Representation of Powder Diffraction Patterns

8.2 Representation of Powder Diffraction Patterns

In a typical experiment the intensity, diffracted by a polycrystalline sample, is mea- sured as a function of Bragg angle, 2 θ. Hence, powder diffraction patterns are usu- ally plotted in the form of the measured intensity, Y , as the dependent variable versus the Bragg angle as the independent variable; see Figs. 8.6 (top) and 8.7a,c. In rare instances, for example, when there are just a few very intense Bragg peaks and all others are quite weak, or when it is necessary to directly compare diffraction pat- terns collected from the same material using different wavelengths, the scales of one or both axes may be modified for better viewing and easier comparison.

200 Hexamethylenetetramine, Cu Kα 3 100 counts)

Y (10

10 5 Bragg angle, 2θ (deg.)

Bragg angle, 2θ (deg.)

Bragg angle, 2θ (deg.)

Fig. 8.6 The powder diffraction pattern 6 of hexamethylenetetramine collected using Cu K α radia- tion and plotted as the measured intensity in counts (top), common logarithm (middle), and square root (bottom) of the total number of registered photon counts versus 2 θ.

6 Powder diffraction data were collected on a Scintag XDS2000 powder diffractometer using Cu K α radiation and cooled Ge(Li) solid state detector. The counting time was 10 s in every point;

the data were collected with a 0 .025 ◦ step of 2 θ.

158 8 The Powder Diffraction Pattern

LaB 6 , Cu Kα Y (arb. units)

LaB 6 , Cu Kα

Y (arb. units)

a 2θ (deg.)

b 1/d 2 (Å) −2

LaB 6 , Cu Kα Y (arb. units)

LaB 6 , Mo Kα

Y (arb. units)

c 2θ (deg.)

d −1 1/d (Å)

LaB 6 , Mo Kα Y (arb. units)

LaB 6 , Cu Kα

(arb. units)

Fig. 8.7 Two powder diffraction patterns of LaB 6 collected using different wavelengths with the scattered intensity plotted versus different independent variables. 7 Each plot contains the same number of Bragg peaks, which can be observed below 2 θ∼ = 140 ◦ when using Cu K α radiation.

When the former is true (i.e., there are few extremely strong Bragg peaks while all others are weak), the vertical axis can be calibrated as a logarithm of intensity (Fig. 8.6, middle) or its square root (Fig. 8.6, bottom). This changes the scale and enables better visualization of the low-intensity features. In the example shown in Fig. 8.6, the middle (logarithmic) plot reveals all weak Bragg peaks in addition to the nonlinearity of the background, and the details of the intensity distribution around

the bases of the strongest peaks. The Y 1 /2 scale is equivalent to the plot of statistical errors of the measured intensities (Sect. 12.3.1), in addition to better visualization of weak Bragg peaks.

7 Powder diffraction data were collected on a Scintag XDS2000 powder diffractometer using Cu K α radiation and cooled Ge(Li) solid-state detector and on a Rigaku TTRAX rotating anode

powder diffractometer using Mo K α radiation with diffracted beam monochromator and scintilla- tion detector. The data were collected with a 0 .02 ◦ step of 2 θ using Cu Kα and with a 0.01 ◦ step using Mo K α radiations.

8.3 Understanding of Powder Diffraction Patterns 159 Various horizontal scales alternative to the Bragg angle, see Fig. 8.7, are usually

wavelength-independent and their use is mostly dictated by special circumstances. For example, d-spacing (Fig. 8.7e) is most commonly used in the time-of-flight (TOF) experiments: according to (6.1), the wavelength is the inverse of the veloc- ity of the particle (neutron). The time-of-flight from the specimen to the detector is therefore, directly proportional to d. This scale, however, reduces the visual resolu- tion in the low d range (equivalent to high Bragg angle range, see (7.9)) when used in combination with X-ray diffraction data. In TOF experiments, the actual resolu- tion of the diffraction pattern is reduced at low d, that is, at high neutron velocities.

The second scale is 1 /d = 2 sin θ/λ (see Fig. 8.7d,f). It results in only slightly reduced resolution at high Bragg angles when compared to the 2 θ scale. Recalling that 1 /d = d ∗ , this type of the plot is a one-dimensional projection of the recip- rocal lattice and it is best suited for direct comparison of powder diffraction data collected using different wavelengths. The similarity of these two diffraction pat- terns is especially impressive after comparing them when both are plotted versus 2 θ (Fig. 8.7a,c).

The third is the q-values scale, where q = 1/d 2 = 4 sin 2 θ/λ 2 , which provides the best resolution at high Bragg angles when compared to other wavelength- independent scales, see Fig. 8.7b. This scale results in the equally spaced Bragg peaks when the crystal system is cubic (see Sect. 14.6). In cases of lower symme- try crystal systems, only certain types of Bragg peaks are equally spaced along the q -axis and in some instances, the q-scaled powder diffraction pattern may be used to assign indices and/or examine the relationships between the lattice parameters of the material with the unknown crystal structure.