*equal numbers of*atoms in the ground state and the first excited state.

*Solution*:

The Boltzmann equation is:

N

**/ N**_{2 }**= [g**_{1 }**/ g**_{2}**]**_{1 }**exp (- E2 – E1) / kT**_{ }
And from the table, g

**= 8 and g**_{2}**= 2**_{1 }
We require the condition that: N

**= N**_{2 }**so:**_{1 }
1 = [8/ 2

**]**_{ }**exp (- E2 – E1) / kT**_{ }
But: E2 = - 13.6 eV and E1 = -3.4 eV, therefore:

1 = 4 exp (-10.2 eV)/ kT

Taking natural logs:

ln (4) = (10.2 eV)/ kT

where: k = 8.6174 x 10

^{-5}eV/K
Solving for T:

T = 10.2 eV/ (ln 4) (8.6174 x 10

^{-5}eV/K)
T = 10.2 eV/ (1.3862) (8.6174 x 10

^{-5}eV/K)
T= 85 388 K or T = 8.54 x 10

^{4}K
2 ) For the Balmer a line (called H- alpha), we know:

E3 – E2 = - 13.6 eV

**(**1/**3**^{2 }^{ }- 1/ 2^{2 }**)**= 1.88 eV
a) From this information calculate the ratio N

**/ N**_{2 }_{1 }
b) Obtain the specific intensity from:

I

**= 2h**_{u}**u**^{3}/ c^{2}[1/ exp (hc/lkT]*Solution*:

N

**/ N**_{2 }**= [g**_{1 }**/ g**_{2}**]**_{1 }**exp (- E2 – E1) / kT**_{ }
From the table in the Dec. 5th post; g

**= 8 and g**_{2}**= 2**_{1 }_{ }

_{}
k = 8.6174 x 10

^{-5}eV/K
N

**/ N**_{2 }**= [8/ 2**_{1 }**]**_{ }**exp (- E2 – E1) / kT**_{ }
N

**/ N**_{2 }**= 4 exp (-1.88 eV)/ (8.61 x 10**_{1 }^{-5}eV/K) (10^{4}K)
N

**/ N**_{2 }**= 4(0.113) = 0.452**_{1 }**= 2h**

_{u}**u**

^{3}/ c

^{2}[1/ exp (hc/lkT]

We need to use consistent cgs units. Planck constant h = 6.62 x 10

^{-27}erg-s
c= 3 x 10

^{10}cm/s
l = hc/ E = (6.62 x 10

^{-27}erg-s) (3 x 10^{10}cm/s)/ 3.0 x 10^{-12}erg
l = 6.62 x 10

^{-5}cm
k=1.38 x 10

^{-16}erg/K
[1/ exp (hc/lkT] =

1/ [exp (6.62 x 10

^{-27}erg-s) (3 x 10^{10}cm/s)/ (6.62 x 10^{-5}cm) (1.38 x 10^{-16}erg/K)( 10^{4}K)
= 0.113

I

**= 2h**_{u}**u**^{3}/ c^{2}[0.113] erg cm^{-2}/s
But

**u**= E/h =
3.0 x 10

^{-12}erg/ 6.62 x 10^{-27}erg-s = 4.53 x 10^{14}/s
So:

I

**= 0.226(6.62 x 10**_{u}^{-27}erg-s) (4.53 x 10^{14}/s)^{ 3}/ (3 x 10^{10}cm/s)^{ 2}
I

**= 1.51 x 10**_{u}^{-4}erg cm^{-2}/s^{}
3)Calculate the transition probability you get using the Einstein equation:

A

**= 6.67 x 10**_{21}^{16}[g f/g2 l^{2}Å]
What possible errors might cause the values to diverge? (Take g = f » 1)

l = 6.62 x 10

^{-7}m = 6.62 x 10^{-5}cm = 6620 Å
With g

**= 8 and g = f = 1**_{2}
A

**= 1.93 x 10**_{21}^{8}[Å^{-2}]
Compare to standard form (see e.g. Wikipedia, “Einstein coefficients”) given in multiple physics papers as:

A

**= [f g1/g2]{ 2 π**_{21}**u**^{3}e^{2}}/ e_{o}m_{e}c^{3 }^{}

A

**= 1.87 x 10**_{21 }^{7 }or: 0.187 (in defined units of 10^{8}s)
Error sources: Imprecise oscillator frequency f

Error in one of the statistical weights.

Error in one of the statistical weights.

_{21.}
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