Cluster decay

Nuclear physics
Radioactive decay Nuclear fission Nuclear fusion
Classical decays Alpha decay · Beta decay · Gamma radiation
Advanced decays Double beta decay · Double electron capture · Internal conversion · Isomeric transition · Cluster decay · Spontaneous fission
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Capturing Electron capture · Neutron capture R · S · P · Rp
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Cluster decay (also named heavy particle radioactivity or heavy ion radioactivity) is a type of nuclear decay in which a parent atomic nucleus with A nucleons and Z protons emits a cluster of N_e neutrons and Z_e protons heavier than an alpha particle but lighter than a typical binary fission fragment (although ternary fission into three fragments produces products which overlap cluster decay). A chemical transformation of the parent nucleus leads to a different element, the daughter, with a mass number A_d = A - A_e and atomic number Z_d = Z - Z_e where A_e = N_e + Z_e. ^[1]

For example:

²²³Ra₈₈ → ¹⁴C₆ + ²⁰⁹Pb₈₂

This type of rare decay mode was observed in radioisotopes that decay predominantly by alpha emission, and it occurs only in a small percentage of the decays for all such isotopes. ^[2]

The branching ratio with respect to alpha decay

B = T_a / T_c

is rather small (see the Table below). T_a and T_c are the half-lives of the parent nucleus relative to alpha decay and cluster radioactivity, respectively.

Cluster decay, like alpha decay, is fundamentally a quantum tunneling process: in order to be emitted the cluster must penetrate a potential barrier. This is a different process than that which precedes light fragment emission in ternary fission, which results from previous nuclear deformation and energy input, as a result of a nuclear reaction (most often from a neutron). By contrast, cluster emission is a type of spontaneous radioactive decay.

Theoretically any nucleus with Z > 40 for which the released energy (Q value) is a positive quantity, can be a cluster-emitter. In practice, observations are severely restricted to limitations imposed by currently available experimental techniques which require a sufficiently short half-life, T_c < 10³² s, and a sufficiently large branching ratio B > 10 ⁻¹⁷.

In the absence of any energy loss for fragment deformation and excitation, as in cold fission phenomena or in alpha decay, the total kinetic energy is equal to the Q-value and is divided between the particles in inverse proportion with their masses, as required by conservation of linear momentum

E_k = QA_d / A

where A_d is the mass number of the daughter, A_d = A – A_e.

Cluster decay exists in an intermediate position between alpha decay (in which a nucleus spits out a He⁴ nucleus), and spontaneous fission, in which a heavy nucleus splits into two (or more) large fragments and an assorted number of neutrons. Spontaneous fission ends up with a probabilistic distribution of daughter products, which sets it apart from cluster decay. In cluster decay for a given radioisotope, the emitted particle is a light nucleus and the decay method always emits this same particle. Nevertheless, for heavier emitted clusters there is practically no qualitative difference between cluster decay and spontaneous cold fission.

History

The first information about the atomic nucleus was obtained at the beginning of the 20th century by studying radioactivity. For a long period of time only three kinds of nuclear decay modes (alpha, beta and gamma) were known. They illustrate three of the fundamental interactions in nature: strong; weak, and electromagnetic. Spontaneous fission became popular soon after its discovery in 1940 by K. Petrzhak and G. Flerov owing to both military and peaceful applications of neutron-induced fission discovered in 1939 by Otto Hahn, Lise Meitner and Fritz Strassmann, employing the large amount of energy released during the process.

There are many other kinds of radioactivity, e.g. cluster decay, proton decay, various beta-delayed decay modes (p, 2p, 3p, n, 2n, 3n, 4n, d, t, alpha, f), fission isomers, particle accompanied (ternary) fission, etc. The height of the potential barrier, mainly of Coulomb nature, for emission of the charged particles is much higher than the observed kinetic energy of the emitted particles. The spontaneous decay can only be explained by quantum tunneling in a similar way to the first application of the Quantum Mechanics to Nuclei given by G. Gamow for alpha decay.

“In 1980 A. Sandulescu, D.N. Poenaru, and W. Greiner described calculations indicating the possibility of a new type of decay of heavy nuclei intermediate between alpha decay and spontaneous fission. The first observation of heavy-ion radioactivity was that of a 30-MeV, carbon-14 emission from radium-223 by H.J. Rose and G.A. Jones in 1984“. ^[3]

Usually the theory explains an already experimentally observed phenomenon. Cluster decay is one of the rare examples of phenomena predicted before experimental discovery. Theoretical predictions were made in 1980, ^[4] four years before experimental discovery. ^[5]

Four theoretical approaches were used: fragmentation theory by solving a Schroedinger equation with mass assymetry as a variable to obtain the mass distributions of fragments; penetrability calculations similar to those used in traditional theory of alpha decay, and superasymmetric fission models, numerical (NuSAF) and analytical (ASAF). Superasymmetric fission models are based on the macroscopic-microscopic approach ^[6] using the asymmetrical two-center shell model ^{[7] [8]} level energies as input data for the shell and pairing corrections. Either the liquid drop model ^[9] or the Yukawa-plus-exponential model ^[10] extended to different charge-to-mass ratios ^[11] have been used to calculate the macroscopic deformation energy.

Penetrability theory predicted eight decay modes: ¹⁴C, ²⁴Ne, ²⁸Mg, ^32,34Si, ⁴⁶Ar and ^48,50Ca from the following parent nuclei: ^222,224Ra, ^230,232Th, ^236,238U, ^244,246Pu, ^248,250Cm, ^250,252Cf, ^252,254Fm and ^252,254No.

The first experimental report was published in 1984, when physicists at Oxford University discovered that Ra²²³ emits one C¹⁴ nucleus out of every billion (10⁹) alpha decays.

Theory

The quantum tunneling may be calculated either by extending fission theory to a larger mass asymmetry or by heavier emitted particle from alpha decay theory. ^[12]

Both fission-like and alpha-like approaches are able to express the decay constant λ = ln 2 / T_c, as a product of three model-dependent quantities

λ = νSP_s

where ν is the frequency of assaults on the barrier per second, S is the preformation probability of the cluster at the nuclear surface, and P_s is the penetrability of the external barrier. In alpha-like theories S is an overlap integral of the wave function of the three partners (parent, daughter and emitted cluster). In a fission theory the preformation probability is the penetrability of the internal part of the barrier from the initial turning point R_i to the touching point R_t. ^[13] Very frequently it is calculated by using the Wentzel-Kramers-Brillouin (WKB) approximation.

A very large number, of the order 10⁵, of parent-emitted cluster combinations were considered in a systematic search for new decay modes. The large amount of computations could be performed in a reasonable time by using the ASAF model. The model was the first to be used to predict measurable quantities in cluster decay. More than 150 cluster decay modes have been predicted before any other kind of half-lives calculations have been reported. Comprehensive tables of half-lives, branching ratios, and kinetic energies have been published, e.g. ^[14] .^[15] Potential barrier shapes similar to that considered within the ASAF model have been calculated by using the macroscopic-microscopic method. ^[16]

Previously ^[17] it was shown that even alpha decay may be considered a particular case of cold fission. The ASAF model may be used to describe in a unified manner cold alpha decay, cluster decay and cold fission (see figure 6.7, p. 287 of the Ref. [2]).

One can obtain with good approximation one universal curve (UNIV) for any kind of cluster decay mode with a mass number Ae, including alpha decay

logT = − logP_s − 22.169 + 0.598(A_e − 1)

In a logarithmic scale the equation log T = f(log P_s) represents a single straight line which can be conveniently used to estimate the half-life. A single universal curve for alpha decay and cluster decay modes results by expressing log T + log S = f(log P_s). ^[18] The experimental data on cluster decay in three groups of even-even, even-odd, and odd-even parent nuclei are reproduced with comparable accuracy by both types of universal curves, fission-like UNIV and UDL ^[19] derived using alpha-like R-matrix theory.

In order to find the released energy

Q = [M − (M_d + M_e]c²

one can use the compilation of measured masses ^[20] M, M_d, and M_e of the parent, daughter and emitted nuclei, c is the light velocity. The mass excess is transformed into energy according to the Einstein's formula E = mc².

Experiments

The main experimental difficulty in observing cluster decay comes from the need to identify a few rare events among an enormous number of background alpha particle. The quantities experimentally determined are the partial halflife, T_c, and the kinetic energy of the emitted cluster E_k. There is also a need to identify the emitted particle.

Detection of radiations is based on their interactions with matter, leading mainly to ionizations. Using a semiconductor telescope and conventional electronics to identify the ¹⁴C ions, the Rose and Jones's experiment was running for about six months in order to get 11 useful events.

With modern magnetic spectrometers (SOLENO and Enge-split pole), at Orsay and Argonne National Laboratory (see ch. 7 in Ref. [2] pp. 188–204), a very strong source could be used, so that results were obtained in a run of few hours.

Solid state nuclear track detectors (SSNTD) insensitive to alpha particles and magnetic spectrometers in which alpha particles are deflected by a strong magnetic field have been used to overcome this difficulty. SSNTD are cheap and handy but they need chemical etching and microscope scanning.

A key role in experiments on cluster decay modes performed in Berkeley, Orsay, Dubna and Milano played P. Buford Price, Eid Hourany, Michel Hussonnois, Svetlana Tretyakova, A. A. Ogloblin, Roberto Bonetti and their coworkers.

The main region of 20 emitters experimentally observed until 2010 is above Z=86: ²²¹Fr, ^221-224,226Ra, ^223,225Ac, ^228,230Th, ²³¹Pa, ^230,232-236U, ^236,238Pu, and ²⁴²Cm. Only upper limits could be detected in the following cases: ¹²C decay of ¹¹⁴Ba, ¹⁵N decay of ²²³Ac, ¹⁸O decay of ²²⁶Th, ^24,26Ne decays of ²³²Th and of ²³⁶U, ²⁸Mg decays of ^232,233,235U, ³⁰Mg decay of ²³⁷Np, and ³⁴Si decay of ²⁴⁰Pu and of ²⁴¹Am.

Some of the cluster emitters are members of the three natural radioactive families. Others should be produced by nuclear reactions. Up to now no odd-odd emitter has been observed.

From many decay modes with half-lives and branching ratios relative to alpha decay predicted with the analytical superasymmetric fission (ASAF) model, the following 11 have been experimentally confirmed: ¹⁴C, ²⁰O, ²³F, ^22,24-26Ne, ^28,30Mg, ^32,34Si. The experimental data are in good agreement with predicted values. A strong shell effect can be seen: as a rule the shortest value of the half-life is obtained when the daughter nucleus has a magic number of neutrons (N_d = 126) and/or protons (Z_d = 82).

The known cluster emissions as of 2010 are as follows: ^{[21] [22] [23]}

Isotope	Emitted particle	Branching ratio	log T(s)	Q (MeV)
Ba114	C12	< 3.4×10−5	> 4.10	18.985
Fr221	C14	8.14×10−13	14.52	31.290
Ra221	C14	1.15×10−12	13.39	32.394
Ra222	C14	3.7×10−10	11.01	33.049
Ra223	C14	8.9×10−10	15.04	31.829
Ra224	C14	4.3×10−11	15.86	30.535
Ac223	C14	3.2×10−11	12.96	33.064
Ac225	C14	4.5×10−12	17.28	30.476
Ra226	C14	3.2×10−11	21.19	28.196
Th228	O20	1.13×10−13	20.72	44.723
Th230	Ne24	5.6×10−13	24.61	57.758
Pa231	F23	9.97×10−15	26.02	51.844
	Ne24	1.34×10−11	22.88	60.408
U232	Ne24	9.16×10−12	20.40	62.309
	Mg28	< 1.18×10−13	> 22.26	74.318
U233	Ne24	7.2×10−13	24.84	60.484
	Ne25		60.776
	Mg28	<1.3×10−15	> 27.59	74.224
U234	Mg28	1.38×10−13	25.14	74.108
	Ne24	9.9×10−14	25.88	58.825
	Ne26			59.465
U235	Ne24	8.06×10−12	27.42	57.361
	Ne25			57.756
	Mg28	< 1.8×10−12	> 28.09	72.162
	Mg29			72.535
U236	Ne24	< 9.2×10−12	> 25.90	55.944
	Ne26			56.753
	Mg28	2×10−13	27.58	70.560
	Mg30			72.299
Pu236	Mg28	2.7×10−14	21.52	79.668
Np237	Mg30	< 1.8×10−14	> 27.57	74.814
Pu238	Si32	1.38×10−16	25.27	91.188
	Mg28	5.62×10−17	25.70	75.910
	Mg30			76.822
Pu240	Si34	< 6×10−15	> 25.52	91.026
Am241	Si34	< 7.4×10−16	> 25.26	93.923
Cm242	Si34	1×10−16	23.15	96.508

Fine structure

The fine structure in 14-C radioactivity of 223-Ra was discussed for the first time by M. Greiner and W. Scheid in 1986. ^[24] The superconducting spectrometer SOLENO of IPN Orsay has been used since 1984 to identify ¹⁴C clusters emitted from ^222-224,226Ra nuclei. Moreover it was used to discover ^{[25] [26]} the fine structure observing transitions to excited states of the daughter. A transition with an excited state of ¹⁴C predicted in Ref. [24] was not yet observed.

Surprisingly, the experimentalists had seen a transition to the first excited state of the daughter stronger than that to the ground state. The transition is favoured if the uncoupled nucleon is left in the same state in both parent and daughter nuclei. Otherwise the difference in nuclear structure leads to a large hindrance.

The interpretation ^[27] was confirmed: the main spherical component of the deformed parent wave function has an i11/2 character, i.e. the main component is spherical.

References

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23. ^ Guglielmetti, A. et al. (2008). "Carbon radioactivity of ²²³Ac and a search for nitrogen emission". J. Phys.: Conf. Series 111: 012050. Bibcode 2008JPhCS.111a2050G. doi:10.1088/1742-6596/111/1/012050. 
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