What are Lichtenberg Figures, and how do we create them?
(Last updated 08/15/08)
Doubly Irradiated "Windblown Lightning" Sculpture
This Captured Lightning® sculpture
was created by irradiating a block of acrylic (Plexiglas) by a beam of
electrons from a 5 million
volt particle accelerator. The electron beam irradiated the left side,
the specimen was then rotated 180 degrees,
and irradiated once more on the opposite side. This created two
independent layers of electrical charge deep inside the specimen. The
rightmost charge layer was then manually discharged, creating a
miniature "lightning storm" within the rightmost charge layer. The
electrical discharges then grew between
the two charge layers, forming a beautiful 3D discharge pattern. The
sculpture is lit from below by blue
LED's. Unlike low resolution laser crystal art, each of our Lichtenberg
specimens
contain unique, and incredibly detailed, natural fractal discharge
patterns, and no two Captured Lightning® sculptures are identical. As
they
branch, the branching discharge channels become increasingly finer and
hairlike, eventually disappearing at the tips. The
smallest discharges may ultimately go to the molecular level.
(Actual size: 3" x 3" x 2")
What Are Lichtenberg figures? Our Captured Lightning® sculptures are technically known as "Lichtenberg Figures".
Lichtenberg figures are branching, tree-like or fern-like patterns that
created by high voltage discharges on the surface of, or within,
electrical insulating materials (dielectrics). The first Lichtenberg figures were actually
2-dimensional patterns formed in dust on the surface of charged insulating plates
in the laboratory of their discoverer, German physicist Georg Christoph Lichtenberg (1742-1799. Professor Lichtenberg made
this observation in the late 1700's, demonstrating
the phenomenon to his physics students and peers, and is reported in
his memoir: Super nova methodo naturam ac motum fluidi electrici
investigandi (Göttinger Novi Commentarii, Göttingen, 1777).
The basic principles
involved in the formation of these electrostatic figures later evolved
to become modern xerography and the science of plasma physics.
Lichtenberg used electrostatic devices to charge the surfaces
of various insulating materials such as resin, glass, or ebonite. He
then
sprinkled a mixture of finely powdered sulfur and red lead (lead
tetroxide) onto the surface. The powdered sulfur was attracted to the
positively charged regions and the red lead to negative regions,
thus making the previously hidden regions of charge clearly visible.
Lichtenberg
also observed that the shapes of the positively and negatively
charged figures were significantly different. Positive figures tended to be star-like with long branches, while negative figures tend to be round or fan-like. By
carefully placing a piece of paper onto the dusted surface, he was able to
transfer these image to the paper, demonstrating what was later to become
the process of Xerography. Drawings of positive and negative figures captured by Dr. Lichtenberg are shown below.
Positive Lichtenberg figure
Negative Lichtenberg figure
Later researchers included Gaston Planté (mid 1850's), Thomas Burton Kinraide (late 1800's), Dr. Carl Edward Magusson, and Dr. Arthur Von Hippel
(1930's+). These researchers used photographic film to directly capture
the light emitted by positive or negative high voltage discharges along
dielectric surfaces. Von Hippel discovered that Lichtenberg figures were actually created through complex interactions between ionized gas (corona
or electrical discharges) and the dielectric surface below. It
was also found that increasing the applied voltage or reducing the
surrounding gas pressure caused the length of the figures to increase.
This
property was used in klydonographs, special recording instruments
that photographically
recorded the size and shape of Lichtenberg figures that appeared during abnormal electrical surges on power lines.Klydonographs allowed lightning researchers and power system designers to
estimate the peak voltage
and polarity
of abnormal transients caused when lightning struck power lines.
A schematic diagram of the main parts of a klydonograph is shown on the leftmost drawing below, along with
examples of "klydonograms" from equal magnitude positive and negative high voltage
transients.
Schematic view of a klydonograph showing the position of the
photographic film and HV electrode. Light from high voltage
discharges creates a permanent photographic record of the event.
From W.W. Lewis, "The Protection of Transmission Systems
Against Lightning", John Wiley & Sons, 1950
Lichtenberg
figures are now known to often occur during electrical breakdown processes
within most gases, insulating liquids, and solid dielectrics.
Lichtenberg figures can be created very quickly (tens of nanoseconds) when dielectrics are
heavily overstressed, or they can grow very slowly , through a series of low energy partial discharges, evolving into partially conductive surface patterns or 3D "electrical trees".
Electrical trees may form on contaminated insulator surfaces, within
dielectrics due to internal defects or voids, or at points where an
insulator has been physically damaged. Considerable pioneering
research on the detailed
behavior of charge storage within dielectrics was performed by Dr.
Bernhard Gross in the middle of the last century. In
the early 1950's, Dr. Gross discovered that internal Lichtenberg figures could be created
within various polymers (plastics) by injecting them with high energy electrons using a linear accelerator (LINAC).
The techniques that we use to make our Captured Lightning® Sculptures builds upon
the original theories and techniques discovered by Dr. Gross. The resulting figures are sometimes called electrical
trees, electron trees, beam trees, or spark trees - we call them Captured Lightning®.
How do we create our Captured Lightning® sculptures?
We have continued to develop and refine irradiation and material
processing techniques to create a truly unique line of 2D and 3D sculptures.
We start with specially cut and polished clear plastic material, (polymethylmethacrylate, or PMMA). This material is commonly called acrylic, or various trade names such as Lucite, Plexiglas, or Perspex. Acrylic
was selected because it has a unique combination of optical clarity and superior electrical
and mechanical properties. Other clear polymers, such as polycarbonate (PC),
polystyrene (PS) , polyethylene terephthalate (PET), and polyvinyl chloride (PVC) also work to varying degrees. Some
of these materials even develop dark, or even black (carbonized), trees.
Our sculptures are created
by injecting acrylic specimens with high velocity electrons.
Electrons
are tiny, negatively charged particles that orbit the nucleus of the atoms that make up all condensed matter.
An electron beam accelerator is used to accelerate
and focus electrons into a high-energy beam. The energy of the accelerated electrons is measured in millions of electron Volts (or MeV). The
LINAC that we use accelerates electrons to a kinetic
energy
of between three and five MeV. At these energies, electrons leaving the accelerator are traveling at relativistic velocities that are between 98.5% and
99.6% the speed of light.
As the specimen is irradiated
by the beam, electrons are driven
deep inside the acrylic. The penetration depth is determined by
the
electron beam's
initial energy, the material's dielectric properties, and its density.
The higher the electron beam energy, the deeper the electrons will penetrate. As
the specimen is irradiated, huge numbers
of electrons accumulate inside the acrylic, creating a cloud-like
layer
of excess negative electrical charge called
a space charge.
Since acrylic is an excellent dielectric, most of the injected electrons cannot escape, so they accumulate under continued irradiation, causing a huge negative space charge to develop inside the specimen. By
carefully changing the orientation of the specimens and passing them
through the beam in two or more passes, complex 3-dimensional space
charge regions can be produced.
As the space charge grows, the resulting electrical field also
increases.
Eventually, the electrical stress from the increasing electrical field overcomes the
dielectric strength of the acrylic, and some of the chemical
bonds that hold the acrylic
molecules
together are ripped apart. This strips away additional free electrons (a process
called ionization). These newly-freed electrons are also accelerated by the
electric field, ionizing even more acrylic
molecules, and creating additional free electrons in a runaway process. Electrically
conductive channels rapidly form within the acrylic as the material undergoes dielectric breakdown.
Once breakdown occurs, the previously trapped charges suddenly rush out, accompanied by a
loud bang(!), and thousands of electrically conductive branches feed current into a brilliant "lightning
bolt" that exits the acrylic. Although
pictures of the discharge seem to suggest that we are injecting a high
voltage into the block, in reality we are removing high voltage already
stored within the block. Dielectric breakdown typically occurs within
an incredibly short amount of time. For example, the electrical discharge within a 2 inch square specimens may only last for 20
billionths of a second! The following image shows a 4 inch square specimen as it was being discharged:
(Photo courtesy of Theodore Gray)
The
escaping lightning bolts leave permanent fingerprints in the acrylic,
forming a branching "lightning fossil" within. The high current
electrical
discharges may reach hundreds or even thousands of
amperes. The hot plasma causes the acrylic to melt and fracture along each path, and higher
current "roots" may slightly char the acrylic.
The
exit point of the discharge appears as a small hole on the surface of
the acrylic. The
discharge point is typically located at a surface defect, or where a
point of
external mechanical stress
has
weakened the dielectric. The defect causes a localized concentration of
the electric field, creating a weak link where the breakdown process
can begin. Interestingly, even though we've injected a huge negative charge into the
specimens, the electrical breakdown process originates from points which
are more positive versus the space charge, and our Captured Lightning® sculptures are
actually "positive" Lichtenberg figures.
Actual discharge current measurements... and a paradox
During
our 2007 production run, we were able to capture the shape of
the current waveform as we discharged a number of 4" x 4" x
3/4" specimens (similar to the specimen above). A special holding
fixture was constructed with copper foil plates that made physical contact with the large surfaces of the
charged acrylic
specimen. A heavily insulated wire connected the pair of foil
plates to the sharp tool which was used to discharge the specimen.
This
wire was also passed through the center of an Ion Physics 50 kA
wideband
current transformer
(CT). The current transformer transformed the short discharge current pulse
that flowed through the wire
into a voltage pulse that could be captured and stored in a high speed
Tektronix digital
storage oscilloscope. The digitized waveform data was subsequently
analyzed using an Excel spreadsheet in order to
recreate the following waveform.
We
found that, for 4" x 4" specimens, the overall discharge
lasted for only 120 nanoseconds (billionths of a second)! For the
specimen shown below, the
peak current was almost 600 amperes, and was seen to consist of four
separate current peaks. Other specimens showed between three and seven
peaks. This suggests that the electrical trees propagated via a series
of advancing waves. Each current peak reflects a surge of newly conducting
channels ("streamers") blasting their way ahead and into previously
untapped reservoirs of
charge in the acrylic, followed by a brief pause, then another
surge, etc. Since the
overall discharge propagated a distance of about 4 inches within 80-120
billionths of
a second, the average streamer velocity was between 8.5 x 105 and
1.3 x 106
meters/second (526 and 790 miles/second!). However, pauses
between successive current surges suggest that the peak streamer
velocity during growth phases was even faster. Surprisingly, the
average streamer velocity within the specimen was comparable to the
velocity of positive streamers in air.
However, this creates a paradox, since the measured velocity is over 800 times the speed of
sound within
PMMA. This is inconsistent with Griffith's theory of crack propagation within solids,which
predicts that the maximum crack propagation speed within a solid
should be limited to the Rayleigh speed (i.e., the speed of sound)
within the material, or 1.614 km/second for PMMA. The
current waveform below clearly demonstrates that the chains of cracks
and gas
channels were fully developed within the acrylic at a rate that's 800
times faster than it should be from classical materials theory. Energy
from the intense internal electrical field is apparently converted into
electronic breakdown of the PMMA and a "wave" of microcracks that
propagate through the charge layer at hypersonic speed. This is
an area ripe for future research. In any event, the discharge process
creates a powerful shockwave (a loud BANG), a brilliant,
miniature, blue-white "lightning" flash, and results in objects of
incredible beauty.
After
the main discharge, there are often tens or hundreds of smaller
secondary electrical
discharges as small pockets of residual charge redistribute
themselves
within the specimen. Larger figures often sparkle and sizzle for
tens of seconds
afterwards, making a sound similar to frying bacon, and intermittent
sparking has been observed 15 - 30 minutes later. These smaller
discharges often sting our fingers when partially discharged
specimens are handled. Click on the
following image to see some high resolution video taken during our
November, 2007 production run showing primary and secondary
discharges.
(Photo and video courtesy of Mike Walker and Theodore Gray) Click on the Above image to see a video clip
of many Lichtenberg figures being discharged
Video clip of a huge 18" Lichtenberg figure being created:
Following
is
another video clip of a larger (18" x 18" x 1") specimen being discharged. This was captured during our 2005
production
run. The estimated potential of the internal charge plane was 2.2
million volts. Because of it's larger size, this specimen had
considerably more
stored electrostatic energy, and the discharge was quite loud and very
bright! The actual discharge, although very brief, saturated the
video camera image sensor. A multitude of secondary discharges can also be observed after
the main discharge. (Video courtesy of Terry Blake, specimen
was owned, and discharged, by Jeff Larson.)
The
rounded, crystalline flakes
that make up the Lichtenberg Figure are actually chains of tiny chonchoidal
fractures. These
shell-shaped
fractures are characteristic of the way noncrystalline
(amorphous)
materials fracture when stressed beyond their breaking point. Since
these tiny fractures reflect light like tiny mirrors, illuminating the
figures through the edges causes the entire figure to glow brilliantly
with the reflected color(s) of the external light source.
Lichtenberg figures are fractals
Lichtenberg figures exhibit branching patterns which
tend
to look similar at various scales of magnification. This self-similar property suggests that Lichtenberg figures can be modeled
using a branch of mathematics called Fractal Geometry. Self similarity is a key property of fractals.
Our Lichtenberg Figures show a range of fractal patterns depending upon
the magnitude of charge injected into the acrylic and how and when the
specimens is discharged. Branching figures are technically called
"dendritic" or "arborescent" (tree-like). If a large amount
of electrical charge is injected into the specimens and it is then
immediately discharged, a very dense dendritic discharge is
created, such as the leftmost figure below.
These dense discharges are quite similar in appearance to ferns or moss
agate. If the level of charge is reduced and the specimen manually
discharged, a more classical, lightning-like or tree-like discharge
results as the
center example below. If premature breakdown occurs as we are
actively irradiating a specimen, tangled "chaotic" discharges occur.
Some specimens show combinations of these basic patterns.
Self
similarity can easily be seen in the following sequence of zooms
from a 12" x 12" Lichtenberg Figure with nominal dendritic discharges.
The branches become finer and
hairlike, ultimately disappearing. Similar fractal patterns are seen in
aerial views of some rivers and their tributaries, branching tree
limbs, and the arteries, veins, and capillaries within your body.
It has recently been discovered that Lichtenberg figures can be modeled using a process called "Diffusion Limited Aggregation" or DLA. A useful macroscopic model that combines an electric field with DLA is called the Dielectric Breakdown Model or DBM.
The dielectric breakdown model appears to describe the branching growth that
characterize the dielectric breakdown process within solids, liquids, and gases.
Solarization and other effects: During
irradiation, the acrylic glows a brilliant blue-white
color. Although radiation chemistry studies suggest that this may be a combination of luminescence or Cherenkov radiation, the reason(s) are not fully understood. You may also notice that our specimens have a
discharge-free zone along all of the outside edges. This is because PMMA
is not a perfect insulator, so some of
the internal charge "leaks away" to the outside surfaces. This reduces the amount of stored charge along the
perimeter to the point where the electrical field is no longer sufficient to break down the
acrylic.
You may also notice that a portion of the acrylic has
an amber or greenish tint. This coloration is called solarization.
Solarization is
thought to be caused by the formation of defects through electron
collisions, high energy x-rays, and the temporary trapping of ionic charges
within the molecular structure of
the PMMA. Solarization normally occurs only in the region between the surface that was
irradiated by the electron beam and the discharge layer. During irradiation, the electrons were initially traveling at
about 99% of the speed of light. As they penetrate the specimen, they collide with acrylic molecules, rapidly come to a stop within a fraction
of an inch. The electrons in the beam have a tremendous
amount of kinetic
energy, and as they suddenly brake to a stop,
they release this energy in the form of heat and powerful X-rays. As the acrylic absorbs electrons and x-rays, various physical and chemical reactions occur that alter its optical properties. Although
the specific causes of solarization are not fully understood, there
is evidence that irradiation creates unstable, or longer-lived
"metastable", compounds that preferentially absorb light at the blue end of
the spectrum (250 - 400 nm). This causes solarized regions to appear initially as a lime green or amber color.Some electrons may become trapped for months,
perhaps years, afterwards, forming color centers which contribute to the solarization.
Many
irradiated specimens initially turn a bright lime green color which,
over a period of minutes to hours, fades to an amber color. This may
take months, or even years, to fade away.
The fading
process can be accelerated by gently heating the block in the
presence of oxygen, or by placing the specimen in bright light or
ultraviolet light. Most
older Lichtenberg figures may no longer show any solarization,
but some show slight
"fogging" from irradiation damage. Some specimens show little initial
solarization, while a small number of specimens appear to permanently
retain
their amber color. Most specimens also exhibit
slight changes in refractive index, especially near the Lichtenberg discharge region. These
behavioral differences are
thought to be due to variations in the acrylic blends used
by various manufacturers, permanent irradiation-induced changes to the
polymeric structure of the acrylic, and residual mechanical stresses near the discharge fractures. Residual
stresses within a sculpture can easily be seen as multicolored regions
near the discharge plane when a sculpture is illuminated by polarized
light and then viewed through a second polarizer.
Natural Lichtenberg figures - fulgurites and lightning discharges
Occasionally, nature also creates "fossilized lightning". Called fulgurites,
these
are hollow and sometimes branching tubes that are formed when the powerful
electrical
current from a lightning strike creates underground discharge channels within poorly conducting sandy or sandy-clay soils. These hollow channels were formed as the intensely hot channel from the lightning arc fused surrounding sand
and soil particles which then cooled to form a solid glassy tube. Some
fulgurites also exhibit fractal characteristics as they split into
smaller diameter root-like branches at further distances from the site
of the main strike.
Lichtenberg figures, sometimes called "lightning flowers" or "skin feathering",
are sometimes formed
beneath the skin of unfortunate humans who have been struck by lightning. The victim will often have one
or more reddish radiating feathery patterns that branch outward from
the entry
and
exit points of the strike:
(From "Lichtenberg Figures Due to a Lightning Strike" by Yves Domart, MD, and Emmanuel Garet, MD,
New England Journal of Medicine, Volume 343:1536, November 23, 2000
Medical terms for this phenomenon include "arborescent lightning burn",
"arborescent erythema", or "keraunographic markings". Although the exact causes are subject to
some debate,
lightning flowers appear to be the result
of damage to small capillaries under the skin, perhaps caused by the
flow of electrical current
from the stroke, or by shock wave bruising from external flashovers just
above
the skin. The arborescent (tree-like)
reddish marks fade away over a
period of
hours
or days. They are recognized by forensic pathologists as clear evidence
that a victim has been struck by lightning. The patient above survived
with no permanent injuries, and the lightning flowers completely faded
two days later. A small Lichtenberg figure
has
also been observed at the point
where a high voltage spark penetrated the skin of an unfortunate (but
surviving) local electrical experimenter who took an accidental "hit"
from a
homemade 60,000 volt Marx Generator.
A similar phenomenon is sometimes seen when
lightning
hits a grassy field, as in this picture where lightning struck a
flagpole, leaving this beautiful 25 foot Lichtenberg figure on the
green of a golf course:
(From "Lightning and Lichtenberg Figures" by Cherington, Olson and Yarnell, Injury, Volume 34, Issue 5, May 2003)
Note how similar the above figure appears to the Lichtenberg figure within this specimen (lit from below by blue LED's):
High voltage discharges to the surface of water can also create Lichtenberg figures. Some very beautiful examples
of
both positive and negative Lichtenberg figures on water surfaces can be seen on Dr. Colin
Pounder's Lichtenberg figures web site.
Natural lightning sometimes creates transient "Lichtenberg Figures" in the sky. Air is an excellent dielectric. And, although the physical breakdown
mechanisms
for air and PMMA are considerably different, the appearance of the
branching
discharges is often quite similar. So it should not be surprising
that the
branching forms of natural lightning also have fractal
characteristics. This similarity can also be observed during "anvil crawler" and horizontal "spider lightning".
Spider lightning follows a positively charged cloud layer, and the
slowly propagating discharges can crawl across the sky for 30-40 miles
- literally spanning from horizon to horizon.
Similar
shapes occur when thunderstorms generate conductive leaders that
propagate downward from a charged cloud to the ground below. When one
of these connects with an unfortunate object on the ground, a high current surge
rushes through the completed path as a Cloud-to-Ground (CG) lightning
strike.
An
exceptional example downward propagating positive leaders was recently
captured by South Dakota lightning
researcher, Tom Warner. Using high speed video imaging equipment, he
was able to capture the downward progression of leaders and the return
stroke from a rare positive lightning bolt. Positive lightning
is a significantly rarer (and considerably more dangerous) form of lightning versus negative lightning. The
"slow motion" video (below) shows the air breaking down, forming
glowing conductive plasma paths (called leaders) that fan
downward from a huge reserve of positive charge within the cloud
above. The glowing positive leaders tips smoothly propagate, unlike
negative lightning
which progresses in a series of jumps (called stepped leaders). Once
one
of the descending leaders finally connects with the Earth below, it
completes the circuit, resulting in a powerful
Positive Cloud-to-Ground (+CG) lightning discharge. The video clip
below was captured at 7200 frames/second (FPS), so the actual elapsed
time in the following clip was only a little longer that three
thousandths of a
second. The estimated speed of the propagating leaders was between
30,000 and 650,000 meters/second. This clip even has a single frame
which captures the beginning of the return stroke from the Earth going
back up the leader channel. And, even at the majestic scale
of natural lightning, you can easily see similarities between the
collection of branching leaders and Lichtenberg Figures.
Positive lightning also has a very long lasting "tail" of
follow-through current which often lasts for many hundreds of
milliseconds after the initial strike
connects to ground. The combination of long propagation distance (often
many miles from the main storm), very high current (up to 300,000
amperes), and long follow through current make positive lightning
exceptionally dangerous. It tends to set fire or kill anything, or
anyone, unfortunate enough to be in its path. More of Tom Warner's
fascinating videos can be seen on YouTube.
Lichtenberg Figures can also be seen at some high energy pulsed
power facilities, where deionized water is often used as a dielectric to briefly store
large amounts of electrical energy. The famous photo below is from Sandia National Laboratory's
mighty Z
Machine, the world's largest pulse generator. After
the completion of a high energy experiment, the water breaks down from the electrical stress,
becoming an electrical conductor that safely dissipates unwanted
residual
energy from the system, and forming Lichtenberg figures that dance along
the water's surface. If you look closely, you'll notice that
many of the radial paths actually trace out high voltage
electrical field lines along the surface of the water. Although
impressive, this display is only dissipating "left over" energy,
representing only a very small fraction (perhaps 5%) of the energy that was
actually used during the previous pulsed power experiment.
(Click for a higher resolution 840 x 554 pixel image, 561 kB)
Holding a Lichtenberg Figure is about the closest
you can
come to holding fossilized lightning in your hand - Captured Lightning® is indeed an apt description. Most
of the acrylic Lichtenberg figures shown on our web site were produced
by
irradiating various acrylic shapes using a 5 MeV Continuous Wave (CW)
LINAC - a 150 kW high power electron beam accelerator called a Dynamitron. A few were created using pulsed linear accelerators at significantly higher beam energies (10
- 15 MeV). Lichtenberg figures are completely
safe - they have been electrically discharged and have no residual radioactivity or X-radiation.
And, as with snowflakes, every Lichtenberg Figure is a one-of-a kind treasure.
Following
are a pair 3-D images that can be rotated 360 degrees so that you can fully
enjoy the beauty of our doubly-irradiated Lichtenberg figures. The irradiation process
results in very complex discharges within and
between the two charge layers. Please wait for the images to
completely
download, then drag your mouse to rotate the images for a full 360
degree view. (Warning: you'll need a Cable or DSL connection to view these since
they are each ~6 MB files and will take quite some time to fully load.)
3D Rotatable Image
3D Rotatable Image
"Heavy Weather"
(Courtesy of Theodore Gray)
"Windblown Lightning"
(Courtesy of Theodore Gray)
Very
few people have actually seen
or held one of these rare objects, and far fewer have ever owned one.
Stoneridge Engineering is proud to be the world's most experienced provider for these beautiful and rare
treasures. We
offer a wide selection of 2D and 3D figures ranging in size from
affordable 2 inch specimens through museum quality figures
as large as 24 inches by 36 inches. Please visit our galleries to see the world's most beautiful Lichtenberg figures:
Everyone is a genius at least once a year.
The real geniuses simply have their bright ideas closer together.
– G.C. Lichtenberg
References and Further Reading: 1. Gross, Bernard, "Irradiation Effects in Plexiglas", Journal of Polymer Science, Volume 27, 1958, Issue 115, Pages 135 - 143 2.
Hashishes. Yuzo, "Two Hundred Years of Lichtenberg Figures", Journal of
Electrostatics, Volume 6, Issue 1 , February 1979, Pages 1-13
3. Chadwick, K. H., "The Effect of Light Exposure on the Optical
Density of Irradiated Clear Polymethylmethacrylate", 1972 Phys. Med.
Biol. 17, Pages 88-93 4.
Chadwick, K. H., and Leenhouts, H. P., "Fading of
radiation-induced optical density in polymethylmethacrylate on oxygen
diffusion", Phys. Med. Biol. 15 No 4 (October 1970), Pages 743-744
5. L. Niemeyer, L. Pietronero*, and H. J. Wiesmann, "Fractal Dimension
of Dielectric Breakdown", Phys. Rev. Lett. 52, 1033–1036 (1984) 6.
Gardner, Donald G., et. al., "Radiation-induced changes in the index of
refraction, density, and dielectric constant of poly(methyl
methacrylate)", Journal of Applied Polymer Science, Volume 11, Issue 7,
July 1967, Pages 1065-1078 7.
Akishin, A.A.; Tseplyaev, L.I., "Edge effect in radiation-charge
dielectric materials", Physics and Chemistry of Materials Treatment, v
31, n 1, Jan.-Feb. 1997, p 30-1. A similar paper is also contained
within the book "Effects of Space Conditions on Materials", Akishin, A.
I., Nova Science Publishers, 2001, ISBN 1590330285
8. Fothergill, J.C.; Dissado, L.A.; Sweeney, P.J.J., "A
discharge-avalanche theory for the propagation of electrical trees. A
physical basis for their voltage dependence", Dielectrics and
Electrical Insulation, IEEE Transactions on, Volume 1, Issue 3 , June
1994, Pages 474 - 486
9. R. A. Galloway, T. F. Lisanti and M. R. Cleland, "A new 5 MeV –300
kW Dynamitron for radiation processing", Radiation Physics and
Chemistry, Volume 71, Issues 1-2, September-October 2004, Pages 551-553
10. Sessler, G.M.. "Charge distribution and transport in polymers",
Dielectrics and Electrical Insulation, IEEE Transactions on, Volume 4 ,
Issue 5 , Oct. 1997
Pages 614 - 628 11. Karczmarczuk, Jerzy, "Dendrites in Nature and in Computer", Foton 84/SPECIAL ISSUE, Spring 2006 12. C. M. Foust, General Electric Revew: Instruments for Lightning Measurements (Includes Klydonograph and Lichtenberg Figures)
13. Watson, Alan and Dow, Julian, "Emission Processes Accompanying
Megavolt Electron Irradiation of Dielectrics", Journal of Applied
Physics, December 1968, Volume 39, Issue 13, pp. 5935-5940
14. Fujimori, S., "Fractal properties of breakdowns", Properties and
Applications of Dielectric Materials, 1988. Proceedings., Second
International Conference on Properties and Applications of , 12-16
Sept. 1988, Pages:519 - 522 vol.2
15. Domart, Yves, M. D., Garet, Emmanuel, M.D., "Lichtenberg Figures
Due to a Lightning Strike", New England Journal of Medicine, Volume
343:1536, November 23, 2000, Number 21, Images in Clinical Medicine 16. H. Hiraoka, "Radiation Chemistry of Poly(methacrylates)", Radiation Chemistry, March 1977
17. Brown, R. G., "Time and Temperature Dependence of Irradiation
Effects in Solid Dielectrics", Journal of Applied Physics,
September 1967, Volume 38, Issue 10, pp. 3904-3907
18. Yu. S. Deev, M. S. Kruglyi, V. K. Lyapidevskii and V. I. Serenkov,
"Mechanism underlying the formation of dendritic or tree-like channels
in a dielectric irradiated with charged particles", Atomic
Energy, Volume 29, Number 4, October, 1970
19. Ebert, Ute and
Arrayas, Manuel, "Pattern Formation in Electric Discharges", p. 270 -
282 in: Coherent Structures in Complex Systems, eds.: D. Reguera et
al., Lecture Notes in Physics 567 (Springer, Berlin 2001)