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 Incoming links: Quantitative powder diffraction using a (2 + 3) surface diffractometer and an area detector by Giuseppe Abbondanza and others

Past exam of the condensed matter physics course of the University of Oxford / 2017 / 1 / d / Solution 2026-10-25

λ = h c / E = 1.54980 \times 1 0^{- 10} m

. Write the amplitude of one conventional unit cell as the face-centred cubic selection rule times the two-atom atomic basis amplitude:

F_{hk l} = [1 + e^{iπ (k + l)} + e^{iπ (h + l)} + e^{iπ (h + k)}] [f_{Ga} + f_{As} e^{iπ (h + k + l) /2}] .

(1)

The first bracket is four when the indices have equal parity and zero otherwise. Among the allowed Bragg reflections, the zinc blende structure factor therefore gives

∣ F_{hk l} ∣^{2} = 16 ⎩ ⎨ ⎧ f_{Ga}^{2} + f_{As}^{2}, (f_{Ga} + f_{As})^{2}, (f_{Ga} - f_{As})^{2}, h, k, l odd, h + k + l \equiv 0 (mod 4), h + k + l \equiv 2 (mod 4) .

(2)

Here the atomic scattering factors are taken as real. For the requested rough estimate set

f_{Ga} ≃ 31

and

f_{As} ≃ 33

. The weak difference reflections remain allowed: unequal species do not have the extra exact extinction of the equal-species diamond limit.

The first six allowed values of

h^{2} + k^{2} + l^{2}

are

3, 4, 8, 11, 12, 16

. Bragg's law gives the measured scattering angle

2 θ = 2 sin^{- 1} (λ h^{2} + k^{2} + l^{2} /2 a)

. For powder diffraction, multiply the squared amplitude by the powder diffraction multiplicity

m_{hk l}

. The last two columns use

I_{111} = 100

hk l 111200220311222400 2 θ (^{\circ}) 27.4693 31.8234 45.6248 54.0830 56.6994 66.5025 m_{hk l} 86122486 ∣ F ∣^{2} 328006465536328006465536 I \propto m ∣ F ∣^{2} 100 0.1463 299.707 300 0.1951 149.854 I \propto m ∣ F ∣^{2} L P 100 0.1068 98.684 67.110 0.03921 21.165

(3)

The first intensity column is the geometry-independent multiplicity-and-structure estimate. The second additionally assumes an unpolarized beam and a common symmetric powder-diffractometer geometry, with Lorentz-polarization factor

L P (θ) \propto \frac{1 + c o s ^{2} ( 2 θ )}{s i n ^{2} θ c o s θ} .

(4)

An overall constant cancels in either normalization. A particular measured intensity set is not fixed by the atomic numbers alone: actual atomic scattering factors decrease with momentum transfer, anomalous scattering and absorption can matter, and the instrument geometry and polarization determine angular corrections. The two columns state the approximation rather than asserting identical peak heights for all instruments.^[ref]

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