Seemann–Bohlin method

Parrish, W.; Langford, J. I.

doi:10.1107/97809553602060000578

International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
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International Tables for Crystallography (2006). Vol. C. ch. 2.3, pp. 52-53

Section 2.3.1.3. Seemann–Bohlin method

W. Parrish^a ^‡ and J. I. Langford^b

^a IBM Almaden Research Center, San Jose, CA, USA, and ^bSchool of Physics & Astronomy, University of Birmingham, Birmingham B15 2TT, England

2.3.1.3. Seemann–Bohlin method

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The Seemann–Bohlin (S–B) diffractometer has the specimen mounted on a radial arm instead of the axis of rotation and a linkage or servomechanism moves the detector around the circumference of a fixed-radius focusing circle while keeping it pointed to the stationary specimen. All reflections occur simultaneously focused on the focusing circle as shown in Fig. 2.3.1.13(a) . The method was originally developed for powder cameras by Seemann (1919) and Bohlin (1920) but was not widely used because of the limited angular range and the broad reflections caused by inclination of the rays to the film. The diffractometer eliminates the broadening and extends the angular range. Diffractometers designed for this geometry have been described by Wassermann & Wiewiorosky (1953), Segmüller (1957), Kunze (1964a , b, c), Parrish, Mack & Vajda (1967), King, Gillham & Huggins (1970), Feder & Berry (1970), and others.

Figure 2.3.1.13| top | pdf |

Seemann–Bohlin method. (a) X-ray optics using incident-beam monochromator. (b) X-ray tube line-focus source showing geometrical relations: γ mean angle of incident beam, β_H inclination of reflecting plane H to specimen surface, θ_H Bragg angle of H plane, t tangent to focusing circle at O. (c) Diffractometer settings for various angular ranges.

The geometry is shown in Fig. 2.3.1.13(b) (Parrish & Mack, 1967). Reflections occur from lattice planes with varying inclinations β_H to the specimen surface. The reflecting position of a plane H is $[\theta _H = \gamma + \beta _H]$ , where γ is the incidence angle and $[4\theta_H]$ the reflection angle. The maximum value of β_H is about 45°. It is essential to align the specimen tangent to FC. This is a critical adjustment because even a small misalignment causes profile broadening and loss of peak intensity.

The source may be the line focus of the X-ray tube [F in Fig. 2.3.1.13(b)] or at the focus of a monochromator [ES in Fig. 2.3.1.13(a)]; in the latter case, the entrance slit at F′ limits the divergent beam reaching the specimen. The source, specimen centre O, and receiving slit RS lie on the specimen focusing circle SFC, which has a fixed radius r. The incidence angle γ is given by $[\gamma=\arcsin(b/2r), \eqno (2.3.1.21)]$ where b is the distance from F or F ′ to O, or 2r sin γ. The γ angle determines the angular range that can be recorded with a given r, decreasing γ decreases 2θ_min. The relationships of specimen position on the focusing circle and the recording range are illustrated in Fig. 2.3.1.13(c). To change the range requires rotation of the X-ray tube axis or the diffractometer around F. The detector must also be repositioned. For forward-reflection measurements, γ is usually $[\leq10^\circ]$ . Extreme care must be used in the specimen preparation to avoid errors due to microabsorption (particle-shadowing) effects, which increase with decreasing γ. The 0° position cannot be measured directly and a standard is used for calibration. The range from 0° to about 15°2θ is inaccessible because of mechanical dimensions. At γ = 90°, only the back-reflection region can be scanned.

The aperture of the beam striking the specimen is $[\alpha _{\rm SB} = 2 \arctan ({\rm ES}_w / 2a), \eqno (2.3.1.22)]$ where ES_w is the entrance slit width and a the distance between F or F′ and the slit. The irradiated specimen length l is constant at all angles, l = 2αr. A large aperture can be used to increase intensity since the specimen is close to F. However, the selection of α is limited if γ is small, and also because of the large flat-specimen aberration.

The receiving-slit aperture varies with the distance of the slit to the specimen $[\alpha _{\rm RS} (^\circ 4 \theta) = 2 \arctan {\rm RS}_w / [2r \sin (2 \theta - \gamma)]. \eqno (2.3.1.23)]$ Consequently, the resolution and relative intensity gradually change across the pattern. The S–B has greater widths at the smaller 2θ's and nearly the same widths at the higher angles compared with the θ–2θ diffractometer. The aperture can be kept constant by using a special slit with offset sides (to avoid shadowing) and pointing the opening to C while the detector remains pointed to O (Parrish et al., 1967). The slit opening is tangent to FC and inclined to the beam and rotates while scanning. The constant aperture slit has $[\alpha _{\rm RS} (^\circ 4 \theta) = 2 \arctan ({\rm RS}_w/2r). \eqno (2.3.1.24)]$

The axial divergence is limited by parallel slits as in conventional diffractometry and the effects are about the same. The equatorial aberrations are also similar but larger in magnitude. The specimen-aberration errors are listed in Table 5.2.7.1 . The flat specimen causes asymmetric broadening; the shift is proportional to $[\alpha ^2_{\rm ES}]$ and increases with decreasing θ. It can be eliminated by making the specimen with the same curvature as r = FC. In this case, one curvature satisfies the entire angular range because the focusing circle has a fixed radius. However, the curvature precludes rotating the specimen. The specimen transparency also causes asymmetric broadening and a peak shift that increases with decreasing θ. For $[\mu h \rightarrow0]$ , the geometric term is the same as for specimen displacement (Mack & Parrish, 1967).

The diffracted intensity is proportional to I₀A(μ h)TB, where I₀ is the incident intensity determined by α, δ, and the axial length L of the incident-beam assembly, A(μh) is the specimen absorption factor, T the transmission of the air path, and B the length L_RS of the diffracted ring intercepted by the slit. The X-rays reflected at a depth x below the specimen surface are attenuated by $[\exp \{- [\mu x \,{\rm cosec} \,\delta + \mu x \,{\rm cosec} (2 \theta - \delta)] \}, \eqno (2.3.1.25)]$ where μ is the linear absorption coefficient. The asymmetric geometry causes the absorption to vary with the reflection angle. The air absorption path varies with the distance O to RS and reaches a maximum at 180° + 2γ. The expression for air transmission includes the radius of the X-ray tube R_T, which is needed only for the case where the X-ray tube focal line is used as F. In a typical instrument with X-ray tube source F and r = 174 mm, the transmission of Cu Kα decreases from 90% at 40°4θ to 62% at 210°, and Cr Kα from 73 to 23% at the same angles.

Some of the advantages of the method include the following: (a) the fixed specimen makes it possible to simplify the design of specimen environment devices; (b) a large aperture can be used and the intensities are higher than for conventional diffractometers; (c) the flat-specimen aberration can be eliminated by a single-curvature specimen; (d) a small γ angle can be used to increase the path length l in the specimen, and hence the intensity of low-absorbing thin-film samples ( $[l = t/\sin\gamma]$ and for γ = 5°, l = 11.5t); (e) the method is useful in thin-film and preferred-orientation studies because about a 45° range of lattice-plane orientations can be measured and compared with conventional patterns. The limitations include (a) the more complicated diffractometer and its alignment, (b) limited angular range of about 10 to 110°2θ for the forward-reflection setting, (c) extreme care required in specimen preparation, and (d) larger aberration errors.

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International Tables for Crystallography (2006). Vol. C. ch. 2.3, pp. 52-53