Technical note Sizing the Stern-Gerlach apparatus

Chapter 1 describes the Stern-Gerlach apparatus; this note shows where its dimensions come from. We start from the deflection formula, work out the separation of the two beams, see which physical constraint limits it and how to get around it, and finally size the vacuum system, the mechanical structure and the detection that follow.

1. Deflection and parameters

The vertical force on the atom is $F_z=\mu_z\,\partial B/\partial z$. For a magnet of length $L$ followed by a free-flight (drift) length $D$, the displacement on the slide is

$$z=\frac{\mu}{m}\,\frac{\partial B}{\partial z}\,\frac{L}{v^{2}}\left(\frac{L}{2}+D\right)$$

with $\mu=\mu_B=9.274\cdot10^{-24}$ J/T, $m_{\text{Ag}}=1.791\cdot10^{-25}$ kg and $v\approx600$ m/s, the most probable velocity of the effusive beam at about 1250 K. Setting $D=0$ recovers the case in which the slide is at the magnet exit.

2. The gap/gradient trade-off

The attainable gradient scales roughly as $B/d$, where $d$ is the gap width. With $d=25$ mm and iron saturation around 2 T, the physical ceiling is about 1 T/cm; since most of the magnetomotive force is spent crossing the gap, in practice one achieves $\partial B/\partial z\approx0.5$ T/cm. A wide gap is therefore safe for the beam but limits the gradient: it is this balance that fixes the separation.

With $\partial B/\partial z\approx0.5$ T/cm, silver at $v\approx600$ m/s and pole pieces of length $L=600$ mm ($D=0$): acceleration $a\approx2.6\cdot10^{3}$ m/s², time of flight $t=L/v\approx1.0$ ms, deflection $z\approx1.3$ mm per beam, i.e. a total separation of about 2.6 mm — fifteen times the splitting observed by Stern and Gerlach in 1922 (~0.2 mm). The beam's excursion inside the gap is only 1.3 mm out of the 12.5 mm available: the beam never approaches the pole pieces.

It is worth setting straight away the ceiling reachable with this geometry: pushing the separation to 7 mm would require $\sim2$ T/cm, i.e. about 5 T at the pole tip with a 25 mm gap, beyond what iron can deliver. With a wide gap the millimetre is therefore the order of magnitude one has to settle for, and it is the one quoted in the chapter.

3. Variant: short magnet plus drift section

To go further it is best to move the length from the field to the drift: the free-flight section after the magnet amplifies the separation without paying the gap penalty. The chamber height (100 mm) leaves room for more massive poles without changing the external drawing, and reducing the gap to 10 mm raises the width-to-gap ratio to about 4, enough to make the field uniform over the useful region.

Parameter	Current design	Variant
Pole length	600 mm	250 mm
Gap	25 mm	10 mm
∂B/∂z	~0.5 T/cm	~1.5 T/cm
Drift length	0	700 mm
Excursion in field	1.3 mm	0.67 mm
Exit angle	—	5.4 mrad
Separation at the slide	2.6 mm	8.9 mm
Ampere-turns	~25 000	~13 000

Check: $a=\mu_B\cdot150/m_{\text{Ag}}=7.77\cdot10^{3}$ m/s²; $t_\text{field}=0.250/600=417$ µs; $z_\text{field}=\tfrac12 a t^2=0.67$ mm; $\theta=a\,t/v=5.4$ mrad; $z_\text{drift}=\theta\cdot0.700=3.78$ mm; $z_\text{tot}=0.67+3.78=4.45$ mm per beam, i.e. 8.9 mm separation. The ampere-turns follow from $N I=B\,d_\text{eff}/\mu_0$, with $d_\text{eff}$ including the double thickness of the stainless wall crossed by the flux ($10+2\times3=16$ mm).

Side benefits: shorter and lighter magnet, halved power coil, parallelism tolerance to hold over 250 mm instead of 600, smaller internal surface to outgas.

4. Vacuum sizing

The working pressure follows from the ratio between the outgassing load, proportional to the internal surface, and the effective pumping speed, limited by the duct conductance; the chamber volume only bears on the initial pump-down time. With an internal surface of about 0.5 m² of stainless steel the gas load is of the order of $10^{-4}$ mbar·l/s; the conductance of the chamber (100 × 50 mm cross-section) is about 40 l/s. An 80 l/s turbomolecular pump gives an effective speed $S_\text{eff}=(1/80+1/40)^{-1}\approx27$ l/s and hence $p=Q/S_\text{eff}\approx4.5\cdot10^{-6}$ mbar, corresponding to a mean free path of several metres.

The criterion to satisfy is $\lambda\gtrsim10\,L$: with $\lambda$ equal to the beam length only 37% of the atoms would survive ($I/I_0=e^{-L/\lambda}$) and those scattered at small angles would create a background masking the split. Mind the unit: the specification is $2\cdot10^{-5}$ mbar; $2\cdot10^{-5}$ bar would be 2 Pa, i.e. $\lambda\approx3$ mm, and the beam would not reach the slide. Pumping from both ends gains almost an order of magnitude.

5. Construction notes

Materials. The crucible is alumina: fused quartz already softens around ~1100 °C, that is within the working range (1100–1200 °C) needed to bring silver's vapour pressure to a useful value — melting (961.8 °C) is not enough. The chamber is AISI 316L stainless steel: cold-worked 304 becomes slightly ferromagnetic and would perturb the magnetic circuit.

Forces and structure. The magnetic pressure $B^2/2\mu_0$ is ~4 bar at 1 T; on a 250 × 40 mm pole face this is about 4 kN of attraction between the poles: rigid spacers are needed, otherwise the gap closes under load. The parallelism of the poles must be held over the whole length. Finally, the gradient must be measured or simulated (FEM), not assumed from an ideal formula, because it is the quantity from which the magnetic moment is obtained and it sets its uncertainty.

Gravity. It is negligible: in ~2.2 ms of flight over the whole path the atoms fall by ~23 µm, i.e. 0.3% of the magnetic deflection.

6. Detection

The silver deposit, only a few atomic layers thick, is developed by exposing the slide to sulphur vapour, which converts it into dark silver sulphide (the 1922 technique). With silver, a hot-wire detector (surface ionisation) is not applicable, because its ionisation potential (7.58 eV) is too high compared with the work function of tungsten. This is why modern teaching laboratories prefer potassium or caesium, which allow the beam profile to be read in real time.