Gamma Decay

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Gamma Decay

We will discuss the process of gamma decay in this chap-ter. As the name suggests, a nucleus in one of its excitedstates can decay to a lower energy excited state or, theground state by the emission of a gamma ray (or, a radia-tive transition). It is possible for a nucleus to decay withoutthe emission of gamma rays; such processes are known asnon-radiative transitions (for example the process of inter-nal conversion). A gamma ray is a high energy (high fre-quency) electro-magnetic radiation and originates from anucleus. It is, therefore, analogous to the emission of lightby atoms. The energies of many gamma rays may actuallyoverlap with the energies of hard x-rays emitted fromsome atoms. Since both are EM-radiations, the only differ-ence is that of the origin. However, most gamma rays havemuch higher energy. For example, typical energies of gam-ma rays are in the range of few hundred keV to 2.5 MeV ormore, and a 1 MeV gamma ray corresponds to a wave-length of 0.0124 Ǻ or, 1240 Fermi.

Normally a nucleus does not remain in an excited state fora long time. The life-time of the excited states, which decayby the gamma emission is very short since the decay oc-curs by the EM-interaction. The typical lifetime is of the or-



der of 10-14 sec. Therefore, we do not have nuclei naturallyoccurring in excited states. However, a nucleus decayingby "- or, #-decay, often leaves the daughter nucleus in oneof its excited states. This, then, leads to the emission of agamma ray. Most of the laboratory gamma sources are ofthis type only. A couple of examples of such gamma de-cays are shown in the diagram below. The gamma transi-tions are shown by the wavy lines and labeled by the typeof radiation emitted.

Since alphaparticles areemitted ingroups withseveral specif-ic energies,they populatedifferent excit-ed states in

the daughter nucleus. The daughter nucleus then emitsgamma rays with specific energies which are correlatedwith the difference of the energies of various alpha groups.Similarly, the end point beta particle energies can also becorrelated with the gamma ray energies. This leads to valu-



able informa-tion about thelow lying en-ergy levels innuclei.

By using themodern accel-erators, it ispossible to

populate a nucleus in a highly excited state whose decayoccurs by the emission of many gamma rays. These gam-ma ray energies carry useful information about the higherlying and high angular momentum states of the nucleusand their quantum numbers. The gamma-ray spec-troscopy, therefore, has become an important modern toolin building the level structure or the decay scheme of a giv-en nucleus.

Lifetime and Decay width

A radiative transition takes a nucleus from an excited stateto a lower energy state without any change in the Z, N val-ues of the nucleus. The emitted gamma ray photon has anobserved energy equal to the energy difference betweenthe two levels minus the recoil energy of the nucleus. This



recoil energy is easy to obtain by momentum conservation.The photon momentum is given by p = Eph / c . The recoil-ing nucleus will have the same momentum but in the op-posite direction. The recoil energy will be given by ER = p2

/ 2m. The energy of the observed photon has to be correct-ed for this recoil loss to obtain the actual gamma ray ener-gy. Quite often, this correction is very small. For a 1 MeVgamma ray and a nucleus in the mass range of 100, the cor-rection is of the order of few eV. One may, therefore, usethe non-relativistic relations safely. This recoil is, however,much larger than the line width discussed below and mayhave important consequences in Mossbauer effect.

The finite lifetime of a decaying level imparts it a finite en-ergy width, which is given by the uncertainty principle, . If

the mean lifetime is denoted by $ and the widthof the level (uncertainty) by % , then

Larger the lifetime, smaller is the width. Thus all excitedlevels will have a finite gamma decay width. An infinitelifetime (a level that does not decay at all, like the groundstate of a stable nucleus) implies zero width. The transitionprobability per unit time or, the decay constant, denoted as



& , is given by

Larger the decay width, higher is the decay probability.The finite decay width results in a finite spread in the ener-gy of the emitted gamma ray. This width is also called asthe natural line width of a given level.

The concept of level width can be extended to include allkinds of decays from a level. Thus, we can define an alphadecay width, a beta decay width, and so on. The totalwidth of a level is then given by the sum of all the individ-ual widths, provided it can decay by all these processes:

ΓTotal = Γα + Γβ + Γγ + ..........

The individual widths for a type of decay are called partialdecay widths, like the alpha decay width, the beta decaywidth etc. If a level decays by only one process, then its to-tal width is given by just one decay width. For a level de-caying by gamma decay only and a decay width of 1 eV,the mean life will be,

sec.



Multipole Expansion of the electric field and Mul-tipole Radiation

A nucleus carries a finite electric charge which is distrib-uted according to the density distribution in the nucleus.These charges set up a charge density and a current densi-ty distribution in the nucleus. An oscillation of the chargesand the currents leads to the emission of gamma rays ofelectric (E) and/or, magnetic (M) nature having variousmulti-polarities. This can be understood very precisely byfollowing the quantum theory of radiation. The classicaltheory of radiation is also sufficient to understand the ba-sics of gamma decay. We will, however, present here abrief discussion of the salient features based on the classi-cal picture of a nucleus as a classical charge density distrib-ution. The potential produced by an arbitrary charge distri-

bution may be written as:

where,

For , so that the integrand may be expand-ed by using the binomial theorem to give,



= V0 + V1 + V2 + ..... + VL + ......

In general, the lth multipole moment is given by

.

Each term is proportional to 1 / rL + 1 . The first term repre-sents the potential due to zeroth order moment, the secondterm is due to dipole moment and so on. One may inter-pret each of these terms as follows:

1. 0 V represents the Coulomb field at r 'of a point charge –a monopole

2. 1 V represents the field of a dipole

3. 2 V represents the field of a quadupole etc.

The monopole term does not lead to any emission of radia-tion. The second term leads to the emission of electric di-pole (E1) radiation. The third term leads to the emission ofelectric quadrupole (E2) as well as magnetic dipole (M1)



radiation. In general, l V leads to the emission of 2l -poleelectric radiation ( El ) or, 1 2l � pole magnetic radiation(M l �1). The l -value defines the multipolarity of the fieldand also the angular momentum carried by the photon.Since the nuclear states have well defined angular momen-tum and parity, the emission of a type of radiation duringa decay is fixed by the conservation laws of angular mo-mentum and parity. Selection Rules for gamma transitionsLet i I and f I be the angular momentum of the initial andthe final states. The conservation of angular momentumdemands that the emitted photon must carry an angularmomentum l , which is given by

.

For example, a transition between the states having Ii = 4 toIf = 2 may have allowed l values 2,3,4,5, and 6. The generalrule is that the transition having the lowest l value domi-nates. However, the E and M nature of the transition is notfixed yet. This is decided by the parity conservation lawwhich is a valid law for EM interaction. According to theparity selection rules, an electric multipole (El) transitioncarries a parity of ( − 1)l . In other words, the parity is posi-tive for even- l and negative for odd- l . Conservation ofparity demands that



πi = πfπl ,

where (i,(f,(l are the parities of the initial state, the finalstate and the l - multipole photon. Therefore, there will beno parity change if l is even and a parity change if l is oddin the case of electric transitions. The situation is reversedin the case of magnetic transitions because a magnetic mul-tipole ( Ml )transition carries a parity of ( − 1)l + 1. Therewill be no parity change if l is odd and a parity change if lis even in the case of magnetic transitions. The followingtable lists the results for various situations:

Since severalmultipoletransitionsmay be possi-ble for a givenpair of initialand final

states, one must be able to decide which will be the mostintense transitions. The table below lists the most intenseof all the allowed values for some situations:

Here, the most prominently observed transitions areshown outside of brackets. For example, when E1, M2, E3



are possible,then E1 willdominate as ithas the lowestmultipolarityand the proba-bility of atransition de-creases rapid-

ly with increasing l - value. However, when M1, E2, M3 arepossible, then M1 and E2 will dominate. This is becausethe magnetic transitions are always weaker than the elec-tric transitions (magnetism is a relativistic effect and veryweak compared to the electric effects). The transition is to-tally forbidden as the photon is an integer spin particle andthe angular momentum will not be conserved in this case.A typical level scheme, containing some large multipoletransitions, is shown below:

It is one of the rare examples where multipolarities fromL=1 to 4 are observed.



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Gamma Decay