Transcript 投影片 1 - NCHU

Chapter 10
Gamma Decay
● Introduction
◎
Energetics of γ Decay
●
Decay Constant for γ Decay
◎
Classical Electromagnetic Radiation
●
Quantum Description of Electromagnetic Radiation
◎
Internal Conversion
§ 10-1 Introduction
1. Most αandβdecays, leave the final nucleus in an excited state. Theses excited
states decay rapidly to the ground state through the emission of one or
moreγrays, which are photons of electromagnetic radiation.
2. Gamma rays have energy difference between the range of 0.1 to 10 MeV.
3. The detail and richness of out knowledge of nuclear spectroscopy depends on
what we know of the excited states, and so studies of γ−ray emission have
become the standard technique of nuclear spectroscopy.
4. Gamma rays are relatively easy to observe (negligible absorption and
scattering in air) and their energies can be measured with high accuracy which
allows good quality determination of energies on nuclear excited states.
5. Spins and parities of nuclear excited states can be deduced
by properties of γdecays.
γ-ray emitting transitions
1. In this figure the vertical distance
between two levels is the energy
difference and the level X having
the higher energy, is drawn above
the one, Y, having the lower
energy.
2. A transition from X to Y normally
involves the emission of a photon,
the energy difference going into
photon energy and energy of the
recoil nucleus.
3. Photon energies involved in this
type of transition usually are of
about 10 keV up to 3 or 4 MeV
and are all referred as γ-rays.
4. γ-rays are emitted when the
nucleus makes a transition from
an excited state to a state of lower
energy.
The best way to study the existence of the heaviest elements, nucleosynthesis in exploding stars, and
other phenomena peculiar to the atomic nucleus is to create customized nuclei in an accelerator like
Berkeley Lab's 88-Inch Cyclotron, then capture and analyze the gamma rays these nuclei emit when
they disintegrate. The Lab's Nuclear Science Division (NSD) has been a leader in building highresolution gamma-ray detectors and was the original home of the Gammasphere, the world's most
sensitive. Now NSD is leading a multi-institutional collaboration to build Gammasphere's successor,
the proposed Gamma-Ray Energy Tracking Array, or GRETA.
http://www.lbl.gov/Science-Articles/Archive/sabl/2007/Feb/GRETINA.html
Energy spectrum observed through
γ-ray emitting transitions
γ-ray spectrum of 257No
Quantum states in 257No and 253Fm
Energy resolutions of
the αparticles are not
as good as γrays.
Intensity against alpha energy for four isotopes, note that the line width is wide and some
of the fine details can not be seen. This is for liquid scintillation counting where random
effects cause a variation in the number of visible photons generated per alpha decay
§ 10-2 Energetics of γDecay
Consider the decay of a nucleus of mass M at rest, from an
initial excited state Ei to a final state Ef :
Ei  E f  E  TR


0  pR  p
Define energy difference between two levels
And using the relativistic relationship
The energy difference ΔE is
The equation (3) has the solution
conservation of total energy
conservation of linear momentum
E  Ei  E f
E  cp
E  E 
E2
(1)
(2)
(3)
2Mc2
1/ 2



E


2
E  Mc  1  1  2

2 
Mc

 

(4)
1/ 2



E


2
E  Mc  1  1  2

2 
Mc

 

(4)
The energy difference ΔE are typically of the order of MeV, while the rest
energies Mc2 are of order A × 103 MeV, where A is the mass number. Thus
ΔE << Mc2 and to a precision of the order of 10-4 to 10-5 we keep only the
first three terms in the expansion of the square root:
(E ) 2
E  E 
2Mc 2
E  E
(5)
The actual γ-ray energy is thus diminished somewhat from the maximum
available decay energy ΔE. This recoil correction to the energy is generally
negligible, amounting to a 10-5 correction that is usually far smaller than the
experimental uncertainty with which we can measure energies.
§ 10-3 Decay Constant for γDecay
In atomic physics the half life of an electromagnetically excited state is of
the order 10-8 second.
In nuclear physics it can range from 10-16 second to 100 years.
Time scale of this sort can be roughly estimated through a semi-classical
consideration.
A burst of gamma rays from space.
http://spaceknowledge.net/wp-content/gallery/nebulea/phot-40f-99-normal.jpg
Consider a point charge with an elementary unit of charge e. This point
charge is accelerated with an acceleration a  axiˆ  ay ˆj  az kˆ .
The radiation power expressed in cgs unit system is:
dE 2 e2a 2

dt 3 c3
with
a2  ax2  ay2  az2
(6)
If a point charge is in a simple harmonic motion then
x(t )  x0 cost; y(t )  y0 cost; z(t )  z0 cost
and x02  y02  z02  R2 where R is the radius of an atom or a nucleus.
2
In such case the acceleration a   R cost
The average radiation power is thus
dE
2 e2 2
2 e2 4 2
e2 R 2 4
2

a 
 R cos t 
3
3
dt
3c
3c
3c3
(7)
This is a classical description of the average radiation power from a point
charge in a simple harmonic motion.
Quantum-mechanically an electromagnetically unstable system would emit a
photon in every mean time interval τ, the average radiation power then is
dE


dt

If we define a decay constant
G 
1

dE
e2 R 2 4 


 G
3
dt
3c

Combining equations (7) and (8):
Since the photon energy
(8)
E  
We may get the following relation:
e2
G  4 3 R 2 E3
3 c
(9)
e2
G  4 3 R 2 E3
3 c
(9)
The decay constant is proportional to the square of the radius of the atom (or nucleus)
under study and is proportional to the cube of the photon energy Eγ.
(1). In an atom take Eγ=1 eV, R = 10-8 cm = 1 Å
(4.801010 )2 (108 )2 (1.601012 )3
G 
 106 second-1
 27 4
10 3
3(1.0510 ) (3 10 )
t1 / 2 
ln 2
G

0.693
7

6
.
93

10
second
6
10
(10)
(2). In a nucleus take Eγ=1 MeV, R = 5 × 10-13 cm = 5 F
(4.801010 )2 (5 1013 )2 (1.60106 )3
15
-1
G 

2

10
second
3(1.0510 27 )4 (3 1010 )3
t1 / 2 
ln 2
G

0.693
16

3

10
second
15
2 10
(11)
Basically we are able to obtain a reasonable estimation for the half life of an
electromagnetically unstable nucleus. It is of the order 10-16 second. In reality the
half life of all unstable nuclei ranges from 10-16 second to 100 years. There are
obviously some important factors which have been left out in the oversimplified
semi-classical picture. We need to explore further on this topic.
Electromagnetic radiation can be treated either as a classical wave phenomenon
or else as a quantum phenomenon.
For analyzing radiations from individual atoms and nuclei the quantum
description is most appropriate, but we can more easily understand the quantum
calculations of electromagnetic radiation if we first review the classical
description.
§ 10- 4 Classical Electromagnetic Radiation
Static distributions of charges and currents give static electric and magnetic fields.
These fields can be analyzed in terms of the multipole moments of the charge (or
current) distribution.
Multipole moments ― dipole moment, quadrupole moment, etc ― give characteristic
fields, and we can conveniently study the dipole field, quadrupole field and so on.
Electric dipole radiation
Electric quadrupole radiation
Electric Dipole Radiation
Put an oscillating dipole along the z direction
p(t )  qz cos(t )kˆ  p0 cos(t )kˆ
with p0  qz
(12)
Then the Poynting vector is:
2

0  p0 2  sin  
S
( E  B) 
cos

(
t

r
/
c
)



 eˆr
0
c  4  r 

1
Within a period the average quantity is
 0 p02 4  sin 2 
 2 eˆr
S  
2
 32 c  r
The total average radiation power is
P 
1  p02 4 


S  dA 
3 
4 0  3c 
(13)
Magnetic Dipole Radiation
Build up an current such that its magnetic
dipole is along the z direction
m(t )  (a 2 ) I (t )kˆ  m0 cos( t )kˆ
(14)
2
with m0  (a ) I0
Then the Poynting vector is:
2

 0  m0 2  sin  
S
( E  B) 
cos

(
t

r
/
c
)



 eˆr
0
c  4c  r 

1
Within a period the average quantity is
 0m02 4  sin 2 
 2 eˆr
S  
2 3 
 32 c  r
The total average radiation power is
P 
1  m02 4 


S  dA 
5 
4 0  3c 
(15)
 0 p02 4  sin 2 
 2 eˆr For electric dipole radiation
S  
2
 32 c  r
 0m02 4  sin 2 
 2 eˆr For magnetic dipole radiation
S  
2 3 
 32 c  r
Same
radiation
patterns
There are three characteristics of the dipole radiation field that
are important for us to consider:
1. The power radiated into a small element of area, in a direction at an
angle θ with respect to the z axis, varies as sin2θ. This characteristic
sin2θdependence of dipole radiation must also be a characteristic result
of the quantum calculation as well. Radiations caused by higher order
multipoles have different power angular distributions.
2. Electric and magnetic dipole fields have opposite parity. Under the
parity transformation, the magnetic field of the electric dipole changes
sign: B(r) = -B(-r) but for the magnetic dipole B(r) = B(-r).
3. The average radiated power is
and
1 4 2
P
p0 for electric dipoles
120 c3
1 4 2
P
m0 for magnetic dipoles
5
120 c
Here p0 and m0 are the amplitudes of the time varying dipole moments.
Without entering into a detailed discussion of electromagnetic theory, the properties
of dipole radiation can be extended to multipole radiation in general.
1. The angular distribution of the 2L-pole radiation, relative to a properly
chosen direction, is governed by the Legendre polynomial P2L(cosθ).
The most common cases are
1
(3 cos 2   1)
2
quadrupole P4 (cos  )  1 (35 cos 4   30 cos 2   3)
8
dipole
P2 (cos  ) 
2. The parity of the radiation field is
 (ML)  (1)L1
 ( EL)  (1)L
Here M is for magnetic
and E is for electric.
3. The radiated power is, using σ= E or M to represent electric and
magnetic radiation
2( L  1)c   
P( L) 

2 
 0 L[(2 L  1)!!]  c 
2 L2
[m( L)]2
(16)
where m(σL) is the amplitude of the (time varying) electric or magnetic
multipole moment.
§ 10- 5 Quantum Description of Electromagnetic Radiation
To carry the classical theory into the quantum domain, we must quantize the
sources of the radiation field, the classical multipole moments.
In equation (16) it is necessary to replace the multipole moments by appropriate
multipole operators that change the nucleus from its initial state ψi to the final
state ψf.
m fi (L)   f mˆ (L) i dv
(17)
This integral is carried out over the volume of the nucleus.
ˆ (L)
m
is the multipole operator which can be obtained by
quantizing the radiation field.
If we regard the equation (16) as the energy radiated per unit time in the form of
photons, each of which has energy   then the probability per unit time for
photon emission is
P( L)
2( L  1)
 
 (L) 


2 

 0L[(2 L  1)!!]  c 
2 L 1
[m fi ( L)]2
(18)
P( L)
2( L  1)
 
 (L) 


2 

 0L[(2 L  1)!!]  c 
2 L 1
[m fi ( L)]2
(18)
The expression for the decay constant can be carried no further until we evaluate
the matrix element mfi(σL), which requires knowledge of the initial and final
wave functions.
We can simplify the calculation by the assumption that the transition is due to a
single proton that changes from one shell-model state to another.
By so doing the EL transition probability is estimated to be
8 ( L  1)
e2  E 
 ( EL) 
 
2
L[(2 L  1)!!] 4ε0c  c 
2 L 1
2
 3  2L

 cR
 L  3
(19)
With R = R0A1/3, we can make the following estimates for some of the lower
multipole orders.
EL transition probability
8 ( L  1)
e2  E 
 ( EL) 
 
2
L[(2 L  1)!!] 4ε0c  c 
2 L 1
 ( E1)  1.0  1014 A2 / 3 E 3
 ( E 2)  7.3  107 A4 / 3 E 5
 ( E 3)  34A2 E 7
 ( E 4)  1.1 10 5 A8 / 3 E 9
2
 3  2L

 cR
 L  3
(19)
where λ is in s-1 and
E in MeV
(20)
The result for the ML transition probability is
8 ( L  1) 
1 
 ( ML) 




p
2
L[(2 L  1)!!] 
L 1
2
  


m c
 p 
2
 e2  E 

 
 40c  c 
2 L 1
2
 3 
2L2

 cR
 L2
(21)
It is customary to replace the factor [μp ̵ 1/(L+1)]2 by 10, which gives the
following estimates for the lower multipole orders:
 ( M 1)  5.6  1013 E 3
 ( M 2)  3.5  107 A2 / 3 E 5
 ( M 3)  16A E
4/3
7
(22)
 ( M 4)  4.5  10 6 A2 E 9
These estimates for the transition rates are known as Weisskopf estimates and are
not meant to be true theoretical calculations. They only provide us with reasonable
relative comparisons of the transition rates.
§ 10- 6 Internal
Conversion
An artist's conception of the blazar
BL Lacertae at it spurts out jets of
charged particles accelerated by
corkscrew magnetic field lines.
Credit: Marscher et al., Wolfgang
Steffen, Cosmovision,
NRAO/AUI/NSF