PFC/JA-88-4
Effect of Scattering Parameters on the Detection
of the Alpha Particle Distribution Function for
CO2 Laser or Millimeter Incident Radiation
Vahala, L.*; Vahala, G.**; Sigmar, D.
Plasma Fusion Center
Massachusetts Institute of Technology
Cambridge, MA 02139
*Old Dominion University, Norfolk, VA 23508
**College of William and Mary,
Williamsburg, VA 23185
February 1988
Submitted to: Nuclear Fusion
This work was supported by the U. S. Department of Energy Contract No. DE-AC0278ET51013. 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.
EFFECT OF SCATTERING PARAMETERS ON THE DETECTION
OF THE ALPHA PARTICLE DISTRIBUTION FUNCTION
FOR CO 2 LASER OR MILLIMETER INCIDENT RADIATION
LINDA VAHALA
Department of Electrical and Computer Engineering
Old Dominion University, Norfolk, VA
GEORGE VAHALA
Department of Physics
College of William and Mary
Williamsburg, VA
23185
DIETER SIGMAR
Plasma Fusion Center
MIT, Cambridge, MA
02139
23508
I
ABSTRACT
Parameters (choice of incident radiation, scattering and orientation angles)
are sought so that the one dimensional alpha particle distribution can be deduced
from the experimental coherent scattering data.
In order to be able to deduce
the
the
alpha
distribution
function,
we
must
have
total
scattering
function
totally dominated by the contribution from the electrons shielding the alphas,
S.(M)~Stot(M)
in
the
dependence in S ,(w)
function.
GHz
frequency
range,
as
well
as
all
the
frequency
is contained only in the one dimensional alpha distribution
Te has disparate effects:
high Te leads to lower S /Stot,
while
leading to more of the w dependence of S 0 being carried by the one dimensional
alpha distribution f,(w/k).
The CO
laser is a passive probe of the plasma because of its very high
2
frequency, but coherent scattering can only be achieved for very acute forward
scattering angles (9 < 1.20).
For electron temperatures Te2> 10 keV, it will be
quite difficult to directly deduce the alpha distribution from Stot(o).
For the two gyrotron sources considered here (2.14 mm at 140 GHz; and 5
mm
at
60
GHz),
the
lower
incident
frequencies
lead
to
concerning penetration, refraction and absorption by the plasma.
serious
questions
Assuming these
problems can be overcome, coherent scattering can be achieved for any forward
(or
backward)
distribution
scattering
from
the
angles.
We
scattering data
find
that one could deduce the
for forward
scattering
alpha
angles, although
the o range for the 5 mm source is quite limited and the associated Stot(w)
has a rapidly decaying amplitude.
For backward scattering angles, there is the
possibility that the alpha distribution could be deduced from Stot from the 5
mm source, although Stot(o) is again rapidly decaying in the limited w range
2
where S(<)
.
3
1.
INTRODUCTION
In an earlier calculation',
incident CO
plasma.
we considered the possibility of employing an
laser as a diagnostic for detecting alpha particles
2
in a fusion
Since the power radiated from a particular plasma species is inversely
proportional
to
the
mass
squared of
that
radiation will be emitted by the electrons.
particles,
we
must
choose
incident
species,
nearly
all' the
observed
Thus, to be able to detect alpha
parameters
so
as
to
achieve coherent
shown
to
occur
scattering from the electrons shielding the alphas.
Theoretically,
Salpeter
scattering2 has
coherent
o(
parameter
E1/k(e)XD > 1. where k=kS-
equilibrium electron density
IkiI
ks1
when
the
i is the effect of the
fluctuations on the incident
ki, and XD is the Debye length.
consideration here,
been
radiation wavenumber
For the typical scattering experiments under
so that
k(6)= kI= 2 ki sin
(1)
where e is the forward scattering angle. Thus, if the incident wavelength is
increased
wavelength
(from
the
millimeter
short
wavelength
gyrotron
sources)
10.6pm
CO 2
laser
the
scattering
to
the
e
angle
longer
can
be
significantly increased and yet remain in the coherent scattering regime.
Now
our
earlier
calculationi
was
restricted
in
scope
orientations which were nearly perpendicular to the toroidal
c=nt/2, where k^a cos 4.
k
to
examining
magnetic field
It was reported there that in a narrow cone around
=0 there existed a very strong lower hybrid resonance peak in the scattering
function
Stot,
with
the
dominant
contribution
arising
from
the
electrons
4
shielding
the
alphas,
SO.
However, several errors 3 had been found
in this
paper', and these have been corrected by us and the corrected results have been
presented at various meetings4 .
unchanged,
It was found that qualitatively the results are
but there were significant quantitative corrections.
These will be
discussed here.
Recently. Hughes and Smith 5 have made the suggestion that scattering data
obtained from particular orientations could give direct information on the alpha
particle distributon function.
In this report, we shall examine this suggestion
quite carefully for both CO 2 and millimeter gyrotron sources.
In particular, we
shall examine this suggestion for a JET-like Maxwellian plasma in which the
temperatures Ti=Te=lO keV,
electron density neo=1.2x1014
cm~3,
deuteron density ni1 =1.185x10 14
cm3,
alpha particle density n 0 =7.5x1011
cm~ 3
since the plasma is assumed to be charge neutral.
For simplicity, we shall
also assume a uniform toroidal magnetic field BO=34 kG.
interest to examine the role of Te on the scattering
consider
both Te=1
keV and 15 keV.
It will be of some
function, and we shall
We find that the scattering results are
quite insensitive to the deuteron temperature Ti.
5
It
can
be
shown''
TO
CONTRIBUTION
PARTICLE
ALPHA
II.
2
from
fluctuation
equilibrium
FUNCTION.
SCATTERING
THE
that
theory
the
alpha
particle contribution to the scattering function is given by
S(k
) =IHe
2
4ncfle
IELI
Here, f(l)
2TC
F'k
f(1)
(2)
__
is the one-dimensional alpha particle distribution function with the
velocity
only
n2
being
dependence
the
in
of
direction
k
(i.e.,
the
velocity
In Eq. (2). He is given
components perpendicular to k have been integrated out).
by
He (k ,()
=
et 11 +
12
,
__k
where r0,, = exp(-kp
(3)
Z
kiiVe
It(kIp) is a typically Bessel function gyroradius term
for Maxwellian electrons, and Z is the Fried-Conte function.
ve is the electron
thermal speed, De the electron gyrofrequency and Pe is the electron gyroradius.
The dielectric function EL(k,<w) is defined by
EL(k,o) = 1 + He(k,w) + Hj(k,w)
+
(4)
G (k,w)
where
Hi(k,)
ri
fio Te
=,2 - -Z
n
rz1a
(.1
1 + --
kTvI
(_-__
i
kI v1 )
(5)
is the anolagous Bessel function gyroradius term for Maxwellian deuterons.
6
Finally G.
is given by
Gd(k,w) =
kv
o -k.v
d
k2
The
scattering
expressions
for
the
electron
and
(6)
+ iS
deuteron
contributions
to
the
function,
Stot (ok) = Se(G),k)
+
Si(o,k)
+
S (cw,k)
will not be presented here since they are somewhat lengthy and not relevant to
the discussion at hand. They are given in Ref. 1.
We now turn to the suggestion made by Hughes and Smiths, that it should be
possible to choose orientations so that the alpha particle distribution function
can be determined directly from the diagnostics.
First, it should be noted that when a certain parmeter regime is chosen
i.e., the incident radiation wavelength Xi, incident scattering angle e, and the
orientation
k=k(e)
relative to the toroidal magnetic
field,
then the
wave number
is fully determined.
Suppose now that over a significant GHz frequency range a parameter regime
can be found in which
(i)
the alpha particle contribution to the scattering function is dominant
SO((k~w) = Stotko)
and,
(ii) that SO itself can be accurately approximated in this frequency range by
S I(k,o)
where
S
(k,co),
7
4neo
-ne T2TrT do
S ((k,w) =C*
-
(7)
Then, in
is some appropriate normalization constant, independent of 0.
and C*
such a
parameter
scattering
regime,
alpha
dependent
frequency
Stot(kw)
function
one-dimensional
a
particle
will
yield
a
direct
function
distribution
of
the
total
measurement
of
the
measurement
f(l
This
occurs
measuerment
is now
/k).
because the complete frequency dependence of the Stot'zS
in the one-dimensional alpha particle distribution function.
retained only
This is the basic idea of Hughes and Smiths, who suggested looking at
are negligible.
orientations where the magnetic effects contained in S
all
these
I He(k.w)
effects
magnetic
the frequency
in
dependent
factor
12 , Eq. (2). and occur because the large scattering volume
EL(k,)
12/
are contained
Now
forces one into a magnetized treatment of the electrons shielding the alphas.
However,
one
if
IHe(k,o))
L(k,)
can
dependence
in
distribution
f(1 )(a/k).
For
results
the
is
2
S
is
a
find
parameter
independent
carried
to be
only
regime
(,
by
the
presented
the
then
of
particles can be described by the slow-down distribution
f'0
(v) =
F
for
lVI
lvi
>
< V,
V + Vc
= 0
,
for
where the normalization constant F0 and v
which
V,
are given by
this
complete
one-dimensional
we shall
here,
in
factor
frequency
alpha
assume that
particle
the alpha
B
F0 =
2 E,
3
V
-
4Tt 1n(1+v"(/vC)
/
for E, = 3.5 MeV
mO
vc is the speed at which the electron drag on the alphas equals that due to the
deuteron drag on the alphas.
It should be noted that for the slow-down distribution, there is an upper
bound to the frequency, wcut. above which there can be no alpha contribution to
Since f
the scattering function.
0
= 0 for
IvI
> vd, it is easily seen that
SO((Gk)=0 for w>wcut(e), where
Wcut(e) = k(e) v.
We
shall
present
2 ki v.
results
for
contribution to it in the GHz range.
(8)
sin(e/2)
the
scattering
function
and
dominant
alpha
Only in this frequency range can the alphas
yield a dominant contribution to the total scattering function.
frequencies w < 1
the
Indeed, for
GHz the electrons shielding the deuterons will yield the
contribution
Si: while typically for 6) > min {wcut, 9-12 GHz} the
electron contribution Se will dominate Stot'
9
INCIDENT
Ill.
A.
The
For
lower
CO
SCATTERING
LASER
2
resonance,
hybrid
around
orientations
AT
10.6jIm
and
orientations
the
scattering
9=TT/2,
k11= 0
with
Stot exhibits a
function
strong lower hybrid resonance at wLH, which has the following simple form in
the cold plasma approximation
2
0LH =
In Fig. 1,
r
22kijp
2
+
we show temperature and orientation effects on Stot [connected or
dashed curves] and S
[unconnected triangles and stars]
laser scattering at forward scattering angle e=0.5.
COLH
resonance
in the
scattering
electron temperature of Te=1
scattering
i
function
comes
function for
for the case of C02
In Fig. la one sees a strong
p=85 0 (**
curves)
at a
low
keV in which almost all the contribution to the
from
However, this resonance is easily
the
electrons
screening
the
alphas,
SO.
destroyed by decreasing the orientation angle
9, as seen in Fig. l a for 9=800 (triangle curves), although with this orientation
we still have Sd=Stot'
The
lower
hybrid
resonance
is also easily
destroyed
by
higher electron
10
temperatures.
From Fig. 1b, one finds that for Te=10 keV and
=850, the lower
hybrid resonance peak in Stot is significantly lowered as well as broadened in
frequency.
Moreover, over most of the GHz range, S.
dominant contribution to Stot.
no longer provides the
for Te=15
This is further exhibited in Fig. 1c
keV.
If instead, one increases the orientation angle 9 to T/2, the lower hybrid
resonance becomes more pronounced than at 9=850, but again the ratio S /Stot
rapidly decreasing
In general, for fixed orientations
with increasing Te.
p. as
the forward scattering angle e. both the lower hybrid resonance
one increases
WLH and the cut-off frequency
decrease
4cut increase, while both Stot and So
and SO/Stot significantly decreases over the GHz frequency range of interest.
B.
Orientations
from
away
the
lower
hybrid
resonance:
9
<<
T/2
ki S0.
In
Figs.
2
and
scattering angle from
3,
we
consider
e=O.5 0 to 1.00,
the
effect
interest,
Fig.
1.
increasing
the
forward
for various Te but for orientations 9
significantly away from 7T/2 so that kH is finite.
with those from
of
it is seen that within
On comparing these results
a large frequency
range of
Stot slightly decreases while So remains unchanged as 9 decreases in
this range of finite k11 [e.g., compare Figs. 1b, 2b, and 3b].
As Te
increases,
11
[Figs. 3a, 3b, and 3c], there is a marked flattening of the deca-y of Stot with
frequency but the contribution from S
On
increasing
the
forward
decreases faster with w.
scattering
angle
from
to
9=0.50
ei.00
the
Salpeter (or coherent scattering) parameter decreases. This results in a further
decreases
in the
total signal
Stot, as one would
expect.
Moreover, as Te
increases, there is significantly less flattening in the frequency decay of Stot
unlike the case with e=0.5
0
[e.g., compare Figs. 3b and 3c).
However one finds
in general that
S,,(k,)
S ,4(k ,w)
Stot(k ,o)
e
=
1.0
Stot(k,(o)
e
=
0.5
which is somewhat unexpected.
C.
Parameter
We now
regime
for
detecting
the
alpha
distribution
function
attempt to find a parameter regime for the incident CO 2 laser
from which a frequency dependent measurement of the scattering function, Stot,
will yield direct information on the one-dimensional alpha particle distribution.
In
particular
the
full
magnetized
alpha
contribution,
SO [Eq. (2)] , will be
compared to that which would be obtained if magnetic effects are neglected.
S * [Eq.
(7)].
For convenience, the normalization constant, C", in Eq.
(7)
is
chosen so that
S. (k ,)z
S., (k ,#'
t
1.8 GHz
C* is strongly dependent on e and Te, but fairly insensitive to the orientation
12
angle P.
The normalization, Eq.
is somewhat
(9).
arbitrarily assigned at the
GHz and any other choice of frequency would be appropriate
frequency w=1.8
provided it is sufficiently large so that the magnetized deuteron response Si at
this frequency is negligible.
right hand side of Eq. (9)
Also, it should be noted that the setting of the
to unity at this frequency is also of no importance.
What is important is whether or not there exists a frequency range in which
so( *(kw,)/S (k,o) is independent of 6.
In Fig. 4, we plot this ratio S O*(k,w)/S (k,&) at forward scattering angle
e=0.5 0 for various orientation angles c
as kg
and electron temperatures Te.
Clearly,
increases (i.e., 9 decreases from t/2) the magnetic effects become less
important.
Also
magnetic
temperature increases.
effects
become
less
important
as
the
electron
From Fig. 4c and 4d, it appears- that for Te> 7 keV and
for 9<45 0 there does exist a frequency range o>5 GHz in which the S */S
curve is flat.
However for 9=45 0 and Te=10 keV we see from Fig. 3b that the
contribution from the alphas to the total scattering function Stot
montonically
decreases from about 86% at o=5 GHz, 71% at 7.4 GHz, to 51% at w=9 GHz.
From our earlier discussion on the Hughes and Smith5 suggestion, it was
shown
that
the one
dimensional
alpha
particle
distribution
function can
be
inferred directly from' the total scattering function data provided there existed
a frequency range such that
:, const.
wt (f)t
with k(e) fixed for given scattering angle 9, and
(10)
13
= stoto)
Sd(6))
0)
It thus appears that for a fusion plasma with Te > 10 keV, the use of the
particle distribution
CO 2 laser as an alpha
difficult since either Eq. (10) or (11)
function diagnostic will
are usually violated.
be quite
Unfortunately, as is
evident from Fig. 3, increasing the forward scattering angle does not help since
it leads to a drastically reduced alpha contribution to the scattering function:
SO /Stot
We have noted that choosing orientations with k11=0 will lead, for low Te,
to a resonance in the scattering function with So=Stot (e.g., Fig. la).
But, as
can be seen from Fig. 4a, the S "(ci))/S (w) curve has a strong dependence on w.
Hence, even under these conditions, it is difficult to directly determine the
alpha particle distribution function since Eq. (10)
As Te increases, the resonance peak
broadens
in
frequency,
with
will
become
orientations
-
Tt/2
since
Relativistic
effects 4' 5
momentum,
since
now
Ocez0ce(p),
in Stot
Se/Stot decreasing.
relativistic effects
gyrofrequency
is not satisfied.
will
the
where
important
the
Doppler
smear
out
gyrofrequency
for
decreases in
Moreover, it
Te L 20
broadening
the
is
amplitude and
keV, especially for
factor
previously
dependent
the relativistic momentum
appears that
on
k1 c/6
-+ 0.
sharp
electron
the
particle's
p=Vmov.
This, in
turn, will further smear out any lower hybrid resonance peak in the scattering
amplitudes.
We are forced
to conclude that there do not appear to be any
significant advantages to be gained by choosing orientations with k ,=0 in the
14
detection of alpha particles.
only for the CO
2
It will also be seen that this conclusion holds not
laser but also for the gyrotron sources.
15
IV.
GYROTRON
Although
SCATTERING
one
is
AT
2.14
necessarily
mm
AND
restricted
5.00
to
very
mm
small
forward
angle
scattering with the CO 2 laser in order to achieve coherent scattering from the
electrons, the very high incident frequency of the CO 2 laser does allow a passive
probing
of
the
plasma.
If
instead one
resorts
to the
longer
wavelength
millimeter radiation, then coherent scattering can be achieved at any forward
(or even backward)
scattering angle.
However, there are now new difficulties
introduced since the incident gyrotron frequency is on the order of the electron
gyrofrequency (or its second harmonic).
question of
incident
beam
penetration
These added complications include the
into the plasma
refraction and absorption by the plasma.
in order, but
scattering
in this report
and the
problems of
A careful study of these questions is
we shall concentrate on the properties of the
functions S. and Stot as functions of scattering angle e. orientation
angle 9 and electron temperature Te.
Preliminary
simplified calculations of
WoskovO seem to indicate that the use of gyrotron sources at 60 GHz (5.00 mm)
and at
140
GHz (2.14
Hughes
and
Smiths
mm)
have
should avoid
also
considered
most
of these new complications.
gyrotron
sources
at
these
two
frequencies.
For orientations
9~T/2,
the
cold
plasma
propagating modes, the X-mode and the 0-mode.
below
the gyrofrequency
elliptically polarized.
dispersion
Qe, one would choose the X-mode (5.00 mm) which is
However,
the elliptical p[olarization introduces a
been calculated for the cold plasma by Bretz 7.
beam
into the
yields two
To propagate into .the plasma
factor into Stot which leads to a reduced final signal.
gyrotron
relation
plasma
between
the
form
This form factor has
If one wants to propagate the
gyroharmonics,
choose the 0-mode (2.14 mm) which is linearly polarized.
then
one
would
16
It
(=0:
should also be remembered
for v > v .
that
for the slow-down
alpha distribution
Thus for frequencies with
CD
=
CL)
>
2ki sin (9/2)
k
there will be no electron-shielded alphas to scatter from.
Sd(k,w) = 0
Thus
for w > kv,
with the cutoff frequency wcut(e)=k(e)
vd.
Hence, in comparing the appropriate
cutoffs for the 2.14 mm and 5.00 mm gyrotron sources at the same forward
scattering angle e, we find
Wcut 2.14 mm = 4.28 x wcut 15MM
(12)
It will be seen that these gyrotron cutoff frequencies will impose restrictions
on the GHz frequency range from which to obtain scattering data, unlike the
situation for the shorter wavelength CO 2 laser where
(A)cut
co
2
>>
cut
1
2.14 mm
and for which there are no such similar limitations in the GHz frequency range.
A.
The
lower
hybrid
resonance,
and
orientations
with
k,,
0
In Fig. 5 we consider the scattering functions for the 2.14 mm gyrotron
radiation at forward scattering angle 8=90 0, and for the same orientations 9
and
temperatures
Te as in Fig. 1 for the CO 2 laser source.
For Te=1
keV (Fig.
Sa), one again sees the strong lower hybrid resonance in Stot at 9=850 with
Sd=Stot.
There
is
a
striking
similarity
between
these
results
and
those
results from the CO2 laser at 9=0.50, (Fig. 1a), up to the cutoff frequency
for
17
the 2.14 mm gyrotron source, wcut(e)= 8.4 GHz.
(Fig.
5b) and 15 keV (Fig. 5c). the resonance peak in the scattering function
broadens and lowered with the S /Stot
one.
As Te increases to 10 keV
ratio now being significantly less than
Again, there is strong similarity with the CO
As the orientation decreases
2
results (Figs. lb and Ic).
to P=80 0, the lower hybrid resonance no longer
occurs in the frequency range 1.8 GHz < w < GCut, as for the case of the CO 2
laser (c.f. Figs. 1 a and Sa).
Again, as Te is increased, the alpha contribution S
to Stot is significantly reduced (c.f., Figs. lb and 5b, 1c and 5c).
On
angles),
increasing
there
the
is even
a
more
scattering
1350
(i.e..
backward
striking similarity
with
the results of CO 2
scattering
angle
to
scattering although the resonance peak in the scattering function at 9=85 0 is
significantly lower in the 2.14 mm gyrotron case (c.f., Figs. 6a and 1a).
similarites continue for both (=80
0
The
and 9=85 0 as can be seen from Figs. 6b and
1b, 6c and 1c.
B.
Orientations
away
from
the
lower
hybrid
resonance:
9
<<
7t/2
kg x 0.
We now consider the effects of different orientations 9 and temperatures
Te on the scattering functions for the 2.14 mm gyrotron for forward scattering
angles e=90
0
and 1350 [Fig. 7 for 9=75 0, Fig. 8 for 9=45 0 , and Fig. 9 for
The results are qualitatively similar to those for the CO
2
=0 0].
laser.
For Te=1 keV, SI=Stot for both 9=900 and the backscatter angle G=1350, but
18
S e=135
>
SIe=90
in the Ghz frequency
range, and changing the orientation
angle has little effect (c.f., Figs. 7a, 8a, and 9a).
As Te
increases,
both
S
and Stot
increase but
is a
SO(O)/Stot(W)
decreasing function of w (c.f., for example Figs. 8a, 8b, and 8c).
C.
Comparison
between
the
2.14
mm
and
5
mm
gyrotron
sources
In Figs. 10 and 11 we contrast the effect of orientation on the 2.14 mm
(140
GHz)
and the 5 mm (60 GHz) gyrotron sources.
At
forward scattering
angle of 9=450 and Te=10 keV, one finds that for both gyrotrons S =Stot, and
decreasing the orientation
from
9=75 0 to
has little effect
9=450
(Fig.
10).
Since for the 5 mm gyrotron the cutoff frequency %cut is only 1.9 GHz, we have
plotted the scattering function for this gyrotron source from c=1
Ghz.
It is
interesting to note that the 2.14 mm source has a larger scattering function
than the 5 mm source, as well as a larger non-zero frequency range [c.f., Eq.
(12)].
However, a disadvantage of these relatively small
forward
scattering
angles is that the cutoff frequency wcut is quite low.
As e increases, wcut also increases.
However, for backscatter angles we
now see that for the 2.14 mm gyrotron S /Stot
and is a monotone decreasing function of 6.
and e=135 0.
is significantly less than one
In Fig. 11, we consider Te=10 keV
As the orientation angle decreases from 9=75 0 to 9=00, the alpha
contribution to the scattering function increases :
19
< S(Q
Sa)
S
( )
10o
9= 750
)
tot
(13)
9
=
450
There is only a small difference between the results for
=450 and 00.
Indeed,
we find that the contribution from the alphas to the scattering function is also
monotonically
decreasing:
2.14 mm : S./So:
w=5 GHz:
ca=7.4
GHz:
w=9 GHz:
For
the
5
mm
gyrotron
e=1350
9=45 0
9=00
85%
88%
70%
75%
53%
59%
source,
however,
the
results
are
somewhat
more
encouraging, since in the frequency range 1.4 GHz < w < 4.4 GHz we see that
Sd/Stot - 0.94 ,
for backscatter angle 8=1350 and for both orientations
and 00, (see Fig. 11b and 11c).
=450
Unfortunately, there are two drawbacks in the
use of the longer wavelength gyrotron, the first being quite serious:
(i)
Stot(c)
is
a
very
Stoto)) 5mm " Stot0)
(ii)
rapidly
I2.14mm
decreasing
function
of
to
with,
typically,
for most of the frequency range,
the cutoff frequency is quite low ocut= 4 . 8 GHz.
Indeed,
One again sees the strong effect of electron temperature on S
Fig. 11d for backscatter angle of e=135 0 and Te=1 keV.
and Stot in
Even for orientations up
20
9=750, S =Stot
for all w up to wcut for both 2.14 mm and 5 mm sources.
Parameter
D.
To
regime
detecting
for
be able to deduce the
the
alpha
function
distribution
function
alpha particle distribution
the
from
scattering data, we must find a frequency range in which both Eq. (10) and (11)
are satisfied simultaneoulsy.
From Fig. 12a-12c, for the 2.14 mm gyrotron at
forward scattering angle e=45 0, it can be seen that Eq. (10),
S*(W)/Sd(w)
=
const., is satisfied for nearly the whole GHz range (up to ocut) for Te > 10 keV
for orientations p < 450.
and
(Figs. 12d-12f).
satisfied:
Similar results hold for the 5 mm gyrotron
is also
From Fig. 10, we also see that in this regime, Eq. (11)
SO()=Stot')).
Thus it appears that one should be able to deduce the one-dimensional alpha
distribution from
the
scattering data
for either gyrotron
source for forward
scattering angles (here shown ate=45 0) and orientations 9< 450.
We now consider the effect of increasing the scattering angle e, Figs.
and
provided
Again,
14.
S *(CO)/S (o)= const.
increaases.
the
However,
orientation
S ( O)/Stot(o)
9
_ 450 and Te >
10
13
keV,
decreases (at fixed w) as e
This decrease is significant for the 2.14 mm source, but not very
significant for the 5 mm gurotron (i.e., if one ignores the much reduced final
signal
amplitude
Stot,
and the effect on the cutoff
example, for 6=900 (with ocut=8. 4 GHz)
frequency
wcut).
For
21
2.14 mm: S /Stt
w=5 GHz:
6)=7.4
GHz:
w=8 GHz:
Thus,
for
information
signal
on the
amplitude
is
9=450
9=00
91%
93%
69%
75%
50%
57%
angles
backscatter
one
_=90 0
dimensional
sufficiently
mm
5
the
alpha
large
for
gyrotron
source can
distribution
experimental
2.14 mm source will have quite limited applicability.
function,
give
detailed
provided
the
while
the
detection,
22
V.
SUMMARY
In this calculation we have considered the possibility of using the short
wavelength CO 2 laser or the longer wavelength gyrotron sources as a diagnostic
for the detection of alpha particles in a JET-like plasma.
In particular we have
examined closely the suggestion of Hughes and Smith5 that judicial choice of
alpha particle
distribution
direct
yield
could
parameters
scattering
information on
the
function.
One of the main advantages of the CO 2 laser is its very
This allows a passive probing of the plasma.
frequency.
dimensional
one
high incident
The price one has to
pay is that coherent scattering can only be obtained for very acute forward
angles
scattering
e
<
On the other hand, the gyrotron sources have
1.00.
incident frequencies in the vicinity of the electron gyrofrequency or its second
harmonic
incident
so that questions of penetration, accessibility and refraction of the
beam
become
very
important.
(These
analyses
have
performed for a hot plasma, and have not been attempted here.)
not
yet
been
The advantage
of the longer wavelength gyrotron sources over the CO 2 laser is that one can
achieve coherent scattering not only for any forward scattering angles but also
for backweard scattering angles (e>iT/2).
Since in the D-T fusion reaction the
alphas are born with 3.5 MeV energy there will effectively arise a cutoff phase
velocity above which there are no alpha particles.
For the gyrotron sources this
will imply a cutoff frequency wcut in the GHz range such that for given e,
f (l)
v) = 0
, for v > wcut/k(e)
where k(e) is related to the incident wave number ki by the relation k(e)=2ki
x
sin(e/2).
x
However for the CO2 laser, because of its higher incident wave
number, the associated wcut is far outside the GHz range and plays no role.
23
In order to be able to determine the alpha particle distribution function
from the scattering data, we find that Eqs.
(10)
and (11)
simultaneously in an appropriate GHz frequency range.
the
electron
must be satisfied
One finds in general that
Te has opposing effects in satisfying Eqs.
temperature
(11): as Te increases, at fixed
(10)
and
e and 9,
decreases , as one expects on physical grounds, while
S ()/Stot(w)
S0(&)/S d(w) -+ const.
Moreover, for S *(0)/S.(()
Thus one must
= const. it is necessary that k,2>0.
stay away from orientations c=pT/2 at which k1 = 0.
On the other hand, only for orientations 9=iT/2,
hybrid
resonance
increases the
peak
peak
in
in the
S
scattering
decreases
while
However SO/Stot decreases as Te increases.
function at
is there a strong lower
wLH.
the resonance
However, as Te
itself
is broadened.
For these orientations, relativistic
effects would become important for Te> 2 0 keV and these would smear out even
further the WLH
resonance.
Thus,
making
measurements
around
perpendicular
orientation seems to have little applicability in determining the alpha particle
distribution function (whether one uses the CO 2 laser or the gyrotron sources).
One of the important diferences between the 2.14 mm and 5 mm gyrotrons
is that
the final signal, Stot(w), is significantly lower for the 5 mm source
in most of the appropriate w range which is also significantly reduced over that
for the 2.14 mm gyrotron.
24
Finally, for orientations with p<Tt/2, one finds that for the 5 mm gyrotron
one can determine the alpha particle distribution for both forward and backward
scattering angles.
However, for the 2.14 mm gyrotron it is more beneficial to
use forward rather than backward scattering angles.
on an electrostatic calculation.
These results are all based
A full electromagnetic calculation is necessary
so as to take proper account of polarization effects.
Acknowledgments
The authors are grateful to Roger Richards of 0. R. N. L for pointing out to
us a mistake in our original numerical code, and to Paul Woskov of M . 1. T for
pointing out to us the shortcoming of treating the electrons shielding the alphas
as unmagnetized.
This work was supported by the Department of Energy.
25
REFERENCES
1.
L. Vahala, G. Vahala, and D. J. Sigmar, Nuci. Fusion 26, 51 (1986)
2.
J. Sheffield, Plasma
Electromagentic
Scattering
of
pointed out
to us a
Radiation,
New
while
Paul
York, Academic Press, 1975;
3.
Roger
Richards
Woskoboinikow
electrons
(MIT)
shielding
(ORNL)
indicated
the
that
as
alphas
it
would
magnetized,
numerical
be
error,
more correct
rather
than
as
to treat
the
unmagnetized.
These errors effect the results for perpendicular orientations more than those
for off-perpendicular orientations.
They were corrected by us and presented at
the annual Sherwood Theory meeting, paper 3A29, (1987)
in April 6-8 at San
Diego.
4.
L. Vahala and G. Vahala,
(1987) : and
paper 3A29, Sherwood Theory Meeting, April 6-8
L. Vahala and G. Vahala, Bull. Am. Phys. Soc. 32, 1896 (1987).
5.
T. P. Hughes and S. R. P. Smith, submitted to Nucl. Fusion
6.
P.
7.
N. Bretz, Princeton University report PPPL-2396 (1986)
Woskov, MIT report PFC/RR-87-16
26
CAPTIONS
FIGURE
Fig.
1
The effect of Te and orientation angle P on the frequency dependence of
the total scattering function Stot(w) and the alpha contribution S ()
for incident C0
curves
are
unconncected
Stot -A-A-A
(a)
for
laser at forward
scattering angle 8=0.50.
: the curve
p=850
* * * is the S
curve.
-"-"- is
The ****
for Stot,
while
the
The AAAA curves are for 9=800:
, while So is the unconnected triangles curve.
For Te=1
provided
2
keV,
there is a sharp
c=85 0 but
this
resonance
lower hybrid resonance in Stot
in
absent
for
=800.
For
both
orientations
S 0()=Stot(j) except near wz1.8 GHz where the deuteron
contribution
dominates.
Tet. S (
relevant
(b) Te=10 keV. (c) Te=15 keV.
',LH
*)/StotD
Te, Eq. (10)
Note that as
-. , and the resonance broadens.
Thus for
is not satisfied, and so one will not be able to
deduce the one dimensional alpha particle distribution function directly
from the scattering data.
Fig.
2
C0 2 scattering.
=60 0. Forward scattering angles: e=1.00
e=0.50 (AAAA). (a) T,=1 keV, (b) Te=10 keV, (c) Te=15 keV.
TeT, Sd(o))/Stot(W
Fig.
3
1
Same as Fig. 2, but for increased k,, ( =45 0).
(
, and
Again, as
27
Fig.
4
The electron temperature effect on S,*(w)/SC(co)
(:Ca)
for various orientations
Forward scattering angle B=0.50.
for CO 2 scattering
p=750, (b) 9=600, (c) 9=450, (d)
=0 0.
Te= 1keV (AAAA) : Te=10 keV (oooo)
Te= 15 keV ("*"*). As Tet , frequency dependence of S I*(w)/SO((O)
As 9 1 , frequency dependence of S *(c)/S (c)
1.
1.
The normalization of
the curves is chosen arbitrarily at o=1.8 GHz.
Fig.
5
SO(M)
and Stot(w)
scattering
6
angle 0=900,
the
2.14
mm
gyrotron
and for 9=85 0 (""*)
source
at
forward
and 9=800(AAAA).
(a)
Te=1 keV, (b) Te=10 keV, (c) Te=15 keV.
These diagrams should be
compared with Fig. 1
Note that the magnitudes of
Stot and S
Fig.
for
for the CO
2
laser.
are very similar, as is the behavior for Tet and for '1.
Same as Fig.
5,
but for backward scattering
angle e=1350
and
still
k1 <<1.
Fig.
7
2.14 mm gyrotron,. 9=750 and e=1350 C"=)
Stot(0) 1e=135 > St o t(O) I e=90 .
Stot T'Sd T but S O/Stot 1.
0=900 (AAAA).
Similarly for S (G).
Typically,
Again, as Te T.
[N.B. compare with the CO 2 curves, Fig. 2]
(a) Te=1 keV, (b) Te=10 keV, (c) Te=15 keV.
28
Fig.
8
Same as for Fig. 7, but for 9=45 0.
Fig.
9
Same as for Figs. 7 and 8, but for 9=00.
Fig.
10 Stot(o) and SO(j)
the 2.14 mm ("""") and 5 mm (AtaL)
for
9=450, Te=10 keV, and (a)
sources.
=750, (b)
gyrotron
For the 5 mm
=450.
0 for w=1 GHz, so Stot and So are presented
gyrotron radiation, Si(G)
for 1 GHz < o. < wcut = 1.8 GHz frequency range.
Note that for the
2.14 mm radiation, the signal strength is considerably greater than for
the 5 mm source, and
for both mm
Notice that
'cut is considerably higher (3.6 GHz).
i.., Eq. (11)
sources, SO(i)eStot(.)
is satisifed at
Te=10 keV.
Fig. 11 Same as Fig. 10, but for backscatter angle 9=1350. and (a) 9=750,
9=450,
(c) 9=0
0
at Te= 10 keV.
The initial fall in Stot is due to the
rapid decay of the deuteron contribution S (.
the
2.14
However.
mm
source,
but
I1I
with w.
Stot(6")
(b)
S (O)/Stot(w)
I
1 for
fo(o)/Stot(r)
S
for the
5 mm
As 91, SO()/Stot(o) T.
source.
Thus, Eq. (10)
is quite well satisfied for the 5 mm source (AAAA), but not that well
satisfied by the 2.14 mm source (****).
should be compared to (a).
In (d), Sd(1)/Stot.
In (d), Te=1
keV, and (d)
29
Fig.
kev (AAAA),
12 Te effects on S O()/Sd(M) at e=450 : Te=1
Te=10
keV
(oooo), Te=15 keV (****).
For 2.14 mm gyrotron :
(a)
=75 0 , (b)
9=600, (c) 9=00,
for 5 mm gyrotron: (d) 9=75 0, (e) p=600, (f)
Te > 10 keV, S "(*)/Sd(Cc)
Provided
=00.
-+ const. as 91 (for 9<60 0 ).
Notice
the low wcut for the sources (ocut=l .9 GHz for 5 mm radiation, and
Wcut=4.6 GHz for 2.14 mm radiation).
Fig.
13 Same as Fig. 12 except that e=900 and the orientations considered are
9=75 0
[(a),
As et, .cutt.
(e)];
9=60 0 [(b),
(f)].
9=450
[(c),- (g); 9=00
Clearly for 9=45 0 and 1, S O*(W)/S .()=
[Cd).
(h)].
const., provided.
Te >_10 keV.
Fig.
14 Same as Fig. 13 except that e=1350. As eT,
. S ()/S
(o)= const.,
%cutt. Again for 9=45 0 and
provided Te>10 keV.
FORWARD SCATT. ANGLE=0 5 DEG, TE=1 KEV
K.B RNGLE
: RR
(80)
mm
,
(85)
DEG
n
-~
,*
*1
*1
-11-
*
-l
1
4
~1
**
-12
-L
s
AAA
C
A
A A A
A
T
T
A AL
A
A
A
13 -A
N
L
o
G
A
I
14
0
*
-15
-
-16
-
-17
-
*
I
I
GE+09
3 .0E+09
I
I
5 .OE+09
2.0GE+09
FREQUENCY
Fig. la
I
9 .E+09
1 .iE+10
FORWARD SCATT. ANGLE=0.5 DEG, TE=10 KEV
Ki.8 ANGLE
: AR
(80)
,
mm
(85)
DEG
-11--
*1
2
*
***
A
'I
A
AA
*
A.
*1
S
A
4
A
T
T
A A A A A\&A
*
A
A
A
*
*
*
*
A
*
*,
F
*A
N
A
-13
A
*
L
0
G
*
1
u
*
A
-14
*
*
-15
I I
I .0E+09
I
I
I
3 .E+09
III
I
I
'
I
I
5 .0E+09
'7 OEi-09
FREQUENCY
Fig. lb
II
I
I
9. OE+09
I
I
I
I
1 .IE+1O
FORWARD SCATT. ANGLE= 0.5 DEG, TE=15 KEV
K.B RNGLE
(80)
: RA
,
EE
(85)
DEG
-11
**
*A
AAA
A
* ****
A A
A A
LLAAAAAAA
*
**A
~
*~
S
C -12
A
T
T
*
AAA
AA A
A *A *
A
A*
F
N
~1
A*
AA
A
*
L
0
G
*
A
0
*
-13
A
A
*
*
*
-14
--
I .OE+09
I
3 .0E+09
I
I
T
7
T
5.0E+09
I
I
7 GE+C
FREQUENCY
Fig. Ic
9
9.GE+09
I
iE+10
C02 SCATT. AT K.B=60 DEG
FORWRRD
-10
SCRTT.
RNGLE:
1.0 DE0
(wo)
TE=1 KEV
0.5 (RR)
*
*
*
-11
--
s
C
A
T
T
A\
**
F
N
**
AA
-12
L
0
******
***
4
kf
t
A.
A
A
A
A
A
A
-14
-
I
1 .0E+09
I
I
I
I
I I
3 .0E+09
I
i
I
I
I
I
I
7 .OE+C 9
5 .OE+09
FREQUENCY
Fig. 2a
9.0E +09
I -. E+10
C02 SCATT. AT K.B=60 DEG, TE=10 KEV
FORWRRO SCRTT.
-10
ANGLE: 1.0 DEG
0.5
(**)
(AR )
-
*
*
*
-Il
-
*
S
C
A
T
T
F
N
* **
-12
A AA
-
A AA
*
L
A
A
A
A
AA
A
*
0
k
AL A
****** A
A
1
0
A
A
A
A
A
A
-13
A
I
I
I.E+09
3 .0E+09
I
I
5 -OE+09
7 .OE+09
FREQUENCY
Fig. 2b
I
I
I
9 .0E+ )9
1 .IE+10
C02 SCATT. AT K.B=60 DEG
FORWRRD SCRTT. RNGLE:
-10
TE=15 KEV
,
i.0 DEG (w*)
0.5 (RR)
-
*
*
*
S
C
A
T
T
LAA
A
A
A
.AA
**A*
A A
-12
A
A
A -6AA
A A
A
A A- A
L
0
G
**
*
~AA
L
0
AA
AA
A
AA
AA
-13
-
-14
-1
I
1.OE+09
3 GE+09
i
I
j
I
I
I
I
5 GOE+09
7
.0E+09
FREQUENCY
Fig. 2c
I
A
I
I
I
9 .OE+09
,
I
1 .1E+10
TE=1 KEV
C02 SCATT. AT K.B=45 DEG
-10
0.5 (AR
1.0 DEG (wo)
FORWRRD SCRTT. ANGLE:
)
*
*
-11
*k
-
S
C
A
T
T
*
*
A
A
A
F
N
*w~
-12
*
-
L
0
G
&
&
£
A
hA
1
0
A
AA
A,
h
*
A
A
A,
A6
A
A
A
-14 -
I .OE+ 39
I I
I
I
I
3.OE+( 39
II
I
5.GE+09
7 .OE+09
FREQUENCY
Fig. 3a
I
I
I
I
I
9 .OE+09
I
1. -E+10
C02 SCATT. AT K.B=45 DEG, TE=10 KEV
FORWARD
SCRTT.
-10
ANGLE:
1.0 DEG (mm)
0.5 (AR)
-
*
*
-11
s
C
A
T
T
F
N
* ***
AA
-12
* * ***
* *
A
A
L
A
0
,A
G
A
AA
A
A
AA
A
AA
*
A
-13
-
A
A
-14
-
1 .OE+C
I
I
I
9
I.
3 OE+09
I
5 .OE+09
I.
7 .OE+09
FREQUENCY
Fig. 3b
I
'I
9 .OE+09
1 .1E+10
C02 SCATT. AT K.B=45 DEG
FORWARD SCRTT. RNGLE:
,
1.0 DEG (mwd
TE=15 KEV
, 0.5 (RR)
I
*
*
*
S
C
A
T
T
A
A A
F
N
A A
-12
****A***********
LsA
h
L
A
AA
L
'A
0
G
*
AA
*
'1
0
A-
*
A***
A
AA
AA
A
A
-13
-
A
A
-14
-
I
I
I.E09
I
I
I
3 .E+09
I
I
I
I
I
I
I
I
I
I
I
5 .E+ 39
7.0E+09
FREQUENCY
Fig. 3c
9 .0E+09
1.1E+10
DEG
TE EFFECTS AT 0.5 DEG, K.B=75
I KEV
## 15 KEV, 00 10 KEV, RA
r02 , TE:
1.4
*
1.3
*
**
1 .2
* *
*
0.9
0
0
0
*
0
0
*
*
0
-
0
0
A
0
0
0
A
0
0
A
-
*
0
A
U
N
m
A 0.7
*
0
A
0.8
*
*
0
0
E
D
*
* *
0
-
*
*0
1.1 N
0
R 1 .0
M
A
L
00000000
0
0
A
0
0
0
A
A 0.6 -
0
A
G
A
L 0.5 .
P
H
A0.4 -
A
A
A
A
0.3
A
A
A
0.2
-
AA
AA
AA
AA
AA&
I
.OE+( 9
3.0E+09
I
5 .E+09
7 .OE+09
FREQUENCY
Fig. 4a
A
A
A
I
9 .0Et09
A
i
1 .iE+10
TE EFFECTS AT 0.5 DEG, K.B=GO DEG
C02
1.5
1
.4
T E:
,
**
15 KEV, 00
10 KEV, RA 1 KEV
-
***
*
1.3
*
1 .2
*
0
0
0
1
1 .0
A
A
oo000000000oooo~~0000
0
i
A
00
A
A
A
0.9
A
A
A
A
A
0.8-
A
A
A
A
A
A
A
A
A
0.6
A
A
A
A
0.5
A
A
A
A
0.4
A
A
-H
A
A
A
A
0.3
A
A
A
A
0.2
0.1
0.0
H
I
I .E+09
I
j
i
I . .
3 .OE+09
I
I I I I
I
5 .CE+09
I
S.0E+09
FREQUENCY
Fig. 4b
I
I
I
9 .E+ 39
1.IE+10
TE EFFECTS AT
C02
, TE:
0.5
mm 15 KEV,
DEG, K.B=45 DEG
00 10 KEV,
RR 1 KEV
1.5 --
1
.4
-
1 .3
**
*
1.2
0
0000000000000000000000000000
0
1.1
-
*AA
A
AA
0.9
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
0. -5
A
0 .4
0.3
0.2
0.2
0.3
I
1..E+09
3 .0E-09
I
7 .OE+09
5 .0E+09
FREQUENCY
Fig. 4c
I
II
9 .OE+09
I
'
1 .1E+10
TE EFFECT2 ATPO 5 DEG, K.B=0 DEG
C2
'E:
CC
10 KEV,
FlCP
I KEV
**
-
.4
a+ i5 KEV,
*
*
0
00 000OO00O
() 0 0
00()oooooooo
0
0
.t
0
e
A A
AA A
A
A&
A
AA
A
A AA
LA
A
-1
A A
AA
A
0.8
A
LA
AL
A
A
A
A
A
A
A
0 .5 4
0.5
0 .4
0 3-
C
-
C0.]
;
I.0E+0G
3 .E+
Os
I
I
i I
I.GE+
5.GF+o~S
'.LE+09
FREQUENCY
Fig. 4d
I
, .COF 1
I
2.14 MM : THETA=90 DEG, TE=1 KEV
K.B RNGLE
: RR (80)
(85)
*
DEG
-10
/
*
*
'I
/
-*
12
*~.
-~t t
-A-
~ * * * * * * *
*
* -*
A
A
A
£
A
A
A
-13
A
A
*
-1
A
4
-16-
-1-
-18
-
-19
-20
I
I
.E+09
3 .OE+09
5 -GE +09
FREQUENCY
Fig. 5a
I
7 .0E+09
9 .OE+09
2.14 MM : THETA=90 DEG, TE= 10 KEV
K. .5 RNGLE
(80)
: qR
DEG
(85)
,
-11
/
\
*
/
/
*
*'
*
*
*
A
*
*
~'
A
S
C -12
A
A
-
A
A
A
A
T
T
A A
A
A
AA
A
A
*
A6
A
A
A
A
A
A
F
N
A
A
A
A
A
*
A
L
0
G
A
A6
0
A
-13
-
*
A
*
-14
I
I.OE+09
3 .E+09
I
5 .0E+9
FREQUENCY
Fig. 5b
I
7 .OE+09
I
9 .E+09
2.14 MM : THETA=90 DEG, TE=15 KEV
K.S RNGLE
: qR
(80)
,
(85)
am
DEG
-4
*
*
tt*
4
A
AA
A
AA*
AA
AA
s
C -12
A
I
*
A
-
A*
AA
*
AA
A A*
*
A
*
A
F
N
*
A
*
*
A
*
L
0
G
A
A
1
0
*
A
-13
*
-14
I
1.GE+09
3 .OE+09
5.GE+09
FREQUENCY
Fig. 5c
7 .OE+09
I
9 .OE+09
2.14 MM : THETA= 135 DEG, TE=1 KEV
K.B RNGLE
:
R (80)
(85i
,
DEG
-9
-10
-11 -
S
C
*
A
T -12
T
AA
A
A
A
bAA
-13
--
A
*
-14
4
--
1
*
I
--6
1*
'I
'I
Il
*
-17 -I
I
.E+09
3 .E+09
7 .OE+09
5 .E+09
FRE QUENCY
Fig. 6a
9.OE+09
1.IE+10
2.14 MM : THETA= 135 DEG, TE= 10 KEV
K.B RNGLE
R (80)
:
,
(85)
*
DEG
-1i
****
A
AAAA
AA
A A A A
A A A A A A
*A**
A\A
A
A
A A
*
AA
*
A
A
A
A
A
*
A
A
F
N
A
*
-13
A
L
0
G
*
4
*
-14
-
-15 II
1.OE+09
I
3 .OE+09
I
I
I
I
I
I
I I
I
5.CE+09
7.OE+09
FREQUENCY
Fig. 6b
I
I I
I
9 .GE+ )9
I
1.
1E
1 IE+10
2.14 MM : THETA=135 DEG, TE=15 KEV
K .B RNGLE
-10
-
-11
-
S
C
A
T
T
(80
: qR
, w*
(85)
DEG
**
*******
**
-12
L
0
G
*
*
it 14
*
*
A
A
*
-13
*
*
-14J
I
i.OE+09
I
I
I
I
3 .E+09
II
I
I
I
I
II
5 .E+09
I
I
7.GE+ )9
FREQUENCY
Fig. 6c
9.OE+09
I . iE+10
2.14 MM; K.B ANGLE=75 DEG ; TE=1 KEV
FORWARO SCRTT. RNGLE
: 90 DEy (RR),
135 DEG (*a;
-1 I I
*
*
S
-12
-
A
T
T
--13 -
-14
I
1.GE+09
3 .E+09
5 .E+G
9
FREQUENCY
Fig. 7a
7.GE+C9
9 .CE+09
2.14 MM
K.B ANGLE=75 DEG; TE=10 KEV
FORWARD SCRTT. ANGLE
-11
90 DEG (RR),
135 DEG (am)
-
*
S
-12
-
A
T
I
*A
*
*
A*-*-*
*
**
*
*
F
N
L
0
G
A
A
A
1
0
-13 -
A
A
A
-14
I
1 .OE+09
3 .OE+09
5 .OE+09
FREQUENCY
Fig. 7b
I
7 .OE+09
I
9 -OE+09
2.14 MM ; K.B ANGLE=75 DEG; TE=15 KEV
FORWARD SCRTT- RNGLE : 90 DEG (RR), 135 DEG (ow)
-10
-
-11
-
*
S
C
A
T
T
*
F
N
* * * ***
*
-12
***
*****--
-
L
0
G
A*
1
u
A
A
A
-13 -
A
A
-14
I
I .E+09
I
I
i
I
I
i
I
I
j
I
3 .OE+09
5 .CE+09
FREQUENCY
Fig. 7c
7 .OE+09
9 .OE+09
2.14 MM; K.B ANGLE=45 DEG ; TE=1 KEV
FORWARD SCRTT. ANGLE
90 DEG (AR),
135 DEG (a*)
C -12 T*-
A
T.
N
*
L
G
-13
-14I.0E+09
3.0E+09
5.0E+09
FREGUENCY
Fig. 8a
7.GE+09
9.CE+09
2.14 MM ; K.B ANGLE=45 DEG; TE=10 KEV
FORWRRD SCRTT. RNGLE
: 90 DEG (RR),
135 DEG (mm)
*
S
C -12
A
T
T
**
**A
F
N
*A
*A
*
*
L
0
G
A
A
-14
I
1.OE+09
I
I
I
I
3 .OE+09
I
5 .OE+09
FREQUENCY
Fig. 8b
I
I
7 .OE+09
I1
9 .OE+09
2.14 MM ; K.B ANGLE=45 DEG; TE=15 KEV
FORWARD SCRTT. RNGLE
90 DEG (RR),
135 DEG (**)
-10 -
-11
-*
S
C
*
A
T
T
F
N
-12
-
L
0
G
I
A
A
A
-13
A
-
A
A
-14
I
I.E+09
I
3 .GE+09
5.GE+09
FREQUENCY
Fig. 8c
7 .OE+09
9 .OE+09
2.14 MM; K.B ANGLE=0 DEG; TE=1 KEV
FORWARD SCRTT. RNGLE
90 DEG (RR),
135 DEG to#)
-11
S -12
T
*
F
N
L
G
C
0
-13-
-141.OE+09
3.OE+09
5.OE+09
FREQUENCY
Fig. 9a
7.OE+09
9.OE+09
2.14 MM; K.B ANGLE=O DEG ; TE=10 KEV
FORWRRD SCRTT. ANGLE
135 DEG (w*)
; 9C DEG (RA),
*
*
S
-12
**
**
A
T
T
A
A*
F
N
*
***
a
AA
L
0
G
**
'
*
**
AA
I
AA
-13
-
A6
i
I
1.OE+09
I
I
I
3 .OE+09
I
I
I
I
I
5 .0E+09
FREQUENCY
Fig. 9b
I
1
II
1
7 .OE+09
I
I
9 .OE+09
2.14 MM; K.B ANGLE=O DEG ; TE=15 KEV
: 90 DEG (RR ) ,
FORWARD SCRTT. ANGLE
-10
-
-11
-
S
C
A
T
T
135 DEG -(tsi
'4
*
F
N
-12
*
**
L
0
C
A*'-
A
*
1
0
A
A*
AA
A
A
A
A
A
-14
-
I
1.GE+09
3 .OE+09
5 .0E+09
FREQUENCY
Fig. 9c
7 .OE+09
.
9 .0E+09
MM SCATT. AT 45 DEG
TE=10 KEV
MM (m)
: 2.14
K.B=75 DEG
,
,
5 MM (RA)
-fl-A
7
-~
A
A
11
-1
A
N
S
C
*
N
A
A
T
*,
N
N
F
N
N
A
-12 ~
L
0
G
0
~1
A
-13
1 .GE+09
I
2 .OE+09
3 .E+09
FREQUENCY
Fig. 10a
I
I
I
I
4 .OE+09
MM SCATT. AT 45 DEG
TE=10 KEV
: 2.14 MM
K.B=45 DEG
5 MM (RR)
(mE)
-11-~
1
A
A
N
A
*
S
N
N
*
A
A
T
T
"I
*
A
11
F
N
A
N
-12
L
0
G
A
0
-13
1 -OE+09
-2 -.E+09
3 .OE+09
FREQUENCY
Fig. 10b
4 GOE+09
K.B=75 DEG
MM SCATT. AT 135 DEG
: 2.14 MM
TE=10 KEV
5 MM (RA)
(xx'
*
-11
AA
*
*
*
****
*
S
A
T
****
*
*
*
**
**
*
A
A
F
N
*
A
A
A
L
0
G
0
-14
-15
-
I
.
i
I
I .OE+09
3 .0E+09
I
~AI
I
I
I
I
5 .E+09
I
I
I
7 .OE+09
FREQUENCY
Fig. Ila
I
I
I
I
9 .E+C
9
1 .IE+10
MM SCATT. AT 135 DEG
-11
-
K.B=45 DEG
5 MM (RR)
: 2.14 MM (**)
KEV
TE=Il
*
A
*~
-12
*
A
**
i***
**
t
*
*
A
-14
-
-15
A
I
I -OE+09
I
I
i
I
I
3 .OE+09
I
I
I
I
5 .OE+09
I
I
I
7 .-E+ 39
FREQUENCY
Fig. lib
9.AE+09
I -lEt10
MM SCATT. AT 135 DEG ,K.B=O DEG
: 2.14 MM (')
TE=10 KEV
, 5 MM
IRR)
*
*
*
-12
AA
A
*
A
S
C
A
T
T
**A
*
A-~
A
**
**
**
F
N
*
t
-13
*
-~
L
0
G
A
-i
*1
2
0
I
-
-14
-
A
E0
1 .OE+09
3 .OE+09
5 .E+09
7 .OE+C
FREQUEN CY
Fig. I c
9 -OE+09
1 . E+1C
MM SCATT. AT 135 DEG
TE=1
KEV
2.14 MM (E
)
,
K.B=75 DEG
5 MM (RR)
-
-11
-12
T -13
N
L
0
G
-14
0
-15
-16
1.OE+09
I
3.GE+09
5.OE+09
7.OE+09
FREQUENCY
Fig. l id
9.GE+09
1.!E+i0
TE EFFECTS AT 45 DEG, K.B=75 DEG
2.14 MM
1.0
*
*
0
, TE: **
15 KEV, 00 10 KEV, RR 1 KEV
*
*
o
*
o
0.9
*
-
0*
*
0
*
*
0.8
*
0
*
N
0
R
M 0.7
A
L
*
*
0
0
A
I
0
E
D 0.6
0
U
N
M
A 0.5
G
A
-1
M
A
G 0 .4
A
L
P
H
A 0 .3
s
A
A
0.2
A
0.1-
A
A
A
A
0.0
-
1.GE+09
~1
I
2 .0E+0 9
I
I
I
I
3. GE+0 9
FREQUENCY
Fig. 12a
I
I
I
4.OE+-09
I
A4
I
I
5.GE+09
TE EFFECTS AT 45 DEG, K.B=60 DEG
, TE:
2.14 MM
1.1
-
1 .0
-
a* 15 KEV, 00 10 KEV, RR I KEV
*
a
*
*
0
0
*
a
A
0.9
*
*
*
0
*
0
0
*
0
A
N
0
R 0.8M
A
L
z 0.7
*
0
A
A
-
E
D
A
N 0.6
A
A
A
0.5
A
A
G
A
A 0.4
L
P
H
A
S 0.3
-
A
'A
A
-
It
0.2
-
0.1
-
0.0 1 .OE+09
I
I
j
2 .OE+09
I
3 .OE+09
FREQUENCY
Fig. 12b
4 .E+09
I
5 .E+09
TE EFFECTS AT 45 DEG, K.B=O DEG
2.14 MM , TE:
1 .04
15 KEV,
m
-
1 .03
1 .02
1 .01
1 .00
0.99 0.98
0-.97
*
**
*
0
0
0
0
00 10 KEV, AR
::
0 0
*
0
0
1 KEV
* * *
0
0
0
0
0
A
A
-
0.96
A
0.95
0.94
0.93
0.92
0.91
0.90
0.89
0.88
0.87
A
A
A
A
0.86
0.85
0.84
0.83
A
A
0.82
0.81
A
0.80
0.79
-
0.78
-
0.77
-
A
A
0.76
0.75
0.74
A
0.730.72
-
A
0.71+
I.IE
I .OE+09
I
I
I
I
2 .OE+09
3 .OE+09
FREQUENCY
Fig. 12c
4 .OE+09
5 .E+09
TE EFFECTS AT 45 DEG, K.B=75 DEG
5.00
I -0 -
MM
, TE:
x*
15 KEV,
00 10 KEV,
AR
I KEV
*
0
*
0
*
*
0.9
0
*
*
*
0 .8
*
0
*
0
A
0
0.7
0
0.6 A
0.5
-
0.4
A
0.3
-
0.2 -
A
A
0.1 -
A
A
A
0.0
I .OE+C
9
1 .2E+09
I .4E+09
I .6E+09
FREQUENCY
Fig. 12d
I .8E+09
2 .E+09
TE EFFECTS AT 45 DEG, K.B=60 DEG
5.00 MM
1 .0
-
TE:
,
*
0
mo 15 KEV,
*
00 10
*
AR I KEV
KEV,
*
*
0
*
0
0
A
0.9
--
0.8
-
*
0
A
0.7
-i
0 .6
A
-i
A
A
0.5
A
A
0.4
A
0
.3-4
0.2
-
0.1
-
0.0 I
I .E+09
I .2E+09
I .4E+09
I .6E+09
FREQUENCY
Fig. 12e
I
I
I
I
1 .8E+C 9
2.GE+09
TE EFFECTS AT 45 DEG, K.B=O DEG
5.00 MM , TE: ** 15 KEV, 00 10 KEV, RR 1 KEV
1 .01
1 .00 -
**
o
a
*
*
a
0
*
*
0
*
0
*
0
0
0.99
0.98
*
A
0.97
0.96
A
N 0.95
0
R 0 .94
M
A 0.93
A
0.92
E
D 0.91
~
A
U 0.90
N
m 0.89
A:
G 0.88
M 0.87
A
-
A
-
G 0.86
A 0.85
A6
p
H 0.84
A
S 0.83
A
0.82
0.81
0.80
A
0.79
0.78
0 .77
A
0.76
I
I .OE+09
I
I
I
I .2E+09
I
III
I
I
I
I
I .4E+09
I
1 .6E+09
FREQUENCY
Fig. 12f
I
I
I
I
I
1 .8E+09
I
I
1
1
2 .OE+09
TE EFFECTS AT 90 DEG, K.B=75 DEG
2.14 MM
1.2
,
TE:
15 KEV,
*m
KEV,
*
*
*
I KEV
**
*
00000
*
0 0
1 .01-0
-
0
0
*
*
0
A
0
0
-
0
A
*
*
*
*
0
*
0
0
0
*
0
*
0
A
-
*
0
o- -
*
0
A
E
D 0.7U
N
m
A 0.6
G
RA
-
1.1
N
0 0.9
R
m
A
L 0.6
00 10
*
0
0
A
*
0
0
0
A
-
0
0
A
M
A 0.5
G
0
A
A
A
L
P 0.4 H
A
A
A
S
0.3
A
-
A
A
0.2 -
A
A
A
A
A
0.1 -
A
A
A
A
A
0.0
A
A
A
A
-
I.OE+09
-I------
--.....
~1~-
3.OE+09
5.OE+09
FREQUENCY
Fig. 13a
7.OE+09
9.OE+09
K.B=60 DEG
TE EFFECTS AT 90 DEG.,
KEV,
2.14 MM , TE: *a 15 KEV, 00 10
RR 1 KEV
1.2
1.1
-
0 0
0 0
00
*
* *0*
00000000000
00000000
0
1 .0
0
000
N
0 0.9
R
M
A
L 0.8
-
AA
-
AA
AA
E
D 0.7
U
N
m
A 0.6
C
AA
AA
-
A
C 0.5
A
A
L
P 0.4
A
A
A
-
A
H-
A
A
S
A4
A
0.3
-
A
A
A
A
A
0.2-
0.1
0.0 -
I .OE+0 9
a
a
i
3.OE+0 .9
5.OE+09
FREQUENCY
Fig. 13b
7 .OE+09
9 .OE+09
TE EFFECTS AT 90 DEG, K.B=45 DEG
ow 15
, TE:.
MM
2 .14
RR I KEV
KEV, 00 10 KEV,
1.3
1 .2
*
0
0
0
0
00
0000
0 00000
0
N 1 -0
0
R
M
A 0.90 8
*
'
A
A
A
A
A
A
L
A
A
E
D
A
A
A
U
N
M
A 0.5
A
A
A
A
1
A
A6
C 0.61
M
A
G
A
A
A
A
0.51
A
A
A
A
A
L
P
H 0.4
A
S
0.3
0.22
0.0 -
II
1.OE+09
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
3 .OE+09
5 .OE+09
FREQUENCY
Fig. 13c
7 .OE+09
I
I
I
I
I
9 .E+09
TE EFFECTS AT 90 DEG, K.B=0 DEG
2.14 MM
, TE:
00 10 KEV,
x* 15 KEV
RR 1 KEV
1 .3
1.2
*
0
*0
*i.0
-.
*
A
*
*
000000o00000000000000ooo
*
0.1
*
0
*0
A
A
AA
A
AA
A
0.9
A
A
A
A
-
A
A
A
A
A
A
A
0.8
A
A
A
A
A
A
A
A
C .7
A
A
0.6
0.5
0.4
0.3
0.2
0.1
0.0
'I
I
.E+09
3 .OE+09
5 .OE+09
FREQUENCY
Fig. 13d
I
7 .OE+09
I
9 .OE+09
TE EFFECTS AT 90 DEG, K.B=75 DEG
5.00 MM , TE: ** 15 KEV, 00 10 KEV, RR 1 KEV
1.I
*
1 .0
0
- S0
00
00 *
o
0.9
0
o
*
*0
0
A
*
0
-
*
0
*
*
*
0
N
0
R 0.8 M
A
L
z 0. E
D
*
0
A
*
0
*
0
*
*
0
*
0
A
0
0
0
0
A
0
U
N0 6
0
16
M1
A
G
A
M 0.5
A
A
A 0.4
L
P
A
H
A
-
S 0.3
A
A
0.2 -
A
A
A
A
0.1
-
A
A
A
A
0.01.OE+09
I
I
I
I
A
A
A
A
A
A
A
I
2 .0E+09
3 .OE+09
FREQUENCY
Fig. 13e
4 .0E+09
TE EFFECTS AT 90 DEG, K.B =60 DEG
5.00 MM
0
1 .0
00 10
A* !5 KEV,
0*
*
0
0
, TE:
*
0000000000
*
*
KEV,
RP
I KEV
*
e
*
0 0
c
- 6
A
A
0 0
A
0.-9
-
A
A
A
0.8
-
A
A
A
A
A
0.7
A
A
A
A6
A
0.5
A
A
0.4
A
A
A
A
0.3 -
A
A
A
0.2 -
0.1 -
0.0 -
1 .OE+09
j
I
2
4 -QE+09
.OE+09
FREQUENCY
Fig. 13f
TE EFFECTS AT 90 DEG, K.B=45 DEG
5.00 MM
.
1 .0
, TE: ** 15 KEV, 00 10 KEV. AR I KEV
-
-
a
A
A
A
A
A
A
A
A
N
0
R
m
A
L
A
A
A
A
A
A
z
0.7
A
A
E
D
A
A
U
N 0.6
m
A
G
A
A
M 0.5
A
G
A
L
P
H
A
S 0.3
0.2
0 1
--
0.0
I
1.OE+09
I
I
i
i
~
I
I
I
i
I
I
I
3 .OE+09
2.CE+09
FREQUENCY
Fig. 13g
I
I
4 .OE+09
TE EFFECTS AT 90 DEG, K.B=0 DEG
5.00 MM
15 KEV,
1*
TE:
,
00 10 KEV,
I KEV
AR
1 .10
1 .09
**
1 .08
1 .07
1 .06
1.05
1 .04
*
*
-
(%oO
O
QQo
0 0 0 0 0
0 0
00 0 J
0
1 .03
1 .02
*0
0
1 .01
1 .00
A
A
0.99
0.98
0.97
0.96
bA
A
A-
0.95
0.94
0.93
0.92
0.91
0.90
0.89
0.88
-
A
A
-
A
-
A
A
0.87
0.86
0.85
A
A
A
0.84
0.83
A
0.82
0.81
A
A
0.80
0.79
A
0.78
0.77
0.76
0.75
0.74
-
0.73
-
0.72
-
A
A
A
-
A
A
0.71
0.70
1.OE+09
A
3 .OE+09
2 .OE+09
FREQUENCY
Fig. 13h
4 .OE+09
TE EFFECTS AT 135 DEG, K.B=75 DEG
, TE: m* 15 KEV. 00 10 KEV, RR I KEV
2.14 MM
1 .5
1 .4
**
1 .3
*
*
*
*
1 .2
*
*
*
*
00
0
0
0
*
*
0
0
0
A
0.9
*
000
0
1 .0
* '4.
0000000
* .0
1.1
*
0
0
A
0
0
A
0.8
A
A
0
0
0
0
0
0
0
A
0.7
0
*
0
0
A
A
0.6 A
A
0.5
A
A
A
0.4
A
A
A
0.3
A
A
0.2
A
-
A
A
0.1
A
-
A
0.0
E+
1.OE+09
3 .0E+09
5 .OE+09
7.0 E+09
FREQUENCY
Fig. 14a
I
I
,
9 OE+09
I
1 .IE+10
TE EFFECTS AT 135 DEG, K.B=60 DEG
2.14 MM , TE:
1 .6
*m
15 KEV, 00 10 KEV, AR
I KEV
-
1 .5
***
1 .4
***
*
-
*
1 .3 00 00o000000000000000000000
*
1 .2
0
*
1.1-
1 .0
0
-
*
A
A
A
A
0.9
A
A
A
A
A
0.4
A
-
A
A
A
A
0.7-
A
A
A
A
A
0.6-
A
A
A
A
A
0.5-
A
A
A
A
A
0.4-
A
A
0.3
-
0.2
-
A
0 .1
0.0
I
I .OE+09
3 .OE+09
5 .OE+09
7 .OE+09
FREQUENCY
Fig. 14b
9 .OE+09
I
I .iE+10
TE EFFECTS AT 135 DEG, K.B=45 DEG
2.14 MM , TE: as 15 KEV,
1
1 .5
-
1.4
-
N
0 1 .2
R
M
A
L
E
D
1 KEV
A
.6
1 .3
I
00 10 KEV,
0 00000
000000000
000
*0
0
-
0
1 .0
-
*AA A
A
AAAA
U
N
M
A 0.8
G
A
A
AA
A
-
A
A
A
A,
As
A
A
M 0.7
A
G
&
£
A
A
A
A 0.6
L
p
H 0.5
A
S
A
A
A
A
A
0.4
0.3
0.2
0.1
-
i
I .E+09
3 .E+09
i
'Ii1
i
5 .OE+09
I
I
I
I
7 .OE+09
FREQUENCY
Fig. 14c
.
i
I
9 .OE+09
i
I
I
I . E+10
TE EFFECTS AT 135 DEG, K.B=O DEG
2.14 MM
,
15 KEV,
TE: o
00 10 KEV,
RR I KEV
1.6 1.5
*************
1 .4
**
0
1 .3
0
*
1-2
00 000 0OO
-.
1 .0
*AA
000000
000000
00 00000 00000000
A A A A A A A A A
A' AL
A
A A
A A
A A
A A
A A
A
0.9
A
0.8
A A
A AA
A A
A A
A
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
-
I
I
1.OE+09
3 .OE+09
5 .OE+09
7 .0E+09
FREQUENCY
Fig. 14d
9 .OE+09
1 .IE+10
TE EFFECTS AT 135 DEG, K.B=75 DEG
, TE
5.00 MM
15 KEV
00
10 KEV,
RR
I KEV
1 .2 *
I .i
*
*
--
* 0 0000000
0
1 .0
*
0000
-
0
A
0.9
0
*
0
0
*
0
0
A
0
0
0
A
*
*
*
*
*
0
0
*
*
0
0
0
0
A
0.7
*
0
A
0.8
*
0
-
0
0
A
0
0
A
0.6
0
0
A
A
0.5
A
0.4
A
A
A
0.3
A
A
0.2
AA
AA
AA
0.1
AA
0.0I
I .E+09
I
I
I
I
2 .E+0
9
.0E+09
FREQUENCY
Fig. 14e
4 .CE+09
5 .E+09
TE EFFECTS AT 135 DEG, K.B=60 DEG
5.00 MM . TE: w# 15 KEV, 00 10 KEV, PR 1 KEV
1 .3 -
1 .2
*
000000000000000000
1.1
00 0000
*
0
0
0000
N i.0
0
R
M
A 0.9
L
A
A
A
A
A
A
A
A
A
L
A
E 0.3
D
A
A
A
U
N 0.7
A
-
A
M-
A
A
G
-
A
A
0.6
A
M
A
G
A
A
A
A
0 .5-
A
L
P
H 0.4
A
S
A
A
A
A
A6
A
A
A
A
0.3
A4
A
A
A
A
0.2
-
0.1
-
0.0
-
I
I
I .OE+09
2 .OE+09
3.GE+09
FREQUENCY
Fig. 14f
I
4 .OE+09
I
5
.0E+09
TE EFFECTS AT 135 DEG, K.B=45 DEG
*a 15 KEV
, TE:
5.00 MM
00 10 KEV,
RR
I KEV
1 .3 -
*
0 0
0 o oo0
oo
*
0
0
o o o o o o0 o o0 o o00oo
0oo o0
0
0
A
A
A
A
0.9
A
A
A
A
A
A
A
0.8
A
A
A
A
A
A
A
A
A
A
A
A
0.6
A
A
A
A
A
A
0.5
A
A
A
0.4-
0.3
-
0.2
0.0
-
I.OE+09
~~1~~~~~~
2.OE+09
I
3.0E+09
FREQUENCY
Fig. 14g
I
I
4.OE +09
5 .0E+09
TE EFFECTS AT 135 DEG, K.B=O DEG
, TE:
5.00 MM
1 .3
am 15 KEV.
00
10 KEV,
RP
I KEV
-
1 .2
*
oo0000000000000000000000
00000000
0
*0
N 1 .0
0
R
ba A A A AAAAA
£
A
M
A 0.9
L
I
A
A
A6
A
A
A6
E 0.8
D
A
A
SA
A
U
A
A
A
A
N 0.7
A
M
A
G
A
- 0.6
M
A
G
A
0.5
L
P
H 0.4
A
S
0.3
0.2
0.1
0.0-
5
i
I .OE+09
I
I
I
2 .OE+09
I
3 OE+0 9
FREQUENCY
Fig. 14h
4
I
4.OE+09
I
I
I
i
5 .OE+09