Neutron techniques

Anderson, I. S.; Brown, P. J.; Carpenter, J. M.; Lander, G.; Pynn, R.; Rowe, J. M.; Schärpf, O.; Sears, V. F.; Willis, B. T. M.

doi:10.1107/97809553602060000594

International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

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International Tables for Crystallography (2006). Vol. C. ch. 4.4, pp. 430-487
https://doi.org/10.1107/97809553602060000594

Chapter 4.4. Neutron techniques

I. S. Anderson,^a P. J. Brown,^a J. M. Carpenter,^b G. Lander,^c R. Pynn,^d J. M. Rowe,^e O. Schärpf,^f V. F. Sears^g and B. T. M. Willis^h

^a Institut Laue–Langevin, Avenue des Martyrs, BP 156X, F-38042 Grenoble CEDEX, France,^bIntense Pulsed Neutron Source, Building 360, Argonne National Laboratory, Argonne, IL 60439, USA,^cITU, European Commission, Postfach 2340, D-76125 Karlsruhe, Germany,^dLANSCE, MS H805, Los Alamos National Laboratory, PO Box 1663, Los Alamos, NM 87545, USA,^eNIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA,^fPhysik-Department E13, TU München, James-Franck-Strasse 1, D-85748 Garching, Germany,^gAtomic Energy of Canada Limited, Chalk River Laboratories, Chalk River, Ontario, Canada K0J 1J0, and ^hChemical Crystallography Laboratory, University of Oxford, 9 Parks Road, Oxford OX1 3PD, England

The first section of this chapter briefly describes the fission and spallation mechanisms for producing neutrons. The general properties of the slow neutron spectra that they produce are discussed and the methods of using beams from steady (fission reactor) sources and pulsed (accelerator-driven) sources are compared. The second section provides an overview of the different kinds of beam-definition devices. Resolution functions are then described in the third section. Scattering lengths for neutrons are tabulated in the fourth section, and scattering and absorption cross sections, isotope effects and correction for electromagnetic interactions are discussed. The fifth section gives tables of the coefficients in analytic approximations to the form factors used in the calculation of the cross sections for magnetic scattering of neutrons. The coefficients for atoms and ions in the 3d and 4d transition series are derived from wavefunctions obtained using Hartree–Fock theory and the coefficients for the rare-earth and actinide ions are obtained by fitting the analytic forms to published form factors calculated from Dirac–Fock wavefunctions. In the final section, the absorption cross sections and 1/e penetration depths of the elements are tabulated for neutrons of 1.8 Å wavelength.

Keywords: absorption; choppers; collimation; cross sections; filters; magnetic form factors; mirrors; monochromators; neutron diffraction; neutron scattering; neutron-beam definition; neutrons; resolution functions; scattering lengths; spin-orientation devices; Zeeman polarizers.

4.4.1. Production of neutrons

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J. M. Carpenter^b and G. Lander^c

The production of neutrons of sufficient intensity for scattering experiments is a `big-machine' operation; there is no analogue to the small laboratory X-ray unit. The most common sources of neutrons, and those responsible for the great bulk of today's successful neutron scattering programs, are the nuclear reactors . These are based on the continuous, self-sustaining fission reaction. Research-reactor design emphasizes power density, that is the highest power within a small `leaky' volume, whereas power reactors generate large amounts of power over a large core volume. In research reactors, fuel rods are of highly enriched ²³⁵U. Neutrons produced are distributed in a fission spectrum centred about 1 MeV: Most of the neutrons within the reactor are moderated (i.e. slowed down) by collisions in the cooling liquid, normally D₂O or H₂O, and are absorbed in fuel to propagate the reaction. As large a fraction as possible is allowed to leak out as fast neutrons into the surrounding moderator (D₂O and Be are best) and to slow down to equilibrium with this moderator. The neutron spectrum is Maxwellian with a mean energy of ∼300 K (= 25 meV), which for neutrons corresponds to 1.8 Å since $[E_n \ ({\rm meV})=81.8/\lambda^2\ ({\rm \AA}^2).]$ Neutrons are extracted in beams through holes that penetrate the moderator.

There are two points to remember: (a) neutrons are neutral so that we cannot focus the beams and (b) the spectrum is broad and continuous; there is no analogy to the characteristic wavelength found with X-ray tubes, or to the high directionality of synchrotron-radiation sources.

Neutron production and versatility in reactors reached a new level with the construction of the High-Flux Reactor at the French–German–English Institut Laue–Langevin (ILL) in Grenoble, France. An overview of the reactor and beam-tube assembly is shown in Fig. 4.4.1.1 . To shift the spectrum in energy, both a cold source (25 l of liquid deuterium at 25 K) and a hot source (graphite at 2400 K) have been inserted into the D₂O moderator. Special beam tubes view these sources allowing a range of wavelengths from ∼0.3 to ∼17 Å to be used. Over 30 instruments are in operation at the ILL, which started in 1972.

Figure 4.4.1.1| top | pdf |

A plane view of the installation at the Institut Laue–Langevin, Grenoble. Note especially the guide tubes exiting from the reactor that transport the neutron beams to a variety of instruments; these guide tubes are made of nickel-coated glass from which the neutrons are totally internally reflected.

The second method of producing neutrons, which historically predates the discovery of fission, is with charged particles (α particles, protons, etc.) striking a target nuclei. The most powerful source of neutrons of this type uses proton beams. These are accelerated in short bursts ( $[\lt]$ 1 µs) to 500–1000 MeV, and after striking the target produce an instantaneous supply of high-energy `evaporation' neutrons. These extend up in energy close to that of the incident proton beam. Shielding for spallation sources tends to be even more massive than that for reactors. The targets, usually tungsten or uranium and typically much smaller than a reactor core, are surrounded by hydrogenous moderators such as polyethylene (often at different temperatures) to produce the `slow' neutrons (E_n < 10 eV) used in scattering experiments. The moderators are very different from those of reactors; they are designed to slow down neutrons rapidly and to let them leak out, rather than to store them for a long time. If the accelerated particle pulse is short enough, the duration of the moderated neutron pulses is roughly inversely proportional to the neutron speed.

These accelerator-driven pulsed sources are pulsed at frequencies of between 10 and 100 Hz.

There are two fundamental differences between a reactor and a pulsed source.

(1) All experiments at a pulsed source must be performed with time-of-flight techniques. The pulsed source produces neutrons in bursts of 1 to 50 µs duration, depending on the energy, spaced about 10 to 100 ms apart, so that the duty cycle is low but there is very high neutron intensity within each pulse. The time-of-flight technique makes it possible to exploit that high intensity. With the de Broglie relationship, for neutrons $[\lambda\, ({\rm \AA})=0.3966t\,({\mu}{\rm s})/L\,({\rm cm}),]$ where t is the flight time in µs and L is the total flight path in cm.
(2) The spectral characteristics of pulsed sources are somewhat different from reactors in that they have a much larger component of higher-energy (above 100 meV) neutrons than the thermal spectrum at reactors. The exploitation of this new energy regime accompanied by the short pulse duration is one of the great opportunities presented by spallation sources.

Fig. 4.4.1.2 illustrates the essential difference between experiments at a steady-state source (left panel) and a pulsed source (right panel). We confine the discussion here to diffraction. If the time over which useful information is gathered is equivalent to the full period of the source Δt (the case suggested by the lower-right figure), the peak flux of the pulsed source is the effective parameter to compare with the flux of the steady-state source. Often this is not the case, so one makes a comparison in terms of time-averaged flux (centre panel). For the pulsed source, this is lowered from the peak flux by the duty cycle, but with the time-of-flight method one uses a large interval of the spectrum (shaded area). For the steady-state source, the time-averaged flux is high, but only a small wavelength slice (stippled area) is used in the experiment. It is the integrals of the two areas which must be compared; for the pulsed sources now being designed, the integral is generally favourable compared with present-day reactors. Finally, one can see from the central panel that high-energy neutrons (100–1000 meV) are especially plentiful at the pulsed sources. These various features can be exploited in the design of different kinds of experiments at pulsed sources.

Figure 4.4.1.2| top | pdf |

Schematic diagram for performing diffraction experiments at steady-state and pulsed neutron sources. On the left we see the familiar monochromator crystal allowing a constant (in time) beam to fall on the sample (centre left), which then diffracts the beam through an angle 2θ_s into the detector. The signal in the latter is also constant in time (lower left). On the right, the pulsed source allows a wide spectrum of neutrons to fall on the sample in sharp pulses separated by Δt (centre right). The neutrons are then diffracted by the sample through 2θ_s and their time of arrival in the detector is analysed (lower right). The centre figure shows the time-averaged flux at the source. At a reactor, we make use of a narrow band of neutrons (heavy shading), here chosen with λ = 1.5 Å. At a pulsed source, we use a wide spectral band, here chosen from 0.4 to 3 Å and each one is identified by its time-of-flight. For the experimentalist, an important parameter is the integrated area of the two-shaded areas. Here they have been made identical.

4.4.2. Beam-definition devices

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I. S. Anderson^a and O. Schärpf^f

4.4.2.1. Introduction

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Neutron scattering, when compared with X-ray scattering techniques developed on modern synchrotron sources, is flux limited, but the method remains unique in the resolution and range of energy and momentum space that can be covered. Furthermore, the neutron magnetic moment allows details of microscopic magnetism to be examined, and polarized neutrons can be exploited through their interaction with both nuclear and electron spins.

Owing to the low primary flux of neutrons, the beam definition devices that play the role of defining the beam conditions (direction, divergence, energy, polarization, etc.) have to be highly efficient. Progress in the development of such devices not only results in higher-intensity beams but also allows new techniques to be implemented.

The following sections give a (non-exhaustive) review of commonly used beam-definition devices. The reader should keep in mind the fact that neutron scattering experiments are typically carried out with large beams (1 to 50 cm²) and divergences between 5 and 30 mrad.

4.4.2.2. Collimators

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A collimator is perhaps the simplest neutron optical device and is used to define the direction and divergence of a neutron beam. The most rudimentary collimator consists of two slits or pinholes cut into an absorbing material and placed one at the beginning and one at the end of a collimating distance L. The maximum beam divergence that is transmitted with this configuration is $[\alpha _{\rm max }=(a_{1}+a_{2})/L, \eqno (4.4.2.1)]$ where $[a_{1}]$ and $[a_{2}]$ are the widths of the slits or pinholes.

Such a device is normally used for small-angle scattering and reflectometry. In order to avoid parasitic scattering by reflection from slit edges, very thin sheets of a highly absorbing material, e.g. gadolinium foils, are used as the slit material. Sometimes wedge-formed cadmium plates are sufficient. In cases where a very precise edge is required, cleaved single-crystalline absorbers such as gallium gadolinium garnet (GGG) can be employed.

To avoid high intensity losses when the distances are large, sections of neutron guide can be introduced between the collimators, as in, for example, small-angle scattering instruments with variable collimation. In this case, for maximum intensity at a given resolution (divergence), the collimator length should be equal to the camera length, i.e. the sample–detector distance (Schmatz, Springer, Schelten & Ibel, 1974 ).

As can be seen from (4.4.2.1), the beam divergence from a simple slit or pinhole collimator depends on the aperture size. In order to collimate (in one dimension) a beam of large cross section within a reasonable distance L, Soller collimators, composed of a number of equidistant neutron absorbing blades, are used. To avoid losses, the blades must be as thin and as flat as possible. If their surfaces do not reflect neutrons, which can be achieved by using blades with rough surfaces or materials with a negative scattering length, such as foils of hydrogen-containing polymers (Mylar is commonly used) or paper coated with neutron-absorbing paint containing boron or gadolinium (Meister & Weckerman, 1973 ; Carlile, Hey & Mack, 1977 ), the angular dependence of the transmission function is close to the ideal triangular form, and transmissions of 96% of the theoretical value can be obtained with 10′ collimation. If the blades of the Soller collimator are coated with a material whose critical angle of reflection is equal to $[\alpha _{\rm max }/2]$ (for one particular wavelength), then a square angular transmission function is obtained instead of the normal triangular function, thus doubling the theoretical transmission (Meardon & Wroe, 1977 ).

Soller collimators are often used in combination with single-crystal monochromators to define the wavelength resolution of an instrument but the Soller geometry is only useful for one-dimensional collimation. For small-angle scattering applications, where two-dimensional collimation is required, a converging `pepper pot' collimator can be used (Nunes, 1974 ; Glinka, Rowe & LaRock, 1986 ).

Cylindrical collimators with radial blades are sometimes used to reduce background scattering from the sample environment. This type of collimator is particularly useful with position-sensitive detectors and may be oscillated about the cylinder axis to reduce the shadowing effect of the blades (Wright, Berneron & Heathman, 1981 ).

4.4.2.3. Crystal monochromators

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Bragg reflection from crystals is the most widely used method for selecting a well defined wavelength band from a white neutron beam. In order to obtain reasonable reflected intensities and to match the typical neutron beam divergences, crystals that reflect over an angular range of 0.2 to 0.5° are typically employed. Traditionally, mosaic crystals have been used in preference to perfect crystals, although reflection from a mosaic crystal gives rise to an increase in beam divergence with a concomitant broadening of the selected wavelength band. Thus, collimators are often used together with mosaic monochromators to define the initial and final divergences and therefore the wavelength spread.

Because of the beam broadening produced by mosaic crystals, it was soon recognised that elastically deformed perfect crystals and crystals with gradients in lattice spacings would be more suitable candidates for focusing applications since the deformation can be modified to optimize focusing for different experimental conditions (Maier-Leibnitz, 1969 ).

Perfect crystals are used commonly in high-energy-resolution backscattering instruments, interferometry and Bonse–Hart cameras for ultra-small-angle scattering (Bonse & Hart, 1965 ).

An ideal mosaic crystal is assumed to comprise an agglomerate of independently scattering domains or mosaic blocks that are more or less perfect, but small enough that primary extinction does not come into play, and the intensity reflected by each block may be calculated using the kinematic theory (Zachariasen, 1945 ; Sears, 1997 ). The orientation of the mosaic blocks is distributed inside a finite angle, called the mosaic spread, following a distribution that is normally assumed to be Gaussian. The ideal neutron mosaic monochromator is not an ideal mosaic crystal but rather a mosaic crystal that is sufficiently thick to obtain a high reflectivity. As the crystal thickness increases, however, secondary extinction becomes important and must be accounted for in the calculation of the reflectivity. The model normally used is that developed by Bacon & Lowde (1948 ), which takes into account strong secondary extinction and a correction factor for primary extinction (Freund, 1985 ). In this case, the mosaic spread (usually defined by neutron scatterers as the full width at half maximum of the reflectivity curve) is not an intrinsic crystal property, but increases with wavelength and crystal thickness and can become quite appreciable at longer wavelengths.

Ideal monochromator materials should have a large scattering-length density, low absorption, incoherent and inelastic cross sections, and should be available as large single crystals with a suitable defect concentration. Relevant parameters for some typical neutron monochromator crystals are given in Table 4.4.2.1 .

Table 4.4.2.1| top | pdf |
Some important properties of materials used for neutron monochromator crystals (in order of increasing unit-cell volume)

Material	Structure	Lattice constant(s) at 300 K a, c (Å)	Unit-cell volume V₀ (10⁻²⁴ cm³)	Coherent scattering length b (10⁻¹² cm)	Square of scattering-length density (10⁻²¹cm⁻⁴)	Ratio of incoherent to total scattering cross section σ_inc/σ_s	Absorption cross section σ_abs (barns)^† (at λ = 1.8 Å)	Atomic mass A	Debye temperature $[\theta_D]$ (K)	$[A\theta^2_D]$ (10⁶ K²)
Beryllium	h.c.p.	a : 2.2856 c : 3.5832	16.2	0.779 (1)	9.25	$[6.5\times10^{-4}]$	0.0076 (8)	9.013	1188	12.7
Iron	b.c.c.	a : 2.8664	23.5	0.954 (6)	6.59	0.033	2.56 (3)	55.85	411	9.4
Zinc	h.c.p.	a : 2.6649 c : 4.9468	30.4	0.5680 (5)	1.50	0.019	1.11 (2)	65.38	253	4.2
Pyrolytic graphite	layer hexag.	a : 2.461 c : 6.708	35.2	0.66484 (13)	5.71	$[>2\times10^{-4}]$	0.00350 (7)	12.01	800	7.7
Niobium	b.c.c.	3.3006	35.9	0.7054 (3)	1.54	$[4\times10^{-4}]$	1.15 (5)	92.91	284	7.5
Nickel (⁵⁸Ni)	f.c.c.	3.5241	43.8	1.44 (1)	17.3	0	4.6 (3)	58.71	417	9.9
Copper	f.c.c.	3.6147	47.2	0.7718 (4)	4.28	0.065	3.78 (2)	63.54	307	6.0
Aluminium	f.c.c.	4.0495	66.4	0.3449 (5)	0.43	$[5.6\times10^{-3}]$	0.231 (3)	26.98	402	4.4
Lead	f.c.c.	4.9502	121	0.94003 (14)	0.97	$[2.7\times10^{-4}]$	0.171 (2)	207.21	87	1.6
Silicon	diamond	5.4309	160	0.41491 (10)	0.43	$[6.9\times10^{-3}]$	0.171 (3)	28.09	543	8.3
Germanium	diamond	5.6575	181	0.81929 (7)	1.31	0.020	2.3 (2)	72.60	290	6.1

^†1 barn = 10⁻²⁸ m².

In principle, higher reflectivities can be obtained in neutron monochromators that are designed to operate in reflection geometry, but, because reflection crystals must be very large when takeoff angles are small, transmission geometry may be used. In that case, the optimization of crystal thickness can only be achieved for a small wavelength range.

Nickel has the highest scattering-length density, but, since natural nickel comprises several isotopes, the incoherent cross section is quite high. Thus, isotopic ⁵⁸Ni crystals have been grown as neutron monochromators despite their expense. Beryllium, owing to its large scattering-length density and low incoherent and absorption cross sections, is also an excellent candidate for neutron monochromators, but the mosaic structure of beryllium is difficult to modify, and the availability of good-quality single crystals is limited (Mücklich & Petzow, 1993 ). These limitations may be overcome in the near future, however, by building composite monochromators from thin beryllium blades that have been plastically deformed (May, Klimanek & Magerl, 1995 ).

Pyrolytic graphite is a highly efficient neutron monochromator if only a medium resolution is required (the minimum mosaic spread is of the order of 0.4°), owing to high reflectivities, which may exceed 90% (Shapiro & Chesser, 1972 ), but its use is limited to wavelengths above 1.5 Å, owing to the rather large d spacing of the 002 reflection. Whenever better resolution at shorter wavelengths is required, copper (220 and 200) or germanium (311 and 511) monochromators are frequently used. The advantage of copper is that the mosaic structure can be easily modified by plastic deformation at high temperature. As with most face-centred cubic crystals, it is the (111) slip planes that are functional in generating the dislocation density needed for the desired mosaic spread, and, depending on the required orientation, either isotropic or anisotropic mosaics can be produced (Freund, 1976 ). The latter is interesting for vertical focusing applications, where a narrow vertical mosaic is required regardless of the resolution conditions.

Although both germanium and silicon are attractive as monochromators, owing to the absence of second-order neutrons for odd-index reflections, it is difficult to produce a controlled uniform mosaic spread in bulk samples by plastic deformation at high temperature because of the difficulty in introducing a spatially homogenous microstructure in large single crystals (Freund, 1975 ). Recently this difficulty has been overcome by building up composite monochromators from a stack of thin wafers, as originally proposed by Maier-Leibnitz (1967 ; Frey, 1974 ).

In practice, an artificial mosaic monochromator can be built up in two ways. In the first approach, illustrated in Fig. 4.4.2.1(a) , the monochromator comprises a stack of crystalline wafers, each of which has a mosaic spread close to the global value required for the entire stack. Each wafer in the stack must be plastically deformed (usually by alternated bending) to produce the correct mosaic spread. For certain crystal orientations, the plastic deformation may result in an anisotropic mosaic spread. This method has been developed in several laboratories to construct germanium monochromators (Vogt, Passell, Cheung & Axe, 1994 ; Schefer et al., 1996 ).

Figure 4.4.2.1| top | pdf |

Two methods by which artificial mosaic monochromators can be constructed: (a) out of a stack of crystalline wafers, each with a mosaicity close to the global value. The increase in divergence due to the mosaicity is the same in the horizontal (left picture) and the vertical (right picture) directions; (b) out of several stacked thin crystalline wafers each with a rather narrow mosaic but slightly misoriented in a perfectly controlled way. This allows the shape of the reflectivity curve to be rectangular, Gaussian, Lorentzian, etc., and highly anisotropic, i.e. vertically narrow (right picture) and horizontally broad (left figure).

In the second approach, shown in Fig. 4.4.2.1(b), the global reflectivity distribution is obtained from the contributions of several stacked thin crystalline wafers, each with a rather narrow mosaic spread compared with the composite value but slightly misoriented with respect to the other wafers in the stack. If the misorientation of each wafer can be correctly controlled, this technique has the major advantage of producing monochromators with a highly anisotropic mosaic structure. The shape of the reflectivity curve can be chosen at will (Gaussian, Lorentzian, rectangular), if required. Moreover, because the initial mosaicity required is small, it is not necessary to use mosaic wafers and therefore for each wafer to undergo a long and tedious plastic deformation process. Recently, this method has been applied successfully to construct copper monochromators (Hamelin, Anderson, Berneron, Escoffier, Foltyn & Hehn, 1997 ), in which individual copper wafers were cut in a cylindrical form and then slid across one another to produce the required mosaic spread in the scattering plane. This technique looks very promising for the production of anisotropic mosaic monochromators.

The reflection from a mosaic crystal is visualized in Fig. 4.4.2.2(b) . An incident beam with small divergence is transformed into a broad exit beam. The range of k vectors, Δk, selected in this process depends on the mosaic spread, η, and the incoming and outgoing beam divergences, $[\alpha _{1}]$ and $[\alpha _{2}.]$ $[ \Delta k/k=\Delta\tau /\tau +\alpha \cot \theta, \eqno (4.4.2.2)]$ where τ is the magnitude of the crystal reciprocal-lattice vector (τ = 2π/d) and α is given by $[\alpha =\sqrt {\;{\alpha {}\/_{1}^{2}\alpha {}\/_{2}^{2}\;+\;\alpha {}\/_{1}^{2}\eta ^{2}\;+\;\alpha {}\/_{2}^{2}\eta ^{2}\;}\over{\alpha {}\/_{1}^{2}\;+\;\alpha {}\/_{2}^{2}\;+\;4\eta ^{2}}}. \eqno (4.4.2.3)]$ The resolution can therefore be defined by collimators, and the highest resolution is obtained in backscattering, where the wavevector spread depends only on the intrinsic Δd/d of the crystal.

Figure 4.4.2.2| top | pdf |

Reciprocal-lattice representation of the effect of a monochromator with reciprocal-lattice vector τ on the reciprocal-space element of a beam with divergence α. (a) For an ideal crystal with a lattice constant width Δτ; (b) for a mosaic crystal with mosaicity η, showing that a beam with small divergence, α, is transformed into a broad exit beam with divergence 2η + α; (c) for a gradient crystal with interplanar lattice spacing changing over Δτ, showing that the divergence is not changed in this case.

In some applications, the beam broadening produced by mosaic crystals can be detrimental to the instrument performance. An interesting alternative is a gradient crystal, i.e. a single crystal with a smooth variation of the interplanar lattice spacing along a defined crystallographic direction. As shown in Fig. 4.4.2.2(c), the diffracted phase-space element has a different shape from that obtained from a mosaic crystal. Gradients in d spacing can be produced in various ways, including thermal gradients (Alefeld, 1972 ), vibrating crystals by piezoelectric excitation (Hock, Vogt, Kulda, Mursic, Fuess & Magerl, 1993 ), and mixed crystals with concentration gradients, e.g. Cu–Ge (Freund, Guinet, Maréschal, Rustichelli & Vanoni, 1972 ) and Si–Ge (Maier-Leibnitz & Rustichelli, 1968 ; Magerl, Liss, Doll, Madar & Steichele, 1994 ).

Both vertically and horizontally focusing assemblies of mosaic crystals are employed to make better use of the neutron flux when making measurements on small samples. Vertical focusing can lead to intensity gain factors of between two and five without affecting resolution (real-space focusing) (Riste, 1970 ; Currat, 1973 ). Horizontal focusing changes the k-space volume that is selected by the monochromator through the variation in Bragg angle across the monochromator surface (k-space focusing) (Scherm, Dolling, Ritter, Schedler, Teuchert & Wagner, 1977 ). The orientation of the diffracted k-space volume can be modified by variation of the horizontal curvature, so that the resolution of the monochromator may be optimized with respect to a particular sample or experiment without loss of illumination. Monochromatic focusing can be achieved. Furthermore, asymmetrically cut crystals may be used, allowing focusing effects in real space and k space to be decoupled (Scherm & Kruger, 1994 ).

Traditionally, focusing monochromators consist of rectangular crystal plates mounted on an assembly that allows the orientation of each crystal to be varied in a correlated manner (Bührer, 1994 ). More recently, elastically deformed perfect crystals (in particular silicon) have been exploited as focusing elements for monochromators and analysers (Magerl & Wagner, 1994 ).

Since thermal neutrons have velocities that are of the order of km s⁻¹, their wavelengths can be Doppler shifted by diffraction from moving crystals. The k-space representation of the diffraction from a crystal moving perpendicular to its lattice planes is shown in Fig. 4.4.2.3(a) . This effect is most commonly used in backscattering instruments on steady-state sources to vary the energy of the incident beam. Crystal velocities of 9–10 m s⁻¹ are practically achievable, corresponding to energy variations of the order of ±60 µeV.

Figure 4.4.2.3| top | pdf |

Momentum-space representation of Bragg scattering from a crystal moving (a) perpendicular and (b) parallel to the diffracting planes with a velocity v_k. The vectors k_L and $[{\bf k}^{'}_{L}]$ refer to the incident and reflected wavevectors in the laboratory frame of reference. In (a), depending on the direction of v_k, the reflected wavevector is larger or smaller than the incident wavevector, k_L. In (b), a larger incident reciprocal-space volume, Δv_L, is selected by the moving crystal than would have been selected by the crystal at rest. The reflected reciprocal-space element, $[\Delta{\bf v}^{'}_{L}]$ , has a large divergence, but can be arranged to be normal to $[{\bf k}^{'}_{L}]$ , hence improving the resolution $[\Delta{\bf k}^{'}_{L}]$ .

The Doppler shift is also important in determining the resolution of the rotating-crystal time-of-flight (TOF) spectrometer, first conceived by Brockhouse (1958 ). A pulse of monochromatic neutrons is obtained when the reciprocal-lattice vector of a rotating crystal bisects the angle between two collimators. Effectively, the neutron k vector is changed in both direction and magnitude, depending on whether the crystal is moving towards or away from the neutron. For the rotating crystal, both of these situations occur simultaneously for different halves of the crystal, so that the net effect over the beam cross section is that a wider energy band is reflected than from the crystal at rest, and that, depending on the sense of rotation, the beam is either focused or defocused in time (Meister & Weckerman, 1972 ).

The Bragg reflection of neutrons from a crystal moving parallel to its lattice planes is illustrated in Fig. 4.4.2.3(b). It can be seen that the moving crystal selects a larger Δk than the crystal at rest, so that the reflected intensity is higher. Furthermore, it is possible under certain conditions to orientate the diffracted phase-space volume orthogonal to the diffraction vector. In this way, a monochromatic divergent beam can be obtained from a collimated beam with a larger energy spread. This provides an elegant means of producing a divergent beam with a sufficiently wide momentum spread to be scanned by the Doppler crystal of a backscattering instrument (Schelten & Alefeld, 1984 ).

Finally, an alternative method of scanning the energy of a monochromator in backscattering is to apply a steady but uniform temperature variation. The monochromator crystal must have a reasonable thermal expansion coefficient, and care has to be taken to ensure a uniform temperature across the crystal.

4.4.2.4. Mirror reflection devices

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The refractive index, n, for neutrons of wavelength λ propagating in a nonmagnetic material of atomic density N is given by the expression $[ n^{2}=1 - {{\lambda ^{2}Nb_{\rm {coh}}}\over{\pi }}, \eqno (4.4.2.4)]$ where $[b_{\rm {coh}}]$ is the mean coherent scattering length. Values of the scattering-length density $[Nb_{\rm {coh}}]$ for some common materials are listed in Table 4.4.2.2 , from which it can be seen that the refractive index for most materials is slightly less than unity, so that total external reflection can take place. Thus, neutrons can be reflected from a smooth surface, but the critical angle of reflection, $[\gamma _{c},]$ given by $[\gamma _{c}=\lambda \sqrt {{Nb_{\rm {coh}}}\over{\pi }}, \eqno (4.4.2.5)]$ is small, so that reflection can only take place at grazing incidence. The critical angle for nickel, for example, is 0.1° Å⁻¹.

Table 4.4.2.2| top | pdf |
Neutron scattering-length densities, Nb_coh, for some commonly used materials

Material	Nb (10⁻⁶ Å⁻²)
⁵⁸Ni	13.31
Diamond	11.71
Nickel	9.40
Quartz	3.64
Germanium	3.62
Silver	3.50
Aluminium	2.08
Silicon	2.08
Vanadium	−0.27
Titanium	−1.95
Manganese	−2.95

Because of the shallowness of the critical angle, reflective optics are traditionally bulky, and focusing devices tend to have long focal lengths. In some cases, however, depending on the beam divergence, a long mirror can be replaced by an equivalent stack of shorter mirrors.

4.4.2.4.1. Neutron guides

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The principle of mirror reflection is the basis of neutron guides, which are used to transmit neutron beams to instruments that may be situated up to 100 m away from the source (Christ & Springer, 1962 ; Maier-Leibnitz & Springer, 1963 ). A standard neutron guide is constructed from boron glass plates assembled to form a rectangular tube, the dimensions of which may be up to 200 mm high by 50 mm wide. The inner surface of the guide is coated with approximately 1200 Å of either nickel, ⁵⁸Ni ( $[\gamma_c]$ = 0.12° Å⁻¹), or a `supermirror' (described below). The guide is usually evacuated to reduce losses due to absorption and scattering of neutrons in air.

Theoretically, a neutron guide that is fully illuminated by the source will transmit a beam with a square divergence of full width $[2\gamma _{c}]$ in both the horizontal and vertical directions, so that the transmitted solid angle is proportional to $[\lambda^{2}]$ . In practice, owing to imperfections in the assembly of the guide system, the divergence profile is closer to Gaussian than square at the end of a long guide. Since the neutrons may undergo a large number of reflections in the guide, it is important to achieve a high reflectivity. The specular reflectivity is determined by the surface roughness, and typically values in the range 98.5 to 99% are achieved. Further transmission losses occur due to imperfections in the alignment of the sections that make up the guide.

The great advantage of neutron guides, in addition to the transport of neutrons to areas of low background, is that they can be multiplexed, i.e. one guide can serve many instruments. This is achieved either by deflecting only a part of the total cross section to a given instrument or by selecting a small wavelength range from the guide spectrum. In the latter case, the selection device (usually a crystal monochromator) must have a high transmission at other wavelengths.

If the neutron guide is curved, the transmission becomes wavelength dependent, as illustrated in Fig. 4.4.2.4 . In this case, one can define a characteristic wavelength, $[\lambda ^{*}]$ , given by the relation $[\theta ^{*}=\sqrt {2a/\rho }]$ , so that $[\lambda ^{*}=\sqrt { {\vphantom2}{\pi }\over{Nb_{\rm coh}\vphantom{\rho}}} \sqrt { {2a}\over{\rho }} \eqno (4.4.2.6)]$ (where a is the guide width and ρ the radius of curvature), for which the theoretical transmission drops to 67%. For wavelengths less than $[\lambda ^{*}]$ , neutrons can only be transmitted by `garland' reflections along the concave wall of the curved guide. Thus, the guide acts as a low-pass energy filter as long as its length is longer than the direct line-of-sight length $[L_{1}=\sqrt {8a\rho }]$ . For example, a 3 cm wide nickel-coated guide whose characteristic wavelength is 4 Å (radius of curvature 1300 m) must be at least 18 m long to act as a filter. The line-of-sight length can be reduced by subdividing the guide into a number of narrower channels, each of which acts as a miniguide. The resulting device, often referred to as a neutron bender, since deviation of the beam is achieved more rapidly, is used in beam deviators (Alefeld et al., 1988 ) or polarizers (Hayter, Penfold & Williams, 1978 ). A microbender was devised by Marx (1971 ) in which the channels were made by evaporating alternate layers of aluminium (transmission layer) and nickel (mirror layer) onto a flexible smooth substrate.

Figure 4.4.2.4| top | pdf |

In a curved neutron guide, the transmission becomes λ dependent: (a) the possible types of reflection (garland and zig-zag), the direct line-of-sight length, the critical angle θ*, which is related to the characteristic wavelength $[\lambda^*=\theta^*{\sqrt{\pi/Nb_{\rm coh}}}]$ ; (b) transmission across the exit of the guide for different wavelengths, normalized to unity at the outside edge; (c) total transmission of the guide as a function of λ.

Tapered guides can be used to reduce the beam size in one or two dimensions (Rossbach et al., 1988 ), although, since mirror reflection obeys Liouville's theorem, focusing in real space is achieved at the expense of an increase in divergence. This fact can be used to calculate analytically the expected gain in neutron flux at the end of a tapered guide (Anderson, 1988 ). Alternatively, focusing can be achieved in one dimension using a bender in which the individual channel lengths are adjusted to create a focus (Freund & Forsyth, 1979 ).

4.4.2.4.2. Focusing mirrors

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Optical imaging of neutrons can be achieved using ellipsoidal or torroidal mirrors, but, owing to the small critical angle of reflection, the dimensions of the mirrors themselves and the radii of curvature must be large. For example, a 4 m long toroidal mirror has been installed at the IN15 neutron spin echo spectrometer at the Institut Laue–Langevin, Grenoble (Hayes et al., 1996 ), to focus neutrons with wavelengths greater than 15 Å. The mirror has an in-plane radius of curvature of 408.75 m, and the sagittal radius is 280 mm. A coating of ⁶⁵Cu is used to obtain a high critical angle of reflection while maintaining a low surface roughness. Slope errors of less than 2.5 × 10⁻⁵ rad (r.m.s.) combined with a surface roughness of less than 3 Å allow a minimum resolvable scattering vector of about 5 × 10⁻⁴ Å⁻¹ to be reached.

For best results, the slope errors and the surface roughness must be low, in particular in small-angle scattering applications, since diffuse scattering from surface roughness gives rise to a halo around the image point. Owing to its low thermal expansion coefficient, highly polished Zerodur is often chosen as substrate.

4.4.2.4.3. Multilayers

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Schoenborn, Caspar & Kammerer (1974 ) first pointed out that multibilayers, comprising alternating thin films of different scattering-length densities ( $[Nb_{\rm {coh}}]$ ) act like two-dimensional crystals with a d spacing given by the bilayer period. With modern deposition techniques (usually sputtering), uniform films of thickness ranging from about twenty to a few hundred ångströms can be deposited over large surface areas of the order of 1 m². Owing to the rather large d spacings involved, the Bragg reflection from multilayers is generally at grazing incidence, so that long devices are required to cover a typical beam width, or a stacked device must be used. However, with judicious choice of the scattering-length contrast, the surface and interface roughness, and the number of layers, reflectivities close to 100% can be reached.

Fig. 4.4.2.5 illustrates how variation in the bilayer period can be used to produce a monochromator (the minimum Δλ/λ that can be achieved is of the order of 0.5%), a broad-band device, or a `supermirror', so called because it is composed of a particular sequence of bilayer thicknesses that in effect extends the region of total mirror reflection beyond the ordinary critical angle (Turchin, 1967 ; Mezei, 1976 ; Hayter & Mook, 1989 ). Supermirrors have been produced that extend the critical angle of nickel by a factor, m, of between three and four with reflectivities better than 90%. Such high reflectivities enable supermirror neutron guides to be constructed with flux gains, compared with nickel guides, close to the theoretical value of m².

Figure 4.4.2.5| top | pdf |

Illustration of how a variation in the bilayer period can be used to produce a monochromator, a broad-band device, or a supermirror.

The choice of the layer pairs depends on the application. For non-polarizing supermirrors and broad-band devices (Høghøj, Anderson, Ebisawa & Takeda, 1996 ), the Ni/Ti pair is commonly used, either pure or with some additions to relieve strain and stabilize interfaces (Elsenhans et al., 1994 ) or alter the magnetism (Anderson & Høghøj, 1996 ), owing to the high contrast in scattering density, while for narrow-band monochromators a low contrast pair such as W/Si is more suitable.

4.4.2.4.4. Capillary optics

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Capillary neutron optics, in which hollow glass capillaries act as waveguides, are also based on the concept of total external reflection of neutrons from a smooth surface. The advantage of capillaries, compared with neutron guides, is that the channel sizes are of the order of a few tens of micrometres, so that the radius of curvature can be significantly decreased for a given characteristic wavelength [see equation (4.4.2.6)]. Thus, neutrons can be efficiently deflected through large angles, and the device can be more compact.

Two basic types of capillary optics exist, and the choice depends on the beam characteristics required. Polycapillary fibres are manufactured from hollow glass tubes several centimetres in diameter, which are heated, fused and drawn multiple times until bundles of thousands of micrometre-sized channels are formed having an open area of up to 70% of the cross section. Fibre outer diameters range from 300 to 600 µm and contain hundreds or thousands of individual channels with inner diameters between 3 and 50 µm. The channel cross section is usually hexagonal, though square channels have been produced, and the inner channel wall surface roughness is typically less than 10 Å r.m.s., giving rise to very high reflectivities. The principal limitations on transmission efficiency are the open area, the acceptable divergence (note that the critical angle for glass is 1 mrad Å⁻¹) and reflection losses due to absorption and scattering. A typical optical device will comprise hundreds or thousands of fibres threaded through thin screens to produce the required shape.

Fig. 4.4.2.6 shows typical applications of polycapillary devices. In Fig. 4.4.2.6 (a), a polycapillary lens is used to refocus neutrons collected from a divergent source. The half lens depicted in Fig. 4.4.2.6 (b) can be used either to produce a nearly parallel (divergence = $[2\gamma _{c}]$ ) beam from a divergent source or (in the reverse sense) to focus a nearly parallel beam, e.g. from a neutron guide. The size of the focal point depends on the channel size, the beam divergence, and the focal length of the lens. For example, a polycapillary lens used in a prompt γ-activation analysis instrument at the National Institute of Standards and Technology to focus a cold neutron beam from a neutron guide results in a current density gain of 80 averaged over the focused beam size of 0.53 mm (Chen et al., 1995 ).

Figure 4.4.2.6| top | pdf |

Typical applications of polycapillary devices: (a) lens used to refocus a divergent beam; (b) half-lens to produce a nearly parallel beam or to focus a nearly parallel beam; (c) a compact bender.

Fig. 4.4.2.6(c) shows another simple application of polycapillaries as a compact beam bender. In this case, such a bender may be more compact than an equivalent multichannel guide bender, although the accepted divergence will be less. Furthermore, as with curved neutron guides, owing to the wavelength dependence of the critical angle the capillary curvature can be used to filter out thermal or high-energy neutrons.

It should be emphasized that the applications depicted in Fig. 4.4.2.6 obey Liouville's theorem, in that the density of neutrons in phase space is not changed, but the shape of the phase-space volume is altered to meet the requirements of the experiment, i.e. there is a simple trade off between beam dimension and divergence.

The second type of capillary optic is a monolithic configuration. The individual capillaries in monolithic optics are tapered and fused together, so that no external frame assembly is necessary (Chen-Mayer et al., 1996 ). Unlike the multifibre devices, the inner diameters of the channels that make up the monolithic optics vary along the length of the component, resulting in a smaller more compact design.

Further applications of capillary optics include small-angle scattering (Mildner, 1994 ) and lenses for high-spatial-resolution area detection.

4.4.2.5. Filters

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Neutron filters are used to remove unwanted radiation from the beam while maintaining as high a transmission as possible for the neutrons of the required energy. Two major applications can be identified: removal of fast neutrons and γ-rays from the primary beam and reduction of higher-order contributions (λ/n) in the secondary beam reflected from crystal monochromators. In this section, we deal with non-polarizing filters, i.e. those whose transmission and removal cross sections are independent of the neutron spin. Polarizing filters are discussed in the section concerning polarizers.

Filters rely on a strong variation of the neutron cross section with energy, usually either the wavelength-dependent scattering cross section of polycrystals or a resonant absorption cross section . Following Freund (1983 ), the total cross section determining the attenuation of neutrons by a crystalline solid can be written as a sum of three terms, $[\sigma =\sigma _{\rm {abs}}+\sigma _{\rm {tds}}+\sigma _{\rm {Bragg}}. \eqno (4.4.2.7)]$ Here, $[\sigma _{\rm {abs}}]$ is the true absorption cross section, which, at low energy, away from resonances, is proportional to $[E^{-1/2}]$ . The temperature-dependent thermal diffuse cross section, $[\sigma _{\rm {tds}}]$ , describing the attenuation due to inelastic processes, can be split into two parts depending on the neutron energy. At low energy, $[E\ll k_{b}\Theta _{{D}}]$ , where $[k_{b}]$ is Boltzmann's constant and $[\Theta _{{D}}]$ is the characteristic Debye temperature, single-phonon processes dominate, giving rise to a cross section, $[\sigma _{\rm {sph}}]$ , which is also proportional to $[E^{-1/2}]$ . The single-phonon cross section is proportional to $[T^{7/2}]$ at low temperatures and to T at higher temperatures. At higher energies, $[E\ge k_{b}\Theta_D]$ , multiphonon and multiple-scattering processes come into play, leading to a cross section, $[\sigma _{\rm {mph}}]$ , that increases with energy and temperature. The third contribution, $[\sigma _{\rm {Bragg}}]$ , arises due to Bragg scattering in single- or polycrystalline material. At low energies, below the Bragg cut-off (λ $[\gt]$ $[2{d}_{\rm {max}}]$ ), $[\sigma _{\rm {Bragg}}]$ is zero. In polycrystalline materials, the cross section rises steeply above the Bragg cut-off and oscillates with increasing energy as more reflections come into play. At still higher energies, $[\sigma _{\rm {Bragg}}]$ decreases to zero.

In single-crystalline material above the Bragg cut-off, $[\sigma _{\rm {Bragg}}]$ is characterized by a discrete spectrum of peaks whose heights and widths depend on the beam collimation, energy resolution, and the perfection and orientation of the crystal. Hence a monocrystalline filter has to be tuned by careful orientation.

The resulting attenuation cross section for beryllium is shown in Fig. 4.4.2.7 . Cooled polycrystalline beryllium is frequently used as a filter for neutrons with energies less than 5 meV, since there is an increase of nearly two orders of magnitude in the attenuation cross section for higher energies. BeO, with a Bragg cut-off at approximately 4 meV, is also commonly used.

Figure 4.4.2.7| top | pdf |

Total cross section for beryllium in the energy range where it can be used as a filter for neutrons with energy below 5 meV (Freund, 1983 ).

Pyrolytic graphite, being a layered material with good crystalline properties along the c direction but random orientation perpendicular to it, lies somewhere between a polycrystal and a single crystal as far as its attenuation cross section is concerned. The energy-dependent cross section for a neutron beam incident along the c axis of a pyrolytic graphite filter is shown in Fig. 4.4.2.8 , where the attenuation peaks due to the 00ξ reflections can be seen. Pyrolytic graphite serves as an efficient second- or third-order filter (Shapiro & Chesser, 1972 ) and can be `tuned' by slight misorientation away from the c axis.

Figure 4.4.2.8| top | pdf |

Energy-dependent cross section for a neutron beam incident along the c axis of a pyrolytic graphite filter. The attenuation peaks due to the 00ξ reflections can be seen.

Further examples of typical filter materials (e.g. silicon, lead, bismuth, sapphire) can be found in the paper by Freund (1983 ).

Resonant absorption filters show a large increase in their attenuation cross sections at the resonant energy and are therefore used as selective filters for that energy. A list of typical filter materials and their resonance energies is given in Table 4.4.2.3 .

Table 4.4.2.3| top | pdf |
Characteristics of some typical elements and isotopes used as neutron filters

Element or isotope	Resonance (eV)	σ_s (resonance) (barns)	λ (Å)	σ_s(λ) (barns)	$[\displaystyle{\sigma _{s}( \lambda /2) }\over {\sigma _{s}( \lambda) }]$
In	1.45	30000	0.48	94	319
Rh	1.27	4500	0.51	76	59.2
Hf	1.10	5000	0.55	58	86.2
²⁴⁰Pu	1.06	115000	9.55	145	793
Ir	0.66	4950	0.70	183	27.0
²²⁹Th	0.61	6200	0.73	< 100	$[\gt]$ 62.0
Er	0.58	1500	0.75	127	11.8
Er	0.46	2300	0.84	125	18.4
Eu	0.46	10100	0.84	1050	9.6
²³¹Pa	0.39	4900	0.92	116	42.2
²³⁹Pu	0.29	5200	1.06	700	7.4

1 barn = 10⁻²⁸ m².

4.4.2.6. Polarizers

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Methods used to polarize a neutron beam are many and varied, and the choice of the best technique depends on the instrument and the experiment to be performed. The main parameter that has to be considered when describing the effectiveness of a given polarizer is the polarizing efficiency, defined as $[ P=(N_{+}-N_{-})/(N_{+}+N_{-}), \eqno (4.4.2.8)]$ where $[N_{+}]$ and $[N_{-}]$ are the numbers of neutrons with spin parallel (+) or antiparallel (−) to the guide field in the outgoing beam. The second important factor, the transmission of the wanted spin state, depends on various factors, such as acceptance angles, reflection, and absorption.

4.4.2.6.1. Single-crystal polarizers

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The principle by which ferromagnetic single crystals are used to polarize and monochromate a neutron beam simultaneously is shown in Fig. 4.4.2.9 . A field B, applied perpendicular to the scattering vector $[\boldkappa]$ , saturates the atomic moments $[{\bf M}_{\nu}]$ along the field direction. The cross section for Bragg reflection in this geometry is $[({\rm d}\sigma /{\rm d}\Omega)=F_{N}({\boldkappa})^{2}+2F_{N}({\boldkappa})F_{M}({\boldkappa})({\bf {}P}\cdot {\boldmu})+F_{M}({\boldkappa}){}^{2}, \eqno (4.4.2.9)]$ where $[F_{N}({\boldkappa})]$ is the nuclear structure factor and $[F_{M}({\boldkappa}) = [({\gamma }/{2})r_{0}]\sum _{\nu }M_{\nu }\,f(hkl)\exp [2\pi (hx+ky+lz)]]$ is the magnetic structure factor, with f(hkl) the magnetic form factor of the magnetic atom at the position (x, y, z) in the unit cell. The vector P describes the polarization of the incoming neutron with respect to B; P =1 for + spins and P = −1 for − spins and $[{\boldmu}]$ is a unit vector in the direction of the atomic magnetic moments. Hence, for neutrons polarized parallel to B $[({\bf P}\cdot {\boldmu}=1)]$ , the diffracted intensity is proportional to $[[F_{N}({\boldkappa})+F_{M}({\boldkappa})]{}^{2}]$ , while, for neutrons polarized antiparallel to B $[({\bf P}\cdot {\boldmu}=-1)]$ , the diffracted intensity is proportional to $[[F_{N}({\boldkappa})-F_{M}({\boldkappa})]{}^{2}]$ . The polarizing efficiency of the diffracted beam is then $[ P=\pm 2F_{N}({\boldkappa})F_{M}({\boldkappa})/[F_{N}({\boldkappa})^{2}+F_{M}({\boldkappa}){}^{2}], \eqno (4.4.2.10)]$ which can be either positive or negative and has a maximum value for $[|F_{N}({\boldkappa})|= |F_{M}({\boldkappa})|]$ . Thus, a good single-crystal polarizer, in addition to possessing a crystallographic structure in which $[F_{N}]$ and $[F_{M}]$ are matched, must be ferromagnetic at room temperature and should contain atoms with large magnetic moments. Furthermore, large single crystals with `controllable' mosaic should be available. Finally, the structure factor for the required reflection should be high, while those for higher-order reflections should be low.

Figure 4.4.2.9| top | pdf |

Geometry of a polarizing monochromator showing the lattice planes (hkl) with |F_N| = |F_M|, the direction of P and $[\boldmu]$ , the expected spin direction and intensity.

None of the three naturally occurring ferromagnetic elements (iron, cobalt, nickel) makes efficient single-crystal polarizers. Cobalt is strongly absorbing and the nuclear scattering lengths of iron and nickel are too large to be balanced by their weak magnetic moments. An exception is ⁵⁷Fe, which has a rather low nuclear scattering length, and structure-factor matching can be achieved by mixing ⁵⁷Fe with Fe and 3% Si (Reed, Bolling & Harmon, 1973 ).

In general, in order to facilitate structure-factor matching, alloys rather than elements are used. The characteristics of some alloys used as polarizing monochromators are presented in Table 4.4.2.4 . At short wavelengths, the 200 reflection of Co_0.92Fe_0.08 is used to give a positively polarized beam [ $[F_{N}({\boldkappa})]$ and $[F_{M}({\boldkappa})]$ both positive], but the absorption due to cobalt is high. At longer wavelengths, the 111 reflection of the Heusler alloy Cu₂MnAl (Delapalme, Schweizer, Couderchon & Perrier de la Bathie, 1971 ; Freund, Pynn, Stirling & Zeyen, 1983 ) is commonly used, since it has a higher reflectivity and a larger d spacing than Co_0.92Fe_0.08. Since for the 111 reflection $[F_{N}\approx -F_{M}]$ , the diffracted beam is negatively polarized. Unfortunately, the structure factor of the 222 reflection is higher than that of the 111 reflection, leading to significant higher-order contamination of the beam.

Table 4.4.2.4| top | pdf |
Properties of polarizing crystal monochromators (Williams, 1988 )

	Co_0.92Fe_0.08	Cu₂MnAl	Fe₃Si	⁵⁷Fe:Fe	HoFe₂
Matched reflection $[\|F_{{N}}\|\sim \|F_{\rm {M}}\|]$	200	111	111	110	620
d spacing (Å)	1.76	3.43	3.27	2.03	1.16
Take-off angle $[2\theta _{{B}}]$ at 1 Å (°)	33.1	16.7	17.6	28.6	50.9
Cut-off wavelength, λ_max (Å)	3.5	6.9	6.5	4.1	2.3

Other alloys that have been proposed as neutron polarizers are Fe_3−xMn_xSi, ⁷Li_0.5Fe_2.5O₄ (Bednarski, Dobrzynski & Steinsvoll, 1980 ), Fe₃Si (Hines et al., 1976 ), Fe₃Al (Pickart & Nathans, 1961 ), and HoFe₂ (Freund & Forsyth, 1979 ).

4.4.2.6.2. Polarizing mirrors

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For a ferromagnetic material, the neutron refractive index is given by $[n_{\pm }^{2}=1-\lambda ^{2}N(b_{\rm {coh}}\pm p)/\pi, \eqno (4.4.2.11)]$ where the magnetic scattering length, p, is defined by $[ p=2\mu (B-H)m\pi /h^{2}N. \eqno (4.4.2.12)]$ Here, m and μ are the neutron mass and magnetic moment, B is the magnetic induction in an applied field H, and h is Planck's constant.

The − and + signs refer, respectively, to neutrons whose moments are aligned parallel and antiparallel to B. The refractive index depends on the orientation of the neutron spin with respect to the film magnetization, thus giving rise to two critical angles of total reflection, γ₋ and γ₊. Thus, reflection in an angular range between these two critical angles gives rise to polarized beams in reflection and in transmission. The polarization efficiency, P, is defined in terms of the reflectivity $[r_{+}]$ and $[r_{-}]$ of the two spin states, $[ P=(r_{+}-r_{-})/(r_{+}+r_{-}). \eqno (4.4.2.13)]$ The first polarizers using this principle were simple cobalt mirrors (Hughes & Burgy, 1950 ), while Schaerpf (1975 ) used FeCo sheets to build a polarizing guide. It is more common these days to use thin films of ferromagnetic material deposited onto a substrate of low surface roughness (e.g. float glass or polished silicon). In this case, the reflection from the substrate can be eliminated by including an antireflecting layer made from, for example, Gd–Ti alloys (Drabkin et al., 1976 ). The major limitation of these polarizers is that grazing-incidence angles must be used and the angular range of polarization is small. This limitation can be partially overcome by using multilayers, as described above, in which one of the layer materials is ferromagnetic. In this case, the refractive index of the ferromagnetic material is matched for one spin state to that of the non-magnetic material, so that reflection does not occur. A polarizing supermirror made in this way has an extended angular range of polarization, as indicated in Fig. 4.4.2.10 . It should be noted that modern deposition techniques allow the refractive index to be adjusted readily, so that matching is easily achieved. The scattering-length densities of some commonly used layer pairs are given in Table 4.4.2.5

Table 4.4.2.5| top | pdf |
Scattering-length densities for some typical materials used for polarizing multilayers

Magnetic layer	N(b + p) (10⁻⁶ Å⁻²)	N(b − p) (10⁻⁶ Å⁻²)	Nb (10⁻⁶ Å⁻²)	Nonmagnetic layer
Fe	13.04	3.08	3.64	Ge
			3.50	Ag
			3.02	W
			2.08	Si
			2.08	Al
Fe:Co (50:50)	10.98	−0.52	−0.27	V
Fe:Co (50:50)	10.98	−0.52	−1.95	Ti
Ni	10.86	7.94
Fe:Co:V (49:49:2)	10.75	−0.63	−0.27	V
Fe:Co:V (49:49:2)	10.75	−0.63	−1.95	Ti
Fe:Co:V (50:48:2)	10.66	−0.64	−0.27	V
Fe:Co:V (50:48:2)	10.66	−0.64	−1.95	Ti
Fe:Ni (50:50)	10.53	6.65
Co	6.65	−2.00	−1.95	Ti
Fe:Co:V (52:38:10)	6.27	2.12	2.08	Si
Fe:Co:V (52:38:10)	6.27	2.12	2.08	Al

For the non-magnetic layer we have only listed the simple elements that give a close match to the N(b − p) value of the corresponding magnetic layer. In practice excellent matching can be achieved by using alloys (e.g. Ti_xZr_y alloys allow Nb values between −1.95 and 3.03 × 10⁻⁶ Å⁻² to be selected) or reactive sputtering (e.g. TiN_x)

Figure 4.4.2.10| top | pdf |

Measured reflectivity curve of an FeCoV/TiZr polarizing supermirror with an extended angular range of polarization of three times that of γ_c(Ni) for neutrons without spin flip, ↑↑, and with spin flip, ↑↓.

Polarizing multilayers are also used in monochromators and broad-band devices. Depending on the application, various layer pairs have been used: Co/Ti, Fe/Ag, Fe/Si, Fe/Ge, Fe/W, FeCoV/TiN, FeCoV/TiZr, ⁶³Ni_0.66⁵⁴Fe_0.34/V and the range of fields used to achieve saturation varies from about 100 to 500 Gs.

Polarizing mirrors can be used in reflection or transmission with polarization efficiencies reaching 97%, although, owing to the low incidence angles, their use is generally restricted to wavelengths above 2 Å.

Various devices have been constructed that use mirror polarizers, including simple reflecting mirrors, V-shaped transmission polarizers (Majkrzak, Nunez, Copley, Ankner & Greene, 1992 ), cavity polarizers (Mezei, 1988 ), and benders (Hayter, Penfold & Williams, 1978 ; Schaerpf, 1989 ). Perhaps the best known device is the polarizing bender developed by Schärpf. The device consists of 0.2 mm thick glass blades coated on both sides with a Co/Ti supermirror on top of an antireflecting Gd/Ti coating designed to reduce the scattering of the unwanted spin state from the substrate to a very low Q value. The device is quite compact (typically 30 cm long for a beam cross section up to 6 × 5 cm) and transmits over 40% of an unpolarized beam with the collimation from a nickel-coated guide for wavelengths above 4.5 Å. Polarization efficiencies of over 96% can be achieved with these benders.

4.4.2.6.3. Polarizing filters

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Polarizing filters operate by selectively removing one of the neutron spin states from an incident beam, allowing the other spin state to be transmitted with only moderate attenuation. The spin selection is obtained by preferential absorption or scattering, so the polarizing efficiency usually increases with the thickness of the filter, whereas the transmission decreases. A compromise must therefore be made between polarization, P, and transmission, T. The `quality factor' often used is $[P\sqrt {T}]$ (Tasset & Resouche, 1995 ).

The total cross sections for a generalized filter may be written as $[ \sigma _{\pm }=\sigma _{0}\pm \sigma _{p}, \eqno (4.4.2.14)]$ where $[\sigma _{0}]$ is a spin-independent cross section and σ_p = (σ₊ + σ₋)/2 is the polarization cross section. It can be shown (Williams, 1988 ) that the ratio $[\sigma _{p}/\sigma _{0}]$ must be $[\ge]$ 0.65 to achieve |P| $[\gt]$ 0.95 and T $[\gt]$ 0.2.

Magnetized iron was the first polarizing filter to be used (Alvarez & Bloch, 1940 ). The method relies on the spin-dependent Bragg scattering from a magnetized polycrystalline block, for which $[\sigma _{p}]$ approaches 10 barns near the Fe cut-off at 4 Å (Steinberger & Wick, 1949 ). Thus, for wavelengths in the range 3.6 to 4 Å, the ratio $[\sigma _{p}/\sigma _{0}\simeq 0.59,]$ resulting in a theoretical polarizing efficiency of 0.8 for a transmittance of $[\sim 0.3]$ . In practice, however, since iron cannot be fully saturated, depolarization occurs, and values of $[P\simeq 0.5]$ with $[T\sim 0.25]$ are more typical.

Resonance absorption polarization filters rely on the spin dependence of the absorption cross section of polarized nuclei at their nuclear resonance energy and can produce efficient polarization over a wide energy range. The nuclear polarization is normally achieved by cooling in a magnetic field, and filters based on ¹⁴⁹Sm (E_r = 0.097 eV) (Freeman & Williams, 1978 ) and ¹⁵¹Eu (E_r = 0.32 and 0.46 eV) have been successfully tested. The ¹⁴⁹Sm filter has a polarizing efficiency close to 1 within a small wavelength range (0.85 to 1.1 Å), while the transmittance is about 0.15. Furthermore, since the filter must be operated at temperatures of the order of 15 mK, it is very sensitive to heating by γ-rays.

Broad-band polarizing filters, based on spin-dependent scattering or absorption, provide an interesting alternative to polarizing mirrors or monochromators, owing to the wider range of energy and scattering angle that can be accepted. The most promising such filter is polarized ³He, which operates through the huge spin-dependent neutron capture cross section that is totally dominated by the resonance capture of neutrons with antiparallel spin. The polarization efficiency of an ³He neutron spin filter of length l can be written as $[ P_{n}(\lambda)=\tanh [{\cal O}(\lambda) P_{\rm {He}}], \eqno (4.4.2.15)]$ where $[P_{\rm {He}}]$ is the ³He polarization, and $[{\cal O}(\lambda)=[^{3}{\rm He}]l\sigma _{0}(\lambda)]$ is the dimensionless effective absorption coefficient, also called the opacity (Surkau et al., 1997 ). For gaseous ³He, the opacity can be written in more convenient units as $[ {\cal O}^{\prime }=p[{\rm bar}]\times l\,{[{\rm {}cm}}]\times \lambda[{\rm \AA }], \eqno (4.4.2.16)]$ where p is the ³He pressure (1 bar = 10⁵ Pa) and $[{\cal O}=]$ $[7.33\times 10^{-2}{\cal O}^{\prime }]$ . Similarly, the residual transmission of the spin filter is given by $[ T_{n}(\lambda)=\exp [-{\cal O}(\lambda)]\cosh [{\cal O}(\lambda) P_{\rm {He}}]. \eqno (4.4.2.17)]$ It can be seen that, even at low ³He polarization, full neutron polarization can be achieved in the limit of large absorption at the cost of the transmission.

³ He can be polarized either by spin exchange with optically pumped rubidium (Bouchiat, Carver & Varnum, 1960 ; Chupp, Coulter, Hwang, Smith & Welsh, 1996 ; Wagshul & Chupp, 1994 ) or by pumping of metastable ³He* atoms followed by metastable exchange collisions (Colegrove, Schearer & Walters, 1963 ). In the former method, the ³He gas is polarized at the required high pressure, whereas ³He* pumping takes place at a pressure of about 1 mbar, followed by a polarization conserving compression by a factor of nearly 10 000. Although the polarization time constant for Rb pumping is of the order of several hours compared with fractions of a second for ³He* pumping, the latter requires several `fills' of the filter cell to achieve the required pressure.

An alternative broad-band spin filter is the polarized proton filter, which utilizes the spin dependence of nuclear scattering. The spin-dependent cross section can be written as (Lushchikov, Taran & Shapiro, 1969 ) $[\sigma _{\pm }=\sigma _{1}+\sigma _{2}P_{\rm {H}}^{2}\mp \sigma _{3}P_{\rm {H}}, \eqno (4.4.2.18)]$ where $[\sigma _{1}]$ , $[\sigma _{2}]$ , and $[\sigma _{3}]$ are empirical constants. The viability of the method relies on achieving a high nuclear polarization $[P_{\rm {H}}]$ . A polarization P_H = 0.7 gives $[\sigma _{p}/\sigma _{0}\approx 0.56]$ in the cold-neutron region. Proton polarizations of the order of 0.8 are required for a useful filter (Schaerpf & Stuesser, 1989 ). Polarized proton filters can polarize very high energy neutrons even in the eV range.

4.4.2.6.4. Zeeman polarizer

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The reflection width of perfect silicon crystals for thermal neutrons and the Zeeman splitting (ΔE = 2μB) of a field of about 10 kGs are comparable and therefore can be used to polarize a neutron beam. For a monochromatic beam (energy $[E_{0}]$ ) in a strong magnetic field region, the result of the Zeeman splitting will be a separation into two polarized subbeams, one polarized along B with energy E₀ + μB, and the other polarized antiparallel to B with energy E₀ − μB. The two polarized beams can be selected by rocking a perfect crystal in the field region B (Forte & Zeyen, 1989 ).

4.4.2.7. Spin-orientation devices

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Polarization is the state of spin orientation of an assembly of particles in a target or beam. The beam polarization vector P is defined as the vector average of this spin state and is often described by the density matrix ρ = $[1\over2]$ (1 + σP). The polarization is then defined as P = Tr(ρσ). If the polarization vector is inclined to the field direction in a homogenous magnetic field, B, the polarization vector will precess with the classical Larmor frequency $[\omega _{{L}}=|\gamma |{B}]$ . This results in a precessing spin polarization. For most experiments, it is sufficient to consider the linear polarization vector in the direction of an applied magnetic field. If, however, the magnetic field direction changes along the path of the neutron, it is also possible that the direction of P will change. If the frequency, $[\Omega]$ , with which the magnetic field changes is such that $[\Omega={\rm d}({\bf {}B}/|{\bf {}B}|)/{\rm d} t\ll \omega _{{L}}, \eqno (4.4.2.19)]$ then the polarization vector follows the field rotation adiabatically. Alternatively, when $[\Omega\gg \omega _{{L}}]$ , the magnetic field changes so rapidly that P cannot follow, and the condition is known as non-adiabatic fast passage. All spin-orientation devices are based on these concepts.

4.4.2.7.1. Maintaining the direction of polarization

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A polarized beam will tend to become depolarized during passage through a region of zero field, since the field direction is ill defined over the beam cross section. Thus, in order to keep the polarization direction aligned along a defined quantization axis, special precautions must be taken.

The simplest way of maintaining the polarization of neutrons is to use a guide field to produce a well defined field B over the whole flight path of the beam. If the field changes direction, it has to fulfil the adiabatic condition $[\Omega\ll \omega _{L}]$ , i.e. the field changes must take place over a time interval that is long compared with the Larmor period. In this case, the polarization follows the field direction adiabatically with an angle of deviation $[\Delta\theta \le 2\arctan (\Omega/\omega _{L})]$ (Schärpf, 1980 ).

Alternatively, some instruments (e.g. zero-field spin-echo spectrometers and polarimeters) use polarized neutron beams in regions of zero field. The spin orientation remains constant in a zero-field region, but the passage of the neutron beam into and out of the zero-field region must be well controlled. In order to provide a well defined region of transition from a guide-field region to a zero-field region, a non-adiabatic fast passage through the windings of a rectangular input solenoid can be used, either with a toroidal closure of the outside field or with a μ-metal closure frame. The latter serves as a mirror for the coil ends, with the effect of producing the field homogeneity of a long coil but avoiding the field divergence at the end of the coil.

4.4.2.7.2. Rotation of the polarization direction

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The polarization direction can be changed by the adiabatic change of the guide-field direction so that the direction of the polarization follows it. Such a rotation is performed by a spin turner or spin rotator (Schärpf & Capellmann, 1993 ; Williams, 1988 ).

Alternatively, the direction of polarization can be rotated relative to the guide field by using the property of precession described above. If a polarized beam enters a region where the field is inclined to the polarization axis, then the polarization vector P will precess about the new field direction. The precession angle will depend on the magnitude of the field and the time spent in the field region. By adjustment of these two parameters together with the field direction, a defined, though wavelength-dependent, rotation of P can be achieved. A simple device uses the non-adiabatic fast passage through the windings of two rectangular solenoids, wound orthogonally one on top of the other. In this way, the direction of the precession field axis is determined by the ratio of the currents in the two coils, and the sizes of the fields determine the angle $[\varphi]$ of the precession. The orientation of the polarization vector can therefore be defined in any direction.

In order to produce a continuous rotation of the polarization, i.e. a well defined precession, as required in neutron spin-echo (NSE) applications, precession coils are used. In the simplest case, these are long solenoids where the change of the field integral over the cross section can be corrected by Fresnel coils (Mezei, 1972 ). More recently, Zeyen & Rem (1996 ) have developed and implemented optimal field-shape (OFS) coils. The field in these coils follows a cosine squared shape that results from the optimization of the line integral homogeneity. The OFS coils can be wound over a very small diameter, thereby reducing stray fields drastically.

4.4.2.7.3. Flipping of the polarization direction

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The term `flipping' was originally applied to the situation where the beam polarization direction is reversed with respect to a guide field, i.e. it describes a transition of the polarization direction from parallel to antiparallel to the guide field and vice versa. A device that produces this 180° rotation is called a π flipper. A π/2 flipper, as the name suggests, produces a 90° rotation and is normally used to initiate precession by turning the polarization at 90° to the guide field.

The most direct wavelength-independent way of producing such a transition is again a non-adiabatic fast passage from the region of one field direction to the region of the other field direction. This can be realized by a current sheet like the Dabbs foil (Dabbs, Roberts & Bernstein, 1955 ), a Kjeller eight (Abrahams, Steinsvoll, Bongaarts & De Lange, 1962 ) or a cryoflipper (Forsyth, 1979 ).

Alternatively, a spin flip can be produced using a precession coil, as described above, in which the polarization direction makes a precession of just π about a direction orthogonal to the guide field direction (Mezei, 1972 ). Normally, two orthogonally wound coils are used, where the second, correction, coil serves to compensate the guide field in the interior of the precession coil. Such a flipper is wavelength dependent and can be easily tuned by varying the currents in the coils.

Another group of flippers uses the non-adiabatic transition through a well defined region of zero field. Examples of this type of flipper are the two-coil flipper of Drabkin, Zabidarov, Kasman & Okorokov (1969 ) and the line-shape flipper of Korneev & Kudriashov (1981 ).

Historically, the first flippers used were radio-frequency coils set in a homogeneous magnetic field. These devices are wavelength dependent, but may be rendered wavelength independent by replacing the homogeneous magnetic field with a gradient field (Egorov, Lobashov, Nazarento, Porsev & Serebrov, 1974 ).

In some devices, the flipping action can be combined with another selection function. The wavelength-dependent magnetic wiggler flipper proposed by Agamalyan, Drabkin & Sbitnev (1988 ) in combination with a polarizer can be used as a polarizing monochromator (Majkrzak & Shirane, 1982 ). Badurek & Rauch (1978 ) have used flippers as choppers to pulse a polarized beam.

In neutron resonance spin echo (NRSE) (Gähler & Golub, 1987 ), the precession coil of the conventional spin-echo configuration is replaced by two resonance spin flippers separated by a large zero-field region. The radio-frequency field of amplitude $[B_{1}]$ is arranged orthogonal to the DC field, $[B_{0}]$ , with a frequency $[\omega = \omega _{L}]$ , and an amplitude defined by the relation $[\omega _{1}\tau]$ = π, where τ is the flight time in the flipper coil and $[\omega _{1} = \gamma {B}_{1}]$ . In this configuration, the neutron spin precesses through an angle π about the resonance field in each coil and leaves the coil with a phase angle $[\varphi]$ . The total phase angle after passing through both coils, $[\varphi =2\omega L/v]$ , depends on the velocity v of the neutron and the separation L between the two coils. Thus, compared with conventional NSE, where the phase angle comes from the precession of the neutron spin in a strong magnetic field compared with a static flipper field, in NRSE the neutron spin does not precess, but the flipper field rotates. Effectively, the NRSE phase angle $[\varphi]$ is a factor of two larger than the NSE phase angle for the same DC field $[{B}_{0}]$ . Furthermore, the resolution is determined by the precision of the RF frequencies and the zero-field flight path L rather than the homogeneity of the line integral of the field in the NSE precession coil.

4.4.2.8. Mechanical choppers and selectors

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Thermal neutrons have relatively low velocities (a 4 Å neutron has a reciprocal velocity of approximately 1000 µs m⁻¹), so that mechanical selection devices and simple flight-time measurements can be used to make accurate neutron energy determinations.

Disc choppers rotating at speeds up to 20 000 revolutions per minute about an axis that is parallel to the neutron beam are used to produce a well defined pulse of neutrons. The discs are made from absorbing material (at least where the beam passes) and comprise one or more neutron-transparent apertures or slits. For polarized neutrons, these transparent slits should not be metallic, as the eddy currents in the metal moving in even a weak guide field will strongly depolarize the beam. The pulse frequency is determined by the number of apertures and the rotation frequency, while the duty cycle is given by the ratio of open time to closed time in one rotation. Two such choppers rotating in phase can be used to monochromate and pulse a beam simultaneously (Egelstaff, Cocking & Alexander, 1961 ). In practice, more than two choppers are generally used to avoid frame overlap of the incident and scattered beams. The time resolution of disc choppers (and hence the energy resolution of the instrument) is determined by the beam size, the aperture size and the rotation speed. For a realistic beam size, the rotation speed limits the resolution. Therefore, in modern instruments, it is normal to replace a single chopper with two counter-rotating choppers (Hautecler et al., 1985 ; Copley, 1991 ). The low duty cycle of a simple disc chopper can be improved by replacing the single slit with a series of slits either in a regular sequence (Fourier chopper) (Colwell, Miller & Whittemore, 1968 ; Hiismäki, 1997 ) or a pseudostatistical sequence (pseudostatistical chopper) (Hossfeld, Amadori & Scherm, 1970 ), with duty cycles of 50 and 30%, respectively.

The Fermi chopper is an alternative form of neutron chopper that simultaneously pulses and monochromates the incoming beam. It consists of a slit package, essentially a collimator, rotating about an axis that is perpendicular to the beam direction (Turchin, 1965 ). For optimum transmission at the required wavelength, the slits are usually curved to provide a straight collimator in the neutron frame of reference. The curvature also eliminates the `reverse burst', i.e. a pulse of neutrons that passes when the chopper has rotated by 180°.

A Fermi chopper with straight slits in combination with a monochromator assembly of wide horizontal divergence can be used to time focus a polychromatic beam, thus maintaining the energy resolution while improving the intensity (Blanc, 1983 ).

Velocity selectors are used when a continuous beam is required with coarse energy resolution. They exist in either multiple disc configurations or helical channels rotating about an axis parallel to the beam direction (Dash & Sommers, 1953 ). Modern helical channel selectors are made up of light-weight absorbing blades slotted into helical grooves on the rotation axis (Wagner, Friedrich & Wille, 1992 ). At higher energies where no suitable absorbing material is available, highly scattering polymers [poly(methyl methacrylate)] can be used for the blades, although in this case adequate shielding must be provided. The neutron wavelength is determined by the rotation speed, and resolutions, Δλ/λ, ranging from 5% to practically 100% (λ/2 filter) can be achieved. The resolution is fixed by the geometry of the device, but can be slightly improved by tilting the rotation axis or relaxed by rotating in the reverse direction for shorter wavelengths. Transmissions of up to 94% are typical.

4.4.3. Resolution functions

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R. Pynn^d and J. M. Rowe^e

In a Gedanken neutron scattering experiment, neutrons of wavevector $[{\bf k}_I]$ impinge on a sample and the wavevector, $[{\bf k}_F]$ , of the scattered neutrons is determined. A number of different types of spectrometer are used to achieve this goal (cf. Pynn, 1984 ). In each case, finite instrumental resolution is a result of uncertainties in the definition of $[{\bf k}_I]$ and $[{\bf k}_F]$ . Propagation directions for neutrons are generally defined by Soller collimators for which the transmission as a function of divergence angle generally has a triangular shape. Neutron monochromatization may be achieved either by Bragg reflection from a (usually) mosaic crystal or by a time-of-flight method. In the former case, the mosaic leads to a spread of [|k_I|] while, in the latter, pulse length and uncertainty in the lengths of flight paths (including sample size and detector thickness) produce a similar effect. Calculations of instrumental resolution are generally lengthy and lack of space prohibits their detailed presentation here. In the following paragraphs, the concepts involved are indicated and references to original articles are provided.

In resolution calculations for neutron spectrometers, it is usually assumed that the uncertainty of the neutron wavevector does not vary spatially across the neutron beam, although this reasoning may not apply to the case of small samples and compact spectrometers. To calculate the resolution of the spectrometer in the large-beam approximation, one writes the measured intensity I as $[I\propto\textstyle\int{\rm d}^3k_i\int {\rm d}^3k_f\, P_i({\bf k}_i)S({\bf k}_i\rightarrow{\bf k}_f)P_f({\bf k}_f), \eqno (4.4.3.1)]$ where $[P_i({\bf k}_i)]$ is the probability that a neutron of wavevector $[{\bf k}_i]$ is incident on the sample, $[P_f({\bf k}_f)]$ is the probability that a neutron of wavevector $[{\bf k}_f]$ is transmitted by the analyser system and $[S({\bf k}_i\rightarrow{\bf k}_f)]$ is the probability that the sample scatters a neutron from $[{\bf k}_i]$ to $[{\bf k}_f]$ . The fluctuation spectrum of the sample, $[S({\bf k}_i\rightarrow{\bf k}_f)]$ , does not depend separately on $[{\bf k}_i]$ and $[{\bf k}_f]$ but rather on the scattering vector Q and energy transfer $[\hbar\omega]$ defined by the conservation equations $[{\bf Q}={\bf k}_i-{\bf k}_f; \quad \hbar\omega={\hbar^2\over 2m}(k^2_i-k^2_f), \eqno (4.4.3.2)]$ where m is the neutron mass.

A number of methods of calculating the distribution functions $[P_i({\bf k}_i)]$ and $[P_f({\bf k}_f)]$ have been proposed. The method of independent distributions was used implicitly by Stedman (1968 ) and in more detail by Bjerrum Møller & Nielson (MN) (Nielsen & Bjerrum Møller, 1969 ; Bjerrum Møller & Nielsen, 1970 ) for three-axis spectrometers . Subsequently, the method has been extended to perfect-crystal monochromators (Pynn, Fujii & Shirane, 1983 ) and to time-of-flight spectrometers (Steinsvoll, 1973 ; Robinson, Pynn & Eckert, 1985 ). The method involves separating [P_i] and [P_f] into a product of independent distribution functions each of which can be convolved separately with the fluctuation spectrum S(Q, ω) [cf. equation (4.4.3.1)]. Extremely simple results are obtained for the widths of scans through a phonon dispersion surface for spectrometers where the energy of scattered neutrons is analysed (Nielson & Bjerrum Møller, 1969 ). For diffractometers, the width of a scan through a Bragg peak may also be obtained (Pynn et al., 1983 ), yielding a result equivalent to that given by Caglioti, Paoletti & Ricci (1960 ). In this case, however, the singular nature of the Bragg scattering process introduces a correlation between the distribution functions that contribute to [P_i] and [P_f] and the calculation is less transparent than it is for phonons.

A somewhat different approach, which does not explicitly separate the various contributions to the resolution, was proposed by Cooper & Nathans (CN) (Cooper & Nathans, 1967 , 1968 ; Cooper, 1968 ). Minor errors were corrected by several authors (Werner & Pynn, 1971 ; Chesser & Axe, 1973 ). The CN method calculates the instrumental resolution function R(Q − Q₀, ω − ω₀) as $[R(\Delta{\bf Q},\Delta\omega)=R_0\exp-{\textstyle{1\over2}} \,\textstyle\sum\limits_{\alpha,\beta}\,M_{\alpha \beta}\,X_\alpha\, X_\beta, \eqno (4.4.3.3)]$ where [X_1] , [X_2] , and [X_3] are the three components of ΔQ, X₄ = Δω, and Q₀ and ω₀ are obtained from (4.4.3.2) by replacing $[{\bf k}_i]$ and $[{\bf k}_f]$ by $[{\bf k}_I]$ and $[{\bf k}_F]$ , respectively. The matrix M is given in explicit form by several authors (Cooper & Nathans, 1967 , 1968 ; Cooper, 1968 ; Werner & Pynn, 1971 ; Chesser & Axe, 1973 ) and the normalization [R_0] has been discussed in detail by Dorner (1972 ). [A refutation (Tindle, 1984 ) of Dorner's work is incorrect.] Equation (4.4.3.3) implies that contours of constant transmission for the spectrometer [R(ΔQ, Δω) = constant] are ellipsoids in the four-dimensional Q–ω space. Optimum resolution (focusing) is achieved by a scan that causes the resolution function to intersect the feature of interest in S(Q, ω) (e.g. Bragg peak or phonon dispersion surface) for the minimum scan interval. The optimization of scans for a diffractometer has been considered by Werner (1971 ).

The MN and CN methods are equivalent. Using the MN formalism, it can be shown that $[{\bi M} = ({\bi A})^{-1}\quad{\rm with}\quad A_{\alpha\beta}=\textstyle\sum\limits_j \chi_{j\alpha}\chi _{j\beta}, \eqno (4.4.3.4)]$ where the $[\chi_{j\alpha}]$ are the components of the standard deviations of independent distributions (labelled by index j) defined by Bjerrum Møller & Nielsen (1970 ). In the limit $[Q\rightarrow]$ 0, the matrices M and A are of rank three and other methods must be used to calculate the resolution ellipsoid (Mitchell, Cowley & Higgins, 1984 ). Nevertheless, the MN method may be used even in this case to calculate widths of scans.

To obtain the resolution function of a diffractometer (in which there is no analysis of scattered neutron energy) from the CN form for M, it is sufficient to set to zero those contributions that arise from the mosaic of the analyser crystal. For elastic Bragg scattering, the problem is further simplified because [X_4] [cf. equation (4.4.3.3)] is zero. The spectrometer resolution function is then an ellipsoid in Q space.

For the measurement of integrated intensities (of Bragg peaks for example), the normalization [R_0] in (4.4.3.3) is required in order to obtain the Lorentz factor. The latter has been calculated for an arbitrary scan of a three-axis spectrometer (Pynn, 1975 ) and the results may be modified for a diffractometer as described in the preceding paragraph.

4.4.4. Scattering lengths for neutrons

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V. F. Sears^g

The use of neutron diffraction for crystal-structure determinations requires a knowledge of the scattering lengths and the corresponding scattering and absorption cross sections of the elements and, in some cases, of individual isotopes. This information is needed to calculate unit-cell structure factors and to correct for effects such as absorption, self-shielding, extinction, thermal diffuse scattering, and detector efficiency (Bacon, 1975 ; Sears, 1989 ). Table 4.4.4.1 lists the best values of the neutron scattering lengths and cross sections that are available at the time of writing (January 1995). We begin by summarizing the basic relationships between the scattering lengths and cross sections of the elements and their isotopes that have been used in the compilation of this table. More background information can be found in, for example, the book by Sears (1989 ).

Table 4.4.4.1| top | pdf | interactive version
Bound scattering lengths , b, in fm and cross sections , σ, in barns (1 barn = 100 fm²) of the elements and their isotopes

Z: atomic number; A: mass number; I(π): spin (parity) of the nuclear ground state; c: % natural abundance (for radioisotopes, the half-life is given instead in annums); b_c: bound coherent scattering length; b_i: bound incoherent scattering length; σ_c: bound coherent scattering cross section; σ_i: bound incoherent scattering cross section; σ_s: total bound scattering cross section; σ_a: absorption cross section for 2200 m s⁻¹ neutrons (E = 25.30 meV, k = 3.494 Å⁻¹, λ = 1.798 Å); i = $[{\sqrt-1}]$ .

Element			$[I(\pi)]$				$[\sigma_c]$	$[\sigma_i]$	$[\sigma_s]$	$[\sigma_a]$
H	1				−3.7390(11)		1.7568(10)	80.26(6)	82.02(6)	0.3326(7)
		1	1/2(+)	99.985	−3.7406(11)	25.274(9)	1.7583(10)	80.27(6)	82.03(6)	0.3326(7)
		2	1(+)	0.015	6.671(4)	4.04(3)	5.592(7)	2.05(3)	7.64(3)	0.000519(7)
		3	1/2(+)	(12.32a)	4.792(27)	−1.04(17)	2.89(3)	0.14(4)	3.03(5)	0

He	2				3.26(3)		1.34(2)	0.00	1.34(2)	0.00747(1)
		3	1/2(+)	0.00014	5.74(7)	−2.5(6)	4.42(10)	1.6(4)	6.0(4)	5333.(7.)
					−1.483(2)	+2.568(3)
		4	0(+)	99.99986	3.26(3)	0	1.34(2)	0	1.34(2)	0

Li	3				−1.90(2)		0.454(14)	0.92(3)	1.37(3)	70.5(3)
		6	1(+)	7.5	2.00(11)	−1.89(5)	0.51(5)	0.46(2)	0.97(7)	940.(4.)
					−0.261(1)	0.257(11)
		7	3/2(−)	92.5	−2.22(2)	−2.49(5)	0.619(11)	0.78(3)	1.40(3)	0.0454(3)

Be	4	9	3/2(−)	100	7.79(1)	0.12(3)	7.63(2)	0.0018(9)	7.63(2)	0.0076(8)

B	5				5.30(4)		3.54(5)	1.70(12)	5.24(11)	767.(8.)
					0.213(2)
		10	3(+)	20.0	−0.1(3)	−4.7(3)	0.144(8)	3.0(4)	3.1(4)	3835.(9.)
					1.066(3)	1.231(3)
		11	3/2(−)	80.0	6.65(4)	−1.3(2)	5.56(7)	0.22(6)	5.78(9)	0.0055(33)

C	6				6.6460(12)		5.550(2)	0.001(4)	5.551(3)	0.00350(7)
		12	0(+)	98.90	6.6511(16)	0	5.559(3)	0	5.559(3)	0.00353(7)
		13	1/2(−)	1.10	6.19(9)	−0.52(9)	4.81(14)	0.034(12)	4.84(14)	0.00137(4)

N	7				9.36(2)		11.01(5)	0.50(12)	11.51(11)	1.90(3)
		14	1(+)	99.63	9.37(2)	2.0(2)	11.03(5)	0.5(1)	11.53(11)	1.91(3)
		15	1/2(−)	0.37	6.44(3)	−0.02(2)	5.21(5)	0.00005(10)	5.21(5)	0.000024(8)

O	8				5.803(4)		4.232(6)	0.000(8)	4.232(6)	0.00019(2)
		16	0(+)	99.762	5.803(4)	0	4.232(6)	0	4.232(6)	0.00010(2)
		17	5/2(+)	0.038	5.78(12)	0.18(6)	4.20(22)	0.004(3)	4.20(22)	0.236(10)
		18	0(+)	0.200	5.84(7)	0	4.29(10)	0	4.29(10)	0.00016(1)

F	9	19	1/2(+)	100	5.654(10)	−0.082(9)	4.017(17)	0.0008(2)	4.018(14)	0.0096(5)

Ne	10				4.566(6)		2.620(7)	0.008(9)	2.628(6)	0.039(4)
		20	0(+)	90.51	4.631(6)	0	2.695(7)	0	2.695(7)	0.036(4)
		21	3/2(+)	0.27	6.66(19)	$[\pm]$ 0.6(1)	5.6(3)	0.05(2)	5.7(3)	0.67(11)
		22	0(+)	9.22	3.87(1)	0	1.88(1)	0	1.88(1)	0.046(6)

Na	11	23	3/2(+)	100	3.63(2)	3.59(3)	1.66(2)	1.62(3)	3.28(4)	0.530(5)

Mg	12				5.375(4)		3.631(5)	0.08(6)	3.71(4)	0.063(3)
		24	0(+)	78.99	5.66(3)	0	4.03(4)	0	4.03(4)	0.050(5)
		25	5/2(+)	10.00	3.62(14)	1.48(10)	1.65(13)	0.28(4)	1.93(14)	0.19(3)
		26	0(+)	11.01	4.89(15)	0	3.00(18)	0	3.00(18)	0.0382(8)

Al	13	27	5/2(+)	100	3.449(5)	0.256(10)	1.495(4)	0.0082(7)	1.503(4)	0.231(3)

Si	14				4.1491(10)		2.1633(10)	0.004(8)	2.167(8)	0.171(3)
		28	0(+)	92.23	4.107(6)	0	2.120(6)	0	2.120(6)	0.177(3)
		29	1/2(+)	4.67	4.70(10)	0.09(9)	2.78(12)	0.001(2)	2.78(12)	0.101(14)
		30	0(+)	3.10	4.58(8)	0	2.64(9)	0	2.64(9)	0.107(2)

P	15	31	1/2(+)	100	5.13(1)	0.2(2)	3.307(13)	0.005(10)	3.312(16)	0.172(6)

S	16				2.847(1)		1.0186(7)	0.007(5)	1.026(5)	0.53(1)
		32	0(+)	95.02	2.804(2)	0	0.9880(14)	0	0.9880(14)	0.54(4)
		33	3/2(+)	0.75	4.74(19)	1.5(1.5)	2.8(2)	0.3(6)	3.1(6)	0.54(4)
		34	0(+)	4.21	3.48(3)	0	1.52(3)	0	1.52(3)	0.227(5)
		36	0(+)	0.02	3.(1.) E	0	1.1(8)	0	1.1(8)	0.15(3)

Cl	17				9.5770(8)		11.526(2)	5.3(5)	16.8(5)	33.5(3)
		35	3/2(+)	75.77	11.65(2)	6.1(4)	17.06(6)	4.7(6)	21.8(6)	44.1(4)
		37	3/2(+)	24.23	3.08(6)	0.1(1)	1.19(5)	0.001(3)	1.19(5)	0.433(6)

Ar	18				1.909(6)		0.458(3)	0.22(2)	0.683(4)	0.675(9)
		36	0(+)	0.337	24.90(7)	0	77.9(4)	0	77.9(4)	5.2(5)
		38	0(+)	0.063	3.5(3.5)	0	1.5(3.1)	0	1.5(3.1)	0.8(2)
		40	0(+)	99.600	1.830(6)	0	0.421(3)	0	0.421(3)	0.660(9)

K	19				3.67(2)		1.69(2)	0.27(11)	1.96(11)	2.1(1)
		39	3/2(+)	93.258	3.74(2)	1.4(3)	1.76(2)	0.25(11)	2.01(11)	2.1(1)
		40	4(−)	0.012	3.(1.) E		1.1(8)	0.5(5)	1.6(9)	35.(8.)
		41	3/2(+)	6.730	2.69(8)	1.5(1.5)	0.91(5)	0.3(6)	1.2(6)	1.46(3)

Ca	20				4.70(2)		2.78(2)	0.05(3)	2.83(2)	0.43(2)
		40	0(+)	96.941	4.80(2)	0	2.90(2)	0	2.90(2)	0.41(2)
		42	0(+)	0.647	3.36(10)	0	1.42(8)	0	1.42(8)	0.68(7)
		43	7/2(−)	0.135	−1.56(9)	0.31(4)	0.5(5) E		0.8(5)	6.2(6)
		44	0(+)	2.086	1.42(6)	0	0.25(2)	0	0.25(2)	0.88(5)
		46	0(+)	0.004	3.6(2)	0	1.6(2)	0	1.6(2)	0.74(7)
		48	0(+)	0.187	0.39(9)	0	0.019(9)	0	0.019(9)	1.09(14)

Sc	21	45	7/2(−)	100	12.29(11)	−6.0(3)	19.0(3)	4.5(5)	23.5(6)	27.5(2)

Ti	22				−3.370(13)		1.427(11)	2.63(3)	4.06(3)	6.43(6)
		46	0(+)	8.2	4.725(5)	0	2.80(6)	0	2.80(6)	0.59(18)
		47	5/2(−)	7.4	3.53(7)	−3.5(2)	1.57(6)	1.5(2)	3.1(2)	1.7(2)
		48	0(+)	73.8	−5.86(2)	0	4.32(3)	0	4.32(3)	8.30(9)
		49	7/2(−)	5.4	0.98(5)	5.1(2)	0.12(1)	3.3(3)	3.4(3)	2.2(3)
		50	0(+)	5.2	5.88(10)	0	4.34(15)	0	4.34(15)	0.179(3)

V	23				−0.3824(12)		0.01838(12)	5.08(6)	5.10(6)	5.08(2)
		50	6(+)	0.250	7.6(6)		7.3(1.1)	0.5(5) E	7.8(1.0)	60.(40.)
		51	7/2(−)	99.750	−0.402(2)	6.435(4)	0.0203(2)	5.07(6)	5.09(6)	4.9(1)

Cr	24				3.635(7)		1.660(6)	1.83(2)	3.49(2)	3.05(8)
		50	0(+)	4.35	−4.50(5)	0	2.54(6)	0	2.54(6)	15.8(2)
		52	0(+)	83.79	4.920(10)	0	3.042(12)	0	3.042(12)	0.76(6)
		53	3/2(−)	9.50	−4.20(3)	6.87(10)	2.22(3)	5.93(17)	8.15(17)	18.1(1.5)
		54	0(+)	2.36	4.55(10)	0	2.60(11)	0	2.60(11)	0.36(4)

Mn	25	55	5/2(−)	100	−3.750(18)	1.79(4)	1.77(2)	0.40(2)	2.17(3)	13.3(2)

Fe	26				9.45(2)		11.22(5)	0.40(11)	11.62(10)	2.56(3)
		54	0(+)	5.8	4.2(1)	0	2.2(1)	0	2.2(1)	2.25(18)
		56	0(+)	91.7	9.94(3)	0	12.42(7)	0	12.42(7)	2.59(14)
		57	1/2(−)	2.2	2.3(1)	0.66(6)		0.3(3) E	1.0(3)	2.48(30)
		58	0(+)	0.3	15.(7.)	0	28.(26.)	0	28.(26.)	1.28(5)

Co	27	59	7/2(−)	100	2.49(2)	−6.2(2)	0.779(13)	4.8(3)	5.6(3)	37.18(6)

Ni	28				10.3(1)		13.3(3)	5.2(4)	18.5(3)	4.49(16)
		58	0(+)	68.27	14.4(1)	0	26.1(4)	0	26.1(4)	4.6(3)
		60	0(+)	26.10	2.8(1)	0	0.99(7)	0	0.99(7)	2.9(2)
		61	3/2(−)	1.13	7.60(6)	$[\pm]$ 3.9(3)	7.26(11)	1.9(3)	9.2(3)	2.5(8)
		62	0(+)	3.59	−8.7(2)	0	9.5(4)	0	9.5(4)	14.5(3)
		64	0(+)	0.91	−0.37(7)	0	0.017(7)	0	0.017(7)	1.52(3)

Cu	29				7.718(4)		7.485(8)	0.55(3)	8.03(3)	3.78(2)
		63	3/2(−)	69.17	6.43(15)	0.22(2)	5.2(2)	0.006(1)	5.2(2)	4.50(2)
		65	3/2(−)	30.83	10.61(19)	1.79(10)	14.1(5)	0.40(4)	14.5(5)	2.17(3)

Zn	30				5.60(5)		4.054(7)	0.077(7)	4.131(10)	1.11(2)
		64	0(+)	48.6	5.22(4)	0	3.42(5)	0	3.42(5)	0.93(9)
		66	0(+)	27.9	5.97(5)	0	4.48(8)	0	4.48(8)	0.62(6)
		67	5/2(−)	4.1	7.56(8)	−1.50(7)	7.18(15)	0.28(3)	7.46(15)	6.8(8)
		68	0(+)	18.8	6.03(3)	0	4.57(5)	0	4.57(5)	1.1(1)
		70	0(+)	0.6	6.(1.) E	0	4.5(1.5)	0	4.5(1.5)	0.092(5)

Ga	31				7.288(2)		6.675(4)	0.16(3)	6.83(3)	2.75(3)
		69	3/2(−)	60.1	7.88(2)	−0.85(5)	7.80(4)	0.091(11)	7.89(4)	2.18(5)
		71	3/2(−)	39.9	6.40(3)	−0.82(4)	5.15(5)	0.084(8)	5.23(5)	3.61(10)

Ge	32				8.185(20)		8.42(4)	0.18(7)	8.60(6)	2.20(4)
		70	0(+)	20.5	10.0(1)	0	12.6(3)	0	12.6(3)	3.0(2)
		72	0(+)	27.4	8.51(10)	0	9.1(2)	0	9.1(2)	0.8(2)
		73	9/2(+)	7.8	5.02(4)	3.4(3)	3.17(5)	1.5(3)	4.7(3)	15.1(4)
		74	0(+)	36.5	7.58(10)	0	7.2(2)	0	7.2(2)	0.4(2)
		76	0(+)	7.8	8.21(1.5)	0	8.(3.)	0	8.(3.)	0.16(2)

As	33	75	3/2(−)	100	6.58(1)	−0.69(5)	5.44(2)	0.060(10)	5.50(2)	4.5(1)

Se	34				7.970(9)		7.98(2)	0.33(6)	8.30(6)	11.7(2)
		74	0(+)	0.9	0.8(3.0)	0	0.1(6)	0	0.1(6)	51.8(1.2)
		76	0(+)	9.0	12.2(1)	0	18.7(3)	0	18.7(3)	85.(7.)
		77	1/2(−)	7.6	8.25(8)	$[\pm]$ 0.6(1.6)	8.6(2)	0.05(26)	8.65(16)	42.(4.)
		78	0(+)	23.5	8.24(9)	0	8.5(2)	0	8.5(2)	0.43(2)
		80	0(+)	49.6	7.48(3)	0	7.03(6)	0	7.03(6)	0.61(5)
		82	0(+)	9.4	6.34(8)	0	5.05(13)	0	5.05(13)	0.044(3)

Br	35				6.795(15)		5.80(3)	0.10(9)	5.90(9)	6.9(2)
		79	3/2(−)	50.69	6.80(7)	−1.1(2)	5.81(12)	0.15(6)	5.96(13)	11.0(7)
		81	3/2(−)	49.31	6.79(7)	0.6(1)	5.79(12)	0.05(2)	5.84(12)	2.7(2)

Kr	36				7.81(2)		7.67(4)	0.01(14)	7.68(13)	25.(1.)
		78	0(+)	0.35		0		0		6.4(9)
		80	0(+)	2.25		0		0		11.8(5)
		82	0(+)	11.6		0		0		29.(20.)
		83	9/2(+)	11.5		185(30.)
		84	0(+)	57.0		0		0		0.113(15)
		86	0(+)	17.3	8.1(2)	0	8.2(4)	0	8.2(4)	0.003(2)

Rb	37				7.09(2)		6.32(4)	0.5(4)	6.8(4)	0.38(4)
		85	5/2(−)	72.17	7.03(10)	6.2(2)	0.5(5)	E	6.7(5)	0.48(1)
		87	3/2(−)	27.83	7.23(12)	6.6(2)	0.5(5)	E	7.1(5)	0.12(3)

Sr	38				7.02(2)		6.19(4)	0.06(11)	6.25(10)	1.28(6)
		84	0(+)	0.56	7.(1.) E	0	6.(2.)	0	6.(2.)	0.87(7)
		86	0(+)	9.86	5.67(5)	0	4.04(7)	0	4.04(7)	1.04(7)
		87	9/2(+)	7.00	7.40(7)	6.88(13)	0.5(5)	E	7.4(5)	16.(3.)
		88	0(+)	82.58	7.15(6)	0	6.42(11)	0	6.42(11)	0.058(4)

Y	39	89	1/2(−)	100	7.75(2)	1.1(3)	7.55(4)	0.15(8)	7.70(9)	1.28(2)

Zr	40				7.16(3)		6.44(5)	0.02(15)	6.46(14)	0.185(3)
		90	0(+)	51.45	6.4(1)	0	5.1(2)	0	5.1(2)	0.011(5)
		91	5/2(+)	11.32	8.7(1)	−1.08(15)	9.5(2)	0.15(4)	9.7(2)	1.17(10)
		92	0(+)	17.19	7.4(2)	0	6.9(4)	0	6.9(4)	0.22(6)
		94	0(+)	17.28	8.2(2)	0	8.4(4)	0	8.4(4)	0.0499(24)
		96	0(+)	2.76	5.5(1)	0	3.8(1)	0	3.8(1)	0.0229(10)

Nb	41	93	9/2(+)	100	7.054(3)	−0.139(10)	6.253(5)	0.0024(3)	6.255(5)	1.15(5)

Mo	42				6.715(2)		5.67(3)	0.04(5)	5.71(4)	2.48(4)
		92	0(+)	14.84	6.91(8)	0	6.00(14)	0	6.00(14)	0.019(2)
		94	0(+)	9.25	6.80(7)	0	5.81(12)	0	5.81(12)	0.015(2)
		95	5/2(+)	15.92	6.91(6)	6.00(10)	0.5(5)	E	6.5(5)	13.1(3)
		96	0(+)	16.68	6.20(6)	0	4.83(9)	0	4.83(9)	0.5(2)
		97	5/2(+)	9.55	7.24(8)	6.59(15)	0.5(5)	E	7.1(5)	2.5(2)
		98	0(+)	24.13	6.58(7)	0	5.44(12)	0	5.44(12)	0.127(6)
		100	0(+)	9.63	6.73(7)	0	5.69(12)	0	5.69(12)	0.4(2)

Tc	43
		99	9/2(+)	(2.13×10⁵a)	6.8(3)	5.8(5)	0.5(5)	E	6.3(7)	20.(1.)

Ru	44				7.03(3)		6.21(5)	0.4(1)	6.6(1)	2.56(13)
		96	0(+)	5.5	0	0	0.28(2)
		98	0(+)	1.9	0	0	< 8.0
		99	5/2(+)	12.7	6.9(1.0)
		100	0(+)	12.6	0	0	4.8(6)
		101	5/2(+)	17.0	3.3(9)
		102	0(+)	31.6	0	0	1.17(7)
		104	0(+)	18.7	0	0	0.31(2)

Rh	45	103	1/2(−)	100	5.88(4)	4.34(6)	0.3(3)	E	4.6(3)	144.8(7)

Pd	46				5.91(6)		4.39(9)	0.093(9)	4.48(9)	6.9(4)
		102	0(+)	1.02	7.7(7) E	0	7.5(1.4)	0	7.5(1.4)	3.4(3)
		104	0(+)	11.14	7.7(7) E	0	7.5(1.4)	0	7.5(1.4)	0.6(3)
		105	5/2(+)	22.33	5.5(3)	−2.6(1.6)	3.8(4)	0.8(1.0)	4.6(1.1)	20.(3.)
		106	0(+)	27.33	6.4(4)	0	5.1(6)	0	5.1(6)	0.304(29)
		108	0(+)	26.46	4.1(3)	0	2.1(3)	0	2.1(3)	8.5(5)
		110	0(+)	11.72	7.7(7)E	0	7.5(1.4)	0	7.5(1.4)	0.226(31)

Ag	47				5.922(7)		4.407(10)	0.58(3)	4.99(3)	63.3(4)
		107	1/2(−)	51.839	7.555(11)	1.00(13)	7.17(2)	0.13(3)	7.30(4)	37.6(1.2)
		109	1/2(−)	48.161	4.165(11)	−1.60(13)	2.18(1)	0.32(5)	2.50(5)	91.0(1.0)

Cd	48				4.87(5)		3.04(6)	3.46(13)	6.50(12)	2520.(50.)
					−0.70(1)
		106	0(+)	1.25	5.(2.) E	0	3.1(2.5)	0	3.1(2.5)	1.
		108	0(+)	0.89	5.4(1)	0	3.7(1)	0	3.7(1)	1.1(3)
		110	0(+)	12.51	5.9(1)	0	4.4(1)	0	4.4(1)	11.(1.)
		111	1/2(+)	12.81	6.5(1)	5.3(2)	0.3(3)	E	5.6(4)	24(3.)
		112	0(+)	24.13	6.4(1)	0	5.1(2)	0	5.1(2)	2.2(5)
		*113	1/2(+)	12.22	−8.0(2)	12.1(4)	0.3(3) E	12.4(5)		20600(400.)
					−5.73(11)
		114	0(+)	28.72	7.5(1)	0	7.1(2)	0	7.1(2)	0.34(2)
		116	0(+)	7.47	6.3(1)	0	5.0(2)	0	5.0(2)	0.075(13)

In	49			4.065(20)	2.08(2)	0.54(11)	2.62(11)	193.8(1.5)
					−0.0539(4)
		113	9/2(+)	43	5.39(6)	$[\pm]$ 0.017(1)	3.65(8)	0.000037(5)	3.65(8)	12.0(1.1)
		115	9/2(+)	957	4.01(2)	−2.1(2)	2.02(2)	0.55(11)	2.57(11)	202(2.)
					−0.0562(6)

Sn	50				6.225(2)		4.870(3)	0.022(5)	4.892(6)	0.626(9)
		112	0(+)	1.0	6.1(1.) E	0	4.5(1.5)	0	4.5(1.5)	1.01(11)
		114	0(+)	0.7	6.2(3)	0	4.8(5)	0	4.8(5)	0.114(30)
		115	1/2(+)	0.4	6.(1.) E	4.5(1.5)	0.3(3) E	4.8(1.5)	30(7.)
		116	0(+)	14.7	5.93(5)	0	4.42(7)	0	4.42(7)	0.14(3)
		117	1/2(+)	7.7	6.48(5)	5.28(8)	0.3(3) E	5.6(3)	2.3(5)
		118	0(+)	24.3	6.07(5)	0	4.63(8)	0	4.63(8)	0.22(5)
		119	1/2(+)	8.6	6.12(5)	4.71(8)	0.3(3) E	5.0(3)	2.2(5)
		120	0(+)	32.4	6.49(5)	0	5.29(8)	0	5.29(8)	0.14(3)
		122	0(+)	4.6	5.74(5)	0	4.14(7)	0	4.14(7)	0.18(2)
		124	0(+)	5.6	5.97(5)	0	4.48(8)	0	4.48(8)	0.133(5)

Sb	51				5.57(3)		3.90(4)	0.00(7)	3.90(6)	4.91(5)
		121	7/2(+)	57.3	5.71(6)	−0.05(15)	4.10(9)	0.0003(19)	4.10(9)	5.75(12)
		123	5/2(+)	42.7	5.38(7)	−0.10(15)	3.64(9)	0.001(4)	3.64(9)	3.8(2)

Te	52				5.80(3)		4.23(4)	0.09(1)	4.32(4)	4.05(5)
		120	0(+)	0.096	5.3(5)	0	3.5(7)	0	3.4(7)	2.3(3)
		122	0(+)	2.60	3.8(2)	0	1.8(2)	0	1.8(2)	3.4(5)
		123	1/2(+)	0.908	−0.05(25)	−2.04(9)	0.002(3)	0.52(5)	0.52(5)	418(30.)
					−0.116(8)
		124	0(+)	4.816	7.96(10)	0	8.0(2)	0	8.0(2)	6.8(1.3)
		125	1/2(+)	7.14	5.02(8)	−0.26(13)	3.17(10)	0.008(8)	3.18(10)	1.55(16)
		126	0(+)	18.95	5.56(7)	0	3.88(10)	0	3.88(10)	1.04(15)
		128	0(+)	31.69	5.89(7)	0	4.36(10)	0	4.36(10)	0.215(8)
		130	0(+)	33.80	6.02(7)	0	4.55(11)	0	4.55(11)	0.29(6)

I	53	127	5/2(+)	100	5.28(2)	1.58(15)	3.50(3)	0.31(6)	3.81(7)	6.15(6)

Xe	54				4.92(3)		3.04(4)			23.9(1.2)
		124	0(+)	0.10		0		0		165.(20.)
		126	0(+)	0.09		0		0		3.5(8)
		128	0(+)	1.91		0		0		< 8.
		129	1/2(+)	26.4						21.(5.)
		130	0(+)	4.1		0		0		< 26.
		131	3/2(+)	21.2						85.(10.)
		132	0(+)	26.9		0		0		0.45(6)
		134	0(+)	10.4		0		0		0.265(20)
		136	0(+)	8.9		0		0		0.26(2)

Cs	55	133	7/2(+)	100	5.42(2)	1.29(15)	3.69(3)	0.21(5)	3.90(6)	29.0(1.5)

Ba	56				5.07(3)		3.23(4)	0.15(11)	3.38(10)	1.1(1)
		130	0(+)	0.11	−3.6(6)	0	1.6(5)	0	1.6(5)	30(5.)
		132	0(+)	0.10	7.8(3)	0	7.6(6)	0	7.6(6)	7.0(8)
		134	0(+)	2.42	5.7(1)	0	4.08(14)	0	4.08(14)	2.0(1.6)
		135	3/2(+)	6.59	4.67(10)		2.74(12)	0.5(5) E	3.2(5)	5.8(9)
		136	0(+)	7.85	4.91(8)	0	3.03(10)	0	3.03(10)	0.68(17)
		137	3/2(+)	11.23	6.83(10)		5.86(17)	0.5(5) E	6.4(5)	3.6(2)
		138	0(+)	71.70	4.84(8)	0	2.94(10)	0	2.94(10)	0.27(14)

La	57				8.24(4)		8.53(8)	1.13(19)	9.66(17)	8.97(5)
		138	5(+)	0.09	8.(2.) E	8.(4.)	0.5(5) E	8.5(4.0)	57.(6.)
		139	7/2(+)	99.91	8.24(4)	3.0(2)	8.53(8)	1.13(15)	9.66(17)	8.93(4)

Ce	58				4.84(2)		2.94(2)	0.00(10)	2.94(10)	0.63(4)
		136	0(+)	0.19	5.80(9)	0	4.23(13)	0	4.23(13)	7.3(1.5)
		138	0(+)	0.25	6.70(9)	0	5.64(15)	0	5.64(15)	1.1(3)
		140	0(+)	88.48	4.84(9)	0	2.94(11)	0	2.94(11)	0.57(4)
		142	0(+)	11.08	4.75(9)	0	2.84(11)	0	2.84(11)	0.95(5)

Pr	59	141	5/2(+)	100	4.58(5)	−0.35(3)	2.64(6)	0.015(3)	2.66(6)	11.5(3)

Nd	60				7.69(5)		7.43(10)	9.2(8)	16.6(8)	50.5(1.2)
		142	0(+)	27.16	7.7(3)	0	7.5(6)	0	7.5(6)	18.7(7)
		143	7/2(−)	12.18	14.2(5) E	$[\pm]$ 21.1(6)	25.(7.)	55.(7.)	80.(2.)	334.(10.)
		144	0(+)	23.80	2.8(3)	0	1.0(2)	0	1.0(2)	3.6(3)
		145	7/2(−)	8.29	14.2(5)	E	25.(7.)	5.(5.) E	30.(9.)	42.(2.)
		146	0(+)	17.19	8.7(2)	0	9.5(4)	0	9.5(4)	1.4(1)
		148	0(+)	5.75	5.7(3)	0	4.1(4)	0	4.1(4)	2.5(2)
		150	0(+)	5.63	5.3(2)	0	3.5(3)	0	3.5(3)	1.2(2)

Pm	61
		147	7/2(+)	(2.62a)	12.6(4)	$[\pm]$ 3.2(2.5)	20.0(1.3)	1.3(2.0)	21.3(1.5)	168.4(3.5)

Sm	62				0.80(2)		0.422(9)	39.(3.)	39.(3.)	5922.(56.)
					−1.65(2)
		144	0(+)	3.1	−3.(4.) E	0	1.(3.)	0	1.(3.)	0.7(3)
		147	7/2(−)	15.1	14(3.)	$[\pm]$ 11.(7.)	25.(11.)	14.(19.)	39(16.)	57(3.)
		148	0(+)	11.3	−3.(4.) E	0	1.(3.)	0	1.(3.)	2.4(6)
		*149	7/2(−)	13.9	−19.2(1)	$[\pm]$ 31.4(6)	63.5(6)	137.(5.)	200.(5.)	42080.(400.)
					−11.7(1)	−10.3(1)
		150	0(+)	7.4	14(3.)	0	25(11.)	0	25(11.)	104(4.)
		152	0(+)	26.6	−5.0(6)	0	3.1(8)	0	3.1(8)	206.(6.)
		154	0(+)	22.6	9.3(1.0)	0	11.(2.)	0	11.(2.)	8.4(5)

Eu	63				7.22(2)		6.75(4)	2.5(4)	9.2(4)	4530.(40.)
					−1.26(1)
		*151	5/2(+)	47.8	6.13(14)	$[\pm]$ 4.5(4)	5.5(2)	3.1(4)	8.4(4)	9100(100.)
					−2.53(3)	−2.14(2)
		153	5/2(+)	52.2	8.22(12)	$[\pm]$ 3.2(9)	8.5(2)	1.3(7)	9.8(7)	312.(7.)

Gd	64				6.5(5)		29.3(8)	151.(2.)	180.(2.)	49700.(125.)
					−13.82(3)
		152	0(+)	0.2	10.(3.) E	0	13.(8.)	0	13.(8.)	735.(20.)
		154	0(+)	2.1	10.(3.) E	0	13.(8.)	0	13.(8.)	85.(12.)
		*155	3/2(−)	14.8	6.0(1)	$[\pm]$ 5.(5.) E	40.8(4.)	25.(6.)	66.(6.)	61100.(400.)
					−17.0(1)	−13.16(9)
		156	0(+)	20.6	6.3(4)	0	5.0(6)	0	5.0(6)	1.5(1.2)
		*157	3/2(−)	15.7	−1.14(2)	$[\pm]$ 5.(5.) E	650(4.)	394.(7.)	1044.(8.)	259000.(700.)
					−71.9(2)	−55.8(2)
		158	0(+)	24.8	9.(2.)	0	10.(5.)	0	10.(5.)	2.2(2)
		160	0(+)	21.8	9.15(5)	0	10.52(11)	0	10.52(11)	0.77(2)

Tb	65	159	3/2(+)	100	7.38(3)	−0.17(7)	6.84(6)	0.004(3)	6.84(6)	23.4(4)

Dy	66				16.9(2)		35.9(8)	54.4(1.2)	90.3(9)	994.(13.)
					−0.276(4)
		156	0(+)	0.06	6.1(5)	0	4.7(8)	0	4.7(8)	33.(3.)
		158	0(+)	0.10	6.(4.) E	0	5.(6.)	0	5.(6.)	43.(6.)
		160	0(+)	2.34	6.7(4)	0	5.6(7)	0	5.6(7)	56.(5.)
		161	5/2(+)	19.0	10.3(4)	$[\pm]$ 4.9(8)	13.3(1.0)	3.(1.)	16.(1.)	600.(25.)
		162	0(+)	25.5	−1.4(5)	0	0.25(18)	0	0.25(18)	194.(10.)
		163	5/2(−)	24.9	5.0(4)	1.3(3)	3.1(5)	0.21(10)	3.3(5)	124.(7.)
		164	0(+)	28.1	49.4(5)	0	307.(3.)	0	307.(3.)	2840.(40.)
					−0.79(1)

Ho	67	165	7/2(−)	100	8.01(8)	−1.70(8)	8.06(16)	0.36(3)	8.42(16)	64.7(1.2)

Er	68				7.79(2)		7.63(4)	1.1(3)	8.7(3)	159.(4.)
		162	0(+)	0.14	8.8(2)	0	9.7(4)	0	9.7(4)	19.(2.)
		164	0(+)	1.56	8.2(2)	0	8.4(4)	0	8.4(4)	13.(2.)
		166	0(+)	33.4	10.6(2)	0	14.1(5)	0	14.1(5)	19.6(1.5)
		167	7/2(+)	22.9	3.0(3)	1.0(3)	1.1(2)	0.13(8)	1.2(2)	659.(16.)
		168	0(+)	27.1	7.4(4)	0	6.8(7)	0	6.9(7)	2.74(8)
		170	0(+)	14.9	9.6(5)	0	11.6(1.2)	0	11.6(1.2)	5.8(3)

Tm	69	169	1/2(+)	100	7.07(3)	0.9(3)	6.28(5)	0.10(7)	6.38(9)	100.(2.)

Yb	70				12.43(3)		19.42(9)	4.0(2)	23.05(18)	34.8(8)
		168	0(+)	0.14	−4.07(2)	0	2.13(2)	0	2.13(2)	2230.(40.)
					−0.62(1)
		170	0(+)	3.06	6.77(10)	0	5.8(2)	0	5.8(2)	11.4(1.0)
		171	1/2(−)	143	9.66(10)	−5.59(17)	11.7(2)	3.9(2)	15.6(3)	48.6(2.5)
		172	0(+)	21.9	9.43(10)	0	11.2(2)	0	11.2(2)	0.8(4)
		173	5/2(−)	16.1	9.56(7)	−5.3(2)	11.5(2)	3.5(3)	15.0(4)	17.1(1.3)
		174	0(+)	31.8	19.3(1)	0	46.8(5)	0	46.8(5)	69.4(5.0)
		176	0(+)	12.7	8.72(10)	0	9.6(2)	0	9.6(2)	2.85(5)

Lu	71				7.21(3)		6.53(5)	0.7(4)	7.2(4)	74.(2.)
		175	7/2(+)	97.39	7.24(3)	$[\pm]$ 2.2(7)	6.59(5)	0.6(4)	7.2(4)	21.(3.)
		*176	7(−)	2.61	6.1(1)	$[\pm]$ 3.0(4)	4.7(2)	1.2(3)	5.9(4)	2065.(35.)
					−0.57(1)	+0.61(1)

Hf	72				7.77(14)		7.6(3)	2.6(5)	10.2(4)	104.1(0.5)
		174	0(+)	0.2	10.9(1.1)	0	15.(3.)	0	15.(3.)	561.(35.)
		176	0(+)	5.2	6.61(18)	0	5.5(3)	0	5.5(3)	23.5(3.1)
		177	7/2(−)	18.6	0.8(1.0) E	$[\pm]$ 0.9(1.3)	0.1(2)	0.1(3)	0.2(2)	373.(10.)
		178	0(+)	27.1	5.9(2)	0	4.4(3)	0	4.4(3)	84.(4.)
		179	9/2(+)	13.7	7.46(16)	$[\pm]$ 1.06(8)	7.0(3)	0.14(2)	7.1(3)	41.(3.)
		180	0(+)	35.2	13.2(3)	0	21.9(1.0)	0	21.9(1.0)	13.04(7)

Ta	73				6.91(7)		6.00(12)	0.01(17)	6.01(12)	20.6(5)
		*180	9(−)	0.012	7.(2.) E	6.2(3.5)	0.5(5)	E	7.(4.)	563.(60.)
		181	7/2(+)	99.988	6.91(7)	−0.29(3)	6.00(12)	0.011(2)	6.01(12)	20.5(5)

W	74				4.86(2)		2.97(2)	1.63(6)	4.60(6)	18.3(2)
		180	0(+)	0.1	5.(3.) E	0	3.(4.)	0	3.(4.)	30.(20.)
		182	0(+)	26.3	6.97(14)	0	6.10(7)	0	6.10(7)	20.7(5)
		183	1/2(−)	14.3	6.53(4)		5.36(7)	0.3(3) E	5.7(3)	10.1(3)
		184	0(+)	30.7	7.48(6)	0	7.03(11)	0	7.03(11)	1.7(1)
		186	0(+)	28.6	−0.72(4)	0	0.065(7)	0	0.065(7)	37.9(0.6)

Re	75				9.2(2)		10.6(5)	0.9(6)	11.5(3)	89.7(1.0)
		185	5/2(+)	37.40	9.0(3)	$[\pm]$ 2.0(1.8)	10.2(7)	0.5(9)	10.7(6)	112.(2.)
		187	5/2(+)	62.60	9.3(3)	$[\pm]$ 2.8(1.1)	10.9(7)	1.0(8)	11.9(4)	76.4(1.0)

Os	76				10.7(2)		14.4(5)	0.3(8)	14.7(6)	16.0(4)
		184	0(+)	0.02	10.(2.) E	0	13.(5.)	0	13.(5.)	3000.(150.)
		186	0(+)	1.58	11.6(1.7)	0	17.(5.)	0	17.(5.)	80.(13.)
		187	1/2(−)	1.6	10.(2.) E	13.(5.)	0.3(3)	E	13.(5.)	320(10.)
		188	0(+)	13.3	7.6(3)	0	7.3(6)	0	7.3(6)	4.7(5)
		189	3/2(−)	16.1	10.7(3)		14.4(8)	0.5(5) E	14.9(9)	25(4.)
		190	0(+)	26.4	11.0(3)	0	15.2(9)	0	15.2(8)	13.1(3)
		192	0(+)	41.0	11.5(4)	0	16.6(1.2)	0	16.6(1.2)	2.0(1)

Ir	77				10.6(3)		14.1(8)	0.(3.)	14.(3.)	425.3(2.4)
		191	3/2(+)	37.3						954.(10.)
		193	3/2(+)	62.7						111.(5.)

Pt	78				9.60(1)		11.58(2)	0.13(11)	11.71(11)	10.3(3)
		190	0(+)	0.01	9.0(1.0)	0	10.(2.)	0	10.(2.)	152.(4.)
		192	0(+)	0.79	9.9(5)	0	12.3(1.2)	0	12.3(1.2)	10.0(2.5)
		194	0(+)	32.9	10.55(8)	0	14.0(2)	0	14.0(2)	1.44(19)
		195	1/2(−)	33.8	8.83(9)	−1.00(17)	9.8(2)	0.13(4)	9.9(2)	27.5(1.2)
		196	0(+)	25.3	9.89(8)	0	12.3(2)	0	12.3(2)	0.72(4)
		198	0(+)	7.2	7.8(1)	0	7.7(2)	0	7.6(2)	3.66(19)

Au	79	197	3/2(+)	100	7.63(6)	−1.84(10)	7.32(12)	0.43(5)	7.75(13)	98.65(9)

Hg	80				12.692(15)		20.24(5)	6.6(1)	26.8(1)	372.3(4.0)
		196	0(+)	0.2	30.3(1.0)	0	115(8.)	0	115(8.)	3080(180.)
		198	0(+)	10.1		0		0		2.0(3)
		199	1/2(−)	17.0	16.9(4)	$[\pm]$ 15.5(8)	36.(2.)	30.(3.)	66.(2.)	2150.(48.)
		200	0(+)	23.1		0		0		< 60.
		201	3/2(−)	13.2			7.8(2.0)
		202	0(+)	29.6		0		0		4.89(5)
		204	0(+)	6.8		0		0		0.43(10)

Tl	81				8.776(5)		9.678(11)	0.21(15)	9.89(15)	3.43(6)
		203	1/2(+)	29.524	6.99(16)	1.06(14)	6.14(28)	0.14(4)	6.28(28)	11.4(2)
		205	1/2(+)	70.476	9.52(7)	−0.242(17)	11.39(17)	0.007(1)	11.40(17)	0.104(17)

Pb	82				9.405(3)		11.115(7)	0.0030(7)	11.118(7)	0.171(2)
		204	0(+)	1.4	9.90(10)	0	12.3(2)	0	12.3(2)	0.65(7)
		206	0(+)	24.1	9.22(5)	0	10.68(12)	0	10.68(12)	0.0300(8)
		207	1/2(−)	22.1	9.28(4)	0.14(6)	10.82(9)	0.002(2)	10.82(9)	0.699(10)
		208	0(+)	52.4	9.50(2)	0	11.34(5)	0	11.34(5)	0.00048(3)

Bi	83	209	9/2(−)	100	8.532(2)	0.259(15)	9.148(4)	0.0084(10)	9.156(4)	0.0338(7)

Po	84

At	85

Rn	86

Fr	87

Ra	88
		226	0(+)	(1.60×10³a)	10.0(1.0)	0	13.(3.)	0	13.(3.)	12.8(1.5)

Ac	89

Th	90	232	0(+)	100	10.31(3)	0	13.36(8)	0	13.36(8)	7.37(6)

Pa	91
		231	3/2(−)	(3.28×10⁴a)	9.1(3)	1	0.4(7)	0.1(3.3)	10.5(3.2)	200.6(2.3)

U	92				8.417(5)		8.903(11)	0.005(16)	8.908(11)	7.57(2)
		233	5/2(+)	(1.59×10⁵a)	10.1(2)	$[\pm]$ 1.(3.)	12.8(5)	0.1(6)	12.9(3)	574.7(1.0)
		234	0(+)	0.005	12.4(3)	0	19.3(9)	0	19.3(9)	100.1(1.3)
		235	7/2(−)	0.720	10.47(3)	$[\pm]$ 1.3(6)	13.78(11)	0.2(2)	14.0(2)	680.9(1.1)
		238	0(+)	99.275	8.402(5)	0	8.871(11)	0	8.871(11)	2.68(2)

Np	93
		237	5/2(+)	(2.14×10⁶a)	10.55(10)		14.0(3)	0.5(5)E	14.5(6)	175.9(2.9)

Pu	94
		238	0(+)	(87.74a)	14.1(5)	0	25.0(1.8)	0	25.0(1.8)	558.(7.)
		239	1/2(+)	(2.41×10⁴a)	7.7(1)	$[\pm]$ 1.3(1.9)	7.5(2)	0.2(6)	7.7(6)	1017.3(2.1)
		240	0(+)	(6.56×10³a)	3.5(1)	0	1.54(9)	0	1.54(9)	289.6(1.4)
		242	0(+)	(3.76×10⁵a)	8.1(1)	0	8.2(2)	0	8.2(2)	18.5(5)

Am	95
		243	5/2(−)	(7.37×10³a)	8.3(2)	$[\pm]$ 2.(7.)	8.7(4)	0.3(2.6)	9.0(2.6)	75.3(1.8)

Cm	96
		244	0(+)	(18.10a)	9.5(3)	0	11.3(7)	0	11.3(7)	16.2(1.2)
		246	0(+)	(4.7×10³a)	9.3(2)	0	10.9(5)	0	10.9(5)	1.36(17)
		248	0(+)	(3.5×10⁵a)	7.7(2)	0	7.5(4)	0	7.5(4)	3.00(26)

4.4.4.1. Scattering lengths

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The scattering of a neutron by a single bound nucleus is described within the Born approximation by the Fermi pseudopotential, $[V({\bf r}) =\left({2\pi\hbar^2\over m}\right) b\delta({\bf r}), \eqno (4.4.4.1)]$ in which r is the position of the neutron relative to the nucleus, m the neutron's mass, and b the bound scattering length. The neutron has spin s and the nucleus spin I so that, if $[I\neq0]$ , the Fermi pseudopotential and, hence, the bound scattering length will be spin dependent. Since s = 1/2, the most general rotationally invariant expression for b is $[b=b_c+{2b_i\over\sqrt{I(I+1)}}\,{\bf s}\cdot {\bf I}, \eqno (4.4.4.2)]$ in which the coefficients [b_c] and [b_i] are called the bound coherent and incoherent scattering lengths. If I = 0, then b_i = 0 by convention.

4.4.4.2. Scattering and absorption cross sections

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When a thermal neutron collides with a nucleus, it may be either scattered or absorbed. By absorption, we mean reactions such as $[(n,\gamma)]$ , (n, p), or (n, α), in which there is no neutron in the final state. The effect of absorption can be included by allowing the bound scattering length to be complex, $[b=b'-ib''. \eqno (4.4.4.3)]$ The total bound scattering cross section is then given by $[\sigma_s=4\pi\left\langle|b|{}^2\right\rangle, \eqno (4.4.4.4)]$ in which $[\langle\,\rangle]$ denotes a statistical average over the neutron and nuclear spins and the absorption cross section is given by $[\sigma_a={4\pi\over k}\langle b''\rangle, \eqno (4.4.4.5)]$ where k = 2π/λ is the wavevector of the incident neutron and λ is the wavelength.

If the neutron and/or the nucleus is unpolarized, then the total bound scattering cross section is of the form $[\sigma_s=\sigma_c+\sigma_i, \eqno (4.4.4.6)]$ in which $[\sigma_c]$ and $[\sigma_i]$ are called the bound coherent and incoherent scattering cross sections and are given by $[\sigma_c=4\pi|b_c|{}^2, \quad \sigma_i=4\pi|b_i|{}^2. \eqno (4.4.4.7)]$ Also, $[b_c=\langle b\rangle, \eqno (4.4.4.8)]$ so that the absorption cross section is given by $[\sigma_a={4\pi \over k}\, b''_c. \eqno (4.4.4.9)]$ The absorption cross section is therefore uniquely determined by the imaginary part of the bound coherent scattering length. It is only when the neutron and the nucleus are both polarized that the imaginary part of the bound incoherent scattering length contributes to the value of $[\sigma_a]$ .

For most nuclides, the scattering lengths and, hence, the scattering cross sections are constant in the thermal-neutron region, and the absorption cross sections are inversely proportional to k. Since k is proportional to the neutron velocity v, the absorption is said to obey a 1/v law. By convention, absorption cross sections are tabulated for a velocity v = 2200 m s⁻¹, which corresponds to a wavevector k = 3.494 Å⁻¹, a wavelength λ = 1.798 Å, or an energy E = 25.30 meV.

The only major deviations from the 1/v law are for a few heavy nuclides (specifically, ¹¹³Cd, ¹⁴⁹Sm, ¹⁵¹Eu, ¹⁵⁵Gd, ¹⁵⁷Gd, ¹⁷⁶Lu, and ¹⁸⁰Ta), which have an (n, γ) resonance at thermal-neutron energies. For these nuclides (which are indicated by the symbol * in Table 4.4.4.1 ), the scattering lengths and cross sections are strongly energy dependent. The scattering lengths of the resonant rare-earth nuclides have been tabulated as a function of energy by Lynn & Seeger (1990 ).

4.4.4.3. Isotope effects

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The coefficients [b_c] and [b_i] in (4.4.4.2) for the bound scattering length depend on the particular isotope under consideration, and this provides an additional source of incoherence in the scattering of neutrons by a mixture of isotopes. If $[\langle\,\rangle]$ is now taken to denote an average over both the spin and isotope distributions, then the expressions (4.4.4.8) for [b_c] , (4.4.4.4) for $[\sigma_s]$ , and (4.4.4.5) for $[\sigma_a]$ also apply to a mixture of isotopes. Hence, if [c_l] denotes the mole fraction of isotopes of type l, so that $[\textstyle\sum\limits_l\,c_l=1, \eqno (4.4.4.10)]$ then, for an isotopic mixture, $[b_c=\textstyle\sum\limits_l\, c_l\,b_{cl}, \eqno (4.4.4.11)]$ $[\sigma_s=\textstyle\sum\limits_l\, c_l\,\sigma_{sl}, \eqno (4.4.4.12)]$ and $[\sigma_a=\textstyle\sum\limits_l\,c_l\,\sigma_{al}. \eqno (4.4.4.13)]$ The bound coherent scattering cross section of the mixture is given, as before, by $[\sigma_c=4\pi|b_c|{}^2, \eqno (4.4.4.14)]$ while the bound incoherent scattering cross section is defined as $[\sigma_i=\sigma_s-\sigma_c. \eqno (4.4.4.15)]$ Hence, it follows that $[\sigma_i\equiv 4\pi|b_i|^2=\sigma_i({\rm spin})+\sigma_i({\rm isotope}), \eqno (4.4.4.16)]$ in which the contribution from spin incoherence is given by $[\sigma_i({\rm spin})=\textstyle\sum\limits_l\, c_l\sigma_{il} = 4\pi\sum\limits_l\, c_l|b_{il}|{}^2, \eqno (4.4.4.17)]$ and that from isotope incoherence is given by $[\sigma_i({\rm isotope})= 4\pi\textstyle\sum\limits_{l\lt l'} \,c_l c_{l'}|b_{cl}-b_{cl'}|{}^2. \eqno (4.4.4.18)]$ Note that for a mixture of isotopes only the magnitude of [b_i] is defined by (4.4.4.16), and its sign is arbitrary. However, for the individual isotopes, both the magnitude and sign (or complex phase) of [b_i] are defined in (4.4.4.2).

4.4.4.4. Correction for electromagnetic interactions

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The effective bound coherent scattering length that describes the interaction of a neutron with an atom includes additional contributions from electromagnetic interactions (Bacon, 1975 ; Sears, 1986a , 1996 ). For a neutral atom with atomic number Z, this quantity is of the form $[b_c(q)=b_c(0)-b_e[Z-f(q)], \eqno (4.4.4.19)]$ where q is the wavevector transfer in the collision, [b_c(0)] and [b_e] are constants, and f(q) is the atomic scattering factor (Section 6.1.1 ). The latter quantity is the Fourier transform of the electron number density and is normalized such that f(0) = Z.

The main contribution to [b_c(0)] is from the nuclear interaction between the neutron and the nucleus but there is also a small electrostatic contribution ( $[\le]$ 0.5%) arising from the neutron electric polarizability. The coefficient [b_e] is called the neutron–electron scattering length and has the value −1.32 (4) × 10⁻³ fm (Koester, Waschkowski & Meier, 1988 ). This quantity is due mainly to the Foldy interaction with a small additional contribution (∼10%) from the intrinsic charge distribution of the neutron.

The correction of the bound coherent scattering length for electromagnetic interactions requires a knowledge of the atomic scattering factor f(q). Tables 6.1.1.1 and 6.1.1.3 provide accurate values of f(q) obtained from relativistic Hartree–Fock calculations for all the atoms and chemically important ions in the Periodic Table. Alternatively, since the correction is small (∼1%), one can often use the approximate analytical expression (Sears, 1986a , 1996 ) $[f(q)={Z \over\sqrt{1+3(q/q_0)^2}} \eqno (4.4.4.20)]$ with $[q_0=\gamma Z^{1/3}]$ . The value γ = 1.90 ± 0.07 Å⁻¹ provides a good fit to the Hartree–Fock results in Table 6.1.1.1 for $[Z\ge20]$ .

4.4.4.5. Measurement of scattering lengths

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The development of modern neutron-optical techniques during the past 25 years has produced a dramatic increase in the accuracy with which scattering lengths can be measured (Koester, 1977 ; Klein & Werner, 1983 ; Werner & Klein, 1986 ; Sears, 1989 ; Koester, Rauch & Seymann, 1991 ). The measurements employ a number of effects – mirror reflection, prism refraction, gravity refractometry, Christiansen filter, and interferometry – all of which are based on the fact that the neutron index of refraction, n, is uniquely determined by [b_c(0)] through the relation $[n^2 =1-{4\pi\over k^2}\,\rho\,b_c(0), \eqno (4.4.4.21)]$ in which ρ is the number of atoms per unit volume. Apart from a small (≤0.01%) local-field correction (Sears, 1985 , 1989 ), this expression is exact.

In methods based on diffraction, such as Bragg reflection by powders or dynamical diffraction by perfect crystals, the measured quantity is the unit-cell structure factor $[|F_{hkl}|]$ . This quantity depends on [b_c(q)] in which q is equal to the magnitude of the reciprocal-lattice vector corresponding to the relevant Bragg planes, i.e. $[q=2k\sin\theta_{hkl}, \eqno (4.4.4.22)]$ where $[\theta_{hkl}]$ is the Bragg angle. In dynamical diffraction measurements, it is usual for the authors to correct their results for electromagnetic interactions so that the published quantity is again [b_c(0)] . In the past, this correction has not usually been made for the scattering lengths obtained from Bragg reflection by powders. However, these latter measurements are accurate only to ±2 or 3% so that the correction is then relatively unimportant.

The essential point is that all the bound coherent scattering lengths in Table 4.4.4.1 with the experimental uncertainties less than 1% represent [b_c(0)] and should therefore be corrected for electromagnetic interactions before being used in the interpretation of neutron diffraction experiments. Failure to make this correction will introduce systematic errors of 0.5 to 2% in the unit-cell structure factors at large q, and corresponding errors of 1 to 4% in the calculated intensities.

Expression (4.4.4.21) assumed that the neutrons and/or the nuclei are unpolarized. If the neutrons and the nuclei are both polarized then [b_c(0)] is replaced by $[\langle b(0)\rangle]$ , which depends on both the coherent and incoherent scattering lengths. If the coherent scattering length is known, neutron-optical experiments with polarized neutrons and nuclei can then be used to determine the incoherent scattering length (Glättli & Goldman, 1987 ).

4.4.4.6. Compilation of scattering lengths and cross sections

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The bound scattering lengths and cross sections of almost all the elements in the Periodic Table, as well as those of the individual isotopes, are listed in Table 4.4.4.1 . As in earlier versions of this table (Sears, 1984 , 1986b , 1992a ,b ), our primary aim, has been to take the best current values of the bound coherent and incoherent neutron scattering lengths and to compute from them a consistent set of bound scattering cross sections. In the present version, we have used the values of the coherent and incoherent scattering lengths recommended by Koester, Rauch & Seymann (1991 ), supplemented with a few more recently measured values, and have computed from them the corresponding scattering cross sections. The trailing digits in parentheses give the standard errors calculated from the errors in the input data using the statistical theory of error propagation (Young, 1962 ). The imaginary parts of the scattering lengths, which are appreciable only for strongly absorbing nuclides, were calculated from the measured absorption cross sections (Mughabghab, Divadeenam & Holden, 1981 ; Mughabghab, 1984 ) and are listed beneath the real parts of Table 4.4.4.1 .

In a few cases, where the scattering lengths have not yet been measured directly, the available scattering cross-section data (Mughabghab, Divadeenam & Holden, 1981 ; Mughabghab, 1984 ) were used to obtain the scattering lengths. Equations (4.4.4.11), (4.4.4.12), and (4.4.4.13) were used, where necessary, to fill gaps in Table 4.4.4.1 . For some elements, these relations indicated inconsistencies in the data. In such cases, appropriate adjustments in the values of some of the quantities were made. In almost all cases, such adjustments were comparable with the stated errors. Finally, for some elements, it was necessary to estimate arbitrarily the scattering lengths of one or two isotopes in order to be able to complete the table. Such estimates are indicated by the letter `E' and were usually made only for isotopes of low natural abundance where the estimated values have only a marginal effect on the final results. Apart from the inclusion of new data for Ti and Mn, the values listed in Table 4.4.4.1 are the same as in Sears (1992b ).

4.4.5. Magnetic form factors

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P. J. Brown^a

The form factors used in the calculations of the cross sections for magnetic scattering of neutrons are defined in Subsection 6.1.2.3 as $[\langle\, j_l(k)\rangle =\textstyle \int\limits^\infty_0\,U^2(r)\, j_l(kr)4\pi r^2\,{\rm d} r, \eqno (4.4.5.1)]$ in which U(r) is the radial wavefunction for the unpaired electrons in the atom, k is the length of the scattering vector, and [j_l(kr)] is the lth-order spherical Bessel function .

Tables 4.4.5.1 –4.4.5.8 give the coefficients in an analytical approximation to the $[\langle \,j_0\rangle]$ magnetic form factors for the 3d and 4d transition series, the 4f electrons of rare-earth ions, and the 5f electrons of some actinide ions. The approximation has the form used by Forsyth & Wells (1959 ) but allowing three instead of two exponential terms: $[\eqalignno{ \langle\, j_0(s)\rangle &=A\exp(-as^2)+B\exp(-bs^2) \cr &\quad +C\exp(-cs^2)+D, &(4.4.5.2)}]$ where s is the value of $[(\sin\theta)/\lambda]$ in Å⁻¹.

Table 4.4.5.1| top | pdf |
〈j₀〉 form factors for 3d transition elements and their ions

Atom or ion	A	a	B	b	C	c	D	e
Sc	0.2512	90.030	0.3290	39.402	0.4235	14.322	−0.0043	0.2029
Sc⁺	0.4889	51.160	0.5203	14.076	−0.0286	0.179	0.0185	0.1217
Sc²⁺	0.5048	31.403	0.5186	10.990	−0.0241	1.183	0.0000	0.0578
Ti	0.4657	33.590	0.5490	9.879	−0.0291	0.323	0.0123	0.1088
Ti⁺	0.5093	36.703	0.5032	10.371	−0.0263	0.311	0.0116	0.1125
Ti²⁺	0.5091	24.976	0.5162	8.757	−0.0281	0.916	0.0015	0.0589
Ti³⁺	0.3571	22.841	0.6688	8.931	−0.0354	0.483	0.0099	0.0575
V	0.4086	28.811	0.6077	8.544	−0.0295	0.277	0.0123	0.0970
V⁺	0.4444	32.648	0.5683	9.097	−0.2285	0.022	0.2150	0.1111
V²⁺	0.4085	23.853	0.6091	8.246	−0.1676	0.041	0.1496	0.0593
V³⁺	0.3598	19.336	0.6632	7.617	−0.3064	0.030	0.2835	0.0515
V⁴⁺	0.3106	16.816	0.7198	7.049	−0.0521	0.302	0.0221	0.0433
Cr	0.1135	45.199	0.3481	19.493	0.5477	7.354	−0.0092	0.1975
Cr⁺	−0.0977	0.047	0.4544	26.005	0.5579	7.489	0.0831	0.1114
Cr²⁺	1.2024	−0.005	0.4158	20.548	0.6032	6.956	−1.2218	0.0572
Cr³⁺	−0.3094	0.027	0.3680	17.035	0.6559	6.524	0.2856	0.0436
Cr⁴⁺	−0.2320	0.043	0.3101	14.952	0.7182	6.173	0.2042	0.0419
Mn	0.2438	24.963	0.1472	15.673	0.6189	6.540	−0.0105	0.1748
Mn⁺	−0.0138	0.421	0.4231	24.668	0.5905	6.655	−0.0010	0.1242
Mn²⁺	0.4220	17.684	0.5948	6.0050	0.0043	−0.609	−0.0219	0.0589
Mn³⁺	0.4198	14.283	0.6054	5.469	0.9241	−0.009	−0.9498	0.0392
Mn⁴⁺	0.3760	12.566	0.6602	5.133	−0.0372	0.563	0.0011	0.0393
Fe	0.0706	35.008	0.3589	15.358	0.5819	5.561	−0.0114	0.1398
Fe⁺	0.1251	34.963	0.3629	15.514	0.5223	5.591	−0.0105	0.1301
Fe²⁺	0.0263	34.960	0.3668	15.943	0.6188	5.594	−0.0119	0.1437
Fe³⁺	0.3972	13.244	0.6295	4.903	−0.0314	0.350	0.0044	0.0441
Fe⁴⁺	0.3782	11.380	0.6556	4.592	−0.0346	0.483	0.0005	0.0362
Co	0.4139	16.162	0.6013	4.780	−0.1518	0.021	0.1345	0.1033
Co⁺	0.0990	33.125	0.3645	15.177	0.5470	5.008	−0.0109	0.0983
Co²⁺	0.4332	14.355	0.5857	4.608	−0.0382	0.134	0.0179	0.0711
Co³⁺	0.3902	12.508	0.6324	4.457	−0.1500	0.034	0.1272	0.0515
Co⁴⁺	0.3515	10.778	0.6778	4.234	−0.0389	0.241	0.0098	0.0390
Ni	−0.0172	35.739	0.3174	14.269	0.7136	4.566	−0.0143	0.1072
Ni⁺	0.0705	35.856	0.3984	13.804	0.5427	4.397	−0.0118	0.0738
Ni²⁺	0.0163	35.883	0.3916	13.223	0.6052	4.339	−0.0133	0.0817
Ni³⁺	0.0012	35.000	0.3468	11.987	0.6667	4.252	−0.0148	0.0883
Ni⁴⁺	−0.0090	35.861	0.2776	11.790	0.7474	4.201	−0.0163	0.0966
Cu	0.0909	34.984	0.4088	11.443	0.5128	3.825	−0.0124	0.0513
Cu⁺	0.0749	34.966	0.4147	11.764	0.5238	3.850	−0.0127	0.0591
Cu²⁺	0.0232	34.969	0.4023	11.564	0.5882	3.843	−0.0137	0.0532
Cu³⁺	0.0031	34.907	0.3582	10.914	0.6531	3.828	−0.0147	0.0665
Cu⁴⁺	−0.0132	30.682	0.2801	11.163	0.7490	3.817	−0.0165	0.0767

Table 4.4.5.2| top | pdf |
〈j₀〉 form factors for 4d atoms and their ions

Atom or ion	A	a	B	b	C	c	D	e
Y	0.5915	67.608	1.5123	17.900	−1.1130	14.136	0.0080	0.3272
Zr	0.4106	59.996	1.0543	18.648	−0.4751	10.540	0.0106	0.3667
Zr⁺	0.4532	59.595	0.7834	21.436	−0.2451	9.036	0.0098	0.3639
Nb	0.3946	49.230	1.3197	14.822	−0.7269	9.616	0.0129	0.3659
Nb⁺	0.4572	49.918	1.0274	15.726	−0.4962	9.157	0.0118	0.3403
Mo	0.1806	49.057	1.2306	14.786	−0.4268	6.987	0.0171	0.4135
Mo⁺	0.3500	48.035	1.0305	15.060	−0.3929	7.479	0.0139	0.3510
Tc	0.1298	49.661	1.1656	14.131	−0.3134	5.513	0.0195	0.3869
Tc⁺	0.2674	48.957	0.9569	15.141	−0.2387	5.458	0.0160	0.3412
Ru	0.1069	49.424	1.1912	12.742	−0.3176	4.912	0.0213	0.3597
Ru⁺	0.4410	33.309	1.4775	9.553	−0.9361	6.722	0.0176	0.2608
Rh	0.0976	49.882	1.1601	11.831	−0.2789	4.127	0.0234	0.3263
Rh⁺	0.3342	29.756	1.2209	9.438	−0.5755	5.332	0.0210	0.2574
Pd	0.2003	29.363	1.1446	9.599	−0.3689	4.042	0.0251	0.2453
Pd⁺	0.5033	24.504	1.9982	6.908	−1.5240	5.513	0.0213	0.1909

Table 4.4.5.3| top | pdf |
〈j₀〉 form factors for rare-earth ions

Ion	A	a	B	b	C	c	D	e
Ce²⁺	0.2953	17.685	0.2923	6.733	0.4313	5.383	−0.0194	0.0845
Nd²⁺	0.1645	25.045	0.2522	11.978	0.6012	4.946	−0.0180	0.0668
Nd³⁺	0.0540	25.029	0.3101	12.102	0.6575	4.722	−0.0216	0.0478
Sm²⁺	0.0909	25.203	0.3037	11.856	0.6250	4.237	−0.0200	0.0408
Sm³⁺	0.0288	25.207	0.2973	11.831	0.6954	4.212	−0.0213	0.0510
Eu²⁺	0.0755	25.296	0.3001	11.599	0.6438	4.025	−0.0196	0.0488
Eu³⁺	0.0204	25.308	0.3010	11.474	0.7005	3.942	−0.0220	0.0356
Gd²⁺	0.0636	25.382	0.3033	11.212	0.6528	3.788	−0.0199	0.0486
Gd³⁺	0.0186	25.387	0.2895	11.142	0.7135	3.752	−0.0217	0.0489
Tb²⁺	0.0547	25.509	0.3171	10.591	0.6490	3.517	−0.0212	0.0342
Tb³⁺	0.0177	25.510	0.2921	10.577	0.7133	3.512	−0.0231	0.0512
Dy²⁺	0.1308	18.316	0.3118	7.665	0.5795	3.147	−0.0226	0.0315
Dy³⁺	0.1157	15.073	0.3270	6.799	0.5821	3.020	−0.0249	0.0146
Ho²⁺	0.0995	18.176	0.3305	7.856	0.5921	2.980	−0.0230	0.1240
Ho³⁺	0.0566	18.318	0.3365	7.688	0.6317	2.943	−0.0248	0.0068
Er²⁺	0.1122	18.122	0.3462	6.911	0.5649	2.761	−0.0235	0.0207
Er³⁺	0.0586	17.980	0.3540	7.096	0.6126	2.748	−0.0251	0.0171
Tm²⁺	0.0983	18.324	0.3380	6.918	0.5875	2.662	−0.0241	0.0404
Tm³⁺	0.0581	15.092	0.2787	7.801	0.6854	2.793	−0.0224	0.0351
Yb²⁺	0.0855	18.512	0.2943	7.373	0.6412	2.678	−0.0213	0.0421
Yb³⁺	0.0416	16.095	0.2849	7.834	0.6961	2.672	−0.0229	0.0344

Table 4.4.5.4| top | pdf |
〈j₀〉 form factors for actinide ions

Ion	A	a	B	b	C	c	D	e
U³⁺	0.5058	23.288	1.3464	7.003	−0.8724	4.868	0.0192	0.1507
U⁴⁺	0.3291	23.548	1.0836	8.454	−0.4340	4.120	0.0214	0.1757
U⁵⁺	0.3650	19.804	3.2199	6.282	−2.6077	5.301	0.0233	0.1750
Np³⁺	0.5157	20.865	2.2784	5.893	−1.8163	4.846	0.0211	0.1378
Np⁴⁺	0.4206	19.805	2.8004	5.978	−2.2436	4.985	0.0228	0.1408
Np⁵⁺	0.3692	18.190	3.1510	5.850	−2.5446	4.916	0.0248	0.1515
Np⁶⁺	0.2929	17.561	3.4866	5.785	−2.8066	4.871	0.0267	0.1698
Pu³⁺	0.3840	16.679	3.1049	5.421	−2.5148	4.551	0.0263	0.1280
Pu⁴⁺	0.4934	16.836	1.6394	5.638	−1.1581	4.140	0.0248	0.1242
Pu⁵⁺	0.3888	16.559	2.0362	5.657	−1.4515	4.255	0.0267	0.1287
Pu⁶⁺	0.3172	16.051	3.4654	5.351	−2.8102	4.513	0.0281	0.1382
Am²⁺	0.4743	21.776	1.5800	5.690	−1.0779	4.145	0.0218	0.1253
Am³⁺	0.4239	19.574	1.4573	5.872	−0.9052	3.968	0.0238	0.1054
Am⁴⁺	0.3737	17.862	1.3521	6.043	−0.7514	3.720	0.0258	0.1113
Am⁵⁺	0.2956	17.372	1.4525	6.073	−0.7755	3.662	0.0277	0.1202
Am⁶⁺	0.2302	16.953	1.4864	6.116	−0.7457	3.543	0.0294	0.1323
Am⁷⁺	0.3601	12.730	1.9640	5.120	−1.3560	3.714	0.0316	0.1232

Table 4.4.5.5| top | pdf |
〈j₂〉 form factors for 3d transition elements and their ions

Atom or ion	A	a	B	b	C	c	D	e
Sc	10.8172	54.327	4.7353	14.847	0.6071	4.218	0.0011	0.1212
Sc⁺	8.5021	34.285	3.2116	10.994	0.4244	3.605	0.0009	0.1037
Sc²⁺	4.3683	28.654	3.7231	10.823	0.6074	3.668	0.0014	0.0681
Ti	4.3583	36.056	3.8230	11.133	0.6855	3.469	0.0020	0.0967
Ti⁺	6.1567	27.275	2.6833	8.983	0.4070	3.052	0.0011	0.0902
Ti²⁺	4.3107	18.348	2.0960	6.797	0.2984	2.548	0.0007	0.0640
Ti³⁺	3.3717	14.444	1.8258	5.713	0.2470	2.265	0.0005	0.0491
V	3.7600	21.831	2.4026	7.546	0.4464	2.663	0.0017	0.0556
V⁺	4.7474	23.323	2.3609	7.808	0.4105	2.706	0.0014	0.0800
V²⁺	3.4386	16.530	1.9638	6.141	0.2997	2.267	0.0009	0.0565
V³⁺	2.3005	14.682	2.0364	6.130	0.4099	2.382	0.0014	0.0252
V⁴⁺	1.8377	12.267	1.8247	5.458	0.3979	2.248	0.0012	0.0399
Cr	3.4085	20.127	2.1006	6.802	0.4266	2.394	0.0019	0.0662
Cr⁺	3.7768	20.346	2.1028	6.893	0.4010	2.411	0.0017	0.0686
Cr²⁺	2.6422	16.060	1.9198	6.253	0.4446	2.372	0.0020	0.0480
Cr³⁺	1.6262	15.066	2.0618	6.284	0.5281	2.368	0.0023	0.0263
Cr⁴⁺	1.0293	13.950	1.9933	6.059	0.5974	2.346	0.0027	0.0366
Mn	2.6681	16.060	1.7561	5.640	0.3675	2.049	0.0017	0.0595
Mn⁺	3.2953	18.695	1.8792	6.240	0.3927	2.201	0.0022	0.0659
Mn²⁺	2.0515	15.556	1.8841	6.063	0.4787	2.232	0.0027	0.0306
Mn³⁺	1.2427	14.997	1.9567	6.118	0.5732	2.258	0.0031	0.0336
Mn⁴⁺	0.7879	13.886	1.8717	5.743	0.5981	2.182	0.0034	0.0434
Fe	1.9405	18.473	1.9566	6.323	0.5166	2.161	0.0036	0.0394
Fe⁺	2.6290	18.660	1.8704	6.331	0.4690	2.163	0.0031	0.0491
Fe²⁺	1.6490	16.559	1.9064	6.133	0.5206	2.137	0.0035	0.0335
Fe³⁺	1.3602	11.998	1.5188	5.003	0.4705	1.991	0.0038	0.0374
Fe⁴⁺	1.5582	8.275	1.1863	3.279	0.1366	1.107	−0.0022	0.0327
Co	1.9678	14.170	1.4911	4.948	0.3844	1.797	0.0027	0.0452
Co⁺	2.4097	16.161	1.5780	5.460	0.4095	1.914	0.0031	0.0581
Co²⁺	1.9049	11.644	1.3159	4.357	0.3146	1.645	0.0017	0.0459
Co³⁺	1.7058	8.859	1.1409	3.309	0.1474	1.090	−0.0025	0.0462
Co⁴⁺	1.3110	8.025	1.1551	3.179	0.1608	1.130	−0.0011	0.0374
Ni	1.0302	12.252	1.4669	4.745	0.4521	1.744	0.0036	0.0338
Ni⁺	2.1040	14.866	1.4302	5.071	0.4031	1.778	0.0034	0.0561
Ni²⁺	1.7080	11.016	1.2147	4.103	0.3150	1.533	0.0018	0.0446
Ni³⁺	1.4683	8.671	0.1794	1.106	1.1068	3.257	−0.0023	0.0373
Ni⁴⁺	1.1612	7.700	1.0027	3.263	0.2719	1.378	0.0025	0.0326
Cu	1.9182	14.490	1.3329	4.730	0.3842	1.639	0.0035	0.0617
Cu⁺	1.8814	13.433	1.2809	4.545	0.3646	1.602	0.0033	0.0590
Cu²⁺	1.5189	10.478	1.1512	3.813	0.2918	1.398	0.0017	0.0429
Cu³⁺	1.2797	8.450	1.0315	3.280	0.2401	1.250	0.0015	0.0389
Cu⁴⁺	0.9568	7.448	0.9099	3.396	0.3729	1.494	0.0049	0.0330

Table 4.4.5.6| top | pdf |
〈j₂〉 form factors for 4d atoms and their ions

Atom or ion	A	a	B	b	C	c	D	e
Y	14.4084	44.658	5.1045	14.904	−0.0535	3.319	0.0028	0.1093
Zr	10.1378	35.337	4.7734	12.545	−0.0489	2.672	0.0036	0.0912
Zr⁺	11.8722	34.920	4.0502	12.127	−0.0632	2.828	0.0034	0.0737
Nb	7.4796	33.179	5.0884	11.571	−0.0281	1.564	0.0047	0.0944
Nb⁺	8.7735	33.285	4.6556	11.605	−0.0268	1.539	0.0044	0.0855
Mo	5.1180	23.422	4.1809	9.208	−0.0505	1.743	0.0053	0.0655
Mo⁺	7.2367	28.128	4.0705	9.923	−0.0317	1.455	0.0049	0.0798
Tc	4.2441	21.397	3.9439	8.375	−0.0371	1.187	0.0066	0.0645
Tc⁺	6.4056	24.824	3.5400	8.611	−0.0366	1.485	0.0044	0.0806
Ru	3.7445	18.613	3.4749	7.420	−0.0363	1.007	0.0073	0.0533
Ru⁺	5.2826	23.683	3.5813	8.152	−0.0257	0.426	0.0131	0.0830
Rh	3.3651	17.344	3.2121	6.804	−0.0350	0.503	0.0146	0.0545
Rh⁺	4.0260	18.950	3.1663	7.000	−0.0296	0.486	0.0127	0.0629
Pd	3.3105	14.726	2.6332	5.862	−0.0437	1.130	0.0053	0.0492
Pd⁺	4.2749	17.900	2.7021	6.354	−0.0258	0.700	0.0071	0.0768

Table 4.4.5.7| top | pdf |
〈j₂〉 form factors for rare-earth ions

Ion	A	a	B	b	C	c	D	e
Ce²⁺	0.9809	18.063	1.8413	7.769	0.9905	2.845	0.0120	0.0448
Nd²⁺	1.4530	18.340	1.6196	7.285	0.8752	2.622	0.0126	0.0461
Nd³⁺	0.6751	18.342	1.6272	7.260	0.9644	2.602	0.0150	0.0450
Sm²⁺	1.0360	18.425	1.4769	7.032	0.8810	2.437	0.0152	0.0345
Sm³⁺	0.4707	18.430	1.4261	7.034	0.9574	2.439	0.0182	0.0510
Eu²⁺	0.8970	18.443	1.3769	7.005	0.9060	2.421	0.0190	0.0511
Eu³⁺	0.3985	18.451	1.3307	6.956	0.9603	2.378	0.0197	0.0447
Gd²⁺	0.7756	18.469	1.3124	6.899	0.8956	2.338	0.0199	0.0441
Gd³⁺	0.3347	18.476	1.2465	6.877	0.9537	2.318	0.0217	0.0484
Tb²⁺	0.6688	18.491	1.2487	6.822	0.8888	2.275	0.0215	0.0439
Tb³⁺	0.2892	18.497	1.1678	6.797	0.9437	2.257	0.0232	0.0458
Dy²⁺	0.5917	18.511	1.1828	6.747	0.8801	2.214	0.0229	0.0439
Dy³⁺	0.2523	18.517	1.0914	6.736	0.9345	2.208	0.0250	0.0476
Ho²⁺	0.5094	18.515	1.1234	6.706	0.8727	2.159	0.0242	0.0560
Ho³⁺	0.2188	18.516	1.0240	6.707	0.9251	2.161	0.0268	0.0503
Er²⁺	0.4693	18.528	1.0545	6.649	0.8679	2.120	0.0261	0.0413
Er³⁺	0.1710	18.534	0.9879	6.625	0.9044	2.100	0.0278	0.0489
Tm²⁺	0.4198	18.542	0.9959	6.600	0.8593	2.082	0.0284	0.0457
Tm³⁺	0.1760	18.542	0.9105	6.579	0.8970	2.062	0.0294	0.0468
Yb²⁺	0.3852	18.550	0.9415	6.551	0.8492	2.043	0.0301	0.0478
Yb³⁺	0.1570	18.555	0.8484	6.540	0.8880	2.037	0.0318	0.0498

Table 4.4.5.8| top | pdf |
〈j₂〉 form factors for actinide ions

Ion	A	a	B	b	C	c	D	e
U³⁺	4.1582	16.534	2.4675	5.952	−0.0252	0.765	0.0057	0.0822
U⁴⁺	3.7449	13.894	2.6453	4.863	−0.5218	3.192	0.0009	0.0928
U⁵⁺	3.0724	12.546	2.3076	5.231	−0.0644	1.474	0.0035	0.0477
Np³⁺	3.7170	15.133	2.3216	5.503	−0.0275	0.800	0.0052	0.0948
Np⁴⁺	2.9203	14.646	2.5979	5.559	−0.0301	0.367	0.0141	0.0532
Np⁵⁺	2.3308	13.654	2.7219	5.494	−0.1357	0.049	0.1224	0.0553
Np⁶⁺	1.8245	13.180	2.8508	5.407	−0.1579	0.044	0.1438	0.0585
Pu³⁺	2.0885	12.871	2.5961	5.190	−0.1465	0.039	0.1343	0.0866
Pu⁴⁺	2.7244	12.926	2.3387	5.163	−0.1300	0.046	0.1177	0.0490
Pu⁵⁺	2.1409	12.832	2.5664	5.152	−0.1338	0.046	0.1210	0.0491
Pu⁶⁺	1.7262	12.324	2.6652	5.066	−0.1695	0.041	0.1550	0.0502
Am²⁺	3.5237	15.955	2.2855	5.195	−0.0142	0.585	0.0033	0.1120
Am³⁺	2.8622	14.733	2.4099	5.144	−0.1326	0.031	0.1233	0.0727
Am⁴⁺	2.4141	12.948	2.3687	4.945	−0.2490	0.022	0.2371	0.0502
Am⁵⁺	2.0109	12.053	2.4155	4.836	−0.2264	0.027	0.2128	0.0414
Am⁶⁺	1.6778	11.337	2.4531	4.725	−0.2043	0.034	0.1892	0.0387
Am⁷⁺	1.8845	9.161	2.0746	4.042	−0.1318	1.723	0.0020	0.0379

Tables 4.4.5.9 –4.4.5.14 give coefficients in the approximation used by Lisher & Forsyth (1971 ) to the $[\langle \,j_2\rangle]$ , $[\langle\,j_4\rangle]$ , and $[\langle\,j_6\rangle]$ form factors for the same series of atoms and ions, again using three rather than two exponential terms, viz for $[l\neq0]$ : $[\eqalignno{ \langle \,j_l(s)\rangle &=As^2\exp(-as^2)+B s^2\exp(-bs^2) \cr &\quad+Cs^2\exp(-cs^2)+Ds^2. & (4.4.5.3)}]$

Table 4.4.5.9| top | pdf |
〈j₄〉 form factors for 3d atoms and their ions

Atom or ion	A	a	B	b	C	c	D	e
Sc	1.3420	10.200	0.3837	3.079	0.0468	0.118	−0.0328	0.1343
Sc⁺	7.1167	15.487	−6.6671	18.269	0.4900	2.992	0.0047	0.1624
Sc²⁺	−1.6684	15.648	1.7742	9.062	0.4075	2.412	0.0042	0.1105
Ti	−2.1515	11.271	2.5149	8.859	0.3555	2.149	0.0045	0.1244
Ti⁺	−1.0383	16.190	1.4699	8.924	0.3631	2.283	0.0044	0.1270
Ti²⁺	−1.3242	15.310	1.2042	7.899	0.3976	2.156	0.0051	0.0820
Ti³⁺	−1.1117	14.635	0.7689	6.927	0.4385	2.089	0.0060	0.0572
V	−0.9633	15.273	0.9274	7.732	0.3891	2.053	0.0063	0.0840
V⁺	−0.9606	15.545	1.1278	8.118	0.3653	2.097	0.0056	0.1027
V²⁺	−1.1729	14.973	0.9092	7.613	0.4105	2.039	0.0067	0.0719
V³⁺	−0.9417	14.205	0.5284	6.607	0.4411	1.967	0.0076	0.0569
V⁴⁺	−0.7654	13.097	0.3071	5.674	0.4476	1.871	0.0081	0.0518
Cr	−0.6670	19.613	0.5342	6.478	0.3641	1.905	0.0073	0.0628
Cr⁺	−0.8309	18.043	0.7252	7.531	0.3828	2.003	0.0073	0.0781
Cr²⁺	−0.8930	15.664	0.5590	7.033	0.4093	1.924	0.0081	0.0631
Cr³⁺	−0.7327	14.073	0.3268	5.674	0.4114	1.810	0.0085	0.0505
Cr⁴⁺	−0.6748	12.946	0.1805	6.753	0.4526	1.800	0.0098	0.0644
Mn	−0.5452	15.471	0.4406	4.902	0.2884	1.543	0.0059	0.0488
Mn⁺	−0.7947	17.867	0.6078	7.704	0.3798	1.905	0.0087	0.0737
Mn²⁺	−0.7416	15.255	0.3831	6.469	0.3935	1.800	0.0093	0.0577
Mn³⁺	−0.6603	13.607	0.2322	6.218	0.4104	1.740	0.0101	0.0579
Mn⁴⁺	−0.5127	13.461	0.0313	7.763	0.4282	1.701	0.0113	0.0693
Fe	−0.5029	19.677	0.2999	3.776	0.2576	1.424	0.0071	0.0292
Fe⁺	−0.5109	19.250	0.3896	4.891	0.2810	1.526	0.0069	0.0375
Fe²⁺	−0.5401	17.227	0.2865	3.742	0.2658	1.424	0.0076	0.0278
Fe³⁺	−0.5507	11.493	0.2153	4.906	0.3468	1.523	0.0095	0.0314
Fe⁴⁺	−0.5352	9.507	0.1783	5.175	0.3584	1.469	0.0097	0.0360
Co	−0.4221	14.195	0.2900	3.979	0.2469	1.286	0.0063	0.0400
Co⁺	−0.4115	14.561	0.3580	4.717	0.2644	1.418	0.0074	0.0541
Co²⁺	0.4759	14.046	0.2747	3.731	0.2458	1.250	0.0057	0.0282
Co³⁺	−0.4466	13.391	0.1419	3.011	0.2773	1.335	0.0093	0.0341
Co⁴⁺	−0.4091	13.194	−0.0194	3.417	0.3534	1.421	0.0112	0.0622
Ni	−0.4428	14.485	0.0870	3.234	0.2932	1.331	0.0096	0.0554
Ni⁺	−0.3836	13.425	0.3116	4.462	0.2471	1.309	0.0079	0.0515
Ni²⁺	−0.3803	10.403	0.2838	3.378	0.2108	1.104	0.0050	0.0474
Ni³⁺	−0.4014	9.046	0.2314	3.075	0.2192	1.084	0.0060	0.0323
Ni⁴⁺	−0.3509	8.157	0.2220	2.106	0.1567	0.925	0.0065	0.0352
Cu	−0.3204	15.132	0.2335	4.021	0.2312	1.196	0.0068	0.0457
Cu⁺	−0.3572	15.125	0.2336	3.966	0.2315	1.197	0.0070	0.0397
Cu²⁺	−0.3914	14.740	0.1275	3.384	0.2548	1.255	0.0103	0.0394
Cu³⁺	−0.3671	14.082	−0.0078	3.315	0.3154	1.377	0.0132	0.0534
Cu⁴⁺	−0.2915	14.124	−0.1065	4.201	0.3247	1.352	0.0148	0.0579

Table 4.4.5.10| top | pdf |
〈j₄〉 form factors for 4d atoms and their ions

Atom or ion	A	a	B	b	C	c	D	e
Y	−8.0767	32.201	7.9197	25.156	1.4067	6.827	−0.0001	0.1031
Zr	−5.2697	32.868	4.1930	24.183	1.5202	6.048	−0.0002	0.0855
Zr⁺	−5.6384	33.607	4.6729	22.338	1.3258	5.924	−0.0003	0.0674
Nb	−3.1377	25.595	2.3411	16.569	1.2304	4.990	−0.0005	0.0615
Nb⁺	−3.3598	25.820	2.8297	16.427	1.1203	4.982	−0.0005	0.0724
Mo	−2.8860	20.572	1.8130	14.628	1.1899	4.264	−0.0008	0.0410
Mo⁺	−3.2618	25.486	2.3596	16.462	1.1164	4.491	−0.0007	0.0592
Tc	−2.7975	20.159	1.6520	16.261	1.1726	3.943	−0.0008	0.0657
Tc⁺	−2.0470	19.683	1.6306	11.592	0.8698	3.769	−0.0010	0.0723
Ru	−1.5042	17.949	0.6027	9.961	0.9700	3.393	−0.0010	0.0338
Ru⁺	1.6278	18.506	1.1828	10.189	0.8138	3.418	−0.0009	0.0673
Rh	−1.3492	17.577	0.4527	10.507	0.9285	3.155	−0.0009	0.0483
Rh⁺	−1.4673	17.957	0.7381	9.944	0.8485	3.126	−0.0012	0.0487
Pd	−1.1955	17.628	0.3183	11.309	0.8696	2.909	−0.0006	0.0555
Pd⁺	−1.4098	17.765	0.7927	9.999	0.7710	2.930	−0.0006	0.0530

Table 4.4.5.11| top | pdf |
〈j₄〉 form factors for rare-earth ions

Ion	A	a	B	b	C	c	D	e
Ce²⁺	−0.6468	10.533	0.4052	5.624	0.3412	1.535	0.0080	0.0522
Nd²⁺	−0.5416	12.204	0.3571	6.169	0.3154	1.485	0.0098	0.0519
Nd³⁺	−0.4053	14.014	0.0329	7.005	0.3759	1.707	0.0209	0.0372
Sm²⁺	−0.4150	14.057	0.1368	7.032	0.3272	1.582	0.0192	0.0319
Sm³⁺	−0.4288	10.052	0.1782	5.019	0.2833	1.236	0.0088	0.0328
Eu²⁺	−0.4145	10.193	0.2447	5.164	0.2661	1.205	0.0065	0.0516
Eu³⁺	−0.4095	10.211	0.1485	5.175	0.2720	1.237	0.0131	0.0494
Gd²⁺	−0.3824	10.344	0.1955	5.306	0.2622	1.203	0.0097	0.0363
Gd³⁺	−0.3621	10.353	0.1016	5.310	0.2649	1.219	0.0147	0.0494
Tb²⁺	−0.3443	10.469	0.1481	5.416	0.2575	1.182	0.0104	0.0280
Tb³⁺	−0.3228	10.476	0.0638	5.419	0.2566	1.196	0.0159	0.0439
Dy²⁺	−0.3206	12.071	0.0904	8.026	0.2616	1.230	0.0143	0.0767
Dy³⁺	−0.2829	9.525	0.0565	4.429	0.2437	1.066	0.0092	0.0181
Ho²⁺	−0.2976	9.719	0.1224	4.635	0.2279	1.005	0.0063	0.0452
Ho³⁺	−0.2717	9.731	0.0474	4.638	0.2292	1.047	0.0124	0.0310
Er²⁺	−0.2975	9.829	0.1189	4.741	0.2116	1.004	0.0117	0.0524
Er³⁺	−0.2568	9.834	0.0356	4.741	0.2172	1.028	0.0148	0.0434
Tm²⁺	−0.2677	9.888	0.0925	4.784	0.2056	0.990	0.0124	0.0396
Tm³⁺	−0.2292	9.895	0.0124	4.785	0.2108	1.007	0.0151	0.0334
Yb²⁺	−0.2393	9.947	0.0663	4.823	0.2009	0.965	0.0122	0.0311
Yb³⁺	−0.2121	8.197	0.0325	3.153	0.1975	0.884	0.0093	0.0435

Table 4.4.5.12| top | pdf |
〈j₄〉 form factors for actinide ions

Ion	A	a	B	b	C	c	D	e
U³⁺	−0.9859	16.601	0.6116	6.515	0.6020	2.597	−0.0010	0.0599
U⁴⁺	−1.0540	16.605	0.4339	6.512	0.6746	2.599	−0.0011	0.0471
U⁵⁺	−0.9588	16.485	0.1576	6.440	0.7785	2.640	−0.0010	0.0493
Np³⁺	0.9029	16.586	0.4006	6.470	0.6545	2.563	−0.0004	0.0470
Np⁴⁺	−0.9887	12.441	0.5918	5.294	0.5306	2.263	−0.0021	0.0583
Np⁵⁺	−0.8146	16.581	−0.0055	6.475	0.7956	2.562	−0.0004	0.0600
Np⁶⁺	0.6738	16.553	−0.2297	6.505	0.8513	2.553	−0.0003	0.0623
Pu³⁺	−0.7014	16.369	−0.1162	6.697	0.7778	2.450	0.0000	0.0546
Pu⁴⁺	−0.9160	12.203	0.4891	5.127	0.5290	2.149	−0.0022	0.0520
Pu⁵⁺	−0.7035	16.360	−0.0979	6.706	0.7726	2.447	0.0000	0.0610
Pu⁶⁺	−0.5560	16.322	−0.3046	6.768	0.8146	2.426	0.0001	0.0596
Am²⁺	−0.7433	16.416	0.3481	6.788	0.6014	2.346	0.0000	0.0566
Am³⁺	0.8092	12.854	0.4161	5.459	0.5476	2.172	−0.0011	0.0530
Am⁴⁺	−0.8548	12.226	0.3037	5.909	0.6173	2.188	−0.0016	0.0456
Am⁵⁺	−0.6538	15.462	−0.0948	5.997	0.7295	2.297	0.0000	0.0594
Am⁶⁺	−0.5390	15.449	−0.2689	6.017	0.7711	2.297	0.0002	0.0729
Am⁷⁺	−0.4688	12.019	−0.2692	7.042	0.7297	2.164	−0.0011	0.0262

Table 4.4.5.13| top | pdf |
〈j₆〉 form factors for rare-earth ions

Ion	A	a	B	b	C	c	D	e
Ce²⁺	−0.1212	7.994	−0.0639	4.024	0.1519	1.096	0.0078	0.0388
Nd²⁺	−0.1600	8.009	0.0272	4.028	0.1104	1.068	0.0139	0.0363
Nd³⁺	0.0416	8.014	−0.1261	4.040	0.1400	1.087	0.0102	0.0367
Sm²⁺	0.1428	6.041	0.0723	2.033	0.0550	0.513	0.0081	0.0450
Sm³⁺	−0.0944	6.030	−0.0498	2.074	0.1372	0.645	−0.0132	0.0387
Eu²⁺	−0.1252	6.049	0.0507	2.085	0.0572	0.646	0.0132	0.0403
Eu³⁺	−0.0817	6.039	−0.0596	2.120	0.1243	0.764	−0.0001	0.0206
Gd²⁺	−0.1351	5.030	0.0828	2.025	0.0315	0.503	0.0187	0.0453
Gd³⁺	−0.0662	6.031	−0.0850	2.154	0.1323	0.891	0.0048	0.0371
Th²⁺	−0.0758	6.032	−0.0540	2.158	0.1199	0.890	0.0051	0.0488
Tb³⁺	−0.0559	6.031	−0.1020	2.237	0.1264	1.107	0.0167	0.0170
Dy²⁺	−0.0568	6.032	−0.1003	2.240	0.1401	1.106	0.0109	0.0463
Dy³⁺	−0.0423	6.038	−0.1248	2.244	0.1359	1.200	0.0188	0.0350
Ho²⁺	−0.0725	6.045	−0.0318	2.243	0.0738	1.202	0.0252	0.0634
Ho³⁺	−0.0289	6.050	−0.1545	2.230	0.1550	1.260	0.0177	0.0351
Er²⁺	0.0648	6.056	−0.0515	2.230	0.0825	1.264	0.0250	0.0409
Er³⁺	−0.0110	6.061	−0.1954	2.224	0.1818	1.296	0.0149	0.0455
Tm²⁺	0.0842	4.070	0.0807	0.849	−0.2087	0.039	0.2095	0.0360
Tm³⁺	0.0727	4.073	0.0243	0.689	3.9459	0.002	−3.9076	0.0502
Yb²⁺	−0.0739	5.031	0.0140	2.030	0.0351	0.508	0.0174	0.0434
Yb³⁺	−0.0345	5.007	−0.0677	2.020	0.0985	0.549	−0.0076	0.0359

Table 4.4.5.14| top | pdf |
〈j₆〉 form factors for actinide ions

Ion	A	a	B	b	C	c	D	e
U³⁺	−0.3797	9.953	0.0459	5.038	0.2748	1.607	0.0016	0.0345
U⁴⁺	−0.1793	11.896	−0.2269	5.428	0.3291	1.701	0.0030	0.0472
U⁵⁺	−0.0399	11.891	−0.3458	5.580	0.3340	1.645	0.0029	0.0444
Np³⁺	−0.2427	11.844	−0.1129	5.377	0.2848	1.568	0.0022	0.0368
Np⁴⁺	−0.2436	9.599	−0.1317	4.101	0.3029	1.545	0.0019	0.0500
Np⁵⁺	−0.1157	9.565	−0.2654	4.260	0.3298	1.549	0.0025	0.0495
Np⁶⁺	−0.0128	9.569	−0.3611	4.304	0.3419	1.541	0.0032	0.0520
Pu³⁺	−0.0364	9.572	−0.3181	4.342	0.3210	1.523	0.0041	0.0496
Pu⁴⁺	−0.2394	7.837	−0.0785	4.024	0.2643	1.378	0.0012	0.0414
Pu⁵⁺	−0.1090	7.819	−0.2243	4.100	0.2947	1.404	0.0015	0.0477
Pu⁶⁺	−0.0001	7.820	−0.3354	4.144	0.3097	1.403	0.0020	0.0513
Am²⁺	−0.3176	7.864	0.0771	4.161	0.2194	1.339	0.0018	0.0374
Am³⁺	−0.3159	6.982	0.0682	3.995	0.2141	1.188	−0.0015	0.0281
Am⁴⁺	−0.1787	7.880	−0.1274	4.090	0.2565	1.315	0.0017	0.0419
Am⁵⁺	−0.0927	6.073	−0.2227	3.784	0.2916	1.372	0.0026	0.0485
Am⁶⁺	0.0152	6.079	−0.3549	3.861	0.3125	1.403	0.0036	0.0732
Am⁷⁺	0.1292	6.082	−0.4689	3.879	0.3234	1.393	0.0042	0.0475

For the transition-metal series, the coefficients of the approximation have been obtained by fitting to form factors calculated from the Hartree–Fock wavefunctions given by Clementi & Roetti (1974 ) in terms of Slater-type functions in the form $[U(r)=\textstyle\sum\limits_{nj}\,N_{nl}\,r^2A_{nlj}\exp(-a_{nlj}\,r) \eqno (4.4.5.4)]$ by using the identity: $[\eqalignno{ &\int^\infty_0\,j_l(kr) r^n\exp(-pr)4\pi r^2\,{\rm d} r \cr &\quad ={\pi^{3/2}\Gamma(n+l+3)k^l \over 2^{l-1}\Gamma(l+3/2)(k^2+p^2)^{(n+l+3)/2}} \cr &\qquad\times {_2}F_1\left({n+l+3\over 2};\, {l-n+3\over 2};\, l+{3\over2};\, {k^2\over k^2+p^2}\right). \cr&& (4.4.5.5)}]$

The form factors have been calculated from these relationships in the range $[(\sin\theta)/\lambda=0]$ to 1.5 Å⁻¹ at intervals of 0.05 Å⁻¹, and the coefficients of the exponential expansion fitted by a least-squares procedure at the calculated points.

For the atoms of the rare-earth and actinide series, the wavefunctions and form factors have been calculated by Freeman & Desclaux (1979 ) and Desclaux & Freeman (1978 ) using Dirac–Fock theory. The constants given in Tables 4.4.5.3 , 4.4.5.4 , 4.4.5.7 , 4.4.5.8 , and 4.4.5.11 –4.4.5.14 have been fitted to the results of these calculations. For the rare-earth ions, the form factors are given in the range $[(\sin\theta)/\lambda=0]$ to 0.5 Å⁻¹ at intervals of 0.5 Å⁻¹ and in the range 0.5 to 1.2 Å⁻¹ at intervals of 0.1 Å⁻¹. For the actinide ions, the calculations extend to 1.5 Å⁻¹. All the values given in the publications cited were included in the fitting procedure. The accuracy with which the exponential expansions fit the theoretical form factors can be judged from the value of the parameter e given in the tables, and defined by: $[e=100\left(\,{\sum_i\delta^2_i \over N}\,\right)^{1/2}, \eqno (4.4.5.6)]$ where $[\delta_i]$ is the difference between the ith fitted point and its theoretical value. The sum is over the N points included in the fitting procedure.

4.4.6. Absorption coefficients for neutrons

| top | pdf |

B. T. M. Willis^h

The cross sections σ discussed in Section 4.4.4 represent the area of each nucleus as seen by the neutron. To calculate the beam attenuation arising from absorption it is more convenient to use the macroscopic cross section $[\Sigma]$ , which is the cross section per unit volume in units of cm⁻¹. $[\Sigma]$ is derived by multiplying σ for the element by the number of atoms per unit volume. Thus, for element j, with density $[\rho _{j}]$ and atomic weight $[A_{j}]$ , $[\Sigma_{j}=N_{A}\rho _{j}\sigma _{j}/A_{j},]$ where $[N_{A}]$ is Avogadro's number.

Table 4.4.6.1 gives the macroscopic absorption cross sections $[\Sigma_a]$ of the elements. They are tabulated for a neutron velocity v = 2200 m s⁻¹, corresponding to a wavelength of 1.80 Å. The cross sections are larger at longer wavelengths (Section 4.4.4 ). Apart from a few exceptions, such as boron and cadmium, the absorption cross section is vastly smaller than for X-rays. The 1/e penetration depth (l) is listed separately – most metals, for example, have a penetration depth of several cm. The data in Table 4.4.6.1 have been derived from the review article by Hutchings & Windsor (1987 ).

Table 4.4.6.1| top | pdf |
Absorption of the elements for neutrons (λ = 1.80 Å)

Atom	Σ_a (cm⁻¹)	l (cm)
H	0.0141	0.288
D	0.0000	6.17
He	0.0001	20.22
Li	3.2698	0.300
Be	0.0009	1.059
B	105.41	0.009
C	0.0004	1.58
N	0.0662	2.14
O	0.0001	5.52
F	0.0003	6.82
Ne	0.0017	8.71
Na	0.0135	10.22
Mg	0.0027	6.11
Al	0.0139	9.48
Si	0.0085	8.45
P	0.0061	8.04
S	0.0208	16.14
Cl	0.9109	0.731
Ar	0.0143	2.03
K	0.0279	18.12
Ca	0.0099	12.35
Sc	1.0906	0.491
Ti	0.3453	1.74
V	0.3658	1.35
Cr	0.2558	1.82
Mn	1.0900	0.789
Fe	0.2174	0.82
Co	3.3440	0.260
Ni	0.4104	0.475
Cu	0.3202	1.00
Zn	0.0730	2.89
Ga	0.1480	2.04
Ge	0.1016	2.07
As	0.2091	2.15
Se	0.4292	1.36
Br	0.1623	3.31
Kr	0.3882	1.97
Rb	0.0041	13.09
Sr	0.0227	7.44
Y	0.0388	3.66
Zr	0.0079	3.42
Nb	0.0640	2.42
Mo	0.1637	1.75
Tc	1.4281	0.542
Ru	0.1886	1.48
Rh	10.544	0.092
Pd	0.4687	1.29
Ag	3.7120	0.249
Cd	116.80	0.008
In	7.4135	0.133
Sn	0.0231	4.87
Sb	0.1689	3.20
Te	0.1386	4.01
I	0.1458	4.36
Xe	0.3904	2.27
Cs	0.2458	3.57
Ba	0.0189	13.39
La	0.2402	2.00
Ce	0.0182	9.60
Pr	0.3333	2.46
Nd	1.4763	0.496
Sm	171.23	0.005
Eu	95.715	0.010
Gd	1474.1	0.000
Tb	0.7334	1.05
Dy	29.731	0.030
Ho	2.0791	0.424
Er	5.1861	0.182
Tm	3.4919	0.269
Yb	0.8513	0.717
Lu	2.5889	0.354
Hf	4.6648	0.195
Ta	1.1434	0.676
W	1.1609	0.681
Re	6.1692	0.143
Os	1.1444	0.442
Ir	30.064	0.032
Pt	0.6843	0.682
Au	5.8181	0.159
Hg	15.146	0.061
Tl	0.1200	2.15
Pb	0.0056	2.68
Bi	0.0009	3.84
Ra	0.1706	2.95
Th	0.2244	1.68
Pa	8.0405	0.118
U	0.3650	1.26
Np	9.0534	0.10
Pu	14.481	0.069
Am	2.5522	0.351

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