2 Estimating the Radii of Nuclei Consider an α-particle that is on a direct collision course with a gold nucleus and its subsequent path. Since the gold nucleus is much more massive than the α-particle we can ignore any recoil of the gold nucleus.
3 Energy Considerations The kinetic energy of the α-particle when it is a long way from the nucleus is Ek. As it approaches the nucleus, due to the Coulomb force, its kinetic energy is converted into electrostatic potential energy.At the distance of closest approach all the kinetic energy will have become potential energy and the α-particle will be momentarily at rest. Hence we have thatwhere the nucleus with atomic number Z has a charge of Ze and the a-particle has a charge of 2e.
4 For a gold nucleus (Z = 79) and an α-particle with kinetic energy 4 For a gold nucleus (Z = 79) and an α-particle with kinetic energy 4.0 MeV we have thatThe distance of closest approach will of course depend on the initial kinetic energy of the α-particle. However, as the energy is increased a point is reached where Coulomb scattering no longer take place. The above calculation is therefore only an estimate. It is has been demonstrated at separations of the order of m, the Coulomb force is overtaken by the strong nuclear force.
5 Measuring Nuclear Masses The measurement of nuclear (isotope) masses is achieved using a mass spectrometer.Positive ions of the element under study are produced in a high voltage discharge tube (not shown) and pass through a slit (S1) in the cathode of the discharge tube. The beam of ions is further collimated by passing through slit S2 which provides an entry to the spectrometer. In the region X, the ions move in crossed electric and magnetic fields.
6 Measuring Nuclear Masses The electric field is produced by the plates P1 and P2 and the magnetic field by a coil arrangement. The region X acts as a velocity selector. If the magnitude of the electric field strength in this region is E and that of the magnetic field strength is B (and the magnitude of the charge on an ion is e) only those ions which have a v velocity given the expression Ee = Bev will pass through the slit S3 and so enter the main body, Y, of the instrument.
7 Measuring Nuclear Masses A uniform magnetic field, B´, exists in region Y and in such a direction as to make the ions describe circular orbits. For a particular ion the radius r of the orbit is given by,Since all the ions have very nearly the same velocity, ions of different masses will describe orbits of different radii, the variations in value depending only on the mass of the ion.
8 Measuring Nuclear Masses A number of lines will therefore be obtained on the photographic plate P, each line corresponding to a different isotopic mass of the element. The position of a line on the plate will enable r to be determined and as B´, e and v are known, m can be determined.IB Outcome- Students should be able to draw a schematic diagram of the Bainbridge mass spectrometer, but the experimental details are not required.- Students should appreciate that nuclear mass values provide evidence for the existence of isotopes.
9 Nuclear Energy LevelsThe α-particles emitted in the radioactive decay of a particular nuclide do not necessarily have the same energy. For example, the energies of the α-particles emitted in the decay of nuclei of the isotope bismuth-212 (a.k.a. thorium-C) have several distinct energies, MeV being the greatest value and MeV being the smallest value. To understand this we introduce the idea of nuclear energy levels.
10 Nuclear Energy LevelsFor example, if a nucleus of bismuth-212 emits an α‑particle with energy MeV, the resultant daughter nucleus will be in its ground state. However, if the emitted α-particle has energy MeV, the daughter will be in an excited energy state and will reach its ground state by emitting gamma photons of total energy MeV.
11 Nuclear Energy LevelsThe existence of nuclear energy levels receives complete experimental verification from the fact that γ-rays from radioactive decay have discrete energies consistent with the energies of the α-particles emitted by the parent nucleus. Not all radioactive transformations give rise to γ-emission and in this case the emitted α-particles all have the same energies.
12 Beta DecayRecall that β- decay results from the decay of a neutron into a proton and that β+ decay results from the decay of a proton in a nucleus into a neutron, i.e.It is found that the energy spectrum of the β-particles is continuous whereas that of any γ-rays involved is discrete. This was one of the reasons that the existence of the neutrino was postulated otherwise there is a problem with the conservation of energy. α-decay clearly indicates the existence of nuclear energy levels so something in β-decay has to account for any energy difference between the maximum β-particle energy and the sum of the γ-ray plus intermediate β-particle energies
13 Beta DecayWe can illustrate how the neutrino accounts for this discrepancy by referring to the figure below showing the energy levels of a fictitious daughter nucleus and possible decay routes of the parent nucleus undergoing β+ decay. The figure shows how the neutrino accounts for the continuous β spectrum without sacrificing the conservation of energy. An equivalent diagram can of course be drawn for β- decay with the neutrino being replaced by an anti-neutrino.
14 The Radioactive Decay Law We have seen in Topic 7 that radioactive decay is a random process. However, we are able to say that the activity of a sample element at a particular instant is proportional to the number of atoms of the element in the sample at that instant. If this number is N we can write thatwhere λ is the constant of proportionality called the decay constant and is defined as ‘the probability of decay of a nucleus per unit time’ and has units of s-1.
17 Finding the Half-Life using the Radioactive Decay Law The radioactive decay law enables us to determine a relation between the half-life of a radioactive element and the decay constant.If a sample of a radioactive element initially contains No atoms, after an interval of one half-life the sample will contain No/2 atoms. If the half-life of the element is T½ then the decay law becomes:
18 Measuring the Half-Life The method used to measure the half-life of an element depends on whether the half-life is relatively long or relatively short. If the activity of a sample stays constant over a few hours it is safe to conclude that it has a relatively long half-life. On the other had if its activity drops rapidly to zero it is clear it has a very short half-life.
19 Elements with Long Half-Lives Essentially the method is to measure the activity of a known mass of a sample of the element. The activity can be measured by a Geiger counter and the decay equation in its differential form is used to find the decay constant. An example will help understand the method.
20 Elements with Long Half-Lives A sample of the isotope uranium-234 has a mass of 2.0 μg. Its activity is measured as 3.0 × 103 Bq. Find its half-life.NOTE: 1 Bq (Becquerel) = 1 decay / second
22 Elements with Short Half-Lives (Hours) For elements that have half-lives of the order of hours, the activity can be measured by measuring the number of decays over a short period of time (minutes) at different time intervals. A graph of activity against time is plotted and the half-life read straight from the graph.Better is to plot the logarithm of activity against time to yield a straight line graph whose gradient is equal to the negative value of the decay constant.
23 Elements with Short Half-Lives (Hours) For elements that have half-lives of the order of hours, the activity can be measured by measuring the number of decays over a short period of time (minutes) at different time intervals. A graph of activity against time is plotted and the half-life read straight from the graph.Better is to plot the logarithm of activity against time to yield a straight line graph whose gradient is equal to the negative value of the decay constant.
24 Elements with Short Half-Lives (Seconds) For elements with half-lives of the order of seconds, the ionisation properties of the radiations can be used. If the sample is placed in a tube across which an electric field is applied, the radiation from the source will ionise the air in the tube and thereby give rise to an ionisation current. With a suitable arrangement, the decay of the ionisation current can be measured.
25 Spherical Chickens in a Vacuum The previous slides are justan outline of the methodsavailable for measuring half-lives and are sufficient for theHL course.Clearly in some cases theactual measurement can be very tricky. For example, many radioactive isotopes decay into isotopes that themselves are radioactive and these in turn decay into other radioactive isotopes. So, although one may start with a sample that contains only one radioactive isotope, some time later the sample could contain several radioactive isotopes.