Short Range Order

When these measurements were planned, it was not expected that a high degree of structured ordering between the diffusing species and vacant sites would be seen at the temperatures used. Ordering has long been observed at low temperatures and is regularly shown in phase diagrams (such as figure 2-F) for the system. The phase boundary for this is

108 commonly given at around 50 𝐾 across a very wide range of concentrations. A phase

transition at this temperature has been shown in neutron scattering measurements by Anderson, Ross, and Carlile (1978a). Further work on this region (marginally above the 50 𝐾

transition) by Blaschko (1984) suggests possible ordering.

The ordering taking place between the diffusing species and vacancies in the lattice is analogous to the ordering seen in binary alloys, as described by Clapp and Moss (1966). This theory predicts several potential ordered structures depending on sign and relative intensity of first and second nearest neighbour potentials (𝑉1 and 𝑉2 respectively). In a binary FCC

system, each of these structures give rise to a peak in 𝑆(𝑄) centred at the (ℎ, 𝑘, 𝑙) index of the relevant superlattice indicated in figure 4-CC.

As SRO was not anticipated, nor could it be easily defined in the preliminary measurements that were made, no direct measurement of 𝑆(𝑄) was performed. As such, little analysis as to the exact nature of the ordering seen has been possible. However, the general form of 𝑆(𝑄)

is given by equation (3-40):

𝑆(𝑄) = 𝑐(1 − 𝑐)

1 +𝑐(1 − 𝑐)𝑉(𝑄) 𝑘𝐵𝑇

Where 𝑉(𝑄) is the Fourier transform of the pair potential to second nearest neighbours. For any specific direction in an FCC lattice, this is given by Bull (2001) as:

4-CC Possible peaks from SRO (FCC) (Clapp & Moss, 1968)

109

𝑉(𝑄) = 4𝑉₁(cos 𝜋ℎ cos 𝜋𝑘 + cos 𝜋ℎ cos 𝜋𝑙 + cos 𝜋𝑘 cos 𝜋𝑙)

+ 2𝑉2(cos 2𝜋ℎ + cos 2𝜋𝑘 + cos 2𝜋𝑙) (4-2)

Figure 4-DD shows possible forms that would result from the three allowed ordered states described by Clapp & Moss for an FCC lattice81_{. The vertical dashed lines represent the value}

of 𝑄 associated with each of the three structures described in figure 4-CC. These have been colour coded to match the 𝑆(𝑄) plot that corresponds to the relevant structure. The

parameters chosen are arbitrary (with the exception of having the correct V2/V1 ratio for each structure) and were simply chosen for illustrative purposes. In the polycrystalline average (as employed here) the peaks in 𝑆(𝑄) may not line up exactly with the predicted

𝑄_ℎ𝑘𝑙 values82_.

Since the measured Lorentzian width from jump diffusion in coherent scattering is inversely proportional to 𝑆(𝑄), the most obvious sign of ordering is an inverted peak the broadening as a function of 𝑄. However, further reduction occurs across the entire measured range. This reduction is heavily dependent on the specific form of the superstructure. Ordering in the

(1,1

2, 0) plane will cause a much greater reduction in the measured width at low Q than

either (1

2, 1 2,

1

2) or (1,0,0). This makes accurate determination of 𝑆(𝑄) across the entire

range of the QENS measurement vital. As the narrowing in this case is not accurately characterised, even the unmasked data for the deuteride sample cannot be considered reliable. As such, the values Arrhenius parameters calculated from these data can only be considered an approximation to the system.

As the peak in 𝑆(𝑄) may not line up with that of Chudley-Elliott broadening, it becomes extremely difficult to accurately locate either in these measurements. The fact that the jump lengths calculated from the masked deuterium data appear to agree well with the expected jump length suggests that these peaks may be very close in 𝑄. This appears to rule out the SRO peak being at (1

2, 1 2,

1

2). Due to the similarity in the peaks from the other possible

81 _{Plot generated from a script originally authored by Dr D.J. Bull} 82 _{This direct relationship between 𝑄}

ℎ𝑘𝑙 and the peak in 𝑆(𝑄) only truly applies for measurements on a single

110 superstructures, only a tentative guess can be made between them from these

measurements.

The previously mentioned work by Blaschko (1984) suggests that, just above the 50 𝐾 phase transition at a concentration of around 𝑐 = 0.7, a structure corresponding to a peak at

(1,1

2, 0) is visible in neutron scattering measurements.

Relatively recent (in terms of the published work for this system) thermal desorption spectra (TDS) taken by Rybalko, Morozov, Neklyudov, and Kulish (2001) appears to show the

persistence of an ordered structure well above the temperatures previously noted across a wide range of initial concentrations. Their work suggests that, even with an initial

concentration as low as 0.54, a sudden shift in concentration that is consistent with a phase transition that has no associated distortion to the host lattice can be seen at around 150 𝐾. As the desorption spectra described start from a temperature above 50 𝐾, it is possible that this ordered structure is not the one previously noted below this temperature. This appears to be consistent with the work of Blaschko (1984) but extends the temperature range of this phase considerably. Since TDS measures loss of concentration, it does not give a direct answer as to whether this phenomenon would persist at higher temperatures should the concentration be maintained (as is the case in this work).

Figure 4-W shows that the apparent location of the peak of 𝑆(𝑄) appears to coincide the 𝑄

range that would correspond to (1,0,0) ordering across the entire temperature range. This type of structure has not previously been reported in this system. Without direct

measurements of 𝑆(𝑄), it is difficult to rule out possible (1,1

2, 0) ordering. This type of

ordering appears in the literature for the mixed (𝛼 + 𝛽) phase region below 150 𝐾, but it has not previously been reported at the temperatures measured in this work (around 300 𝐾

higher).

While it is not possible to draw a solid conclusion on the exact form of ordering present, the evidence for a high level of ordering at temperatures up to 470 𝐾 (and possibly beyond) is conclusive. This strongly suggests that previous assumptions about the strength of the interactions between the diffusing particles may not accurately describe the system and that

111