274 Ion traps
different valuesωzωmωc(Exercise 12.4).12.7.2 Mass spectroscopy of ions
The determination of the mass of ions by means of eqn 12.20 for Paul
trapping has been mentioned earlier. Alternatively, measurement of the
ratio of cyclotron frequencies of two different species of ion in the same
Penning trap gives their mass ratio:
ωc′
ωc=
eB/M′
eB/M=
M
M′
. (12.26)
This assumes the simplest case with two species of equal charge, but the
ratio of the charges is always known exactly. Superconducting magnets
give very stable fields so that the cancellation ofBin the above equa-
tion introduces very little uncertainty and in this way masses can be
compared to better than 1 part in 10^8.12.7.3 The anomalous magnetic moment of the electron
The advantages of the Penning trap have been exploited to make precise
measurements of the magnetic moment of the electron (confined in the
same way as ions but with a negative voltage on the end caps). From
an atomic physics perspective, this can be viewed as a measurement of
the Zeeman effect for an electron bound in a trap rather than one bound
in an atom (Dehmelt 1990), but the splitting between the two magnetic
statesms=± 1 /2 is the same in both situations, corresponding to a
frequency ∆ω=gsμBB/=gseB/ 2 me. Measurement of this frequency
gives the gyromagnetic ratio for spings. To determineBaccurately
they measured the cyclotron frequencyωc=eB/meand found the ratio
∆ω
ωc=
gs
2. (12.27)
The relativistic theory of quantum mechanics developed by Dirac pre-
dicts thatgsshould be exactly equal to 2, but the incredibly precise
measurement by Van Dycket al. (1986) found
gs
2=1.0011596521884(4).
The accuracy is better than 4 in 10^12. Often this is quoted as a mea-
surement ofg−2 for the electron and the difference from 2 arises from
quantum electrodynamic (QED) effects. For the electron the theoretical
calculation gives
gs
2=1+
α
2 π+A 2
(α
π) 2
+A 3
(α
π) 3
+A 4
(α
π) 4
+.... (12.28)
The very detailed calculations give the coefficientsA 2 =− 0 .328478965,
A 3 =1.17611 andA 4 =− 0 .99. The numerical value of this expression