1
Masers
In thermal equilibrium at any tempertaure, there are always fewer atoms in an
excited state than in the ground state. Masers occur when an external agent
pumps electrons into an excited state, thus inverting the population.
Detailed balance requires that the number of transitions from level 2 to level
1 equal the number of transitions from level 1 to level 2.
N2 (A21 + B21 M + C #) = N1 (P + B12 M + C ")
where A and B are the Einstein coe¢cients, M is the mean intensity in the
maser transition, the C "# are collisional excitation and de-excitation rates,
and P is the pump rate.To simplify, letzs take C "= C #; and g1 = g 2 so that
B12 = B21 = B: Then
N2
P + BM + C
=
(1)
N1
A + BM + C
Now let N = N2 + N 1 ; and N 2 = xN: Then
x
P + BM + C
P + BM + C
=
)x=
1¡x
A + BM + C
P + 2BM + 2C + A
(2)
and
N (P ¡ A)
P + 2BM + 2C + A
If the pump gets very strong, then ¢N ! N :
The transfer equation is
¢N = N 2 ¡ N 1 = N (2x ¡ 1) =
(3)
dIº
= j º ¡ ·Iº
d`
where
jº = N 2 A21
hº
4¼
and
· º = (N 1 ¡ N 2 ) B
So
hº
4¼
dIº
hº
hº
= N2 A21
¡ (N 1 ¡ N 2 ) B Iº
d`
4¼
4¼
and
Jº = M =
1
4¼
Z
Iº d- = f Iº
where the factor f = 1 if the radiation is isotropic. Now using our results (2
and3) for the populations, we have
µ
¶
dIº
hº
P + BM + C
(P ¡ A) BIº
=N
A
+
d`
4¼
P + 2BM + 2C + A
P + 2BM + 2C + A
1
Now assume that the pump is strong (P À A); and we obtain
µ
¶
dIº
hº
BIº
=N
+²
d`
4¼ 1 + 2C=P + 2Bf I=P
(4)
where
²
=
NA
hº
4¼
·
=
hº
NA
4¼
"
=
µ
¶
1 + C=P + B (f ¡ 1) I=P
A
1¡
1 + (2Bf I + 2C ) =P
1 + (2Bf I + 2C ) =P
¸
BI=P
A
¡
1 + 2C=P + 2Bf I=P 1 + (2Bf I + 2C) =P
1 + C=P + B (f ¡ 1) I=P
A [1 + C =P + B (f ¡ 1) I=P + BI=P ]
¡
2
1 + (2Bf I + 2C) =P
P
[1 + (2Bf I + 2C ) =P ]
"
#
hº 1 + C=P + B (f ¡ 1) I=P
A [1 + C=P + Bf I=P ]
NA
¡
4¼
1 + (2Bf I + 2C) =P
P [1 + (2Bf I + 2C) =P ]2
and is much less than the ¤rst term, as we show below.
(I used the expansion
µ
¶
1
1
1
y
³
´ =
=
1¡
1+x+y
1+x
1+x
(1 + x) 1 + 1+y x
to ¤rst order in y; for y ¿ 1)
The equation of transfer (4) is of the form
dIº
Iº
= ®0
+"
d`
1 + II º
s
with
hº
®0 = N
4¼
and
Is =
µ
µ
B
1 + 2C=P
P + 2C
2Bf
¶
¶
When I ¿ Is ; the intensity grows exponentially:
I+
"
= I0 e®0`
®0
so the assumption ®0 I À " is soon satis¤ed. In this regime small changes in
gain ® 0 ` produce large changes in the intensity, and the intensity is independent
of the pump strength P: We should expect strong variablilty as ® 0 ` changes.
But for Iº À Is À "=® 0; Iº grows linearly. The maser saturates. In this
regime small changes in gain produce only small changes in the intensity, and
I / P:
2
#
As Iº approaches Is ; we must solve the full equation (neglecting "), which
takes the form
(1 + y) dy
= ®0
y
d`
with y = I=Is: Integrating, we have
y
ln
ln y + yjy 0
= ®0`
Iº
Iº ¡ I0
+
I0
Is
= ®0`
Plot for Is =I0 = 10: The plot shows Iv =I0 versus ®0 `: The red line is the
exponential.
18
16
14
12
y10
8
6
4
2
0
1
2
x
3
4
Notice that the saturation intensity Is increases with the pump strength P:
Letzs look at the emission from a spherical cloud of radius R and uniform
density and temperature. Then a line of sight the through the cloud has path
length
p
` = 2 R 2 ¡ d2
3
and the intensity is
³
´
p
I (d) / exp (®0 l) = exp 2® 0 R2 ¡ d2
Near the center of the cloud, d=R ¿ 1;
·
µ
¶¸
1 d2
I (d) / exp 2® 0 R 1 ¡
2 R2
= e2®0Re ¡®0d
2
=R
The intensity reaches half its maximum value where
® 0 d2 =R = ln 2 = 0:7
or
p
d
0:7
0:8
= p
=p
R
®0 R
®0 R
This value is ¿ 1 if the optical depth through the center of the cloud is large.
Thus maser sources appear small on the sky. Thus we may suspect a maser
when (a) the intensity is very large and (b) the source is very small.
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