Glossary for Small-Angle Scattering

Adrian R. Rennie
Materials Physics
Uppsala University
Uppsala, Sweden

This glossary of terms is intended to provide a brief guide to jargon, abbreviations, acronyms and the technical expressions used in the description of small-angle scattering experiments and related fields. By necessity, the account of each topic is brief. In some cases, an equation is quoted or a quantity is defined mathematically but these descriptions can not replace a full text book account of the subject! Many users come from different scientific disciplines and for this reason the explanations are kept at a simple level.

The aim has been to describe the terminology used to describe instruments and experiments. A few particular materials are mentioned and explained as experienced users may assume that the purpose will be evident just from mentioning a material!

The terms are arranged alphabetically but many links are provided within this glossary so that further clarification of a subject can be found. The explanation is at an introductory level. The experienced reader will have to excuse some simplifications.

Note. In order to use notation that is standard, equations have been set using the symbol definitions for HTML 4. If this is not available to your browser, the equations may be confusing. As most browsers can display these symbols, this method was chosen in place of using large numbers of graphic elements for equations.

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A

Absorption

Absorption refers to that fraction of the beam that is neither transmitted nor scattered by the sample. This attenuation of the beam intensity must be considered when calculating the scattered intensity on an absolute scale. If there is significant absorption of X-rays from an intense source, this can cause heating and beam damage. Materials that absorb strongly may be used as shielding in the construction of scattering instruments.

Anode

The anode forms part of a generator that is the most common source of X-rays for laboratory scattering experiments. Electrons are accelerated to impinge on the anode, which then emits a spectrum that is characteristic of the metal that forms the anode (for example, copper or molybdenum).

Anomalous Small-Angle X-Ray Scattering, ASAXS

Anomalous small-angle X-ray scattering (SAXS) exploits the variation of scattering cross-section near absorption edges to vary the contrast between different components in a sample. It requires measurements at different energies (wavelengths). In some respects, it is similar to contrast variation used in small-angle neutron scattering (SANS) experiments.

Aperture

An aperture is used to define the size and shape of a beam in a scattering experiment. The angular divergence, that must be well-defined in small-angle scattering instruments, is often controlled by apertures made from absorbing material at either end of a collimator that may simply be an evacuated flight path. The apertures may be adjustable, for example by replacing pieces of absorber with different size holes or by using adjustable jaws.

Area Detector

Area Detector - see position sensitive detector.

Attenuator

An attenuator is a device or material that is inserted in a beam to reduce the intensity. This may be calibrated such that the attenuation factor is known to allow comparison between different measurements.

Avalanche Photodiode, APD

An avalanche photodiode is a detector based on a pin diode with a large reverse bias so that it can detetect single photons.

Azimuthal Angle

The azimuthal angle, Φ is often taken as varying in the plane of the detector in small-angle scattering experiments. It can be used as a polar co-ordinate about the centre of the beam and is useful if the distribution of scattering is not isotropic. The variation of intensity with Φ at a given momentum transfer, Q can be used to calculate order parameters and can be modelled for particular types of anisotropy such as those arising from uniform deformation of spheres or from applied magnetic fields. There is no particular convention as to which direction is taken for Φ = 0 although it is often convenient to use a principal axis that is related either to the anisotropy of the scattering, the sample or the detector (such as horizontal or vertical).

B

Background

An important part of the data reduction can be assessment of the background and subtracting this in an appropriate way from the data for each sample. Particularly at large momentum transfer, the signal may be dominated by a number of factors that do not arise from the structure of interest within the sample. What is considered as background may vary between experiments and disciplines. Factors that may be important are solvent scattering, scattering from an empty sample holder or windows, electronic noise that comes from the detector or detector read-out and background radiation. In the case of neutrons, incoherent scattering from the sample may also be considered as part of the background.

Beam Damage

Beam damage refers to the physical and chemical changes that can occur in a sample because of exposure to the radiation that is being used for a scattering measurement. This arises from absorption and subsequent heating or ionisation of the sample material. This is more often a problem with synchrotron sources than with other less intense radiation.

Beam Stop

A beam stop is a piece of absorbing material that is placed in front of the detector so that the direct beam does not reach the detector. Many detectors cannot count the high flux that is present in the direct beam and this would distort the signal and possibly cause damage to the detector. For position sensitive detectors the beam stop is adjusted so that only the region around direct beam is obscured and scattering can be measured at wider angles. A beam stop is also sometimes provided to block the line of direct beam so that it cannot cause irradiation outside an experimental enclosure.

Bending Magnet

Bending magnets form parts of storage rings and synchrotrons - they deflect the electron beam so that it circulates in a continuous path. When a high-energy electron beam is deflected by a magnet, it emits synchrotron radiation. Modern facilities use insertion devices to provide more brilliant radiation sources.

Beryllium, Be

Beryllium is a light metal with a low atomic number (4) and consequently has a low electron density. This gives low absorption and scattering for X-rays and so thin sheets of beryllium are often used as windows for X-rays. Beryllium has a high scattering cross-section for neutrons and it is sometimes used as a filter to remove short wavelength neutrons from a beam. A cooled polycrystalline block of several cm length that is surrounded by a neutron absorber will diffract short wavelengths away from the beam direction but allow wavelengths larger than 4 Å (energies below about 5 meV) to be transmitted.

Bio-SAXS

Bio-SAXS refers to instruments or experiments that are optimised for the study of biological molecules. These studies often require measurements of dilute solutions and so facilities to measure the small differences between the sample solution and pure solvent or buffer are important. Analysis of the data can often exploit the known composition and sequence of proteins and their monodisperse character.

Bonse Hart Camera

If two crystals are set so that they both diffract a particular wavelength, they can be used to create a very high-resolution small-angle scattering camera. When a sample that scatters is placed between the two crystals, it will broaden the beam from the first crystal. A rocking scan of the second crystal can measure this broadening and thus determine the small-angle scattering. The geometry can be used for both X-rays and neutrons. The resolution is usually much higher than that achieved with a pin-hole camera. An advantage, particularly for neutron scattering, is that the resolution in momentum transfer is almost independent of beam size. If large samples and crystals are available, using a large beam area can compensate for a low flux. In practice, the crystals are often cut with channels so that multiple Bragg diffraction from each crystal provides a beam with very narrow spread.

Boron Trifluoride (BF3) Detector

A means to detect thermal or cold neutrons that uses primarily the nuclear reaction:

n + 10B -> 7Li + 4He + 2.3 MeV

A convenient material has been the gaseous compound boron trifluoride BF3. The released energy ionises the gas. The ionisation can be detected readily and even the position of detection can be determined using a suitable array of electrodes connected to a high voltage supply and amplification circuits. These detectors have been used less frequently in modern instruments because of the lower detection efficiency than 3He at short wavelengths and difficulties associated with the reactivity and toxicity of BF3.

Brilliance

The brilliance is an important figure of merit for synchrotron radiation sources. It describes the number of photons emitted per unit area per unit solid angle per unit wavelength per unit time. As small-angle scattering experiments require well-collimated beams with defined angle and wavelength, the brilliance is more important than simply the total flux that is the number of photons emitted per unit area per unit time.

C

Cadmium, Cd

Cadmium is a soft metal that has a high absorption cross-section for thermal neutrons. 113Cd has a high neutron capture cross-section (over 20000 x 10-28 m2 i.e. 20000 barns) and with a natural abundance of about 12% even natural cadmium acts as a good absorber of thermal and cold neutrons. This reaction is associated with strong γ-ray emission at 558.5 keV. Cadmium is sometimes used to make apertures or provide shielding of components in small-angle neutron scattering experiments. Cadmium and its compounds are toxic and needs to be handled with care. As it is difficult to provide good shielding for the hard γ emission, the minimum possible cadmium should be exposed to neutron beams. Other absorbing elements such as boron and lithium are usually preferred.

CCD, Charge Coupled Device

CCD is an abbreviation for charge coupled device. These are used as multipixel photon detectors. Most are sensitive to visible photons and so to use them for X-rays or neutrons, converters such as scintillator screens, are usually needed. As the CCD is often small (order cm) with thousands of pixels in each direction, optical coupling from a large scintillator screen can provide some optimisation of solid angle of detection and pixel resolution. A CCD detector is an example of an integrating detector, the intensity recorded in each pixel is usually read-out at the end of an exposure.

Chopper

A chopper is used to provide a pulsed beam or to provide shorter pulses from a source that is already pulsed. These are used in neutron scattering experiments to allow time-of-flight measurements. This method can be used to measure simultaneously a broad range of momentum transfer when using a 'white beam' or for calibration of monochromatic beams.

Circular Average

If data is recorded on a two-dimensional detector for samples that display no anisotropy in the scattering with respect to the azimuthal angle, it is convenient to average the data in rings of given ranges of momentum transfer, Q. These rings are centred at Q = 0, where the direct beam would hit the detector. This averaging provides data with higher statistical accuracy. It is often simpler to calculate models of scattering for these smaller data sets. In some data reduction packages this is average is known as a radial average.

Coherent Scattering

If scattering occurs such that the phase of the wave is preserved, interference effects can be observed either with other waves that are scattered from the same (coherent) source or with the residue of the incident wave. The coherent scattering can provide information about structure as it arises from interference between waves from different scattering objects. This idea is common to light, X-ray and neutron scattering. For neutrons, the possibility of spin changes causes an extra possibility of spin incoherent scattering that destroys interference. The coherent scattering length is therefore important in determining the intensity and distribution of small-angle scattering. Incoherent scattering is often treated as part of the background.

Cold Source

The wavelength of a neutron depends on the energy. Multiple inelastic scattering can be used to change the energy distribution to provide more neutrons in the desirable range. As the long wavelengths that are desired for small-angle neutron scattering experiments correspond to a distribution at low temperature, the neutrons from a reactor or a spallation source are often 'moderated' by multiple scattering in a material with a high density of hydrogen or deuterium maintained at a temperature of about 20 K.

Collimation

The collimation part of a small-angle scattering instrument usually involves apertures or slits that are separated by an evacuated path to define the angular divergence of the beam that is incident on the sample.

Contrast Variation

Contrast variation is the term used to describe changing isotopic labels of particular components in a sample to allow identification of neutron scattering from those components or to highlight particular aspects of the structure. This often exploits the large difference in coherent scattering length of normal hydrogen (1H) and deuterium (2H or D).

Convolution

The convolution of two functions can be considered as the smearing of one function by another. For example scattering from a sample is broadened by a number of instrument resolution factors. The mathematical expression of this would be that the convolution, C of the function A(x) with the function B(x) could be written as:

C(x) = A * B = ∫-∞ A(u-x) B(u) du

This expression is the convolution integral and the function C(x) depends only on the variable x.

Convolution Theorem

A simple relation between Fourier transform of of the convolution of two functions is important as it can be used to derive many simple results in small-angle scattering. It states that the Fourier transform of a convolution of two functions is equal to the product of the Fourier transform of the two functions. If the Fourier transform of A(r) is FT[A(r)] = A'(Q) and the FT[B(r)] = B'(Q) then for the convolution, C(r), of A(r) and B(r),:

C(r) = A(r) * B(r)

has a Fourier transform, C'(Q) given by:

C'(Q) = FT[C(r)] = FT[ A(r) * B(r) ] = FT[A(r)] . FT[B(r)] = A'(Q) . B'(Q)

For example, this relation can be used to deduce the scattering from interacting particles that have density distributions A(r) and correlations between particles described by B(r). In the approximation that these are separable, the intensity as a function of Q is described by the product of the form factor and the structure factor.

Copper, Cu

Copper is used as an anode in many laboratory X-ray experiments. The Cu radiation has a wavelength of about 1.54 Å (0.154 nm) that is convenient for many SAXS experiments.

Count-rate

The count-rate is the number of counts per unit time. Many detectors can provide a linear response to the intensity only for a limited range of count-rates. The detector dead time will limit the maximum count rate that can be recorded either in individual pixels of position sensitive detectors or on the entire detector.

Critical Angle

The critical angle is the grazing angle of incidence below which total reflection of X-rays or neutrons is observed. For X-rays and neutrons, the refractive index for most materials is very slightly less than 1 and so total reflection is observed when the beam penetrates from a vacuum or air in to the material. As the difference of the refractive index to unity is small, in contrast to visible light, the critical angle is small and usually less than 1 degree for most materials and wavelengths of about 0.1 to 1 nm.

Cross-section

An important measure of scattering is the probability (or fraction of an incident beam) that is scattered. This is known as the scattering cross-section, often given the symbol σ or Σ. More information is provided by the differential scattering cross-section. This latter function describes the angles and energy changes involved in the scattering.

D

Dark Count

The dark count corresponds to the signal recorded by the detector when it is not exposed to the scattering, for example with a strong absorber in the sample position. This will contribute as one component to the background that is observed in a scattering measurement. The background may come from electronic noise and ambient radiation that does not arise from the source used for scattering. The different origin of this signal can imply that the normalisation of this component of the signal may sometimes depend on time rather than integrated flux in the incident beam.

Data Fitting

In the context of small-angle scattering, 'data fitting' is jargon that is used to describe the comparison of measured data with intensity distribution that is expected for physical models of structure and composition of the sample. This process may involve several different stages. Calculation of scattering from proposed structures can be compared with the data. These may need to be corrected (convoluted) with instrument resolution. The parameters of a model such as size and composition may need to be refined in an iterative process to optimise the fit of the calculation with the measured data. This may involve conventional least-squares minimisation, Monte Carlo models and genetic algorithms. In general even if a 'model' agrees well with experimental data, care is necessary, as it may not be a unique solution. 'Data fitting' may therefore include constraints imposed from multiple data sets and from other experimental techniques.

Data Reduction

Data reduction is the term that is used to describe the processing of raw data such as counts in individual pixels on a detector to a form that is useful for analysis by modelling or inversion procedures. Reduction will usually involve making allowance for a number of factors. These include normalising each measurement for the incident flux, for example by using the counts recorded with a beam monitor. Further steps involve subtraction of instrument and sample background, making allowance for the sample transmission, as well as correcting for the uniformity of the efficiency of the detector. The data may also be scaled so that it is place in absolute units of intensity and momentum transfer.

Dead Time

See detector dead time.

de Broglie Relation

The de Broglie relation states that the momentum of a particle, p is related to the wave vector, k, of the corresponding wave by:

p = h k / 2π

where h is Planck's constant (6.626 x 10-34 J s). As the amplitude of |k| is 2π/λ then

|p| = h / λ

and this can be used to derive expressions for the momentum, velocity and wavelength of particles/waves. For example the velocity, v, of a neutron of a given wavelength is:

|v| = |p| / mn = ( h / mn) (1/λ)

where mn is the mass of the neutron (1.6749 x 10-27 kg). Thus:

v λ = 3956 m s-1 Å.

Deconvolution

Deconvolution is the inverse of convolution. For example, a common problem arises with measured data that are smeared with an instrument resolution function. Comparison with simple models would require deconvolution. The slit smearing that arises from the collimation of Kratky or Bonse-Hart cameras will substantially broaden peaks and even alter significantly the slope of scattering curves. Deconvolution may not be a simple numerical process as it will require knowledge of both the measured data and the resolution function from the momentum transfer, Q=0 upto a value where both functions can be considered equal to zero. These data may be limited by the beam stop at small Q and experimental noise at large Q. Extrapolation formulae may be used to account for missing data but these may introduce artefacts in the results.

Decoupling Approximation

The separation of the description the scattered intensity for a concentrated solution or dispersion in to two independent functions: the form factor that describes the size and shape and the structure factor that is dependent on the spatial correlations between the individual scattering objects, relies on there being no interdependence of these quantities. This assumption is known as the decoupling approximation and is often valid for monodisperse particles with weak interactions and in the case of spherical symmetry.

Detector Dead Time

The detector dead time is the amount of time after one detection event that the detector is unable to respond to another incident particle (photon or neutron). This effect can arise from physical factors such as the duration of ionisation in a gas detector or from electronics that creates pulses of specific length for each detection event. If the radiation flux on the detector is high, some radiation will not be counted, as the events are too close. This can give rise to losses of measured intensity.

Detector Efficiency

The efficiency of a detector is the ratio of the number of particles that are counted to the number that arrive at the detector. Knowledge of the relative efficiency of detectors may be important in the data reduction, for example, if the incident beam and scattering are measured on different detectors. The wavelength dependence of the efficiency may be significant and it is important to allow for this if instruments operate over a wide range of conditions.

Differential Scattering Cross-section

The differential scattering cross-section describes the amount of scattering per unit solid angle, Ω, and per unit energy transfer, E, at a given scattering angle, θ, and given energy transfer. This is often written as d2Σ / dΩdE. For the case of elastic scattering, this quantity is simpler and there is no energy dependence. It can be written as dΣ / dΩ. This can be related to a physically measurable quantity such as a flux of particles (number per unit area per unit time) on a detector at a particular position defining θ and the structure and contrast within the sample. The energy variation depends on dynamics within the sample.

Diffusion

This is the random thermal motion of molecules or colloidal particles that gives rise to intermixing and collisions. The process is characterised by a random walk that represents a solution of a differential equation known as the diffusion equation. The mean square displacement x2 rather than the displacement, x increases linearly with time, t. The diffusion constant, D is given by:

D = x2 / t

This behaviour is known as Fick's law.

Dynamic Light Scattering

Dynamic light scattering (in contrast to static light scattering is sometimes also called photon correlation spectroscopy (PCS). The correlation of scattered photons is measured. This can be used to determine the dynamic behaviour of particles or polymer molecules at length scales determined by the reciprocal of the scattering vector, Q = (4π/λ) sin(θ/2) and if there is a simple exponential decay characterised by a relaxation time, τ, the diffusion coefficient, D and hence hydrodynamic radius, can be determined as follows:

D = 1 / τ Q2

The same technique can also be used with X-rays: see X-ray photon correlation spectroscopy.

E

Elastic Scattering

If a particle such as a photon or neutron is scattered, it may exchange energy with the scattering object (sample). If the scattered particle gains or loses energy the scattering is described as inelastic. When the scattered particle retains the same energy, the scattering is elastic. If the scattering is elastic, it is clear that the momentum of the incident and scattered particles has the same magnitude (but the direction is different). If the wave vector (which is proportional to the momentum) is given by k, then this is written as:

|k1| = |k2|

where the subscripts 1 and 2 refer to the incident and scattered waves respectively. The symbol |k| indicates the magnitude of k which is (2π/λ) where λ is the wavelength.

There is, in general, still a difference between the incident and scattered wave vectors. This is described by the vector Q:

Q = k2 - k1

If the scattering interaction is elastic then the vector Q will be given by:

|Q| = 2 |k| sin (θ/2)

where θ is the scattering angle.

Electron Density

In X-ray scattering, the product of the number density of electrons and the Thomson length for the electron can be used as an approximation to the scattering length density provided that wavelength is far from an absorption edge. The scattered intensity is then sometimes described in units of electron density.

Event Mode Data Recording

Event mode data recording provides a record of every scattering event (for every detected neutron or photon) rather than summed or averaged data that has been recorded for a long measurement interval. The data that are recorded with a detector location and time can be subsequently analysed with a resolution that is chosen after the experiment and/or changes in the sample with time may be observed.

F

Filter

Filters are absorbers that are used to remove an undesired component in a beam of radiation. This may restrict the wavelength range that is transmitted so that the beam is more monochromatic or even remove a different type of radiation. For example, it is often desirable to reduce the γ-ray contamination in a neutron beam by use of a lead filter or to remove Cu kβ radiation from the spectrum emitted by an X-ray tube with a copper anode with e.g. a nickel filter .

Fitting

See Data Fitting.

Flux

Flux is a number (of particles) per unit area per unit time. When evaluating the absolute intensity of scattering, it is important to normalise for the flux that is incident on the sample. This is one step in the 'data reduction'.

Form Factor

Under some approximations it is possible to describe the intensity distribution, I, of the scattering from a dispersion or solution as the product of two terms that depend on the momentum transfer, Q. These are the form factor, P(Q) and the structure factor, S(Q):

I(Q) = constant P(Q) S(Q).

The form factor P(Q) is determined by the size and shape of individual particles or molecules. The structure factor, S(Q) depends on the correlation and separation of the particles or molecules. The constant term will depend on the contrast. The assumptions in making this separation in to two terms involve radial symmetry and lack of interdependence of P and S, for example caused by variation of size (see decoupling approximation). In studies of scattering from dilute solutions, dilute dispersions or other uncorrelated objects, it can often be assumed that S(Q) = 1 for all Q.

G

Gaussian Size Distribution

A Gaussian distribution of size has a probability P(R) that there is an object with size R given by:

P(R) = exp[ -{(R-R0)/√2 σ}2 ] [1 / (σ √(2π))]

where the mean size is R0 and the variance of the distribution is σ2. The expression for P(R) is normalised, so that the integral of probability over the entire distribution is 1, by the factor (σ √(2π)). The value of σ describes the polydispersity of size.

GIXOS

GIXOS is an acronynm for Grazing Incidence X-ray Off-specular Scattering.

Grazing Incidence Small-Angle Neutron Scattering, GiSANS

Small-angle scattering of neutrons measured when the beam impinges at a low angle of incidence on a surface or interface provides information about lateral structure in the plane of the interface and near surface structure.

Grazing Incidence Small-Angle X-ray Scattering, GiSAXS

Small-angle scattering of X-rays measured when the beam impinges at a low angle of incidence on a surface or interface provides information about lateral structure in the plane of the interface and near surface structure.

Guide

See Neutron guide.

Guinier Plot

A Guinier plot is a way of displaying scattering data (static light scattering, small-angle neutron scattering or small-angle X-ray scattering) for particles and polymers. Data is plotted as:

ln I(Q) vs. Q2

where Q is the momentum transfer or scattering vector given by:

Q = (4π/λ) sin(θ/2)

and λ and θ are the wavelength and the scattering angle respectively.

The data should lie on a straight line for QRg less than 1 and the gradient is -Rg2/3. The intercept is proportional to the particle or molecular mass if an extrapolation of the data to zero concentration is made. This can be compared with a Zimm plot. The radius determined from this plot is the radius of gyration, Rg and for a spherical particle of radius R this is related by:

R2 = (5/3) Rg2

Rg is sometimes described as a Guinier radius.

H

Hard X-rays

The term hard X-rays is used for high energy, short wavelength X-rays. There is no precise definition as to what constitutes hard (or soft) X-rays. It is probably widely understood that X-rays with energies of a few hundred eV or even 1 to 2 keV are soft X-rays but these terms are best used as relative expressions as they can not be relied upon as clearly defined. In some contexts it might only be energies above 50 keV that are considered as 'hard'.

Helium (3He) Detector

A widely used means to detect thermal or cold neutrons uses the nuclear reaction:

n + 3He -> 3H + 1H + 0.76 MeV

The released energy ionises the gas that is usually a mixture of 3He and other materials. The ionisation can be detected readily and even the position of detection can be determined using a suitable array of electrodes connected to a high voltage supply and amplification circuits.

Hutch

Hutch is the term that is used to describe the radiation enclosure around instruments at synchrotron facilities. These are usually large, lead-lined rooms that are interlocked so that entry is not possible when the beam is on and the radiation shutter is open.

Hydrodynamic Radius

This is the radius of a particle or polymer molecule in solution that is determined from a measurement of mobility or diffusion, for example in viscosity or dynamic light scattering experiments. The diffusion coefficient, D is related to the viscosity, η and the hydrodynamic radius, RH by:

D = kBT / 6 πη RH

where kB is the Boltzmann constant and T is the absolute temperature.

I

Image Plate

An image plate (sometimes written as imaging plate) allows integral intensity in two dimensional X-ray scattering patterns to be recorded. These use a photostimulable phosphorescent material to record the integrated scattering. After exposure the plate is read and the intensity determined. The incident radiation excites electrons in a 'phospor' material that is later stimulated to emit light by scanning with a lower energy light source, usually a laser, that provides sufficient energy to release the trapped, excited electrons to produce a luminescence with an intensity that depends on the original scattered intensity and can be detected, for example with a photomultiplier. The plates can be reused and the scanner may be built in to the scattering instrument.

Incoherent Scattering

For neutrons, the change of spin orientation during a scattering process will give rise to incoherent scattering with no interference between waves. The incoherent cross-section does not contribute to the structural information that is derived from coherent scattering.

Indirect Fourier Transforms

Scattering can often be considered as measuring the Fourier transform of the density distribution in a sample. Although it is attractive to consider analysis by performing an inverse transform, this is limited in practice by noise in the data and poor termination of the data that can not extend to zero or infinity in momentum transfer, Q. Fitting functions such as splines that are constrained to appropriate values at zero and large Q and then transforming those functions can provide an interesting route to obtaining radial distribution functions from scattering data.

Inelastic Scattering

Inelastic scattering describes the condition that energy is transferred between the incident radiation and the sample. This contrasts with elastic scattering. Inelastic scattering may be measured to investigate the dynamic behaviour of samples.

Insertion Device

Insertion devices are used to provide better radiation sources than bending magnets at synchrotron radiation facilities. Examples are undulators and wigglers.

Invariant

The 'invariant' or 'scattering invariant', G, is an integral function of the intensity or differential scattering cross-section, dΣ(Q)/dΩ that allows simple comparisons of contrast and composition to be made for samples that can be considered as two-phase systems. G is defined as:

G = ∫0 Q2 (dΣ(Q)/dΩ) dQ

where Q is the momentum transfer. In evaluating this integral, it is important that background is carefully subtracted. An appropriate extrapolation of dΣ/dΩ to Q = 0 may also be needed.

For a system of two phases with a scattering length density difference, Δρ (= ρ1 - ρ2) and a volume fraction φ and (1-φ) for each component,

G = 2π2 (Δρ)2 φ(1-φ)

J

K

Kapton

Kapton is a trade name of the DuPont Company for polyimide films. This is a durable, high strength, impermeable polymer and has been widely used as a window material for X-rays. The films can support a vacuum and give rise to relatively low absorption and scattering of X-rays.

Kratky camera

A small-angle scattering X-ray scattering instrument with the incident beam collimation defined with a slit made using parallel blocks. This can be compared with e.g. a pinhole camera. Use of a slit allows data to be collected faster than with a point source and so can be valuable in laboratory experiments. The data is however smeared by the range of angles in one direction and this needs to be incorporated in the analysis.

Kratky Plot

A Kratky plot displays small-angle scattering data as Q2I(Q) vs Q where Q is the momentum transfer and I is the intensity.

L

Lead (Pb) Filter

In design of small-angle neutron scattering instruments it is often desirable to minimise the γ radiation that reaches the sample and detector. Lead is fairly transparent to neutrons but can absorb a broad band of γ radiation. Pieces of lead (Pb) are sometimes used as filters if the instrument can not be designed so that it is out of line with the source using neutron guides.

Lead is also used as shielding material around X-ray instruments.

Light Scattering

Light scattering is used as an experimental technique to determine the size and/or mobility of colloids and polymers. It can be divided in to two different techniques: static light scattering in which the angular distribution of scattered intensity is measured to determine the size of scattering objects and dynamic light scattering in which a correlation function of scattered photons is measured and is often used to determine mobility and to deduce a hydrodynamic size.

Linear Detector

A linear detector is an example of a position sensitive detector in one dimension. These are, for example, conveniently used with a Kratky camera.

Lithium, Li

Lithium is used in some applications as a detector for neutrons or as an absorber. 6Li has a thermal neutron capture cross-section of 940 x 10-28 m2 (i.e. 940 barns) The natural abundance of 6Li is about 7.5%. This reaction is associated with γ-ray emission at 2 MeV. The reaction can be written as:

6Li + n -> 3H (2.74 MeV) + 3He (2.05 MeV)

As a detector, lithium is used commonly in a crystal or glass doped with cerium that scintillates so as to provide photons of visible light that can be counted, for example, with photomultipliers.

Log-Normal Distribution

The Log-Normal size distribution, P(R) is skewed with a higher probability for large particles. It is defined for R > 0 by the following probability distribution:

P(R) = exp[ -{(ln R - A)/√2 σ}2 ] [1 / (R σ √(2π))]

The expression for P(R) is normalised, so that the integral of probability over the entire distribution is 1, by the factor (R σ √(2π)). The mean of this distribution, Rm is given by:

Rm = exp[ (A + σ2)/2 ]

Lupolen

Lupolen is a trade name for polyethylene produced by BASF but in the context of small-angle X-ray scattering refers to pieces of this material that are used as secondary standards for intensity. The scattering from such samples is placed on a defined scale by calibration either against a primary standard or by making a measurement of the intensity (flux) of the primary beam that can be compared directly with the observed scattering. The samples then provide a quick means to check reproducability or performance of an instrument.

M

Mask

In analysis or reduction of scattering data, it is sometimes useful to exclude or mask certain pixels from a detector or regions of momentum transfer from analysis. This need arises because of systematic errors, for example that a region on a position sensitive detector is obscured by a beam stop or that particular pixels are known not to count linearly or are 'noisy'. The list of exclusions used in data reduction software is sometimes called a mask.

Metadata

Extra information about the sample, measuring conditions, instrument configuration, etc. is known as a metadata. This may be important for analysis of the scattering data that may be influenced strongly by parameters such as concentration, temperature and instrument resolution. Data formats and storage for small-angle scattering data will usually make provision to record both scattering data and appropriate metadata.

MIEZE

MIEZE is an acronym for 'Modulated Intensity with Zero Effort'. This uses a resonance spin-echo spectrometer design with high frequency flippers and no magnetic components between the sample and the detector. It is used for quasielastic small-angle neutron scattering.

Moderator

The wavelength of neutrons depends on their energy. Multiple inelastic scattering can be used to change the energy distribution of neutrons. If the scattering object (moderator) is at room temperature and the neutrons reach an energy equilibrium, these are called thermal neutrons. The wavelength is of order 0.1 nm. For small-angle scattering it is often desirable to use longer wavelengths and a cryogenically cooled moderator or cold source is often used. Thermal moderators are needed to provide a suitable neutron energy spectrum to operate continuously a nuclear reactor.

Molecular Mass

If the concentration and contrast of a polymer (or biological macromolecule) in a solution is known, the intensity of scattering that is extrapolated to zero momentum transfer can be used to estimate the molecular mass.

The molecular mass of a polymer is defined as for other compounds as the mass in grams of a mole (Avogadro's number) of molecules. It is of particular importance in polymers as many properties depend on the length or degree of polymerisation and thus the molecular mass of the molecules. Most synthetic polymers will have a distribution of molecular mass and it is often important to characterise this. It is common to distinguish various different averages of the molecular mass such as the number average molecular mass and the weight average molecular mass.

Molybdenum, Mo

Molybdenum is sometimes used as an anode in laboratory X-ray experiments. The Mo radiation has a wavelength of about 0.711 Å (0.0711 nm).

Momentum Transfer Vector

The difference between the incident (k1) and scattered wave (k2) vectors is described by the difference vector Q:

Q = k2 - k1

If the scattering interaction is elastic then the vector Q will is:

|Q| = 2 |k| sin (θ/2)

where θ is the scattering angle. This is equivalent to:

|Q| = (4π/λ) sin (θ/2)

where λ is the wavelength.

Monitor

A monitor is a detector that is intended to record the incident flux on the sample either as a low efficiency detector in the incident beam or in other ways.

Monochromator

A crystal monochromator is used to diffract a single wavelength in a particular direction to provide a monochromatic (single wavelength) incident beam for a scattering experiment.

Monodisperse

A material is described as monodisperse if it consists of particles or polymer molecules of a single size or mass. This contrasts with polydisperse samples with a range of sizes. Distributions of sizes are important in small-angle scattering as the variation of intensity with momentum transfer for monodisperse samples of e.g. spherical particles can show sharp minima that will be smeared if the the sample is polydisperse.

Moving Slit Attenuator

If a beam has a spatial intensity or flux distribution and it is necessary to use an attenuator to determine the flux in the primary beam because the count-rate would be too high for a detector, then it is sometimes convenient to use a moving slit attenuator. These have been used frequently with slit collimation small-angle X-ray scattering instruments such as Kratky cameras. A narrow slit that is long in the direction perpendicular to the collimation slit is used with an oscillatory motion at constant speed. In this way all parts of the collimated beam are sampled with an attenuation by the same factor. This type of attenuator is also used on some slit collimated instruments that are used to measure specular reflection of neutrons.

Multidetector

The term multidetector is used to describe either an array of single detectors for radiation, or commonly in small-angle scattering (SAXS or SANS), a position sensitive detector.

Multiple Scattering

If a sample is thick or has a high contrast then there is a possibility that a photon or neutron that enters the sample may be scattered more than once. In small-angle scattering experiments that can be thought as spreading the incident beam over a small range of angles, multiple scattering will increase the spread. Multiple Scattering from a sample will usually complicate the analysis of the data. In general there will be smearing or convolution of the function that describes the scattering from a thin or dilute sample with itself. A simple estimate of the extent of multiple scattering can be made if the component of the attenuation, As (= 1 - Ts) due to scattering can be determined. Ts is found from a transmission measurement but somtimes a separate allowance for absorption must be made. The probability of two-fold scattering will be approximately As2.

N

Neutron Energy

The kinetic energy, E, of a neutron is simply mn v2/2 where mn is the neutron mass and v the velocity. The momentum, p=mnv, of a neutron is h/λ and so:

E = (h/λ)2 / 2mn.

Thus neutrons with a wavelength of 1 nm (10 Å) have an energy of about 0.82 meV and this changes as 1/λ2.

Neutron Guide

A neutron guide is a wave-guide for neutron beams. Metal or multilayer coatings on glass can provide total reflection of long wavelength neutrons up to angles of 0.5 degrees or more. Pipes (usually rectangular) that are coated in this way can therefore be used to enhance transport of neutrons from the source to the instrument. For pinhole SANS instruments, neutron guides are often used to form adjustable collimation. If neutron guides are inserted after the source aperture, the divergent beam is brought closer to the sample. Inserting guides relaxes the angular resolution but increases the flux at the sample.

Neutron Reflection

The technique of neutron reflection (NR) uses specular reflection of neutrons to investigate the structure and composition of flat interfaces.

Neutron Source

Neutron sources used for SANS experiments are usually research reactors or spallation neutron sources. Research reactors provide neutrons by means of fission reactions. A spallation source is usually a beam of protons that impinges on a heavy metal target to cause nuclear disintegration and 'spallation' of neutrons. As long wavelengths are often desirable for small-angle scattering, the neutrons are usually moderated by a cold source.

Neutron Velocity

Neutrons have a low velocity, v that can be calculated from the wavelength λ using the following equation:

v λ = 3956 m s-1 Å.

The result is derived readily from the de Broglie relationship and a knowledge of the mass of the neutron. Thus cold neutrons of 10 Å wavelength have a velocity of about 396 m s-1.

NeXus

NeXus is an acronym that identifies a common data format for neutron, X-ray and muon science. NeXus data files should contain information (metadata) about the instrument configuration as well as a description of the sample along with recorded counts for the measured scattering. NeXus files are usually written in an hdf format. Dictionaries are defined so that parameters can be described in a standard way so as to promote compatibility between data files measured at different facilities. There is more information at www.nexusformat.org.

Nickel Filter

On laboratory X-ray sources with a copper anode it is sometimes desirable to restrict the wavelength band with a filter rather than a monochromator. A thin nickel foil can be used to absorb Cu kβ radiation with a wavelength of 1.392 Å and transmit the Cu kα1 and Cu kα2 radiation (1.540 and 1.544 Å) as the absorption edge for Ni is at 1.49 Å.

NR

NR is an abbreviation for neutron reflection.

NSE

NSE is an abbreviation for neutron spin-echo.

Nuclear Reactor

A nuclear reactor sustains a fission reaction. A thermal neutron is absorbed by the nucleus of a fissile isotope such as 235U that then splits and produces more neutrons as well as energy and daughter isotopes. Provided that a sufficient fraction of the neutrons that are emitted are moderated to low enough energies and reabsorbed in the fuel, the chain reaction can continue. Research reactors are used as sources of neutrons for scattering experiments.

Number average molecular mass or molecular weight, MN

The distribution of size of particles and polymers in most synthetic materials (polydispersity) implies that different averages of the size will give different values. The number average is defined as the sum of niMi divided by the sum of ni where ni is the number of molecules in the distribution with mass Mi. The number average molecular mass is the quantity measured by determination of colligative properties such as osmotic pressure. The number average radius of particles is simple evaluated from micrographs. If the full distribution is recorded any average can be calculated.

O

Order Parameter

In systems such as liquid crystals that display partial order, for example only in orientation but not in position, it is useful to use a measure of the extent of ordering. Physicists define order parameters to describe these states. A simple example would be a description of the orientational order in a nematic phase which is described by an average of the angle that the particles or molecules make with the director that is the overall (average) direction of orientation . Several different averages can be taken to provide an indication of the width of the distribution of orientations about this angle. The first order parameter is known as P2 and is defined as:

P2 = < (1/2 (3 - cos2Φ) >

where Φ is the angle that each particle or molecule makes to the director and < > indicates an average over all particles. Other order parameters would take averages over higher powers of cos Φ. (These are other even Legendre polynomials.). The order parameters are often chosen with scaling and offsets so that perfect order would give a value of 1 and no orientation would give an order parameter of zero.

P

PCS, Photon Correlation Spectroscopy

See Dynamic light scattering.

Pin Diode

Pin diodes are photodiode X-ray detectors that are frequently used in SAXS experiments as monitors or transmission detectors. It is named after the assembly of p-type, intrinsic, and n-type semiconductors in the device.

Pinhole 'camera'

A Pinhole 'camera' is a small-angle scattering instrument with incident beam collimation defined by point-like apertures rather than slits. This can be compared with e.g. a Kratky camera.

Polarisation

The term polarisation is used slightly differently in the fields of X-ray and neutron scattering. As with visible light, the waves that represent X-rays can be considered as coupled oscillations of the electric and magnetic field vectors. The direction of the electric field oscillation is used to describe the plane of polarisation of the wave. Laboratory sources of X-rays from electrons hitting an anode provide unpolarised (or more strictly randomly polarised) X-rays. Synchrotron radiation is polarised in the plane of the ring although X-ray optical devices can be used to provide, for example, circularly polarised beams. Corrections to allow for the systematic effects of polarisation are small at low angles but become much more significant when measurements are made at wide angles.

In the case of neutron scattering, polarisation describes the direction of the spin of the neutron. Neutron beams are usually unpolarised but absorption or diffraction of specific spin states can be used to make polarisers and analysers. These are useful to investigate magnetic structures.

Polydispersity

The distribution of molecular mass of polymers or colloidal particles is described as the polydispersity. Polydispersity will cause sharp features, such as the minima in the form factor of spheres, in scattering curves to be smoothed as the measured intensity corresponds to a weighted average of the scattering from the different size objects.

The details of the shape size of the distribution will depend on the preparation and any fractionation but, particularly for polymers, a simple measure of the distribution is often quoted as MW/MN where MW is the weight average molecular mass and MN is the number average molecular mass. For particles it is common to use the width of Gaussian or Log-normal distributions to describe polydispersity. A sample that consists of a single size or molecular mass is described as monodisperse.

Porod Law

The Porod Law describes scattering that is in an intermediate region between small momentum transfer, Q and large Q where the scattering is usually dominated by the background. When 1/Q is signicantly larger than the characteristic size of scattering objects, the intensity that arises due to scattering from sharp interfaces, I(Q) varies as Q-4 and is proportional to the ratio of the surface area to the volume.

Porod Plot

A Porod plot allows a check to be made whether data conforms (within a given region of momentum transfer, Q, with the Porod law that intensity, I varies as Q-4. The term is used for a plot of either I Q4 vs Q or I Q4 vs Q4. If there is no extra background scattering and the data are in absolute units, a straight line fit can provide the specific surface area.

Position Sensitive Detector

A position sensitive detector can record where radiation or a particle hit the detector. Both one and two dimensional position sensitive detectors can be built for X-rays and neutrons.

Q

Quasi-elastic scattering

Quasi-elastic scattering is inelastic scattering with a small energy transfer so that it appears to give a broadened elastic scattering peak rather than different peaks with distinct energies. This scattering often arises from slow motion such as diffusion and is measured in spin-echo spectroscopy, X-ray photon correlation spectroscopy and dynamic light scattering experiments.

Q-vector

See momentum transfer. This quantity is often written as Q. Many different symbols are also found for this quantity in the literature and they include q, h, k, s and κ. The quantity s is also sometimes used to designate (Q / 2 π). This is found particularly in literature on SAXS and care must be taken to identify definitions in a given article or book or expected in software.

R

Radial Average

See Circular Average.

Radiation Damage

See beam damage.

Radius of Gyration, Rg

Radius of gyration, Rg is used widely to describe the size of synthetic polymers and biological macromolecules. The term is also sometimes used as the result of a fit in a Guinier plot. The Guinier radius is the same as the radius of gyration of a polymer.

For a polymer molecule Rg can be defined in terms of the distribution of distances (in any direction) ri of each monomer in the molecule from the centre of gravity of the molecule:

Rg2 = Σi ri2

It can be shown that a freely jointed chain of N links of length a has the following, Rg:

Rg2 = N a2 / 6

In practice N may not be the number of monomers as there may be a finite persistence length or stiffness of a polymer chain and a may correspond to the length of several monomers that extend in one direction.

Random Phase Approximation

The random phase approximation has been an important idea in the study of polymers using SANS. If there is no significant interaction of labelled and unlabelled polymer molecules (for example deuterated and normal hydrogenous molecules), then the random correlations between the molecules implies that the single molecule scattering will be observed even at high concentrations. The sample composition for studies of bulk polymers can be optimised for contrast (for example with a volume fraction of 0.5 for each species). In practice there may be some differences of interaction between hydrogenous and deuterated polymers but many studies of bulk samples can benefit from use of high concentrations. In general for solutions there will be a significant change with concentration.

Reactor

See Nuclear Reactor.

Read-out Noise

The read-out noise corresponds to the signal recorded by a detector when it is read that does not arise from the detection of radiation. For some detectors such as those based on CCD devices or image plates this may be significant This will contribute as one component to the background that is observed in a scattering measurement. The origin of this signal from the read-process implies that the normalisation of this component of the signal should not depend on the exposure time or the integrated flux on the sample.

Research Reactor

A research reactor is a nuclear reactor that is designed for research. Some of these are optimised to produce neutron beams for scattering experiments. These are usually optimised for high neutron flux rather than energy production with a small core.

Resolution

The term 'resolution' is used in a number of different ways in scattering experiments. It can refer to the spread, dQ, in the momentum transfer vector, Q at each point. As Q depends on both angle and wavelength, dQ will have components that arise from the spread in wavelength, the collimation of the incident and the angular resolution of the detector.

In crystallography, the word resolution is sometimes used to describe the precision in distance in real space to which a structure is determined. This will depend on the largest Q at which Bragg reflections are measured. Resolution when used in this way is proportional to 1/Qmax. This latter use of the term resolution is less common in small-angle scattering but is sometimes encountered.

Rheo-SANS and Rheo-SAXS

The terms rheo-SANS and rheo-SAXS are used to describe simultaneous measurement of scattering with neutrons or X-rays and rheological measurements. These will employ a rheometer or other flow geometry as the sample environment. Such experiments are used to determine structural changes in dispersions, solutions and melts of polymers under flow.

Rotating Anode

A rotating anode is used in an X-ray generator to provide higher intensity beams. The heat from the incident electrons is spread across a rotating target and can be more efficiently cooled than when using a fixed anode target.

RPA

RPA - see random phase approximation.

S

Sample Changer

A sample changer is a device that can automatically change samples on the measuring stage of a scattering instrument. This may simply be a translation stage with a number of pre-aligned samples. These devices are common in small-angle scattering experiments as measurement times are often fairly short and it is convenient to run an automated sequence of measurements. This type of equipment also facilitates comparison of sample and background as well as allowing simple checks of reproducibility and changes with time.

Sample Environment

The term sample environment is used to describe the sample holder and the equipment that is used to maintain the sample under specific conditions such as defined temperature, pressure, humidity etc. Many complex sample environments are now used to study chemical reactions, samples under deformation (for example in rheometers), under the influence of applied electric and magnetic fields and many other systems.

SANS

See Small-Angle Neutron Scattering.

SAXS

See Small-Angle X-ray Scattering.

SAXS-WAXS

SAXS-WAXS refers to the simultaneous measurement of small-angle and wide-angle X-ray scattering. This allows assessment of changes in local order as well as well as long-range structure.

Scattering Angle

The scattering angle is the angle between the incident beam and the direction at which the scattering is being measured. In experiments that use a position sensitive detector, there will be simultaneous measurements at a range of different scattering angles. In this glossary we have used θ to designate the scattering angle.

Other definitions are also encountered: in particular studies of diffraction from crystalline materials often use 2θ as the scattering angle and designate half this angle (i.e. θ) as the Bragg angle. This definition is also used widely in studies of specular reflection where θ is used for the grazing angle of incidence that is equal to the angle of reflection. The total deflection of the beam is then 2θ. It is important to establish what convention is being used to define the angle when calculating the momentum transfer in a scattering experiment!

Scattering Length

The scattering length for an atom or nucleus determines the amplitude of the wave that is coherently scattered. For X-rays this depends on the number of electrons in the atom and the X-ray energy. For neutrons the scattering length is a nuclear property that may vary significantly between isotopes of the same element. The scattering length for an individual electron can be approximated as the Thomson length and hence for X-rays, the scattering depends on the electron density.

Scattering Length Density

The contrast between components in a sample is function of the difference in the scattering length density, ρ, of the materials. For a molecule this can be obtained from the sum of the scattering lengths, bi, of all the atoms or nuclei (depending on whether the calculation is for X-rays or neutrons) divided by the molecular volume, vmol:

ρ = Σ bi / vmol.

If there are just two components in a scattering sample, a and b, the intensity, I, is proportional to the square of the scattering length density difference:

I = constant (ρa - ρb)2.

Scintillator

Scintillation is the process of absorption of radiation and consequent emission of a photon. In the context of scattering instruments, a scintillator is used to convert X-rays or neutrons to photons in the visible spectrum that can be detected readily with, for example, photomultiplier tubes, image plates or charge coupled device detectors.

Sector Average

A sector average contrasts with a full circular average for data recorded on a two-dimensional position sensitive detector. If the data is anisotropic and varies with azimuthal angle, it can be useful to compare the scattered intensity as function of momentum transfer in particular directions or sectors.

SERGIS

SERGIS stands for 'spin-echo resolved grazing incidence scattering'. This is the GiSANS equivalent of SESANS.

SESAME

SESAME is an acronym for spin-echo scattering angle measurement.

SESANS

SESANS is an acronym for spin-echo small-angle neutron scattering. Neutrons that are deflected by scattering take a slightly different path though a region of high magnetic field. As neutrons have spin they will precess and the small changes in precession can be detected and used to determine the scattering using the spin-echo method to detect small changes the path of neutrons in regions of homogeneous magnetic field and hence changes in scattering angle. SESANS complements USANS in providing information about larger length scales than conventional SANS.

Shielding

Radiation shielding is used to prevent the unwanted escape of scattered radiation or the direct beam from the enclosure of a scattering instrument by absorption. This serves to provide a safe working environment for the experimentalist. For X-ray radiation it is normal to use heavy metal such as lead although for laboratory sources, lighter metal enclosures can sometimes be adequate. When using neutron beams it may be necessary to use both absorbers for neutrons and material that will further absorb the resulting γ radiation that arises from many neutron capture reactions such as those of boron or cadmium.

A further use of shielding may be to reduce the background radiation that reaches the detector used for the scattering measurements.

Shutter

A shutter is an absorber that can be inserted in the beam so as to block the radiation. The shutter may be used to make the instrument safe for an intervention such as a change of sample or to protect the sample and/or the detector. Some instruments may be equipped with more than one shutter for these different purposes.

Silver Behenate

Silver behenate is frequently used as a standard sample to calibrate the wavelength or momentum transfer on small-angle scattering instruments as it has a lattice parameter of 5.838 nm and a first Bragg peak at 1.076 nm-1. Higher order peaks are usually also readily visible.

Size Distribution

The size distribution describes the probability of finding particles of different size in a sample. The distribution function can be expressed in terms of a dimension such as the radius of a particle or in terms of the mass, for example of a polymer. As these are not linearly related, it is important to describe how the distribution is defined. Often a model function for the scattering will need to be convoluted with a size distribution in order to adequately 'fit' measured data. A Gaussian function is sometimes used as a simple symmetric distribution but other commonly used models for size distributions are log-normal distributions and for polymers the Schulz-Flory distribution.

Small-Angle Approximation

The small-angle approximation is the condition that an angle, θ (for example the scattering angle) is sufficiently small that sin θ approximately equals θ (in radians). A comparable approximation for cos θ is that cos θ = 1 - θ2/2. This approximation can lead to considerable simplification in the theory and interpretation of small-angle scattering.

Small-Angle Neutron Scattering

Small-angle neutron scattering, often known as SANS, is used to study structure in materials on the scale of 1 to many nm. Typical samples include bulk polymers, polymer solutions and in colloidal dispersions, biopolymers, biomaterials, magnetic materials, ceramics, etc. T he principles are similar to elastic or static light scattering but relies on scattering contrast from nuclei. Different isotopes can scatter quite differently. SANS can be used to investigate blends of chemically identical but differently labelled materials such as the size of polymers in the bulk. SANS is also important as a tool to study opaque and multi-component systems. Typically experiments are performed with 'cold' neutrons with a wavelengths of about 0.2 to 2 nm. The small angles of measurement are needed to probe the large dimensions in materials and nanostructures.

Small-Angle X-ray Scattering

Small-angle X-ray scattering, often known as SAXS, is used to study structure in bulk polymers, polymer solutions, colloidal dispersions, ceramics, metals, biological molecules and many other materials. The principles are similar to elastic or static light scattering or small-angle neutron scattering. Typically experiments are performed with a wavelength of approximately 0.1 to 0.2 nm. Small angles of measurement are needed to probe nanostructures in materials such as metals, ceramics, colloids and polymers.

Soft X-rays

X-rays with long wavelengths and energies of a few hundred eV or even 1 to 2 keV are sometimes called soft X-rays. Higher energy X-rays are known as hard X-rays but these expressions are best used as relative terms as they can not be relied upon to provide a clear definition. In some contexts it might only be energies above 50 keV that are considered as 'hard'.

Spallation Neutron Source

A spallation source is usually a high-energy beam of protons that impinges on a heavy metal target so as to cause nuclear disintegration and 'spallation' of neutrons. Electron beams have also been used. These can operate either continuously or, more commonly, as a pulsed source that facilitates time-of-flight measurements.

Specular Reflection

Reflection of X-rays or neutrons from an interface in the specular direction (angle of incidence equal to the angle of reflection) can be considered as a special form of small-angle scattering to probe the density profile perpendicular to the interface. As the refractive index of materials is close to unity, the reflectivity is only significant for small momentum transfer. If there is significant scattering from lateral structure this will arise in other directions and is known as GiSAXS or GiSANS.

Spherical Form Factor

The form factor, P(Q) that describes the scattering from a sphere of uniform density is simply expressed in terms of the radius, R, and the momentum transfer vector, Q as:

P(Q) = [3 ( sin(QR) - QR cos(QR) ) / QR3]2

This function is normalised to unity at Q = 0. The Guinier radius, Rg is related to the actual radius by Rg = (3/5)1/2 R.

Spin

For most purposes of small-angle scattering, the neutron can be considered as a spin 1/2 particle with a defined magnetic moment. The neutron will therefore interact with external magnetic fields and can be scattered from magnetic structures. The first property is exploited in the spin-echo method that is used to provide high-resolution spectroscopy and enhanced small-angle scattering by means of SESANS. Neutrons can provide data about magnetic structures such as domains in magnetic materials and flux-line lattices in superconductors. Magnetic scattering combined with polarised beams can also be used to enhance the contrast in a range of metal samples such as steels.

Spin-echo

This is a measuring technique that is used for neutron scattering. The precession of the neutron spin in a magnetic field can be determined by measuring the change of polarisation. It is particularly convenient to compare precession after spin-flip in two regions with the same magnetic field, as the change in polarisation (the echo signal) is then proportional to the change in velocity or the difference in path length of the neutron in the field. This has been used to measure small energy transfers for dynamic measurements that complements dynamic light scattering and X-ray photon correlation spectroscopy. The spin labelling or use of precession to measure small changes is now exploited also in SESANS and SERGIS measurements.

sSAXS

sSAXS is an abbreviation for scanning SAXS. It describes measurements made at many points in a sample that are used to provide maps of the microstructure and anisotropy at different places in a sample. Applications have included studies of a range of biological material such as bone and wood. With modern synchrotron radiation facilities micro focus beams of a few μm size are easily used.

Static Light Scattering

A light scattering technique to measure the size (radius of gyration) and molecular mass of polymers, micelles and colloidal particles. The total scattered intensity I is measured as a function of angle θ. For polymers the data is often interpreted by means of a Zimm plot. For solid objects, such as colloidal particles, it is more usual to use a Guinier plot.

Stroboscopic Measurements

Cyclic processes (for example strain) can be studied even though they occur too fast for adequate scattering data to be collected at a particular point in the cycle by using stroboscopic measurements. Data from different intervals within a cycle is collected in different memory channels and data counted for sufficient cycles are summed together. This procedure works if the sample can be cycled reproducibly between different states.

Structure Factor

Under some approximations it is possible to describe the scattered intensity, I, from a dispersion or solution as the product of two terms that depend on the momentum transfer, Q. The form factor is designated P(Q) and the structure factor, S(Q):

I(Q) = constant P(Q) S(Q).

The structure factor S(Q) depends on the correlation and separation of the particles or molecules. The form factor P(Q) is determined by the size and shape of individual particles or molecules. The constant term will depend on the contrast. The assumptions in making this separation in to two terms involve radial symmetry and lack of interdependence of P and S, for example caused by variation of size. Care should be taken in the literature as in some disciplines, the term structure factor is sometimes used to describe the total scattering from materials. This is not uncommon in studies of polymers.

Synchrotron Radiation

Synchrotrons and electron storage rings are used as brilliant sources of X-ray radiation. Deflection of a beam with energy in the range of order one or a few GeV by magnetic fields such as those used to make the electrons circulate in a closed path causes the emission of X-rays. Modern facilities dedicated to studies with X-rays often use a synchrotron to accelerate the electrons that are then injected in to a 'storage ring'. The radiation is polarised in the plane of the ring. Facilities may use insertion devices such as wigglers and undulators to improve the characteristics compared to the emission from bending magnets.

T

Thomson Length

The classical expression for cross-section, σ of electrons for photons according to Thomson is:

σ = (8π/3) (e2 / me c2)2

where e is the electronic charge, me is the electron mass and c is the velocity of light. This gives a value of 6.65 x 10-29 m2. This is sometimes known as 0.665 barns. The square root of this cross-section is the Thomson length and can be taken as an approximation for the scattering length of electrons for X-rays.

Time-of-Flight, ToF

Time-of-Flight or ToF is a means to determine the wavelength of neutrons in a pulsed beam. The time of detection at a known distance from a pulsed source is recorded and this can be converted to a velocity. The wavelength can be derived from the velocity using the de Broglie relationship.

TISANE

TISANE is an acronym for time dependent small-angle neutron scattering experiments. This is a modified stroboscopic measuring technique that uses a pulsed beam, usually with a fast chopper, to enhance the time-resolution.

ToF

ToF is an abbreviation for Time-of-Flight.

Transmission

The transmission is the fraction of the incident beam that passes through the sample without absorption or scattering and continues in the original forward direction. This will have values between 0 and 1. Determination of this quantity is usually important for data reduction to account correctly for background scattering and to place data on an absolute scale of intensity.

U

Ultra Small-Angle Neutron scattering (USANS)

This refers to small-angle neutron scattering in a range of Q that is lower than that accessible with a pinhole camera and often exploits Bonse-Hart scattering geometry.

Ultra Small-Angle X-ray scattering (USAXS)

This refers to small-angle X-ray scattering in a range of Q that is lower than that accessible with a pinhole camera and often exploits Bonse-Hart scattering geometry.

Undulator

An undulator is a type of insertion devices in an electron storage ring that gives rise to a periodic vertical deflection of the beam. These are used to enhance the emittance of synchrotron radiation in particular bands of wavelength. These devices generally provide more brilliant sources than bending magnets and can be compared with wigglers that provide radiation over a broader range of wavelengths.

V

Vanadium

Vanadium has been a traditional material to use in neutron scattering to provide a uniform, angle independent scattering pattern as the cross-section for neutron scattering is almost entirely incoherent. For the purpose of small-angle scattering, it suffers from several disadvantages. First, the grain structure and consequent contrast between regions of different density causes small-angle scattering that can be observed varying with momentum transfer. Secondly, the incoherent cross-section is rather small and so long measuring times are required because the total solid angle observed in small-angle scattering experiments is low.

It is sometimes thought that vanadium could provide a more reliable standard for scattering intensity than other materials but as it is prone to absorb hydrogen to varying extents, samples unless carefully prepared and annealed, will display significant systematic variations of intensity as well as some of the inelastic scattering effects observed with other materials.

Velocity Selector

A velocity selector is a mechanical device that provides a helical path for neutrons that is surrounded by absorbing material. When it is continuously rotating only neutrons within a defined range of velocity can pass through. This is frequently used as a low-resolution monochromator for neutrons.

VSANS

VSANS is an acronym for very small-angle neutron scattering. This generally exploits focussing methods to resolve lower angles in scattering than conventional SANS instruments. The limit of low momentum transfer is however higher than that in USANS experiments.

W

WAXS - wide-angle X-ray scattering

The abbreviation WAXS is often used for wide-angle X-ray scattering that can be compared with small-angle X-ray scattering. Scattering in the wide-angle range investigates local order rather than the large structures and long-range correlations that are measured in SAXS experiments. Combined measurements with multiple detectors to cover both ranges are sometimes called SAXS-WAXS.

Weight average molecular mass or molecular weight, MW

The distribution of particle and polymer sizes in most synthetic materials (polydispersity of molecular mass) implies that different averages will give different values. The weight average is defined as the sum of niMi2 divided by the sum of niMi where ni is the number of molecules in the distribution with mass Mi. The weight average molecular mass is the quantity measured in a small-angle scattering or elastic light scattering experiment. The weight average molecular mass will always be bigger than the number average molecular mass for polydisperse materials.

White Beam

The term white beam is used as an analogy with light to describe a beam of radiation with many different wavelengths or energies. For visible light, the addition of a broad range of different wavelengths or colours will appear to be white. In scattering experiments the momentum transfer depends on the wavelength but experiments can sometimes be performed with detectors that distinguish different energies and so exploit white beams.

Wiggler

An wiggler is a type of insertion device in an electron storage ring that gives rise to a periodic lateral deflection of the beam. These are provided to enhance the flux of synchrotron radiation in broad bands of wavelength. These insertion devices generally provide more brilliant sources than bending magnets. They can be compared with undulators.

X

X-Ray Energy

The energy, E of an X-ray photon is simply related to the wavelength λ by:

E = h c / λ

where c is the velocity of light and h is Planck's constant. Often at X-ray facilities, the energy is expressed in units of eV or keV rather than Joules and so the resulting energy will be divided by the electronic charge, 1.602 x 10-19 C and this gives:

E λ = 12.4 keV Å.

X-Ray Free Electron Laser, XFEL

This is a linear accelerator for electrons equipped with devices that create oscillating magnetic fields designed for coherent emission of high-energy photons.

X-Ray Photon Correlation Spectroscopy, XPCS

This is analogous to dynamic light scattering (or PCS) and exploits coherent X-ray scattering to measure dynamics of samples. The range of time and space that are probed complements that of studies made with visible light, as the momentum transfer for X-rays, even at low angles, is usually much larger than that for visible light.

X-Ray Reflectivity

X-Ray reflectivity (XRR) is a technique that uses specular reflection of X-rays to investigate the structure and composition of flat interfaces.

XRR

See X-Ray reflectivity.

X-Ray Sources

Sources for small-angle X-ray experiments in a laboratory are generators with either fixed or rotating anode targets. Synchrotron radiation is also widely used.

Y

Z

Z-average molecular mass or molecular weight, MZ

The distribution of polymer sizes in most synthetic materials (polydispersity of molecular mass) implies that different averages will give different values. The Z-average is defined as the sum of niMi3 divided by the sum of niMi 2 where ni is the number of molecules in the distribution with mass Mi. This quantity can be compared with other averages of the molecular mass such as the weight average and the number average.

Zimm Plot

A way of plotting scattering data (static light scattering, small-angle neutron scattering or small-angle X-ray scattering) for polymers. Intensity, I(Q), is plotted as:

c / I(Q) vs. Q2

or

c / I(Q) vs. Q2 + const. x c

where c is the concentration and Q is the scattering vector given by:

Q = (4π/λ) sin(θ/2)

where λ is the wavelength and θ is the scattering angle.

The data should lie on a straight line for QRg less than 1 and the gradient divided by the intercept is Rg2/3. Rg is the radius of gyration. The intercept is inversely proportional to the molecular mass if an extrapolation of the data to zero concentration is made. This can be compared with a Guinier plot that is frequently used for solid particles (colloids).

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Please address any comments or report any errors to Professor Adrian R. Rennie at:

Materials Physics
Department of Physics and Astronomy
Uppsala University
Uppsala, Sweden

Adrian.Rennie@physics.uu.se

© Copyright Adrian R. Rennie, 2012, 2013, 2015, 2016

The material in this glossary is made available without any warranty. The material must not be copied or redistributed without permission.

Last Updated 06 January 2016

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