Modifications and Considerations for the Deuteride Sample

Scattering from deuterium contains considerable components from both its coherent and incoherent cross sections (𝑋𝑠_𝑐𝑜ℎ and 𝑋𝑠_𝑖𝑛𝑐). Therefore, the assumption made for the hydrogen case (total scattering can be reasonably be approximated by treating it as purely incoherent) is not valid for measurements on deuterium. It is therefore necessary to consider the contributions from both coherent and incoherent scattering.

Ross, Kemali, and Bull (1999) shows that the coherent quasielastic scattering from a diffusing lattice gas can be described by a modified form of the Chudley-Elliott model where the measured Lorentzian is of the form:

𝑆𝑐𝑜ℎ(𝑄, 𝜔) =

1 𝜋

𝑆(𝑄)Γ′(𝑄)

57 and the broadening predicted by the incoherent model Γ_𝐶𝐸 (equation (3-36)) for any real concentration 𝑐 is replaced with the modified function Γ′(𝑄):

Γ′(𝑄) = Γ𝐶𝐸(𝑄)

𝑐(1 − 𝑐)

𝑆(𝑄) _(3-39)

Here, 𝑆(𝑄) is the static structure factor of the diffuse scattering from the lattice gas as given in the Mean Field Limit by Clapp and Moss (1968):

𝑆(𝑄) = 𝑐(1 − 𝑐)

1 +𝑐(1 − 𝑐)𝑉(𝑄)_𝑘

𝐵𝑇 (3-40)

Where 𝑉(𝑄) is the Fourier transform of the mean field real space inter-atomic energy potential 𝑉(𝑟).

It can be seen that, in a system where the particle interaction 𝑉(𝑄) is negligible, or in the limit of high temperature (𝑇 → ∞), 𝑆(𝑄) simply becomes 𝑐(1 − 𝑐). In either of these cases, substituting equation (3-40) into equations (3-38) and (3-39) gives a concentration

dependence in the amplitude of the measured Lorentzian that is independent of 𝑄:

𝑆𝑐𝑜ℎ(𝑄, 𝜔) =

𝑐(1 − 𝑐) 𝜋

Γ′_(𝑄)

(Γ′_(𝑄))2_{+ 𝜔}2 _(3-41)

and cancels the concentration dependence of the Lorentzian broadening, leaving:

Γ_coh(𝑄) = Γ′_{(𝑄) = Γ} 𝐶𝐸(𝑄)

(3-42) Equation (3-42) shows that, in the case of a non-interacting system (or in any system that can be reasonably approximated as such), in the limit of low concentration (as assumed by the standard form of the Chudley-Elliott model), the broadening from coherent scattering is identical to that from incoherent scattering. Thus, it is possible to fit the broadening seen in the total scattering of such a system using the unmodified Chudley-Elliott model (equation (3-36)).

Returning to equation (3-39) it follows that, for any system where particle interactions cannot reasonably be neglected, measured broadening in coherent scattering has a

58 dependence on concentration (that is independent of 𝑄) of the form 𝑐(1 − 𝑐). Broadening is further reduced around values of 𝑄 where peaks occur in the static structure factor 𝑆(𝑄). It has been noted that this phenomenon is qualitatively similar to De Gennes narrowing (De Gennes, 1959; Sinha & Ross, 1988).

In any real QENS experiment, the total scattering function is measured. This can be expressed as the sum of the contributions from coherent and incoherent scattering weighted by the relative magnitudes of their cross sections (𝑋𝑠𝑐𝑜ℎ and 𝑋𝑠𝑖𝑛𝑐):

𝑆(𝑄, 𝜔) = ( 𝑋𝑠𝑐𝑜ℎ

𝑋𝑠_𝑐𝑜ℎ+ 𝑋𝑠_𝑖𝑛𝑐) 𝑆𝑐𝑜ℎ(𝑄, 𝜔) + (

𝑋𝑠_𝑖𝑛𝑐

𝑋𝑠_𝑐𝑜ℎ+ 𝑋𝑠_𝑖𝑛𝑐) 𝑆𝑖𝑛𝑐(𝑄, 𝜔) _(3-43)

Similarly, in this work, it has been assumed that the broadening from any individual form of translational diffusion (having a characteristic jump length 𝑙 and mean residence time 𝜏) seen in the total scattering can be approximated as a linear combination of its coherent and incoherent parts48_: Γ(𝑄) = ( 𝑋𝑠𝑐𝑜ℎ 𝑋𝑠𝑐𝑜ℎ+ 𝑋𝑠𝑖𝑛𝑐 ) Γ_𝑐𝑜ℎ(𝑄) + ( 𝑋𝑠𝑖𝑛𝑐 𝑋𝑠𝑐𝑜ℎ+ 𝑋𝑠𝑖𝑛𝑐 ) Γ_𝑖𝑛𝑐(𝑄) (3-44) Expanding this expression with the concentration dependent forms of Γ_𝑐𝑜ℎ(𝑄) and Γ_𝑖𝑛𝑐(𝑄)

given in equations (3-39) and (3-37) yields:

Γ(𝑄) = ( 𝑋𝑠𝑐𝑜ℎ 𝑋𝑠𝑐𝑜ℎ+ 𝑋𝑠𝑖𝑛𝑐 )𝑐(1 − 𝑐) 𝑆(𝑄) ℏ 𝜏(1 − sin(𝑄𝑙) 𝑄𝑙 ) + ( 𝑋𝑠𝑖𝑛𝑐 𝑋𝑠𝑐𝑜ℎ+ 𝑋𝑠𝑖𝑛𝑐 ) (1 − 𝑐)ℏ 𝜏(1 − sin(𝑄𝑙) 𝑄𝑙 ) (3-45)

Since the fractions relating to the relative contributions from the relevant cross sections can be calculated explicitly from known values, these can be reduced to constant for each case. Factorising and rearranging equation (3-45) then gives:

Γ(𝑄) = ( 1 𝑆(𝑄) 0.7317𝑐 + 0.2683) (1 − 𝑐) ℏ 𝜏 (1 − sin(𝑄𝑙) 𝑄𝑙 ) _(3-46)

59 If it is possible to consider the system to be free of contributions from the static structure factor 𝑆(𝑄), then it is possible to fit the Lorentzian broadening using the form:

Γ(𝑄) = 𝐴 (1 −sin(𝑄𝑙)

𝑄𝑙 ) _(3-47)

where,

𝐴 = (0.7317𝑐 + 0.2683)(1 − 𝑐)ℏ

𝜏 (3-48)

A further complication to this is described in work by Cook, Richter, Hempelmann, Ross, and Züchner (1991), where it is shown that a single form of translational motion with a

characteristic timescale 𝜏 has independent timescales associated with its coherent (𝜏𝑐𝑜ℎ) and

incoherent (𝜏_𝑖𝑛𝑐) scattering that are related to mobility and tracer correlation factors (𝑓_𝑚

and 𝑓_𝑡) via Haven’s ratio𝐻_𝑅:

𝜏_𝑐𝑜ℎ 𝜏_𝑖𝑛𝑐 =

𝑓_𝑡

𝑓_𝑚 = 𝐻𝑅 _(3-49)

Their work proposed a method using spin polarisation analysis for separating the coherent and incoherent scattering from deuterium dissolved in niobium to acquire direct

measurements of each component49_.

The interpretation of this in a single measurement of the total scattering is problematic.

Fitting an unknown number of components to measured data is a classic “ill posed problem”

in mathematics. In this work, a Bayesian model selection50 _{approach has been taken. This}

method is widely used for distinguishing whether “one or more” Lorentzian components are

necessary to achieve an accurate fit to any dataset. It also provides a good guide to the widths of these components where they are sufficiently different. Even with high resolution data, it is likely that similar features will be indistinguishable and can be fitted accurately by functions that represent their arithmetic mean. Applying this logic to equation (3-46) gives:

49 _{This method was not employed in this work.} 50 _{See section 3.9.4}

60 Γ(𝑄) = ( 1 𝑆(𝑄) 0.7317𝑐 + 0.2683) (1 − 𝑐) ℏ 〈𝜏〉_𝐻𝑅 (1 − sin(𝑄𝑙) 𝑄𝑙 ) (3-50)

Where 〈𝜏〉𝐻𝑅 is the measured residence time that represents the mean of the two

components weighted by Haven’s ratio.

The application of this logic to the measurements in this work (as well as how it affects the analysis and outcomes in this document) is discussed in the sections about the

measurements on deuterium (section 4.5).