Abstract
This chapter gives a short introduction into neutron scattering instrumentation to allow the non-specialist reader to acquire the basics of the method necessary to understand the technical aspects in the topical articles. The idea is not to go into details but to elaborate on the principles as general as possible. We start with a short discussion of neutron production at large-scale facilities. We then present the main characteristics of neutron beams and show how these can be tailored to the specific requirements of the experiment using neutron optical devices and time-of-flight discrimination. This will allow us in the final section to present a non-exhaustive selection of instrument classes. Emphasis will be given to the design aspects responsible for resolution and dynamic range, as these define the field of scientific application of the spectrometers.
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Notes
- 1.
We use the constants as given by the National Institute of Standards and Technology (http://physics.nist.gov/cuu/Constants/): \(h = 6.62606896 \times 10^{-34}\) Js and \(m=1.674927211 \times 10^{-27}\) kg. As \(e = 1.602176487\times 10^{-19}\) C, one Joule is equivalent to \(6.241 509 65 \times 10^{18}\) eV. A wealth of interesting information on neutron and on neutron instrumentation can be found in the Neutron Data Booklet [3].
- 2.
Bragg scattering from a sample provides the necessary information about its atomic structure. If the same Bragg reflection is used for monochromating the beam, then the sample becomes a technical instrumental device.
- 3.
It is evident that the physics involved in moderation changes drastically along the cascade from MeV to meV. At high energies the collisions are inelastic, that is, they involve excitations of the nuclei that scatter the neutron. At low energies the equilibrium is attained via the exchange of low-frequency excitations of the moderator with the neutron. The character and spectrum of these excitations influence the moderation. In the case of H2O and D2O, the collisions involve, for example, predominantly rotational motions around the center of mass. This is the reason why it is the mass of H or D that counts for the moderation efficiency and not that of the molecule.
- 4.
If the dimensions of the moderator are small, as in the case of a cold source, then H2 may be a useful and easier - to - handle alternative to D2 [6], despite the fact that D2 would offer the optimum cold spectrum.
- 5.
The nuclear reactions following the impact of the proton beam depend on the proton energy. Below an excitation energy of 250 MeV, the boiling-off of neutrons is dominant. Most of them have energies in the 2 MeV range. This boiling-off is the main neutron production channel even for 1 GeV proton beams as normally only part of the proton energy is deposited in the target nuclei. There is, however, also an appreciable amount of faster neutrons. Their spectrum reaches up to the incident proton energy. These very fast neutrons require extremely heavy shielding around the target. The neutrons and the remaining excited nucleus engage in a cascade of secondary decay processes. Per useful neutron about 30 MeV of energy have to be evacuated in the case of spallation, compared to about 200 MeV in the case of fission.
- 6.
In practice, the spectrum will differ from the ideal Maxwell–Boltzmann distribution. This is due to leakage of fast neutrons or incomplete moderation processes. These corrections become more important in the case of pulsed sources. The detailed shape of the source matters for the flux transmitted to the guides [7].
- 7.
There are naturally exceptions to this. Low-energy excitations are populated at temperatures very much lower than the lowest moderation temperature. Their investigation requires a high relative energy resolution \(\Delta E/E\) using, for example, a backscattering spectrometer (see Section 3.4.2).
- 8.
Throughout this chapter we will denote the nominal values by the subscript zero and the standard deviations or full widths half maximum (FWHM) by the prefix \(\Delta\).
- 9.
If the interaction with the sample produces a deterministic change in one of the variables, e.g., if it does not change the spin state of the neutron, then we can use this variable as a label to encode one of the others, e.g., the energy. This principle is at the origin of the spin - echo technique (see Section 3.4.8).
- 10.
This is equal to the number of neutrons scattered per second into a solid angle element \({\rm d} \Omega\) with energies comprised between \(E_f\) and \(E_f + {\rm d} E_f\) and normalized to the incoming flux.
- 11.
The scattering function as defined here does still contain the interaction potential of the neutron with matter. In other words, it contains the scattering length \(b_l\) of the nuclei and has, therefore, the unit of area per energy.
- 12.
If we observe the scattered wave \(\psi_{\rm out}\) at the detector, we must include in Eq. (3.25) the description of the scattered beam as a function of \(\vec{k}_f\). This includes as outlined above the probability of a neutron \(\vec{k}_f\) reaching the detector and leads us to an expression equivalent to Eq. (3.21).
- 13.
One may ask why we do not use energy - sensitive detection. This would avoid tailoring the outgoing beam and thus increase the overall luminosity of the experimental set-up. Energy - sensitive detection requires a correlation between the detector output signal and the neutron energy. A typical energy - sensitive detector is a gas chamber for charged particles. The principle works because the energies of the particles are orders of magnitude larger than the ionization energies required to leave a trace of their trajectory in the gas. For a similar technique to work with thermal neutrons, we would require a system that possesses trackable excitations restricted to the sub-meV range as any meV excitation would completely ruin the energy sensitivity. Such systems are difficult to conceive. An alternative to tracking the trajectory via local inelastic excitation of a medium is a system based on elastic scattering. Incoming neutrons are scattered into different angular regions depending on their energies. The correlation between angle and energy is based on the Bragg-law (see Section 3.3.5) and the scattering can be produced by oriented powders surrounded by position - sensitive detectors. Some pioneering work is currently carried out in this area. The problem with such devices lies with the fact that they are quite cumbersome and difficult to realize for large-area detectors.
- 14.
A more detailed discussion of the different filters can, for example, be found in [20].
- 15.
For a historical account of the early experiments with neutrons, see [21].
- 16.
The appropriate choice of the multi-layer material allows constructing a guide that in a magnetic field allows to reflect only one spin component of the material. It thus can be used as a polarizing device.
- 17.
In practice, it turns out that due to imperfections the transmission can be well described by a Gaussian distribution.
- 18.
The shape of \(p(\lambda)\) will naturally reflect the shape of the distributions \(p(\theta)\) and \(p(d)\). For simplicity we may assume both to be described by Gaussian functions.
- 19.
A good overview of focusing Bragg optics is given in [26].
- 20.
This effect cannot be described by the Bragg equation, as it was derived on the assumption of an infinitely large homogeneously illuminated scattering volume.
- 21.
The task of pulsing the beam would be considerably more difficult, if we were dealing with lighter particles (e.g., electrons) of the same kinetic energy and spread out over similar guide cross sections.
- 22.
This relies on the fact that the transmission of both collimators and crystals is a very good approximation Gaussian (see Sections 3.3.5 and 3.3.4, as well as [36]).
- 23.
We have used the notation \(\vec{q} = \vec{Q} - \vec{\tau}_{\rm hkl}\) with \(\vec{\tau}_{\rm hkl}\) indicating the closest reciprocal lattice vector.
- 24.
On IN16 at the ILL this 1Â \(\mu\)eV resolution is accompanied by a Gaussian profile of \(p(\hbar\omega)\). This is not a trivial fact as the Darwin distribution arising from primary extinction is not Gaussian.
- 25.
In the case of a pulsed source, this function can in principle be assured by the source itself. In most cases it is, however still necessary to shape the pulse.
- 26.
In that case \(P_{hkl}(\alpha,\beta) = \delta(\alpha-\alpha_0)\delta(\beta-\beta_0)\).
- 27.
The shape of the gauge volume depends on the scattering angle. It is a rectangular solid with a square base in the case of \(2\theta=90^\circ\).
- 28.
If there were means of moderating the neutron spectrum efficiently down to even lower temperatures, that is, of achieving high neutron flux at wavelengths ranging from 10 to 1000Â Ã… , large objects could in principle be investigated at wider angles. In practice one would, however, reach limits due to the high absorption and finally weak penetration (see Section 3.4.7) of long-wavelength neutrons.
- 29.
We will assume in the following circular apertures. In practice one often uses the guide opening with a rectangular aperture in front of the sample. This introduces an asymmetry in the divergence contribution to the resolution.
- 30.
For a nice example see [51].
- 31.
- 32.
This is a direct consequence of special relativity. The Dirac equation
$${\bf H} \psi = (c \cdot \vec{\alpha} \cdot \vec{\bf p} + \beta m c^2) \psi$$((3.86))is relativistically invariant if, and only if,
$${\bf H}^2 \psi = c^2 ( \vec{\bf p} \cdot \vec{\bf p} + c^2m^2) \psi.$$((3.87))\(\vec{\bf p} = -i \hbar \vec{\nabla}\) is the momentum operator. This implies that the coefficients \(\alpha_i\) and \(\beta\) satisfy the anti-commutation relations
$$\alpha_i\alpha_j + \alpha_j\alpha_i = 2 \delta_{ij} {\bf 1} ,$$((3.88))$$\alpha_i \beta + \beta \alpha_i = 0 ,$$((3.89))$$\beta^2 = {\bf 1},$$((3.90))with \({\bf 1}\) denoting the unit matrix and \( i,j = x,y,z\). It can be shown that at least four dimensions are required to represent such an algebra. A possible choice for the \((4 \times 4)\) matrices is given by
$$\alpha_i = \left[\begin{array}{cc} 0 & \sigma_i\\ \sigma_i & 0 \end{array}\right],\, \beta = \left[\begin{array}{cc} {\bf 1} & 0\\ 0 & -{\bf 1} \end{array}\right],$$((3.91))with \(\sigma_i\) denoting the \((2 \times 2)\) Pauli matrices. In vacuum the only symmetry breaking direction is give by the momentum \(\vec{p}\) of the neutron. Without loss of generality the \(\hat z\)-direction can be choosen to coincide with \(\hat p\). The four components of the Dirac wave function then correspond, respectively, to particle and anti-particle of positive (spin aligned parallel to \(\hat p\)) and negative (spin aligned anti-parallel to \(\hat p\)) helicity.
- 33.
This contribution adds to the nuclear interaction described by the Fermi pseudo-potential introduced in Eq. 3.26
- 34.
As the magnetization of the neutron gas is very small, we are not obliged to make a distinction in between the magnetic field \(\vec{H}\) and the magnetic flux density \(\vec{B}\) outside magnetic materials.
- 35.
There is a whole area of neutron scattering devoted to creating stable polarized beams in field-free regions generated either via super-conducting (cryo-pad) or \(\mu-metal\) shielding (\(\mu\)-pad). We will not further discuss this technique here, but refer to the literature [57].
- 36.
The rotation angle of the polarization is a function of the time the neutrons spend within the field. It thus depends on the neutron wavelength. Such devices have, therefore, diminishing performance for broad wavelength bands.
- 37.
To get an idea about field strength: The earth magnetic field has a strength of about 0.5 Gauss, which is equivalent to 0.05 mT. A strong permanent neodymium magnet (Nd\(_2\)Fe\(_{14}\)B) used in hard-disk drives can reach values above 1 T, and strong superconducting laboratory magnets achieve nearly 20 T in continuous operation. The highest magnetic fields (\(B > 10^{8}\)) Tesla that are observed in the universe stem from neutron stars. The question of whether in these stars the magnetic field could move only with the charged particles, leaving the neutrons behind, is a matter of current scientific debate.
- 38.
For a single neutron \(|\vec{P}|\) is preserved. The depolarization of the beam arises from the fact that for every neutron with polarization vector \(\vec{P}\) we find another one in the beam with \(\vec{P}^\prime = - \vec{P}\).
- 39.
In principle, one could equally work with two coils with opposite field directions. In practice, one prefers often the version with identical fields and a \(\pi\)-flipper. In that case, the two coils can be put into series assuring identical currents. In addition, opposite field directions would make it more difficult to create homogeneous field components along the trajectory.
- 40.
In resonance spin-echo spectroscopy, the two precession solenoids are substituted by two pairs of radiofrequency coils. As this does not change the basic principle, we refer the reader to the literature for further insight into this very interesting technique [58].
- 41.
We assume for the moment that the sample does not alter the polarization of the beam. This is the case for nuclear coherent and isotope incoherent scattering.
- 42.
In practice, this is done by a small extra coil in one of the spectrometer arms.
- 43.
Strictly speaking, we get the cosine-transform of the wavelength distribution. For symmetric functions the cosine transform is, however, identical to the Fourier transform.
- 44.
We may choose in principle any fixed value for the energy transfer for the spin-echo condition [59]. This is, for example, useful when measuring the width of an excitation with high precision using a combination of spin-echo and three-axis techniques. This is realized at the TRISP instrument at the FRM-II in Munich [60].
- 45.
As outlined before this is true for symmetric functions. As we are, however, interested in small energy transfers, \(S(\vec{Q}, \hbar\omega)\) is symmetric for all but the lowest temperatures.
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Schober, H. (2009). Neutron Scattering Instrumentation. In: Liang, L., Rinaldi, R., Schober, H. (eds) Neutron Applications in Earth, Energy and Environmental Sciences. Neutron Scattering Applications and Techniques. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-09416-8_3
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