Ultrahigh magnetic fields produced in a gas .. puff Z pinch

Ultrahigh magnetic fields produced in a gas..puff Z pinch
F. S. Felber, F. J. Wessel ,a) N. C. Wild, and H. U. Rahman b )
JAYCOR, P. O. Box 85154, San Diego, California 92138
A. Fisher
University of California at Irvine, IrlJine, California 92717
C. M. Fowler
Los Alamos National Laboratory, Los Alamos, New Mexico 87545
M. A. Liberman and A. L. Velikovich
Institute for Physical Problems, Moscow J17334, USSR
(Received 14 March 1988; accepted for publication 28 June 1988)
C:0ntrolled, ultrahigh axial :naglletic fields have been produced and measured in a gas-puff Z
pmch. A O.S-MA. 2-cm-radlus annular gas-puff Z pinch with a 3-min repetition rate was
imploded radially onto an axial seed field, causing the field to compress. Axial magnetic field
com~ressions up to 180 and peak ~agnetic fields up to 1.6 MG were measured. Faraday
rotation of an Argon laser (5154 A) in a quartz fiber on-axis was the principal magnetic field
diagnostic. Other diagnostics included a nitrogen laser interferometer, x-ray diodes, and
magnetic field probes, The magnetic field compression results are consistent with simple
snowplow and self-similar analytic models, which are presented here. Even small axial fields
help stabilize the pinches, some of which exhibit several stable radial bounces during a current
pulse. The method of compressing axial fields in a gas-puff Z pinch is extrapolable to the order
of 100 MG. Scaling laws are presented. Potential applications of ultrahigh axial fields in Z
pinches are discussed for x~ray lasers, inertial confinement fusion, gamma-ray generators, and
atomic physics studies.
!. INTRODUCTION
A general method for producing controlled ultrahigh
magnetic fields up to the order of 100 MG has been proposed
recently. I The general method involves imploding a plasma
in which a magnetic field has been entrained. The imploding
plasma compresses the magnetic field to high strengths. This
paper reports details of the experimental test at the 1-2-MG
level and high repetition rate of an embodiment of this method using a gas-puff Z pinch. A brief report has been pllb~
lished earlier.2
In our experiments, a gas-puff Z pinch produces a radially imploding plasma in which an axial magnetic field has
been entrained. First, a seed, or initial, magnetic field is generated in a gas that has been puffed into the region between a
cathode and anode. When a voltage is applied between the
cathode and anode, the resulting electrical discharge ionizes
the gas, producing a plasma. The axial current through the
plasma column produces an azimuthal magnetic field. The
force of the azimuthal magnetic field on the axial-currentcarrying plasma causes the plasma to implode radially. If the
magnetic Reynolds number of the plasma is much greater
than 1, then the axial seed field will be entrained and compressed by the plasma during the implosion. The magnetic
Reynolds number is the ratio of the time for diffusion of a
magnetic field out of the plasma to the implosion time of the
plasma.
UntH recently, the method that has generated the highest magnetic fields in a controlled manner has been explosive
a)
h)
Present address: Department of Physics, U. C. Irvine, Irvine. CA 92717.
Present address: Department of Physics, U. C. Riverside, Riverside. CA
92521.
3831
J. Appl. Phys. 64 (8), 15 October 1968
flux-compression generators,3,4 The highest controlled
fields. 15-25 MG, were produced by this method over twenty years ago. 5 ,6 The explosive generation !l}ethod is ultimately limited to field strengths much lower than 100 MG by
energy requirements and flux diffusion losses. 7 The method
tested here, on the other hand, may be extrapolable to much
higher fields. Fields up to 42 MG were indicated by this
method on a 7.5-MA pulsed power generator. 8
Another major advantage of the gas-puff Z pinch method of producing ultrahigh magnetic fields, which greatly facilitated our experiments, is the high repetition rate. Since
this method involves no explosives or material deformations,
it is possible to produce many successive plasma pinches
without breaking the vacuum.
If the diagnostic measurements are monitored remotely
and the probes survive from shot-to-shot, then the only limitation on the repetition rate of diagnosed experiments is the
repetition rate ofthe Z pinch device. The repetition period of
our megagauss facility was < 3 min, and could have easily
been reduced, if desired. Even the repetition period of much
larger gas-puff Z pinch devices, such as PROTO II at Sandia
National Laboratories or PITHON at Physics International, with an order-of-magnitude higher field-producing
capability, is typically only several hours to one day.
In general, the requirements for producing ultrahigh
magnetic fields may be summarized as speed, high conductivity, and large initial field. Implosions must be fast, of the
order of tens ofns for 100 MG, in order to avoid destruction
of the containment volume by shocks and instabilities. High
conductivity is needed either on the containment surface or
in the volume in order to inhibit diffusion during the implosion. Generally, high initial fields are required because flux
0021-8979/88/203831-14$02.40
© 1988 American Institute of Physics
3831
compressions are typically limited to the order of a few
hundred.
The methods described in Ref. 1 are expected to produce
higher magnetic fields than flux-compression generators.
The methods involve entraining an axial magnetic field in a
plasma, and thea imploding the plasma. The plasma will be
imploded under conditions of high magnetic Reynolds number (Rm;p 1), so that diffusion is impeded. The plasma win
be at low pressure, however, so that compression is unimpeded by the plasma. Moreover, the axial field can grow
much stronger than the azimuthal field, because the pinch
overshoots the equilibrium radius during implosion.
In the experiments reported here, the gas-puff Z pinch
uses a fast valve to produce a supersonic gas puff. A seed field
or initial magnetic field is created by a Helmholtz coil.
A schematic illustration of a gas-puff Z pinch with a
frozen-in axial magnetic field is shown in Fig. 1. The nozzle
for the gas puff ejects a gas into the region between the cathode and the anode, and the axial field coils produce an axial
magnetic field before the electrical discharge occurs. The
electrical discharge creates the azimuthal field B",. The radial implosion compresses the axial magnetic field B z to high
field strengths. With this method it is the azimuthal magnetic field that is in effect compressing the longitudinal magnetic field through the plasma medium. Under ideal conditions
the plasma has a pressure very much lower than the azimuthal magnetic field pressure, however, so that it does not
significantly affect the implosion. The only purpose of the
plasma is to entrain the longitudinal magnetic field so that it
can be compressed by the azimuthal field. Because one field
is compressing another, the limitation on magnetic field
strength that can be achieved is set by the magnitude of the
azimuthal magnetic field that can be created, which is limited in turn by the current that can be driven through the
plasma. It will be shown in the next section that using this
method of magnetic field compression produces fields about
two-four times higher than just the field produced by the
current alone. With sufficient electrical pulsed power available, the limitation on field strengths that can be achieved by
this method may be mainly set by the magnitude of the initial
field that can be created by conventional means before compression. The next section presents some of the models that
were useful in predicting and understanding the performance of the gas-putT Z pinch in producing ultrahigh magnetic fields.
The experimental test of the gas-puff Z pinch method of
producing controlled ultrahigh (> 1 MG) magnetic fields
was performed at the gas-puff Z pinch facility at the U niversity of California at Irvine. This facility is described in Sec.
III. The current into the Z pinch had a quarter-period rise
time of about 1 ps and a maximum of S 0.5 MA The highest
magnetic fields that could be measured with a Faraday rotation diagnostic ranged lip to 1.6 MG, and were produced
from a 9-kG seed field, implying that field amplifications of
more than 100 were observed. The experimental results, including evidence of unusual stability in gas-puff Z pinches
with axial magnetic fields, are presented in Sec. IV. Discussion of the results is presented in Sec. V, and conclusions and
potential applications in Sec. VI.
3832
J. Appl, Phys., Vol. 64, No.8. i 5 October 1988
AXIAL
FIELD
COilS
aAS
PUFF
NOZZLE
FIG. I. Gas-puff Z pinch with frozen-in magnetic field.
II. MODELS
Several simple analytic models have been used to support the experiments on ultrahigh magnetic field production. These include a snowplow model and self-similar models of annular and solid Z pinches.
Modeling magnetic fields entrained in plasmas depends
critically on the magnetic Reynolds number, which serves as
a figure of merit for the suitability of the plasma for producing ultrahigh fields. For purposes of magnetic field compression, the magnetic Reynolds number
Rm == TdlTimp
(1)
is the ratio of the characteristic diffusion time of a magnetic
field out of a compression volume to the implosion time of
the volume. The characteristic diffusion time is
(2)
where L is the thickness of plasma confining the field, and
(3)
is the diffusivity. The usual expression for conductivity
gives 9
Rm ;::; 1.2 X 1O- 6 L 2T!/2(eV)/ATimp
,
(j
(4)
where A is the Coulomb logarithm, and T. is the electron
temperature in eV. Since R"., ;p 1 is desirable for efficient
compression, the best plasmas are hot, thick plasmas imploding rapidly_ This consideration suggests that entraining
magnetic fields in imploding plasmas is a method of field
compression that works better at higher field strengths for
which higher powers are needed for implosion, because plasmas are hotter and confine the fields better.
The experiments reported in this paper were power limited, and were performed near the threshold at which good
field compression is possible. The maximum currents of
$ 0.5 MA could only produce an annular plasma sheath of
thickness L;::; 1-2 mm and temperature To < 10 eV during
most of the implosion. Moreover, the implosion time was
long compared to large pulsed power machines. The
quarter-period rise times of the current of ;;:; 1 f-ts gave an
implosion time T,mp ;C; 100 ns. Thus, R", for these experiFelber et al.
3832
ments is estimated from Eq. (4) to be
Rm $1-3.
As might be expected with this value of Rm, good compressions were achieved only with difficulty. Analytical and numerical calculations of magnetic diffusion in implosions
with finite R m were presented in Refs. 10 and 11. N evertheless, in some shots field compressions greater than 100 were
measured. We reemphasize that these high compressions
can be more easily achieved as this method is scaled to higher
electrical powers and higher magnetic field strengths, because R m will be higher.
The snowplow model assumes infinite conductivity and
infinite R m , thereby overestimating peak compressed fields.
Nevertheless, it provides a simple and reasonably accurate
upper bound on the maximum fields that can be expected
under a wide variety of implosion conditions. The usual
snowplow model 12 is modified here to describe the implosion
of a cylindrical plasma shell pinching an axial magnetic field.
The axial current in the plasma shell generates an azimuthal
magnetic field outside the shelL The azimuthal magnetic
field causes the shell to implode and compress the axial magnetic field in the center.
The equation of motion of the plasma shell is
Md2~ =
dt -
_2177l[_1_(2I(I»)2
8ir
cr
+ E~
817'
(1- ,46)],
r
(5)
where M is the mass of the shell, ris the radius, I is the length,
and 21frl is the area. At the initial radius, r = r o' the axial
magnetic field Eo is equal inside and outside the shell.
In dimensionless variables the equation of motion is
d R2 = ~ ( - f 2 U)
dT
R
where we have defined
+ ~ (! _
T=.t /t""
f.l=.M /1,
2
R=.r/ro,
a=:I;"t!,/pc2 ri,
R2
R
4»),
(6)
b=. (crollo/2Irn)2,
and have let I=.Imf( T), where 1m is the maximum current
occurring at t = t m . The parameter a is a measure of the
external force on the pinch per unit mass, and b is a measure
of the degree to which the axial field impedes the implosion.
The qualitative behavior of the pinch can be deduced by
examining Eq. (6). Substantial compression of the axial
field will occur only for b < 1. The pinch will implode relatively unimpeded until R ;:::;b l12 , when the outward pressure
of the axial field becomes comparable to the inward pressure
of the azimuthal field. The turnaround radius for constant
current is shown in Fig. 2. In an earlier paper13 it was shown
that a self-similar pinch with a Gaussian density profile has a
turnaround radius given by the dashed curve in Fig. 2, if
kinetic pressure effects are ignored. Figure 2 shows that in
the high compression regime, a diffused self-similar pinch
has the same turnaround radius as the cylindrical-shell
snowplow pinch.
As an example, a constant current of 0.4 MA through a
cylindrical shell with an initial radius of 1 cm will cause a
hundred-fold compression of a 17-kG magnetic field to 1.7
I
I
I
I
/
/
I
I
I
YLiNORICAlSHELL PINCH
0.1
b={ Cf oBo/2!m)2
FIG. 2. Normalized turnaround radius vs b ofa cylindrical-shell, COllstantcurrent pinch (solid curve) from Eg. (6) and ora Gaussian-density-profile,
constant-currellt pinch (dashed curve) from Ref. 12.
For a sinusoidal current pulse shape, such as
1 = 1m sine 17'T /2) ,
which is a close approximation to our experiments, the turnaround radius has a shallow minimum as a function of a, as
shown in Fig. 3. At least for a range of values of b between
about 0.01 and 0.1, the minimum turnaround radius occurs
for a;:::; 4. For smaner values of a, the shell is too massive and
slow, reaching turnaround well after the peak of the current
pulse and the pinching force has passed. For larger values of
a, the shell lacks inertia, and bounces hard off the compressed magnetic field early in the current pulse. Over the
range of values of b shown in Fig. 3, the solutions of Eq. (6)
show that the optimal implosion time to reach minimum
radius is within a few percent of T;:::; 1. !.
Figure 4 shows the time history of a radial compression
by a factor of 10, corresponding to an axial field compression
of 100, according to the snowplow model. To relate this example to the experiments, we take = 2 em as an initial
radius, and 1m = 0.47 MA as the peak current. Then
b = 0.045 implies Eo = 10 kG by the definition of b. Therefore, the snowplow model suggests that a lO-kG axial field
can be compressed to as high as 1 MG under the conditions
of this example.
Because the effective implosion time is much shorter
than the quarter-period rise time, the magnetic Reynolds
number is improved. Also, the pinch bounces hard off the
axial field owing to the strong inverse radial dependence of
'0
0.25 f
0.2
~b= 0.0971
0.15
,,---
O.()E.125
-j
(L 1
0.05
EG. 3. Normalized turnaround radius vs a for several values of b.
MG.
3833
I
GAUSSIANDENS!TY PINCH
J. Appl. Phys., Vol. 64, No.8, 15 October 1988
Felber et sl.
3833
1.0
1.0
E
..........
...
0
JI
0.8
II
c::
IZ
W
IE:
fh
.J
0.8
a:
:!:)
u
0
-<
a:
Q
w
N
Q
0.4
0.2
:Ii
c::
0.2 0
<
-<
::E
0
N
..l
...J
IX:
W
0.4
Z
a==4
Z
b=O.045
0
0
0.4
O.S
1.2
NORMALiZeD TiME, T=t/to
0
FIG. 4. Normalized radius and normalized current vs time for a = 4,
b = 0.045.
the axial field pressure. In the experiments a short burst of x
rays is observed at the bounce, showing that some thermaIization of kinetic energy is occurring upon stagnation of the
pinch on the axial field.
Using the solutions of Ref. 13, we find that if the kinetic
pressure is negligible compared to the axial field pressure,
then the normalized effective radius at ofthe pinch at turnaround is
In J..a; = (\
CO
oBO)2 (J.. - 1) .
21
a;
(7)
This turnaround radius was plotted as the dashed line in Fig.
2. It agrees with the snowplow model in the compression
regime of interest.
An axial magnetic field is compressed to achieve higher
field strengths than could be achieved with an azimuthal
field alone. For a diffuse pinch, the effect is even greater.
From the self-similar solution of Ref. 13, the peak value
of B¢ is found to occur at r = 1.24a, and to have a magnitude
of only
(8)
or about half the value for an annular pinch at radius a. The
reason is that a fraction of the axial current flows outside the
radius of peak azimuthal field and does not contribute to the
peak field.
From Eq. (7), the peak value of the axial field is given by
(9)
The ratio of fields in Eqs. (8) and (9) shows that for diffuse
pinch~s the maximum axial field can be about four times
greater than the maximum azimuthal field.
Because the pinch in our experiments was annular, the
magnetic Reynolds number is more closely approximated by
the snowplow model than the diffuse-pinch model. The solutions of both the snowplow and diffuse-pinch models indi3834
J. Appl. Phys., Vol. 64, No.8, 15 October 1988
cate that the radius of the pinch oscillates after imploding to
some turnaround radius. The turnaround radii for both
models and for constant current were shown in Fig. 2 (A
normalized turnaround radius greater than 1 in this figure
indicates that the current is not high enough to cause the
plasma to pinch from its initial radius.) These oscillations,
which were discussed in Ref. 13, are a consequence of the
combined effects of the inward pinching pressure of the azimuthal magnetic field and the outward pressure of the axial
field.
The stability analysis of Ref. 13 showed that in the absence of an axial magnetic field, the period of radial oscillations of the pinch is much greater than the e-folding time of
the fastest growing sausage instability mode, so that it was
unlikely to observe oscillations of a pinch without an axial
magnetic field. The experiments confirm this by showing the
pinch without an axial field going unstable and developing
hot spots after stagnating on axis. Noting that the period of
radial oscillations of a pinch is shortened by an axial field,
however, Ref. 13 concluded that "the oscillations of a selfsimilar pinch should be observable before the pinch becomes
distorted if B z is comparable to D.p." The experiments also
confirm this earlier conclusion, as will be shown in the following sections. Our experiments show that sufficient stability is given to an annular pinch to observe several oscillations, or bounces, of the pinch when the final axial field is
comparable to the final azimuthal field. In practice, this criterion suggests that to give good stability to a pinch about 5
kG of initial axial field should be provided for each megampere of current. For example, in our experiments at < 0.5
MA, a few kG ofinitial axial field were all that was needed to
observe several stable bounces of the pinch for certain gases.
To produce a tenfold radial compression of a Z pinch,
which corresponds to a hundredfold amplification of an axial field, Eq. (9) shows that for either an annular or a diffuse
Z pinch, the required current is
/z(23MA)
(~)(
1 mm
D
100 MG
),
where at' is the final radius and B is the final magnetic field.
Th~ electrical power and energy scale as f2 for rapid
implosions. The stored electrical energy required to produce
a final magnetic field amplitude B in a volume having radius
0, therefore, is
EZ(6~~)C ~mrCooBMGr
The efficiency from stored electrical energy to axial magnetic field energy is expected to be of the order of 20%. As an
example, to produce a lOO-MG field in a pinch of radius 1
mm requires a current of order 20 MA and an electrical
energy of order 6 MJ/cm oflength delivered to the pinch.
These energies and powers might be produced more readily
in the future with capacitor banks and fast opening switches
than with water-line generators.
In both the diffuse-pinch and snowplow models, the
compression of the Z pinch was assumed to be adiabatic,
meaning that shock losses, radiation, end losses, and thermal
and electromagnetic diffusion losses were ignored. A magnetohydrodynamic a.!lalysis was performed to learn the effects
Felber et al.
3834
of shocks on the liner compression. 14 Radiation, end losses,
and thermal diffusion losses are plasma energy sinks. But
since the plasma energy is much less than the field energy,
the principal effect of these losses will be to depress the temperature and the magnetic Reynolds number, but not so
much as to appreciably affect field confinement. Electromagnetic diffusion losses are negligible as long as Rm » 1 in
the plasma. Numerical computations of the magnetohydrodynamics of a closely related imploding plasma model with
axial magnetic field show that good confinement and com~
pression can be achieved even with Rm ;;;: 1. 10
m. EXPERIMENTAL APPARATUS
All experiments were performed on the UCI gas-puff Z
pinch facility. IS In the gas-puff Z pinch device, a slidinghammer fast-gas valve l6 injects an annular gas puff between
two vacuum electrodes driven by a capacitor bank (12 pF)
and parallel plate transmission line (total system inductance
;:::: S8 nH, quarter-period rise time:::::; 1.25 #s). Normally the
bank is charged to 32 kV to deliver a peak current of 470 kA.
To accommodate the magnetic field experiments, the Z
pinch was modified with: (1) Helmholtz coils to inject the
axial, seed magnetic field; (2) high-resistivity anode-cathode (A-K) electrodes to enhance rapid field diffusion into
the A-K gap; and (3) diagnostics to measure the high magnetic fields. Figure 5 displays a schematic illustration of the
experimental apparatus. Figure 6 displays the experiment
diagnostics.
The Helmholtz coils consist of two coils (7 turns each,
No. 18 gauge formvar-insulated copper wire, l.S-cm coil
spacing) wound on a G-1O form (lO.4-em 00) and reinforced with fiberglass cloth and epoxy. Machined into the G10, between the coils, were several slots to anow diagnostic
radial viewing of the pinch. The coils were energized by a 68
pF (13-kJ, 65-fls-to-peak-current) capacitor bank, and survived repeated shots to 40 kG, although our experiments
were normally performed at fields less than 18 kG,
The cathode electrode was fabricated from a high
strength, (3D matrix) carbon-fiber composite. 17 The largest
concentration of fibers, 50%, lie in the z direction; the x and
y directions each have 25%. As a result the electrical conductivity is largest in the direction which carries the pinch
current and lowest in the perpendicular direction to inhibit
CAPAC
;~------"'-
FARADAY
PROBE BEAM
"-TO
VACUUM
PUMP
FIG. ;, Schematic iilustration of the gas-puff Z pinch.
3835
J. Appt. Phys., VoL 64, No.8, 15 October 1988
PHOTODIODE
FARADAYROTATED BEAM
M
7
POLARIZER
C ROGOWSKI
M~____~-=~C~O~IL__-1POLAROID
~ {gRD S
FilM
PLASM
TVPE
PINCH
101
N2
S
lASER.
3311 A
0.6 MW, 5 nil
ARGOI\I. LASER,
5141 A,
2 W CW
S - DENOTES BEAM SPliTTER
M - DENOTES MIRROR
FIG. 6. Experiment diagnostics.
eddy currents that would exclude the injected magnetic
field. Nevertheless, during the final stages of pinch implo~
sion the injected magnetic field is frozen into the electrodes
with the resultant effect of causing the compressed field to
bow radially near the electrodes. Bowing reduces the length
of the region of highest field compression. We have observed
this bowing with interferograms and estimated the effect on
the compressed magnetic field. Equally important is the
"zipper effect," which is a tapered radial constriction of the
plasma column that propagates along the pinch axis during
the final stages of compression. Both effects will be discussed
in subsequent sections.
The cathode electrode also serves as a nozzle to collimate the gas puff into an annular flow between the A~K gap.
The design of the nozzle is not characterized by a Mach
number, as is common in other Z pinch devices. Instead, in
our design the mean radius and gap spacing of the nozzle
annulus was varied to adjust the mass-pee-unit length, M / L,
and hence the timing of the pinch relative to the maximum
current; our experiments used nozzles with 1- and 2-cm annular radii and gap spacings up to 60 mils, These nozzles
were simple to fabricate and produced an acceptable amount
of divergence of the gas column. A 6.5-mm-diam hole is
bored along the axis of the nozzle to facilitate access for the
fiber used in the Faraday diagnostic. Gas does not flow
through this hole.
The grounded-anode electrode is axially displaced from
the cathode by I em. It is fabricated from 0.7 -em-thick stainless-steel honeycomb, Mounted at the anode center is a 2cm-diam (POCO) graphite washer, also with a O,65-cmdiam hole for diagnostic access.
The repetition rate of the ucr Z pinch is approximately
one shot every 3 min, although most of our experiments were
performed at much longer intervals to allow an opportunity
to analyze data and to adjust experimental parameters. The
limits on the repetition rate are primarily due to the power
dissipation of the various power supplies and charging networks, although ohmic heating in the gas-puff'valve, Helmholtz coils, and discharge electrodes is also a concern.
Table I displays the range of parameters over which exFelber et al.
3835
TABLE I. Z-pinch operating parameters for magnetic field compression
experiments.
Parameter
Nozzle radius, em
Nozzle gap spacing, mils
Pinch charging voltage, kV
Pinch current, MA
Injected B, field, kG
Working gas
Gas pressure, psig
Operating
range
Optimal
values
1,2
2-60
2
32
25-32
32
0.47
9
0.31-0.41
0-18
H z,He,CH.,N 2•
CO2.Ar,Kr,Xe
0-120
Kr
80
periments were actually performed. In many cases the data
obtained enabled us to qualitatively evaluate the pinch performance and effectiveness in compressing the axial B z field.
The best results and highest values of compressed field were
obtained for the operating parameters listed in the last column of this table.
Standard experiment diagnostics include: dB /dt (Edot) magnetic probe, vacuum x-ray diode (XRD), MachZehnder laser interferometer, 18 and Faraday rotation.
The B-dot probe is used to measure the time derivative
of the pinch current and can be integrated to give the total
pinch current. When used in conjunction with the XRD,
these two diagnostics provide data to evaluate the "hardness," reproducibility, and timing of the pinch.
The interferometer uses a pulsed nitrogen laser
(A = 337.1 nm, S-nsFWHM) as the light source to illuminate the plasma column radially; the magnified image is recorded on standard Polaroid film (Type 107). By delaying
the firing of the laser with respect to the pinch, the interferometer is used to evaluate the density, radius, and dynamics
of the plasma column during implosion. Exact timing of the
laser with respect to the pinch is determined by electronically adding the laser pulse to the B-dot trace on a fast oscilloscope. In the final stages of field compression the high density of the plasma column reduces the resolution of the
interferometer due to diffraction and scattering of the nitrogen laser beam. At these times the resolution was improved
by blocking the reference beam of the interferometer to obtain a shadowgram of the plasma column.
To measure the magnitude of the compressed magnetic
field, we used the Faraday effect. In this technique a magneto-optically active medium, immersed in a magnetic field,
rotates the plane of polarization of a polarized laser beam in
proportion to the length of the medium and strength of the
magnetic field. Our initial attempts to measure Faraday rotation used the plasma as the active medium. However, we
discovered that because the plasma column was hollow, the
on-axis plasma density was insufficient to rotate the plane of
polarization of the laser beam significantly during column
implosion. Furthermore, just before peak field compression,
the transmitted laser intensity was severely reduced.
To alleviate these difficulties, we mounted a fused-silica
quartz fiber on the pinch axis, as has been done successfully
in other high-field experiments. 19 In this technique the polarization plane of the transmitted laser beam is rotated by
an angle,
3836
J. Appl. Phys., Vol. 64, No.8, 15 October 1988
a¢(t)
= vi !lBU),
where 'V = Verdet constant (degrees/MG cm), I = effective
length of the fiber (em), !lB (t) = change in magnetic field
intensity (MG). Since I ~Bis the Hneintegral of the magnetic field, we are purposely underestimating the peak magnetic
field. By analyzing the laser beam through a polarizer, the
laser intensity will oscillate as a function of time as
f(t) = fo sin2[~tP(t) -
¢o] ,
where fo is the initial transmitted laser intensity, and ¢o is the
initial angle of the polarizer measured with respect to the
laser plane of polarization.
The fiber was a conventional fiber-optic cable, 35 cm
long, 0.725 mm diam, with a 25-Jlm-thick, opaque (polyvinyl chloride) cladding that reduced stray light from the
pinch. The Verdet constant of the fiber was calibrated using
an alternate high-field (~150 kG) facility at San Diego
State University (c.r., Fig. 7) and was found to have approximately the same value as that reported in the literature. 5
The polarized probe beam for the Faraday diagnostic was
the green line (514.7 nm) of a 2-W (Coherent) argon-ion
laser. The detector was a photodiode (Hamamatsu 81722,
l.S-ns rise time) with a 1.0-nm bandpass filter. All fiber experiments were performed with the 2-cm-diam nozzle installed. After the fiber was installed on the axis of the pinch,
the Faraday diagnostic was recalibrated by measuring the
strength of the injected magnetic field and comparing with
independent magnetic probe measurements. Excellent
agreement was found with the earlier calibration. The Faraday diagnostic is discussed in greater detail in Ref. 20.
IV. EXPERIMENTAL RESULTS
This section summarizes the results of our experiments,
in which axial magnetic seed fields were compressed to ultrahigh magnitudes. First, the qualitative and quantitative behavior of the pinch with an injected field, as measured using
the B-dot, XRD, and laser interferometer, will be discussed.
Then measurements of the compressed field by the Faraday
diagnostic will be described.
To briefly summarize, the effects on the pinch of increasing injected magnetic field strength, at fixed mass-per-
.-
01
III
"
"""
"$<l
50
40
iiJ
..I
~
30
z
«
z 20
0
I-
-<
1=1.1c:m
0.0. = 0.80 m m
10
I-
0
c:
0
0
50
100
150
APPLIED FIELD, B(kG)
FlG. 7. Quartz fiber calibration curve.
Felber et at.
3836
unit length, MIL, were to (1) delay the onset of the pinch,
(2) decrease the x-ray intensity at pinching, (3) increase the
final compressed radius of the pinch, and (4) stabilize the
plasma coiumn before and after the pinch. By slightly decreasing the value of MIL as the initial axial field was increased, we noted a distinct improvement in the pinch quality and output x-ray intensity. However, the general trends
noted above were unchanged.
Figures 8(a)-8(e) display oscilloscope traces of pinch
data, including B-dot (upper left), and iaser interferogram
(right) traces, that demonstrate the effects noted above.
These data were measured for a krypton pinch at fixed MIL
and increasing seed magnetic field (increasing implosion parameter b). In these shots the pinch occurs approximately
200-400 ns prior to maximum current. In the B-dot trace the
sharp dip in the wavefonn (negative slope) is characteristic
of a rapid drop in current, and the subsequent rapid rise in
the B-dot trace (positive slope) is characteristic of a resumption of the current and increase in the radius of the plasma
column that normally follows the pinch. As shown here, the
effect of increasing the initial axial field is manifested as a
broadening of the dip in the XRD trace, suggesting a slower
and softer construction of the current channel at pinching.
The instant of maximum compression of the current
channel is coincident with the peak in the XRD trace. As the
initial field increases, the x-ray intensity at peak compression
decreases. The largest decrease is measured at small values
ofinitial injected field (c.f., Fig. 9). A fourfold reduction in
XRD signal results by increasing the initial injected field
2
6
8
10
iNJECTED FIELD
STRENGTH, 13 0 (kG)
FIG. 9. Normalized x-ray intensity as a function of injected axial field.
Pinch parameters: krypton, 70 '= 2 em, a-3.
from 0 to 3 kG. A higher initial field reduces the final plasma
density and temperature at peak compression, which reduces the x-ray yield.
The laser interferograms of Fig. 8 provide additional
insight into the r-z profile of the plasma-compressed field at
the time of maximum compression. In these photos the
pinch axis is vertical, and the laser views the pinch from the
side. As a function of increasing i.njected field, these interferograms display the trend toward larger final radius and a
more uniform outer boundary of the plasma column. At low
injected fields the centrai region ofthe column at peak compression appears as a diffuse, high-density plasma core. At
high injected fields the internal structure of the column becomes visible at peak compression, demonstrating the existence of a high density plasma annulus. Interferograms taken earlier in the compression phase revealed distinct inner
and outer boundaries of the plasma column which were identified with the J o and l z current sheaths. As the implosion
progressed towards maximum compression the spacing between these sheaths decreased until they could no longer be
resolved.
If we neglect diffusion losses and assume that the initial
radius Yo of the plasma column is defined by the annular
radius of the cathode nozzle, then the laser interferometer
provides a rough indication of the final compressed axial
field strength as Bf = Bo{roIYf) 2, where Yf is the radius measured at peak compression. Using the data provided by the
interferograms of Fig. 8, we have estimated the aVerage outer
radius of the plasma column and hence, the compression
ratio BflBo• as a function of the implosion parameter h.
These estimates are displayed in Fig. 10 as vertical error bars
which characterize the uncertainty associated with measuring the column radius. At low values of b, the uncertainty is
greater, since
t:.(B/Bo) ~2(BfIBo)(b.rflrf) ,
FrG. 8. B-dot (upper left), XRD traces (lower left), and interferograms
(right) as a function of initial injected axial field; (a) Eu ,-~ 0, (b) Eo = 1.5
kG, (e) Bo = 3 kG, (d) B,,= 6 kG, (e) Sn = 9 kG. Pinch parameters:
krypton, initial radius 2 em, acceleration parameter a ~ 3. B-dot and XRD
sCl>le "hanges: (a) 5 VIdiv, (b)-(el 2 VIdiv, (d)-(e) 1 VIdiv.
3837
J. Appl. Phys., Vol. 64, No.8, 15 October 1988
where t:.rf is an approximately constant uncertainty associated with measuring rf . The interferograms may be underestimating the peak compression because the interferometer
is incapable of resolving the inner radius of the plasma column at peak compression.
Except for the vertical error bars, which represent interferogram data, aU of the data points in Fig. 10 are inferred
from the Faraday rotation diagnostic. No correction factors
Felberetal.
3837
1. 6
300~--~----~---'----'----'
\'--THEORY a-4
I
o HELIUM, a- 63
iW
• ARGON, a-a
" KRYPTON, a-3
If XENON, a-2.4
z,...
~C]
\···t'. \ \
.~ ~ \ I "
~ \.
'L
g
100
0
o
~
-
....lm
J.
-
.
0.04
0.06
ARGON, a~6
'" KRYPTON. 1'1-3
eXENON, a~2.4
0,8~.
g
';ii
>'
,1~
§!
I
-0
x...l
"
;
~
0.4
<:
--
0.08
0.1
FIG. 10. Magnetic field compression vs implosion parameter for several
pinch gases and values of acceleration parameter a. Data points from Faraday rotation diagnostic. Dashed line is optimal compression (at a = 4) expected from simple theoretical model. Vertical error bars and beRt fit (dotted line) are inferred from laser interferograms of krypton pinch. For all
data points, nozzle radius = 2 cm.
have been added, such as to account for a higher maximum
magnetic field than the line-averaged value given by the Faraday diagnostic.
Since a:::::A is close to the optimal value of the acceleration parameter over the range of compression of interest, this
value is chosen in Eq. (6) to find the upper bound on the
maximum magnetic field compression as a function of the
implosion parameter b in Fig. to. According to this figure,
lower initial fields can be compressed not only to higher ratios EmaJBo, but also to higher absolute magnitudes. Of
course, for sufficiently high compressions, this curve becomes invalid owing to the neglect of kinetic pressure in the
snowplow model.
The maximum magnitudes of magnetic fields measured
by the Faraday diagnostic are shown for several different
gases and values of parameter a in Fig. 11. The data represent the same shots as in Fig. 10, but displayed as peak fields
rather than as peak compressions. The data in these figures
are representative Faraday data of the best reliability.
A remarkable result ofinjecting even a small amount of
axial field, Eo';:;; 1 kG, was the stabilizing effect on the pinch
before and after pinching, Normally, with Bo = 0 kG the
i.mploding plasma displayed an asymmetic radial profile approximately 75 ns before pinching, which is characteristic of
sman amplitUde Raleigh-Taylor instability growth. After
pinching, the column became highly unstable and disassembled, usually in less than 25 ns. By comparing Figs. 8(a)8 (e) we see that the column profile with increasing injected
field becomes smooth and uniform compared to the zero
field case.
This trend is confirmed by the time-integrated pinhole
photographs of Fig. 12. In this figure the intensity and contrast of sequential photos were increased approximately ten
3838
~
~I.l..
LASER INT,ERFEROGRAMS
0.02
~
0HEl!UM' 1'1-63
<: •
<w
/" .~_
(KR~PTON4
o
<::\!:
:::l: .....
N
~.... "~
'1."~'"
I··· .. '8
l·rr. . .:"
1.2
.----.,..~--..,~--:r-::,,;__-.,
J. Appl. Phys., Vol. 64, No.8, 15 October 1988
w
0..
oo
3
6
$;I
12
INITIAL AXIAL MAGNETiC
FIELD, Bo (kG)
FIG. 11. l'eak axial magnetic field vs initial axial magnetic field for several
pinch gases and values of acceleration parameter a. Data points from Faraday rotation diagnostic. Same data as in Fig. 10.
times from Figs. 12 (a) to 12 (f). These pinhole photographs
also reveal a lack of hot spots and unstable behavior as the
injected field strength increases.
In earlier experiments the hydrodynamic (time-topinch) and electrical (time-to-maximum-current) time-
lal
Idl
{hI
iel
lei
m
FIG. 12. Pinhole photographs of an argon pinch for increasing values of
injected axial field. (a) Bo = 0 kG, (b) Bo = 2 kG, (cl 8 0 .~ Ii kG, (d) Bo
~~ 8kG, (el Bo'~ 12kG, (f) 8" = 15kG. Pinchparamctcrs:argon,r" = 2
em, a~6.
Felber et al.
3838
scales were mismatched, so that the pinch occurred 500-800
ns prior to maximum current. This was achieved by using
low Z (atomic number) gase.'j such as N z , He, or H2 or by
installing the l-cm-diam nozzle. In these experiments we
observed multiple, periodic radial bounces of the plasma column due to the strong compressional force that remains after
the first pinch. Each bounce was evinced by a dip in the Bdot trace and a burst ofx rays. By adjusting M / L the bounce
period couid be alternately increased (at higher M / L) or
decreased. With high Z gases fewer bounces were observed
since the dissipative mechanisms, such as resistivity and radiation, were more effective.
In preparation for the Faraday rotation diagnostic, we
evaluated the qualitative effects of installing a 750-f.lm-diam
quartz fiber on the pinch axis. To facilitate this investigation,
we modified the experimental apparatus to allow the quartz
fiber to be inserted or replaced on sequential shots after being
destroyed by the pinch discharge without the need to break
vacuum.
At values of injected fields, Bo < 6 kG, the fiber was
completely severed at the location of the pinch by a single
discharge. At intermediate injected fields, 6 kG < Bo < 9 kG,
the fiber survived several machine firings. Later analysis of
the fiber revealed that although the outer cladding of the
fiber remained intact, the quartz core was shattered with the
greatest damage at the pinch center, Z = 5 mm, and decreasing toward the electrodes. At higher iI\jected fields, Bo> 9
kG, the damage to the fiber was minimal. We only observed
a small decrease in the transmitted laser probe beam from
shot-to-shot, perhaps due to fiber degradation. B-dot and
XRD traces showed that the fiber affected the pinch dynamics by broadening the B-dot traces at pinching and decreasing the XRD signal. In the range of operating parameters in
which the fiber survived, specifically 6 kG < Bo < 9 kG, we
measured the largest values of compressed magnetic field.
Although the fiber appeared to be isolated from the
pinch at values of Bo > 9 kG, laser interferograms and shadow grams taken at various times during field compression
revealed the presence of a plasma sheath in contact with the
fiber surface. This sheath is probably caused by breakdown
at the fiber surface of the large induced azimuthal electric
fields in the presence of ionizing radiation.
The dynamics of this sheath are revealed in the series of
interferograms shown in Figs. 13(a)-(d). The dark central
region of each interferogram corresponds to the location of
the quartz fiber on the pinch axis. These images of a krypton
discharge were taken in a region of sufficiently large implosion parameter, b~O.07, so that the dynamics of the fiber
sheath and plasma column could be separately imaged.
In Fig. 13(a) (measured at - 60 ns with respect to
minimum column radius) sheath breakdown appears to be
occurring at the bottom of the fiber. In subsequent photos
the radius of the sheath is seen to expand, reaching a thickness of the order of flx = 0.4-0.8 mm at maximum column
compression in Fig. 13 ( c) .
Also evident in these interferograms is the so called
"zipper effect" and bowing of the plasma column. As the
plasma column implodes, the minimum column radius is
located at the bottom of the fiber. After maximum compres3839
J, Appl. Phys" Vol. 64, No.8, i 5 October 1988
ial
[bl
Idl
!
_0,5
em
t
FIG. 13. Interfcrograms of a pinch taken at various times during compres,
sion: (a) -,60 IlS. (b) - 40 ns, (e) 0 ns, (d) + 40 ns. Pinch parameters:
krypton, Do =~ 9 kG, ro = 2 em, a~6, b-O.07.
sion the minimum column radius is located at the top of the
fiber. This zipper effect arises from asymmetries in the profile of the initial breakdown path through the gas column,
and is well known in gas-puff Z pinches. Normally, in Z
pinches without an injected field the zipper effect is only
observed prior to the maximum column compression and
disappears after the pinch owing to the rapid disassembly of
the pinch column. With an injected field, however, the column structure is preserved, and we can see the plasma column bounce off the central region of compressed axial field.
Also apparent in Fig. 13 (c) is the bowing of the plasma
column at maximum compression. This bowing is due to
entrainment of the axial magnetic field lines in the discharge
electrodes on the timescale of pinch implosion. In this particular interferogram, the ratio of the maximum to minimum
radius is approximately 1.25" Both the zipper effect and bowFelber et al.
3839
ing decrease the effective axial length of the compressed field
region and must be considered in interpreting magneto-optic
Faraday measurements. We will discuss these effects in more
detail in the next section.
Ear Her it was mentioned that the high rate of increase of
axial magnetic field induces a large azimuthal electric field
which may cause breakdown at the surface of the on-axis
optical fiber. When this breakdown occurs, a conducting
plasma sheath is formed, which inhibits further diffusion of
the axial field into the fiber. The rapid increase of magnetic
pressure outside the conducting sheath drives a shock into
the fiber. As the shock propagates into the fiber, apparently
it affects the optical properties of the fiber in such a way that
the laser loses its polarized quality in the region of the fiber
traversed by the shock. This depolarization, which is discussed in the next section, causes the amplitude of the Faraday signal osciHations to decay over a time comparable to the
shock transit time in the fiber, about 100 us.
Typical Faraday traces for an argon pinch, displayed in
Fig. 14, demonstrated this effect. In these sequential shots
the injected magnetic-field strength was large enough to protect the quartz fiber from severe damage by the pinch, and
the polarizer angle was alternately changed from O· to 45° to
90°. These traces also demonstrate the high degree of shatto-shot reproducibility of our experiments,
For each of the polarizer settings in Fig. 14, we measured final compressed field strength on the order of 500 kG
(compression of ~ 56). The measurement of higher values
was apparently prevented by the rapid onset oflaser depolarization (denoted in this figure by the "depolarization" arrow) which persisted for approximately 50--100 fiS following
POLARiZER
AN G L E IIIIII!IIPPP8"I1IIlI
90°
the pinch. All Faraday rotation traces displayed this characteristic behavior and typically provided a 160-200-ns window, in which we could reliably measure Faraday rotation,
and after which it was difficult to interpret further field compression or decompression.
To confirm the effects of shock-induced depolarization,
we protected the quartz fiber with a small diameter alumina
tubing (o.d,-2 mm), This tubing inhibited breakdown on
the fiber surface and hence, shock coupling.
The Faraday-measured signal for the protected fiber is
displayed in Fig. 15, and reveals a complete absence of shock
effects. The pinch conditions for this shot were identical to
those of the previous figure. In this figure, the reduced field
strength is attributed to the interaction between the plasma
column and the larger diameter of the alumina tubing, which
limits the final radius of the plasma column. Interaction with
the alumina tubing also cools the pinched plasma and results
in a rapid decay of the compressed field to about 100 kG. The
slower decay of field strength after 400 ns is characteristic of
a decrease in the discharge current and in the pinching force.
The arrow in Fig. 15 denotes the time-of-maximum discharge current,
The largest Faraday rotations were measured in krypton discharges, although several xenon discharges gave comparable results. In all we documented 14 shots in which the
final field strength exceeded 1 MG.
A representative Faraday trace for our highest field
compression in krypton is displayed in Fig. 16 where the
injected field strength is 9 kG. In this shot the strength of the
final compressed field is 1.6 MG, correspondi.ng to a field
compression ratio of approximately 180. Also seen in this
figure is the shock depolarization of the Faraday signal.
The Faraday-measured field compressions for numerous shots in He, Ar, Kr, and Xe are displayed in Fig. 10 and
demonstrate the trend toward increased compression ratios
as the implosion parameter b decreases. The best agreement
with the theoretical curve is observed at b~O.04 for the Kr
and Xe discharges, presumably since the parameter a ~ 3 is
POl.ARiZER
ANGLE 450
0°
0.6
oC!1 0,5
I
_:I:
1- ....
wOO
2:
(!1Q
0.3
,
,..
/
0,4
0.3
<..l 0.2
~~
u- 0.1
0"
-:::E
1- ....
~ro
C!J
200 400
~w
!l- 0.0
600
TIME ( i1s)
FIG. 14. Faraday rotation traces for three different polarizer settings, (a)
90', (b) 45', (cl 0'. and corresponding magnetic field. Pinch parameters:
argon, Bn = 0 kG, ro = 2 em, a~6, b-O.037.
3840
•
<:3 0.1
()
{)
MAXIMUM PINCH
CURRENT
0.2
J. Appl. Phys., Vol. 64, No, 8.15 October 1988
0
200 400 600
800
T!ME (ns)
FIG. 15. Faraday rotation trace for a quartz fiber protected by an alumina
tube (3 mm diam). Pinch parameters: argon, Eo = 9 kG, ro = 2 em, a~6,
b~O.037.
Felber et al.
3840
O.S
POLARIZE
ANGLE
.....
e
45°
0.4
t)
(fj
;:)
1.5
O(!l
0
-::2:
t--....,
a:
wm
:z
'I(
0.1
<..J
IJ..
0.2
1.0
(!lo
~~
0.3
0.5
0.0
i
-120
o
-300 -200
-40
0
40
60
TIME (na)
100
TIME (ns)
FIG. 16. Faraday rotation trace and Held strength as a function of time.
Pineh parameters: krypton, Bo = 9 kG. ro ,= 2 em, a~ 3. b~O.037.
closest to the optimum value a:=::4 predicted by Fig. 3. At
lower values of implosion parameter, the experimental data
indicate field compressions substantially less than the theoretical predictions. This disagreement is attributed to the
earlier onset of shock-induced depolarization in the Faraday
diagnostic. If this depolarization could have been avoided,
higher strengths might have been measured.
Another effect of surface breakdown on the quartz fiber
that was observed may be related to diffusion of the compressed magnetic flux into the fiber. Correlating times between the Faraday-measured field strength from Fig. 16 and
pinch radius, as measured from interferograms and shadowgrams from Fig. 17, shows that the minimum pinch radius
occurs approximately 100 ns before the maximum field. This
time delay may be caused by diffusion of field through the
plasma sheath into the fiber. This effect, discussed in the next
section, suggests that higher fields might have been produced on-axis if the fiber had not been there.
v. DISCUSSION AND ANALYSIS
The purpose of our experiments was to demonstrate the
compression of an axial magnetic field using an annular gaspuff Z-pinch plasma as the driver. The Z-pinch plasma is
itself compressed by an azimuthal magnetic field generated
by an axial current on the outer surface of the plasma column. We start initially with a hollow plasma shell with an
injected axial magnetic field Bo ranging from 0 to 18 kG. The
injected axial field applies a restoring force to the imploding
plasma, which increases as the radius oftlle pinch decreases,
reaching a maximum value at peak compression.
An important goal was to obtain the highest value of
compressed magnetic field in a given range of machine parameters: maximum current 1m = 0.47 MA, charging voltage Vc = 32 kV, rise time-to-peak-current tm = 1.25 fhs. To
achieve this goal it was necessary to match the pinch time, t p'
closely to t m • In this way, the acceleration parameter a was
made dose to the optimum value, a = 4. Matching of hydro3841
-80
J. Appl. Phys., Vol. 64, No.8, 15 October 1988
FIG. 17. Pinch radius vs time measured from laser interferograms and shadowgrams. Pinch parameters: krypton, Bo ~c 9 kG, ro= 2 em, a-3,
b-O.037.
dynamic and electrical times was achieved using a nozzle
with an appropriate radius and throat width, so that the optimal amount of gas was injected between the pinch electrodes.
The key results of our experiments are summarized in
Fig, 10, which shows the maximum magnetic field compression
Bmax I Bo
versus
implosion
parameter
b = (croBoI2I", )2 for different gases. We observed the highest compressions for the high Z gases, Kr and Xe, which He
dose to the optimum curve predicted by theory (with the
value of a = 4). The He and Ar data lie wen below the optimum curve, because these data are characterized by a larger
value of a caused by a less-than-optimal mass-per-unitlength. In the low Z gas pinches, the column lacks inertia,
and bounces hard off the compressed magnetic field early in
the current pulse.
The maximum field strengths we measured wereBf <. 1.6
MG. These measurements were made by Faraday rotation
and were consistent with measurements of the final radius by
interferograms or shadowgrams. The problem with measuring the radius using laser interferograms was poor resolution
of the inner radius of the plasma column at the final stage of
implosion.
To measure magnetic fields in the compressed Z pinch,
we have considered the following diagnostics: Faraday rotation, synchrotron radiation, electron- and ion-beam deflectiOll, magnetic probes, and Zeeman effect.
The diagnostic we selected for implementation on the Z
pinch was Faraday rotati.on. Faraday rotation has been successful as a diagnostic for measuring field strengths up to
about 10 MG in flux-compression generator experiments.
The Faraday effect relies on the rotation of the polarization
of polarized light passing through a magneto-optic material
immersed in a magnetic field. A quartz fiber seemed to be a
suitable magnetooptic material to give a reasonable number
of rotations of polarization for the 1-2-MG fields expected in
the experiments.
Our measurement of the compressed magnetic field using the Faraday diagnostic encountered several experimenFelber et 8/.
3841
tal difficulties, foremost among which was shock-induced
depolarization of the probe laser beam near peak compression. We were only able to measure field strengths in excess
of 1 MG when the time to maximum field compression was
less than the depolarization time.
We have considered both shock and thermal effects as
possible causes for depolarization of laser light within the
fiber. Thermal effects might be caused by radiation depositing heat within the fiber. The possibility of thermal effects
causing the depolarization was rejected, however, because
thermal relaxation times by diffusion of heat axially within
the fiber are much greater than a millisecond, and the polarization recovers in only tens of microseconds. This consideration leaves shock as a possible cause of the depolarization.
A model is suggested here to explain, with reasonable
agreement, how a shock might be responsible for the depolarization of laser light within the fiber. The fiber has a
diameter of 750 11m. It consists of a 650-pm-diam quartz
core and a 50-,um-thick cladding comprising a high-opticalindex layer and an outer layer of plastic. The entire fiber has
very low electrical conductivity. The model proposes that
the high rate of increase of axial field B z within the fiber
produces a strong azimuthal electric field Ee. This electric
field, aided by the x radiation present, as described below,
causes an electrical breakdown at the surface of the fiber. It
may be the surface itself or tenuous plasma near the surface
that provides the breakdown path.
Evidence for this breakdown is seen in Fig. 13, which
compares interferograms showing the outer plasma surface
of the fiber expanding during breakdown. After breakdown,
a dense layer appears just outside the original fiber diameter.
The breakdown layer is assumed to be highly conductive so
that magnetic field Hnes can only penetrate the fiber by diffusion. From the moment of electrical breakdown then, a difference of magnetic pressure builds up across the breakdown
layer. This sudden pressure difference generates a shock at
the surface of the fiber which propagates radially inwards.
The model supposes that the polarization of light becomes
randomized in any part of the fiber traversed by the shock.
Evidence for stress-induced depolarization has been reported elsewhere. 21
Simple estimates support the features of this model. In
mks units, the azimuthal electric field at the surface of the
fiber at radius ro before breakdown is
When breakdown occurs at a field strength approaching 100
T, the doubling time of the field is the order of 100 ns, as seen
in Fig. 16, for example. This estimate suggests Ez ::::::}09 T/s,
and an electric field at the fiber radius of ro = 375 pm of Ee
::::;2 kV/ern. This estimate approaches the breakdown fields
of insulators in high radiation environments, lending further
support to our suggestion that breakdown is occurring at the
fiber surface as the fields reach megagauss levels.
In addition to the plasma generated at the fiber surface,
the high x-radiation level present during the pinch implosion
can also lead to an enhanced electrical conductivity in the
fiber dielectric material itself. We have observed typical integrated x-ray energy outputs of = 1 J during the last 50 ns of
3842
J. Appl. Phys., VoL 64, No.8, 15 October 1988
pinch implosion and field compression. The average photon
energy of 0.5 keY allows for a 15-;tm penetration into the
polymer fiber jacket. Assuming about one-half of the radiation is absorbed by the on-axis fiber, the total radiation dose
deposited in the outer 15-flm layer of the fiber is calculated
as
Q5Xl~ Mg
E
r = Adp =
(0.33 cm 2 ) (l5X 10- 4 cm)(2g/cm3 )
= 5.1 X 109 erg/g
=
5.1 X 107 rad ,
where A i.s the area of the exposed fiber, d is the penetration
depth, and p is the mass density of the polymer. One fad is
equivalent to 100 erg/g of deposited energy E. The so-called
"prompt" induced conductivity22 is dependent on the dose
rate and is given by
r
a
= Ky = 3 X 10-- 18 1-'=3 X 10- 18
=3X 10-
3
(5.1 X 107 /50 X 10- 9 )
mho/em,
where K = 3 X 10- 18 mho s/rad em is the coefficient for a
polymer similar to Teflon. This value of u is only about two
orders of magnitude lower than for copper, which shows
that the conductivity of the fiber jacket itse1fis high enough
to interact with the external compressed B-fie1d and allow
for shock generation.
Once breakdown occurs at the fiber surface, then a
shock of magnitude S 100 kbar is driven radially inwards by
the rapidly rising magnetic pressure outside the surface. 1f
the shock speed Vs is at least about equal to the sound speed
of a few km/s, then the shock will reach the center of the
fiber in about 100 ns. This time is comparable to the time
during which the apparent depolarization oflaser light within the fiber occurs, as shown in Fig. 14. Moreover, the recovery of the polarization occurs over tens of microseconds.
This recovery time allows several hundred shock transit
ti.mes across the fiber for the shock to attenuate sufficiently
to restore the polarization-preserving property of the fiber.
Immediately after breakdown occurs at time tb at the
fiber surface, this model supposes that the surface of the
shock is at radius
r,
.
ro,
= {ro-vsCt-tb)'
t>tb
If the light in the fiber at radius greater than r, becomes
depolarized, then the intensity oflaser light observed beyond
the polarizing filter will be
r;
( ro0)
2
1= 2"" If) cos (}F(t)
r;:;-r;
[0
0 ") -,
+ (--,-
ro
2
where 10 is the constant intensity oflight passing through the
fiber. The first term represents the intensity of light in the
center of the fiber that has not yet been traversed by the
shock, but in which the polarization has been rotated by an
angle OF through the Faraday effect. The second term represents the intensity oflight that has passed through the region
traversed by the shock, been depolarized, and then attenuated by half in the polarized filter. Figure 16 displays a typical
oscilloscope trace of intensity from the Faraday diagnostic
that agrees with the time dependence predicted by this semiempirical model, assuming v, ::::; 3 X 105 cm/s.
Felber et a/.
3842
If this model of shock-induced depolarization of the
Faraday fiber is correct, then the intensity oscillations observed as the signal approaches 10 /2 still represent valid rotation of the polarization by the Faraday effect. These oscillations have been induded in our analysis of the magnitude
of the compressed magnetic field.
A correction to the field strength measured by the Faraday diagnostic should be made to account for the inward
bowing of the sides of the plasma column at peak compression that was observed in numerous interferograms and shadowgrams. Both ends of the pinch appeared flared compared to the center, which was relatively more constricted.
We suggest that the primary cause of this bowing effect is the
partial entrainment of magnetic-field lines in the anode and
cathode.
Even though the cathode is made of carbon-carbon
composite and the anode of a stainless-steel honeycomb
mesh (to reduce conductance), these materials are stilI good
conductors on the timescale of the pinch implosion. If the
velocity of the diffusion of field lines through the cathode
and anode cannot keep pace with the velocity of the pinch
implosion, then the pinch should develop an inward bowing
at the center, as we have observed. The reason is that the field
lines bend most, and are consequently most intensified, at
the ends of the plasma, producing a higher radial pressure
outwards there.
As mentioned earlier, most of the radial compression of
the pinch occurs on a timescale of less than a few hundred
nanoseconds, even though the quarter-period rise time of the
current is about a microsecond. This means that magnetic
field lines permeating the anode and cathode must diffuse
through radial distances of 1-2 cm in a few hundred ns to
avoid bowing of the pinch. The magnetic diffusivity of stainless steel is 5.8 X 103 cm 2/s and of graphite is 3.6 X 103 cm 2
Is. 5•23 On a timescale of a few hundred ns then, the magnetic
field diffuses less than 1 mm through the anode or cathode
materials.
The inward bowing of the magnetic field affects the
measurement of its magnitude by the Faraday rotation diagnostic. The Faraday rotation of the polarization of a laser
beam propagating through a fiber on the axis of the pinch is a
measure of the line-averaged magnetic field only.
The interferograms, such as those shown in Figs. 8 and
13, suggest that the magnetic field lines are parabolic, if they
are confined by the plasma sheath. A simple analysis that
relates the maximum magnetic field on the midplane of the
pinch B m to the line-averaged magnetic field B measured by
the Faraday rotation diagnostic shows that the ratio is independent of the length of the pinch. The ratio depends only on
the ratio of the flared radius ao at the ends of the pinch to the
constricted radius a at the midplane as
Bm =2(~+ tan-![(ao/a) -,:]
B
ao ·
r(ao!a) - 1] L -
112)-1
This ratio is plotted in Fig. 18. In the limit of severe bowing,
in which aola> 1, the maximum field is related to the lineaveraged field by
Bm
:::;-±- (ao/a)
I
i2jj
(ao>o).
11'
3843
J. Appl. Phys .. Vol. 64, No.8, 15 October 1988
S
I III
"-
l(
"
<II
E
ill
2
0
0
4
12
8
16
20
110 / 1'1
FIG. 18. Magnetic-field correction factor for plasma column bowing.
There is some evidence that the inward bowing of the
pinch is more pronounced in implosions in which the initial
magnetic field is smaller. This is to be expected, because, as
explained in Sec. II and Fig. 4, the minimum constricted
radius is smaller for smaller initial magnetic fields.
The bowing observed in our experiments suggested that
a typical field correction factor for our experimental conditions was Bm IE,;:;; 1.2. The Faraday field measurements reported in this paper, however, were line-averaged magnetic
fields B, and were not increased by this correction factor.
Consideration of shock-induced depolarization, delayed diffUSIon of the magnetic field through the sheath surrounding the on-axis fiber, and plasma column bowing suggest that higher fields were present than were measured
using the Faraday rotation diagnostic. Admittedly, plasma
measurements using invasive techniques such as probes,
wires, etc., that are inserted into the piasma channel. are difficult to interpret. Nevertheless, the measurements were highly reproducible shot after shot and on different occasions.
Therefore, our confidence in these results is high.
VI. CONCLUSIONS
A novel method for producing controlled ultrahigh
magnetic fields has been tested. The method involves imploding an axial magnetic field by a gas-puff Z pinch. The
method worked much as expected. However, on at least ten
shots we achieved compressions higher than 150. Our expectation of only a hundredfold compression did not anticipate
the unusual degree of stability that we observed in pinches
imploding onto axial fields.
Relying on 5-ns resolution interferograms and good
shot-to-shot reproducibility, we frequently observed multiple radial bounces of the Z pinch, with the pinch maintaining
good axial symmetry throughout. In one case we observed
five stable radial bounces of a He pinch before the current
pulse decayed and the pinch dissipated. Our results suggest
that only about 5 kG of initial axial field is needed to stabilize
a Z pinch for each megampere of peak current, because this
produces a final axial field comparable to the final azimuthal
field.
Our measurements offield magnitudes were made using
a Faraday rotation diagnostic with an on-axis magnetooptic
fiber. The Verdet constant of the fiber was wen matched to
the measured fields; each MG offield produced 3000 rotation
on about a IOO-ns timescale, which was easily resolvable.
Felber et al.
3843
The fiber survived shots in which the initial axial field was at
least about 6-9 kG. The intensity oscmations corresponding
to rotations damped out on most of the shots, however. We
attributed this effect to shock-induced depolarization of
light within the fiber. Some interferograms show evidence of
electrical breakdown and plasma sheath formation at the
surface of the fiber. The plasma sheath apparently inhibited
further diffusion of magnetic field into the fiber, and produced a surface agai.nst which a rising magnetic field could
generate a shock. Thus, the Faraday diagnostic might have
been failing before the highest fields could be reached within
the fibeL Of course, surface breakdown of specimens on axis
should be considered not only for diagnostics but for certain
applications of high magnetic fields as well.
Another effect that probably caused an underestimate
of peak fields achieved was bowing of the plasma column.
The bowing, caused by entrainment offield lines in the cathode and anode, resulted in higher fields at the center of the
pinch than at the ends. The Faraday diagnostic only measures line-averaged magnetic fields, however, rather than
peak fields.
Finally, owing to the relatively low currents at which
these initial experiments were performed, the magnetic
Reynolds number was only marginally greater than one.
Therefore, confinement of fields during compression was
not as good. as it might be in experiments at higher currents
and shorter pulse widths. Since this method of ultrahigh
field compression, unlike magnetic flux compression generators, can be extrapolated to the order of 100 MG,l we still
expect that currents of > 20 MA will be sufficient to reach
100 MG. Moreover, this method offield production is repetitive. OUf megagauss facility had a repetition rate of one
shot per 3 min. Facilities that deliver ten times as much current have repetition rates of at l.east once per day.
Applications of ultrahigh fields include particle acceleration,24 plasma, equation-of-state, and material property
studies, 4 macropartic1e acceleration, and atomic physics
studies at high-energy densities. A recent suggestion is to use
ultrahigh magnetic fields together with wail confinement to
decrease ignition energy thresholds in fusion reactions by
orders of magnitude. 25 A related possibility is a combination
of direct induction current heating of an imploding plasma
together with particle confinement and heat insulation by an
ultrahigh magnetic field. Another recent suggestion is the
generation of collimated beams of high-energy gamma radiation from a high-energy electron beam incident on an ultrahigh magnetic field. 26
An application that should be of considerable current
interest is the use of ultrahigh magnetic fields in x-ray lasers.
Besides the pronounced stability given to a pinch by an axial
field, our experiments also disclose that a fiber on the pinch
axis remains isolated from an imploding annular plasma
even at peak compression, if the initial field is sufficiently
high. This result suggests that a plasma pinch in an x-ray
laser can implode stably without unwanted interactions of
the pinch with a lasant on axis. Moreover, the initial magnetic field can be varied to tune the "hardness" of the implosion.
3844
J . .t\PPl. Phys., Vol. 64, No. B, 15 October 1988
We find that about 15-20 kG of initial field per megampere
of peak current should provide good stabilization of a gaspuff Z pinch, good isolation of a lasant on axis, and good
conversion of kinetic energy to x rays.
ACKNOWLEDGMENTS
We gratefully acknowledge help on the experiments by
Dr. A. Ron of the Technion Institute, Israel, Dr. J. Davis of
San Diego State University, and Dr. N. Rostoker and E.
Ruden of the University of California at Irvine. This work
has been supported by the U.s Department of Energy, Office
of Basic Energy Sciences, Division of Advanced Energy
Projects under Grant No. DE-FG03-84ER13302.
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3844