Decay Rates

fhe differential decay rate AW^a(T) will be defined to be the probability that a state a will decay into state ¡3 in the time interval [T/2, -T/2] divided by the total time T (hence a rate). Formally

Under most experimental circumstances, when a decaying system is isolated from the apparatus, the measurement is made over a time interval long compared to the internal time scale of the system, and we may therefore take the limit as T —♦ oo. This limiting rate is denoted AWpa,

Later, we will see that our calculations of the Sßa(T/2,-T/2) can always be expressed in the form

Cross Section

To treat photon scattering from atoms we will need to calculate the differential cross section. This is defined experimentally as target target beam

Fig. 3.2 Drawing of an idealized scattering process showing the differential solid angle Afi and the scattering angle 8.

beam

Fig. 3.2 Drawing of an idealized scattering process showing the differential solid angle Afi and the scattering angle 8.

where the quantities are defined with the help of Fig. 3.2. Note that the cross section has the units of an area. In most experiments, the target is larger than the beam, as illustrated in Fig. 3.2, so that the number of scattering centers in the path of the beam per unit area is

If the particles in the beam have charge eo, then the number of beam particles incident per second can be determined from the beam current where j is the beam current. For photon beams, the quantity no is determined indirectly from an analysis of how the beam is produced.

Theoretically, we evaluate the cross section assuming one scattering center, and a number of particles incident per second determined by the velocity v and a density derivable from one particle in volume L3 (consistent with the box normalization introduced in Chapter 2). Hence

pi _ target density(p)x target length mc mass of each scatterer

=i where A is the area of the beam and Avt is therefore the volume swept out by the beam in time t (see Fig. 3.3). Scattering differs from a decay in that there

We are now ready to calculate the electromagnetic decay of an atom! The calculation is so simple, it's almost an anticlimax.

If the initial state a and the final state ¡3 are not identical, which is always the case for a decay, (/3|l|a) = 0, and to first order in perturbation theory the 5-matrix element is

where the superscript (1) reminds us that this is the first order expression only and we have assumed that (0|i/(oo,-oo)|0) = 1, which is usually true to first order. The interaction Hamiltonian Hj was given in Eq. (3.5), and the states were defined in Eq. (3.6). For one-photon decay of an initial atomic state a into a final atomic state b and a photon with energy ujn and polarization A, the states are

where |1„a) = a^jO) is the one-photon state with frequency un and polarization A. Hence

Sba = -i f dt{b, lnA|Ujl (-^A(re,t) • Ve) UA\a,0) , (3.55)

where pe = —¿Ve and Ve • A(re, t) = 0 were used to obtain the simplified form <3.55). Since the states are direct products, the matrix element (3.55) reduces immediately to the product of two terms, an atomic matrix element expressed as an integral over re and an EM matrix element:

Sba = ~ T dtei,E"~E•>* - /d3re {Vi(re)VeVa(re)} • (lnA|A(reit)|0> , J-oo m J

where Ea and Eb are the energies of the two atomic states. Now, taking the matrix element of the field operator between the vacuum and a one-photon state gives a non-zero result! This is the origin of electromagnetic decay. We get

<lnA|A(re,t)|0> = £ ««'*<0|anA <¿,,10) . (3.57)

Inserting this expression into (3.56) gives

where the decay amplitude fba is

The next step is to reduce fba and compute the decay rate. In most atomic decays, the energy of the emitted photon, which is equal to uin = Ea - Eb, is much less than 1 /R, where R is the size of the atomic system, and hence the maximum range of the integral over re. In this case, the dipole approximation e~ikn rc ^ l

is extremely good. Introducing matrix elements of the momentum operator,

we can write the dipole approximation to the decay amplitude in the following reduced form:

TO \J2u)n and the differential decay rate becomes en 'Pba

27r p4,

2un rri

Summing over all final photon states to get the total a —> b decay rate gives

2uinm

where, in the second step, we took the continuum limit (3.44). Eliminating the ¿-function by integrating over the magnitude of k and using k — u gives

We leave the calculation here, assuming that applications of this result are familiar from previous studies.

Note that quantization of the EM field has given a natural explanation for decay [ (lnx|A(re, t)|0) ^ 0 ] and the normalization of the decay rate is uniquely predicted by the theory. Also, note how energy conservation (u> = Ea - Eb) arises naturally.

3.4 THE LAMB SHIFT

We search for additional effect due to the quantization of the electromagnetic field. Imagine ourselves back in the late 1940's. The Lamb shift has been discovered.* Everyone believes it is due to field quantization. Can we calculate it? H. A. Bethe did [Be 47], and it is said that he did it on a train, while returning from a conference.

The Lamb shift was measured by W. E. Lamb and W. E. Retherford in 1947 using microwave techniques [LR 47]. It is the splitting between the 25i/2 and 1P\j2 states, which are degenerate to order {v/c)2 (and even exactly to all orders when the Dirac equation is used). The S-state is higher than the P-state by about 1060 MHz. A diagram of the energy levels of hydrogen-like atoms is shown in Fig. 3.5.

To calculate the shift in energy of a bound state, we use second order perturbation theory. The derivation of the energy shift starts from the equation

(Ho + A Hi) |q) - (40) + + A242) + ■■ Ma) , (3.69)

AEc where A is a parameter which keeps track of the orders of perturbation theory but is eventually set to A = 1. The derivation of the formula for the energy shift in the general case is identical to that from ordinary nonrelativistic, non-degenerate, bound state perturbation theory, so we will not repeat the steps here. We obtain the usual result, valid to second order:

The task is to evaluate AEa to second order, i.e., to order e2.

First, note that (a\Hi\a) — 0, because the only such term which might be non-zero, the A2 term in Hi, is normal ordered. Hence its vacuum expectation value is zero, and

Thus the entire contribution comes from the sum in (3.70).

'For a review of the early experiments see [La 51],

where \(3) — \b, 1*.) is a direct product of an atomic state b and a one-photon state with momentum k. In this section we denote / by ft,a to emphasize that it depends on atomic states a and b and is (nearly) independent of the photon states. Hence A Ea reduces to j fba I 2

Note that a — b is included in the sum over b. Even when a = b, |P) ^ |a), because \(3) has one photon and |a) does not.

The low energy contributions to the sum (3.73) can be estimated using the dipole approximation for fba, Eq. (3.60). We obtain

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