HOM, 1/14/05; DVB 014-Jan-9, 01-Dec-17, 013-Oct-16 Beta Spectrum Goal: to investigate the spectrum of β rays emitted by a 137 Cs source. The instrument used is a so-called 180 o magnetic spectrometer that separates β rays of different momenta by using a magnetic field. 1 Introduction 1.1 Beta decay The continuous spectrum of β rays (electrons) from a radioactive source is explained by the fact that with each β, also a neutrino (υ) is emitted, and the total decay energy is shared between them. The decay energy equals the largest possible β energy, T β,max. One nucleus with that exhibits beta decay is 137 Cs whose decay scheme is shown in fig.1. 137 7 / 55Cs 5% T β,max = 1176 kev 30 y 95% 11/ T β,max = 511.6 kev kev.5 ms Eγ = 661.7 kev Fig.1: decay scheme of 137 Cs (from Ref. [LED78]) 137 56Ba The main β decay branch is to a meta-stable excited state of 137 Ba, which in turn decays by γ emission to the ground state. We ignore the much less probable (5%) β decay branch directly to the ground state. Competing with the γ decay of the meta-stable level is a process called Internal Conversion, in which the excitation energy is transferred to an atomic electron, which emerges with a discrete energy. The ratio between conversion electrons and gammas is called the conversion coefficient α. In our case, α = 0.09. In about 80% of the cases, a K-electron is converted. The rest mostly involves an L I electron, and the shells L II, L III, M, etc. play a minor role. The binding energy E B of a K (L I ) electron in 137 Ba is 37.4 kev (6.0 kev). The energy of the emitted conversion electrons is then T conv = E γ E B. We expect two sharp peaks in the observed spectrum, corresponding to internal conversion with K- or L-shell electrons. 1. Charged particles in magnetic fields A particle of charge e, and momentum p, in a uniform magnetic field B travels on a circle with radius ρ. The following relationship holds p( kev / c) 0.998 B( Gauss) ( cm) (1) In our spectrometer, the radius ρ is fixed (at 15.0 cm), and B is varied, to select a given momentum. 1
The mass of the electron is m = 511 kev/c. The kinetic energy is measured in kev. These units are non-s.i., but very practical (setting the speed of light c = 1). To convert energy to momentum, and vice versa, we use the relations T p m m, p T mt. () As an example for eqs. 1 and, an electron of T = 600 kev has a momentum of p = 986.5 kev/c. The field necessary to achieve a bending radius of 15 cm is B = 19 Gauss. 1.3 Fermi Theory The continuous β spectrum was explained by Fermi [FER34] by postulating a new, elusive particle, the neutrino (υ). In this theory, the main features of the β spectrum are understood in terms of the density of states in phase space. This leads to an expression for the probability to observe a β of a given momentum p. N( p) dp const. F( Z, T) p ( T T) dp (3) The constant contains information on the decay probability, but does not depend (much) on the energy. The factor F is called the Fermi-function and takes into account the effect of the Coulomb field. An approximate expression for the Fermi-function for 137 Cs can be found in [VEN85]: m F( Z, T) A B, (4) T with the constants A = 5.0, and B = 3.67. We see that when we plot (N(p)/p /F(Z,T)) 1/ versus T, a straight line should result that intercepts the T-axis at T max. This is called a Kurie plot. Introductions to the theory of β decay and information on internal conversion can be found, for instance, in [SIE55], and [SEG5]. Task: draw approximately the expected β spectrum showing the two continuous decays and the two conversion lines. Make a table with the kinetic energies and the momenta of the K- and L I -conversion lines. Spectrometer In the 1950 s the measurement of β activity in nuclei was an important part of the nuclear research in the IU Physics Department, and a group headed by L.M. Langer made many important contributions to the field. The spectrometer used in our lab is a piece of history of physics at IU. It is a somewhat smaller version of the original device [LAN48] built in 1948. The smaller (ρ = 15 cm), newer device is also described in the literature [BRU50]. The principle of operation of a magnetic β spectrometer is simple. The main trajectory is a semi-circle that starts at the source and ends at the detector. A slit of width max
s, just in front of the detector (see fig.) selects the particles with the desired radius of curvature (or, for a given field, the desired momentum). The field is shaped radially in a Fig.: detector and slit assembly. The slit is set with a knurled knob near the detector. Looking towards the magnet (against the electrons), turning the knob clockwise closes the slit. The scale reads 0 when the slit is closed. Turning the knob counterclockwise, past 5 small divisions, to 15, opens the slit to. mm. Continue counterclockwise to 10 (4.7 mm), 5 (7.3 mm), 0 (9.9 mm), and 15 (1.4 mm, fully open). Do not open the slit beyond 1.4 mm, or the mechanism may disengage. clever way such that source particles emitted in a range of angles are all focused onto the detector slit. The momentum resolution δp is determined by the source thickness, and its lateral size, and by the slit width s. If we assume that the latter is the only contribution to resolution, it is easy to show from eq. 1 that p s (5) p Task: Estimate the slit width needed to separate the K- and L I -conversion line? The detector consists of a piece of plastic scintillator (fig.) glued to a light guide that is coupled to a RCA 8575 photomultiplier. Pulses from the photomultiplier are sent to a discriminator which selects for counting only pulses above an adjustable threshold. This is needed in order to discriminate against the small, but frequent noise pulses. Since the pulse height also depends on the energy of the β particles, it is important to set the threshold carefully just above the noise. The inside of the spectrometer is evacuated and a thermocouple gauge measures the pressure in the spectrometer chamber. Task: how good does the vacuum have to be? Using fig.3, calculate the energy loss ΔE along the semi-circular path of a 100, 300, and 500 kev electron as a function of pressure. Determine the pressure for which the fractional loss, p/p, from the residual gases would equal the momentum resolution with a 1 mm slit (eq.5). What is the relevance of this calculation to your experiment? Fig.3: energy loss of electrons in various materials. To suppress the large variation in de/dx, arising from the number of electrons in the material, (de/dx)/ρ is plotted. Solid lines are for collisions, dashed lines for radiation loss. 3
3 Measurements 3.1 Magnetic field Mount a Hall probe inside the magnet gap. Measure the magnetic field as a function of current from 10 to 100 ma and back. Pay particular attention to the effects of reversing the direction of change in the current (i.e. going up vs. down). In normal operation the red wire is connected to the + pole of the supply. Plot B versus I and determine the amount of hysteresis. Investigate the effects of changing decades on the current supply. Do you want to record your data as a function of current or magnetic field? 3. Counting β particles Set the slit to fully open (one full turn from fully closed), the detector high voltage to -1700V, and the magnet current to 40 ma. Measure the count rate as a function of discriminator threshold, with and without the baffle. The baffle is a brass rod that can be inserted into the beam to block all β particles. Plot the two data sets and decide on a discriminator setting just above the noise. Confirm that the upper level discriminator is high enough so that you are able to see the conversion lines mentioned in section 3.3 (i.e. collect preliminary data from approximately 0 ma through to 90 ma in steps of 5 to 10 ma in order to understand the main features of the spectrum; you may wish to use this rough spectrum to help guide you in selecting current settings and counting times in your determination of the endpoint in section 3.4). The gain of a photomultiplier is affected by magnetic fields: does the fringe field from the spectrometer have an effect on your measurement? Can you think of a way to study this? 3.3 Conversion lines It is not possible to place the Hall probe in the main trajectory. However, we may assume that the measured B is proportional to the field at the main trajectory. In other words, the measured B is proportional to the momentum (see eq.1). The appropriate proportionality constant is found by measuring B for a known momentum. Fortunately, we have the conversion lines for which we know the momentum. Measure with good resolution (s < 3 mm) the spectrum in the neighborhood of the conversion peaks (7 87 ma). Choose the step size to correspond to the momentum resolution calculated with eq.5. Determine the field at the K conversion peak, and the desired calibration constant. From now on, you take data versus β momentum. Can you see the L I conversion peak? Do you get the same calibration constant, using that peak? Does this calibration depend on slit size? (Why or why not?) Is the intensity ratio between the two conversion peaks as expected? 3.4 Continuous spectrum Adjust the slit width to something on the order of 1/3 the value you determined in task 1, and then collect a complete spectrum (from above the L conversion line to well below the peak in the continuous spectrum). Thus, you obtain a measurement of the number of count N(p) as a function of momentum. Use your own judgment to determine the step size (in magnetic field) and counting times needed to accurately determine the 4
endpoint momentum (and therefore energy) of the continuous spectrum, but you should try for statistical accuracy better than about 5%. Do the steps need to be uniform in current, if not, where do you want to make small steps and where do you want to make large steps? Make a table of p, N(p), T, F(Z,T) (eq.4), and (N(p)/p /F(Z,T)) 1/. Construct the Kurie plot, (N(p)/p /F(Z,T)) 1/ versus T. Discuss to what extent the data represent a straight line that intercepts the T-axis at T max. Consider how the choice of slit size would affect your data for determining T max?, and determine whether you want to repeat the measurement with a smaller slit size. Estimate the conversion coefficient α K from the areas under the continuous spectrum and the conversion line. The figure below gives a rough indication of the overall layout of the spectrometer. 4 References [BRU50] J.A. Bruner and F.R. Scott, A High-Resolution Beta-Ray Spectrometer, Rev. Sci. Instr. 1, 545 (1950) [FER34] E. Fermi, Z. f. Physik 88 (1934) 161 [LAN48] L.M. Langer and C.S. Cook, A High-Resolution Nuclear Spectrometer, Rev. Sci. Instr. 19, 57 (1948) [LED78] C.M. Lederer and V.S. Shirley, Table of Isotopes, Wiley, New York 1978 [SEG5] [SIE55] E. Segré, Nuclei and Particles, Benjamin, Reading 1977, p.410ff. (QC776.S4) K. Siegbahn, Beta- and Gamma-Ray Spectroscopy, North-Holland, Amsterdam 1955. (QC771.S57) [VEN85] P. Venkataramaiah et al., J. Phys. G: Nucl. Phys. 11, 359 (1985) 5 More references on beta spectra from IU s past L.M. Langer and H. Clay, Beta Spectra of Forbidden Transitions, Phys. Rev. 76, 641 (1946) L.M. Langer, Note on the preparation of Beta-Ray Sources, Rev. Sci. Instr. 0, 16 (1949) L.M. Langer and R.D. Moffat, A Precise Determination of the Energy of the 137 Cs Gamma Radiation, Phys. Rev. 78, 74 (1950) L.M. Langer and F.R. Scott, The Measurement of the Magnetic Field in a Nuclear Spectrometer, Rev. Sci. Instr. 1, 5 (1950) 5
L.M. Langer and R.D. Moffat, The Twice-Forbidden Transition of 137 Cs and the Law of Beta- Decay, Phys. Rev. 8, 635 (1951) G.A. Graves, L.M. Langer and R.D. Moffat, K/(L+M) Internal Conversion Ratios for M4 Transitions, Phys. Rev. 88, 344 (195) E.A. Plassmann and L.M. Langer, Beta Spectrum of Radium E, Phys. Rev. 96, 1593 (1954) 6