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University of Michigan Physics 441- 442
May, 2005 Advanced Physics Laboratory
Alpha Ray Spectroscopy
1. Introduction
In radioactive α decay , a nucleus with Z protons and a total of A nucleons decays to a lower energy
state with (Z-2, A-4) by emission of an “α particle”, a tightly bound nuclear fragment comprising 2
protons and 2 neutrons. Many heavy radioactive nuclei are members of alpha decay chains, where a
succession of these decays takes an unstable nucleus through a number of unstable intermediate
states on the way down to nice stable heavy, like lead. It was in the study of these processes that
Rutherford became the first person to observe the alchemists dream of “transmutation of the
elements”.
In this experiment, you will measure alpha-ray energies with a “solid state” particle detector. Using
a small sample of Po
210
, which emits a single alpha line, you can study the basics of the detector
response, and establish an energy calibration. With a calibrated detector, the remainder of the lab is
two separated studies:
a. The practical matter of the energy loss of charged particles in matter is studied by measuring the
alpha particle energy as a function of the air pressure in the detector, converting this into a
measure of energy loss per effective distance in air. This is compared to the classic theory of dE/dx
and total particle range. This part of this experiment is described in great completeness in
Melissinos, Sec. 5.5.3.
b. The nuclear physics of the alpha-decay sequence is studied by resolving several separate lines in
the spectrum of Th228. This line energies can be related to the mass differences between the
intermediate nuclei in the Th228 sequence, and the areas of the peaks to the relative populations
and thus the half-lives of intermediate species. If time permits, study of the Th228 spectrum vs.
pressure gives further information about dE/dx vs. E.
2. Alpha Rays
Find a good summary of nuclear physics and read it: Many “modern physics” texts include a chapter
level summary discussion of nuclear physics, Eisberg and Resnick (Chap 16) or Tipler are
particularly good. The classic reference is still Evans, which is surprisingly accessible.
In Bohr’s model for atomic line spectra, an electron makes a transition between quantized energy
levels, and the energy lost by the electron appears in a photon of specific frequency. Although we
tend to think of this as a change in state of a single electron, it is useful to think of the process as a
change in the configuration of the whole atom: the nucleus and its collection of electrons find a
lower energy state, and the excess energy is carried away by a photon.
Now consider the nucleus, a multi-particle collection of protons and neutrons held together by the
“strong” interaction, whirling about themselves in bound configurations with stationary wave-
functions and quantized energies. There is no fixed attractive center (the nucleus does not have
nucleus!), but the net effect of all the nucleons is to create an average nuclear potential in which the
bound states of the individual nucleons are arranged in a shell scheme reminiscent of atomic
structure. The configuration of all of the nucleons in this level structure is a collective nuclear state
with a quantized energy. For large nuclei, the spectrum of the multi-particle states can be very
complicated. In addition, with two different kinds of fermions, the Exclusion Principle allows 4
particles in each state, and changes of state can therefore include changes in particle identity.
a. The Curve of Binding Energy
The strong interaction is independent of charge: the strong force between a neutron and a proton is the
same as that between two neutrons or two protons. However, the electric Coulomb repulsion between
the like signed protons is still there, and a net destabilizing force for the nuclear state. This is the
reason that A > 2Z when Z is large: the excess of neutrons contributes enough strong binding energy
to overcome the large internal Coulomb repulsion. This trend is evident in the famous “curve of
binding energy”, which shows the average binding energy per nucleon as a function of the atomic
weight.
Figure 1. Binding energy per nucleon as a function of atomic mass. Note
the change in horizontal scale at A=30. (Evans)
At low A, the addition of nucleons leads to a net increase in binding. At the hot centers of stars, the
thermal energy is sufficient to push nuclei and nucleons together past the net Coulomb repulsion. This
lead to the building up of complex nuclei from hydrogen; the binding energy is released as heat and
light. The details of the nuclear force (particularly a strong spin dependence) lead to certain preferred
configurations, akin to closed shells in atomic physics, such as the tightly bound helium nucleus, A=4,
a.k.a., the α particle. Above iron, the binding trend turns downward. Stars thus burn nuclei up to
iron, whereupon the burning stops, and if the mass is sufficient, gravitational collapse leads to a
supernova explosion, where all the elements heavier than iron are made in the intense nuclear flux.
As the curve shows, for nuclei above iron, it is energetically favorable to move to a smaller value of
A. This happens in nature by emission of an α particle, which reduces the Coulomb repulsion in the
c. Alpha Decay Sequences
In many cases, the nucleus left after an α−decay is itself unstable, leading to an α−decay sequence, as
seen for the Thorium series in Fig. 3. We tend to think of this as a trip though various distinct
elements as arranged in the periodic table; however, an alternative interpretation is that the entire
chain is a set of metastable excited states of the final nucleus. In this view, Fig. 3 shows a set of
excited states of lead, and the energies of the α’s are simply the corresponding line spectra as the
excited states decay to the ground state, as in atomic physics.
Many of the intermediate states have short half-lives, relative to the age of the earth, and would not
exist now, except that ongoing decay sequences starting from long-lived states are continually
replenishing the supply. The relative populations of all the states in the sequence are determined by the
“related-rate” problem involving all of the half-lives, and the steady state solution is known as secular
equilibrium. A sample will be in secular equilibrium after a time long compared to the longest lifetime
In the decay chain.
Figure 3 The 4n series which includes Th
228
(Tipler)
2. Interaction of Charged Particles with Matter
You will detect α−rays by converting their energy to an electrical signal. The first step in this
conversion is ionization.
a. The Rate of Energy Loss or “Stopping Power”
A charged particle passing through bulk matter loses energy by ionizing the atoms in its path. Each
ionization is the result of a peripheral inelastic collision, and the process can be treated statistically
to derive an average energy loss per unit length. As first worked out by Neils Bohr, for the case
where the projectile energy is large compared to the ionization energy, the energy loss rate is
2 4 2
e 2
v 4
N ln
v I
e
e
m dE z e
dx m
!
where z and v are the charge and velocity of the projectile particle, and N e
and Iare the density of
electrons and average ionization potential in the target material. The derivation is straightforward
and elegant, and you should check it out in the references. The essential conclusions are that the rate
of energy loss is proportional to the local density of electrons and the square of the projectile charge,
and inversely proportional to the projectile energy.
The density of electrons is Z times the density of atoms, which is simply related to the mass density,
thus
e 0
Z
N N
A
=! , where N
0
is Avogadro’s number
The Bohr formula can be rewritten as
2 4 2
0 2
4 Z v
N ln
v A I
e
e
dE z e m
dx m
b. The Mass Thickness
Since the ratio Z/A is approximately constant, and the ionization energy appears only in the
logarithm, the strongest dependence of dE/dx on the target material is through the target density.
This makes sense: the projectile slows down fastest in dense materials. However, it is a trivial sort of
dependence, and it is useful to factor it out, giving
2 4 2
0
2
4 N Z v
ln
( ) v A I
e
e
dE dE z e m
d d x m
where ξ = ρ x is the mass thickness , with units of g/cm
2
. Expressed in this way, the energy loss
formula has a certain universal behavior, to first order independent of the material, and depending
most strongly on the velocity of the projectile. The mass thickness is frequently the experimentally
convenient variable. For instance, in cosmic ray physics, given a flux of cosmic rays per cm
2
incident on the top of the atmosphere, the flux in a given cm
2
at the earth depends on the total
amount of air in between, which depends on the total mass of air in the column. The actual path
length is irrelevant, since the density is changing as a function of altitude. What you really want to
know is the mass, e.g., g/cm
2
of air. Sometimes the symbol x is maintained in the formalism with
the implicit understanding that the units are g/cm
2
; be careful about this.
c. The Bethe-Bloch Formula
Bohr’s theory of energy loss was worked over by Bethe and others to account for the complications
of quantum mechanics and relativity; this finally resulted in the famous Bethe-Bloch formula:
2 4 2 2 2
0 max
2 2 2 2
4 N Z 2 c Q 2
ln
( ) c A I ( 1 )
e
e
dE dE z e m
dz d x m
! " "
" "
where Q max
is maximum energy transfer from an electron to the alpha. This equation predicts the
behavior shown in Fig. 4 below. The curve is steep at low energies, but flattens out above βγ ~ 5,
where all particles are “minimum ionizing particles”, with an approximately universal rate of energy
loss of - dE/dx ~ 1-2 MeV per g/cm
2
, in all materials.
Figure 5 Schematic representation of the "Bragg Curve", the mean rate of energy loss per length, as a
function of the distance. Notice that the rate of energy loss is highest near the end of the range. From Knoll.
3. The Detection and Measurement of the Alpha Energy
One way to detect α’s would be to let them pass through a gas between the plates of a capacitor. In
the electric field of the capacitor, the electrons and ions created by the α would drift to the plates,
and be recorded as a current pulse. This is called, appropriately, an ionization detector. This idea
works, but it is limited in practice by the fact that the current or voltage signals are very small, so
that sensitive electrometers, careful control of grounding, etc. are required to measure it. A better
technique is to build a semiconductor analog of the ionization chamber, measure the pulses with a
charge-integrating amplifier, digitize the pulse heights, and send to a histogramming program. We
outline the basic ideas below; further detail is given in the discussion of procedure in Sec. 4.
a. Solid-State Detectors (Required Reading: Leo 10.1-10.5.)
In a semiconductor, the equivalent of the ionization energy is the band-gap energy to promote an
electron from the valence to the conduction band. In Si at room temperature, E g
= 1.1 eV, compared
to ~15 eV to ionize a gas. A charged particle moving through Si therefore creates more ionization
and a larger signal.
When n-type and p-type silicon are put in contact, creating a p-n junction, the flow of the two
different free charges across the boundary creates a depletion zone, an electrically neutral area near
the junction where an internal electric field sweeps out any free charge. By reverse biasing the
junction, the depletion zone can be made large, ~ hundreds of microns. If an energetic charged
particle ranges out in the depletion zone, an amount of ionization proportional to the particle’s initial
energy will be created there, and swept out. By plating metallic ohmic contacts on the outer surfaces
of the crystal, it is possible to both apply the bias and collect the free charge from the depletion
zone, so that the whole assembly is a high gain, solid state version of the capacitive ionization
chamber.
In practice, the solid-state capacitor trick can be accomplished in a simpler way with a surface
barrier detector. Here, a layer of metal, such as gold, is plated onto n-type silicon. The contact
potential creates an electric field at the boundary, and the interface, called a Schottky barrier that has
many of the same properties as a p-n junction and a depletion zone that can be made millimeters
thick
This experiment uses a surface barrier detector. It reaches full depletion at a reverse bias of about 40
V. Note that a small amount of leakage current flows at reverse-bias, even without a source. A
small part of this is from defect “recombination centers” in the detector, but most of it is typically
due to unwanted surface films with high, but non-infinite, resistance, which conduct small amounts
of wayward current around the detector. This can be minimized by keeping the detectors clean. With
a source in place, the reverse current should be dominated by the average value of the ionization
current.
b. Modular Electronics
You can find more detail in Knoll, Chapter 4. Briefly, the issues in our setup are as follows. The
charge signal from the barrier detector is collected with a charge-integrating preamplifier. The charge
on the detector flows into an operational amplifier with a large capacitance in the feedback loop so
that the charge signal is converted to a voltage signal. The large capacitance of the amplifier insures
that the capacitance of the detector, which varies with bias voltage and temperature, does not affect the
charge-to-voltage gain. This signal is sent to a linear amplifier with RC-CR pulse shaping: the signal
is differentiated to remove baseline shifts, and then integrated to remove high frequency noise. Some
of the meanings of this will become clear when you look at the real signals below. The time-constant
of the RC-CR circuit is called the “shaping time”, and sets the ability of the apparatus to resolve pulses
close together in time. The overall amplification allows increased resolution in the electronics for
small energy differences, but the contributions to the resolution from noise are amplified as well.
Finally, the amplifier output is sent to an analog-to-digital converter (ADC) in the small black box,
and the output of that is sent over the serial bus to the computer, which runs software with the plotting,
histogram, and analysis package. It’s a pulse-height-analysis: PHA. In the pre-computer days, the
histogram function was done in hardware with a special purpose instrument like a big oscilloscope,
which could record data in many channels (remarkable at the time), and hence was called a Multi-
Channel-Analyzer. The jargon MCA has stuck, and that’s why our device is called a “Pocket-MCA”.
For testing and calibration, we will also use a pulser, which outputs voltage pulses of controllable
magnitude. The modularity of the electronics allows you to insert these test pulses into any stage of
signal chain.
4. Experimental Procedure
a. Preliminaries
First become familiar with the use of the oscilloscope, pulser, and pulse height analyzer (PHA). Note
that all of the modular electronics expects output signals to be terminated in 50- 100 Ω. If you plug the
boxes into each other, you are fine. If you plug into the oscilloscope, which is high impedence
looking in, you need to add a 50 Ω termination in parallel using a “tee” connector.
Connect the Ortec 480 pulser attenuated output to the oscilloscope with 50 Ω termination, and
examine the pulses while varying the pulser amplitude, polarity, and attenuation. At the same time
become familiar with the oscilloscope controls, including the voltage sensitivity, time scale, and
triggering. Note the pulse height, pulse width, rise time and decay time.
i)
252
Cf Spectrum
Obtain a low activity californium α source (
252
Cf). [Be careful in handling the source, as it is
mounted on a very thin film and is very fragile.] Mount it as close as possible to the solid-state
detector as shown in Fig. 6. Make sure the front side of the source is facing the front side of the
detector and the source is well aligned with the detector. Put the lid back on. Open the valve
between the vacuum pump and chamber, and close the valve which bleeds air into the chamber.
Turn on the vacuum pump and press firmly around the edges of the lid to seat the lid against the
O-ring. The vacuum gage should show the pressure decreasing rapidly. Connect the detector
output to the preamp input (remember to remove your test input from the pulser!) and look at the
output of the preamp with the scope. You should see a rather well defined grouping of positive
pulse heights ~several millivolts from the α’s, plus some very small pulses due to detector and
amplifier noise. Note the size of the pulses, width, etc.
Check that the vacuum is below about 5 mm. Raise the bias voltage slowly while observing the
pulse height and noise. Do not exceed 40 V bias.
Connect the output of the preamp to the input of the main amplifier and run its output into the
scope and MCA inputs as before. Set the gain to give a unipolar positive output signal of about 6
V. Make a plot of the α pulse height vs. bias voltage. As the bias is raised, more and more of the
charge produced in the detector is collected. At bias voltages >30 V or so, essentially all the
charge is collected and the pulse height will saturate. Make a rough plot of pulse height vs. bias;
choose a bias corresponding to about 95% of saturation (or the maximum output from the bias
supply) and use this for all subsequent measurements.
Connect the output of the main amplifier to the PHA input. Adjust the gains so the californium α
line appears at about 60% of full scale. Once you have understood the basics, you should take
some time to understand the further use of the program to analyze the data. Learn how to define a
“Region of Interest” (ROI), and get the centroid, width, and area of a peak. In addition to energies,
record the count rate.
You should print out pulse height spectra for this and subsequent runs and label all prominent
features. Turn these in with your lab report.
ii) Use of Californium and Pulser for Calibration of Energy vs. Channel Number
The Californium peak will give you the PHA channel corresponding to an α with energy 6.2 MeV.
Write down the channel number corresponding to your energy centroid. You now need to establish
how the peak channel varies with energy. As a working hypothesis we assume the dependence is
linear so we can write
0
E k(n - n )
!
where n is the channel number and the n 0
term allows for the possibility of an offset. The best way
to determine n 0
is with a pulser. The pulser can also be used to establish whether the electronics is
in fact linear. It is very difficult to prove that the solid-state detector itself has a linear response
with energy; however, it is expected to be quite linear if the depletion region is sufficiently thick.
Turn off the bias voltage, let the vacuum chamber up to atmospheric pressure, and remove the α
source. Connect the attenuated output of the pulser to the test input of the preamp. Choose an
attenuation setting on the pulser to give a reading just above the channel number of the
252
Cf peak
with a dial setting of 10 on the pulser. Record the peak channel for dial settings of 0.50, 1.00,
2.00, 3.00, …, 10.0. You may need to lower the Threshold to see the lowest point. Make a plot of
the peak channel vs. dial settings. Do a fit to the data and use this to determine n 0
You can then determine k in units of keV per channel from the position of the peak for the
252
Cf
α’s and use the equation above to convert from channel number to energy. Learn how to input the
calibration into the MCA program so you can change the scale from channels to MeV.
c. Measurement of the Polonium- 210 α Energy and Estimation of the Source Activity
Replace the
252
Cf source with a
210
Po button source. [The half life of
210
Po is quite short so try to
find a source that isn’t too “old”.] Place the source as close as possible to the detector and make sure
they are well aligned. Measure the detector–source spacing and the diameter of the detector so you
can calculate the solid angle the detector subtends. Take a spectrum and obtain the number of counts
in the peak and the live time. From the observed rate for the source and the detector geometry,
estimate the activity of the source in units of Curies, where 1 Ci = 3.7× 10
10
decays/sec. Compare this
with the expected activity calculated from the nominal activity and age of the source.
d.
228
Th α - particle Spectrum and the 4n Sequence
Now, we will examine the level spectrum of the excited states of the lead nucleus. You will collect a
multi-peak spectrum from a sample of
228
Th, a relatively long-lived member of the
232
Th radioactive
decay chain (Fig. 3). You will use your calibration from the californium “standard” to measure the
energy of each line. You can relate these to the expected energies in the 4n series and observe for
yourself the transmutation of the elements.
Before starting, note that the overall gain of your signal chain may drift with time. You may wish to
re-establish the Cf standard and check k and n 0
before doing this part.
Replace the Po α source with a
228
Th α source. Handle the source with care; it is extremely
fragile! Don’t touch the foil. Pump down and record the spectrum. Make sure the 8.78 MeV peak is
within the range of the PHA. (You should see 6 peaks.) Define ROI’s for each peak, and get the
centroid, width(FWHM), and number of counts for each peak.
e. Range and Stopping Power of Alpha Particles in Air (if a 3 week experiment)
Varying the pressure in the chamber varies the effective thickness of air between the α source and the
detector. By measuring the final detected α energy vs. pressure, you can measure the energy loss vs.
mass thickness, your own version of the Bragg curve, and also the effective range of α’s in air. This
measurement, including typical data, is described in detail in Melissinos I, Sec. 5.5.3.
[NOTE: In the procedure below, the measurement is done with a polonium source. However, you
would be better off with a thorium-228 source, since these sources have much higher activities. They
also have the advantage that you can do several energies at once.]
(2) It takes about 3 eV to produce an electron-hole pair in the silicon detector. Use this to estimate
how many electrons are produced in the detector by the polonium α’s. From this, estimate the
ideal energy resolution if it were determined only by statistical fluctuations in the number of
electrons produced. Compare this with actual resolution from your spectrum. What are some
additional effects that could worsen the energy resolution?
(3) Make a good-sized diagram showing the decay chain for
228
Th. On it, show all α energies,
lifetimes, and decay probabilities. Make a table of your α energies for
228
Th. Identify each with
α’s in the decay chain worked out above. How do the measured energies compare with those
expected?
(4) Explain why the α peak has a longer tail on the low side than the high side.
(5) We make the hypothesis that the output pulse height of the detector is a linear function of α
energy. Is there evidence for or against this hypothesis in your data? Explain.
(6) What is secular equilibrium? Assuming secular equilibrium for the
228
Th, what would you
predict for the ratios of the counts in the α peaks for
228
Th? Compare this with the observed
ratios. Is the
228
Th source in secular equilibrium? If not, try to explain why.
5. References
R. Eisberg and R. Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles
R.D. Evans, The Atomic Nucleus
G. F. Knoll, Radiation Detection and Measurement
Leo, Techniques for Nuclear and Particle Detection
Melissinos, Experiments in Modern Physics
Pocket MCA Manual
Tipler, Modern Physics
ORTEC manual AN34, Experiments in Nuclear Science
