For the electron of a one-electron hydrogen-like ion of nuclear charge

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Condensed Matter, Materials & Soft Matter · Crystal structure, bonding and diffraction

Solid State Physics, Ch Diffraction as measurement: form factors, thermal motion and refinement

Problem

For the

1s

electron of a one-electron hydrogen-like ion of nuclear charge

Z

, the Fourier transform gives the closed-form atomic form factor

f(Q)=\dfrac{1}{\left[1+\left(\dfrac{Qa_0}{2Z}\right)^2\right]^{2}},

where

a_0=0.529\,\mathrm{\AA}

is the Bohr radius and

Q=|\mathbf{Q}|

. ((a)) Confirm that

f(0)=1

whatever

Z

is, and say why. ((b)) For atomic hydrogen (

Z=1

), evaluate

f(Q)

Q=2.5\,\mathrm{\AA^{-1}}

and express it as a percentage of

f(0)

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