Stern-Gerlach Experiment

Transcription

1 Stern-Gerlach Experient HOE: The Physics of Bruce Harvey This is the experient that is said to prove that the electron has an intrinsic agnetic oent. Hydrogen like atos are projected in a bea through a agnetic field which varies in intensity in a direction perpendicular to the bea. The atos behave like sall agnetic dipoles and are deflected to for two beas depending on whether they align parallel or anti-parallel to the agnetic field. There is an excellent description of the odern version of this experient which can be downloaded fro the apparatus anufacturer PHYWE SYSTEE GBH. There is another excellent article in Physics Today, Deceber 2003 which explains how the experient was originally perfored to confir the original Bohr odel. Bea of atos S N S N agnetic field of varying flux density The splitting of the bea into two coponents confirs that the atos behave neither as agnetic dipoles, current loops or charged gyroscopes. agnetic dipoles and current loops should align parallel producing a single deflected bea. Charged gyroscopes should precess aintaining their angle with the field and produce a broadening of the bea. The twofold splitting confirs that there is no gyroscopic action and that the atos align parallel and anti-parallel with equal probability. We can account for this behaviour with our unified theory. We can analyse the effect of placing the hydrogen ato in a agnetic field perpendicular to its orbital plane. We restrict ourselves to the case of circular orbits threaded by n quanta of agnetic flux in a background agnetic field of flux density B perpendicular to the orbit. We recall that the quantised energy levels of the hydrogen ato were derived by solving the equations: 4 π ε 0 r + 4 π2 r 2 ν 2 = 0 () 2 ν e n Φ 0 = 2 2 π2 r 2 ν 2 (2) We odifying the first equation by dividing each side by r to give the equation relating electric force to centrifugal force. Then add a ter for the agnetic force Bev. For parallel alignent the Bev force is outward in the sae direction as the centrifugal force attepting to increase flux linkage. The velocity is v = 2π r ν. We then solve the equations: 4 π ε 0 r π2 r ν 2 + B e 2π r ν = 0 (3) These equations only approxiate to the real ato, describing a odel in which an electron of ass orbits a proton of infinite ass. Solving equation (2) for ν gives: ν = n e Φ 0 2 π 2 r 2 Copyright Bruce Harvey 997/2008 Page of 5

2 This is sufficient to deterine the quantised behaviour of the electron orbits for when we substitute for ν in the expressions for angular oentu and agnetic oent, the r s and s cancel: L = 2π r 2 ν = 2π r 2 n e Φ 0 2 π 2 r = n e Φ 0 2 π = n ħ On substitution of Φ 0 = 2e h the expression reduces to n h = n ħ a ultiple of the reduced Planck constant h- 2π bar. Siilarly: µ = I A = e ν π r 2 = e n e Φ 0 2 π 2 r 2 π r2 = n e2 Φ 0 = e 2 n ħ To find the energy, we substitute for ν in equation (3) this into equation () and change the variable fro radius r to curvature c = r giving: e 2 e 2 4π ε 0 r 2 + n2 e 2 Φ 2 0 π 2 r 3 + B e2 n Φ 0 π r = 0 4π ε 0 c 2 + n2 e 2 Φ 2 0 π 2 c3 + B e2 n Φ 0 π c = 0 After dividing by c, this is a quadratic in c producing two solutions. Selecting the solution which does not reduce to zero when B 0 and expanding the square root using the binoial theore we obtain the solutions: r = 8n 2 Φ 2 (π + (π π n 3 Φ 3 0 ε 2 0 B) 2) 0 ε 0 r = 4n 2 Φ 2 0 ε 0 4n Φ 0 ε 0 B Fro this we can calculate the energy: E = 2 4 π ε 0 r = e 2 32 n 2 Φ 2 0 ε e2 n Φ 0 2 π B + There are two ters, the first giving the energy of the ato when B = 0 and the second giving the coupling energy. Now the orbiting electron generates a agnetic oent µ = n e2 Φ 0 2 and the coupling energy is as π expected: E = µ B : µ = n e2 Φ 0 e 2 E = + 32 n 2 Φ 2 0 ε 2 µ B 0 = n e ħ 2 The effect of parallel alignent has been to increase the energy of the orbital syste aking it less negative. The interesting thing is that the energy contained in the orbital agnetic field E ag = 2 E kinetic = 4 E is reduced. The noral effect of parallel alignent of a current loop is to increase the flux threading the loop and so the torque exerted by the agnetic field on the current loop is always directed towards achieving parallel alignent. It does this through a siple otor action in which 2 E is supplied by the circuitry aintaining the current in the loop (and the current of the electroagnet generating the agnetic field) to do E of echanical work and increase the energy content of the flux threading the loop by E. This action is not possible in the ato for two reasons: the flux threading the orbit is quantised and cannot increase and Page 2 of 5 Copyright Bruce Harvey 997/2008

3 there is no external circuitry to supply the energy. The result is that there is no preference for parallel alignent. Indeed, the coupling action is quite different fro that between a current loop and a agnetic field. To understand the nature of the coupling, we ust take into account the change in H-flux µ 0 H 0 A treading the orbit. Here H 0 is the agnetic intensity of the agnetic field and A the area of the orbit expressed as a vector. Electroagnetic theory casually describes the ef in a circuit being equal to the rate at which flux is cutting the circuit. In fact, the physical flux we describe by its flux density B never cuts the circuit. What cuts the circuit is the atheatical artefact we describe as agnetic intensity H with the result that µ 0 (H 0 A) eerges fro (or is adsorbed into) the conductor increasing or decreasing the flux threading the loop. Once we understand this, we realise that although the flux threading the orbit cannot change the ef generated by a change in µ 0 (H 0 A) is still felt. In fact the ef can be explained by the fact that the Bev force depends on the velocity of the electrons relative to the H-flux and that as H-flux cuts the circuit, the Bev force includes a coponent in the direction of the conductor so that suing over all the conduction band electrons gives the ef. The coupling action between our ato and the agnetic field is provided by the ef-like coponent of the Bev force doing work to increase the energy of the orbital syste. Our basic assuptions about the relationship between the total, potential, kinetic and orbital agnetic energies are preserved. So the syste is still doinated by the orbital echanics. This leads to a very interesting discovery which could explain how orbits ight align anti-parallel to the field. In the noral situation, aligning a current loop (with field H L) parallel to a agnetic field H 0 increases the flux Φ = A µ 0 (H 0 + H L) da threading it thereby increasing the energy content of the agnetic field generated by the current loop. But in the atoic syste, the coupling energy increases the energy of the syste which becoes less negative. The Virial Theore then tells us that the kinetic energy of the electron is reduced. In our unified theory, the kinetic energy of the electron is shared equally between the orbital agnetic field and the reaining part of the agnetic field which oves with the electron. If we now consider anti-parallel alignent, the coupling energy is negative and the energy of the orbital syste becoes ore negative increasing the kinetic energy of the electron. In parallel alignent, the ef-like coponent of the Bev force does E work to increase the energy of the orbital syste. The orbit oves outwards requiring 2 E of work to be done against the electric force. The extra E coing fro a reduction in the kinetic energy of the electron half of which is stored in the flux threading the orbit. In anti-parallel alignent, the ef-like coponent of the Bev force acts in the opposite direction and adsorbs E of work to decrease the energy of the orbital syste. The orbit oves inwards and the electric potential does 2 E of work. The extra E providing the increase in the kinetic energy of the electron half of which is stored in the flux threading the orbit. Each of these actions is able to flip the orbit, so as the hydrogen like atos in the bea enter the agnetic field, they are flip to the nearest alignent, either parallel or anti-parallel resulting in the division of the bea in two. When we are dealing with real atos, orbital echanics has another effect. The electron does not orbit the proton. They both orbit about their cobined centre of gravity. atheatics allows us to reduce the proble to an equivalent one in which an electron of reduced ass + orbits a proton of infinite ass. The Bohr theory uses the reduced ass of the electron, but when we are dealing with a syste which is part echanical, part electrical and part electroagnetic, this introduces sall errors. If r is the distance between the electron and the proton, the ass of the electron and the ass of the proton, then the radius of the electron's orbit is + r and the radius of the proton's orbit is + r. We coe to the sae conclusion as Bohr that the quantisation applies to the whole of the angular oentu and this can be derived if the whole of the kinetic energy is equipartitioned. Equations () and (3) now becoe: Copyright Bruce Harvey 997/2008 Page 3 of 5

4 2 ν e n Φ 0 = 2 (2 π2 ( + r) 2 ν 2 4 π ε 0 r + 4 π 2 ( + r) 2 ν π 2 ( + r) 2 ν 2 2 r + π 2 ( + r) 2 ν 2 ) (4) ( + B e 2π r) + ν = 0 (5) In this, r is the distance between the electron and the proton. We can siplify these equations and write then ore neatly introduce a constant λ = + : 2 ν e n Φ 0 = 2 2λ π2 r 2 ν 2 (4) 4 π ε 0 r 2 + 4λ π2 r ν 2 + B e 2π r ν = 0 (5) Solution of equations (4) and (5) follows exactly as above, but with a sprinkling of λ s to produce the results: L = n e Φ 0 π = n ħ µ = λ n e2 Φ 0 λ e 2 E = + 32 n 2 Φ 2 0 ε 2 µ B 0 = λ e 2 n ħ It is custoary for authors at this point to write: "We leave this as an exercise for the student" The agnetic oent cannot be easured accurately with the Stern-Gerlach apparatus because the deflection depends on the velocity of the atos and the easureent of the agnetic field both of which are difficult to accurately deterine. The odel answers quoted by PHYWE SYSTEE GBH for their apparatus clais a result within 2.5% of the accepted value. ore accurate deterination is ade by using two sets of Stern-Gerlach agnets. The first splits the bea and allows atos of either parallel or antiparallel alignent to be selected. That bea is then passed through a radio frequency agnetic field which can be tuned to flip the atos. A second set of Stern-Gerlach agnets then analyses the bea. The frequency depends on the strength of the agnetic field and was originally understood as the frequency with which the orbit would precess if the syste behaved as a charged gyroscope. The frequency is called the Laror frequency after its discoverer. This understanding is hard to reconcile with the probability distribution odel of quantu echanics, but an alternative explanation is that photons of the Laror frequency have the correct energy to flip the ato fro one state to the other. The theory we have advanced so far indicates two stable states of the orbit, one aligned parallel to the agnetic field and the other anti-parallel. The difference in energy between these two states is: E = 2 B λ n e2 Φ 0 For hydrogen atos in their ground state, n =. We would expect the flip fro anti-parallel to parallel which requires an increase in energy to be associated with the adsorption of a photon of frequency of: ν = E h = E 2 e Φ 0 Coparing this with the original interpretation, the Laror angular velocity is ω 0 = B 2 e corresponding to a Page 4 of 5 Copyright Bruce Harvey 997/2008 = B e

5 frequency: ν 0 = ω 0 2π = B e 4 π We ust conclude that the flip corresponds to the adsorption or eission of 2 photons of frequency. ν 0 Copyright Bruce Harvey 997/2008 Page 5 of 5