The experiment performed by Otto Stern and Walther Gerlach in 1922 provided very convincing evidence of two important consequences of Modern Quantum Mechanics

1. Space quantization can occur even in non-periodic systems.
2. That some particles have an intrinsic angular momentum, and therefore magnetic moment.

The experiment did not really produce any new ideas but dramatically confirmed ideas developed indirectly from Spectroscopy and the Zeeman Effect. The Stern Gerlach experiment was in principle simple and its results were clear. It removed many of the lingering doubts that Quantum Mechanics is true.

The Original Experiment

Stern and Gerlach generated a beam of neutral silver atoms by evaporating silver from an oven. The process was performed in a vacuum so that the silver atoms moved without scattering. The atoms were collimated by slits and sent through a region with a large non-uniform magnetic field.

A magnetic field non-uniformity [(¶B_z)/( ¶z)] produces a force [(¶B_z)/( ¶z)]m_z on a magnetic moment where m_z is the component of the magnetic moment m in the z direction.

The silver atoms were thus deflected and allowed to strike a cold metallic plate. After about 8-10 hours the number of condensed silver atoms was large enough to show a visible trace. The trace showed 2 marks showing that the silver atoms had 2 possible components of m_z.

This would not have been expected with classical physics since this would have predicted that the z component of m would have been

Even the original Schrödinger theory predicts an odd number of possible states. This could explain m_z having, for example, 3 values: -m, 0 and +m

The obvious 2 states shown by the experiment is evidence that something is missing in the original Schrödinger Theory. The missing idea is that electrons have an intrinsic spin (a spin which cannot be removed) and that the angular momentum can be written in the form:

This Experiment

We include two improvements to the original experiment that allow it to be done more easily.

1. Potassium is used instead of silver because:

   1. it is easier to evaporate (63.6^°C for K instead of 961.9^°C for Ag)

   2. The low temperature means that the Potassium atoms are moving at a lower speed and are more easily deflected.

   3. Potassium has a single valence electron outside its closed shells and the magnetic moment is more obviously due to the single valence electron.

   4. Potassium is easier to ionize than silver and is thus easier to detect electrically.

2. A hot wire detector is used instead of a cold metallic plate. The hot wire ionizes a fraction of the potassium atoms which strike it. The positive recoiling ions are then collected on a nearby negative electrode and the small ionization current is measured by an electrometer. This allows modern electronics to be used with its high sensitivity for small currents and fast reaction time.

Theory

One result of Schrödinger's Quantum Theory is that a system existing in a region of space with a unique axis of symmetry (such as the z axis) will have a wavefunction which can be expressed in the form of a product

The angle f is the azimuthal angle measured by rotating around the axis of symmetry. The r, q and f form a set of spherical coordinates. Read Eisberg and Resnick pages 256-259 for the solution of the equation.

The function F(f) must repeat after f is increased by 2p and consequently F(f) has a solution:

The number of such spatial states is always odd, since m is any integer from -j to +j where j depends upon the form X(r,q). In our case of potassium, j = 0 and so only one state (m = 0) is possible.

However the intrinsic spin (introduced by the Dirac improvement to Schrödinger's Theory) may also be included as an extra product term if the space has a unique axis of symmetry. The total angular momentum about the z axis then includes that of the intrinsic spin [S\vec]

In our case of potassium, the intrinsic spin causes the single m = 0 angular momentum to be split into two states with angular momenta of ±¹/₂(^h/_2p). The regular Schrödinger Theory can only predict an odd number of states and so the number of states is an important test for the existence of intrinsic spin.

The spin is measured by an effect on the associated magnetic moment. The intrinsic magnetic moment predicted for an electron has two possible z components:

The suggestion of intrinsic spin by Goudsmit and Uhlenbeck and the original Stern-Gerlach experiment are discussed in Eisberg and Resnick, pages 296-302.

The Atoms

Potassium atoms have 19 protons and either 10 or 12 neutrons. The electrons are:

n = 0 shell		filled with 2
n = 1 shell		filled with 8
n = 3	l = 0 subshell	filled with 2
	l = 1 subshell	filled with 6
	l = 2 subshell	empty
n = 4	l = 0 subshell	- - one valence electron

thus 18 of the electrons form a closed core with the same configuration as in the Argon atom. The 19th electron sees a potential very similar to that of a single proton and thus:

1. The energy levels of the 19th electron (if excited above n = 4, l = 0) are somewhat similar to those of a Hydrogen atom.

2. The magnetic moments of the electrons in the core cancel to give no contribution to the total magnetic moment of the atom. The nuclear magnetic moments are 0.391 and 0.215 nuclear magnetons for ₁₉K³⁹ and ₁₉K⁴¹ and since a nuclear magneton is about [1/ 2000] of a Bohr magneton, the only significant magnetic moment is that of the valence electron.

3. The valence electron is in an l = 0 state and so it has no spatial (orbital) magnetic moment. The atom therefore has a magnetic moment approximately equal to the intrinsic magnetic moment of an electron.

The above was strongly suspected from spectroscopic studies. The idea of an ad-hoc ``intrinsic magnetic moment" is however rather unsatisfactory. The Stern Gerlach experiment provided a direct method for finding the number of states and measuring the intrinsic magnetic moment. The observation of an even number of states then confirms the concept of ``intrinsic" spin.

Magnetic Field

The experiment sends a beam of potassium atoms in the x direction through a strong magnetic field gradient. The magnetic field causes a force

The magnetic poles are designed so that there is a region in which B is in the horizontal z direction and [(Ñ)\vec] B is both large and approximately constant. Hence the force is simply:

The assumption of intrinsic spin thus predicts that the atomic beam will split into 2 groups in the z direction.

Classically (pre-quantum theory), the spin of the atom may be at any angle to the z axis. The component of the magnetic moment along the z axis is then any value between +¹/₂m₀ and -¹/₂m₀ where m₀ is the Bohr magneton.

The probability of the magnetic moment being between angles q and q+ dq is equal to

This fraction of the atoms will give magnetic moments between:

z Displacement

An atom moving with velocity v in the x direction will be acted on by a force -m_B [(Ñ)\vec] B for a time t = d₁ / v, where d₁ is the distance travelled by the atom in the magnetic field. The acceleration along the z-axis (the direction of [(Ñ)\vec] B) will be

From that point to the detector, [v\vec]_z remains constant, so that the deflection, [s\vec], at the detector is:

The Stern-Gerlach experiment, like many experiments utilizing molecular beam techniques, is limited to a certain extent by the fact that the atoms or molecules in a beam issuing from an oven at absolute temperature T do not have a unique velocity. The velocity distribution of the beam intensity is

Now if s_a is the deflection for an atom of velocity a, then

Each value of m_B gives rise to a pattern similar to Fig. , but with a different value for s_a and therefore for s_max. The two values of m_B given by quantum theory produce two patterns, one as shown, and one its mirror image. Classically an infinite number of values of m_B occur, and the resultant composite pattern could be a roughly bell-shaped curve centered at s = 0.

Figure 1: Deflection pattern for atoms having a unique value of m_B. Undeflected beam width, a, is s_a/10.

Summary of Predictions of z direction distributions

1. The classical (pre-quantum theory) predicts a uniform distribution between the limits. This is smeared by the thermal velocity distribution.

2. The Schrödinger Quantum Mechanics (no spin) predicts an odd number of groups. In the case of Potassium it predicts one group.

3. The Dirac Theory predicts intrinsic spin. In the case of Potassium it predicts two groups.

The Apparatus

Potassium Oven

A slug of potassium has been sealed into the oven by a steel plug and sealed with a copper gasket. The oven is heated by a nichrome heater and requires little power to keep it hot since it is thermally isolated on 4 mounts.

The oven is operated at 120^° C (above the melting point at 62.3^°C) and delivers a hot potassium vapor beam by means of a small hole covered by two stainless steel precision slits.

The heater has been carefully located with respect to the slits so that the slit area is kept hotter than the bulk storage chamber and is therefore kept clean.

Oven Heater

The oven heater is driven from a stabilized 5 volt 5 Amp DC power supply. The potassium atomic beam intensity depends upon the vapor pressure of the potassium and this depends critically upon the temperature. The stabilized heater thus ensures a steady beam after a warm-up period.

Monitoring the Oven Temperature

The temperature of the oven is monitored by an iron-constantan thermocouple which passes through the oven mounting face. The thermocouple is connected to a voltmeter calibrated in ^°C. The approximate vapor pressure of potassium is 10^-6 Torr at 63^°C and 10^-3 Torr at 161^°C.

Vacuum System

A pressure of less than 5×10^-6mm Hg (Torr) is necessary for a good signal since the potassium atoms can be easily scattered. The system is pumped by a fore-pump and a 3 stage oil diffusion pump.

A water cooled cold trap is installed to reduce the unwanted migration of oil vapor. The vacuum is monitored by two triode vacuum gauges-one on either side of the magnet box. Only one gauge is needed for normal operation-the other has been used when locating leaks.

Migration of unwanted potassium through the vacuum system from the oven to the detector is reduced by baffles and by running the oven at low output. The baffles force any potassium atom travelling from the oven to the detector to strike at least one surface before reaching the detector unless the atom is travelling along the prescribed path of the potassium beam.

Tempered pyrex glass pipe forms the major part of the vacuum system and allows visual monitoring of the position of the beam gate, the temperature and position of the detector, and the possible accidental contamination of the oven end of the system by potassium or pump vapors. Connection between pipe and mating surfaces is made using a cast metal flange, a fiber ring to cushion the glass from the flange, and a neoprene O-ring which forms a vacuum seal between the pipe and the flat mating surface.

Beam Gate

The beam gate is attached to the long vertical baffle seen in Fig. 2 in the cross tube. It is able to pivot and stay in position, either open or closed, since its center of gravity is higher than the pivot point. The gate is moved from one position to another by a hand-held magnet outside the vacuum system, and rests against the glass in both positions, either fully open or fully closed. When the gate is closed, the oven end of the system is virtually separated from the detector end of the system.

The overall layout of the potassium beam path is shown in Fig. 3.

Figure 2:

Figure 3:

Vacuum System Maintenance

The maximum operating pressure of the Stern-Gerlach apparatus is approximately 5.0 ×10^-6mm of Hg. At higher pressures the potassium beam is scattered by the residual gases and no beam is detected.

The present system, when uncontaminated will maintain a vacuum around 1.0 - 2.0×10^-6mm of Hg with the oven off.

Due to the finite pumping speed through the pole piece box, the pressure in the detector end will be slightly higher than the pressure in the oven end.

It is important to remember that:

1. the ion gauges are never to be operated at pressures above 1.0 × 10^-4mm of Hg.

2. the grid current can be reduced by a factor of [10] to lengthen filament life; (in this case the pressure reads lower by factor of [10] also).

3. the diffusion pump must have a constant cold water supply.

4. the diffusion pump must also always be on for normal operation.

   The pressure will rise rapidly if the diffusion pump is shut off.

5. the diffusion pump back pressure will always be greater than the pressure of the main system

6. the mechanical forepump must operate continuously to evacuate the gases pumped by the diffusion pump.

   The rubber hose from the forepump to the diffusion pump should be free from crimps or sharp bends that would restrict the flow of gas inside.

7. The instructor should be consulted for a vacuum system shutdown.

Triode Ion Gauges

The 1949 ion vacuum gauge contains a plate, a grid, and two tungsten filaments, one of which is a spare. The filament is operated at a voltage such that all of the emitted electrons are drawn to the grid. The grid is operated at a positive potential (V_p). Electrons from the filament are accelerated to the positive grid. They bombard and ionize some of the molecules of any gas present in the tube. The resulting positive ions are attracted to the negative plate and constitute the plate current (I_p). The ratio of this current to the grid current (I_c) is proportional to the gas pressure for pressures below about 0.0001mm Hg (10^-4 Torr). The pressure meter reads correctly for a 10 mA grid current.

Nonuniform Magnetic Field

A cross section of the pole tips is shown in Fig. . The iron pole tips consist of a .218 inch diameter convex half cylinder spaced approximately 0.155 inch (3.97mm) from a .500 inch (1.27cm) diameter concave half cylindrical groove.

The concave circle, if extended, intersects the convex circle at the ends of its vertical diameter. These two curves are magnetic equipotentials of the magnetic field that would be present if the pole tips were replaced by two current carrying wires centered on the intersection points of the circles. The magnet is designed to approximate this field, which has a reasonably homogeneous gradient in the region the beam is allowed to pass through.

Figure 4: View of the pole unit looking ``down the beam path" with one end piece removed. The extension of the concave circle (r₁ = .25 inches) intersects the ends of the vertical diameter of the convex circle (r₂ = .218 inches). The dimensions of the spacings are h₁ = .055 inches and h₂ = .100 inches.

Field strength. A magnet capable of producing 0.4 Tesla (4000 gauss) across a .750 inch (1.91 cm) gap with a face 4 inches (10.16 cm) in diameter will produce the best results (this corresponds to about 7000 ampere turns), although respectable Stern-Gerlach patterns can be obtained with half that strength. A 7000 ampere turn magnet produces a gradient [(Ñ)\vec] B of a little over 100 Tesla/meter.

Hot Wire Detector

The hot wire detector is an important advance over the techniques used in the original Stern-Gerlach experiment. With it, the beam can be continuously sampled and experimental runs can be completed in a short time. The detector is a hot pure tungsten filament wire, .005 inch in diameter, surrounded by a collecting cylinder kept at a voltage of about 15 volts below the wire. The collecting cylinder has narrow slits to admit the potassium atoms. The wire is heated by a variable stabilized voltage supply which ``floats" at the bias voltage of 15 volts.

The detector can be moved laterally, by means of a micrometer, from outside the system; its position is indicated by the scale on the micrometer drive. The micrometer shaft passes through an O-ring into the vacuum chamber.

Principle of Operation

The potassium atoms ionize when heated at the surface of wire. A voltage placed across the collector and the hot wire will collect the positive ions if the collector is negative with respect to the hot wire; the small current is detected by a picoammeter or electrometer.

The collector and the hot wire together are similar to a vacuum diode with a directly heated cathode. In the detector, the ``plate" is always negative with respect to the ``cathode." In fact, if the bias voltage were reversed, the detector would act like a conducting diode; the collector would collect the electrons from the potassium and, in addition, it would collect electrons from the electron gas surrounding the hot wire.

The picoammeter current obtained is dependent on the filament temperature. A low filament current causes a low temperature and hence a low ionization rate and hence sluggish picoammeter readings. A high filament current will shorten the lifetime of the filament. DO NOT EXCEED 2.0 AMPS AT ANY TIME. A current of 1.5 amps is recommended.

The ionization rate increases temporarily whenever the filament temperature is increased slightly as absorbed potassium atoms are boiled off. For this reason you should wait for about 5 minutes after changing the filament current before trusting the picoammeter readings.

The setting of the bias voltage is not critical. A voltage of 15 volts is recommended.

A micrometer is provided to sweep the detector across the beam. DO NOT EXCEED 220 mils. The micrometer screw rod forms part of the seal in maintaining a good vacuum. If the rod is pulled out farther than 220 mils the rod could slip out of the O-ring and this could destroy the vacuum and may damage the diffusion pump oil.

Operating Conditions

1. Heat the oven at 20 W to an operating temperature of 120^°C. Back off in heater current to ~ 2.5 A when close to 120^° so you don't overshoot. Stabilize the temperature at 120^°. It is very important to make the actual measurements with a stable oven temperature.

2. The recommended hot wire current is ~ 1.5 A. The detector wire should be baked at ~ 2 A for 10 min. Expect a cool down of ~ 5 min. before stable operation is achieved.

3. This experiment requires a good vacuum. Good signals will be obtained when the starting pressures before the oven bake are £ 2×10^-6 Torr.

4. The signal current at the undeflected beam peak with a good vacuum can be as high as 10 pA. Depending on the conditions of the system it may be less.

Procedure

1. Set up according to the above conditions and scan for the signal with the magnet off. Once you know where to look you should do a first pass to locate the peak.

2. At the peak, optimize the detector hot wire current for maximum useful signal. The voltage bias is not critical.

3. Do a fine scan ( ~ 0.010" steps) over the peak. Each point will probably require both closed and open shutter readings. Check reproducibility of the data.

4. Do the same with the magnet on at ~ 2.0 A.

5. Possible additional studies include a different oven temperature and/or a different magnet setting.

Physics Analysis and Questions

1. Calculate the magnetic field gradient from your data assuming the predicted magnetic moment for Potassium. This will involve making some geometry measurements and deciding on a procedure for analyzing the data.

2. Derive an expression for the most probable velocity of the potassium atoms in the beam. Evaluate this quantity numerically at the measured operating temperature. Compare to the most probable value of the velocity in the oven. Why are they different?