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Chapter 10 Gamma Decay Introduction Energetics of γ Decay Decay Constant for γ Decay Classical Electromagnetic Radiation Quantum Description of Electromagnetic Radiation Internal Conversion

Chapter 10 Gamma Decay

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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. - PowerPoint PPT Presentation

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Page 1: Chapter 10 Gamma Decay

Chapter 10

Gamma Decay

● Introduction

◎ Energetics of γ Decay

● Decay Constant for γ Decay

◎ Classical Electromagnetic Radiation

● Quantum Description of Electromagnetic Radiation

◎ Internal Conversion

Page 2: Chapter 10 Gamma Decay

§ 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.

Page 3: Chapter 10 Gamma Decay

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.

γ-ray emitting transitions

Page 4: Chapter 10 Gamma Decay

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 high-resolution 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 Gamm

a-Ray Energy Tracking Array, or GRETA.

http://www.lbl.gov/Science-Articles/Archive/sabl/2007/Feb/GRETINA.html

Page 5: Chapter 10 Gamma Decay

γ-ray spectrum of 257No

Quantum states in 257No and 253Fm

Energy spectrum observed through γ-ray emitting transitions

Page 6: Chapter 10 Gamma Decay

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

Energy resolutions of the αparticles are not as good as γrays.

Page 7: Chapter 10 Gamma 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 :

Rfi TEEE conservation of total energy

ppR

0 conservation of linear momentum

Define energy difference between two levels fi EEE

And using the relativistic relationship cpE

The energy difference ΔE is 2

2

2Mc

EEE

(1)

(2)

(3)

The equation (3) has the solution

2/1

22 211

Mc

EMcE (4)

Page 8: Chapter 10 Gamma Decay

2/1

22 211

Mc

EMcE

(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:

2

2

2

)(

Mc

EEE

EE (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.

Page 9: Chapter 10 Gamma Decay

§ 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

Page 10: Chapter 10 Gamma Decay

Consider a point charge with an elementary unit of charge e. This point charge is accelerated with an acceleration . kajaiaa zyx

ˆˆˆ

The radiation power expressed in cgs unit system is:

3

22

3

2

c

ae

dt

dE (6)2222

zyx aaaa with

If a point charge is in a simple harmonic motion then

tztztytytxtx cos)( ;cos)( ;cos)( 000 22

020

20 Rzyx and where R is the radius of an atom or a nucleus.

In such case the acceleration tRa cos2

The average radiation power is thus

3

422224

3

22

3

2

3cos

3

2

3

2

c

RetR

c

ea

c

e

dt

dE

This is a classical description of the average radiation power from a point charge in a simple harmonic motion.

(7)

Page 11: Chapter 10 Gamma Decay

Quantum-mechanically an electromagnetically unstable system would emit a photon in every mean time interval τ, the average radiation power then is

If we define a decay constant

1G

dt

dE(8)

Combining equations (7) and (8): Gc

Re

dt

dE

3

422

3

Since the photon energy E

3234

2

3 ERc

eG We may get the following relation: (9)

Page 12: Chapter 10 Gamma Decay

3234

2

3 ERc

eG (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

1-6310427

31228210

second10)103()1005.1(3

)1060.1()10()1080.4(

G

second 1093.610

693.02ln 762/1

G

t

(2). In a nucleus take Eγ=1 MeV, R = 5 × 10-13 cm = 5 F

1-15310427

36213210

second102)103()1005.1(3

)1060.1()105()1080.4(

G

second 103102

693.02ln 16152/1

G

t

(10)

(11)

Page 13: Chapter 10 Gamma Decay

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.

Page 14: Chapter 10 Gamma Decay

§ 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

Page 15: Chapter 10 Gamma Decay

Electric Dipole Radiation

Then the Poynting vector is:

recrtr

p

cBES ˆ)/(cos

sin

4)(

122

00

0

Within a period the average quantity is rerc

pS ˆ

sin

32 2

2

2

4200

The total average radiation power is

34

1P

3

420

0

c

pdAS

(13)

Put an oscillating dipole along the z direction

ktpktqztp ˆ)cos(ˆ)cos()( 0 qzp 0with

(12)

Page 16: Chapter 10 Gamma Decay

Magnetic Dipole Radiation

Build up an current such that its magnetic dipole is along the z direction

(14)ktmktIatm ˆ)cos(ˆ)()()( 02

with 02

0 )( Iam

Then the Poynting vector is:

Within a period the average quantity is

recrtrc

m

cBES ˆ)/(cos

sin

4)(

122

00

0

rerc

mS ˆ

sin

32 2

2

32

4200

The total average radiation power is

(15) 34

1P

5

420

0

c

mdAS

Page 17: Chapter 10 Gamma Decay

rerc

pS ˆ

sin

32 2

2

2

4200

rerc

mS ˆ

sin

32 2

2

32

4200

For electric dipole radiation

For magnetic dipole radiation

Same radiation patterns

Page 18: Chapter 10 Gamma Decay

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 203

4

012

1p

cP

for electric dipoles

205

4

012

1m

cP

and for magnetic dipoles

Here p0 and m0 are the amplitudes of the time varying dipole moments.

Page 19: Chapter 10 Gamma Decay
Page 20: Chapter 10 Gamma Decay

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

)1cos3(2

1)(cos 2

2 P

)3cos30cos35(8

1)(cos 24

4 P

dipole

quadrupole

2. The parity of the radiation field is1)1()( LML

LEL )1()( 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

222

20

)] ([]!)!12[(

)1(2) ( Lm

cLL

cLLP

L

where m(σL) is the amplitude of the (time varying) electric or magnetic multipole moment.

(16)

Page 21: Chapter 10 Gamma Decay

§ 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.

(17)dvLmLm iffi )(ˆ)(

This integral is carried out over the volume of the nucleus.

)(ˆ Lm 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

212

20

)]([]!)!12[(

)1(2)()( Lm

cLL

LLPL fi

L

(18)

Page 22: Chapter 10 Gamma Decay

212

20

)]([]!)!12[(

)1(2)()( Lm

cLL

LLPL fi

L

(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

LL

cRLc

E

e

LL

LEL 2

212

0

2

2 3

3

4]!)!12[(

)1(8)(

(19)

With R = R0A1/3, we can make the following estimates for some of the lower multipole orders.

Page 23: Chapter 10 Gamma Decay

EL transition probability

93/85

72

53/47

33/214

101.1)4(

34)3(

103.7)2(

100.1)1(

EAE

EAE

EAE

EAE

where λ is in s-1 and E in MeV

(20)

LL

cRLc

E

e

LL

LEL 2

212

0

2

2 3

3

4]!)!12[(

)1(8)(

(19)

Page 24: Chapter 10 Gamma Decay

The result for the ML transition probability is

22212

0

22

p

2

p2 2

3

41

1

]!)!12[(

)1(8)(

LL

cRLc

E

c

e

cmLLL

LML

(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:

926

73/4

53/27

313

105.4)4(

16)3(

105.3)2(

106.5)1(

EAM

EAM

EAM

EM

(22)

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.

Page 25: Chapter 10 Gamma Decay
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§ 10- 6 Internal Conversion

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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