1

Absorption Cross Sections(Mihalas Chapter 4)

Bound-bound transitions(Einstein relations)

Bound-free transitions(Einstein-Milne relations)

2

Bound-Bound Transitions

• Balance transitions between two levels

• Upwards absorption• Downwards by

spontaneous or stimulated emission

j

i

hνij

3

Energy Absorption Rate: Bij = Einstein coefficient for absorption

4

Return Rates: Aji=Einstein coeff. spontaneous em. Bji=Einstein coeff. stimulated em.

(isotropic)

(same direction as Iν)

5

*Thermodynamic Equilibrium: Iν=Bν

6

b-b Transfer Equation

7

b-b Transfer Equation

8

Transition Rates

• Calculate or measure cross section

9

10

Transition Rates: Hydrogen

• H transitions n'→n

11

Einstein-Milne Relations forBound-Free Transitions

12

Einstein-Milne Relations:energy and velocity

h d mV dV Replace v dv increment in recombination term with this.

13

Einstein-Milne Relations:#photoionizations=#recombinations

F v

G v

h

c

2 3

2

14

Denominator Term:Maxwell dist. ne(v); Saha eq. n0/n1

ng

g

h

m kTe e p m

nm

kTe v hG v

g h p

g m v G ve

e

h

kT

m v

kTv

e

m v

kT

h

kT

0

1

2 3 2

2

3 2

2 2

02

12 2

2 2

24

1

2 41

2

2

/

/

…….=1. ……

15

b-f Transfer Equation

T.E.: αν=pνhν , ne=Maxwellian, n0*/n1

*=Saha, G(v)/pν above …