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ELECTR OM A GNETIC SPECTRUM

Energy (eV)

μ eV meV eV keV MeV Te V

10 -6 10 -3 1 10 3 10 6 10 9

3

Public domain image (source: NASA)

DIPOLE RADI A TION

Rate from Fermi’ s Golden Rule + Density of states:

2 ⇡

W = ~

| h f | V ˆ

| i i | 2

⇢ ( E f ) =

ω 3 2

2 ⇡ c 3 ~ | h r ˆ i |

sin 2

✓ d ⌦

Integrating over angles:

λ ( E 1) =

4 e 2 ω 3

3 ~ c 3

| h r i | 2

4

DIPOLE APPR O XIM A TION

In deriving the dipole emission formula we only kept lowest order expansion:

A ~ a a † e i ~ k · ~ r ~✏

k ≈ a † (1 + i ~ k · ~ r + . . . ) ~✏

k → a † ~✏ k

This yields the typical dipole emission pattern:

W a sin ✓ 2 d ⌦

In QM the angular distribution is related to the photon angular momentum

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BE Y OND THE DIPOLE

Higher order terms in the expansion give rise to gamma emission

with dif ferent angular-dependence pattern

and higher angular momentum for the gamma photon emitted

~ † i ~ k · ~ r

† X ( i ~ k · ~ r ) `

A a a k e

~✏ k ≈ a k ` !

`

~✏ k

Each l term contributes to a dif ferent decay rate.

6

MU L TIPOLE RADI A TION

Electric multipole

8 ⇡ ( ` + 1 ) e 2

✓ E ◆ 2 ` +1 ✓ 3 ◆ 2

D ˆ E 2 `

A ( E ` ) =

` [(2 ` + 1) ! ! ] 2 ~ c ~ c ` + 3

D | ~ r ˆ | E ⇡ R 0 A 1 / 3

c | ~ r |

Rates: A ( E 1) = 1 . 0 ⇥ 10 14 A 2 / 3 E 3 A ( E 2) = 7 . 3 ⇥ 10 7 A 4 / 3 E 5 A ( E 3) = 34 A 2 E 7

A ( E 4) = 1 . 1 ⇥ 10 - 5 A 8 / 3 E 9

7

MU L TIPOLE RADI A TION

Magnetic multipole

8 ⇡ ( ` + 1 ) e 2

E 2 ` +1 ✓ 3 ◆ 2

D ˆ E 2 ` - 2 ~ ✓ 1 ◆

Rates: λ ( M 1) = 5 . 6 ⇥ 10 13 E 3 λ ( M 2) = 3 . 5 ⇥ 10 7 A 2 / 3 E 5 λ ( M 3) = 1 6 A 4 / 3 E 7

λ ( M 4) = 4 . 5 ⇥ 10 - 6 A 2 E 9

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WHICH TRANSITION?

The lowest multipole dominates:

Lower multipoles decay faster (higher rates)

Electric multipoles are faster than magnetic multipoles

➡ Why don’t we always only observe electric dipole (E1) radiation?

9

SELECTION RULES

The multipole l is related to the gamma angular momentum

the angular momentum must be conserved in gamma decay

Possible l : | I f — I i | ≤ ` g ≤ I f + I i

Parity: (-1) l for Electric and (-1) l-1 for Magnetic: parity must be conserved, Π γ = Π i Π f

10

WHICH TRANSITION?

The lowest permitted multipole dominates

Electric multipoles are more probable than the same magnetic multipole by a factor 100 ) ( E l )

) ( M l )

⇡ 10 2

Emission from the multipole l +1 is 10 -5 times less probable than the l -multipole emission

) ( E, l + 1)

⇡ 10 - 5 ,

) ( M, l + 1)

⇡ 10 - 5

) ( E l ) ) ( M l )

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WHICH TRANSITION?

Combining the two rules:

λ ( E, l + 1)

⇡ 10 — 3 ,

λ ( M, l + 1)

⇡ 10 — 7

λ ( M l ) λ ( E l )

Thus E2 competes with M1

But M2 does not compete with E1

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INTERNAL CONVERSION

In some cases energy is not released in the form of gamma photons, but carried away by an electron:

A X ⇤ ! A X +

+ e -

Z Z

This process is called Internal Conversion

It is the only process possible, when selection rules do not allow any of the multipole transitions:

e.g. even-even nuclides, decay from a 0 + level

the photon cannot have zero angular momentum.

13

MIT OpenCourseWare http://ocw.mit.edu

22.02 Introduction to Applied Nuclear Physics

Spring 2012

For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .