Solution

ID: past-exam-of-the-mphil-in-micro-and-nanotechnology-enterprise-course-of-the-university-of-cambridge/2019/paper-1/section-b/8/iv/solution

There is a verified inconsistency in the printed data: the original PDF says , whereas the relativistic de Broglie wavelength at is approximately :
Use consistent length units, , and distinguish literal arithmetic from a physically consistent estimate. The familiar approximate Scherzer defocus is , or about with . Slightly different optimum-phase conventions use in place of , giving about ; both are underfocus by a few tens of nanometres. The manufacturer's tabulated values confirm that scale.[ref]
Using the question's additional numerical relation exactly as supplied, with , gives
For the physically appropriate rounded wavelength this is . The CTF itself gives an independent check: its first nonzero zero satisfies , so implies . The plotted condition is therefore a little less underfocused than the usual Scherzer setting and has a first-zero spacing near . The nominal and approximate optimization formula should not be treated as exactly reproducing every plotted parameter.
If the printed is retained literally, the approximate Scherzer formula instead gives , the supplied relation gives , and the plotted first zero requires . These do not describe a consistent microscope. The physically consistent comparison is approximately to optimum underfocus versus from the plot; the wavelength correction is explicit.

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