Bryn Mawr College
Department of Physics
Undergraduate Teaching Laboratories
Relativistic Beta Spectroscopy in a Magnetic Field
Introduction
A radioactive source that emits  (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
,
(1)
which means a neutron has decayed into a proton
.
(2)
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2
Or, the nucleus can emit an antielectron and a neutrino according to
,
(3)
in which case a proton has decayed into a neutron;
.
(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 n e or electron neutrino n 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 E0 = mc 2 is a relativistic invariant that measures the "length" of the
energy-momentum four-vector: E02 = E2 - p2c 2 where E is the total energy and p is the
momentum. The total energy is E = K + E0 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 = g m v . In the theory section this evB =
dp/dt is used with the energy-momentum four-vector to show that B2 = 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 B2 / K versus K yields a straight line with slope a and intercept b.
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  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 '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  sources: sodium-22, (Na-22 or
22
11
Na which means Z = 11 and A = 22), thallium-204 (Tl-204 or
or
36
17
Cl), and technetium-99 (Tc-99 or
99
43
Tl), chlorine-36 (Cl-36
204
81
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  radiation is detected. However, the  '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  's will
be absorbed and only the very low-level  radiation escapes. That said, we need to be
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
æ tö
R(t) = R0 expç- ÷,
è tø
where
(5)
t is the time constant and R0 is the strength at t = 0. Show that the half-life T1/ 2 is
T1/ 2 = (ln 2)t
(6)
and that equation 1 can be rewritten
R(t) = R0 2
t
T1 /2
.
(7)
The specific decays for these four samples are:
Tl ®
204
81
204
82
Pb + e + n e ,
(8)
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5
Ru + e + n e ,
(9)
Ar + e + n e ,
(10)
Tc ®
99
44
36
17
Cl ®
36
18
22
11
Na ®
99
43
and
22
10
Ne + e+ + n e .
(11)
Note that the first three are negative  decay (electron emission) and the fourth is
positive  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, '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
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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 0.5 - 1.5") 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 s 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 sec/cm, trig norm & input ch 1) and identify
the  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  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. 2.44 [for 2:44]) then press STORE.
Press NO.
Enter the date (e.g. 18-8-09 [for the 18th of August 2009])
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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 2048. 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  's from chlorine-36. Note that there is rapidly accumulating
garbage in the first 100-200 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?])
Relativistic beta spectroscopy
8
The channel number where the count "reaches zero" corresponds to Kmax. (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 708.6 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 10.62 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.
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.
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.
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 B2/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 = qv 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 =
dp
dt
(14)
Relativistic beta spectroscopy
12
where the relativistic form for momentum is
p = g mv
(15)
and  = (1 - v2/c2)-1/2. The general solution using equations (12)-(15) is very complicated
since d/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 d/dt term is zero
and
dp
dv
= gm
dt
dt
(16)
dv
= a
dt
(17)
The acceleration is
This gives
F = g ma
(18)
and now the magnitudes of equation (13) and equation (18) can be set equal to give
e vB = g ma
(19)
v2
R
(20)
which, along with
a =
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13
and the magnitude of equation (15),
p = g mv
(21)
p = e RB
(22)
yield
Now, the energy momentum relation is
Eo = E2 - p2c 2
(23)
E = K + Eo
(13)
K 2 + 2EoK = p2c 2
(14)
K 2 + 2EoK = e 2R 2B2c 2
(15)
2
which, along with
gives
This all yields
or
æ 1 ö
æ 2E ö
B2
= ç 2 2 2 ÷K + ç 2 2o 2 ÷
èe R c ø
èe R c ø
K
(16)
Knowing the radius of the orbit R (2 inches), the slope gives e and the intercept, gives Eo.
expt15_betaSpec_2013.docx