PFC/JA-85-18
DETECTION OF ALPHA PARTICLES BY CO2 LASER SCATTERING
Linda Vahala and George Vahala
Department of Physics - College of William and Mary
Williamsburg, VA 23185
Dieter J. Sigmar
Plasma Fusion Center
Massachusetts Institute of Technology
Cambridge, MA
02139 U. S. A.
JUNE 1985
This work was supported by the U.S. Department of Energy Contract
No. DE-AC02-78ET51013. Reproduction, translation, publication, use
and disposal, in whole or in part by or for the United States government is permitted.
By acceptance of this article, the publisher and/or recipient acknowledges the U.S. Government's right to retain a non-exclusive,
royalty-free license in and to any copyright covering this paper.
DETECTION OF ALPHA PARTICLES BY CO2 LASER SCATTERING
Linda Vahala and George Vahala
Department of Physics
College of William & Mary
Williamsburg, VA 23185
and
D. J. Sigmar
Plasma Fusion Center
Massachusetts Institute of Technology
Cambridge, MA 02139
U. S. A.
ABSTRACT
It is shown that the alpha particle contribution to the scattered
power can be dominant in the coherent scattering of CO 2 laser in a
Maxwellian plasma.
The optimal forward scattering angle is around
1.0* with detection of the electron density fluctuation wavenumbers
k i > > k 11 (relative to the toroidal magnetic field).
Because of the
strong dependence of the scattered signal on the alpha particle
temperature and the alpha distribution function, it seems feasible that
CO2 laser scattering, with clever heterodyne techniques, could give
detailed local information on fusion alphas.
-2-
1. INTRODUCTION
Recently, there has been considerable experimental interest [1-5] in
the use of CO2 laser scattering to investigate certain collective effects
in plasmas.
In the scattering of electromagnetic waves from a charged
particle, the radiated power is proportional to (particle mass)- 2 .
the scattering is principally from the electrons.
Thus,
If collective or ion
information is required then one must resort to long wavelength lasers so
as to probe the plasma around the electrons.
Also, since one wishes to
use the laser as a passive diagnostic tool, the incident laser frequency
should be chosen much higher than the electron plasma frequency wpe or
the electron gyrofrequency
%e.
As a result the plasma is optically thin,
with nearly all the incident power passing straight through the plasma.
A ruby laser with (short) wavelength 6.9 x 10-5 cm is standardly used to
study single-particle aspects of electron behavior in fusion plasmas.
The longer wavelength CO2 laser (wavelength 1.06 x 10-3 cm) can be used
to study collective aspects of electron behavior, but very small forward
scattering angles are required.
Unfortunately, lasers with longer
wavelength than the CO2 laser are not presently available at the required
high incident power densities.
In this paper we investigate the possibility of using CO2 laser
scattering to detect alpha particles by calculating the scattering function
S (kw) of a magnetized plasma, as a function of various models for the
alpha particle velocity distribution.
Since the theoretical formulation of electromagnetic scattering in
plasma has been presented in detail in, for example, Sheffield [61 and in
Slusher and Surko [11 we shall be very brief about the fundamentals and
-3-
concentrate on the application.
A preliminary report not allowing for
the important effects of the magnetic field was written by Hutchinson, et
al. (7].
2. COHERENT SCATTERING FROM ELECTRON DENSITY FLUCTUATIONS
The scattering geometry is shown schematically in Fig.1.
The back-
ground plasma is assumed Maxwellian for both electrons and ions, with
temperatures Ti - Te - 10 keV and densities neo - 1.2 x 1014 cmx 3 and
nio
-
1.196 x 1014 cm-3 .
Because the plasma scattering volume V is rather
localized in the plasma, we assume a uniform toroidal magnetic field with
Several alpha particle distribution functions are considered
B0 - 50 kG.
using various effective "temperatures" Ta with density n ao
Thus for the CO2 laser, with incident frequency t3
-
7.5 x 1011cm-3.
- 2.8 x 101 4 Hz, we have
i >-Qa(1)
>> weWp > > 2e>
wI>wpi
For alpha particle temperatures of interest here (Ta > 50 keV), we
find that the alpha particle Larmor radius p. > > L - V
scattering length.
3
, the plasma
Thus we can treat the alphas as unmagnetized.
electrons and ions are treated as magnetized.
The
The electrons are assumed
to be nonrelativistic (though this assumption becomes less justified for
Te > 10 keV).
Because of the high incident laser frequency, Eq.(1),
there is negligible attenuation of the incident laser beam so that all
electrons in the scattering volume V interact with the same incident
laser electric field E1 .
pI
m2V2
A
8fe 2
Moreover, the incident laser power is such that
2
-4-
(A is the cross section area of the laser beam, ve is the electron thermal
speed) so that the plasma is not perturbed by the incident laser.
This
permits us to treat the equilibrium background plasma in the linear
approximation, using the Rostocker superposition principle for dressed
test particles.
It is readily shown [1, 6] that the power scattered into the solid
angle dQ is
dw
Ps df ---
r
neo L d
dw
-
S(k,
)
(2)
2w
2ls
where P1 is the incident laser power, and r2
-
8 x 10- 2 6
=2 is the
Thomson cross section.- It is to overcome this very small Thomson scattering
cross section that one must resort to high power lasers.
S(kc)
is the
electron spectral density correlation function or scattering function
S(kw) =
|n
y(k,w)12
.2(3)
neo
with perturbed electron density nel, and < > represents an ensemble
average over initial particle phase space coordinates.
and w -
wS -
IkS|
=
k-
I
wI are the wave vector and frequency shift due to the
incident wave-plasma interaction.
here
k
= IkI| so that
|k|
For the scattering under consideration
- 2 kI sin 6/2 (see Fig.1).
Following standard procedures [1, 6] for a thermal magnetized plasma
with unmagnetized alphas under the longitudinal approximation, one readily
finds that the electron spectral density correlation function can be
separated into 3 parts
-5-
S (kw)- Se (k,W) + Sj MkW) + Sa (kw)
(4)
where, for Maxwellian distributions,
1 + H 2 + GiGa
eSL (k,w)
2
2n1/2
e
I
-.W
Si (k, w) -
Ik IlIve
"
I' (k2 p2) e
I m - CO
I He 2
nio
2%1/2
SLI 2
neo
Ik R vi -e
(
(w ye)2)
k25)2
.-k 2 P
(k2 p2)
I
exp
2
kkIkV
I
m-
-
J (6)
O
and
|a 2
-
2
[1 + Ce Z(Ce)]
4nao
2%1/2
W2
exp(
I EL 2
neo
IkIva
2 a
The dielectric function is
eL - 1 + He(k,w) + Hi(k,W) + Ga(k, w)
(8)
with
He - a2
e-k p
I
(k2p2)
1 +
Z(
k ive
Hi = a2
nie
neo
Te E
'
e
I
(k2p )
1 + -
Ti
(9)
k lve
Z
k qvi
(10)
k lvi
and for the unmagnetized Maxwellian alphas
4
Gn
nao
- _oneo
Te
Ta
1 + Ca Z(Ca)
(11)
-6-
The gyroradii
e2
,
Ve/2S
v2/ 2 2
P
and
1/2
W
2n
Te
4pa
neo
Ta
C a21/2
e
with the plasma dispersion function for Im C > 0
40
1
2
f
e-x
dx
z ()=---
7rI/2
x - C
and analytic continuation for Im C
<
0.
The so-called Salpeter parameter
(12)
a = (k I Xge)~I
where the electron Debye length XDe
=
ve/21/2 4Ipe*
Since the dielectric function EL + 1 as a + 0, the Salpeter parameter
is an indication of the strength of collective processes within the
plasma.
Se (kw)is due to both free electrons moving in their unperturbed
gyro-orbits as well as from electrons shielding these test particles electrons.
Si (k,w) and Sa (k,ii) are contributions from the scattering off electrons
shielding the ions and alphas, respectively.
be important we require a
-
For collective effects to
1 and only under these conditions can one
observe any alpha particle effects on the scattered radiated power.
Since, for C02 laser scattering and Te - 10 keV,
0.0124
a-
(13)
sin 9/2
-7-
we will require very small forward angle scattering:
e 4 1.40.
However,
using heterodyne techniques and accurate optics the experimentalist can
detect down to scattering angles
(31 of 6 - 0.5.
3. ALPHA PARTICLE CONTRIBUTION TO SCATTERED RADIATION
To investigate the importance of alpha particles on the total
scattered power we plot Sa (kw),Eq.(7), and the total scattering density
S (k,w) for various forward scattering angles 8 and angles 0 with
k
- k cos 0, kj - k sin *.
is 5 GHz
(
The experimental frequency range of interest
It is typically found that for o 4 3 - 5 GHz, the
w e 25 GHz.
ion contribution Si (kw) dominates the total scattering density S (kw);
but for higher frequencies Si becomes negligible.
In Fig.2, S (,w)
6 -0.50,
and Sa (kw) are shown as a function of w for
For this very small forward scattering the Salpeter
- 45*
=
Nevertheless, for kj - ki, we find that the alpha
parameter a - 2.85.
particle contribution to the scattered power is negligible: Sa < < S
For w > 4 GHz, the electrons (free and screening) dominate the scattering
density Se = S with a slow decay of the spectral density with frequency W.
Decreasing the alpha particle temperature from Ta
=
300 keV to 200 keV
has little effect on the total spectral density S (kw).
On increasing
*
collective parameter a
Sa < < S -
Fig.3.
with frequency.
in 0.
*
to
=
=
85* so that k1 < < ki but still keeping the
2.85 (i.e., 8 - 0.5*), it is still seen that
However, the spectral density S rapidly decreases
For w > 6 GHz, Sa is very little affected by the increase
-8-
However, for scattering angle 6 -1.0* (with a - 1.42) and kg < < ki,
we find a very high resonance peak at w - 21 GHz, Fig.4.
Moreover, for
w > 6 GHz the alpha contribution totally dominates the spectral density
function Sa (k,w) - S (k,w) with a plateau in S for 7 GHz 4 w 4 17 GHz at
Ta
-
500 keV.
still
On decreasing the alpha particle temperature to 300 keV we
find Sa = S for w > 6 GHz,
but now the resonance amplitude has
decreased by nearly two orders of magnitude.
This resonance can be
identified with the lower hybrid resonance, whose dispersion relation
for a cold plasma is given by
2
k2
m
2
+--
me
(14)
k21
1 + 5e2
2
There is also a significant decrease in S for w > 13 GHz due to decreasing
This is further illustrated in Fig.5.
Ta.
Sa < < S for w > 10 GHz.
Ta
=
Note that at Ta
-
100 keV,
increasing k1 /k 1 with 6 - 1.0* and
Moreover,
200 keV leads to a slightly reduced spectral density S with Sa = S,
Fig.6.
As can be seen from Eq.(14) the lower hybrid resonance frequency
also increases with increasing k1 /k 1 and this accounts for the absence of
any resonance structure in S for w < 17GHz.
In Figs 7-9 we consider the effect of decreasing the Salpeter
- 0.5*)
parameter a - 2.85 (6
(k1 - 1.73 k 1 ) at Ta
=
to a
200 keV.
=
0.95 (6
=
1.5*)
for
= 60*
As the forward scattering angle 6
increases, the plateau height in the spectal density S occurs for frequencies w > 9 GHz -
which typically
first decreases till e - 1.10 and
then increases by an order of magnitude by 6 = 1.5.
In most cases
Sa < < S and only for a small frequency range is the alpha contribution
to the total spectral density significant (see Fig.9 for 6
-
1.1* in the
-9-
frequency range 7 GHz 4 w 4 10 GHz).
For 6
=
1.0* but k I < < k1 with
*
=
890, we find a very strong lower
hybrid wave resonance at w - 7 GHz, with the total spectral density S
being totally dominated by the alpha contribution Sa in the complete
The effect of alpha particle temperature is
frequency range of interest.
to increase the detectable frequency signal range from wCax - 9 GHz for
- 20 GHz for Ta - 300 keV (Fig.10).
Ta
-
to
*-
75,
so that kp/k 1 increases by a factor of about 20 from its value
at
*-
89,
the lower hybrid resonance is pushed into an undetectable
50 keV.to qx
frequency range with Sa < S for all frequencies, Fig.11.
If one decreases
*
With the electron
contribution giving a major contribution to the total spectral density S
we start to see the beginnings of a plateau in S.
the scattering angle to 6 - 0.5* at Ta
=
If one now decreases
50 keV we find a much reduced
signal even at 0 - 89* (but still the alpha contribution dominates the
spectral density), Fig.12.
However, at
*=
75* the alpha contribution Sa
Note again the plateau that forms in S -
is now negligible.
this being
due to the electron contribution.
It is also of interest to consider an alpha particle distribution
function of the form
f
(v)
-
v 2 exp (-v2 /v2)
modelling a loss effect for lower energies.
(k11
<
(15)
For Ta - 300 keV,
*
-
85*
< k 1 ) we again find that for 0 - 1.0* the alpha contribution dominates
the spectral density function, with the lower hybrid resonance level now
being approximately ten times higher than that for the Maxwellian alpha
distribution (see also Fig.3).
Increasing or decreasing the scattering
-10-
angle from
e-
1.08 leads to a reduced alpha contribution to S, Fig.13.
4. SUMMARY AND CONCLUSION
For a Maxwellian plasma with Ti - Te = 10 keV, we find that as the
forward scattering angle e increases from 6 - 0.5* to 1.1* the Salpeter
parameter a decreases as well as the electron contribution Se to the
total spectral density S.
Sa increases.
As
On the otherhand the alpha contribution
8 further increases from 6 - 1.10 to 1.5* the electron
contribution to S now increases and becomes more important than the slowly
increasing alpha particle contribution.
The background ion contribution to S is negligible for frequencies
w > 5 GHz.
The optimal observing angle 0 is that which is nearly perpendicular
to the toroidal magnetic field (0 > 85*).
Under these circumstances the
alpha particle contribution to S (kw) totally dominates the electron
contribution.
A resonance peak, corresponding to the lower hybrid resonance, is
also seen for k0 < < kj.
The use of a loss-cone type distribution for the alpha particles,
Eq.(15), rather than a Maxwellian leads to an enhanced alpha contribution
to S (kw).
This sensitivity to the shape of fa is precisely the purpose
of this diagnostic.
In conclusion, it appears that with suitable choice of local
oscillators with which to perform the heterodyne detection in the frequency
range 5 GHz 4 w < 25 GHz one should be able to detect alpha particle
-11-
contributions to the scattered power and thereby infer the shape of alpha
For a 100 MW CO2 laser
distribution function.
Ps ddw = 3 x 10~
S (k,w) d~dw
We further suggest that by appropriate focusing of these linear oscillators,
one should be able to localize the scattering plasma volume V under
consideration and due to the easy access to the tokamak for near forward
scattering one may be able to probe different vertical chords through the
plasma cross section rather than just the "diameter" chord illustrated in
Fig.1.
ACKNOWLEDGEMENTS
This work was performed under contract with U.
S.
Department of
Energy at William and Mary, Oak Ridge National Laboratory and Massachusetts
Institute of Technology.
We thank J. Sheffield for stimulating discussions.
-12-
REFERENCES
[11
SLUSHER, R. E., SURKO, C. M., Phys. Fluids, 23 (1980) 472.
[2]
SURKO, C. M., SLUSHER, R. E., Phys. Fluids, 23 (1980) 2425.
[31
SLUSHER, R. E., et al., Phys. Fluids, 25 (1982) 457.
[4]
KASPAREK, W., HOLZHAUER, E., Phys. Rev., A27 (1983) 1737.
[51
VALLEY, J. F., SLUSHER, R. E., Rev. Sci. Instrum., 54 (1983) 1157.
[6]
SHEFFIELD, J. "Plasma Scattering of Electromagnetic Radiation"
Academic Press, New York, 1975.
[7]
HUTCHINSON, D. P., VANDERSLUIS, K. L., SHEFFIELD, J., SIGMAR, D. J.,
"Feasibility of Alpha Particle Measurement by C0 2 Laser Thomson
Scattering", ORNL-TM9090 (1985).
-13-
FIGURE CAPTIONS
Figure 1
Schematic diagram of the laser scattering geometry.
Incident
beam of wavenumber kI, and scattered wavelength kS.
e is
forward scattering angle.
k = kg - kI, and $ is the angle
between k and the toroidal magnetic field BO.
Figure 2
the
k
=
2ki sin 0/2.
Plot of the spectral density function S (k,W) vs. W= Wg for frequency range 1 GHz < w < 27 GHz for
parameter a = 2.85) and 4
spectral density for Ta
=
=
45* (k-
k1 ).
e-
(q
0.5* (Salpeter
SItot is the total
200 keV with SI being the alpha
a
particle contribution to SI
SII is for Ta
= 300 keV.
a=30kV
tot*
It should be noted that S = 10-12 will yield a scattered power
density PS = 3 x 10-15 watts for a 100 MW laser (see Eq. 16).
Experimentally (see Ref. 3) power levels of 5 x 10-13 watts
have been readily detected in CO2 laser scattering.
Figure 3
S (k,w) vs.
Figure 4
S (k,w) vs. w for 0 = 1.0* (a
w for 8 = 0.5* but $
Ta = 300 keV and II: Ta
=
=
85* (ki > > k 1j).
=
1.43) and $ = 85* for I:
500 keV.
Notice that S. ~ Stots
and the appearance of a resonance at the lower hybrid frequency.
Figure 5
The effect of decreasing Ta on S (k,w) for 0 = 1.0*,
Figure 6
Effect of ki/kq on S (k,w) for 8
Figure 7
The effect of decreasing the collective Salpeter parameter a
= 1.0* and T.
by increasing 8 = 0.5* (curves I) to
S (k,w).
e-
=
4 = 85.
200 keV.
0.7* (curves II) on
Notice that Stot decreases but the alpha particle
contribution Sa is enhanced.
4 = 60* and Ta = 200 keV.
-14Figure 8
The trend of Fig. 7 continues as a decreases due to increasing
forward scattering
I), 0
0.7* (curves
0.8*
=
0 - 0.9* (curves III).
(curves II),
Figure 9
angle 0 =
Further increase of 0 = 1.1* to 0 - 1.5* at 4
=
60* and
Ta = 200 keV.
Figure 10
The effect of decreasing Ta
on S (k,w) for k1 > > k11 (4
and forward scattering angle 0 = 1.00.
=
890)
Notice the high lower
hybrid resonance peak in S (kw). For these optimal angles
Sa = StotFigure 11
The effect of decreasing ki/kH ratio on S (k,w for 0 = 1.00
and low Ta = 50
$
=
akeV.
For 4
S1 - Sto,
890,
a
tot'
but
at
75* the alpha particle contribution is significantly
reduced, SH
a < Sttot*
Figure 12
The effect of decreased scattering angle 0
with Ta = 50 keV.
=
0.5* on S (k,w)
For $ = 89* we still find that the alpha
particle contribution dominates S (k,w) but the signal amplitude
is very much reduced over that for 0
$ = 75*,
1.0* (see Fig. 11).
=
the alpha particle contribution
For
is negligible,
SH
tot*
a < < Stot
Figure 13
The scattering density S (k,w) for a loss-cone type alpha
particle distribution function, Eq.
$ - 85.
As the collective parameter increases:
(SI < Stot),
(SIII < S
Ta = 300 keV and
15.
9
).
=
1.00 (SII
S
)
=
0
=
1.2*
0.8*
Note that the spectral density amplitude
is increased over that for a Maxwellian alpha distribution function.
---- w
Bo
k,
ks
E)
-- *
--
-- w
k = ks-kl
A
4
Figure
1
16I
-11
10
S
-
3
-13
1/2
454
~=
-'14
10
0
200 keV -I
U-300 keV ---U
-615
13
5
7
9
11
13
15 17 19 21
w(GHz)
Figure
2
23 25 27 29
-10
10
i
-
I
I
I
I
I
I
i
I
i
E=1/2 0
I
200 keV--t
' 300 keV---f_
T
-11
10
-
=
-
:162
s
-s
10
s
t-
\,
\
-14
10
I
1
i
-15
10
1
3
5
7
9
11
13
15 17 19 21 23 25 27 29
w (GHz)
Figure
3
-10
10
11
I
3
Sl
stot
"
-613
10
10
~ =85 0
1614
::- T
-
300 keV-I
500keV--I
10S'
I
I
3
5-
7
9
11
13
I
15 17 19 21 23 25 27 29
w (GHz)
Figure
4
10
160
1
1
e:= 10
p=85 0
-a= 100 keV-I
keV---H_
161,1200
-12
10
*\
3
-a13
10'
Sa \\\\tot7S
-14
0
-
10
of
1
3
5
7
9
11
13
j
z
15 17 19 21 23 25 27 29
GHz)
Figure
5
-610
10'*
Tc =200 keV
161" -
~
t
S '
00a
1612
6-
3
60
-I
-
-1l3
10
~ \ stotl
10-,14
stot'
-'15
1
3
5
7
9
11
13 15 17 19 21
wj(GHz)
Figure
6
23 25 27 29
10
p=600
Ta=200 keV
. o-0
O.V-'5
-
10-
-
stot
-12\s
0
I
3
-13
10-
10
1
3
5
7
9
11
13
15 17
w (GHz)
Figure 7
19 21
23 25 27 29
110
.
.60*
TX=20o keV
.
-11(=
0.90--
1
.- -.
l
0.7*--0
0.8*- -I
tj'-.tot
101'.-$
SiI
-.
*.
m
.
10
-15
1
3 5
7
29
911 13 1517 19 21 232-5-27
w (Gtiz)
Figure
8
10
110
1
600
To 200 keV
0= T
12
10
-12
-
-
.- 1
-
--
T-
,
-5'
S
stoH
-13
10
S
\i IsaCI
14
101
13
,5
7
9
11
13
1572
15 17 19 21
w(GHz)
Figure
9
23 25 27 29
10
e=1
0
=:89 0
(300 keV-I
T= 100 keV ---.
50 keV-----]ff
1-9
-
10'
I'
-10
10
-
l
a
S
*San:
S,,
a
15
131
114
-
-142
10
1
r
3
S
I
tot ..*-
e
S
-'
15
S
S-, a z
-
I
0
0
0
I
5
7
tot
9
I'
-
11
I
13 15 17
w(GHz)
Figure 10
I
I
I
19 21 23 25 27 29
I
I
I
10
'a
*
~
e=1 0
TU=50 keV
-
890
101-11
I
75---I
I
S
-12
\
-
10
13
3C,
Ii
I
Stot"
~1
101
iti
r-'- t I'
1061
10
-14
F.-
1
3
5
7
9
11
I
I
I
I~
I
I
I
F
i
I
I
13 15 17 19 21 23 25 27 29
(. (GHz)
Figure 11
-1 0~
10
E)=1/2 0
Ta=50 keV
890 -- 1
75! ---11
I
10
=-.
-
S.4
-12
s/r
10 *
stoe"W
-
I
'4
113
~
3-
-
1-14
I
-15
-
1016
-
10- 16
~ Sa Stot
1061
1
I
A
3 5
7
9
11
13 15 17 19 21 23 25 27 29
eGHzl
Fi ure 12
10
110
1
Loss Cone
P= 850
1.2 =11
1.0---
0.8
-
~ .---.
..
/
-
keV
0-0
- --
%-
1612
3..Ta=300
3
0
00
0
*)
0.
-13
10
la30
'00
*.*0
~0
0
,61
13
5
7
9
11 13
15 17
wo (GHZ)
Figure
13
19 21 23 25 27 29