BETA DECAY. Beta decays are nuclear processes mediated by the weak interaction. The three types of beta decay are

Transcription

1 BETA DECAY Beta decays are nuclear processes mediated by the weak interaction. The three types of beta decay are a. The conversion of a neutron to a proton along with the emission of an electron and an anti-neutrino (negative beta decay) b. The conversion of a proton to a neutron along with the emission of a positron and a neutrino (positive beta decay or positron emission) c. The capture of an orbital electron by a nuclear proton, which becomes a neutron along with the emission of a neutrino (orbital electron capture) Negative beta decay can take place irrespective of whether the neutron is free or bound inside a nucleus. However positive beta decay and orbital electron capture is possible only for protons bound inside the nuclei (other than that of hydrogen). Free protons cannot undergo positive beta decay or orbital electron capture because the processes require a net energy intake, and, hence cannot occur spontaneously. Negative beta decay can take place in a nucleus if the process leads to a product nucleus whose binding energy is greater than that of the original nucleus, thereby producing a more stable nucleus. In this case, the process leads to a net release of energy and the energy of the product nucleus is less than that of the original nucleus. When negative beta decay takes place in a nucleus, the atomic number of the nucleus increases by one and the mass number remains unchanged. When positive beta decay takes place in a nucleus, the atomic number of the nucleus decreases by one and the mass number remains unchanged. Orbital electron capture also decreases the atomic number of the nucleus by one while the mass number remains unchanged. When an electron of an inner shell is captured by the nucleus, the vacancy created is filled by an electron of the outer shell, along the emission of x-ray photons (characteristic x rays), the energy of which equals the energy difference between the inner and outer energy levels of the atom.

2 Negative beta decay was discovered by Becquerel in Positive beta decay was first observed by Frederic and Irene Joliot Curie in 1934 and orbital electron capture by Luis Alvarez in The interaction of neutrinos with matter is very weak and hence it is very difficult to observe them. Therefore, when beta decay was first discovered, only the electron (beta particle) could be observed. The existence of a fourth particle was predicted by Pauli in an attempt to account for energy conservation during beta decay. This particle was given the name neutrino by Fermi, who developed a theory to explain beta decay. Energy spectrum of beta decay and the neutrino When beta decay was first discovered, it was thought to be a three particle process i. e. or The kinetic energy of the emitted electron should be the difference between the energies of the nuclei X and Y. For example in the energy difference between Bi and Po nuclei is 1.16 MeV and all emitted electrons are supposed to have a kinetic energy of 1.16 MeV leading to a discreet energy spectrum. However, the observed energy spectrum is continuous, with energies ranging from 0 to MeV. For the beta decay of a free neutron, we obtain a similar spectrum with a maximum KE of MeV. To explain this discrepancy, Pauli proposed that an additional particle is emitted along with the electron. The KE is shared by the electron and the additional particle called neutrino. The KEs of both the electron and the neutrino can range from 0 to the maximum possible value, subject to the condition that the total remains the constant maximum possible value which equals the difference in the energies of initial and final nuclei. The energy difference between the initial and final states is called the Q value of the process. It is equal to the kinetic energy of the emitted particles. In the decay of a free neutron, the Q value is

3 ( ) For a neutron at rest, the Q value is the sum of the kinetic energies (T) of the final state particles As the proton is much more massive than the electron and the neutrino, almost all the KE is shared by the electron and the neutrino. The mass of the neutrino can be estimated from the Q value ( ) ( ) ( ) But the observed value of the maximum KE is MeV. Thus it is concluded that the mass of the neutrino is zero up to the accuracy of the measured value. i. e. neutrino is assumed to be massless. It travels at the speed of light. For a nuclear beta decay process the Q value is (assuming neutrino is massless) In electron capture process, the vacancy of the captured electron is filled by another electron from a higher energy level. The extra energy is emitted as x-rays and the energy of the emitted x-rays equals the atomic binding energy of the electron B e. Thus to calculate Q value And the Q value is The existence of the neutrino can also be inferred from the conservation of linear momentum. If only one particle (electron) is emitted in beta decay, the direction of recoil of the final nucleus and the direction of the emission of electron should be exactly opposite to each other. However, it was observed that these two are not exactly opposite to each other. Thus, to account for conservation of linear momentum, another particle also should be emitted so that the total momentum vector of the electron and the neutrino is exactly opposite to the direction of recoil of the final nucleus. The spin of the neutrino can be determined by considering the conservation of angular momentum. In the decay process

4 the neutron, the proton and the electron are spin half particles. The total spin could be conserved only if the neutrino also is a spin half particle. Similarly, the anti-neutrino also is a spin half particle. Conservation of charge demands that the neutrino is electrically neutral. Electric charge: Fermi theory of beta decay Fermi s theory of beta decay is based on Pauli s neutrino hypothesis. According to Fermi, the electron and the neutrino does not exist inside the nucleus and are created at the time of the beta decay process. The interaction responsible for this process is the weak interaction, whose strength is small compared to that of the strong and electromagnetic interactions, and hence can be tackled using quantum mechanical perturbation theory. As the KE of the electron is of the same order of its mass, the electron is to be treated as a relativistic particle. The neutrino, which travels at the speed of light, also, is relativistic. The transition probability for beta decay is given by Fermi s golden rule Here is the matrix element of the interaction potential between the initial and final states and is the density of final states. If the density of final states is large, i. e. if there are a large number of possible final states, the process is more likely to occur. The matrix element is given by where and are the initial and final states. is the initial nuclear state, but contains the final nucleus, the electron and the neutrino. Thus Here g is the coupling constant of weak interaction which is a measure of the strength of the weak force. The operator V is a combination of vector and axial vector terms. Hence, the interaction term responsible for the beta decay process is called V-A (vector-axial vector) interaction term. As the electron and neutrino are emitted as free particles, as a first approximation, we take the matrix element as the nuclear matrix element

5 The number of electrons in unit volume with momenta between p and p+dp is given by ( is the volume of a spherical shell in momentum space. is included to make the number a dimensionless quantity). Similarly, the number of neutrinos with momenta between q and q+dq is As neutrinos travel at the speed of light, their energy E and momenta q are related by E=qc. Therefore, and the density of final states is. Substituting these values, the transition probability is given by where C is a constant for a particular decay process. The number of electrons emitted, with momenta between p and p+dp is The total energy released during the decay Q is the sum of the KE of the electron and the energy of the neutrino qc. i. e. Or The total energy of the electron is the sum of its KE and mass energy. But as the electron is a relativistic particle its total energy is given by Or -. Therefore Thus the number of electron emitted with momenta between p and p+dp is ( ) This function vanishes at and at. Its plot is shown below, which agrees with the observed energies of electrons emitted in the beta decay process.

6 However, the agreement between Fermi s theory and observations is not complete. One of the reasons is the electromagnetic interaction of the electrons. To account for this Fermi added a correction term where Z is the atomic number of the product nucleus. Again some of the nuclei, contrary to what is predicted by the nuclear matrix element, did not give rise to any beta decay. Such decays are called forbidden decays. To account for these, a correction term called shape factor was included along with the matrix element. Thus, the number of emitted electrons is proportional to a) the statistical factor b) the Fermi function and c) the nuclear matrix element along with the correction for forbidden decays.. i. e. Experimental tests Femi theory can be experimentally verified by a)observing the shape of the beta decay spectrum b) calculating the total decay rate and hence half-lives and c) measuring the mass of the neutrino. According to Fermi theory,. The plot of against should give a straight line. Such a plot is called a Fermi-Curie plot and is shown below. The plot can be used to determine the Q value. The total decay rate can be calculated by integrating over momenta from 0 to p max and the resulting integral is called Fermi integral. The half-lives can be calculated from the decay

7 rate using the formula s and agree with the observed values.. The calculated values of half-lives range from 10-3 s to The Fermi theory is based on the assumption that the neutrino is massless. The upper limit on neutrino mass can be calculated from the formula ( ) and it is found the mass of the neutrino is zero within an accuracy of 1 kev. The 1 kev limit is there because atomic masses have been measured with this error bar. By measuring beta decays with small Q values the mass limit has been lowered to the order of a few tens of ev. Angular momentum and parity selection rules If we assume that the nuclear decay process takes place at the origin (r = 0), the orbital angular momentum of the emitted particles can be made to zero. Thus the angular momentum of electrons and neutrinos is solely due to their spin. As both electrons and neutrinos are spin half particles, their spin can be either parallel or anti-parallel. If the spins are parallel, the total spin is 1 and if the spins are anti-parallel the total spin is 0. When the spins are parallel, the spin of the nucleus changes by one (Gamow-Teller decay) and when the spins are antiparallel, the spin of the nucleus does not change (Fermi decay). The parity of the system does not change as the change in orbital angular momentum is zero. Thus the selection rules for beta decay can be written as For example is a Fermi decay whereas is a Gamow-Teller decay. can occur in both ways and is called a mixed decay. Decays such as in which the number of neutrons and protons get interchanged are called mirror decays. In such decays the nuclear wave functions remain unchanged during the decay process. This is because Fermi decay is independent of the meson cloud distribution inside the nucleus. The hypothesis that Fermi interactions of the nucleons are unchanged by the surrounding mesons is called the conserved vector current (CVC) hypothesis. Forbidden decays Forbidden decays are beta decay processes whose probability is small compared to other beta decays. Such decays occur when the initial and final states of allowed decay has opposite parities, and hence cannot occur. Decays with are called first forbidden decays. Their probability of occurrence is small compared those of allowed decays. In a first forbidden decay, the total angular momentum change is either 0 or 1. Therefore, the change in spin angular momentum can be either 0 or 1 or 2. Thus the selection rules are

8 For example is a first forbidden decay. If allowed and first forbidden decays cannot occur, second forbidden decays occur. In this case, the parity change is zero and the probability is even smaller. For a total angular momentum change of 2, the change in spin can be 0,1,2 or 3. However, the decay in which spin change is either 0 or 1 with no parity change is an allowed decay. So in a second forbidden decay, the spin change is either 2 or 3. Thus the selection rules are When allowed decays, first forbidden decays and second forbidden decays are not possible third and further forbidden decays can occur. However, their probability is very small. The reason for the very small probabilities of forbidden decays is that it very difficult to create an electron or a neutrino in a state with etc. This is because, for a 1MeV electron, the typical angular momentum is of the order of and state requires an angular momentum of order, the probability of which is very small. The probabilities of states are even smaller. As the half-lives depend on the total decay rate, the half-lives of such decays are different from those of allowed decays by many orders of magnitude. Thus, the beta decay half-lives are distributed through a large range from 10-3 s to s. Neutrinos Neutrinos are charge less, massless spin half particles. As they are massless, they travel at the speed of light. As they have half integral spin, they obey Fermi-Dirac statistics. They are not subject to strong or electromagnetic forces. They belong to the class of particles called leptons. There are three generations of neutrinos called electron neutrino (first generation), muon neutrino (second generation) and taon neutrino (third generation). The neutrinos emitted in the ordinary nuclear beta decay are electron neutrinos. Antineutrinos are the anti-particles of neutrinos. As neutrinos do not have electric charge, neutrinos and anti-neutrinos are distinguished by their direction of spin or helicity given by As neutrinos travel at the speed of light, their helicity is a constant. Neutrinos have spin and momentum vectors opposite to each other. Anti-neutrinos have spin and momentum vectors in the same direction. i. e. when viewed from behind, the neutrino spins anti-clockwise and the anti-neutrino spins clockwise. i. e. the neutrino is left handed and the anti-neutrino is right handed. As neutrinos are not subject to strong and electromagnetic interactions, their interaction with matter is very feeble and it is very difficult to detect them. The only processes in which the neutrinos interact with matter are called inverse beta decays.

9 And Parity violation in beta decay Parity (P) is the operation in which all coordinates of a coordinate system gets reflected. Charge conjugation (C) is the operation in which particles and anti-particles are interchanged. Time reversal (T) is the operation in which the direction of time is reversed. P, C and T are said to be conserved, if the equations governing a physical system remains unchanged after the respective operation. According to the CPT theorem, all systems described by quantum field theory are invariant under the combined operation of CPT. Parity is conserved in processes involving strong and electromagnetic interactions. On the suggestion of Lee and Yang, Wu et. al. showed that parity is not conserved in weak interactions by observing the beta decay of 60 Co. The spins of the 60 Co atoms were aligned in a particular direction, by aligning the electric dipole moments in a magnetic field at low temperature. The parity operation was accomplished by reversing the direction of the magnetic field. It was observed that the electrons were emitted in a preferred direction i. e. the direction opposite to the magnetic field, thereby violating parity. Thus it was concluded that parity is not conserved in weak interactions. It was also observed that the preferred direction for positrons is along the magnetic field. i. e. particles and anti-particle behave differently in beta decay, thereby violating the C symmetry. However, in ordinary beta decay, the combination CP is conserved. Neutrino mass Neutrinos are considered to be massless in Fermi theory. However, recent observations have revealed that neutrinos have a small non-zero mass of the order of or less than an ev. Neutrinos are produced in the sun during nuclear fusion and their rate can be the calculated by observing the power output of the sun. The number of neutrinos that should reach the earth can be calculated from this. However, it was observed that only one third of what is expected are observed. This is explained by a phenomenon called neutrino oscillation. As neutrinos travel through space, they can change their flavour (i. e. electron neutrino, muon neutrino or taon neutrino). i. e. the three types of neutrinos can change into each other. Thus only one third of the neutrinos remain as electron neutrinos, one third being muon neutrinos and the other one third being taon neutrinos. This is possible only if the neutrinos have nonzero mass. Thus it is concluded that the mass of neutrinos is non zero.