16.451 Lecture 3 16/9/2003 Recall Penning Trap orbits......

• cylindrical coordinates: (,,z); • B = constant along z• radial () and axial (z) electric field

2

22

2

2/

d

zVVwithVE o

zzd

V

m

ez

Bd

V

m

e

zo

o

22

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A superposition of three motions for a given particle energy near the center of the trap:

1. circular orbits around the magnetic field at the cyclotron frequency c’ = eB/m - m 2. vertical oscillations (along z) at the axial frequency z

3. slow circular orbits in the horizontal plane at the magnetron frequency m = z2/ 2 c

Bv:forceLorentz

EqF

1

2

12

zd

VE o

Detection schemes:

• axial and cyclotron oscillation frequencies are proportional to (e/m). • trapped ions absorb power when driven at a resonance frequency

Examples from early paper on proton/electron mass ratio: Phys. Rev. Lett. 47, 395 (1981)

-- precision was already so high that there is a correction required to account for the number ofparticles in the trap! (m/m correction to mp/me = -1.2 x 10-10/ particle; for one species, -2.2 x 10-7 )

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State of the Art today:

State of the art today:

precise proton-antiprotonmass comparison to test matter/antimatter symmetry.

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Antihydrogen experiments at CERN:

main goal is trapping for both antiprotons and positrons to encourage atomformation – use a cylindrical Penning trap:

potential minimum traps antiprotons – maximum traps positrons...

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Big news: Sept. 2002

Great website!!!!

http://athena.web.cern.ch/athena/

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Back to the Proton... Magnetic Moment ()

For the electron, which is a true pointlike spin- ½ particle (as far as we know), quantum electrodynamics predicts that the magnetic moment is almost exactly equal to the Bohr magneton:

TeVm

e

eB /1078.5

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This is usually expressed in terms of a “spin g-factor”: B sg

with g = 2 and s = ½ for the electron spin.

The exact measured value is:e/B = 1.001 159 652 187 0.000 000 000 004

The tiny discrepancy is referred to as the electron’s anomalous magnetic moment or alternatively as the value of “(g-2)”. It arises from the electron’s interaction withvirtual particles in the vacuum, and can be calculated from first principles in QED.Agreement between theory and experiment is staggeringly good – better than 10-10

Bottom line: a pointlike spin- ½ particle should have a g factor very close to 2.

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BE.

Now for the proton:

By analogy, a pointlike proton should have a magnetic moment given by the nuclearmagneton, N:

TeVm

e

pN /1015.3

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that is, we expect:Np sg

with g = 2 and s = ½ for the proton spin.

The exact measured value is:

which implies a g-factor of about 5.586 --- a huge discrepancy with the predictionfor a pointlike object...

p/N = 2.792847337 0.000000029

Conclusion: the proton must have an internal structure that accounts for its magnetic moment discrepancy

(quark model prediction: 2.793 N, based on 3 pointlike, spin – ½ constituents!)

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Measurements: I Early Days

Stern – Gerlach effect:

)( BUF

Deflection in an inhomogeneousmagnetic field is proportional tothe magnetic moment.

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• H2 molecule exists in two angular momentum states:

Ortho-H2 (J = 1) and Para-H2 (J = 0)

• room temperature gas is ¾ (J = 1) and ¼ (J = 0) (unpolarized, random orientations)

• (J = 1) component has proton spins parallel and electron spins coupled to zero

• magnetic moment points along the spin direction for protons: (ortho-H2) = 2 p

• beam should separate into 3 separate components in a nonuniform B – field, corresponding to mJ = (1, 0, -1) with separations proportional to p

Basic idea ... 9

Some details ...

• had to account for magnetic moment associated with molecular rotation – complications reduced by limiting rotational excitations at low temperature (90 K)

• temperature, pressure and geometry dependence of the lineshape carefully modelled

• results relied on absolute magnetic field map

not a huge signal!

Result: p = 2.46 0.07 N

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Modern measurement, II: spectroscopy of atomic hydrogen

• hydrogen atom ground state has a hyperfine structure due to interaction ofthe electron and nuclear spins

• total angular momentum of the atom:

pe ssIJF

general case, Krane 16.3 – atomicelectron angular momentum is J, nuclear angular momentum is I

state me mp mF

1 ½ ½ 1

2 ½ -½ 0

3 - ½ -½ -1

4 - ½ ½ 0

Addition of angular momentum:

Total angular momentum haseigenvalues F = 0, 1. F=1 has3 magnetic substates (1, 0, -1)and F = 0 has only 1 (mF = 0)

See Krane, 16.3 and 16.4

(Clever trick allows for a comparison of states without knowing the B field exactly!)

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peF mmm

Hyperfine splitting of hydrogen atomic states in zero external field:

E

)1,0,1(,1 FmF

0,0 FmF

eV108.5 6 JIE

origin of the effect: magnetic field of electron couples to magnetic momentof the nucleus (proton), affecting the total energy via BE

)1()1()1( JJIIFFJI

Expectation value:

(We will work this out later!!!)

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Energy levels split in an external magnetic field:

pe

pe

pe

pe

energy

e spin flip (microwave)

p flip (RF)

p flip (RF)

Fundamental accuracy limit: all transition frequencies are proportional to B

Solution: ratio of p flip to e flip transition frequencies, measured simultaneously in the same magnetic field, gives p/e independent of B!

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Precision double resonance experiment:

Result: p/e measured toa relative accuracy of 10-8 !

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Double resonance – microwaves showabsorption dip when RF frequency isjust right to flip proton spins. Statesare “filtered” beforehand so that onlythe double resonance can lead to a dipin the microwave signal.

Summary and follow up:

• We have identified some of the frontiers of modern nuclear science, which wewill explore in more detail as the course progresses.

Follow up: print out pages 14 – 27 of the DOE Long Range Plan for Nuclear Science, and read through it, focusing on the topics mentioned in the lectures.(Don’t worry about the parts we have not discussed yet, and don’t let the details scare you!)

• We have examined some of the basic static properties of the proton – spin, mass, magnetic moment... and we have seen how they are measured. (see Krane, 16.4)

Follow up: as an exercise, think about the particle orbits in a Penning trap and try to work out the 3 frequencies discussed in the notes.

General: explore some of the links on the web page, particularly the Athena experimentat CERN. Think about the precision experimental techniques that we have discussedand summarize for yourself the “tricks” that made these impressive results possible.

Homework: next week!

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