Energy Spectrum of Beta-Decay

You are required to formulate a piece of investigative report specializing in the field of nuclear physics (Specifically on the concept of shielding from radioactive decay, magnetic deflection of Beta decay products, energy and velocity calculation of Beta decay products.)

The concept of beta decay unites kinds of nuclear transformations: electron (β‾) decay, positron (β?) decay and electron capture. About 900 beta-radioactive isotopes are known. Only about 20 of them are natural, the rest were obtained by artificial methods. The overwhelming majority of these isotopes are subject to β‾- decay. One electron is emitted in each event of  β‾- decay. Double  beta decay, with two electrons(positrons) emitted in each event, is theoretically possible, but has not been observed experimentally as yet (Detlafand I?A?vorski??, 1980).

In beta decay the energy spectrum of the emitted electrons or positrons is continuous, extending from E=0 to E=E₀, where the quantity E₀ is called the end-point energy of the beta spectrum.

The average energy of the electrons emitted by heavy nuclei is E?≈ E₀/3; for natural β‾-radioactive elements E?=0.25 to 0.45 MeV. For light nuclei the energy spectrum of the electrons (positrons) is more symmetrical: E?≈0.5E₀. The half-lives for beta decays are within a wide time range: from 2.5 X 10^-2 second to 4 X 10¹² years, which is incommensurably longer than the characteristic nuclear time (  ?10^-21 to 10^-22 second). This indicates that beta decay is due to weak interaction. Beta decay is commonly accompanied by the emission of gamma rays which have a discrete energy spectrum (Detlafand I?A?vorski??, 1980).

(Figure 1: Energy specturum of Beta decay electron)

An electron antineutrino is emitted together with the electron in beta decay, and an electron neutrino together with positron. The interaction between the electron neutrino (antineutrino) and the nuclei is negligibly small in comparison with the interaction between the nucleons in the nucleus (nuclear interaction). In beta decay the electron (positron) and electron antineutrino (neutrino) have spins equal in magnitude and opposite in direction. Hence the change in the spin of the nucleus equals zero. The continuous spectrum of beta decay is due to the different distributions of energy between the electron (positron) and the electron antineutrino (neutrino), the total energy of the two particles being equal to E₀ (Detlafand I?A?vorski??, 1980).

According to the current concept, the electron (positron) and the electron antineutrino (neutrino) do not exist in atomic nuclei but are formed at the instant of emission from the nucleus as a result of weak interaction between the nucleons in the nucleus. Since new particles are produced in the beta decay, the methods of non-relativistic quantum mechanics are inapplicable to this process and the problem is dealt with the methods of quantum field theory.

Decay Constant

In the theory of beta decay, the production of an electron and an electron neutrino (positron and electron neutrino) is treated as the result of the interaction between a nucleon of the nucleus and the electron (positron) and neutrino fields. Besides the production of e‾ and υ?_e (or e? and υ_e) particles, ₀n¹ is changed to ₁p¹ (or, on the contrary, ₁p¹à ₀n¹). The intensity of this interaction is characterised by the weak interaction constant g (coupling constant of the nucleon and electron-positron fields);  g≈1.4 X 10^-49 erg-cm³. The probability of beta decay is characterised by the nuclear matrix element of the transition |H_ik| containing: a wave function of the nucleon in the initial state i; wave functions of the nucleon, electron (positron) and  electron neutrino (anti-neutrino) in the final state k; interaction energy corresponding to the transition i →k; and, finally, a quantity determining the density of the number of final states of the system. The selection rules for beta decay establish a considerably higher probability of allowed transitions and a low probability of so-called forbidden beta transitions (Detlafand I?A?vorski??, 1980).

Of essential significance in the study of beta decay is the analysis of the energy spectrum N(E) , where N is the number of emitted electrons (or positrons). According to their N(E) distribution curves, beta spectra are divided into allowed (Fermi-Spectra) and forbidden spectra. Forbidden spectra, in turn are distinguished by their degree of forbiddenness. For allowed beta spectra, assuming that mass of the neutrino equals zero

where p and E = momentum and energy of the electrons in units of m_ec and m_ec²

m_e= rest mass of the electron

E₀= end-point, or maximum, energy of the electrons (or positrons) of the beta spectrum.

The function F(Z,E) takes account of the influence of the nuclear field on the shape of the N(E) curve. For forbidden beta spectra, N(E) contains a factor which depends on E₀, E and the degree of forbiddenness.

To decide whether a given beta spectrum belongs to the Fermi or forbidden kind, a Fermi-Curie plot is made. Thus

Where N_exp(E) is the observed curve of the beta spectrum. For the Fermi beta spectrum, K(E) is a straight line which intersects the axis of abscissas at E=E₀. Deviation of K(E) from a straight line indicates that the given beta  spectrum is of the forbidden kind.

The decay constant λ for beta decay is

λ =  C   = CF (Z,E₀

The factor C is determined in the theory of beta decay as

Where g=weak interaction constant (coupling constant)

|H_ik|=nuclear matrix element

Since λ=ln 2/T_1/2, where T_1/2 is the half-life, then

The product F(Z,E₀)T_1/2=T_0.5red is called the reduced half-life.

It depends only on the character of the interaction between the nucleons of the nucleus and the electron-neutrino field. Reduced half-life values, obtained from experimental data, enable |H_ik| to be determined.

According to their T_{0.5 red} values, the types of beta decay can be classified as follows:

• Those with log T_{5 red}≈3.5 are superallowed transitions; they include: beta decay of the neutron, ₁H³, and transitions between mirror nuclei [(N-Z=1 for the initial nucleus and N-Z=-1 for the final one), as well as transitions for which N-Z=±1 for one isobar and N-Z=±2 for the other (where N is the number of neutrons in the nucleus)]; for superallowed transitions, the | H_ik | values are near to the maximum values;
• Those with logT_5red≈5 are normally allowed transitions, and
• Those with logT_1/2red≈9,≈13, ≈18 are forbidden transitions of the 1^st, 2^nd and 3^rd orders of forbiddenness, respectively. The last cases correspond to a sharp decrease in the probability of beta decay, due to large changes in the angular momentum of the nucleus and, frequently, to the change in the parity of its states.

The decay constant λ_E for an allowed electron capture is

λ_E =  |H_ki|² F_E(ε₀,Z)

where  F_E(ε₀,Z)=2Π ()³ [ε₀  + 1 –  ]²

ε₀ =  .

This formula does not take into account the relativistic effects which become appreciable when the value of E₀ approaches the rest energy of the electron, equal to 0.511MeV.

The state of the quantum system is said to be even if the corresponding wave function is not reversed in sign when the signs of all the coordinates of the particles in the system are changed (in inversions); otherwise it is said to be odd. The conservation of the sign of the wave function upon a space inversion can also be characterised by even parity P (where P = +1). If, upon changing the signs of the coordinates of the wave function, the sign of the latter is reversed, then  the parity is said to be odd (P=-1). The conception of the parity of state and of the wave function is related to the symmetry of space, i.e. to the equivalence in space of the right- and left-hand directions, upward and downward, etc.

Any system of particles can be in a state with a definite parity if the number of particles in it remains constant or is changed by an even number.

It follows from the properties of the Schrödinger equation that if the energy of a particle (or system of particles) is conserved constant, then its parity is also conserved (parity conservation law). A system of particles before and after beta decay should correspond to wave functions of the same parity [nucleus before decay; nucleus, beta particle and (anti-neutrino) neutrino after decay].

It has been established that in weak interactions which, in particular, bring about beta decay, the parity of the wave function of the system can change in decay (non-conservation of parity). Along with the state described, for instance, by an even function, an odd state can also appear. This phenomenon was first discovered in the beta decay of K mesons. In essence, this is the decay of a K meson into two or even three pi mesons. In beta decay, the violation of parity conservation is manifested in the asymmetry of the spatial directions of the electrons emitted by the nuclei: fewer electrons are emitted in the direction of spin of the nuclei than in the opposite direction.  It follows from this asymmetry that there is a definite relationship between  the direction of spin of the particle and the direction of its motion in space. The spins of a neutrino and antineutrino should always be oriented parallel or antiparallel to the direction of their motion: the antineutrino is oriented along the direction of motion and the neutrino in the opposite direction (longitudinal polarization). If spin is likened to rotation, then the motion of the antineutrino corresponds to the motion of right-hand spin and that of the neutrino to left-hand spin. Accordingly distinction is made between right-hand helicity of the antineutrino and left-hand helicity of the neutrino.

Maximum research results were established from the evidences found from experiments in which the radiation from radioactive compounds was allowed to pass through magnetic or electric fields. While performing such an experiment it was found that the gamma rays passed through the fields without any deflection. But unfortunately, the alpha and beta rays deflected to opposite directions. Thus it proves that alpha and beta are charged while the gamma ray is not. 

While shooting from a gun, a bullet penetrates depending on the composition of the material of the bullet used  and the composition of the material of the target. For example, a pellet from an air gun will be stopped by a few millimetres of wood, whereas a bullet from a high powered rifle will pass through many millimeters of steel. We may say it is the similar in case of  ionizing radiation. Different ionizing radiations have different penetrating ability (α-, β- or γ-radiation). The examples below give such an idea about the penetration of the radiation from the radioactive sources.

Alpha particles which come from radioactive sources have a range of less than 10 cm in air. In condensed matter medium (e.g. water or tissue) the range becomes even shorter (the difference in range is approximately 1/1000, due to the difference in density) – almost less than 100 mm. Logically, it can be understood that α -particles will not be able to penetrate even clothing. We can sit on a radioactive source and will not cause any harm to ourselves if it were only the α –particles that were emitted . But if the source happens to be inside our body, all the energy is deposited in the body and may give a large radiation dose. Aleksandr Litvinenko in 2006 with Po-210 of such an incident.  Po-210 emit an alpha particle with energy 5.3 MeV

Beta-particles have an energy of 1 MeV and its range is about 5 mm in soft tissue. Almost all the β-particles have an energy less than 1 MeV. And hence all the β-particles coming from environmental sources, are stopped by our clothing

Gamma-radiation can penetrate tissue in our body and can even concrete in it. We have,   50% of the γ-rays from Cs-137, having an energy of 0.662 MeV and can penetrate a water layer of about 9 cm., known as half-value layer. Fortunately γ-radiation can be easily measured, be the source be outside or inside the body. Moreover, isotopes emitting γ-radiation can be used as medical diagnostic.

The binding energy of a nucleus determines which, if any, of these three decay processes is energetically possible. For example, in the decay of the ⁶⁰Co₂₇ nucleus into the ⁶⁰Ni₂₈ nucleus, a β‾-particle is emitted, i.e.,

⁶⁰Co₂₇→ ⁶⁰Ni₂₈ + e‾ + υ?_e ,

releasing an energy of 1.56 MeV in the form of kinetic energies of the three products. An example of the β?- particle emission in the decay of ²²Na₁₁, i.e.,

²²Na₁₁→²²Ne₁₀ + e?+υ_e +0.55 MeV

Further, an example of electron capture is the decay of the ³⁷Ar₁₈ nucleus into the ³⁷Cl₁₇ nucleus after an atomic electron is absorbed, i.e.,

³⁷Ar₁₈ + e¯→³⁷Cl₁₇ + υ_e.

The energy released in this reaction is 0.0186 MeV.

A nucleus lying above the region of stability, i.e. , region of neuron excess,

becomes stable by reducing the number of neurons in it. In other words, such a nucleus disintegrates in a manner that its N/Z ratio gets reduced. This is achieved through the β‾-decay where a neuron inside the nucleus changes into a proton, simultaneously creating and emitting an electron and a neutrino.

Similarly, a nucleus lying below the stability region becomes stable by increasing its N/Z ratio. Thus, such a nucleus decays by emitting a β?- particle or by electron capture. In the β?-decay, a proton inside the nucleus changes into a neutron, simultaneously creating and ejecting a positron and an antineutrino. In electron capture, an atomic orbital electron of a nucleus combines with a proton of the nucleus to change the proton into a neutron and a neutrino(Mani and Mehta, 1990).

To study the energetics of the beta-decay process, we write the mass-energy  equations for this process in terms of the atomic masses:

^AM_Z = ^AM_Z+1 + Q/c² (β¯- decay), (1)

^AM_Z = ^AM_Z-1 +2m_e+ Q/c² (β?- decay), (2)

^AM_Z = ^AM*_Z-1 + Q/c² (electron capture). (3)

In these equations, the neutrino mass does not appear because, a neutrino (and also an antineutrino) is massless. Further, Q is the energy available in the decay. As can be noted, the electron mass does not appear on the right-hand side of (1). His is because an electron is needed for the atom  ^AM_Z+1 to become neutral. On the right-hand side of the β?- decay equation, there are two electron masses. Of these, one is due to the positron the nucleus emits and the other is due to the extra electron (in the atom ^AM_Z-1) the atom releases to make itself neutral. In electron capture eq (3), an orbital electron is captured by the nucleus, and thus the atom ^AM_Z-1 produced is neutral but in an excited state, indicated by the superscript asterisk (Mani and Mehta, 1990).

Let us consider a nucleus containing Z protons and N neutrons. If we attempt to separate this nucleus into individual protons and neutrons, a certain amount of work has to be done because these particles are held together by attractive forces. In other words, energy must be supplied to the nucleus to separate it into its individual constituents and move them to infinity with each constituent being at rest. This energy necessary to separate all the particles is called the binding energy of the nucleus. After the separation, if the nucleons interact and coalesce to form a nucleus, their energy must be less than what it was when they were separated. According to the Einstein mass energy relation, their combined mass in the coalesced state must be less than the sum of their masses when they are far away from one another. If we denote the loss in mass by Δmc² which is the binding energy, i.e., the binding energy (BE) is

BE = Δmc².

If M, M_n, and M_p are the masses of a nucleus (^AX_Z), a neutron, a proton, respectively, then the mass defect Δm is

Δm= NM_n + ZM_p – M

And the binding energy of the nucleus is

BE =(NM_n + ZM_p – M)c².

The energy of the state of infinite separation of nucleons at rest is taken to be zero.

It is more convenient  to work with the mass of an atom than with that of the nucleus. To do this, we add to and subtract from the above equation the mass of Z electrons. Thus,

BE=(NM_n + ZM_p + ZM_e-M – ZM_e)c².

Now, ZM_p + ZM_e= Z(M_p+ M_e). The quantity (M_p+ M_e) is just the mass of one atom of hydrogen (M_H) since the binding energy of an electron is negligible as compared to (M_p+ M_e)c². Similarly, (M(^AX_Z)+ ZM_e) is the mass of the atom ^AX_Z (M_A). Thus, the binding energy in terms of the atomic masses is

BE=(NM_n+ ZM_H-M_A)c².

The binding energy and mass defect are not peculiar to a nucleus. An electron in an atom, an atom in a molecule, and a molecule in a crystal each has a binding energy, but it is usually so small that its mass equivalent can be neglected. The mass equivalent of the binding energy of a nucleus is not negligible in comparison to the total mass of the nucleus because the nuclear forces are very strong.

The expression for the average binding energy per nucleon B? is

B?=BE/A= ((ZM_H + (A-Z)M_n– M)/A)c²,

where M is the mass of the atom. When B? is plotted for all values of A for the known nuclei, we get the curve as shown:

 

( figure 2: B? is plotted for all values of A for the known nuclei)

A special element is that of Strontium-90. When Plutonium and Uranium undergoes a nuclear fission reaction, Strontium is produced as a by-product. As a matter of fact, large number of ⁹⁰Sr is produced during the nuclear weapon tests in 1950 and 1960. Though natural Strontium has no radio-active properties but its isotopes have radio-active properties. They undergo beta decay. It has a decay energy of 0.546 MeV and is distributed among an anti-neutrino, an electron and the ⁹⁰Y (Yttrium-90) isotope (“Strontium-90 Half-Life, Properties, Beta-Decay, Uses | Chemistry Learner”, 2018).

The underlying equation is

⁹⁰Sr₃₈→⁹⁰Sr₃₉ + ⁰e_-1. It has a half-life of about 28.8 years which means that it takes 28.8 years to decay to the half of its initial amount. Strontium decays to Yttrium-90, having a half-life of 64 years. ⁹⁰Y then decays to a stable substance known as Zirconium (Zr) (“Strontium-90 Half-Life, Properties, Beta-Decay, Uses | Chemistry Learner”, 2018).

The beta particles radiated by Yttrium are of high energy whereas that of Zirconium is of moderate energy(“Radiation Protection | US EPA”, 2018).

Strontium is a soft and shiny coloured metal, quite similar to lead. It radiates only beta decay and not gamma decay. Its chemical properties are similar to that of Calcium and has the tendency to precipitate in the teeth and bones of an animal(US EPA, 2018)

It has a very hazardous effect on our health; when accidently consumed along with water, it causes inhalation problem. If it is accidently being consumed, it is easily taken up by bones for its Calcium like properties. Once absorbed it continues to deteriorate our health owing to its long half-life period. It also causes cancer of bones, Leukemia and softening of tissues around our bone marrow (US EPA, 2018).

References:

Detlaf, A. A. and I?A?vorski??, B. M.(1980). Handbook of physics.Translated from the Russian by Nicholas Weinstein, 3rd ed. London: Mir Publications.p. 1131.

Mani, H. S. and Mehta, G. K.(1990).Introduction to Modern Physics. New Delhi:  Affiliated East West Press PVT Ltd.ISBN 10: 8185095736 / ISBN 13: 9788185095738
Strontium-90 Half-Life, Properties, Beta-Decay, Uses | Chemistry Learner. (2018). Retrieved from https://www.chemistrylearner.com/strontium-90.html

US EPA. (2018). Radiation Protection | US EPA. [online] Available at: https://www.epa.gov/radiation [Accessed 24 Jul. 2018].

Radiation Protection | US EPA. (2018). Retrieved from https://www.epa.gov/radiation