Download Beta Decay Spectroscopy

Beta Decay Spectroscopy
Revision of 1/16/2012
I.
INTRODUCTION
Beta decay of a nucleus takes place when it emits either an electron (β − ) or a positron (β + ). In general,
radioactive emission occurs when a nucleus makes a transition from one energy state to another of lower
energy, releasing a discrete amount of energy that is carried off by the emitted particle. Because of energy
conservation, particles emitted as a consequence of the same process should thus have the same energy.
This was observed to be true for alpha and gamma emission and for some beta processes. However, for
most beta particle emission, while the maximum energy observed in the beta spectrum corresponded to the
amount that should be released according to the transition occurring, most of the betas had lower energies.
This seemed to violate conservation of energy. In 1930 it was proposed that a second particle, a neutrino,
having zero mass and zero charge, was emitted along with the beta. This particle carries an energy equal to
the difference between the energy of the beta and the energy released by the nucleus. The existence of the
neutrino has been largely substantiated by measuring the recoil of the parent nucleus during a beta decay
and, more directly, by using large scintillation counters to observe neutrino-induced reactions in the vicinity
of high-intensity reactors. Beta decay may be symbolized by n → p + β − + ν̄ or p → n + β + + ν, where n,
p, and ν are neutrons, protons, and neutrinos, respectively.
There is also a type of nuclear process in which monoenergetic electrons can be emitted. Just like atoms,
nuclei can exist in a number of excited states. And, similar to an atomic transition, when a nucleus decays
to its ground state, the difference in energy ∆Enucleus between its excited state and ground state is carried off
by a photon. Because nuclear energies are much higher than atomic energies, this photon has an energy that
is typically in the MeV range, that is, it is a gamma ray.
Occasionally, however, a different emission process occurs. The inner 1s (or “K shell”) orbital electrons
of an atom have a finite overlap with the nucleus. Thus they are weakly coupled to any nuclear processes,
including a decay from a nuclear excited state to the ground state. In such a decay, then, the energy difference
∆Enucleus can sometimes be carried off, not by a gamma, but by one of the K shell electrons of the atom. In
this case, the energy of the ejected electron is ∆Enucleus minus the binding energy of the K electron; typical
K-electron binding energies are on order of 30 keV. Thus this internal conversion process is characterized
by emitted electrons of a single energy. In our experiment, the conversion electrons can thus be used as a
calibration point for the spectrometer.
II.
THEORY
The details of the theory of beta decay have no classical analogue. Using the formalism of quantum
mechanics, it is possible to calculate the probability that an emitted beta particle has a momentum between
p and p + dp. This probability is proportional to dp (if dp is small enough) and is a function of p. Therefore,
we label this probability I(p)dp. Using quantum mechanics, one can predict that
I(p)dp = CF(Z, p)(Emax − Eβ )2 p2 dp
(1)
where
C depends only on the transition and is a constant.
F(Z, p), the “Coulomb factor,” takes into account the effect of the nuclear electrostatic field. Here, Z
is the atomic number of the nucleus. For energetic betas F ∼ 1.
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Emax is the maximum energy of the spectrum considered, i.e., the energy of the beta plus that of the
neutrino.
Eβ is the kinetic energy of the beta particle, which has momentum p. Since the energies involved are
on order of the rest energy m0 c2 , the correct relativistic expression for energy should be used.
III.
THE BETA RAY SPECTROMETER
A.
Theory
The force on an electron of charge −e moving with velocity v in a magnetic field B is given by F =
−ev × B. It follows that if v is perpendicular to a uniform field B, the particle will move in a circular path
in a plane perpendicular to the field.
Because the path is circular, the electron undergoes a centripetal acceleration of magnitude ac = v2 /r.
This acceleration is caused by the magnetic force so that, by Newton’s second law, we can write
F = evB = mac =
mv2
r
(2)
or, solving for the radius r,
r=
p
mv
=
eB eB
(3)
where p = mv is the momentum. If r is known, it is then simple to calculate the momentum p of a particle
by measuring the field B that is necessary to bring it into the detector. Although we have derived this result
using classical mechanics, the same result is found if relativistic mechanics is used. This is important, since
the electrons are generally traveling at speeds comparable to that of the speed of light c.
A beta ray spectrometer uses the fact that r is proportional to p to determine the particle’s momentum.
In the type of spectrometer used in our lab, a beta particle is deflected through an angle of 180◦ between
the beta-ray source and the detector spaced a distance 2R apart. As shown in Fig. 1, only particles with
momentum p = eBR will enter the detector; those with momentum less than or greater than this value will
miss the detector. By varying the magnetic field B, the intensity of particles with different momenta can be
measured.
y
Momentum too
high. Electrons
miss detector.
Momentum too
low. Electrons
miss detector.
B
Source
R
These particles have
the correct momentum
to reach the detector.
x
Detector
xL
xR
FIG. 1: Particle paths through the spectrometer. The magnetic field is perpendicular to the plane of the page, and the
distance between the centers of the source and detector is 2R.
An important performance characteristic of a spectrometer is its resolution, which is a measure of the
spread of the momenta of electrons that can enter the detector from a source of mono-energetic electrons. In
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Fig. 1, suppose those electrons that just enter the left edge of the detector, at a distance xL from the source,
have momentum pL , while those just entering the right edge of the detector, at position xR , have momentum
pR . From Eq. 3, we can write (since x = 2r)
pL
eB
pR
xR = 2
eB
xL = 2
(4)
(5)
from which we have
2
(pR − pL ).
eB
xR − xL =
(6)
Now electrons of momentum p that enter the center of the detector follow a path of radius R so that, by
Eq. 3,
1
R
=
eB
p
(7)
Then, using the fact that xR − xL equals the detector width w and defining ∆p = pR − pL , Eqs. 6 and 7 give
w=
2R
∆p.
p
(8)
We can then define the resolution of the spectrometer as
w
∆p
=
.
p
2R
(9)
Notice the important fact that the resolution defined in this way is a constant, independent of p; for our
spectrometer ∆p/p is approximately 0.1 or 10%. This means that the width ∆p of a peak due to monoenergetic electrons is always 10% of their momentum p.
The resolution computed in Eq. 10 is due to the finite width w of the detector. We also need to include the
finite width of the source and the fact that the beta rays are emitted from the source in every direction, not
just directly forward as Fig. 1 would suggest. Including these effects, the total resolution of the spectrometer
can be shown to be
∆p wsource + wdetector α 2 + β 2
=
+
.
p
2R
2
(10)
Here, wsource and wdetector are the widths of the source and the detector, respectively. The angles α and β ,
shown in Fig. 2, are related to the fact that the beta rays from the source are emitted in all directions. α is the
maximum angle, measured in the plane perpendicular to B, at which electrons can be emitted and still make
it to the source. As shown in the figure, α is determined by a baffle inserted in the electrons’ path. Because
α appears only quadratically in the resolution equation, the baffle slit can be quite large without negatively
affecting the resolution. The angle β is similar to α, but is measured out of the plane. It is set by the height
of the detector.
Note again that even in this more involved expression, the resolution depends only on the geometry of
the spectrometer and is thus independent of the momentum of the electrons.
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α α
Baffle that
determines α.
S
D
FIG. 2: In this figure, all the electrons have the same momentum, but are emitted in different directions. The angle α is
defined as the largest angle from the forward direction for which an electron can reach the detector.
B.
Apparatus and Experimental Procedure
The construction of the β -ray spectrometer is shown in Fig. 3. It consists of a Cs137 source and a GeigerMüller end window counter. A baffle is used to define the path of the emitted β -particles. The whole
apparatus is evacuated to a low pressure, and is mounted between the poles of an electromagnet.
Before taking data, the magnet should be calibrated using a Hall probe inserted into the position of the
baffle slit. You should think carefully about magnetic hysteresis, the fact that the magnetic field depends on
the history of the current. The basic data consists of spectrometer counts vs. current (which can be related
to magnetic field via your calibration). It’s always good to start with a coarse run to make sure things are
working, and to decide where you might need more points for good resolution. As in any experiment, you
should also perform a preliminary analysis early on, to make sure you’re on track.
FIG. 3: Details of the β -ray spectrometer.
The decay scheme of 137 Cs is shown in Fig. 4. There are actually two possible beta decays of 137 Cs.
5.6% of the time, 137 Cs beta decays directly to the ground state of 137 Ba, with a maximum beta energy of
1.18 MeV. This decay is relatively rare, and we will ignore it in our analysis. In the more common beta
decay, which occurs 94.4% of the time, the 137 Cs nucleus decays to an excited state of the 137 Ba nucleus,
which we denote as 137 Ba∗ . With a half-life of 2.55 minutes, the Ba nucleus then decays to its ground state.
In 85% of these decays a gamma ray is emitted. Because photons are uncharged, this gamma is not detected
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by our spectrometer. In the other 15% of decays an internal conversion (IC) electron is given off. 8.2%
of the decays give off a K-shell IC electron with energy 624.2 keV; 1.5% of the decays give off an L-shell
(2s or 2p) IC electron with energy 655.7 keV; and in the remaining 5% of the decays, other IC electrons
are given off with varying energies, leading to very small peaks. The resolution of our spectrometer is not
good enough to separate the K- and L-shell IC electrons, and so they will appear in the spectrum as a single,
somewhat asymmetrical peak with a maximum at 624.2 keV.
137
Cs (t1/2 = 30.07 y)
β − , 0.514 MeV (94.4%)
137
β , 1.176 MeV (5.6%)
−
Ba * (t1/2 = 2.55 min)
γ's and conversion electrons;
see table below
137
Ba
Particle
Gamma γ
Energy
661.6 keV
% of decays
85.1
IC electron
IC electron
IC electrons
624.2 keV
655.7 keV
8.2
1.5
5.2
Notes
Not detected by
spectrometer
K-shell IC
L-shell IC
Other IC electrons, too
weak to detect
FIG. 4: Decay scheme for Cs137 .
The peak due to the IC electrons occurs at some current corresponding to a particular field B. Thus, one
may calculate the effective radius of curvature of the beta particle over its trajectory from Eq. 3 where p will
be the momentum corresponding to an energy of 624.2 keV when using the Cs137 sample. This measured
reff may differ slightly from the nominal 1.5 inches of the spectrometer due to construction tolerances and
slight inhomogeneities in the field.
IV.
ANALYSIS
The beta ray spectrometer measures the number of particles per second n that enter the detector slit. This
count rate is related to, but is different from, the intensity I(p) defined in Eq. 1. We can think of I(p) as
follows. Beta particles are emitted from the source in every direction, with a wide range of momenta p. The
rate of beta particles that are emitted with momenta in the narrow range p to p + dp is then given as I(p)dp.
Note that for small dp the rate is proportional to dp. That is, if we double the range of momenta we are
willing to accept, the count rate will double as well.
To see how the count rate n is related to the intensity I we refer back to Eq. 10. There, we saw that the
ratio dp/p was constant, so that dp = K p. This means that if the momentum of the particles is doubled,
for the same intensity of the source the spectrometer lets in twice as many particles. To correct for this
effect, and to get a number that is proportional to the true intensity, we need to divide the count rate n by the
momentum p, that is,
I(p) ∝ n(p)/p
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We can now turn to the analysis of our data, using the experimentally measured values of n(p) and the
basic equation for beta decay, Eq. 1. We can rearrange this equation as
1/2
I(p)
1
√
= Emax − Eβ .
C p2 F(Z, p)
(11)
We saw earlier that, for energetic betas the factor F(Z, p) is approximately equal to one, and C is also a
constant. Substituting I(p) ∝ n(p)/p into Eq. 11 then gives
K
n(p)
p3
1/2
= Emax − Eβ .
(12)
where K is a constant. Thus if we plot [n(p)/p3 ]1/2 vs. Eβ we should get a straight line with an x-intercept
of Emax . This is called a Kurie (or Fermi) plot.
Experimentally, the plot deviates from a straight line at low values of the energy. This is because of
the finite thickness of the source and mounting. It is evident that the source has to be very thin to avoid
interfering with the motion of an appreciable number of particles.
An expression for the energy as a function of the magnetic field is needed. By means of the previous
calibration of the field, the energy may be expressed as a function of the current. Then, if
evB = mv2 /r
and
Eβ = mc2 − m0 c2 = ET − E0 ,
where Eβ = kinetic energy, E0 = rest mass energy, and ET = the total energy of the particle, we have
r
v 2
E
m0
=
= 1−
ET
m
c
s mv2 m0 c2
ET 2
1q
rB =
=
Eβ (Eβ + 2E0 )
−1 =
ev
ec
E0
ec
Thus
Eβ = −E0 +
q
E02 + (ecBr)2 .
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