teaching and education
Incoherent neutron scattering from multielement materials
^{a}University of Delaware, Newark, DE 19716 , USA, and ^{b}Center for Neutron Research, NIST, 100 Bureau Drive, Gaithersburg, MD 208996102, USA
^{*}Correspondence email: charles.glinka@nist.gov
In a neutron diffraction measurement, including smallangle scattering, there is generally a featureless (i.e. Qindependent) component due to This scattering contains no information about the atomic structure or structure on any scale. There may also be featureless scattering that arises from atomic disorder in multielement materials. This scattering is sometimes referred to as compositional or mixture However, this designation is misleading. A much better designation is diffuse coherent scattering. Here the differences and distinguishing characteristics of visàvis diffuse coherent scattering due to atomic disorder are delineated and demonstrated experimentally.
Keywords: neutron scattering; diffuse scattering; incoherent scattering; atomic disorder; isotopes; nuclear spin; Laue monotonic scattering; isotope exchange; deuterated solvents.
1. Introduction
Most textbooks on neutron scattering discuss the concept of e.g. Bacon, 1962; Squires, 1978; Roe, 2000). Such discussions are adequate for introducing the concepts of nuclear spin incoherence and isotopic incoherence. Most materials, however, consist of more than one atomic species and hence the question arises as to how to calculate the incoherent contribution to the scattering from such materials.
in detail only for materials consisting of a single atomic species (When considering the neutron scattering from a multielement material, whether it be a compound, a solid or liquid solution, or a molecular solid or liquid, it is important to distinguish between the Q independent, similar to the true that arises from the isotopic and nuclear spin distributions.
and any diffuse coherent scattering that may be present and is related to the degree of atomic disorder in the material. In the extreme case of complete atomic disorder, there will in general be a component of the coherent scattering that is essentially1.1. Scattering formalism
The scattering N atoms in a volume V is given in the Born approximation by
from a system ofwhere the angle brackets denote averaging over an ensemble of equivalent systems.^{1} In this expression, b_{i} is the scattering length of the atom at location r_{i}. If all of the atoms are of a single atomic species (i.e. a single element), then the ensemble average can clearly be written as
because there is no correlation between an atom's location and the isotope or nuclear spin state at that location. What may be less obvious is that equation (2) also applies when there is more than one type of atomic species in the sample. In that case, there is, in general, a correlation between a given location and the type of atom at the location. However, even though a given site may be more likely to be populated by one type of atom than another, there is still no correlation with a particular isotope or nuclear spin state of that atomic species at that site. Hence equation (2) remains valid even for multielement materials.
For the same reasons the average of the product of scattering lengths can be written as
Only the first term in equation (4) contains information about the arrangement, or structure, of the atoms. This term is called the coherent scattering:
The second term in equation (4) contains no structural information and is called the incoherent scattering:
To reduce equation (4) further to a useful form for computations, one must indicate how many of each type of atom is present in V. If there are, say, m elements represented, and N_{j} atoms of element j, then the fraction of atoms of type j is
Now if all the atoms are disordered, then
Substituting equation (8) into equation (4) leads to
Notice the similarities, and differences, between equations (9) and (4). The first and third terms of equation (9) correspond to the first and second terms of equation (4), and represent coherent and respectively. It is the middle term in equation (9) that is new, arising from the presence of more than one element, and, more importantly, the assumption that the atoms are randomly distributed among the available sites. This term is Q independent and thus is sometimes referred to as compositional or mixture incoherent (Cotton, 1991; Brûlet et al., 2007) scattering. A better designation, however, is diffuse coherent scattering: diffuse because of the lack of Q dependence, and coherent because this term provides information about the structure of the system, namely, that the atoms are disordered. It may at first seem odd to refer to disorder as a type of structure, yet that is exactly what it is. The essence of the assumption leading to equation (9) is that there is no correlation between the type of atom occupying a given site and the type occupying any other site, which is, in fact, a strong statement about the structure of the system: quite strong, for as we shall see, it does not apply to most real materials.
2. Specific examples
2.1. NaCl
Consider polycrystalline NaCl, a system with two atomic species (A and B), which has a facecenteredcubic with ρ = 2.165 g cm^{−3}, molecular weight M_{W} = 58.44 and N/V = 1/v_{m} = 2.231 × 10^{22} molecules cm^{−3} (v_{m} is the molecular volume). For such a material, the coherent from equation (5) becomes
where G_{hkl} = ha* + kb* + lc* is a vector, is the for the hkl Bragg reflection and the sum is over the atoms in the For the NaCl rock salt structure,
The bound coherent scattering lengths, 〈b〉 = b_{c}, are b_{c}(Na) = 3.63 fm and b_{c}(Cl) = 9.566 fm (Sears, 1992).
The incoherent cross sections for each element are (Sears, 1992)
(1 barn = 10^{−28} m^{2}). Hence the corresponding is
For hypothetical disordered NaCl, in which the sites in the rock salt structure are occupied at random by either Na or Cl, the first term in equation (9) gives
and
In addition, there is the diffuse coherent scattering [second term in equation (9); this diffuse scattering due to site disorder is called the Laue monotonic scattering (Warren, 1969)],
as well as the
These two cases are summarized in Fig. 1 (where the multiplicity factors for the individual Bragg peaks have been ignored to emphasize the structure factors).
2.2. H_{2}O
From equation (6), the macroscopic for light water is
where and n_{j} is the of atoms of type j.
Using the values in Table 1, and n_{H} = 2 (ρN_{A}/MW) = 6.69 × 10^{22} atoms of H per cm^{−3} (n_{O} = 3.35 × 10^{22} per cm^{−3}; N_{A} is Avogadro's number) yields
This ^{−1} for 5 meV neutrons at 290 K, and ∼7.7 cm^{−1} for 1 meV neutrons at 290 K (Brookhaven National Laboratory, 1976, hereafter denoted BNL 325).
is calculated using the bound scattering lengths for the nuclei. The actual for water depends on the incoming neutron energy and the water temperature. The measured from water is, for example, ∼5.7 cm

What about the other Qindependent term in equation (9)? Should the middle term in equation (9)^{2} be added to the result obtained in equation (18) to give the `total incoherent scattering'? The answer is no, because the assumption leading to equation (9) that any atom is equally likely to occupy any available site does not apply to a molecular liquid like water. [For equation (9) to apply to water, all possible molecular permutations (H_{2}O, HO_{2}, H_{3} and O_{3}) would have to be present in the liquid.]
Another way to obtain this result [equation (18)] for the and one that will give additional insight when we consider H_{2}O/D_{2}O mixtures next, is to treat the water molecule as the primary scattering entity. This approach is valid at low Q where the internal structure of the molecule is unresolvable (Qr_{m} << 1, where r_{m} is any intramolecular distance).
We start again from equation (1) and proceed as before to equation (4),
where N_{m} is the number of molecules in volume V,
and
Hence
and
2.3. H_{2}O/D_{2}O mixtures (not including H/D exchange)
Since H_{2}O/D_{2}O mixtures are used extensively to control scattering contrast in aqueous solutions, this is an important case to consider. At low Q, we can again treat the individual water molecules as the primary scattering entities (thereby ignoring their internal structure) as discussed in the previous section. We start again from equation (1), written as
where N_{m} is the number of molecules in the volume V, r_{i} is the position (e.g. the center of mass) of molecule i and b_{i} is the scattering length for the molecule.
If we assume there is no correlation between a site, r_{i}, and the type of molecule, H_{2}O or D_{2}O, at that site, then equation (25) can be developed as was done in arriving at equation (9), i.e.
where v_{m} is the volume of one molecule, and φ_{H2}_{O} (φ_{D2}_{O}) is the of H_{2}O (D_{2}O). Equation (26) shows explicitly that the flat `background' seen at low Q from such mixtures consists of a combination of diffuse coherent scattering (second term) and true (third and fourth terms).
From the scattering lengths and cross sections tabulated by Sears (1992),
or, in terms of the total
per molecule,Notice that equation (26) is the basis for the highconcentration labeling technique used to study the conformation of polymer chains in mixtures of protonated and perdeuterated chains in the melt (Akcasu et al., 1980). Equation (26) can be extended to larger molecules by including the molecular form factor in the second term, which is another demonstration that this term represents coherent, not incoherent, scattering.
2.4. H_{2}O/D_{2}O mixtures (including H/D exchange)
For water there is exchange of H and D. For this reason, the second (coherent scattering) term in equation (26) is reduced in real water, as pointed out by Arleth & Pedersen (2000). To show this explicitly we begin again from equation (20):
In this case we have three types of molecules to consider: H_{2}O, D_{2}O, and HDO or DHO (HDO and DHO are indistinguishable in terms of their scattering lengths, hence there are three and not four types of molecules to consider). Hence
where f_{H2O}, f_{D2O} and f_{HDO} are the fractions of H_{2}O, D_{2}O and HDO (or DHO) molecules in the mixture, respectively.
The number of H atoms in the mixture is , where N_{m} is the number of molecules and is the of H_{2}O that is mixed with a of D_{2}O. Similarly . Hence when the mixture is at chemical equilibrium,
Thus,
Similarly,
The terms and are given in equations (30) and (31), respectively; and
Substituting equations (36), (37) and (38) into (33) gives, after some manipulation,
For comparison, equation (26) can be written as
The only difference between equations (39) and (40) is in the diffuse coherent scattering term (the last term in each), which is reduced by a factor of two when H/D exchange is included. It is instructive to note that the `true' terms in both (40) and (39) are the same, as they must be since this scattering does not depend on where the atoms are located.
The coherent [fourth term in (39) and (40)] and incoherent [second and third terms in equations (39) and (40)] contributions to the total lowQ scattering from H_{2}O/D_{2}O mixtures are plotted in Fig. 2.
The reduction in the diffuse coherent scattering when H/D exchange is included in the calculation begs the question, where does the diffuse scattering go? To understand this, we compare the Qdependent coherent scattering terms in equations (39) and (40), which are proportional to 〈b〉^{2}:
without H/D exchange, and
with H/D exchange. Then
and
Hence the reduction in the diffuse coherent scattering due to H/D exchange is not accompanied by a corresponding increase in the intermolecular Qdependent coherent scattering. The reduction in diffuse coherent scattering therefore likely appears (although not shown here) in the intramolecular coherent scattering (at larger Q), which we have neglected in this treatment.
3. Demonstration experiment
To demonstrate the distinction between diffuse coherent scattering and true _{2}. This molecular material was chosen because titanium is one of only a few elements with a negative coherent scattering length. In addition the isotopic and nuclear spin for both oxygen and titanium are small compared to most elements. As a result, if there were a diffuse scattering term [the middle term in equation (9)] in the it would dominate the measured SANS and be easily identified by putting the scattering on an absolute scale. However, for such a term to exist in the for TiO_{2}, the positions of the titanium and oxygen atoms must be completely uncorrelated, which is physically not the case.
we have measured the smallangle neutron scattering (SANS) from titanium dioxide, TiOFrom equation (18) the incoherent for TiO_{2} is
From the ρ = 4.23 g cm^{−3}) and molecular weight (79.9 atomic mass units) of TiO_{2}, the densities are n_{Ti} = 3.19 × 10^{22} cm^{−3} and n_{O} = 2n_{Ti}. The elemental incoherent cross sections are (Sears, 1992) σ_{incoh,Ti} = 2.87 barns and σ_{incoh,O} ≃ 0 barns. Hence
(The additional Qindependent scattering that would arise if the Ti and O atoms were completely disordered is given by the second term of equation (9), which we call Σ_{c,Laue} as in the NaCl example:
where 〈b_{Ti}〉 = b_{c,Ti} = −3.44 fm and 〈b_{O}〉 = b_{c,O} = 5.80 fm (Sears, 1992). The disordered scattering term is three times larger than the true and hence should be readily apparent from the scale of the Qindependent SANS.
For the SANS, measurements a 2 mmpathlength quartz cell was filled with a coarse TiO_{2} powder. The cell was weighed before and after filling to estimate the bulk density of the powder in the cell. This was found to be 0.99 g cm^{−3}. Hence the cross sections per unit volume given in equations (46) and (47) should be multiplied by 0.99/4.23 for comparison with the scattering from this particular sample. The measurements were made on the 30 m SANS instrument on neutron guide NG7 at NIST using a wavelength of 6 Å and a sampletodetector distance of 1 m. The particle size of tens of micrometres does produce SANS at very low Q, which decays roughly as Q^{−4}. At larger Q, still in the SANS region, the scattering becomes essentially flat at a level that was put on an absolute scale by measuring the neutrons per second incident on the sample. In addition to the scattering from the sample in its cell, scattering from the empty cell was also measured and subtracted taking into account room background and the transmission of the sample. The resulting SANS for the TiO_{2} is shown in Fig. 3 along with the scattering from a reference sample of D_{2}O treated in the same way. The Qindependent scattering is slightly lower, perhaps as a result of overestimating the bulk density of the sample, than the calculated level of but far below the level expected from the sum of incoherent plus disordered coherent scattering.
4. Discussion
The ), is present in both Xray and neutron scattering. Because atomic disorder scattering may be nearly Q independent, like the authors of neutron scattering papers and texts have in some cases referred to this scattering as compositional or mixture However, this designation blurs the distinction between true which has no structural information, and what is more properly referred to as diffuse coherent scattering, which does contain structural information pertaining to the degree of atomic disorder.
discussed here, which arises solely from the lack of any correlation between an atom's location and that atom's nuclear spin state or nuclear isotope, is unique to neutron scattering. There is no analog in Xray scattering. Scattering that arises from atomic disorder, such a Laue monotonic scattering (Warren, 1969The salient point is that incoherent cross sections are simply additive. Hence the macroscopic
for a material is readily calculated fromwhere N is the number of atomic species in the material, n_{j} is the of atomic species j and σ_{incoh,j} is the tabulated for element j. For any material with an appreciable amount of hydrogen, a good approximation for the bulk incoherent is simply, Σ_{incoh} ≃ n_{H}σ_{incoh,H}, since the incoherent for hydrogen is so much larger than that for other elements.
Caveat
The coherent and incoherent cross sections tabulated by Sears (1992) are calculated from the bound scattering lengths for nuclei. The actual cross sections depend on the incoming neutron energy and sample temperature, especially for light elements. For the tabulated cross sections are generally a lower limit. The measured from hydrogen, for example, can be considerably larger than its bound value. For example,
[from tables of scattering lengths given by Sears (1992)] and
[measured for 1 meV, or 0.9 nm, neutrons at 290 K (BNL 325)].
Footnotes
^{1}Equivalent systems refer to all possible configurations of the atoms, including their nuclear isotopes and spin states, which may affect the scattering of a neutron beam incident on a sample.
^{2}The middle term in equation (9) integrated over dΩ for H_{2}O would be , where n is the number of atoms per unit volume. For f_{H} = 2/3, f_{O} = 1/3, , and n = 3(3.35 × 10^{22}) atoms cm^{−3}, Σ_{Laue} = 0.255 cm^{−1}.
Acknowledgements
Helpful discussions with B. Hammouda, D. Mildner, J. Barker and R. Cappelletti and correspondence with A. Brûlet are gratefully acknowledged.
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