Abstract
Scattering methods are powerful tools used for the examination of condensed matter. They offer the possibility to analyze particles without disturbing their natural environment. Small angle neutron scattering (SANS) can also be used to analyze solid and liquid systems, phase transformations, germination, growth flaws and defects, as well as generally any inhomogeneity occurring in a range of 10–1000 Å. The most common experimental SANS methods and patterns are reviewed and explained.
Keywords
- SANS
- methods
- patterns
1. Introduction
1.1. Material science issues accessible through small angle scattering of X-rays (SAXS) and neutrons (SANS)
The fundamental properties of the neutron make it a powerful tool for materials science investigations. It is useful to introduce neutrons by comparison with X-rays because both probes are used for diffraction investigations, and more workers are familiar with X-ray methods. X-rays can be obtained with a generator tube or, in the past few years, at a synchrotron X-ray laboratory.
Neutrons for neutron beam research arise either from nuclear fission in reactors or from the spallation process. In the latter, accelerator-produced high-energy charged particles yield a large number of energetic neutrons from collisions with heavy-atom nuclei in the accelerator target.
X-ray diffraction was utilized for the first time in establishing the atomic structure of the crystals. Later, this method was used in other applications, extended to the study of imperfections in crystals, of the size of the crystallites, and even to the study of amorphous objects. From the classical discovery by Laue, in 1912, that X-rays are diffracted on their passage through crystals, the development of the applied domain of this technique was due to the subsequent progress of the X-ray diffraction theory and the improvement of experimental methods. The first observations on small angle X-ray scattering (SAXS) were performed in the beginning of the 1930s [1], and its domain was developed later.
In the following years, the small angle X-ray scattering technique has known to be an important development, proved by a great number of publications over the years. SAXS technique was developed from the necessity to observe long lattice spacing in crystals, in comparison with wavelengths of the X-rays used in structural analyses. Such distances are encountered in certain minerals and certain complex molecules such as polymers or proteins. For example, in the study of macromolecular crystals, the systems for X-ray diffraction should be extended to include the very small angles. The fundamental equation which describes the X-ray diffraction in the crystalline substance is
In ordinary crystals, especially those of inorganic matter, most inter-lattice distances observed are on the same magnitude with the wavelength of utilized X-rays; therefore the used
The first device of this type, initially called “neutron spectrometer,” was built for the first time in 1945 at Argonne National Laboratory in USA. Since then, many instruments with different shapes and conception were built around the world, and they are mostly known as “neutron diffractometer.” The investigation of condensed matter with the help of neutrons is not a substitute for classical X-ray diffractions, but it is a completion of these techniques, because the special nature of neutron interaction with the substance allows the neutron diffraction to provide information that cannot be revealed by other means. The spectrum of neutrons emitted by a reactor is dependent on the moderator temperature, and in most reactors, the spectrum peak is around 1.5 Å. This spectrum is Maxwellian and has a maximum which depends on the temperature. Therefore, for
There are many processes that lead to scattering length variations, extended to zones up to 100 Å. Among these, there are the processes in solid phase namely aging, precipitating processes in solid phase, repeated thermal shock, neutron irradiation, the processes in mixed systems, for example the catalysts, where the two coexisting solid crystalline components are of enormous interest, as well as the ferofluids or the polymers in which neither one component is crystalline. The application of SAXS and SANS to material science has its origin from the experiments performed by Guinier in 1938, to study different types of aluminum alloys with various components such as Al–Cu and Al–Ag. He proved the existence of extremely small precipitate zones (so-called Guinier–Preston (GP) zones) as the first step in solidification of the precipitate of these alloys [3]. The GP zones represent submicroscopic regions of the matrix in which are produced a concentration of the initial solvate element. From one point of view, the GP zones could be considered as pre-precipitates because their formation represents a preliminary stage of precipitations set up from ultra-saturated matrix, but from another point of view it could be considered “clusters,” in the sense that represent preferential associations of atoms.
In Al–Cu alloys, it has a diameter of about 80 Å and a thickness of 3–6 Å. With an average composition of 90% Cu, it comes out that the GP zones in these alloys are predominant constituted from Cu atoms. The physical nature and structural characteristics of the formed precipitates are dependent on temperature and heating duration because the precipitates are formed by germination and growing up processes, based on diffusion phenomena. The most important modification of alloy proprieties occur in the initial stages of the aging, with low temperatures and short heating durations when microstructural transformations are to a scale size under the resolution limit of optical microscopy (submicron precipitates). All these phenomena could be emphasized by SAXS and SANS methods. Regrouping of various atoms species during the precipitation process modifies the dispersion strength of X-rays or neutrons in different zones of aged alloy, due to the existence of an important density difference between the precipitated zones and the rest of alloy matrix. Among the structural processes produced through thermal treatment, which is applied to solid crystalline materials, the spinodal transition is something special [3]. The first phases of coherent precipitation took place through phase separation inside the crystalline lattice. Phase separation could start only through a nucleation process followed by nuclei growing. The germination of a new phase in a solid solution needs that in some regions the solid solution should considerably modify its composition. To be transformed into stable germs, these regions should have a specific dimension (about 10 Å). In these conditions, the interface between the germ and the solid solution (matrix) shows a distinct structural discontinuity, and therefore it has a free positive energy which inhibits the germination. This germination difficulty is more evident in spinodal decomposition. In this case, the matrix regions, where the composition is modified, became more extended than classical germs and are named clusters. The interface between the matrix and the regions with modified composition became diffusive and does not present anymore a distinct structural discontinuity characterized by a positive free energy. To produce regions with modified composition, extended on large volumes, long-range composition fluctuations are necessary that would tend to decompose the original solid solution. The probability of these long-range composition fluctuations is much more increased when the temperature of original solid solution is situated close to the alloy “spinodal temperature.” Above the spinodal temperature
Another important field of study is offered by complex disordered materials such as ceramics, clays, cement, or glasses. Among these, the importance of cement in modern society cannot be underestimated. Cement is found in concrete structures everywhere such as buildings, roads, bridges, dams, and even for conditioning radioactive wastes. There is no escape from the impact of cement in our everyday life. Due to its great importance in the following, we will point only on its applications. Cement, as it is commonly known, is a mixture of compounds made by burning limestone and clay together at very high temperatures ranging from 1400 to 1600°C. Water is the key ingredient, which when mixed with cement forms a paste that evolves as a hard product. The water causes the hardening of the cement by a process called hydration. Hydration is a chemical reaction in which the major compounds in cement form chemical bonds with water molecules to yield hydrates or hydration products.
The heterogeneous nature and chemical complexity of the cement make the characterization of this system difficult. In spite of the inherent difficulties of the system, considerable understanding of the nature of cement and concrete has been obtained through the use of a wide variety of tools.
Studies of the cement pore liquid as a function of time, X-ray diffraction investigations, electron microscope, and NMR experiments have all contributed to a better understanding of the behavior of this material.
The greater penetrating power of neutrons (in comparison to X-rays), the opportunities for in situ measurement of hydrating pastes, the ability to use very long wavelength incident neutrons and the unique opportunities to change contrast offered by D2O–H2O exchange make SANS attractive as a means to probe the internal microstructure of cement pastes. The issue to be resolved, however, is the interpretation of the small angle scattering data. While the actual measurements are generally straightforward, the complexity of phases, contrasts, object morphologies, and size ranges forms a barrier to simple interpretations.
Small angle neutron scattering was first used to study the microstructure of cements in the early 1980s by Allen and coworkers [4]. They measured the small angle neutron scattering from a set of cement specimens hydrated at different water to cement ratios. They also examined the effect of soaking the specimen in heavy water and the change in SANS obtained when the specimen was dried. In their analysis of the SANS data, they assumed a simplified model of the cement hydration reactions. In this model, the hydrated compounds C6AFH12, C4AH13, C3ACSH12, and C3S2H2.5 and the hydroxides Mg(OH)2, Ca(OH)2, NaOH, and KOH in appropriate amounts were the products of cement hydration. Each of these compounds represented a different volume fraction of the cement paste. The strength of the SANS signal from the specimen in the small
The SANS is presumed to be caused by a size distribution of objects of a known shape. To calculate the form factor for objects of known shape we can use Equation 17.
The parameters describing the object and its size distribution are then varied to minimize the difference between the calculated scattering and the data. Spheres, cylinders, and disks were considered by Allen and coworkers, and it was ultimately concluded that the SANS from their cement paste specimens was created by a distribution of water-filled spherical pores approximately 5 nm in diameter and a smaller component of pores with diameters of about 10 nm. SANS from the specimen that had been dried and not rewetted was interpreted as the one due to scattering from a much broader distribution of pore sizes with a peak of about 5 nm but extending to much larger diameters.
SANS from the specimens immersed in D2O showed that the heavy water rapidly exchanged with the water in the pores and with the water in the surrounding C–S–H gel. The total porosity was of the order 1% of the volume.
All these are only a few examples that illustrate the large field of SAS application, where the immediate observation of the phenomena produced in materials lead to solving many practical problems and to extend the theoretical knowledge as well.
2. Physical principle of SANS
SANS is a method used to obtain information regarding the shape, dimensions, and internal configuration of the zones with scattering density significantly different from the average value, having relatively small dimensions (up to 1000 Å), unevenly distributed in some environment, where the distance arrangement is present or not. In a considered sample, all scattering centers participate in the scattering process. The scattering centers could be atoms, molecules, or particles of the analyzed system. System response could be a spatial distribution of scattered radiation reducible to a succession of maximums, determined by the existence of distance arrangement in a material, or a succession of overlapping maximums over a continuous distribution when the distance arrangement is accompanied by disordered processes, such as amorphization, precipitation in solid phase, flaws, etc., or by a continuous distribution which present one or two isolated maximums like in the case of liquids (in this case the maximum is owed to the next close vicinity of each scattering center) or a continuous curve in case of SANS. Because X-ray scattering is similar to neutron scattering, in the following presentation I will start with the theoretical principles of SAXS accordingly adapted for SANS. It is well known that a diffraction image of a sample could be easily described in terms of reciprocal space or Fourier space [1]. If we consider
The X-ray diffraction theory is based on the fact that
where
Let us consider a particle bathed in an X-ray beam; thus all the electrons are wave scattering sources. If the scattering direction is the same as the incident ray, we can say that the scattered rays are all in phase and, if the scattering angle increases, the phase difference between the different scattered waves will also increase. Therefore, the amplitude of the resulting scattered wave will have values bigger or smaller in accordance only with the existing phase differences. This happens for a scattering angle about 2
If
This can be demonstrated in the following manner. Let us consider the functions
Thus, the Fourier transform of
Using the notations
we replace the elementary differentials
Giving the dimensions of the region where
where
The total amplitude of scattered radiation will then be
and the scattered intensity, the product between the amplitude
The intensity scattered by an electron is
which is a three-variable function, the incident intensity, the scattering angle, and the distance of observation;
We define the structure factor of the environment as the report between the total scattered amplitude and the radiation amplitude scattered by a single electron in the same conditions:
The scattered intensity will then be
The term “
Small angle neutron scattering is used where the X-rays cannot provide the desired information either of the lack of scattering contrast or of severe absorption of the studied material. The choice of thermal neutrons as a means of substance investigation was owed to their wavelengths and their energies corresponding to the inter atomic distances and excitation energies of condensed matter. Neutron absorption from the matter is very low, thus resulting in the samples that can have thicknesses larger than in the X-rays case which are strongly absorbed with an increase in the volume of analyzed samples. We can say that SANS allows the investigation of materials in various conditions, such as in containers, furnaces, cryostats, etc. Due to its magnetic moment, the neutron provides unique possibilities to the study of magnetic structures, magnetic moment distribution, and magnetic excitations. Being a nuclear propriety, the nuclear scattering amplitude can be considerably different between various isotopes of a specific chemical species. For example, the big difference between the coherent scattering lengths of hydrogen and deuterium lead to the usage of phase contrast in the study of hydrogenate materials, allowing a good resolution in analyzing polymers and biological substances in general. Because thermal neutrons interact very low with the matter, this interaction can be theoretically treated on the base of first Born approximation [5, 6]. We will show further that the variation of the scattering density is very important in the study of condensed matter through SANS. In the following, we will consider the “static approximation”; the scattering process is produced only at nuclear level on fixed targets letting those targets unchanged after the collision, and without considering the polarization effects.
Thus, we can write the coherent elastic scattering differential cross section on a single atom:
where
where the integration is performed over the whole sample volume
The scattering length density
where
For a specific distribution of scattering length, the cross section can be analytically calculated with the help of Equations (13) and (15). Because the cross section values are not known everywhere in reciprocal space,
Using Equation (13) we obtain
where the integral is extended over the volume
where
that is valid in the case of binary systems, without contributions from the interferences between particles.
The interference term from Equation (16) that was neglected in Equation (18) is the Fourier transform
where
The
where
where
with
For spheres of radius
to the term proportional with
For rotational ellipsoids, the Guinier approximation coincides with the extension of scattering functions until
This shows the average decrease in the scattering function at large
3. Experimental methods used in the study of condensed matter by SANS
The goal of SANS method is to measure
Therefore, to measure
3.1. The method I (θ ) at constant λ
This method is based on monochromatic (
When the cylinder is rotating with the angular speed
Monochromatic neutron beam separation from the thermal neutron beam provided by the reactor is realized through its Bragg reflection on a monocrystal. If we note the distance between the crystalline planes with
where
In other words, if a monocrystal is used to select neutrons with a specific wavelength
Analyzing the Bragg relation, we can see that by using of monochromator crystals, only monochromatic neutrons with the wavelength of
This observation suggests a simple way to eliminate the second-order reflections, utilized mostly when the monochromatic neutrons desired to be extracted from the reactor emergent beam have wavelengths relatively large, for example, in the range
Frequently, the experimental pattern used to obtain monochromatic neutron beams in this way contains a Soller collimator which spatially delimitates the incident neutron beam coming out from the reactor, the monochromator crystal, and the second Soller collimator which delimitates the diffracted beam. Simultaneous rotations of the second collimator and the crystal with angular speeds staying in a ratio of 2:1 allows to extract monochromatic neutron beams of different wavelengths from the continuous spectrum of the incident thermal neutron beam. The monochromatic neutrons are recorded by a detector placed after the second collimator rotating in the same time with it.
The devices built on this principle, the so-called neutron spectrometer with crystal, allow the determination of total cross section by the measurement of sample transmission in accordance with neutron wavelength. To study the angular distribution of scattered neutrons by a certain target and implicitly to measure the differential cross sections, a pattern with a third Soller collimator between the sample and detector is used. To investigate the neutron scattering of different wavelengths, the whole system built from the second collimator, the studied sample, the third collimator, and the neutron detector should be rotated around the first fixed axis that is going through the middle of the monochromator crystal and together with it; the angular speed ratio of the crystal and the mentioned system should be 1/2. For a certain position of the monochromator crystal and the second collimator, the third collimator and the detector could be rotated around the second fixed axis (that is going through the middle of the sample) allowing the measurement of sample differential cross section for thermal neutrons having the wavelengths determined by the monochromator preceding the sample. This experimental pattern is named neutron spectrometer with two axes. Many experimental systems use optical patterns derived from the two axis spectrometer described earlier. Anticipating the discussion in the next section, we would mention that from the simple presentation of the optical pattern of considered spectrometer arises a specific limitation of its efficiency. Actually, to measure
Two crystal systems were used as monochromators in spectrometric assemblies long time ago, first in X-ray physics and then in neutron physics. To discuss the utilization manner of two monocrystals systems in SANS, we have to reveal the optical characteristics of these systems. To simplify the explanation, we will limit our case to consider that the two monocrystals are ideal identical crystals, from the point of view of their internal crystalline structure and their cutting method (has the same active reflecting planes). A two-monocrystal system could work in two optical patterns: parallel (Fig. 3) and antiparallel (Fig. 4).
In the parallel pattern, any selected wavelength from the incident polychromatic beam through Bragg reflection on the first crystal leads to the propagation of the double reflected beam in parallel direction with the incident beam. Therefore, this pattern is not dispersive (does not spatially separate the beams of different wavelengths). In the second optical pattern, the Bragg double reflected beam is propagated on a direction oriented toward the region containing the radiation source and is named antiparallel. To notice that the wavelength change of the selected radiation beam through double reflection is realized along two directions depending on the wavelengths, revealing the dispersive feature of the antiparallel pattern. In the case of the parallel pattern, the thermal neutron beam falling on the first monochromator crystal under the incident angle
If a sample is interposed between the two crystals, then the beam reflected by the first crystal will be scattered by the sample under the
To detect the scattered beam, we have to rotate the second monochromator crystal together with the detector by an angle
Because the sample is scattering, the neutrons under different angles
If we have a sample between the two crystals, then the reflected beam by the first crystal will be scattered by the sample under a
The neutrons scattered by the sample
To notice that this pattern is the most frequently used due to its dispersive feature this allows a better measurement of the interest neutrons.
3.2. The method I (λ ) at constant θ
Another method is to make the observation only on a certain scattering direction 2
where
Neutrons of different wavelengths will need different time to cover the same distance
Here,
In the case of steady-state reactors, the neutron beam leaving the reactor channel meets an obturator in its way which opens the way only for very short time intervals. In this manner, the detector “notices” a pulsed neutron beam with a determined time length, for a specific obturator–detector distance, by the neutron energetic spectrum. The detector is connected with a time of flight analyzer, which can be unleashed by the obturator opening. To the pulsed reactors, the obturator function is undertaken by the neutron source itself. The time of flight analyzer start-up command is given by the neutron source when the pulse is accomplished. In every SANS experiment, the main goal is to determine with a good precision and in a reasonably short time the
For any
4. Experimental patterns
4.1. The method I (Q ) at constant λ
As it was discussed in Section 3, from among the optical patterns, the one with two crystals and the sample situated between them is much more appreciated. This pattern was used in many reactors, and a typical description of it is given in Fig. 9.
The monochromator group built by
We consider that both crystals are copper monocrystals identically cut, with the reflecting plane orientation in the (002) direction and a mosaic divergence structure of 14 minutes. The
The neutron beam is geometrically built up by a 3 m-long vacuum collimator. The maximal beam dimension in the sample is 2 cm diameter. The distance from the sample to detector can vary from 1 to 5 m. The spectrometer is provided with an automatic sample changer controlled by a computer. The scattering vector range lies between 5 × 10−3 Å−1 <
Measurement instrument control and experimental data collection are performed by a computer.
4.2. The method I (λ ) at constant θ
This method is used mostly for the neutron pulsed sources. This type of spectrometer built on the time of flight principle is installed to the IBR-2 pulsed nuclear reactor at Joint Institute for Nuclear Research (JINR) in Dubna, Russian Federation. The name of the spectrometer is YuMo [7], given from its constructor, the scientist Yu. M. Ostanevich (Fig. 11). The reactor fuel is plutonium oxide, and uses liquid sodium as a coolant. The reactor reflector is composed of a static part and a rotational one with constructive and movement characteristics that determine the issue of fast neutron pulsed flux. The produced neutrons are moderated in water and through the horizontal beam tubes that surround the reactor active core in the experimental hall.
In the current working conditions, the reactor provides a maximum flux of thermal neutrons of 1016 neutrons/cm2s per pulse at the moderator surface. The frequency of pulse repetition is 5 Hz.
The chopper placed after the neutron moderator plays the main role to remove the neutrons from other reactor pulses and to assure that the neutrons from the interested pulse arrive at the detector. The neutron tube is made by steel pipes of different diameters and is kept under vacuum of about 10−2 Torr to avoid neutron absorption.
It is connected to the central vacuum system which is controlled from reactor control room and is provided with a mechanical beam stop that automatically closes the beam. A complete cycle of closing–opening the beam stop lasts about 6 minutes. The chopper is followed by a collimator with discrete variable aperture (C1) that assures a geometrical pre-shape of the neutron beam and by the monitor whose signals are used for the normalization of the data obtained from different reactor pulses. The next collimator (C2) assures the geometrical profile of the incident neutron beam in the sample and defines its dimensions. The collimator (C1) is composed of four cylinders of different diameters (100 mm, 80 mm, 60 mm, and 40 mm), but with the same length which can be remote-controlled to be aligned in the neutron beam by a rotational movement in accordance with the experimental requirements of the beam geometrical pre-shape. The collimator (C2) which limits the geometrical profile of the incident neutron beam in the sample has a remote-controlled variable opening (28 mm, 14 mm, and 7 mm). The holder with the collimation orifices of the (C2) collimator are made from boron polyethylene having an insignificant scattering cross section in comparison with the absorption cross section. All collimators and the most important shielding elements are covered with a mixture of polyester and boron carbonate or boron acid. The sample holder having six slots is connected to a special remote-controlled mechanism of sample changing that allows each sample to be introduced in the neutron beam according to the experimental program. The sample is placed at 18.74 m from the moderator surface. The sample holder is provided with a cooling and warming system that allows adjusting the sample temperature between 20 to 150°C.
To record and analyze the scattered neutrons in accordance with their time of flight, the experimental equipment is provided with a measurement system having two detectors. The detectors are functionally identical but different by dimensions that assure their efficiency in observing the
The detector room is located after the samples and is made from a steel pipe of 1200 mm diameter and 12 m length having the inside surface covered with a cadmium layer of 0.5 mm thickness. Inside the room are mounted two rails for the movement of the electric-driven detectors trucks. The incident neutron beam is transmitted through the central hole of the detectors. At a distance of about 1.5 m from the reactor side, the detector is placed on a truck-driven mechanism in the vanadium standard (a vanadium metal sheet of about 0.3 mm thickness) that allows the introduction of the beam according to a program. This standard is used for the obtained experimental data calibration. The incident beam detector placed after the annular detectors measures the spectral distribution of the incident neutrons in the sample and forwards the normalization to this spectra of the scattered neutron spectra.
5. Conclusions
To summarize, we can say that small angle scattering is the collective name given to the techniques of SANS and SAXS scattering. In each of these techniques, radiation is elastically scattered by a sample, and the resulting scattering pattern is analyzed to provide information on the size, shape, and orientation of some components of the sample. They offer the possibility to analyze particles without disturbing their natural environment. The type of sample that can be studied by SAS, the sample environment that can be applied, the actual length scales that can be probed, and the information that can ultimately be obtained all depend on the nature of the radiation used. For example, SAXS cannot be used to study thick samples or samples requiring complex containers, while SANS can penetrate deeper in the condensed matter. SANS is produced by heterogeneities in matter. If these are randomly oriented, every atom pair contributes to the scattering of a sample. Inhomogeneities of sizes larger than atomic distances (10–1000 Å) produce scattering patterns with
A scattering experiment sees a scattering length density; in the case of X-rays, this is simply the electron density. The absolute density is not important but, in contrast, the difference between the particle and the surrounding medium.
The result of the experiment is the Fourier transformation of the contrast distribution. By comparing the experimental data with the theoretically calculated intensities or by a Fourier back transformation, we receive information about the contrast distribution, e.g., the mass density distribution of one particle. In the case of concentrated systems, we receive combined information about the single particle and the interaction between different particles.
In most cases, the sample and/or the sample environment are rather bulky.
Therefore, SANS instruments usually have to be large themselves in order to yield the desired resolution. Small SANS instruments can only serve a very limited number of applications. For reasons of intensity, a relatively large beam divergence, i.e., a beam cross section larger than the sample size is accepted as well as wavelength resolutions Δλ/λ of up to about 20%.
SANS and SAXS techniques are complementary; however, they share several similarities. Perhaps the most important of these is the fact that, with minor adjustments to account for the different types of radiation, the same basic equations and “laws” (for example, those due to Guinier and Porod) can be used to analyze data from any of the two techniques. This is a great advantage and one that has certainly eased the transition from one technique to another.
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