4.1. In a Stern-Gerlach experiment a well-collimated beam of silver atoms in the

Question

4.1. In a Stern-Gerlach experiment a well-collimated beam of silver atoms in their ground state ('S, 2) emerges from an oven inside which the atoms are in thermal equilibrium at temperature T. The beam enters a region, of length I, in which there is a strong magnetic field B and a gradient of field aB/oz perpendicular to the axis of the beam. After leaving this region the beam travels a further distance /' in a field-free region to a detector. Show that in the plane of the detector the deflection s, of those atoms which had the most probable speed a in the oven is = ± ti.ca (12 + 211'). where is the Bohr magneton. Evaluate s, for T 1.400 K. BR 300 Tm1/ 01 m.

Explanation / Answer

Purpose The purpose of this experiment is to perform a version of one of the most im-

portant experiments in the development of quantum physics, and to derive from the results

information about the quantum properties of angular momentum, the magnetic moment of

potassium atoms, and the Maxwell-Boltzmann distribution.

1 PREPARATORY QUESTIONS

1. Sketch the expected plot of beam intensity versus lateral deflection in a Stern-Gerlach

experiment with a beam of atoms in a state with j = 1. The same for the case of

j = 3

2 .

2. Make an energy level chart of the magnetic substates of the electronic ground state of

the potassium atom in a weak magnetic field and in a strong magnetic field. Take into

account the effects of the electronic and nuclear magnetic moments. In the light of this

chart predict how many intensity peaks you will see in this experiment and explain

your prediction.

3. What would be the result of passing one of the deflected beams from this experiment

through a second inhomogeneous field that is identical to the first except for being

rotated 90 around the axis?

4. How is the intensity of atoms with velocity between v and dv in the beam related to

the density of atoms with velocity between v and dv in the oven?

5. Derive equation (8) from the preceding results.

6. Derive equation (21) from (19) and (20) in Appendix I.

WHAT YOU WILL MEASURE

1. The angular momentum quantum number of the ground state electronic configuration

of potassium atoms.

2. The magnetic moment of the potassium atom.

3. The temperature inside the oven from which the atomic beam emerges into vacuum.

2 INTRODUCTION

The following sketch of the history of the Stern-Gerlach experiment is based on the much

more complete account by B. Friedrich & D. Herschbach in Daedalus, 127/1, 165 (1998).

The discovery of the Zeeman effect (1896) and its theoretical interpretation demonstrated

that atoms have magnetic dipole moments. However, no constraint was placed on the ori-

entation of the moments by the ”classical” explanation of the normal Zeeman effect, in

which the spectral lines of some elements in a magnetic field are split into three compo-

nents. Bohr’s theory (1913) of the hydrogen atom assumed circular orbits and required

the quantization of angular momentum and, by implication, quantization of the associated

magnetic moment. Sommerfeld (1916) generalized the Bohr theory to allow elliptical orbits

described by three quantum numbers: n, k, and m. The number n = 1, 2, 3..., called the

principal quantum number, corresponded to the quantum number of of the Bohr theory.

The number k = 1, 2, 3..n defined the shape of the orbit which was circular for k = n. The

number m = k, k + 1, ..., k 1, +k,m 6= 0, determined the projection of the vector an-

gular momentum on any prescribed axis, a consequence of the theory that was called space

quantization. Sommerfeld showed that his theory could account for the fine structure of the

hydrogen atom (now expained in terms of spin-orbit coupling) when relativistic effects on

the motion in the elliptical orbits were considered. The Sommerfeld theory also provided an

alternative explanation of the normal Zeeman effect. Nevertheless, the question remained

as to whether space quantization really occurs, e. g., whether the projections of the angular

momentum and its associated magnetic moment on an axis defined by the direction of an

imposed magnetic field are quantized.

Otto Stern proposed (1921) a defintive experiment to decide the issue. It would consist

of passing a beam of neutral silver atoms through an inhomogeneous magnetic field and

observing how the beam was deflected by the force exerted by the field on the magnetic

dipole moments of the atoms. The detector would be a glass plate on which the silver atoms

in the deflected beam would be deposited. Since the silver atom has one valence electron,

it was assumed that k = n = 1 and m = ±1 in the ground state. If the magnetic moments

were randomly oriented, then the distribution of deflections would decrease monotonically

on either side of zero deflection, reflecting a random distribution of the dipole orientations.

If space quantization was a reality, then the beam should be split into two distinct beams

corresponding to the parallel and anti-parallel alignments of the magnetic moments with

respect to the direction of the inhomogeneous magnetic field. Stern was clumsy with his

hands and never touched the apparatus of his experiments. He enlisted Walther Gerlach, a

skilled experimentalist, to collaborate in the experiment.

Stern predicted that the effect would be be just barely observable. They had difficulty in

raising support in the midst of the post war financial turmoil in Germany. The apparatus,

which required extremely precise alignment and a high vacuum, kept breaking down. Finally,

after a year of struggle, they obtained an exposure of sufficient length to give promise of an

observable silver deposit. At first, when they examined the glass plate they saw nothing.

Then, gradually, the deposit became visible, showing a beam separation of 0.2 millimeters!

Apparently, Stern could only afford cheap cigars with a high sulfur content. As he breathed

on the glass plate, sulfur fumes converted the invisible silver deposit into visible black silver

sufide, and the splitting of the beam was discovered.

The new quantum mechanics of Heisenberg, Shr¨odinger, and Dirac (1926-1928) showed that

the orbital angular momentum of the silver atom in the ground state is actually zero. Its

magnetic moment is associated with the intrinsic spin angular momentum of the single

valence electron the projection of which has values of ±¯h

2 , consistent with the fact that the

silver beam is split in two. If Stern had chosen an atom with L = 1, S = 0, then the beam

would have split into three, and the gap between the m=+1 and m=-1 beams would have

been filled in, and no split would have been visible! Vol. II, chapters 34 and 35, and Vol. III,

chapters 5 and 6 of the Feynman Lectures gives a lucid explanation of the quantum theory of

the Stern-Gerlach experiment. Platt (1992) has given a complete analysis of the experiment

using modern quantum mechanical techniques. Here we present an outline of the essential

ideas.

2.1 THEORY OF ATOMIC BEAM EXPERIMENTS

Within the framework of classical mechanics one can show that an electron in a circular orbit

has an angular momentum L~ = mr2 and an associated magnetic moment µ = e

2meL~ ,

where m and e are, respectively, the mass and charge of the electron, and r and are the

radius and angular velocity of the orbital motion. In a magnetic field B~ the atom will be

acted on by a torque µ × B~ which causes L~ to precess about the direction of B~ with some

fixed value of the projection µz = |µ|cos of its magnetic moment along the direction of the

field. The atom will also have a potential energy µ · B~ , and if the field is inhomogeneous

such that at a certain point it is in the z direction and varies strongly with z, then the atom

will be acted on by a force Fz = z(µ · B~ ) = µz

Bz

z which may have any of a continuous

set of values from |µ|

Bz

z to +|µ|

Bz

z One would then expect a monoenergetic beam of

atoms, initially randomly oriented and passing through an inhomogeneous magnetic field, to

be deflected in the +z and z directions with a distribution of deflection angles that has a

maximum value at zero deflection and decreases monotonically in either direction. This is

not what is observed. Instead, an atomic beam, passing through such a field, is generally

split into several distinct beams, implying that the sideways force deflecting the beam is

restricted to certain discrete values.

According to quantum mechanics, an atom can exist in a steady state (i.e. an eigenstate of

the Hamiltonian) with a definite value of the square of the magnitude of its total angular

momentum, F~ · F~ and a definite component Fz of its angular momentum in any particular

direction such as that of the z axis. Moreover, these quantities can have only the discrete

values specified by the equations

F~ · F~ = f(f + 1)¯h2 (1)

and

Fz =

where f, the angular momentum quantum number, is an integer or half integer, mf , the

magnetic quantum number, can have only the values f, (f1), ..., +(f1), f ,and ¯h = h

2 .

The magnetic moment associated with the angular momentum is:

µ = gf e

2mecF~

where g , called the g-factor, is a quantity of the order of unity and characteristic of the

atomic state. The projection of µ on the z axis can have only one or another of a discrete set

of values µz = gfmfµB where µB = e¯h

2mec (= 0.92731x1020 erg/gauss) is the Bohr magneton.

In the presence of an inhomogeneous magnetic field in the z direction the atoms will be acted

on by a force which can have only one or another of a discrete set of values mf gfµB

Bz

z .

When a monoenergetic beam of such atoms, distributed at random among states with 2f +1

possible values of mf , passes through an inhomogeneous magnetic field, it is split into 2f + 1

beams which are deflected into ±z directions with deflection angles corresponding to the

various possible discrete values of the force. Thus, if a beam of atoms of some particular

species were observed to be split into, say, 4 beams in a Stern-Gerlach experiment, then one

could conclude that the angular momentum quantum number associated with the magnetic

moment responsible for the deflection is 41

2 = 3

2 .

Turning now to the present experiment in which a beam of potassium atoms passes through

an inhomogeneous field, we note first that the total angular momentum is the sum of the spin

and orbital momenta of the electrons and nucleons. The electronic ground state of potassium

is designated as 2S1/2, which means that the total orbital angular momentum of the electrons,

L, is equal to 0 (i.e. the atom is in an S-state), the fine-structure multiplicity of higher states

(i.e. those with non-zero orbital angular momentum) due to spin-orbit interactions is 2 (one

unpaired electron with spin 1/2), and the total angular momentum J~ = L~ + S~ = ¯h

2 .The

magnetic moment associated with the spin of the electron is gsµBS~

¯h where S~ is the spin

angular momentum, and gs = 2.002319304 is the gyromagnetic ratio of the electron. The

nuclear angular momentum (total spin and orbital momenta of the nucleons lumped into

what is called nuclear ”spin”) of 39K (the most abundant isotope of potassium) is ~I = 3

2h¯

, and the nuclear magnetic moment is gnµBI

~

¯h , where gn is much smaller than the me

mp 1

1836 .

In field free space the interaction between the magnetic moments associated with the total

electronic angular momentum J~ = L~ + S~ and nuclear angular momentum ~I causes them

to precess with a frequency of the order of 100 MHz around their sum F~ = J~ + ~I which

is the total angular momentum of the atom. According to the rules for combining angular

momenta the quantum number of the sum, is f = i ± j = 1 or 2. With each combination

there is associated a magnetic moment whose value can be calculated by matrix mechanics

or, more simply, by the ”vector” model, as explained in Melissinos and other texts.

Potassium atoms emerging into a field-free region from an oven at a temperature of 200

will be

1. almost exclusively in the ground electronic state,

2. nearly equally distributed among the two hyperfine states with f = 1 and f = 2,

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