{c}

Let $v$ be the source <velocity> towards the absorber, positive for approach along the <photon> direction. A first-order <Lorentz transformation> of the <photon> <energy> gives $E'=E+vp$; hence, for $|v|\ll c$,
$$
\delta E\simeq E_\gamma\frac{v}{c}.
$$
Moving the source shifts its <emission line> through the stationary <absorption line>. A minimum in transmitted <photon> counts marks their strongest overlap. The absorber is the narrow <energy> discriminator; the downstream <particle detector> only needs to count the transmitted <gamma rays>.

For a thin absorber with unsplit <Lorentzian line shapes>, the overlap is a <convolution> of the source and absorber profiles. The combined <full width at half maximum> is $W=\Gamma_s+\Gamma_a$. A simple expected transmission model is
$$
\frac{I(v)}{I_0}=1-\frac{A}{1+\left[2E_\gamma(v-v_0)/(cW)\right]^2},\qquad 0<A<1.
$$
Here $I_0$ is the off-resonance intensity, $v_0$ encodes a relative shift of the two transition <energies>, and $A$ includes the resonant absorber strength and $f_s f_a$. The convolution result can be checked by multiplying the <Fourier transforms> of two <Cauchy distributions>: their exponential decay constants add. The width of the absorption dip in <velocity> is $cW/E_\gamma$, rather than the width of one individual <spectral line> converted to <velocity>.

For the teaching system with $\Gamma_s=\Gamma_a=\Gamma_E$ and $v_0=0$, the dip's <full width at half maximum> is $2.9979\,\mathrm{mm\,s^{-1}}$. Stipulate $A=0.25$ at zero <temperature> and $10{,}000$ expected off-resonance counts per equal-duration bin: the central bin then has $7{,}500$ expected counts, and the half-depth bins have $8{,}750$. These are model expectations, without sampled counting noise. Holding all other parameters fixed while heating both lattices to $k_BT=2\hbar\omega$ multiplies $A$ by $[f(T)/f(0)]^2$. This isolates how a falling <recoil-free fraction> weakens the dip.

\Image[/mossbauer-resonant-absorption-velocity-scan.png]
{title=Mechanical velocity scans an absorption line much narrower than detector resolution}
{description=Original ideal transmission model, with schematic source, absorber and counting detector. Both curves use the same stipulated <energy> widths; heating changes only $f_s f_a$. Dashed markers show the cold curve's half-depth positions. This is a new analytic teaching scan, not a historical measurement or a fit. The upper scale converts <velocity> to <Doppler effect> <energy> shift.}

The thin-absorber model is deliberately limited. A thick absorber can saturate and broaden the dip; unresolved <hyperfine structure> can create multiple components. Differences in chemical surroundings can shift the centers, and uncontrolled motion can smear them. These effects must be modeled before interpreting an observed width as a <natural linewidth>. Their sensitivity also makes the <Mössbauer effect> useful: a small displacement of a narrow dip can reveal a small change in the local nuclear environment.


 Mössbauer velocity scan (source code)