Lab manual
KF7: β-spectrum and Fermi-Kurie plot
Supervisor: Julius Scherzinger
February 11, 2014
The purpose of this laboratory exercise is to determine the energy released in the
process (the Q-value) for a β-transition by studying the decay of the phosphorus isotope
32 P →32 S + e− + ν,
¯ and to verify Fermis Theory of the weak decay. The measurements
are done by a plastic scintillation detector in combination with a multi channel analyzer,
and the result is compared with the theoretically calculated value.
Learning outcome
2.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Prepare BEFORE the lab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Introduction to laboratory exercise
4.1 β-decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 Q-value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Fermi’s Theory of beta decay and the rewriting of the Fermi Golden Rule . . .
The detector and the experimental equipment
Preparatory questions
Learning outcome
Laborationens mål är, i enlighet med kursplanen för FYSC12: Kärnfysik och Reaktorer, 7.5
hp, att studenten efter genomgången laboration skall:
• känna till grundläggande egenskaper hos beta-sönderfall;
• kunna beskriva huvuddragen i omskrivningen av Fermis Gyllende regel, inklusive de
approximationer som gjorts;
• förstå hur – och varför – ett Fermi-Kurie-histogram görs för att bestämma Q-värdet för
ett beta-sönderfall;
• känna till huvudprinciperna för hur en plastscintillator-detektor fungerar;
• kunna diskutera lämplig uppställning och tillvägagångssätt för experimentet;
• kunna – med hjälp av handledaren – hantera ett radioaktivt preparat och använda en
plastscintillator (inklusive kalibrering) för mätning av sönderfallspartiklars energi;
• värdera experimentella resultat.
In accordance with the syllabus of FYSC12: Nuclear Physics and Reactors, 7.5 credits, the
goal of the laboration exercise is that the student – after completed the lab – shall:
• be familiar with the basic properties of beta-decay;
• be able to describe the main features of the rewriting of the Fermi Golden Rule, including approximations made;
• understand how – and why – a Fermi-Kurie plot is done in the purpose of determine
the Q value for a beta-decay;
• know the main principles of how a plastic scintillator detector works;
• be able to discuss the proper setup and procedures of the experiment;
• with the help of the supervisor handle radioactive samples and use a plastic scintillator
(including calibration) to measure the decay particle energy;
• evaluate experimental results.
• Supervisor: Julius Scherzinger, office: B213
• Email: [email protected], please contact me for any questions regarding
the lab.
• We start the lab 8:30 in hall D with some theory, and then we continue in hall L311.
• Bring: This lab manual, Krane, calculator, solutions (or an attempt to a solution) for the
preparatory questions.
• NO FOOD or DRINKS in L311
Prepare BEFORE the lab
• Read Krane chap. 7.3, 7.6 (3 pages), 9.1 and 9.2.
• Read the lab manual
• Do the preparatory questions in the lab manual
Reports should be sent to [email protected] (checks for plagiarism), as
an attachment (NOTE: IN .PDF FORMAT! No .doc, .docx, .dvi or .m-files!) Send the report
within 10 working days from the lab exercise.
The following should be included in the report. It does not necessarily have to follow the
order listed here. You can write the reports in groups of two, or alone. The report can be
written in Swedish or English, and must be computer written; min. 5 pages, max. 15 pages
(including figures and spectra). Remember to state references if you use any.
• Introduction: purpose/goal of the laboratory exercise.
• Theory: Q-value, β-decay, Fermi’s Theory and the rewriting of Fermi Golden Rule.
Describe the detector principles.
• Experimental setup.
• Procedure: measurements (figures of all measured spectra; discuss the shape of the
spectra), calibration of the detector and the construction of the Fermi-Kurie plot.
• Results: figure of Fermi-Kurie plot and calculation of the Q-value.
• Conclusions: discuss the result, compare result to the theoretical value, discuss sources
of error and approximations made.
• All preparatory questions should be answered in the report, preferably included in the
text, and not as a list.
Introduction to laboratory exercise
The purpose of this laboratory exercise is to determine the energy released in the process
(the Q-value) for a β-transition by studying the weak decay of the isotope 32 P →32 S + e− + ν¯
(QUESTION 1). This is done by measuring the energy of the electrons from the decay. Since
the end point energy of the spectrum cannot be determined due to background and noise,
the spectrum needs to be transformed to a Fermi-Kurie plot by rewriting the Fermi Golden
The measurements are done by a plastic scintillation detector in combination with a Multi
Channel Analyzer (MCA). The detector is calibrated before the phosphorus measurement.
The data is then collected and analyzed to be able to construct the Fermi-Kurie plot, from
which the Q-value easily can be determined. The experimental Q-value is compared to the
theoretically calculated value – if the measured value is in agreement with theory, Fermi’s
Theory of weak decay can hence be verified.
There are three different kinds of β decay:
β− in this process, a neutron is converted to a proton with the emission of an electron and
anti-neutrino, n → p + e− + ν¯ (QUESTION 2). This decay occurs mainly for neutron
rich nuclei; it can also happen for free neutrons.
β+ in this process, a proton is converted to a neutron with the emission of a positron and
¯ This decay occurs mainly for proton rich nuclei, and the
a neutrino, p → n + e+ + ν.
proton must be bound to a nucleus (QUESTION 3).
E.C. (Electron Capture) if the energy of the excited nucleus is not high enough for a β±
decay, an electron near the nucleus can be captured by a proton, leaving the atom in an
excited state, where the hole is filled by an outer electron, giving characteristic X-rays
(the energy of the X-ray is the binding energy of the captured electron).
β decay is a weak decay, and occur via an intermediate process: a quark changes flavor
by emitting a W-boson that decay fast (lifetime of W ∼ 10−24 s) to a lepton pair, see Fig. 1. As
can be seen in the figure, this is not a point like decay. To be able to use Fermi’s Theory and
the Golden Rule, one of the requirement is that the decay can be treated as a perturbation,
i.e. it must be point like. How is this solved (QUESTION 4)?
Figure 1: A Feynman diagram showing β decay on quark level.
The released energy in a nuclear reaction is called the Q-value. It can be expressed as the
difference in the nuclear binding energy before and after the decay:
Q β − = [ m N ( X ) − m N ( X 0 ) − m e − m ν ] · c2
where X is the mother nucleus, X 0 is the daughter, and m N denote the nuclear masses. Calculate the Q-value for the process we are looking at in this laboratory exercise, 32 P →32
S + e− + ν¯ (QUESTION 9).
Fermi’s Theory of beta decay and the rewriting of the Fermi Golden Rule
In this exercise we want to determine the Q-value of phosphorus-32 decaying into sulfur.
The electrons and neutrinos from this spontaneous decay will share the Q-value as kinetic
energy. The neutrinos cannot be detected, but we can measure the kinetic energy of the
electrons and receive an energy spectrum. We can in principle determine the Q-value from
the spectrum; the highest recorded energy. But the count rate near the end point energy is
small due to noise and background, so the value cannot be determined from the spectrum
with high enough accuracy. To calculate a better approximation of the Q-value from the
electron count rate, the Fermi Theory is used to construct a Fermi-Kurie plot, giving us a
linear function, which can be extrapolated. The intersection with the x-axis is the Q-value; in
this way the value can be determined more precisely. In order to understand the Fermi-Kurie
plot, we have to rewrite Fermi’s Golden Rule.
Fermi’s Theory of β decay is based on the treatment of the decay as a perturbation. Quantum mechanical calculations yield the Fermi Golden Rule, which allow us to calculate the
decay probability, λ (number of decays per unit time), for a point like interaction influenced
by a weak perturbation.
| Vf i | 2 ρ ( E f )
where the first term, Vf i , is the expectation value, i.e the matrix element of the transition
operator V. This term gives the relative probability for a specific transition (QUESTION 6).
The second term, ρ, is the density of final states. A transition is more likely to occur
if there is a large number of accessible final states. To calculate this, we need to determine the number of accessible final states for the electron, dne and neutrino, dnν . The direction of the
q electron and neutrino momenta is not important,
q only the absolute values,
p = | p¯ | =
p2x + p2y + p2z for the electron, and p = |q¯| =
q2x + q2y + q2z . The number of
final states containing both an electron and neutrino, dn = dne · dnν , with momenta in the
intervals ( p, p + dp) and (q, q + dq), is described by a spherical shell with radius p (and q),
and thickness dp (dq), where the volume is 4π p2 dpV/¯h. Hence (QUESTION 7):
dn ∝ q2 dqp2 dp
We can express the number of final states as
ρ( E f ) =
= N ( Te )dTe
dE f
where N ( Te ) is the number of electrons as a function of the electron kinetic energy, Te , which
hence can be written as
N ( Te ) ∝ p2 q2
By using (QUESTION 8) and rewriting
Q = Te + Tν = Te + qc
Ee2 = p2 c2 + m2e c4 = ( Te + me c2 )2
we can find expressions for q and p (this will be shown in more detail during the lab) – the
ingredients in N ( Te ) – and finally describe
N ( Te )
N ( Te ) p
Q − Te ∝
Ee f
where F ∝ f /p2 , f = 1.3604A2 + 0.1973A + 0.0439, and A = pc/me c2 is introduced in the
function to correct for the Coulomb interaction between the electron from the decay and the
daughter nucleus. The Q-value can now easily be obtained by plotting Q − Te against Te ,
which is the Fermi-Kurie plot.
The detector and the experimental equipment
A plastic (QUESTION 9) scintillator is used for detecting the radiation. The incident radiation interacts with the plastic molecules and excites them. A molecule can be excited in two
ways: firstly, electrons in the atoms can be excited to higher states (∼ 1eV); and secondly,
the atoms in the molecule can vibrate to higher excitations (∼ 0.1eV). The molecule deexcite
with the emission of visible light, which is not energetic enough to excite further atoms; the
material is transparent to its own radiation, see Fig. 2.
Figure 2: Energy levels in plastic molecules.
The light from the molecular deexcitation strikes a photocathode and release one or more
photoelectrons per photon, called secondary electrons, via Compton scattering. See Fig. 3 for
the experimental setup. The secondary electron are accelerated and multiplied in a photomultiplier tube (PMT) containing dynodes at different potential. The PMT hence produce
an output voltage pulse, where the amplitude of the pulse is proportional to the number of
scintillation events, which in turn is proportional to the energy deposited by the primary
ionization. To keep this proportionality, the transparency mentioned earlier is necessary.
The PMT only delivers a few electrons per event. The signal therefore needs to be am8
plified. Firstly, the signal is converted from a current to a voltage pulse in the preamplifier,
with a typical size of ∼ mV. Secondly, the pulse is amplified in the main amplifier, where the
signal go from ∼ mV to ∼ V. The pulse is then analyzed in he MCA, where it is digitized
and stored in channels which can be displayed on a computer screen.
To be able to translate the detected signal to the electron energy, the detector must be
calibrated with the help of detecting radiation with known, discrete, energies from 207 Bi and
137 Cs.
Figure 3: The experimental setup.
Remember to save all the spectra!
1. Theory session
Table 1: Electron energies
Shell Ee (MeV)
Table 2: Ingredients in Q − Te (complete this table in Excel)
N ( Te ) Te (MeV) Ee (MeV) p (MeV/p)
Q − Te
eq. 8
N ( Te ) p
Ef f
2. Measure 207 Bi in 15 min for calibration of the detector – measure two times: the second
time with an Al-plate between the sample and the detector to stop the electrons
3. Subtract the second spectrum from the total (left with only internal conversion, I.C.,
electrons with known energies)
4. Measure 137 Cs for 15 min
5. Calculate the weighted mean of the electron energies from I.C. by < T >= Σ( E · I )/ΣI,
where T is the electron energy, and I the intensity, see Table 1
6. Calculate the calibration curve according to a straight line: Te = k · ChNr + m
7. Measure 32 P for 1h
8. Measure background for 1h and subtract background from 32 P electron spectrum
9. Construct the Fermi-Kurie plot by calculating the ingredients in Q − Te by – preferable
in Excel – completing table 2
10. Calculate the Q-value and compare to theory
11. Plot all spectra (using the data text file)
Preparatory questions
1. Why is 32 P a suitable sample to use in this laboratory exercise?
2. How does the electron energy spectrum look like for a β− decay? Why?
3. Why must the proton undergoing β+ decay be bound to a nucleus?
4. Can we treat β− as a point like interaction? Why? Hint: calculate the range the Wboson travels before it decay. The W mass is ∼ 80GeV/c2 .
¯ Hint: you can neglect the
5. Calculate the Q-value for the process 32 P →32 S + e− + ν.
neutrino mass and the difference in binding energy of the electrons, why?
6. Write down the general expression for the matrix element Vf i . Optional: Write also the
expression in the specific case of β decay in terms of the wave functions. Hint: assume
that the electron and neutrino are free particles after the decay; in addition we know
that pr << 1 (p is the electron momentum, and r the radius).
7. Why can we neglect all the constants in the derivation?
8. Why can we neglect the recoil energy of the daughter particle?
9. Why is plastic a good material for a scintillator?