Determination of lattice parameters Perhaps the most common use of the powder method

5.3.4.2 Determination of lattice parameters Perhaps the most common use of the powder method

is in the accurate determination of lattice parame- ters. From the Bragg law we have the relation a D p

N/ can be used to determine the lattice parameter of a

material. Several errors are inherent in the method, however, and the most common include shrinkage of the film during processing, eccentricity of the speci- men and the camera, and absorption of the X-rays in the sample. These errors affect the high-angle diffrac- tion lines least and, consequently, the most accurate parameter value is given by determining a value of

Figure 5.16 (a) and (b) Phase-boundary determination

a from each diffraction line, plotting it on a graph using lattice parameter measurements .

against an angular function 1 of the cos 2 -type and then ° .

The determination of precision lattice parameters is camera at various temperatures or by quenching the of importance in many fields of materials science, par-

powder sample from the high temperature to room ticularly in the study of thermal expansion coefficients,

temperature (in order to retain the high temperature density determinations, the variation of properties with

state of solid solution down to room temperature) and composition, precipitation from solid solution, and

then taking a powder photograph at room temperature. thermal stresses. At this stage it is instructive to con-

sider the application of lattice parameter measurements

5.3.4.3 Line-broadening

to the determination of phase boundaries in equilibrium Diffraction lines are not always sharp because of diagrams, since this illustrates the general usefulness

various instrumental factors such as slit size, specimen of the technique. The diagrams shown in Figures 5.16a

condition, and spread of wavelengths, but in addition and 5.16b indicate the principle of the method. A varia-

the lines may be broadened as a result of lattice strain tion of alloy composition within the single-phase field,

˛ in the region of the crystal diffracting and also its

, produces a variation in the lattice parameter, a, limited dimension. Strain gives rise to a variation of the since solute B, which has a different atomic size to

interplanar spacing d and hence diffraction occurs the solvent A, is being taken into solution. However,

at the phase boundary solvus this variation in a ceases, because at a given temperature the composition of the

(5.12) ˛ -phase remains constant in the two-phase field, and

the marked discontinuity in the plot of lattice parameter versus composition indicates the position of the phase

of the crystal diffracting the X-rays is small, 2 then boundary at that temperature. The change in solid solu-

this also gives rise to an appreciable ‘particle-size’ bility with temperature may then be obtained, either by

broadening given by the Scherrer formula taking diffraction photographs in a high-temperature

(5.13) 1 Nelson and Riley suggest the function

2 The optical analogue of this effect is the broadening of cos 2 cos 2

C diffraction lines from a grating with a limited number of lines.

The characterization of materials 141 where t is the effective particle size. In practice this

(see Chapter 8), and the aggregation of lattice defects size is the region over which there is coherent diffrac-

(see Chapter 4).

tion and is usually defined by boundaries such as dislo- cation walls. It is possible to separate the two effects

5.3.4.5 The reciprocal lattice concept by plotting the experimentally measured broadening ˇ

The Bragg law shows that the conditions for diffraction depend on the geometry of sets of crystal planes. To simplify the more complex diffraction problems, use is made of the reciprocal lattice concept in which the

5.3.4.4 Small-angle scattering sets of lattice planes are replaced by a set of points, The scattering of intensity into the low-angle region

this being geometrically simpler. ° ⊳ arises from the presence of inhomo-

The reciprocal lattice is constructed from the real geneities within the material being examined (such as

lattice by drawing a line from the origin normal to small clusters of solute atoms), where these inhomo-

the lattice plane hkl under consideration of length, Ł geneities have dimensions only 10 to 100 times the

d , equal to the reciprocal of the interplanar spacing wavelength of the incident radiation. The origin of

d hkl . The construction of part of the reciprocal lattice the scattering can be attributed to the differences in

from a face-centred cubic crystal lattice is shown in electron density between the heterogeneous regions

Figure 5.17.

Included in the reciprocal lattice are the points ticles afford the most common source of scattering;

and the surrounding matrix, 1 so that precipitated par-

which correspond not only to the true lattice planes other heterogeneities such as dislocations, vacancies

with Miller indices (hkl) but also to the fictitious and cavities must also give rise to some small-angle

planes (nh, nk, nl) which give possible X-ray scattering, but the intensity of the scattered beam will

reflections. The reciprocal lattice therefore corresponds

be much weaker than this from precipitated particles. to the diffraction spectrum possible from a particular The experimental arrangement suitable for this type of

crystal lattice and, since a particular lattice type is study is shown in Figure 5.13b.

characterized by ‘absent’ reflections the corresponding Interpretation of much of the small-angle scatter

spots in the reciprocal lattice will also be missing. It data is based on the approximate formula derived by

can be deduced that a fcc Bravais lattice is equivalent Guinier,

to a bcc reciprocal lattice, and vice versa.

2 I D Mn 2 I e 2 2 ε A simple geometrical construction using the recipro- R / 2 ] (5.14) cal lattice gives the conditions that correspond to Bragg

where M is the number of scattering aggregates, or on the origin of the reciprocal lattice, then a sphere of particles, in the sample, n represents the difference in

number of electrons between the particle and an equal points which correspond to the reflecting planes of a volume of the surrounding matrix, R is the radius of

stationary crystal. This can be seen from Figure 5.18,

gyration of the particle, I e is the intensity scattered

in which the reflecting plane AB has a reciprocal point at d Ł . If d Ł lies on the surface of the sphere of radius

the wavelength of X-rays. From this equation it can

be seen that the intensity of small-angle scattering is zero if the inhomogeneity, or cluster, has an electron Ł d D 1/d hkl D (5.16)

density equivalent to that of the surrounding matrix, even if it has quite different crystal structure. On a

plot of log 10 I as a function of ε 2 , the slope near the

origin, ε D 0, is given by

2 / 2 ⊳R 2 log

10 e

which for Cu K˛ radiation gives the radius of gyration of the scattering aggregate to be

RD 0.0645 ð P 1/2 nm

It is clear that the technique is ideal for studying regions of the structure where segregation on too fine

a scale to be observable in the light microscope has occurred, e.g. the early stages of phase precipitation

1 The halo around the moon seen on a clear frosty night is the best example, obtained without special apparatus, of the

scattering of light at small angles by small particles. Figure 5.17 fcc reciprocal lattice .

142 Modern Physical Metallurgy and Materials Engineering circle corresponds to the powder halo discussed previ-

ously. From Figure 5.19 it can be seen that the radius of the sphere describing the locus of the reciprocal lattice point (hkl) is 1/d ⊲hkl⊳ and that the angle of devi-

⊲hkl⊳

which is the Bragg condition.