Download Experiment 20. Beta Decay

Experiment 20. Beta Decay
Updated RWH February 13, 2017
1
Objectives
By measuring the beta energy spectrum of promethium-147 (147 Pm) you will estimate the
maximum energy of the beta particle in this three-body decay. In addition, you will use the
electron spectrometer to measure the momentum-energy relationship for electrons as a means
of testing the laws of special relativity.
2
Safety issues
In this experiment we are using radioactive sources. All sources are of low intensity and well
shielded. Nevertheless, you should avoid any unnecessary exposure by keeping away from the
sources (r −2 dependence) and limiting the handling time. If not in use, radioactive sources
should stay in the lead container provided. Another safety concern is the high voltage power
supply used to power the detector. Under no circumstances switch on the high voltage power
supply if the detector unit is not under vacuum. If you notice any exposed or damaged electrical
wires, please notify laboratory staff immediately.
20–2
3
S ENIOR P HYSICS L ABORATORY
Introduction
3.1
Beta decay
Radioactivity can be viewed as a particular type of nuclear reaction in which the product particle(s) are not emitted promptly. The average time interval between emissions can be longer
than 1020 years, or as short as 10−15 s. In this experiment the time interval between beta decays
is typically on the order of milliseconds.
At the beginning of the 20th century the phenomenon of radioactive decay was investigated in
great detail and classified into three categories: alpha, beta and gamma. In the course of this
work it was discovered that alpha and gamma were both two-body decays: an alpha particle
(or gamma photon) was emitted and a recoil nucleus produced. By using the conservation
of energy and momentum it can easily be shown that the sharing of the disintegration energy
between the alpha particle (or gamma photon) and the recoil nucleus is unique. Because of the
mass of the nucleus, most of the energy goes into the alpha (or gamma) and a tiny fraction goes
into the recoil nucleus.
The situation with beta decay is quite different because the process is a three-body decay. The
two possible decay paths are as follows:
A
Z XN
A
→Z+1
YN −1 + e− + νe
(β − )
A
Z XN
A
→Z−1
YN +1 + e+ + νe
(β + )
−
A
A process similar to β + decay is electron capture A
Z XN + e →Z−1 YN +1 + νe ; see section 5.
As well as the beta particle (e− or e+ ), there must be the product nucleus of course. However
there is a mysterious third particle: the electron neutrino or electron antineutrino (depending on
whether it is a positron or electron decay). The neutrino is a neutral particle with very small or
zero rest mass; see Section 6.1 It is a fermion like the electron (it has a spin of 1/2) and belongs
to the lepton family of particles shown below.
e− µ− τ −
νe νµ ντ
The three columns denote particles of different flavours. In the second and third columns we
have the muon and its neutrino and the tau particle and its neutrino. All the neutrinos (in the
second row) are uncharged and their counterparts in the top row are charged negatively. For
each of these six particles there is an antiparticle with opposite charge; the antiparticles to the
neutrinos have zero charge.1
The kinematics of a three-body decay means that the sharing of the disintegration energy among
the three particles varies from decay to decay. Most of the disintegration energy goes to the
two light particles and not to the heavy recoil nucleus. Conservation of momentum guarantees
1
The antineutrinos observed so far all have right-handed helicity (i.e., only one of the two possible spin
states has ever been seen), while the neutrinos are left-handed. Because antineutrinos and neutrinos are
neutral particles it has been suggested that they are actually the same particle, i.e., a Majorana particle
(http://en.wikipedia.org/wiki/Majorana fermion).
B ETA D ECAY
20–3
that the energy is (effectively) shared between the electron and neutrino. Most of the time the
energy is shared equally, but in a small fraction of decays the electron will carry away most
of the energy and, likewise, there will be a small fraction of cases in which the neutrino takes
most of the energy. This results in a distribution of electron energies, as shown in Fig. 20-1.
Fig. 20-1 Typical energy distribution of electrons in beta decay.
3.2
Energy, mass, and momentum in classical and relativistic theories
The electrons in this experiment are moving at speeds that are a significant fraction of the speed
of light. As a result, this allows us to investigate some of the aspects of special relativity versus
classical physics. In particular, we will use the velocity of electrons in a perpendicular magnetic
field to find their momentum and energy dependence. We first recall some equations from both
theories:
Ek =
p2
2m
(Newton)
E 2 = (Ek + E0 )2 = (pc)2 + E02
(1)
(Einstein)
(2)
where E is the total energy, Ek is the kinetic energy of the particle, p is its momentum, m is its
mass, E0 = mc2 is its rest-mass energy, and c is the speed of light in vacuo.
The momentum of a charged particle with velocity v perpendicular to a uniform magnetic field
is given in both theories by the equation:
p = qrB
(3)
where q is the charge of the particle, r is the radius of its circular path, and B is the strength
(induction) of the magnetic field2 .
2
See appendix 11 for a more detailed explanation and derivation of the equations listed above, as well as the
natural and SI units used in high energy physics.
20–4
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S ENIOR P HYSICS L ABORATORY
Apparatus
The electrons are detected using a surface-barrier semiconductor detector with a depletion layer
of 500 µm. The amplified pulses from this detector are processed by a UCS30 computer spectrometer to produce an electron energy spectrum. Energy calibration of the UCS30 is carried
out using the internal electron conversion spectrum of 207 Bi; see Section 5.
In the beta particle spectrometer used later in the experiment, the electrons are accelerated by a
perpendicular magnetic field. Electrons within a narrow momentum range are counted by the
UCS30 spectrometer and their energy measured. This allows a direct comparison of electron
energy and momentum, enabling the electron’s mass to be measured and the effects of special
relativity to be observed.
The surface barrier detector is mounted in one corner of the aluminium vacuum box shown in
Fig. 20-2. Opposite the detector is the vacuum sealed radioactive source holder. There are two
radioactive sources marked 207Bi and 147Pm that can be mounted in this holder. If you are
using the permanently-installed 90 Sr source, the source holder should be plugged with the plug
marked EMPTY. The EMPTY plug should also be inserted when measurements are not being
taken (or at the end of the day) so as to prolong the life of the detector which is susceptible to
damage by radiation.
A vacuum has to be established in the box since the electrons lose energy by ionisation in any
medium they travel through. At atmospheric pressure, a few cm of air (say 5 mg/cm2 ) will
degrade the energy of electrons enough to make energy measurements inaccurate. The detector
itself also requires a vacuum. It can be damaged if its 100 V bias voltage is applied before a
reasonable vacuum is established. An interlock cuts off the bias voltage if the pressure in the
aluminium box is greater than 0.6 torr.
Fig. 20-2 Construction of the vacuum chamber in the beta particle experiment.
B ETA D ECAY
4.1
20–5
Operating procedure — PLEASE READ
1. The radioactive sources being used in this experiment are very weak, but to prevent any
unnecessary irradiation, the unused sources should be kept in their slots in the lead block.
2. To change the radioactive source (or insert a source in place of the EMPTY holder) the
power supplies must first be turned off before turning off the vacuum pump and opening
the air inlet valve to allow air to fill the vacuum chamber.
3. Starting up: Close the air inlet, start the vacuum pump and when the pressure is in the
green region of the meter (corresponding to DC volts > 4 V) turn on the bias voltage and
adjust to 100 V.
4. Collecting data: Switch on the UCS30 computer spectrometer and start up the UCS30
application on the computer. Choose the Mode → PHA (Amp In) menu and click on
Settings → High voltage/Amp/ADC to open the dialog box. The following settings should
be used:
• High Voltage: Off
• Amp In Polarity: positive (pulse polarity)
• Coarse Gain: 16
• Fine Gain: 1
• Conversion Gain: 2048 (number of channels)
• LLD: 50 (lower level discriminator)
• ULD: 2048 (upper level discriminator)
• PeakTime: 4µs (pulse width)
5. Saving spectra: The spectra can be saved in various formats, such as spectrum files
(*.spu) or tab-separated ASCII (*.tsv), using the File → Save menu
6. Shutting down: Turn off the power supplies and vacuum pump, and open the air inlet
valve (knurled knob) so that the pump’s oil is not sucked into the vacuum region. Ensure
that the EMPTY plug is inserted into the exchangeable source holder. Turn off the power
at the power point and check that the battery-operated magnetometer is also turned off.
5
Internal electron conversion
The 207 Bi spectrum is used for calibration purposes, and this enables us to examine some of
the physics of the internal electron conversion process, since the four lines in the spectrum are
due to this process.
Fig. 20-3 shows the decay scheme of 207 Bi. It decays mostly by the process of orbital electron
capture. This decreases the atomic number Z one unit whilst leaving the mass number A
unchanged. (So does positron emission (β + emission) but an extra 2mc2 of energy is needed
to create the positron). Orbital electron capture (called EC in the nuclear data tables), is possible
because the orbital electrons’ wave functions are non-zero in the nucleus. It cannot happen in a
20–6
S ENIOR P HYSICS L ABORATORY
Fig. 20-3 Branching ratios for the decay 207 Bi.
nucleus deprived of its orbital electrons and is, of course, impossible unless the process releases
energy. As expected, a neutrino is released in the process:
207
e− +207
83 Bi →82 Pb + νe .
The 207 Pb is left after this process in various excited states which can de-excite by gamma
ray cascades. We are particularly concerned here with the 1064 keV and 570 keV gammas. In
many cases the nucleus will give the transition energy, not to a gamma ray but to an orbital
electron, a process known as internal electron conversion. This again is only possible with a
nucleus ‘clothed’ with at least some orbital electrons. The electrons must obey the conservation
of energy
Ek = hν − EB ,
(4)
where Ek is the kinetic energy measured outside the atom, hν is the quantum energy of the
γ had there been a gamma transition and EB is the binding energy of the electron emitted.
One would expect K-shell electron conversion to be more likely (providing it is energetically
possible) because the wave function for an electron in the s-state (n = 1) has an antinode at
the nucleus. The s-state electrons further out in the atom have similar wave functions except
for the effects of electron screening which tends to spread the wave function out and cut down
its amplitude at the nucleus. For heavy atoms, only two out of the eight L-shell electrons are in
s-states so we expect L-shell conversion to be no more probable than K-shell conversion.
The surface barrier detector used in this experiment has insufficient resolution to distinguish Lshell lines from M-shell, etc lines further out in the atom. This is why we see for each transition,
only two peaks (the composite nature of a second peak can be observed if enough pulses are
collected), the lower energy one due to K-shell electrons and the other due to (L+M+N+...)shell electrons. The binding energy EB of K-shell electrons of the product atom Pb is 88.0 keV
and the weighted mean of the L-shell electron binding energy is 14.3 keV.
B ETA D ECAY
6
20–7
The Kurie plot
Fig. 20-4 Energy level diagram for 147 Pm.
The β spectrum being analysed in this experiment is that of 147 Pm. Unlike the line spectrum
of 207 Bi, the emitted spectrum of 147 Pm is continuous, extending from a low-energy cutoff (set
by the level discriminator in the computer spectrometer) up to some maximum energy Emax 3 .
For 147 Pm, Emax /c2 is nearly equal to the mass difference between 147 Pm and the end product
147 Sm, the difference being the small recoil energy of 147 Sm. We are not able to measure
directly this recoil energy or the neutrino’s energy, only the beta particle’s energy. If it is close
to Emax then we know that the electron has been given most of the available energy and the
neutrino very little. It is understandable that this occurs only rarely, the probability decreasing
as the beta energy approaches Emax (224 keV).
To determine Emax we need to extrapolate the beta spectrum to the background level. Rather
than simply plotting counts C(N ) against E, the conventional practice is to plot a special
function K(p) (called the Kurie4 function) against p. The theory is given in many text books
and in Kurie’s papers (see the reference list).
According to the theory, the Kurie function is defined as;
K(p) =
P (p)
p2 F (Z, p)
(5)
where P (p) is the probability of getting an electron momentum of p in small range ∆p and
F (Z, p) is the Fermi function. The latter allows for the Coulomb attraction of the nucleus for
the beta particle as it leaves the nucleus and depends, as expected, on the atomic number Z of
the product nucleus.
3
Beta decay is frequently followed by gamma decay, i.e., the product nucleus is created in an excited state. This
does occur in 147 Pm (see Fig. 20-4) but with extremely low probability.
4
After Franz N. D. Kurie who first use it to analyse beta decay spectra. This function can be derived from
Fermi’s model of beta decay. In the case of so-called allowed beta transitions it should be a straight line.
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S ENIOR P HYSICS L ABORATORY
In this experiment we are interested in electron energy cutoff rather than momentum. Accordingly we use the energy form of the Kurie function (see [3], page 234)
K(E) =
s
P (E)
,
pcEF (Z, p)
(6)
where E is the total energy of the electron (kinetic energy plus rest energy). P (E) can be
approximated by C(N ), the count in channel N . In 147 Pm the Fermi function F (Z, p) varies
slowly in the region of Emax so we set it to a constant.
The electron momentum p and total energy E are calculated from Equation (2) to give
v
u
u
K(Ek ) ∝ t
C(N )
q
.
(Ek + E0 ) Ek2 + 2Ek E0
(7)
The rest energy of the electron is E0 = 511 keV and Ek is calculated from the calibration
equation. The constant Fermi function F (Z, p) has been omitted for simplicity.
6.1
Neutrino mass
The possibility of a non-zero neutrino mass has been intriguing physicists for many years now.
Currently there is an international collaboration planning a new experiment to investigate the
end point of the tritium β-decay spectrum called KATRIN. The KATRIN experiment is designed to measure the rest mass of the electron neutrino directly with a sensitivity of 0.2 eV/c2 .
7
Prework
1. An electron is moving perpendicular to a uniform magnetic field (B = 0.1 T) in a circular
path of radius 0.10 m. The rest-mass energy of the electron is 0.511 MeV. Using special
relativity, calculate the
• momentum,
• total energy,
• kinetic energy, and
• speed of the electron.
Express your results in both SI and natural (using e and c) units; see Section 11.2.
2. Why is a semiconductor detector that is used to detect α particles of kinetic energy 5 MeV
not suitable for detecting 1 MeV β particles?
3. Calculate the energies of the four 207 Bi internal conversion electrons used in the calibration process; see Section 5.
C1 ⊲
B ETA D ECAY
8
20–9
Calibration
1. Follow the startup procedure given in section 4.1 with the 207 Bi calibration source inserted into the vacuum chamber, and setup the UCS30 computer spectrometer to acquire
data. Note that the software can display counts on a linear or log scale using the ‘log’
button at the top of the screen, and the scaling can be adjusted using the button on the
right of the screen.
2. Acquire data for ≈30 minutes. You should see narrow peaks in the spectrum near
channels 730, 850, 1500 and 1610 corresponding respectively to electrons of energies
481.65 keV, 555.35 keV, 975.64 keV and 1049.34 keV (see the spectrum on page 1; the
origin of the narrow peaks is described in Section 5). These peaks appear on the highenergy slope of the main broad peak, which has a maximum around channel 260. This
background ‘continuum’ arises because a high energy electron can pass through the detector and deposit just a portion of its energy in the detector’s depletion layer.5
The count C(N ) in a given channel N of the computer
spectrometer is subject to statisp
tical fluctuations with a standard deviation σ = C(N ). The fractional error in C(N )
therefore decreases as counting progresses and the spectrum becomes smoother.
3. Save the spectrum in *.spu format and make a hard copy using File → Print.
4. Set the region of interest (ROI) for each of the four above-mentioned peaks by clicking
the right mouse button and selecting ROIs → Set ROI from the popup menu. Select a
symmetric region about the peak for determining the position of each line, using click and
drag with the left mouse button; do NOT include the wings of the line or the neighbouring
background continuum.
The program calculates a number of parameters for each region, but the only one that’s
relevant here is the Centroid. You can delete and re-select the ROI to obtain an estimate
of the uncertainty in position.
5. The narrow peaks in the spectrum are due to internal conversion electrons (Section 5)
and we use their known energies to calibrate the spectrometer. A linear plot of energy
against channel number will therefore have the form
Ek = aN + b,
(8)
where Ek is the kinetic energy of the electron corresponding to channel number N , and
a and b are constants. Use QtiPlot to determine a and b and their errors. We assume that
Equation (8) applies throughout the energy range covered in this experiment.
Question 1: Why are the emission lines near 1 MeV so weak in comparison with those around
500 keV? In light of footnote (5) and Section 4, why do we even detect the lines near 1 MeV?
C2 ⊲
5
For electrons of energy 1 MeV, the detector should have a depletion layer ≈2 mm thick to allow almost all
electrons to deposit all their energy inside the detector.
20–10
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S ENIOR P HYSICS L ABORATORY
Beta particle energy spectrum
1. Following the procedure given in section 4.1, insert the 147 Pm source and acquire its
spectrum for ∼30 min using the same gain settings as before. Because we are only
interested in the high-energy tail of the energy distribution (Fig. 20-1) the lower level
discriminator (LLD) has been set to remove the lower energies.
2. Export the spectrum as an ASCII file using File → Save as Tab separated file (*.tsv).
3. Open QtiPlot and use File → Import ASCII to load your data. Set Separator as \t\t (two
TAB characters), set Ignore first to 18 lines to exclude the header information and set
Endline character as LF (Unix).
4. Convert channel numbers to keV using the calibration equation, calculate the Kurie function (Equation 7) and plot K(Ek ) against Ek over the range, say, 180–240 keV.
You will notice an abrupt change in slope at Ek ∼ 220 keV; this is due to the rare
occurrence of two β particles being detected almost simultaneously6 .
5. We ignore the spurious counts due to double-counting of betas and simply fit a straight
line to the Kurie plot over the range ∼180–210 keV and hence obtain an estimate for
Emax from the x-intercept. The uncertainty in Emax can be determined from the errors
in the slope and intercept of the fitted straight line.
Compare your result with the accepted value Emax = 224 keV.
C3 ⊲
10 Relativistic momentum-energy relation for electrons
10.1 Preamble
In this experiment a magnetic spectrometer is used to select electrons of known momentum
from a 90 Sr beta source (Emax = 2.28 MeV) and measure their energy. The beta source is
mounted in the aluminium box shown in Fig. 20-2. The decay scheme of 90 Sr is shown in
Fig. 20-5 and the relationship being tested is a rearrangement of Equation (2), namely
E0 = mc2 =
(pc)2
Ek
−
.
2Ek
2
(9)
10.2 Procedure
1. Following the setup procedure given in section 4.1, insert the EMPTY plug in the radioactive source holder and establish a vacuum. Set the bias voltage as before and start
up the UCS30 spectrometer.
6
The newly installed (2016) 147 Pm sample is sufficiently active that two betas can occasionally be recorded
within the 4µs acceptance window (PeakTime in Section 4.1) of the detector. In this case the UCS30 spectrometer
records the sum of the two energies, in principle up to 448 keV.
B ETA D ECAY
20–11
Fig. 20-5 Energy level diagram for 90 Sr.
2. Turn on the magnetometer and check that the magnetic field probe is firmly located in
the hole provided for it near the preamplifier. Set the magnet current to give B ≈ 200 G7
on the 19,999 range; note that the magnetometer is correctly calibrated ONLY on this
→
−
range. The direction of B is vertical and the force on the electrons is given by
→
−
→
−
−
F = q→
v × B.
(10)
The motion of the electrons is therefore circular, with radius r = 0.042 m determined by
the geometry of the vacuum chamber.
3. Acquire the spectrum until the peak is well formed —around 150–200 counts in the
peak is sufficient— and save the spectrum. As for the calibration, use Set ROI to select
the region of interest (ROI) to include only points within the narrow peak; record the
Centroid channel number.
4. Repeat the above measurement for magnetic fields increasing in 100 G intervals up to
a maximum B = 1100 G. At the higher magnetic fields you will notice an additional
broad peak appearing at channel ≈260 and a plateau to higher energies. As noted earlier
(section 8), this broad peak (and plateau) arise because most high energy electrons will
go straight through the detector’s depletion layer and deposit only a small fraction of
their energy.
5. Switch off the magnetometer when not in use.
7
The magnetometer reads in gauss (G), an old unit of magnetic field strength, where 1 G = 10−4 T. With no
applied current there is a small meter reading due to residual magnetism in the iron core of the electromagnet.
20–12
S ENIOR P HYSICS L ABORATORY
6. Enter the values of B and Centroid channel number into QtiPlot. Use your calibration
equation to convert Centroid channels to Ek in keV, and Equation (3) to determine pc,
also in keV. Assume a 1% error in B; with 150–200 counts in the spectral peak, the
uncertainty in the Centroid positions should be ≤ 1 channel. The uncertainty in the
radius r is not known; if significant, it will show up as a systematic error in the fit.
7. Plot (pc)2 /(2Ek ) versus Ek and make a weighted linear fit to the data, recording the
slope and intercept. Comment on the agreement (or otherwise) with the expected values
from equation (9).
Question 2: Calculate the speed of the electrons you detected at B = 1000 G in terms of the
speed of light c.
C4 ⊲
11 Appendix
11.1 Classical (Newton’s) theory
Definition of energy and momentum:
mv 2
2
→
−
−
p = m→
v,
E = Ek =
(11)
(12)
−
where E is total energy, Ek is kinetic energy, and →
p denotes momentum of a body with mass
→
−
m moving with a speed v .
→
−
If this body has a charge q and is moving perpendicular to magnetic field B , then its trajectory
−
−
will be circular with radius →
r , where →
r vector is directed towards the centre of the circle.
By comparing acceleration from circular movement and electromagnetic force we can write
equation:
−
−
−
−
→
−
−
→ d→
r
d(m→
v)
d(→
v)
v2 →
p
−
=
=m
=m
.
(13)
F = q→
v ×B =
dt
dt
dt
r r
By taking absolute values of all vectors (we know the directions) in Equation (13), we have
qvB = m
v2
⇒ mv = qrB ⇒ p = qrB.
r
(14)
11.2 Relativistic (Einstein’s) theory
According to special relativity theory, energy and momentum can be expressed as follows:
E = Ek + mc2 = γmc2
→
−
−
p = γm→
v
(15)
(16)
B ETA D ECAY
20–13
p
where γ = 1/ 1 − (v/c)2 and c denotes the speed of light in vacuum.8
Energy and momentum are closely related9 because both contain the factor γm. Therefore
E = γmc2 =
pc2
,
v
(17)
and the particle speed in terms of energy and momentum is
v
pc
= .
c
E
(18)
Thus, the energy may be written
E = γmc2 = q
mc2
1−
(pc)2
E2
.
(19)
Solving for E, we arrive at the expression relating the total energy (E), momentum (p), and
mass (m) of particle:
E 2 = (pc)2 + (mc2 )2 .
(20)
Very often mc2 is called rest mass energy and denoted by E0 . Using this convention we write
pc =
q
E 2 − E02 .
(21)
In nuclear and elementary particle physics we often use electron volts10 (symbol eV) as a unit
of energy. Equation (20) invites a convenient unit for momentum of a particle, electron volt
divided by the speed of light (symbol eV/c). Also for mass of particle we can use electron volt
divided by squared speed of light (symbol eV/c2 ). By defining those units we save the trouble
of dividing by c or c2 when calculating momentum or mass from the energy equation (20). Also
the specific use of elementary charge e expressed in coulombs can be avoided in most cases.11
Now using similar reasoning as in classical physics we will derive relativistic equation for
movement of charged particle in uniform, perpendicular magnetic field:12
−
−
−
−
→
−
→
−
r
d(γm→
v)
d(→
v)
v2 →
d→
p
−
=
= γm
= γm
.
F = q→
v ×B =
dt
dt
dt
r r
(22)
By taking absolute values of all vectors in equation (22), we have
8
By definition of the metre, the speed of light is exactly c = 2.99792458 × 108 m/s.
It is usually represented as the invariant magnitude of the energy-momentum 4-vector in 4D Minkowski space.
10
The electron volt (symbol eV) is the kinetic energy acquired by an electron in passing through a potential
difference of 1 V in vacuum. 1 eV=1.602 176 487(40)×10−19 J.
11
These units do not formally belong to international system of units (SI), but are accepted for use especially in
high energy or elementary particle physics.
→
−
−
12
Here we are using the fact that the electromagnetic force F is perpendicular to the velocity →
v of the particle
and the factor γ is constant in time.
9
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S ENIOR P HYSICS L ABORATORY
qvB = γm
v2
⇒ γmv = qrB ⇒ p = qrB.
r
(23)
This equation is identical to the classical one!
References
[1] G.F. Knoll, Radiation Detection and Measurement, 2000, 3rd ed, New York:Wiley
(539.77 96A)
[2] William E. Burcham, Nuclear Physics: an Introduction, 1963, Longmans (539.7 29)
[3] Robert Howard, Nuclear Physics, 1963, Wadsworth Pub Co (539.7 113)
[4] C.M. Lederer et al., Table of Isotopes, 1978, 7th ed, New York:Wiley (DG 49055)
[5] E.J. Konopinski & G.E. Uhlenbeck, 1935, Phys Rev 48, 7
[6] F.N.D. Kurie, J.R. Richardson & H.C. Paxton, 1936, Phys Rev 49, 368
[7] F.N.D. Kurie, 1948, Phys Rev 73, 1207