Stern-Gerlach Experiment: Discovering Quantum Spin with Potassium Atoms, Lab Reports of Physics

Download Stern-Gerlach Experiment: Discovering Quantum Spin with Potassium Atoms and more Lab Reports Physics in PDF only on Docsity! (revised 12/31/07) STERN-GERLACH Advanced Laboratory, Physics 407, University of Wisconsin Madison, Wisconsin 53706 Abstract 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 con- firmed 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. 1 The Original Experiment Stern and Gerlach generated a beam of neutral silver atoms by evaporat- ing 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 ∂Bz ∂z produces a force ∂Bz ∂z µz on a magnetic moment where µz is the component of the magnetic moment µ 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 µz. This would not have been expected with classical physics since this would have predicted that the z component of µ would have been µz = µ cos θ where cos θ could have all values from −1.0 to +1.0. Even the original Schrödinger theory predicts an odd number of possible states. This could explain µz having, for example, 3 values: −µ, 0 and +µ The obvious 2 states shown by the experiment is evidence that something is missing in the original Schrödinger Theory. The missing idea is that elec- trons have an intrinsic spin (a spin which cannot be removed) and that the angular momentum can be written in the form: ~S = 1 2 h̄~σ. The component of ~S in any specified direction, say the “z” or third direction must then have eigenvalues ±1 2 h̄. 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: (a) it is easier to evaporate (63.6◦C for K instead of 961.9◦C for Ag) 2 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 19K 39 and 19K 41 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 ` = 0 state and so it has no spatial (or- bital) 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 apparatus sends a beam of neutral potassium atoms in the x direction through a strong magnetic field gradient transverse to the beam. The field exerts a deflecting force on the magnetic moments of the moving atoms, proportional to the magnetic field gradient given by: ~F = −~∇(potential of mag. mom. in field B) = −~∇(~µ · ~B). If the magnetic field gradient is taken to be in the z direction then the z component of the force is given by: Fz = − ∂ ∂z (−µzB) = µz ∂B ∂z . Since µz is quantized and independent of x, y, and z, and since the magnetic poles are designed so that there is a region of constant ∂B ∂z , the beam will split into two distinct beams, the force given by: Fz = ±µB ∂B ∂z . 5 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 2 µ0 and −12µ0 where µ0 is the Bohr magneton. The probability of the magnetic moment being between angles θ and θ + dθ is equal to = area of section of sphere between θ and (θ + dθ) full area of sphere = (rdθ) (2πr sin θ) 4πr2 = sin θ dθ 2 . Hence the number of atoms with the z component between µ and µ+ dµ is proportional to sin θ dθ. This fraction of the atoms will give magnetic moments between: eh̄ 4m cos θ and eh̄ 4m cos(θ + dθ) i.e., between: µ and µ− dµ where: dµ = − d dθ ( eh̄ cos θ 4m ) dθ = eh̄ 4m sin θ dθ. Hence dµ is also proportional to (sin θ dθ). The number of atoms between µ and µ − dµ is thus proportional to dµ. The classical prediction is therefore that the z components of the magnetic moment will be uniformly distributed from − eh̄ 4m to + eh̄ 4m . z Displacement An atom moving with velocity v in the x direction will be acted on by a force in the z direction µz ~∇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 ~∇B) will be ~az = µz ~∇B M 6 where M is the atom’s mass; the velocity and deflection of the atom in this direction as it leaves the magnetic field will be: ~vz = ~azt = µz ~∇B d Mv and ~sz ′ = ~azt 2/2 = µz ~∇Bd21 2Mv . From that point to the detector, ~vz remains constant, so that the deflec- tion, ~s, at the detector is: ~s = ~vzdz/v + ~sz ′ = µz(~∇B) [d21 + 2d1d2] 2Mv2 where d2 is the distance between the magnet exit and the detector. For fixed v, ~s is proportional to µz, and can be used to determine the distribution of µB. Quantum mechanically, µz can be only +µB or −µB, so s will take on only two values, +s0 and −s0. Half of the atoms in the beam will arrive at each position. Classically the atoms would be distributed uniformly among all values of s between +s0 and −s0. 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 I(v)dv = 2I0(v/α) 3 e−(v/α) 2 d(v/α), where I0 is the total beam intensity, and α = √ 2kT/M is the most proba- ble velocity of atoms in the oven (but not the beam). k is the Boltzmann constant. Now if sα is the deflection for an atom of velocity α, then s/sα = (α/v) 2, and changing variables in the equation above from v/α to s/sα yields the following relation for the deflection pattern produced by a narrow beam of thermal atoms having moment µB: I(s)ds = aI0 s2α s3 e−(sα/s)ds 7 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 turbo pump which vents to a mechanical forepump. The vacuum is monitored by two triode vacuum gauges-one on either side of the magnet box. Migration of unwanted potassium through the vacuum system from the oven to the detector is reduced by additional slits and by running the oven at low output. The slits 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 possible accidental contamination of the oven end of the system by potassium or pump vapors. 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 well separated from the detector end of the system. The overall layout of the potassium beam path is shown in Fig. 3. 10 V E R T IC A L B A F F L E (R C A 1 94 9) T R IO D E I O N G A U G E S O V E N A SS E M B L Y W A T E R L IN E C O L D T A P T O M E C H A N IC A L F O R E P U M P H E A T E R D IF F U SI O N P U M P D R A IN L IN E 12 0 V A C P O L E P IE C E B O X D E T E C T O R A SS E M B L Y T O D IF F U SI O N P U M P O U T L E T T H E R M O C O U P L E V A C U U M G A U G E M E C H A N IC A L F O R E U M P Figure 2 11 POTASSIUM BEAM PATH What does the Stern-Gerlach apparatus look like from the inside? WITHOUT FIELD WITH FIELD Figure 3 12 r1 r2 h 1 h 2 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. 15 the wire. The collecting cylinder has a narrow slit aperture to admit the potassium atoms. The wire is heated by a variable stabilized voltage supply at about 1.4 A 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 microme- ter 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 temper- ature. 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 A AT ANY TIME. A current of 1.4 A is normally used. The ionization rate increases temporarily whenever the filament temper- ature 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 used to move the detector across the beam. DO NOT EXCEED a micrometer setting of 325 mils. The micrometer screw rod forms part of the seal in maintaining a good vacuum. If the rod is pulled out farther than 350 mils the rod could slip out of the O-ring and this could destroy the vacuum and may damage the turbo pump. Operating Conditions 16 1. Heat the oven at 75 V rms to an operating temperature of about 110◦C. Back off the heater voltage to ∼ 35 V when close to 110◦ so you don’t overshoot. Stabilize the temperature at the normal operating temper- ature of about 110◦. It is very important to make the actual measure- ments with a stable oven temperature. 2. The recommended hot wire current is 1.4 A. Sometimes the detector wire has to be baked at ∼1.7 A for 10 min. to clean the wire. Consult with the instructor. 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. Estimate the expected width of the beam at the detector using the slit information in Fig. 3. You need to know the expected width of the beam since you are going to scan with the hot wire detector and you need to know what step size to use or you may miss the signal altogether. 2. 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. 3. At the peak, see if you can reduce the detector wire current without losing appreciable signal. This will result in a longer lifetime of the detector before it becomes contaminated with potassium. The bias voltage is not critical. 4. Do a fine scan over the peak. Each point may require both closed and open shutter readings. However if the background is stable you may only have to take a closed shutter reading at the beginning and at the end of a scan. Your data should be reproduceable. 17