Bryn Mawr College Department of Physics Undergraduate Teaching Laboratories Relativistic Beta Spectroscopy in a Magnetic Field

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1 Bryn Mawr College Department of Physics Undergraduate Teaching Laboratories Relativistic Beta Spectroscopy in a Magnetic Field Introduction A radioactive source that emits β (beta) particles (high-energy electrons) having a range of energies is used in an evacuated chamber. The chamber is put in a magnetic field causing the charged electron to have a circular orbit due to the q v! " B! force. For each value of the magnetic field B, electrons of the right energy will have the right circular orbit and will find their way to the detector. One measures the detected energy as a function of the magnetic field. This is a good experiment from the technical point of view because several aspects have to be working simultaneously: the vacuum system, the magnetic field, and the nuclear physics detection, display and analysis system. This is a good experiment from the theoretical point of view because you have to put together special relativity, F! = m a!, F! = q v! " B!, the geometry of a circular orbit, and nuclear physics to understand what is going on. It's an undergraduate physics education all rolled into a single experiment. You can measure, independently, the charge q = e and rest mass m of the electron. This is unusual. Most experiments give only ratios of fundamental constants. Until the early 1990's, this was the experiment used to determine an upper limit on the rest mass of the neutrino. Some Brief Theory A nucleus X emits a relativistic electron and an antielectron neutrino and ends up as a different nucleus Y according to A ZX " A Z+1 Y + e + # e, (1) which means a neutron has decayed into a proton n " p + e + # e. (2)

2 Relativistic beta spectroscopy 2 Or, the nucleus can emit an antielectron and a neutrino according to A ZX " A Z#1 Y + e + $ e, (3) in which case a proton has decayed into a neutron; p " n + e + # e. (4) Note that charge, lepton number, and spin are all conserved. This is a Weak Interaction process. Thus the total A = number of protons (Z) + number of neutrons (N) is the same in X and Y but Z = the number of protons increases or decreases by one. The electron e or positron e and the electron antineutrino " e or electron neutrino " e produced in the Weak Interaction decay above together share a certain amount of kinetic energy K max. So, the observed electron (positron) energies span a range of energies from zero to K max. (The most likely fate of the neutrinos is that they will travel in a straight line for the rest of the age of the universe.) At these energies, special relativity must be used, although as is pointed out in the more detailed theory section, some Newtonian relations turn out to be okay because the electrons have circular trajectories and their speeds are not changing. The rest mass E 0 energy-momentum four-vector: E 0 2 = mc 2 is a relativistic invariant that measures the "length" of the = E 2 " p 2 c 2 where E is the total energy and p is the momentum. The total energy is E = K + E 0 where K is the kinetic energy. In a magnetic field B, an electron of charge q = e experiences a force of magnitude F = evb where v is the electron's speed and we have assumed the particle enters the field with a velocity perpendicular to the field. Newton's second law is F! = d p! /dt and this is still relativistically valid so long as p! is taken as p! = " m v!. In the theory section this evb = dp/dt is used with the energy-momentum four-vector to show that B 2 = ak 2 + bkwhere a and b are constants that involve the charge e and mass m of the electron and the radius of the circular orbit. Thus B 2 /K versus K yields a straight line with slope a and intercept b.

3 Relativistic beta spectroscopy 3 Not only does the result reaffirm our faith in the Lorentz force and in special relativity, but we also obtain both the mass and charge of the electron. This is unusual. Usually only ratios of fundamental constants can be measured. Preliminaries and the beta sources Look at the brass chamber and remove the lid. Note the position of the detector and the position across a diameter where the sample will go during the experiment Caution: Keep all sharp objects well away from the sources. You don't want to puncture the thin metal seals. Always wear one glove when handling the β sources. The thallium sample in particular, is chemically potentially dangerous should the thin metal film peel off or get punctured. Handle these samples by holding the disks by a diameter with the glove hand. Keep them away from your face. The β 's come from a 3 mm (which is large) region in the center as stated on the certificates. Using the non-glove hand, pull the glove off from the wrist towards the fingers, turning the glove inside out in the process without touching any other part of the glove with the non-glove hand. Dispose of the gloves. Wash and dry your hands thoroughly after handling the thallium sample You are going to identify the sources so put on a pair of disposable lab gloves. The lead container next to the wall that contains the four β sources: sodium-22, (Na-22 or 22 11Na which means Z = 11 and A = 22), thallium-204 (Tl-204 or Tl), chlorine-36 (Cl-36 or Cl), and technetium-99 (Tc-99 or Tc). Do not touch the sample yet. These samples have very low-level "conventional" radiation doses. Use the Geiger counter to confirm that an inch from each sample, very little γ radiation is detected. However, the β 's can travel several meters in air and several millimeters in water (i.e., humans). They cannot penetrate normal clothing or glasses or any solid for that matter. A piece of paper will eliminate them. So, if the samples are in a plastic bag or a small plastic box, the β 's will be absorbed and only the very low-level γ radiation escapes. That said, we need to be

4 Relativistic beta spectroscopy 4 very careful not to touch the active surface in the center of these disks. In addition, thallium, and its oxides (which develop as the material is exposed to air) is very toxic and should be handled carefully. Read the fact sheets. The chlorine-36 and technetium-99 isotopes purchased in 2000 are from Isotope Products Laboratories. The fact sheets are labeled "NIST (National Institutes of Standards and Technology) Traceable Certificate Beta Standard Source." The sodium-22 and thallium-204 isotopes purchased in 2007 are from Eckert & Ziegler Isotope Products. The fact sheets are labeled " Certificate of Calibration Beta Standard Source." Note the half-life and the purchase date. How strong are the sources now? Do you really need to do a calculation for the chlorine-36 and technium-99 samples bought in 2000? Why do you suppose those samples were not replaced in 2007? The strength is given by $ R(t) = R 0 exp " t ' & ), (5) % # ( where " is the time constant and R 0 is the strength at t = 0. Show that the half-life T 1/2 is T 1/2 = ( ln 2)" (6) and that equation 1 can be rewritten R(t) = R 0 2 t T 1 /2. (7) The specific decays for these four samples are: Tl! Pb + e +! e, (8)

5 Relativistic beta spectroscopy Tc! Ru + e +! e, (9) 36 17Cl! Ar + e +! e, (10) and 22 11Na! Ne + e + +! e. (11) Note that the first three are negative β decay (electron emission) and the fourth is positive β decay (positron emission) TWO IMPORTANT NOTES: (1) Never open the chamber if the detector is powered. (2) Ensure that the bias supply voltage is slowly ramped down to zero before turning off the NIM bin Some electrical connections and turning things on Start with the chlorine-36 source. Place the source up against the detector. You can push it in the same circular indent that houses the detector. Place it with the shiny side facing the detector. Fewer, if any, β 's come out the side with the printing. Put the brass lid back on the chamber. Note the coaxial connector from the detector to the Ortec Model 142A Preamplifier. The preamp has a permanent long grey power cable that plugs into the rear of some module that is, in turn, in the NIM BIN. It doesn t matter which module it is plugged into. The BIAS input of the 142A Preamplifier is connected to the A output of an Ortec Model 428 Detector Bias Supply via a digital voltmeter. Notice that the cable connector going to the bias supply is very different from a normal BNC connector. It is called an SHV ( safe

6 Relativistic beta spectroscopy 6 high voltage connector and is identifiable by the long white dielectric that surrounds the central conductor. Take it out from the bias supply and look at it. Put it back in. Make sure the Bias Supply is turned to zero (both A and B) and set to positive. Turn on the scope, NIM BIN power supply, and the DMM. (Make sure the lid is on the chamber.) Slowly raise the detector supply to 90 volts. Note that this less than one full turn of the HV power supply potentiometer. Watch the DMM as you are turning the knob. Note that the "E" output of the preamplifier is connected to the input of an Ortec 570. On the 570, set the "course gain" to 500. The top potentiometer ("GAIN ") is the fine gain. The label says 0.5 to 1.5 and the potentiometer goes from about four to about 15. Set it to 10, which should be a fine gain of about 1.0. Set the "shaping time" to its minimum value of 0.5 µs and the switch just below it to its middle position, "PZ ADJ." Note that to change these switches, you actually pull out a little before changing the position. Last, but not least, the "pos/neg" switch should be set to positive. Look at the amplifier s output on a scope (2 V/cm, 1 µsec/cm, trig norm & input ch 1) and identify the β signals. The scope triggering can be tricky. The brass lid must be placed on the sample chamber. (Why?) Note on the scope that there is a range of β energies. The "unipolar output" on the Ortec 570 Amplifier is connected both to the scope and to the "ADC IN" connector on the back of the Canberra Series 35 Plus. The "ADC IN" switch on the back of the Canberra next to the input should be in the "EXT" position. Turn the Canberra on. The switch is on the back near the power cord. Again, you must pull the switch out a little before rotating it up. You need to engage in a little dialog with the machine before it starts doing anything: Enter the time (e.g [for 2:44]) then press STORE. Press NO. Enter the date (e.g [for the 18th of August 2009])

7 Relativistic beta spectroscopy 7 Press STORE. Press YES. Turn Memory to 1/1 and press clear data. Turn Memory back to 1/4. Two commands are used to collect data. If the MCA is not collecting data (red light on COLLECT out), pressing COLLECT causes it to start taking data (red light on). If the MCA is collecting data, pressing COLLECT stops it from collecting data. Wait for a moment each time you press COLLECT. When the unit is not collecting data, pressing CLEAR DATA clears the screen (and wipes out the data). Note the current value of "PSET(L)" on the Canberra. This is the length of time the MCA acquires a spectrum. The "L" stands for "live" time and you should read about this on a separate page in the binder ("Adjusting the Preset Run Time on the Canberra MCA's"). If the run time is less than 60 s or so, you may wish to reset this to a longer period. It doesn't matter if it is too long for initial quick-and-dirty spectra. You can stop the MCA from acquiring data at any time. The Canberra Series 35 Plus should be set to 1/4 memory and an ADC gain of Set the "LLD" (lower-level) potentiometer on the MCA to about zero to start with. The MCA and the spectrum Take a spectrum of the β 's from chlorine-36. Note that there is rapidly accumulating garbage in the first channels or so. This slows down the MCA's acquisition and you want to eliminate it. Use the LLD (lower level discriminator) on the MCA to do so. You can Collect, stop Collect, vary the LLD, Clear, etc., until you have zeroes in the appropriate number of channels. Adjust the 570 amplifier so the whole spectrum is on scale on the MCA but with a clear region of high energy where there are zero counts. Go back and forth between the linear and log scales on the MCA but make all decisions about scaling on the log scale. Obtain a good spectrum. This will only take a few minutes. (Note the time duration. Use "L" time, not "T" time on the MCA. [Why?])

8 Relativistic beta spectroscopy 8 The channel number where the count "reaches zero" corresponds to K max. (Why?) Note the uncertainty in this energy value. Go back and forth between a vertical range of "256" and "log." In addition to the upper-energy cutoff, measure the number of counts in the vicinity of the maximum in the spectrum, the total number of counts, and sketch the spectrum in your notebook. In determining the total number of counts, pay special attention to the low-energy cutoff for the integral, which must be above the garbage that you have now blanked out. Determine the total count rate (counts per second). You cannot now change any of the gains in the apparatus. Remember though that you must ramp down the voltage to zero before opening the chamber and exposing the detector to light. Leave the chlorine-36 spectrum in the first quarter of MCA memory and look at the spectrum of technetium-99 in the second quarter, and thallium-204 in the third quarter. Remember to use gloves when handling the thallium sample and remember to dispose of the gloves properly. K max for technetium-99 is 294 kev; for chlorine-36 it is kev, and for thallium-204 it is 763 kev. Do the spectra exhibit maximum electron enerergies that reflect these relative values for K max? Download these spectra to the computer using the instructions provided. The beta spectrum and its energy calibration The calibration of the MCA will be K = Ax + B where x is the channel number and K is the kinetic energy. Once you have decided on settings for the electronics, they must remain the same for the duration of the experiment. K max for each element is provided on a fact sheet. You can plot the three values of K max against their channel numbers to get A and B. The general shape of each of these curves is given in the text An Introduction to Particle Physics (equation on page 311) by Griffiths. That equation, plotted on page 312 models the spectrum for lone neutron decay (equation 2 above) and in this case, the cutoff (the maximum kinetic energy for the electron) is just the difference between the rest mass of the neutron and the proton. In our case, K max involves the complexities of the difference in binding energies between the "before" and "after" nuclear states.

9 Relativistic beta spectroscopy 9 The Gaussmeter: Magnetic field measurement At this point, you have to look to the future and determine where in the chamber you are going to measure the magnetic field. Carefully inspect the Walker Scientific gaussmeter. This is a very expensive and delicate instrument for measuring magnetic fields very accurately. Spend some time looking at the instruction manual and identify the parts. Carefully remove the plastic cover on the gaussmeter probe. Measure the fields in the two standard magnets next to the meter to verify the meter s calibration. Note the dependence of an accurate reading on the orientation of the probe. ################################################################ Before removing the chamber lid, slowly turn the high voltage to zero. ################################################################ Remove the chamber lid and note the hole from the outside that extends deep into the chamber. This is for measuring the magnetic field. Note how far it is from the center of the chamber to the center of the detector and the center of the sample. (Are these distances the same?) This is the radius at which you will eventually measure the magnetic field. The magnetic field is inhomogeneous in the radial direction but ought not to be too inhomogeneous in the angular direction. Insert the probe into the hole between the vacuum lead and the signal lead. (Don t turn the magnetometer on; you re not going to use it here.) The place where the probe measures the field is a small area about 2 mm from the end of the probe. Determine some procedure for allowing you to insert the probe such that you are measuring the field at the appropriate radius of the chamber. Put the gaussmeter aside when you have determined where you will insert it when field measurements are made. The vacuum system Set up the vacuum system and evacuate the chamber. You may need to press down on the lid to make the seal. Pressure is read with a Hastings Model 760 Vacuum Gauge.

10 Relativistic beta spectroscopy 10 Atmospheric pressure is 760 Torr and a vacuum is zero. A Torr is a mm of mercury. When turning the pump off, open the thumb-screw holding the glass tube in the chamber and let air into the system. Do this reasonably soon after turning the pump off in order to bleed the pump (i.e., let air into it). Turn the pump off and bring the chamber up to atmospheric pressure. Place the chlorine-36 sample in the holder 180 degrees from the detector. Make sure it isn t going to become dislodged when you move the sample chamber. Use a bit of tape if necessary but again, don't cover the active region of the disk. Evacuate the chamber. Carefully, watching the electrical leads and the glass parts, gently place it in the magnetic field. (Continue to pump on the chamber during this transfer.) It rests in a wood V with the BNC cable up and towards you and the vacuum line down and towards you. The chamber lid is to the left. Why does it matter which way in the chamber goes? Make sure the chamber is concentric with the pole faces of the magnet. The Magnet Investigate the magnet and it's Agilent power supply. The most efficient way to learn to use this multi-purpose, high-current, super-stable supply is to get an experienced operator to show you. Page 38 in the manual provides directions for constant current operation. Note various exposed leads. Be careful. We will use fairly high dc currents (up to about 3 amperes). It is important that the leads from the magnet to the Agilent supply are such that the electrons emitted from the chlorine sample travel in the appropriate clockwise or anticlockwise direction, depending on your perspective. At last: The Experiment Observe the chlorine-36 spectrum as a function of magnetic field. You need to determine the appropriate range of currents on the magnetic field power supply. You need to determine how long each run should be.

11 Relativistic beta spectroscopy 11 Measure the magnetic field as accurately as possible. You can measure the field each time both at the beginning and the end of the run or you can note the magnet current for each run and calibrate it later with the gaussmeter. It s probably best to do both. Either way, make sure you note the magnetic field setting very carefully. Determine the slope and intercept from B 2 /K versus K. Develop a procedure to determine reasonable uncertainties in the slope and intercept. This is important. From the slope and intercept and the radius of the electron orbit (2.0 inches), you can (a) verify the theory presented below and (b) obtain the rest mass and the charge of the electron independently. Watch your units. There are 10 4 gauss in one Tesla. Background Spectrum Time permitting, investigate the background spectrum. You might want to take a spectrum with the chamber out of the magnetic field or with the sample removed or both. Why might either of these cases give a spectrum? Where do you think the background is coming from? Some Brief Theory The Lorentz force F! is! F = q! v x! B (12) for charge q, velocity! v and magnetic field! B. We assume that! v is perpendicular to! B. The magnitudes are related by F = q vb (13) Newton s Law is! F = d! p dt (14)

12 Relativistic beta spectroscopy 12 where the relativistic form for momentum is! p = " m v! (15) and γ = (1 - v 2 /c 2 ) -1/2. The general solution using equations (12)-(15) is very complicated since dγ/dt as well as d v! /dt is not zero but fortunately, for particles entering the field perpendicularly, v does not change, only the direction of v! changes. That is, the particle goes in a circle. When equation (15) is inserted into equation (14), the dγ/dt term is zero and d! p dt = " m d! v dt (16) The acceleration is d! v dt =! a (17) This gives! F = " m! a (18) and now the magnitudes of equation (13) and equation (18) can be set equal to give e vb = " m a (19) which, along with a = v2 R (20)

13 Relativistic beta spectroscopy 13 and the magnitude of equation (15), p = " mv (21) yield p = e RB (22) Now, the energy momentum relation is which, along with E o 2 = E 2 " p 2 c 2 (23) gives E = K + E o (13) K 2 + 2E o K = p 2 c 2 (14) This all yields K 2 + 2E o K = e 2 R 2 B 2 c 2 (15) or K = " 1 % " 2E $ ' K + o % $ ' (16) # e 2 R 2 c 2 & # e 2 R 2 c 2 & B 2 Knowing the radius of the orbit R (2 inches), the slope gives e and the intercept, gives Eo. expt15_betaspec_2013.docx