Past exam of the materials science course of the University of Cambridge / 2016 / Paper 1 / 3

Measure ring centers and radii against the printed scale, rather than comparing raw squared radii. A representative reading of the four rings is $R\simeq 6.0,7.0,10.6,13.2\mathrm{mm}$, with broad photographic lines. For a flat plate at distance $L$,With $L=16\mathrm{mm}$ the ${\sin }^2\theta$ ratios are approximately $1:1.33:2.67:3.67$, givingFor identical atoms at the four conventional FCC positions,The structure factor is $4f$ for indices all even or all odd, and zero for mixed parity. Thus, for example, 100, 110 and 210 are systematic absences. The allowed squared-index sums begin $3,4,8,11,12,16,\dots$. Only the first four rings fit within the displayed field; 222 and later allowed reflections are not forbidden merely because they are outside the image. The central shadow and its holder are not extra diffraction peaks.

Take ${\lambda }_{\mathrm{Fe}}\simeq 0.1937\mathrm{nm}$ for the unresolved iron $K_{\alpha }$ doublet. The underlying tabulated $K_{\alpha 1}$ and $K_{\alpha 2}$ wavelengths are approximately $0.193604$ and $0.193998\mathrm{nm}$.^[ref]

Using the inferred $a\simeq 0.348\mathrm{nm}$, Bragg's law gives the following approximate cone angles:The exact angles inherit the lattice-parameter uncertainty. Five reflection families are kinematically possible with this longer wavelength: $h^2+k^2+l^2\le (2a/{\lambda }_{\mathrm{Fe}}{)}^2\simeq 12.9$. The next FCC family, 400, cannot satisfy Bragg's law.

A cylindrical Debye-Scherrer camera records paired curved lines where each diffraction cone intersects the film. Both forward and backward cones can be intercepted. The last three families are back reflections here, unlike the rings recorded on the original forward flat plate. Film position is proportional to camera angle along the equatorial circumference, rather than to $L\tan 2\theta$; the incident and transmitted beam holes must also be distinguished from reflections. A sufficiently resolved exposure may separate the characteristic doublet, but the schematic does not claim to predict line intensities.

For any indexed cubic reflection,Use ${\lambda }_{\mathrm{Mo}}\simeq 0.07107\mathrm{nm}$ for the unresolved molybdenum $K_{\alpha }$ doublet; the tabulated components are approximately $0.0709300$ and $0.0713590\mathrm{nm}$. These are external spectral constants, not an assertion about the unavailable historical Data Book.^[ref]

Reading broad ring centers in the supplied image more closely gives representative radii $5.96,6.99,10.63,13.23\mathrm{mm}$. Their corresponding estimates are about $0.3471,0.3473,0.3478,0.3480\mathrm{nm}$. Their small scatter understates possible common scale and camera-distance errors. In particular,for a first-order signed perturbation; independent uncertainties are combined in quadrature instead. A reading uncertainty of order $0.2\mathrm{mm}$, plus the printed-scale and setup uncertainty, justifies a deliberately modest precision:Nickel is a plausible identification, since it is FCC with room-temperature edge about $0.352\mathrm{nm}$.^[ref] The source photograph alone does not chemically identify the specimen; nearby elemental lattice constants and the scale uncertainty preclude a rigorous unique assignment. The estimate is not reported with artificial precision inherited from pixel fitting.