Neutron radiography and tomography are imaging procedures that are based on the different absorption properties of the substances. An object is irradiated by a neu­tron beam; the intensity of the neutrons is measured on the other side of the object

6000 7000 8000 9000 10,000 11,000 12,000 Energy (keV) Figure 10.8 A prompt gamma activation spectrum of a standard cement sample using an HPGE detector with Compton suppression. (Thanks to Dr. Zsolt Rtsvay, Department of Nuclear Research, Institute of Isotopes, Budapest, Hungary, for the spectrum.)

(Figure 10.9). The neutrons are detected using a LiF + ZnS detector (as discussed in Section 14.5.5). The measuring time is between a few seconds to even a day, depending on the resolution and the neutron flux. When the neutron beam is paral­lel, the size of the object and the image is the same; the resolution is 100—300 pm. If the beam is divergent, a magnified image is obtained; the resolution is about 10 pm.

Neutron radiography and tomography can be combined with PGAA (see Section 10.2.2.2). In this way, the chemical composition of the objects can also be analyzed (Figure 10.10).

10.2.2.1 Neutron Scattering/Diffraction

Scattering methods are based on the interaction of neutrons with nuclei to study the microscopic structure, kinetic processes, and magnetic field (see Section 5.5.3). Most applications, which are interesting for chemists, were made in the first two fields and commenced in the 1970s, when the first high-flux research reactors were built.

The diffraction is quantitatively described by the well-known Bragg formula (Figure 10.11):

As = (n)A = 2d sin 0 (10.14)

where d is distance in a chemical system determined by the structure, A is the wavelength, 0 is the angle of incidence at which the intensity maximum occur,

Figure 10.9 A scheme of neutron radiographic/tomographic system. (Thanks to Dr. Zsolt Rtsvay, Department of Nuclear Research, Institute of Isotopes, Budapest, Hungary, for the spectrum.)

Energy (KeV)

Figure 10.10 Left: A neutron radiogram of objects within a lead container (wall thickness: 7 mm). A: M6 iron bolt; B: U3O8 powder, C: three copper balls. Right: Prompt gamma spectra of the object in the lead container. There is an aluminum rod in the container which cannot be seen well on the radiogram, but its spectrum is well-seen. (Thanks to Dr. Laszlo Szentmiklosi and Dr. Zsolt Rtsvay, Department of Nuclear Research, Institute of Isotopes, Budapest, Hungary, for the radiogram and the spectrum.)

Chemical structure, e. g.,
lattice or condensed matter

and n is a small integer (at the first interference, n is 1), As is the difference of the path of the interfering waves.

On the basis of the Bragg formula (Eq. (10.14)), when we know the wavelength and the intensities at different angles, it is possible to calculate the distance charac­teristic of the different chemical structures. This is true for any electromagnetic or particle radiation; e. g., X-ray diffraction and electron diffraction are used for struc­tural analysis.

In structural research, the application of the probe particle (neutron, electron, or photon) is provided by its (energy-dependent) wavelength: it should be comparable to the characteristic size of the microstructure to be investigated. The neutron velocities produced in a nuclear reactor follow the Maxwell distribution determined by the temperature of the nuclear reactor zone, passing the neutrons through a cooled moderator; thereafter a velocity selector (between rotating slits), ~5 X 10—n —10—9m wavelength, usually can be obtained, which is calculated using Eq. (4.93). Assuming a thermal neutron with 0.01 eV energy:

д. H_ = 66256 X 10-34 J/S mv 1.67 X 10227 kg X v

and about 10210 m is given for the de Broglie wavelength.

The incident neutron is represented by a transversal plane wave:

ф(к) = ^0 exp(i к ~r) (10.16)

where ф0 is a constant amplitude, k = 2n n /A, and n is the unit vector in the direction of propagation. The upper arrows mean vectors. The moving front of the wave interacts with nucleus j at point rj of the sample with probability defined by the scattering cross section (oj) and may generate around the nucleus a spherical wave bj exp[ — i k r ]/r with amplitude bj specific of the nucleus. The wavelengths

Figure 10.12 Principles of neutron diffraction (bold letters designate vectors).

of the incident and scattered waves are equal; thus, the scattering process is elastic. Because of its length dimension, bj is called “scattering length” and

aj = 4nbj2 (10.17)

At the applied neutron energies, the phase of the scattered waves is «bk for the individual nuclei.

The scattering length of some isotopes/elements is listed in Table 10.5.

Far from the jth nucleus, the front of the spherical waves can be expressed by plane waves (Figure 10.12):

фj(k0) = bj exp(- i!'[R — j

The scattered wave from a system of N nuclei (in vacuum) is equal to:

ф(к’) = exp(— i к’ Rbj exp(i к’ rj) (10.19)

and the scattering intensity is the thermodynamic average of the squared scattered waves:

(ф*{к ’)ф(к ‘2j = ^ exp(— i к 0 R )exp(— i к’R )^2jbj exp(i к ГТ])^2кьк exp(i к ГГк))

=(ZjHj exp(— i к ‘к —кк d)

(10.20)

where the complex conjugate is denoted by * and the summation is made for

jji = 1, … ,N. к к к

Considering that к0 = к + Q, the intensity will be divided into two terms:

і(кк0)=i(k)+/(Q) = exp(— i к ‘io —кк d

(10.21)

1 exp(— iQfrj —^кі)

where І(к ) is a contribution to the direct beam, and the I(Q ) is the actual scattered intensity, where Q is the scattering vector (Figure 10.12). The magnitude of the scattering vector is:

4n $

Q = T sin 2 (10.22)

where $ is the scattering angle (Figure 10.12).

Considering that the bs are independent of one another and the coordinates, we obtain:

I(Q) = N <b2> 1 (b)2^.^ exp(— i QPj —2 ])) For a condensed phase in thermal equilibrium:

where (1/VN) ФN is the N-particle spatial correlation function.

Because the scattered intensity depends on the difference of two-particle coordi­nates, in isotropic systems, the averaging is reduced to the Fourier transform of the pair-correlation function:

g(r)exp(— i Q 7)d 7 — S(Q) — 1 (10.25)

V

From here,

I(Q) = N{(b2) — (b)2 1 (b2)S(Q)} (10.26)

In this equation, S(Q) is called the “structure factor”; it provides information on the interference caused by the spatial distribution of the scattering nuclei. (b2) — (b)2 is the incoherent part of the scattered intensity, and (b)2S(Q) is the coherent part of the scattered intensity and is determined by the nuclear spin.

The practical applications of this result are promoted by introducing the concept of scattering length density (p), defined as b/v, where b is the sum of scattering lengths in a sufficiently small volume (v). For example, the volume of a water molecule (vw) is:

Vw = Vw/Na (10.27)

where Vw and NA are the molar volume of water and Avogadro’s number, respec­tively. The scattering length of a water molecule is calculated from the scattering length of the atoms:

b — 2Ьн 1 bo

The scattering length density of water is:

Pw — bw/vw (10.29)

A dissolved molecule with a certain scattering length (bs) and volume (vs) in aque­ous solution is seen by the neutrons only if the excess scattering length

Abs — bs — vspw (10.30)

or the scattering contrast

Abs/vs = APs = Ps — Pw (1°.31)

is not equal to zero.

It can be shown that the scattered intensity from mesoscopic inhomogeneities caused by molecular systems (association colloids, macromolecules, polymers, and biological structures) in the solution is described by the same expression as before,

if the bs are replaced by the Fourier transform of the spatial distribution of the scattering contrast:

By dividing I(Q) by the sample volume (V), we obtain the macroscopic scattering cross section (dS/dO, given in cm—1 units):

N

dS/dO = V{(B X (Q)B(Q)) — (B X (Q))(B(Q)) + (B X (Q))(B(Q))S(Q)}

(10.33)

On the basis of the neutrons discussed here, neutron scattering is applied for the structural studies in condensed phases. Since the neutrons are scattered by the nuclei, substances containing light elements are also studied. As seen in Table 10.5, the scattering length can be different for the isotopes of the same elements. This is especially important for the isotopes of hydrogen, 1H and 2H. The great difference in the scattering lengths of hydrogen and deuterium provides the possibility of studying hydrogen compounds, such as biological or other organic molecules, in which the exchangeable hydrogen atoms can be investigated.

During the application of neutron scattering, the intensity of the scattered neu­trons (in other words, the macroscopic scattering cross section (dS/dO)) is plotted as a function of the scattering vector (Q). Models elaborated for B(Q) and S(Q) are fitted to the experimental scattering patterns, and the reliability of the fitting para­meters is judged by the quality of the fit. In Figure 10.13, the neutron-scattering patterns from micellar solutions of sodium alkyl sulfates are plotted, together with the best-fit curves and the squared deviations of the experiment and theory. The systematic study with different alkyl chain length results in reliable structural data and in a direct proof of the electrostatic potential acting among the ionic micelles.

One of the most important technical features of neutron scattering is contrast variation. By varying the hydrogen isotope composition (hydrogen/deuterium ratio) of water (Table 10.5), pw can have a varied range, which is wide enough to cover the scattering contrast of most components of organic molecules. Therefore, certain parts of the molecules can be excluded from the observed scattering patterns (Figure 10.14). The method is unique and is used mainly in biological systems.

In condensed phases, many important kinetic processes are random and occur over a relatively long time scale. Rotational jumps of a molecule and the diffusion of a particle in liquid are two examples of the types of motion that contribute to the quasi-elastic component of the inelastic spectrum, where the sum of the changes in the energy of the neutrons scattered in a particular direction (ДЕ) equals zero.

Figure 10.13 SANS patterns from 0.0729 mol/dm3 solutions of sodium alkyl sulfates of different chain lengths (open symbols) with best-fit curves (solid lines). Residual squares are plotted with solid symbols connected with dotted lines.

Source: Adapted from Vass et al. (2000), with permission from American Chemical Society.

Figure 10.14 Scattering length density of water and proteins at different H:D ratios obtained by small-angle neutron scattering.

The intensity I(Q, AE) scattered in the angle defined by Q, stemming from diffus­ing molecules in a liquid, is described as follows:

h

-DQ2

KQ, AE) = A— ^ h 2 (10-34)

n [(AE)2] 1 2Ldq2

where A is the amplitude defined by the scattering amplitude b and apparative con­stants, h is the Planck constant, AE is the energy change of the neutron (inelastic component), and D is the self-diffusion coefficient of the molecule. Quasi-elastic neutron scattering provides a precise tool for determining the self-diffusion coeffi­cient in the bulk phase of liquids. Figure 10.15 shows the decomposition of a quasi-elastic neutron-scattering spectrum of a micellar solution. The solvent mole­cules were found in two kinetic states: along with the major (~94%) component moving with the bulk-phase self-diffusion coefficient, a slower water component (~6%) could also be observed. The slow component was assumed to form the hydrate sphere of the micelles.