= Gravity deflection of an atomic beam
{title2=$y=gL^2/(2v^2)$}
<Immanuel Estermann>, <O. C. Simpson> and <Otto Stern> used gravitational deflection to measure atomic speeds. Stern describes scanning a hot-wire <surface ionization detector> through a <cesium> beam and recording <ion> <electric current>.<The Method of Molecular Rays by Otto Stern>[The method of molecular rays, lecture 12 December 1946, pp. 9–11: collaborators, falling beam and ion-current readout]{ref} Their 1947 paper reports <potassium> and <cesium> beams; a deficit of slow arrivals increased with oven <pressure>, supporting a source-slit <collision> explanation.<The Free Fall of Atoms and the Measurement of the Velocity Distribution in a Molecular Beam of Cesium Atoms by Immanuel Estermann, Oliver Cecil Simpson, and Otto Stern>[Estermann, Simpson and Stern, Physical Review 71 (15 February 1947), p. 238, abstract: gravity deflection, modified Maxwellian distribution and pressure-dependent slow-atom deficit]{ref}
First consider an original point-launch model, with horizontal <speed> $v>0$, zero initial vertical <velocity>, flight distance $L$, and uniform <gravitational acceleration> $g$. Downward displacement $y$ follows from flight time $t=L/v$:
$$
y=\frac{gL^2}{2v^2}=\frac{C}{v^2},\qquad C=\frac{gL^2}{2}.
$$
Slow particles fall farther. However, a detector scan counts arrivals per interval of height, not per interval of <speed>. <Change of variables for probability densities> supplies the <Jacobian determinant>:
$$
p_y(y)=g\!\left(\sqrt{C/y}\right)\frac{\sqrt C}{2y^{3/2}}
=\frac{b^2}{y^3}e^{-b/y},\qquad b=aC=\frac{mgL^2}{4k_BT},\qquad y>0.
$$
Here $g(v)$ denotes the <effusive beam speed distribution>; the unadorned constant $g$ in $C$ denotes <gravitational acceleration>. The <integral> of $p_y$ is one: substituting $u=b/y$ reduces it to $\int_0^\infty ue^{-u}\,du=1$. Its <mode> is $y=b/3$. Simply mapping the most probable arrival <speed> into a fall distance instead gives $2b/3$. The peaks differ because equal height bins cover unequal speed intervals.
\Image[/effusive-beam-speeds-and-gravity-profile.png]
{title=<Effusion> changes the arrival speed distribution, and a detector-height scan requires another <Jacobian determinant>}
{description=Original calculated distributions with chosen $m=133\,\mathrm u$, $T=500\,\mathrm K$, $L=1.50\,\mathrm m$ and $g=9.81\,\mathrm{m\,s^{-2}}$. These are not historical measurements.}
For those chosen parameters, $b\simeq176.54\,\mu\mathrm m$, so the detector-height <probability density> peaks near $58.85\,\mu\mathrm m$. A detector covering only $0<y<Y$ accepts the fraction
$$
F(Y)=\int_0^Yp_y(y)\,dy=e^{-b/Y}\left(1+\frac bY\right).
$$
At $Y=0.80\,\mathrm{mm}$ this is about $97.9\%$, even before allowing for imperfect detection. Discarding the remainder changes the observed normalization.
Real slits transmit a range of launch heights and vertical <velocities>. Their geometrical response must be combined with the gravitational deflection and detector width; the point-launch formula is not an exact reduction of the historical apparatus. In the point-launch model, an initial vertical <velocity> $w$ would add $wL/v$ to $y$. Source <collisions>, detector efficiency and alignment can also distort a profile. Varying oven <pressure> helps distinguish such source losses from an incorrect thermal distribution; inferring a source <temperature> from the peak alone ignores both selection and the detector's coordinate transformation.
Back to article page
