Single-order and multiple-order reflections

Helliwell, J. R.

doi:10.1107/97809553602060000577

International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

pdf | chapter contents | chapter index | related articles

International Tables for Crystallography (2006). Vol. C. ch. 2.2, pp. 27-29

Section 2.2.1.3. Single-order and multiple-order reflections

J. R. Helliwell^a

^a Department of Chemistry, University of Manchester, Manchester M13 9PL, England

2.2.1.3. Single-order and multiple-order reflections

| top | pdf |

In Laue geometry, several relp's can occur in a Laue spot or ray. The number of relp's in a given spot is called the multiplicity of the spot. The number of spots of a given multiplicity can be plotted as a histogram. This is known as the multiplicity distribution. The form of this distribution is dependent on the ratio λ_max/λ_min. The multiplicity distribution in Laue diffraction is considered in detail by Cruickshank, Helliwell & Moffat (1987).

Any relp nh, nk, nl (n integer) will be stimulated by a wavelength λ/n since d_nhnknl = d_hkl/n, i.e. $[{\lambda\over n}=2{d_{hkl}\over n}\sin\theta.\eqno(2.2.1.5)]$ However, d_nhnknl must be $[\gt]$ d_min as otherwise the reflection is beyond the sample resolution limit.

If h, k, l have no common integer divisor and if 2h, 2k, 2l is beyond the resolution limit, then the spot on the Laue diffraction photograph is a single-wavelength spot. The probability that h, k, l have no common integer divisor is $[\eqalignno{Q&=\left[1-{1\over2^3}\right]\left[1-{1\over3^3}\right]\left[1-{1\over5^3}\right]\ldots \cr &=0.832\ldots\,. &(2.2.1.6)}]$ Hence, for a relp where d_min $[\lt]$ d_hkl $[\lt]$ 2d_min there is a very high probability (83.2%) that the Laue spot will be recorded as a single-wavelength spot. Since this region of reciprocal space corresponds to 87.5% (i.e. 7/8) of the volume of reciprocal space within the resolution sphere then 0.875 × 0.832 = 72.8% is the probability for a relp to be recorded in a single-wavelength spot. According to W. L. Bragg, all Laue spots should be multiple. He reasoned that for each h, k, l there will always be a 2h, 2k, 2l etc. lying within the same Laue spot. However, as the resolution limit is increased to accommodate this many more relp's are added, for which their hkl's have no common divisor.

The above discussion holds for infinite bandwidth. The effect of a more experimentally realistic bandwidth is to increase the proportion of single-wavelength spots.

The number of relp's within the resolution sphere is $[{4\over3}\ {\pi d^{*3}_{\max}\over V^*},\eqno (2.2.1.7)]$ where $[d^{*}_{\max}]$ = 1/d_min and V* is the reciprocal unit-cell volume.

The number of relp's within the wavelength band λ_max to λ_min, for $[\lambda_{\rm max} \lt 2/d^*_{\rm max}]$ , is (Moffat, Schildkamp, Bilderback & Volz, 1986) $[{\pi\over4}\ {(\lambda_{\max}-\lambda_{\min})d^{*4}_{\max}\over V^*}.\eqno (2.2.1.8)]$ Note that the number of relp's stimulated in a 0.1 Å wavelength interval, for example between 0.1 and 0.2 Å, is the same as that between 1.1 and 1.2 Å, for example. A large number of relp's are stimulated at one orientation of the crystal sample.

The proportion of relp's within a sphere of small d* (i.e. at low resolution) actually stimulated is small. In addition, the probability of them being single is zero in the infinite-band-width case and small in the finite-bandwidth case. However, Laue geometry is an efficient way of measuring a large number of relp's between $[d^{*}_{\max}]$ and $[d^{*}_{\max}/2]$ as single-wavelength spots.

The above is a brief description of the overall multiplicity distribution. For a given relp, even of simple hkl values, lying on a ray of several relp's (multiples of hkl), a suitable choice of crystal orientation can yield a single-wavelength spot. Consider, for example, a spot of multiplicity 5. The outermost relp can be recorded at long wavelength with the inner relp's on the ray excluded since they need λ's greater than λ_max (Fig. 2.2.1.3 ). Alternatively, by rotating the sample, the innermost relp can be measured uniquely at short wavelength with the outer relp's excluded (they require λ's shorter than λ_min). Hence, in Laue geometry several orientations are needed to recover virtually all relp's as singles. The multiplicity distribution is shown in Fig. 2.2.1.4 as a function of λ_max/λ_min (with the corresponding values of δλ/λ_mean).

Figure 2.2.1.3| top | pdf |

A multiple component spot in Laue geometry. A ray of multiplicity 5 is shown as an example. The inner point A corresponds to d and a wavelength λ, the next point, B, is d/2 and wavelength λ/2. The outer point E corresponds to d/5 and λ/5. Rotation of the sample will either exclude inner points (at the λ_max surface) or outer points (at the λ_min surface) and so determine the recorded multiplicity.

Figure 2.2.1.4| top | pdf |

The variation with M = λ_max/λ_min of the proportions of relp's lying on single, double, and triple rays for the case $[\lambda_{\rm max} \lt 2/d^*_{\rm max}]$ . From Cruickshank, Helliwell & Moffat (1987).

References

Cruickshank, D. W. J., Helliwell, J. R. & Moffat, K. (1987). Multiplicity distribution of reflections in Laue diffraction. Acta Cryst. A43, 656–674.Google Scholar

Moffat, K., Schildkamp, W., Bilderback, D. H. & Volz, K. (1986). Laue diffraction from biological samples. Nucl. Instrum. Methods, A246, 617–623.Google Scholar

International Tables for Crystallography (2006). Vol. C. ch. 2.2, pp. 27-29