Detection of ordering

5.6.2 Detection of ordering

The determination of an ordered superlattice is usually done by means of the X-ray powder technique. In a disordered solution every plane of atoms is statistically identical and, as discussed in Chapter 4, there are reflections missing in the powder pattern of the material. In an ordered lattice, on the other hand, alternate planes become A-rich and B-rich, respectively, so that these ‘absent’ reflections are no longer missing but appear as extra superlattice lines. This can be seen from Figure 5.13: while

Rays lⲐ2 out of phase but amplitude not the same

Plane A′ Plane B Plane A

Figure 5.13 Formation of a weak 1 0 0 reflection from an ordered lattice by the interference of diffracted rays of unequal amplitude.

256 Physical Metallurgy and Advanced Materials

3000 Degree of order

Domain size (Å)

Degree of order 0.6 Domain size

1000 10 000 100 000 t (min)

Figure 5.14 Degree of order ( ×) and domain size (◦) during isothermal annealing at 350 ◦

C after quenching from 465 ◦

C (after Morris, Besag and Smallman, 1974; courtesy of Taylor & Francis).

the diffracted rays from the A planes are completely out of phase with those from the B planes their intensities are not identical, so that a weak reflection results.

Application of the structure factor equation indicates that the intensity of the superlattice lines is proportional to 2 |F 2 |=S (f A −f B ) 2 , from which it can be seen that in the fully disordered alloy, where S = 0, the superlattice lines must vanish. In some alloys, such as copper–gold, the scattering factor difference ( f A −f B ) is appreciable and the superlattice lines are therefore quite intense and easily detectable. In other alloys, however, such as iron–cobalt, nickel–manganese and copper–zinc, the term ( f A −f B ) is negligible for X-rays and the superlattice lines are very weak; in copper–zinc, for example, the ratio of the intensity of the superlattice lines to that of the main lines is only about 1:3500. In some cases special X-ray techniques can enhance this intensity ratio; one method is to use an X-ray wavelength near to the absorption edge when an anomalous depression of the f -factor

occurs which is greater for one element than for the other. As a result, the difference between f A and

f B is increased. A more general technique, however, is to use neutron diffraction, since the scattering factors for neighboring elements in the Periodic Table can be substantially different. Conversely, as Table 4.4 indicates, neutron diffraction is unable to show the existence of superlattice lines in

Cu 3 Au, because the scattering amplitudes of copper and gold for neutrons are approximately the same, although X-rays show them up quite clearly. Sharp superlattice lines are observed as long as order persists over lattice regions of about 10 −3 mm, large enough to give coherent X-ray reflections. When long-range order is not complete the superlattice lines become broadened, and an estimate of the domain size can be obtained from a measurement of the line breadth, as discussed in Chapter 4. Figure 5.14 shows variation of order S and domain size as determined from the intensity and breadth of powder diffraction lines. The domain sizes determined from the Scherrer line-broadening formula are in very good agreement with those observed by TEM. Short-range order is much more difficult to detect but nowadays direct measuring devices allow weak X-ray intensities to be measured more accurately, and as a result considerable information on the nature of short-range order has been obtained by studying the intensity of the diffuse background between the main lattice lines.

Physical properties 257

Figure 5.15 One unit cell of the orthorhombic superlattice of CuAu, i.e. CuAu 11 ( from Pashley and Presland, 1958–9; courtesy of the Institute of Materials, Minerals and Mining).

0.05 µ (a)

0.05 µ (b)

Figure 5.16 Electron micrographs of (a) CuAu 11 and (b) CuAu 1 ( from Pashley and Presland, 1958–9; courtesy of the Institute of Materials, Minerals and Mining).

High-resolution transmission microscopy of thin metal foils allows the structure of domains to be examined directly. The alloy CuAu is of particular interest, since it has a face-centered tetragonal structure, often referred to as CuAu 1 below 380 ◦

C and the disordering tem- perature of 410 ◦

C, but between 380 ◦

C it has the CuAu 11 structures shown in Figure 5.15. The (0 0 2) planes are again alternately gold and copper, but halfway along the a-axis of the unit cell the copper atoms switch to gold planes and vice versa. The spacing between such periodic anti-phase domain boundaries is five unit cells or about 2 nm, so that the domains are easily resolvable in TEM, as seen in Figure 5.16a.

258 Physical Metallurgy and Advanced Materials The isolated domain boundaries in the simpler superlattice structures such as CuAu 1, although not

in this case periodic, can also be revealed by electron microscopy, and an example is shown in Figure 5.16b. Apart from static observations of these superlattice structures, annealing experiments inside the microscope also allow the effect of temperature on the structure to be examined directly. Such observations have shown that the transition from CuAu 1 to CuAu 11 takes place, as predicted, by the nucleation and growth of anti-phase domains.

Worked example

The X-ray diffractometer data given below were obtained from a partially ordered 75 at.% Cu/25 at.% Au alloy (a-spacing = 0.3743 nm), using CuK α radiation (λ average = 0.15418 nm). Using this and the other information provided, calculate the ordering parameter S for this alloy.

Diffraction Integrated intensity Peak

Lorentz polarization factor

Bragg’s law: λ = 2d sin θ, λ = 0.15418 nm sin θ

= = 1.335, so θ 2d 100

2d = 0.3743 = 2.672, so θ 200 = 24.325 (LPF) 100 = 44.28; (LPF) 200 = 9.29. Plot f Cu and f Au vs sin θ λ and read off appropriate values. This gives:

f Cu = 24.6 for (1 0 0)

f Cu = 19.4 for (2 0 0)

f Au = 71.0 for (1 0 0)

f Au = 60.2 for (2 0 0)

F 100 = S( f Au −f Cu ) and F 200 =f Au + 3f Cu . S (degree of order) P A −C = A 1 −C A , where P A = probability of A sites filled by A atoms, and C A = atom

fraction of A atoms.

I 100

S 2 (f Au −f Cu ) 2 × (LPF) 100

I 200 = (f Au + 3f Cu ) 2 × (LPF) . 200

Physical properties 259 Substitute in expression:

so S = 0.77.