= Lamb-Mössbauer factor
{c}
{title2=$f$}
= Recoil-free fraction
{synonym}
The <Lamb-Mössbauer factor> $f$ is the fraction in the coherent elastic line. In the harmonic-lattice approximation it is a <Debye–Waller factor>. Write the displacement of the emitting or absorbing <atomic nucleus> relative to its equilibrium site as $\mathbf u$, put $u_\parallel=\widehat{\mathbf k}\cdot\mathbf u$, and define the angular <wavenumber> $k=E_\gamma/(\hbar c)$. Choose the equilibrium site so that $\langle u_\parallel\rangle=0$. Its directional displacement <variance> is $\sigma_\parallel^2=\langle u_\parallel^2\rangle$. A harmonic <solid> in <thermal equilibrium> has a <Gaussian distribution> for this displacement, whose <characteristic function> gives the elastic amplitude
$$
\left\langle e^{iku_\parallel}\right\rangle=\exp\left(-\frac{k^2\sigma_\parallel^2}{2}\right).
$$
Taking the squared magnitude gives the intensity fraction
$$
f=\left|\left\langle e^{iku_\parallel}\right\rangle\right|^2=\exp\left(-k^2\langle u_\parallel^2\rangle\right).
$$
The factor $1/2$ belongs to the amplitude exponent; it disappears when the amplitude is squared. For an isotropic <solid>, $\langle u_\parallel^2\rangle=\langle|\mathbf u|^2\rangle/3$, so an equivalent expression is $f=\exp[-k^2\langle|\mathbf u|^2\rangle/3]$. Directional and three-dimensional mean squared displacements must not be substituted for one another without this factor.
An <Einstein solid> approximation replaces the vibrational spectrum by one positive <angular frequency> $\omega$. For the directional <quantum harmonic oscillator>, with the same effective <mass> $M$ as in $E_R$,
$$
\langle u_\parallel^2\rangle=\frac{\hbar}{2M\omega}(2\bar n+1),\qquad \bar n=\frac{1}{e^{\hbar\omega/(k_BT)}-1}.
$$
Here $k_B$ is the <Boltzmann constant> and $T$ is <temperature>. This follows by taking the <expected value> of $u_\parallel^2$ in each oscillator <energy eigenstate>, $\hbar(n+1/2)/(M\omega)$, and averaging with the <Boltzmann distribution>. Substitution gives
$$
f(T)=\exp\left[-\frac{E_R}{\hbar\omega}\coth\left(\frac{\hbar\omega}{2k_BT}\right)\right],\qquad f(0)=\exp\left(-\frac{E_R}{\hbar\omega}\right)<1.
$$
The zero-<temperature> limit retains <zero-point energy> and a nonzero displacement <variance>. Cooling therefore increases the <recoil-free fraction> towards a limit below one for finite $E_R/(\hbar\omega)>0$. At high <temperature>, the exponent tends to $-2E_Rk_BT/(\hbar\omega)^2$. This is a single-frequency approximation to a harmonic lattice's coherent elastic weight, not a complete description of all zero-net-<energy> processes of an isolated oscillator. Real vibrational spectra, directional binding and anharmonic motion can change its predictions.
Extend the stipulated teaching system by choosing $\hbar\omega=0.020\,\mathrm{eV}$, so $\eta=E_R/(\hbar\omega)=0.1$. Then $f(0)=0.90484$, while $f=0.80542$ at $k_BT=\hbar\omega$. If source and absorber are independent, the fraction of events eligible for recoil-free emission followed by recoil-free absorption contains the product $f_s f_a$; this gives $0.81873$ for two zero-<temperature> copies. It is not the total probability of absorption, which also depends on the number of resonant <atomic nuclei>, geometry and <spectral line> overlap.
\Image[/mossbauer-recoil-linewidth-and-lattice-fraction.png]
{title=Free recoil separates the lines; lattice binding supplies an elastic fraction}
{description=Original model using only the stipulated teaching parameters. Left: peak-normalized <Lorentzian line shapes> of the isolated <atomic nucleus>, separated by $2E_R=0.004\,\mathrm{eV}$; each physical width is $10^{-7}\,\mathrm{eV}$, with an inset resolving one peak. Right: <Lamb-Mössbauer factor> in the <Einstein solid> approximation for three choices of $E_R/(\hbar\omega)$. Every zero-<temperature> intercept is below one. No historical trace is reproduced.}
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