Beta Decay Beta Decay

Beta Decay Beta Decay

spectrum and lifetime spectrum and lifetime

†

W

= 2p

h f

V f

dn

dE ( e

,n

)

wave function of the parent nucleus

product wave function of the daughter nucleus, electron, and antineutrino

Let us first calculate the density of final states

†

r p

+ p r

+ p r

= 0 T

+ T

+ T

= Q = E

†

r p

,E

†

r p

,E

†

r p

, E

†

dn

dE ( e

,n

) ^{= V} _h

_dE ^d Ú ^p

^dp

^dW

^p

^dp

^dW

If we are interested in electrons emitted with an energy between E_e and E_e+dE_e, the variation does not affect the electron observables. Hence one gets:

†

dn

dE ( e

,n

) ^{= V}

2

dW

dW

h

p

2

dp

p

dp

dE

If we neglect the very small nuclear recoil energy, for constant electron energy we obtain

†

d

dE = d

dE

Beta Decay Beta Decay

spectrum and lifetime (cont.) spectrum and lifetime (cont.)

†

dn

dE ( e

,n

) ⁼ _4p ^V

h

c

p

2

( E - E

)

^{1- m}

^c

E - E

( )

^dp

The final expression for the density of final states of en electron emitted with a given energy and momentum (integrated over all angles) is:

Now we need to calculate the interaction matrix element:

†

M

≡ Ú y

( r r

, r r

, K r r

) ^y

*

r r

e^-

;Z

( ) ^y

^{( ) r r}

¥ V

y

r r

, r

r

, K r r

( ) ^{d r r}

^{d r r}

^{, Kd r r}

^{d r r}

d r r

The neutrino wave function can be written as:

†

1

V exp i r k

r r

( )

For the electron, a plane wave approximation is too crude, and one has to consider the distortion of the wave function caused by the interaction with the electromagnetic field of the nucleus. Quantitatively, the main effect is to alter the magnitude of the electron wave function at the origin:

†

y

(0;Z)

@ 1 V

2 ph

1- e

= 1

V F Z, p (

) ^{, h ≡ ±} _hv ^Ze

e^-

Fermi function positive (negative) sign used for b

(b

) decay

The Fermi function slightly distorts the beta spectrum shape.

Beta Decay Beta Decay

spectrum and lifetime (cont.) spectrum and lifetime (cont.)

†

W

( ) p

^dp

^{= M}

' 2

2p

h

c

F Z (

, p

) ^p

( ^{E - E}

)

^{1- m}

^c

E - E

( )

^dp

Depends on nuclear wave functions

Beta Decay Beta Decay

Influence of the neutrino mass Influence of the neutrino mass

†

W

p

e^-

( )

p

2

F Z

, p

e^-

( ) ^{µ M}

'

( E - E

)

If we assume that the nuclear matrix element is totally independent of p_e, and for vanishing neutrino mass, one gets

Fermi-Kurie plot

The intercept with the energy axis is a convenient way to determine the Q-value!

This procedure applies to allowed transitions. (For forbidden transitions, there is an additional p_e dependence of |M’|…

Fermi-Kurie plot for the allowed beta decay in

66

Ga

†

0 ⁺ Æ 0 ⁺

total energy (in m_ec²) electron

scattering within the source

Deviations around the endpoint due to nonzero neutrino mass…

Hypothetical case of

beta decay with non-

vanishing neutrino mass

(

H decay; m

c

=30 eV)

Beta Decay Beta Decay

Total half-life Total half-life

†

W

= M

2 p

_h

^c

^{F Z} (

^{, p}

) ^p

( ^{E - E}

)

0 pe-(max)

Ú ^dp

This integral can be expressed as:

†

W

= m

c

2 p

_h

^{F Z}

^{, w}

2

-1

( ) ^M'

^w

^{- 1 w (}

^{- w )}

1 w₀

Ú ^wdw

where and w₀ is the reduced max. electron energy.

†

w = E

/ m

c

If we assume that the matrix element does not depend on w, and after taking out the strength g of the weak interaction, one obtains:

†

fT = 0.693 2 p

_h

g

m

c

M ' ˆ

, M ' ˆ

≡ gM'

†

f Z (

,w

) = F Z (

, w

-1 ) ^w

^{- 1 w (}

^{- w )}

1 w₀

Ú ^{wdw f-function}

electrons

positrons

Beta Decay Beta Decay

Total half-life Total half-life

From the expression for fT, it is possible to determine the strength g of the beta- decay process, if one knows how to determine the reduced matrix element. As will be discussed later, for superallowed transitions, the matrix

element is so the fT values should be identical.

†

0⁺ Æ 0⁺

†

2

3088.6(2.1) s

†

g = 0.88 ¥ 10 ^-4 MeV fm ³

†

G = g m

c

h 1.026 ¥10

or, introducing the dimensionless constant G:

Beta Decay Beta Decay

Microscopic picture Microscopic picture

On a more fundamental level, beta decay of hadrons can be viewed as the transformation of one type of quark to another through exchange of charged weak currents (W bosons carry net charges; Z boson is neutral - it is the source of neutral weak current).

n

p e

n _

W

e

n

e

n

Z

The flavor of quarks is conserved in strong interactions. However, weak interactions change flavor! For example:

†

u Æ d + e

+ n

†

d Æ u + e

+ n

†

n Æ p + e

+ n

fi (udd) Æ (uud) + e

+ n

Beta Decay Beta Decay

Microscopic picture Microscopic picture

When a quark decays, the new quark does not have a definite flavor. For instance:

†

u Æ d'= d cosq

+ ssin q

Cabibbo angle

However, the observed weak transitions are between quarks of definite flavor. The strong-interaction quark eigenstates

†

d' s' b' Ê

Ë Á Á Á

ˆ

¯

˜ ˜

˜ =

U

U

U

U

U

U

U

U

U

Ê

Ë Á Á Á

ˆ

¯

˜ ˜

˜ d

s b Ê

Ë Á Á Á

ˆ

¯

˜ ˜

˜

This means that the observed beta-decay strength in reactions is modified by the mixing angle.

†

u d Ê Ë Á Á

ˆ

¯ ˜

˜ c s Ê Ë Á Á

ˆ

¯ ˜

˜ t b Ê Ë Á Á

ˆ

¯ ˜

˜

†

u d' Ê Ë Á Á

ˆ

¯ ˜

˜ c s' Ê Ë Á Á

ˆ

¯ ˜

˜ t b' Ê Ë Á Á

ˆ

¯ ˜

˜

are different from weak interaction eigenstates).

Cabibbo _Kobayashi- Maskawa (CKM) matrix

For nuclear beta-decay, we are mainly concerned with the transition between u- and d-quarks. As a result, only the product

enters into the process.

†

G

= G

cosq

Beta Decay Beta Decay

Microscopic picture Microscopic picture

What are the consequences of parity violation in beta decay?

†

h =

s r ⋅ r p

p ^helicity

The eigenvalue of h is v/c. For a massless particle, the eigenvalues of h can be only +1 or -1. In general, the particle with

•h>0 is called “right-handed”

•h<0 is called “left-handed”

Experimentally,

†

h( n

) ª -1, h( n

) ª +1, h(e

) = mv/c

All the leptons emitted in beta-decays are left-handed and all antileptons - right-handed!

The operators that are scalars, pseudoscalars and tensors produce leptons of both helicities under a parity transformation. Only vector operators V and axial vector operators A can accommodate the observed result. Furthermore, since V and A are of different parity, they must appear in a linear combination. This leads to the V-A theory of beta decay. In principle, both V and A parts should be characterized by different coupling constants, G_V and G_A, respectively.

The vector current is known to be a conserved quantity (CVC hypothesis)

†

V

= 0 four-divergence

For the axial vector current, there is not such a relation. The four-divergence of A (a pseudoscalar!) does not vanish. The pion is a pseudoscalar particle. Hence the weak interaction is modified in the presence of strong interactions. This leads to a partially conserved axial-vector current (PCAC) hypothesis:

†

A

= a f

a constant the pion field

Beta Decay Beta Decay

Microscopic picture (cont.) Microscopic picture (cont.)

How to relate G_V and G_A?

†

g

≡ G

G

= f

g

M

c

pion decay constant pion-nucleon coupling constant Goldberger-Trieman relation

Experimentally, g

=-1.259

This value is close to obtained from the relation above. It is a nice confirmation of the PCAC

†

g_A ª 1.31

Matrix elements Matrix elements

†

V

ª g d ^r r

- r r

( ) ^d ( ^{r r}

^{- r r}

) ^{d ( r r}

^{- r r}

^{) O (n Æ p) ˆ zero-range}

The nuclear operator transforming a neutron into a proton must be one body in nature. Hence it must involve the isospin raising or lowering operators.

In the non-relativistic limit, the vector part may be represented by the unity operator times and the axial-vector part by a product of and s. (A proper derivation requires manipulation with Dirac 4-component fuctions and g matrices!)

†

t

†

t

†

V

Æ G

t

( j) + g

r

s ( j) ⋅ r t (j)

[ ]

j=1 A

Â

Fermi decay, carries zero angular momentum

Gamow-Teller decay, carries one unit of angular momentum