

$[\sigma = 1-\Delta]$

and

$[R_{\rm in} = \sigma R_{\rm out}.]$

The volume of a spherical layer is

$[\Delta V = {{4}\over{3}}\pi (R_{\rm out}^{3}-R_{\rm in}^{3}) = {{4}\over{3}}\pi R_{\rm out}^{3}(1-\sigma^{3}) = V(1-\sigma^{3}),]$

where V = $[{\textstyle {4 \over 3}}\pi R_{\rm out}^{3}]$ is the volume of the whole particle.

Figure 5
Hollow sphere.

Calculations of G(Z) for such a hollow sphere can be found in Appendix B. $[G(\xi)]$ has two distinct regions. For a hollow sphere we have two characteristic sizes. At distances 2 [greater-than or equal to] $[\xi]$ 2 - $[\Delta]$ , hollow-sphere behaviour is indistinguishable from that of a solid sphere (see also Figs. 6 and 8, Appendix B). All autocorrelations are apparently limited by two radii. For $[\Delta]$ $[\xi]$ 0 we have initial decay due to the disappearance of ` $[\Delta]$ -layer' correlation.

Figure 6
Density autocorrelation function (dotted line) and SESANS correlation function (solid line) for hollow spheres with $[\Delta]$ = 0.3.

Figure 7
Regions of analytical representation of the correlation function $[\gamma(\chi)]$ for hollow spheres.

Figure 8
Example of layer projection where $[\chi]$ runs from $[\beta]$ to $[\alpha]$ covering the region where two hollow spheres interact as solid spheres.

5. Conclusion

SESANS gives direct information about the structure of particles in real space without an intermediate Fourier transform. Reciprocal space is now completely eliminated from SESANS theory. Explicit relations between the SESANS correlation function and the particle autocorrelation function and pair correlation function are found. It is shown that two-dimensional SESANS has no advantages compared with one-dimensional SESANS. An analytical SESANS correlation function is found for Gaussian coils, and solid and hollow spheres. Experimental results for a dilute solution of polystyrene spheres in water are satisfactorily fitted by an analytically calculated function for non-interacting spheres. In the case of non-interacting particles of unknown shape, the SESANS correlation function can be used to calculate the total scattering length and Guinier radius.

Appendix A

Moments calculation

Equation (20) can be rewritten in terms of p(R):

$[G_{p}(Z) = 2\int\limits_{Z}^{\infty}{{\gamma (R) R\,{\rm d}R}\over{(R^{2}-Z^{2})^{1/2}}} = 2\int\limits_{Z}^{\infty}{{p(R)\,{\rm d}R}\over{R(R^{2}-Z^{2})^{1/2}}}.]$

Using $[\langle Z^{n+1}\rangle\propto \langle R^{n}\rangle]$ we multiply both sides of the equation by Z³ and integrate over dZ:

$[\int\limits_{0}^{\infty}G_{p}(Z)Z^{3}\,{\rm d}Z = 2\int\limits_{0}^{\infty}Z^{3}\,{\rm d}Z\int\limits_{Z}^{\infty}{{p(R)\,{\rm d}R}\over{R(R^{2}-Z^{2})^{1/2}}}.]$

After swapping the integration order we have

$[\int\limits_{0}^{\infty}G_{p}(Z)Z^{3}\,{\rm d}Z = 2\int\limits_{0}^{\infty}{{p(R)\,{\rm d}R}\over{R}}\int\limits_{0}^{R}{{Z^{3}\,{\rm d}Z}\over{(R^{2}-Z^{2})^{1/2}}}.]$

Using

$[\int\limits_{0}^{R}{{Z^{3}\,{\rm d}Z}\over{(R^{2}-Z^{2})^{1/2}}} = {{2R^{3}}\over{3}},]$

we finally have

$[\int\limits_{0}^{\infty}G_{p}(Z)Z^{3}\,{\rm d}Z = {{4}\over{3}}\int\limits_{0}^{\infty}p(R)R^{2}\,{\rm d}R.]$

In a similar way, the other moments of Z and R can also be calculated.

Appendix B

Hollow-sphere calculations

Calculations for G(Z) below are for one particle. The correlation function of a hollow sphere is known analytically (Glatter, 1979):

$[\eqalign{ &\gamma(\chi) =\rho^{2}V \cr\!&\times \cases{ 1-{{3}\over{4}}\chi+{{1}\over{16}}\chi^{3} & $2 \geq \chi \geq 2-\Delta$\cr {{3}\over{8}}{{(1-\sigma^{2})^{2}}/{\chi}}-\sigma^{3}+{{3}\over{4}}\sigma^{2}\chi-{{1}\over{16}}\chi^{3} & $2-\Delta \geq \chi \geq 2-2\Delta$\cr {{3}\over{8}}{{(1-\sigma^{2})^{2}}/{\chi}} & $2-2\Delta \geq \chi \geq \Delta$ \cr (1-\sigma^{3})-{{3}\over{4}}(1+\sigma^{2})\chi+{{1}\over{8}}\chi^{3} & $\Delta \geq \chi \geq 0$} }]$

or in a short notation

$[\eqalign{ \gamma(\chi) =\hskip.2em& \rho^{2}V\cr &\!\times\cases{\gamma_{0}(\chi) & $2 \geq \chi \geq 2-\Delta$\cr \gamma_{1}(\chi)-\sigma^{3}\gamma_{0}({{\chi}/{\sigma}}) & $2-\Delta \geq \chi \geq 2-2\Delta$\cr \gamma_{1}(\chi) & $2-2\Delta \geq \chi \geq \Delta$\cr \gamma_{0}(\chi)+\sigma^{3} [\gamma_{0}({{\chi}/{\sigma}})-2] & $\Delta \geq \chi \geq 0$} }]$

where $[\gamma_{0}(\chi)]$ is a correlation function of a homogeneous sphere [see equation (29)] and $[\gamma_{1}(\chi)]$ = $[{\textstyle{3 \over 8}} (1-\sigma^{2})^{2} / \chi ]$ .

$[G(Z) = 2\int\limits_{Z}^{R_{\rm out}}{\rm d}R\,{{\gamma(R)R}\over{(R^{2}-Z^{2})^{1/2}}}.]$

The reduced form is

$[G(\xi) = 2R_{\rm out}\int\limits_{\xi}^{2}{\rm d} \chi \,{{\gamma(\chi)\chi}\over{(\chi^{2}-\xi^{2})^{1/2}}}.]$

Since in four different $[\chi]$ regions $[\gamma(\chi)]$ is defined by different analytical expressions, we decompose the integration into four corresponding regions. The further procedure is trivial. As a result of four different regions we have `layer projections' for $[\beta]$ [greater-than or equal to] $[\chi]$ $[\alpha]$ :

$[{}_{\alpha}^{\beta}\!G(\xi) = 2R_{\rm out}\int\limits_{\alpha}^{\beta}{\rm d} \chi \,{{\gamma(\chi)\chi}\over{(\chi^{2}-\xi^{2})^{1/2}}}.]$

Or in short notation (used below):

$[{}_{\alpha}^{\beta}\!G(\xi) = \int\limits_{\alpha}^{\beta}\gamma.]$

This operation is schematically depicted in Fig. 8. After integration we have the following set of functions:

$[\cases{ 2 \geq \xi \geq 2-\Delta \cr\cr G(\xi) = {}^2_\xi\!G(\xi)}]$

$[\cases{2-\Delta \geq \xi \geq 2-2\Delta \cr\cr G(\xi) = \int\limits^2_\xi\gamma \cr = \int\limits^2_{2-\Delta}\gamma+\int\limits^{2-\Delta}_{2-2\Delta}\gamma \cr = \int\limits^{2}_{2-\Delta}\gamma_0+\int\limits^{2-\Delta}_{2 -2\Delta} [\gamma_1(\chi)-\sigma^3\gamma_0 (\chi/\sigma ) ] \cr\cr = {}^2_{2-\Delta}\! G_0(\xi)+{}^{2-\Delta}_{2-2\Delta}\!G_1(\xi)-{}^{2-\Delta}_{2-2\Delta}\!G_\sigma(\xi)}]$

$[\cases{2-2\Delta \geq \xi \geq \Delta\cr\cr G(\xi) = \int\limits^2_\xi\gamma\cr = \int\limits^2_{2-\Delta}\gamma+\int\limits^{2-\Delta}_{2-2\Delta}\gamma+\int\limits^{2-2\Delta}_\xi\gamma \cr = \int\limits^2_{2-\Delta}\gamma_0+\int\limits^{2-\Delta}_{2-2\Delta}[\gamma_1(\chi)-\sigma^3\gamma_0 (\chi/\sigma)]+\int\limits^{2-2\Delta}_\xi\gamma_1(\chi) \cr\cr = {}^2_{2-\Delta}\!G_0(\xi)+{}^{2-\Delta}_{2-2\Delta}\!G_1(\xi)-{}^{2-\Delta}_{2-2\Delta}\!G_\sigma(\xi)+{}^{2-2\Delta}_\xi\!G_1(\xi) \cr = {}^2_{2-\Delta}\!G_0(\xi)-{}^{2-\Delta}_{2-2\Delta}\!G_\sigma(\xi)+{}^{2-\Delta}_\xi\!G_1(\xi)}]$

$[\cases{\Delta \geq \xi \geq 0 \cr\cr G(\xi) = \int\limits^2_\xi\gamma = \int\limits^2_\Delta\gamma+\int\limits^\Delta_\xi\gamma = \int\limits^2_\Delta\gamma+\int\limits^\Delta_\xi\gamma \cr = \int\limits^2_\Delta\gamma+\int\limits^\Delta_\xi \{\gamma_0(\chi)+\sigma^3 [\gamma_0(\chi/\sigma)-2]\} \cr\cr = {}^2_{2-\Delta}\!G_0(\xi)-{}^{2-\Delta}_{2-2\Delta}\!G_\sigma(\xi)+{}^{2-\Delta}_{\Delta}\!G_1(\xi)+{}^\Delta_\xi\!G_0(\xi)\cr \quad+{}^\Delta_\xi\!G_\sigma(\xi)-2\sigma^3(\Delta^2-\xi^2)^{1/2}}]$

where

$[{}^{\beta}_{\alpha}\!G_{0}(\xi) = 2R_{\rm out}\int\limits_{\alpha}^{\beta}{\rm d}\chi\,{{\gamma_{0}(\chi)\chi}\over{(\chi^{2}-\xi^{2})^{1/2}}} = 2R_{\rm out}[\Gamma(\beta,\xi)-\Gamma(\alpha,\xi)],]$

in which

$[\eqalign{\Gamma(\alpha,\xi) =\hskip.2em& {{1}\over{16}}(\alpha^{2}-\xi^{2})^{1/2}\left[16+{{\alpha^{3}}\over{4}}+{{3}\over{8}}\alpha(\xi^{2}-16)\right]\cr &\!+ {{3}\over{128}}\xi^{2}(\xi^{2}-16)\ln[\alpha+( \alpha^{2}-\xi^{2})^{1/2}],}]$

and where

$[{}^{\beta}_{\alpha}\!G_{\sigma}(\xi) = \sigma^{4}\,{}^{\beta/\sigma}_{\alpha/\sigma}\!G( \xi / \sigma )]$

and

$[\eqalignno{{}^{\beta}_{\alpha}\!G_{1}(\xi) =\hskip.2em& 2R_{\rm out}\int\limits_{\alpha}^{\beta}{\rm d}\chi\,{{\gamma_{1}(\chi)\chi}\over{(\chi^{2}-\xi^{2})^{1/2}}} \cr=\hskip.2em& 2R_{\rm out}{{3}\over{8}}(1-\sigma^{2})^{2}\ln\left[{{\beta+(\beta^{2}-\xi^{2})^{1/2}}\over{\alpha+(\alpha^{2}-\xi^{2})^{1/2}}} \right].&\hfill\llap{(35)}}]$

These explicit functions can be directly used for fitting experimental data.

Acknowledgements

This work is a part of the research program of the `Stichting voor Fundamenteel Onderzoek der Materie' (FOM), which is financially supported by the `Nederlandse Organisatie voor Wetenschappelijk Onderzoek' (NWO). We would like to thank Oktay Uca for discussions.

References

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research papers

Real-space interpretation of spin-echo small-angle neutron scattering

1. Introduction

2. SESANS concept

3. SESANS: from reciprocal to real space

3.1. Two-dimensional SESANS and anisotropy

4. Isotropic media formed by non-interacting particles

4.1. Total scattering length

4.2. Guinier radius

4.3. Normalization

4.4. Eigenfunction and Gaussian coil

4.5. Sphere

4.6. Hollow sphere

5. Conclusion

Appendix A

Moments calculation

Appendix B

Hollow-sphere calculations

Acknowledgements

References