Transcript Chapter9_new

Chapter 9
Beta Decay and Weak Interactions
● The Scope of Weak Interactions
◎ Some Review of β-decay
●
Fermi’s Golden Rule
◎
Fermi Theory of β-Decay
● Nonconservation of Parity
◎
The Mass of the Neutrino
§ 9-1 The Scope of Weak Interactions
Beta decay is actually one type of weak interactions which are relatively less in
interaction strength when compared to the electromagnetic interaction.
Eventually electromagnetic and weak interactions are unified together as the
electro-weak interaction.
1. Beta decay is the process by which complex nuclei return towards the line of
stability by emitting electrons or positrons, or by electron capture:
( Z , A)  ( Z  1, A)  e    e
( Z , A)  ( Z  1, A)  e   e
(1)
( Z , A)  e   ( Z  1, A)  e (electron capture)
2. Weak interaction decays of hadrons. In addition to some β-decays such as
K+ → π 0 + e+ + νe, many decays occur which are also due to weak interactions:
K      0
K           ,etc.
(2)
3. A few decays do not involve hadrons: for example
   e   e   
(3)
4. Some gauge spin-1 boson, ( W+,W-, Z0), are involved in the weak interactions.
According to types of particles involved in a decay process we may classify them into
four kinds of decay:
1. Leptonic hadron decays: The weak interaction decays of
hadrons that involve only leptons in the final state.
      
(4)
2. Semi-leptonic decays: The weak interaction decays that have
both leptons and hadrons.
  n      
(5)
3. Non-leptonic decays: The weak interaction decays that involve only hadrons.
K      0
(6)
4. Pure leptonic decays: Decays or interactions involving leptons alone.
   e   e   
(7)
Beta decay is clearly a semi-leptonic decay since in this type of decay
both leptons and hadrons are all involved.
n  p  e  νe
(8)
Form hadrons
The three generations of charged and neutral leptons
All the leptons believed to exist are listed in the table shown above.
They obey Fermi-Dirac statistics and each lepton has an anti-lepton partner.
Hadrons
§ 9-2 More on β-decay
The apparent process in β-decay is the conversion of
nucleus (Z,A) into nucleus (Z+1,A) and an electron (e-):
( Z , A)  (Z  1, A)  eIt is actually the conversion of a bound neutron
(n) into a bound proton (p).
n  p  e
β- - decay
For some proton-rich nuclei they can frequently undergo
β-decay in which a positron (e+) is emitted.
p  n  e
β+ - decay
A proton-rich nucleus can capture an atomic electron and thereby
change a proton into a neutron.
e  (Z , A)  (Z  1, A)
or e  p  n electron capture
The electron is captured usually from the K-shell but can be from the L, M,
N, or even higher shell. In this process the electron in annihilated.
(38%)
(19%)
(43%)
This experimental energy spectrum is
from G. J. Neary, Proc. Phys. Soc.
(London), A175, 71 (1940).
Electrons emitted
through a process called
the internal conversion
Internal conversion is a process by which a
nuclear excited state decays by the direct ejection
of one of its atomic electrons; it occurs normally
in competition with photon emission.
Early mysteries concerning the β- decay:
1. The kinetic energy spectrum of the
emitted electrons is continuous.
2. Consider the following decay. It seems
to violate the law of angular
momentum conservation.
14
6
C147 N  e0 1 1/2
?
There is no way that
angular momentum can be
conserved in the decay.
In 1930 Pauli made a hypothesis
that provided a solution to these
difficulties and that has satisfied all
experimental tests.
He proposed that an electrically
neutral particle of spin 1/2 is created
and emitted at the same time as the
electron (or positron) in β-decay.
Wolfgang Ernst Pauli
Austrian–Swiss physicist
(1900–1958)
The particle is called a neutrino (symbol,
ν) and it can take a share of the energy
because in β-decay there is now a threebody configuration of the final state.
It turns out that nature has three kinds of neutrinos
each with its own antiparticle and mass.
( e , e )
( μ , μ )
( τ , τ )
We need to label the neutrinos in nuclear
β-decay in the following fashion:
β- - decay
( Z , A)  ( Z  1, A)  e   e
β+ - decay
( Z , A)  ( Z  1, A)  e    e
Electron capture
e  ( Z , A)  ( Z  1, A)   e
In order to consider the energetics of β-decay, we need to know the mass of the
neutrino emitted in β-decay. It is known to be less than 18 eV and since this is
small compared to the total energy released in most β-decays we shall assume
that the mass is zero. In fact the neutrino mass would be of considerable
significance in cosmology and in theories of elementary particles.
Free nucleon decays:
(1). A free neutron undergoes β-decay with a mean life time τ = 898 seconds.

n  p  e  νe
Qβ = [Mn – (Mp+me)]c2
= [939.573 – (938.791 + 0.511)] MeV
= 0.782 MeV > 0
This process is certainly energetically possible since Qβ = 0.782 MeV
which is larger than zero.
(2). Consider the case for a free proton:
Qβ = [Mp – (Mn+me)]c2
= [938.791 – (939.573 + 0.511)] MeV
p  n  e  νe
= – 1.293 MeV < 0
This process is energetically impossible since Qβ = – 1.293 MeV
which is smaller than zero.
This is a fortunate situation as the stability of protons (on the time scale of >> 1014 years)
is essential to the existence of the universe and of ourselves.
(3). Electron capture in a hydrogen atom
e-  p  n  ν e
Qβ = [(Mp+me) – Mn]c2
= [(938.791+0.511) – 939.573] MeV
= – 0.271 MeV < 0
This process is unlikely to happen as
its Qβ value is seen smaller than zero.
Furthermore the safety of the proton
against electron capture follows
immediately from the existence of
free neutron decay.
Energy conditions in β-decay and electron capture in
terms of nuclear masses, M(Z, A):
  : ( Z, A)  ( Z  1,A)  e   ν e
Qβ = [M(Z,A) – M(Z + 1, A) – me]c2
>0
  : ( Z, A)  ( Z  1,A)  e   ν e
Qβ = [M(Z,A) – M(Z – 1, A) – me]c2
>0
EC : ( Z, A)  e   ( Z  1,A)  ν e
QEC = [M(Z,A) – M(Z – 1, A )+ me]c2
>0
If a condition is satisfied, then the appropriate decay is possible and the excess energy available is
shared as kinetic energy among the products in a manner which conserves linear momentum.
Note that electron capture
can sometimes occur when
β+-decay is impossible.
§ 9-3 Fermi’s Golden Rule
In order to describe the Fermi theory which successfully explains continuous β-decay
spectra (both for β- and β+ decays) we need to know the Fermi’s Golden rule.
A true stationary state lives forever. The expectation values of physical observables,
computed from the wave function of a stationary state, do not change with time.
In particular, the expectation value of the energy is constant in time. The energy of
the state is precisely determined, and the uncertainty in the energy, ΔE vanishes.
E 
E2  E
2
(9)
The Heisenberg relation, ΔEΔt ≥ h/4π, then implies that Δt = ∞.
Thus a state with an exact energy lives forever; its lifetime against decay is infinite.
Suppose that a quantum mechanical system is subject to a weak perturbing potential
Vˊ, in addition to the original potential V (Vˊ<< V). In the absence of Vˊ, we can solve
the Schrödinger equation for the potential V and find a set of eigenstates Ψn and
corresponding eigenvalues En.
If the weak additional potential Vˊ is included then the states are approximately, but not
exactly, the previous eigenstates Ψn of V.
This weak additional potential permits the system to make transitions between the
“approximate” eigenstates Ψn .
Ψi
Ei
E  Ei  E f
Ψf
Ef
The energy difference ΔE must
appear as radiation emitted in a
decay. A photon is emitted to
carry the energy ΔE.
Ei
Ψi
E  Ei  E f
A nonstationary state has a nonzero
energy uncertainty ΔE. This quantity
is often called the “width” of the state
and is usually represented by Γ.
Ef
Ψf
The lifetime τ of this state can be estimated from the uncertainty principle by
associating τ with the time Δt during which we are permitted to carry out a
measurement of the energy of the state.
Thus
 /
(10)
The decay probability or transition probability λis inversely related to the mean
lifetime τ.

1

(11)
If we have the following knowledge then it would be possible to calculate λ or
τdirectly from nuclear wave functions.
1. The initial and final wave functions Ψi and Ψf, which we regard as
approximate stationary states of the potential V.
2. The interaction V’ that causes the transition between states.
The calculation of λ is too detailed to be covered in this lecture but I will merely
state the result, which is known as the Fermi’s Golden Rule:
2 ' 2

V fi  ( E f )

(12)
'
*
The quantity Vfi’ has the form: V fi   f V ' i dv
and is sometimes called the matrix element of the transition operator V’.
The quantity ρ(Ef) is known as the density of final states. If there is a larger
density of states near Ef, there is a larger transition probability.
§ 9-4 Fermi Theory of β-Decay
Enrico Fermi (September 29, 1901 –
November 28, 1954) was an Italian
physicist most noted for his work on the
development of the first nuclear reactor,
and for his contributions to the
development of quantum theory, nuclear
and particle physics, and statistical
mechanics. Fermi was awarded the Nobel
Prize in Physics in 1938 for his work on
induced radioactivity and is today regarded
as one of the top scientists of the 20th
century.
In order to construct a theory for β-decay some of the matters are to be
considered:
(a). The electron and neutrino do not exist before the decay process, and
therefore we must account for the formation of these particles.
(b). The kinetic energy of an emitting electron is of the order of several
MeV which is comparable to the rest mass of an electron (0.511 MeV).
The rest mass of a neutrino is practically zero and moves with the
speed of light. The electron and neutrino must be treated relativistically.
(c). The continuous distribution of electron energies must result
from the theory.
In 1934, Fermi developed a successfully theory of β decay based on Pauli’s neutrino
hypothesis. By treating the decay-causing interaction as a weak perturbation, Fermi’s
Golden Rule can be applied for the calculation of the transition rate λ.

with
V   V ' i dv and
'
fi
*
f
2 ' 2
V fi  ( E f )

dn
 (E f ) 
dE f
(12)
is the density of the final state
Fermi did not know the mathematical form of V for β decay that would have
permitted calculations using equation (12).
Instead, he considered all possible forms consistent with special relativity and
eventually came up with correct one that can be used in his theory.
There are five possible mathematical operators from which either one has to be
tested through experiment and proved acceptable in Fermi’s theory.
Five possible mathematical forms:
(I). Vector (V) (II). Axial vector (A) (III). Scalar (S) (IV). Pseudoscalar (P) (V). Tensor
It actually took 20 years (and several mistaken conclusions) for
the V-A form to be deduced.
Therefore the operator which is used to compute the interaction matrix element
should have the following form:
Oˆ X  VˆV  VˆA
(13)
Since the final state include three different types of particles (a daughter nucleus,
an electron and a neutrino) we need in our calculation three wave functions to
identify a final state through a β-decay process.
For β-decay, the interaction matrix element then has the form


V fi'  g   *f  e** OX i dv
i
the initial nuclear wave function
f
the final nuclear wave function
e
the wave function of the electron

the wave function of the neutrino
(14)
The value of constant g determines the strength of the interaction.
Next we need to know how to describe the density of final state, namely
 (E f ) 
dn
?
dE f
The density of states factor determines (to lowest order) the shape of
the beta energy spectrum.
To find the density of states, we need to know the number of final
states accessible to the decay products.
In a decay an electron (or positron) is emitted with momentum p
and a neutrino (or antineutrino) with momentum q.
Since we are interested in the shape of the energy spectrum, and thus
the directions of p and q are irrelevant here in our discussion.
p  q  P nucleus  0
conservation of linear momentum
In a momentum space with axes labeled by px, py, and pz the locus of the points
representing a specific value of

p p p p
2
x
2
y
pz

2 1/ 2
z
p
py
is a sphere of radius p.
px
More specifically, the locus of points representing momenta in the range
dp at p is a spherical shell of radius p and thickness dp, thus having
volume 4πp2dp.
If the electron is confined to a box of volume V then the number of final electron
states dne, corresponding to momenta in the range p to p+dp, is
4p 2 dpV
dne 
h3
(15)
Similarly, the number of neutrino states is
4q 2dqV
dn 
h3
(16)
and the number of final states which have simultaneously an electron
and a neutrino with the proper momenta is:
2 2 2
2
(
4

)
V
p
dpq
dq
d 2n  dne dn 
h6
(17)
The electron and neutrino wave functions have the usual free-particle form,
normalized within the volume V:
1 i p r / 
 e (r ) 
e
V
1 i qr / 
 (r ) 
e
V
(18)
For an electron with 1 MeV kinetic energy, p = 1.4 MeV/c and p/(h/2π) = 0.007 fm-1.
Thus over the nuclear volume, pr << 1 and we can expand the
exponentials, keeping only the first term:
ei pr /   1 
ipr
 ....  1

(19)
iq  r
 ....  1

This approximation is known as the allowed approximation.
e i q r /   1 



V fi'  g   *f e** OX i dv
From the equation (14)



g
g
*
V  g     OX i dv   f OX i dv  ( M fi )
V
V
'
fi
*
f
* *
e 

where M fi   *f O X i dv is the nuclear matrix element
With electron’s momentum being within p → p + dp and neutrino’s
momentum being within q → q + dp, the partial decay rate under the
allowed approximation is:
2 dn
2 ' 2 dn 2 g 2
d 
V fi

M
fi

dE f
 V2
dE f
2 2 2
2
2 ( 4 ) V p dpq dq
2 g 2

M fi
2
 V
 6 dE f
2
2
2 2
dq
2 p dpq

g (4 )

h 6 dE f
therefore
2
2
2 2
p
dpq
dq
d 
g (4 ) 2

h6 dE f
(20)
The final energy Ef is just E f  Ee  E  E e  qc , and so dq/dEf = 1/c at fixed Ee.
As far as the shape of the electron spectrum is concerned, all of the factors in the
equation (20) that do not involve the momentum can be combined into a constant
C, and the resulting distribution gives the number of electrons with momentum
between p and p + dp is:
N ( p)dp  Cp2q 2dp
(N(p)dp ~ dλ)
(21)
N(p) = Cp2q2 here is the total number of electrons with a momentum magnitude p.
If Q is the decay energy, then ignoring the negligible nuclear recoil energy,
2 2
2 4
2
Q  Te Q  p c  me c  mec
q

c
c
(22)
and the spectrum shape is given by
C 2
2
p
(
Q

T
)
e
c2
C
 2 p 2 [Q  p 2c 2  me2c 4  mec 2 ]2
c
N ( p) 
(23)
N ( p) 
C 2
2 2
2 4
2 2
p
[
Q

p
c

m
c

m
c
]
e
e
2
c
(1) p = 0
N(0) = 0
(2) Q = Te
N(pmax) = 0
(23)
at the endpoint
Expected electron momentum
distribution with Q = 2.5 MeV
More frequently we are interested in the energy spectrum, for electrons with kinetic
energy between Te and Te + dTe. With c2pdp = (Te + mec2) dTe, we have
N (Te ) 
C 2
2 1/ 2
2
2
(
T

2
T
m
c
)
(
Q

T
)
(
T

m
c
)
e
e
e
e
e
e
5
c
(1) Te = 0
N(Te) = 0
(2) Te = Q
N(Te = Q) = 0
Expected energy distribution
with Q = 2.5 MeV
at the endpoint
(24)
64
29
64
29


64
Cu 35 
30 Zn 34

Cu 35 64
28 Ni36
(38%)
(19%)
Other 43% goes to the electron capture.
1. In this figure the general shape of spectra is evident, but there are systematic
differences between theory and experiment. These differences originate with the
Coulomb interaction between the β particle and the daughter nucleus.
2. From the more correct stand point of quantum mechanic, we should instead refer to
the change in the electron plane wave brought about by the Coulomb potential inside
the nucleus.
The effect of the nuclear Coulomb field can be accounted for by introducing
an additional factor, the Fermi function F(Z’,p) or F(Z’, Te) which modifies
the β-decay spectrum where Z’ is the atomic number of the daughter nucleus.
x
F ( Z ' , Te ) 
1  e x
x  2Z ' c / v
(25)
for   - decay and α is the fine structure constant.
In terms of momentum:
N ( p) 
C 2
2
p
(
Q

T
)
F ( Z ' , p)
e
2
c
(26)
In terms of energy:
N (Te ) 
C 2
2
1/ 2
2
2
(
T

2
m
c
T
)
(
Q

T
)
(
T

m
c
) F ( Z ' , Te )
e
e
e
e
e
e
5
c
(27)
Under the assumption of allowed approximation we have the following relation:
(Q  Te ) 
N ( p)
p 2 F ( Z ' , p)
(28)
If we plot N ( p) / p 2 F ( Z ' , p)
against Te there should be a
straight line which intercept
the x axis at the decay energy Q.
Such a plot is called a Kurie plot.
Kurie plot gives us a
convenient way to determine
the decay endpoint energy.
endpoint energy
There are some cases in which the nuclear matrix element Mfi vanished in the
allowed approximation since higher order terms in the expansion for electron and
neutrino wave functions are omitted.
1 i p r / 
1
e ( r ) 
e

V
V
therefore
and

 (r ) 
1 i q r / 
1
e

V
V

M fi    *f  e** OX i dv  0
In such cases we must take the next terms of the plane wave expansion, which
introduce yet another momentum dependence.
Such cases are called forbidden decays, a somewhat misleading name for that
they are not really forbidden at all.
These decays are less likely to occur and tend to have longer half-lives.
The degree to which a transition is forbidden depends on how far we must take the
expansion of the plane wave to find a non-vanishing nuclear matrix element.


M fi    *f e** OX i dv
 *


p  re
q  r



   f 1  i
  1 i
     OX i dv




 


Thus the first term beyond the 1 gives first-forbidden decays, the next term gives
second-forbidden, and so on.
The complete β spectrum then includes three factors:
1. The statistical factor p2(Q-Te)2, derived from the number of final states
accessible to the emitted particles.
2. The Fermi function F(Z’, p), which accounts for the influence of the nuclear
Coulomb field.
3. The nuclear matrix element │Mfi│2, which accounts for the effects of particular
initial and final nuclear states and which include an additional electron and
neutrino momentum dependence S(p,q) from forbidden terms.
ft1 / 2  0.693
2 3 7
2
5 4
e
g m c M fi
2
§ 9-5 Nonconservation of Parity
The parity operation consists of reflecting all of the coordinates of a system : r → -r.
If the parity operation gives us a physical system or set of equations that obeys the
same laws as he original system, we conclude that the system is invariant with respect
to parity.
The original and reflected systems would both represent possible states of nature, and
in fact we could not distinguish in any fundamental way the original system from its
reflection.
In fact, there are three different “reflections”.
1. Parity Operation (P).
the spatial operation r → -r.
2. Charge Conjugation Operation (C). replacing particles with antiparticles
3. Time Reversal Operation (T).
reverse the time direction t → -t.
In this figure three process under
the P, C, and T operations are
shown.
Vectors that change sign under P
are called true or polar vectors.
─ position, velocity, force, electric
field.
Vectors that do not
change sign under P are
called pseudo- or axial
vectors.
─ angular momentum,
magnetic field , torque.
In this figure, each reflected image
represents a real physical situation that
we could achieve in the laboratory, and
we believe that gravity and
electromagnetism are invariant with
respect to P, C, and T.
One way of testing the invariance of the nuclear interaction to P, C, T would be to
perform the series of experiments described in this figure.
In each case we could compare the probability of the reversed reaction with that of
the original, and if the probabilities proved to be identical, we could conclude that P,
C, T were invariant operations for the nuclear interaction.
We may use the method encircled to test
the parity conservation in a laboratory.
In such kind of arrangement the spin of the
decaying particle A does not change
direction under P. The original experiment
shows particle B emitted in the same
direction as the spin of A, while the
reflected experiment shows B emitted
opposite to the spin of A.
We may simply align the spins of some
decaying nuclei and look to see if the
decay products are emitted equally in both
directions or preferentially in one direction.
The parity conservation was found to be correct in many experiments and was
considered a fundamental principle of nature. It was not until 1950’s the validity of
parity conservation was questioned due to the θ-τpuzzle.
At that time there were two particles, called θandτ, which appeared to have
identical spins, masses, and lifetimes but decay into states of different parities!
θ π π
 π  π 
Since the decays were governed by a process similar to nuclear β-decay, Lee and
Yang, in 1956, suggested that θandτwere the same particle (today called a K meson)
which could decay into final states of different parities if the P operation were not
an invariant process for β-decay.
Shortly after the proposal of Lee
and Yang, C.S. Wu and her coworkers completed a delicate
experiment demonstrating that the
parity conservation was violated in
the β-decay of 60Co.
They aligned the 60Co spins by a
magnetic field at very low
temperature, T ~ 0.01 K. Reversing
the magnetic field direction reversed
the spins and in effect accomplished
the P operation.
The observed fact was that at least
70% of the β particles were
emitted opposite to the nuclear spin.
Improved results of C.S, Wu
and co-workers on the parity
violation in the 60Co β-decay.
§ 9-6 The Mass of the Neutrino
Near the endpoint of the β spectrum, the
neutrino energy approaches zero. If the
neutrino has rest mass then Eν ~ mνc2 and
our previous calculation of the statistical
factor for the spectrum shape is incorrect.
The neutrino kinetic energy can
be treated nonrelativistically, so
that q2 = 2mνTν and

N ( p)  p Q  p c  m c  me c
2
2 2
4 4
e
dN
0
dp
if mν = 0
dN

dp
if mν > 0
2

1/ 2
or N (Te )  Te2  2Te mec 2  (Q  Te )1 / 2 Te  mec 2 
1/ 2
We can therefore study the limit on neutrino mass by
looking at the slope at the endpoint of the spectrum
as indicated in the figure.
N → 0 as p (or Te) →0 , bad statistics!!
Experimental
Determination of the
neutrino mass from
the β decay of tritium
─The End ─