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 are created by high voltage discharges along the surface, or within, electrical insulating materials (called 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. He reported his findings 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.

Dr. Lichtenberg used electrostatic devices to charge the surfaces of various insulating materials such as resin, glass, or ebonite (hard rubber). He then sprinkled a mixture of finely powdered sulfur and red lead (lead tetroxide) onto the surface. The powdered sulfur (being slightly negatively charged through friction) was attracted to the positively charged regions, and the red lead was preferentially attracted to the negative regions. This made previously hidden regions of charge trapped on the surface 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 shorter, round, or fan-like. By carefully pressing a piece of paper onto the dusted surface, he was able to transfer these images onto the paper, demonstrating what was later to become the process of Xerography. Drawings of positive and negative figures that he are shown below.

Positive Lichtenberg figure	Negative Lichtenberg figure

Notable 19th and 20th century researchers included Gaston Planté (mid 1850's), French artist and scientist Etienne Leopold Trouvelot and Thomas Burton Kinraide (late 1800's), and Dr. Carl Edward Magnusson, and Dr. Arthur Von Hippel (1930's+). These later researchers used photographic film to directly capture the faint light emitted by positive or negative high voltage discharges along dielectric surfaces. Dr. Von Hippel discovered that Lichtenberg figures were actually created through complex interactions between ionized gas (corona or small propagating electrical sparks, called streamers) 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 high voltage 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 occur during various electrical breakdown processes within most gases, insulating liquids, and solid dielectrics. Lichtenberg figures may 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 often form on contaminated insulator surfaces or within dielectrics due to internal defects or voids, or at points where an insulator has been physically damaged, and they can eventually cause a flashover and complete electrical failure of the insulator.

The first report of Lichtenberg figures being created inside clear plastic by directly injecting electrons was from physicists Arno Brasch and Fritz Lange in the late 1940's. Electrons are tiny, negatively charged particles that orbit the nucleus of the atoms that make up all condensed matter. At their laboratory at AEG (Berlin, Germany), they used a 2.4 million volt Marx Generator to provide high energy pulses to drive an electron beam accelerator. An article about their research and the accelerator (which they called a "Capacitron") appeared in the March 10, 1947 issue of LIFE Magazine. The Capacitron could deliver 3 million volts to generate a beam of electrons that had an incredibly high peak current of up to 100,000 amperes. The glowing region of ionized air created by the exiting high-current beam of electrons resembled a bluish-violet rocket flame. A complete set of previously unpublished B&W pictures, including Lichtenberg figures inside a clear block of plastic, has recently become available online, as has another article with color pictures from the April, 1951 issue of Popular Mechanics.

Formal research on the detailed behavior of charge storage and movement within dielectrics was first conducted by Dr. Bernhard Gross in the early 1950's. Dr. Gross confirmed that internal Lichtenberg figures could be created within a variety of polymers and glasses by injecting them with high energy electrons using a linear accelerator (LINAC). The techniques that we use to make our sculptures build upon the theoretical work and techniques originally developed by Gross, Brasch, and Lange. The resulting Lichtenberg figures are sometimes called electrical trees, electron trees, beam trees, or spark trees - we call ours Captured Lightning® sculptures.

How do we create our Captured Lightning® sculptures?

Over many years, we have refined irradiation parameters and material fabrication processes to create a truly unique line of beautiful 2D and 3D sculptures. We start with specially cut and polished specimens of a clear polymer, polymethylmethacrylate (or PMMA). This material is commonly known as acrylic, or by various trade names such as Lucite, Plexiglas, or Perspex. PMMA has a unique combination of high 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. However, most of these materials form dark gray or black trees instead of the sparkling mirror-like fractures seen within acrylic.

We inject our acrylic specimens with high velocity electrons from a high energy particle accelerator. The energy of the accelerated electrons is measured in millions of electron volts (or MeV). The LINAC that we most often use use accelerates electrons to a typical kinetic energy of between three and five MeV. At these energies, electrons exiting the accelerator are traveling at relativistic velocities - between 98.5% and 99.6% of the speed of light.

When an acrylic specimen is irradiated, the relativistic electrons are driven deep inside. Penetration depth is determined by the energy of the electron beam, the target material's dielectric properties, and its atomic density. The higher the initial electron beam energy, the deeper the electrons will penetrate. As the specimen is irradiated, huge numbers of electrons accumulate inside, creating a cloud-like layer of excess negative electrical charge called a space charge. Since acrylic is an excellent insulator, the injected electrons become trapped. As they accumulate, they form a plane of negative space charge deep inside the specimen. By passing specimens through the beam in two or more passes, or by rotating them as they are irradiated, complex 3-dimensional space charge regions can be produced. As the space charge grows, the resulting internal electrical field increases dramatically. Eventually, the immense electrical stress 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.

Within billionths of a second, electrically conductive channels form within the acrylic as the material suddenly undergoes dielectric breakdown. When breakdown occurs, the previously trapped charge suddenly rushes out, accompanied by a loud bang(!), as thousands of electrically conductive branches feed their current into the main "lightning bolt" that exits the acrylic with a brilliant flash. Although pictures of the "Tree of Fire" discharge seem to suggest that we are injecting high voltage into the block, in reality we are removing the excess charge that was previously trapped within the block. The dielectric breakdown process occurs within an incredibly short amount of time. For example, the electrical discharge within a 2 inch square specimen may only last for less than 60 billionths of a second! The following image shows a 4 inch square specimen as it was being discharged:

(Photo courtesy of Theodore Gray)

The miniature lightning bolts leave their fingerprints in the acrylic, forming a complex, branching, and permanent "lightning fossil" within. The current within the electrical discharge is typically hundreds, or even thousands, of amperes. The hot plasma within the discharge causes the acrylic to melt and fracture along each path, and higher current "roots" may even char the acrylic slightly. The exit point of the discharge appears as a small crater 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 concentrates the electric field, creating a small region where the breakdown process begins. Although we inject a huge amount of negative charge into our specimens, electrical breakdown actually originates from points which are more electrically positive (versus the space charge layer), so our Captured Lightning® sculptures are actually "positive" Lichtenberg figures!

Actual discharge current measurements... and a paradox

During our 2007 and 2009 production runs, we were able to capture and record the shape of current waveforms when 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 made physical contact with the large surfaces of the charged acrylic specimen. A heavily insulated wire connected the pair of foil plates to a pointed tool which was used to discharge the specimen. The wire was also passed through the center of an Ion Physics 50 kA wideband current transformer (CT). The CT transformed the discharge current pulse that flowed through the wire into a voltage pulse that could be captured and stored within 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 discharge lasted for less than 120 billionths of a second (120 ns). For the specimen shown below, the peak current was almost 600 amperes, and consisted of four discrete current peaks. Other specimens showed between three and seven distinct current peaks.

This suggests that the electrical trees may propagate via a series of advancing waves, where each current peak reflected a surge of newly conducting channels ("streamers" and "leaders"). New channels apparently blasted their way into previously untapped reservoirs of charge within the acrylic, briefly paused, then surged again, etc. The average discharge velocity was between 8.5 x 10⁵ and 1.3 x 10⁶ meters/second (526 and 790 miles/second, or around 0.3% the speed of light). However, pauses between successive current surges suggest that the peak discharge velocity during propagating phases was significantly faster. Surprisingly, the average velocity within the specimen was actually 10-100 times faster than the velocity of positive lightning leaders in air. This is thought to be due to the extremely high electrical field (estimated to be in the range of 10-20+ million volts/cm) at the tips of the propagating discharges within the acrylic.

However, the high streamer velocity creates a paradox, since it 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 is limited to the Rayleigh speed (i.e., the speed of sound) or 1.614 km/second for PMMA. The current waveform clearly demonstrates that the chains of cracks and gas channels developed at a speed that was almost three orders of magnitude faster than it should be from classical materials theory. We suspect that that potential energy (from the intense internal electrical field) causes intrinsic ("electronic" breakdown) of the PMMA, generating a detonation wave of microcracks that propagates as a wave through the charge layer at hypersonic speed. This is an area ripe for future research. A Russian researcher, Yu N Vershinin, has termed the process of energy exchange between the electrical field and propagating fractures as "electronic detonation". Not surprisingly, the discharge process also generates a powerful shockwave (a loud BANG!), and a brilliant, miniature, blue-white "lightning" flash.

After the main discharge, there are often hundreds of smaller secondary electrical discharges as small pockets of stranded charge redistribute themselves within the specimen. Larger figures sparkle and sizzle for many seconds, making a sound similar to frying bacon. Intermittent sparking has been observed over 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 an 18" x 18" x 1" specimen being discharged during our 2005 production run. Before discharging, the estimated potential of the internal charge plane was 2.2 million volts. Because of its size, this specimen had considerably more stored electrostatic energy, and the discharge was quite loud and extremely 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 conchoidal fractures. These shell-shaped fractures are characteristic of the way noncrystalline (amorphous) materials fracture when stressed beyond their breaking point. These tiny fractures reflect light like tiny mirrors, so illuminating a figure through the edges causes the entire figure to glow brilliantly with the reflected color(s) of the external light source.

Lichtenberg figures are fractals

(Click for larger image) Figure 1. Densely Dendritic Discharges (higher charge density)

(Click for larger image) Figure 2. Dendritic Discharges (lower charge density)

(Click for larger image) Figure 3. Chaotic Discharges (prematurely discharged while being irradiated)

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.

Solarization, fluorescence, and birefringence:

You may also notice that a portion of the acrylic has an amber tint. This is called solarization. Solarization is thought to be caused by the formation of structural changes and defects through electron collisions, high energy x-rays, and the temporary trapping of excess charge within the molecular structure of the PMMA. Solarization usually occurs within the region between the surface that was irradiated by the electron beam and the discharge layer. During irradiation, electrons are initially traveling at about 99% of the speed of light. As they penetrate the specimen, they collide with acrylic molecules, rapidly coming 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 very powerful X-radiation. As the acrylic absorbs electrons and x-rays, various physical and chemical reactions occur that may alter its physical and 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 some or all of the clear PMMA to turn green, brown, or amber. Some electrons may become trapped for months, perhaps years afterward, creating color centers which may also contribute to the solarization.

Many irradiated specimens initially turn a bright lime green color which, over a period of minutes to hours, fades to amber. The solarized region may take months, or even years, to fade away. Fading can often be accelerated by gently heating the block in the presence of oxygen, or by leaving the specimen in bright light for an extended period of time. As oxygen diffuses into the PMMA from the outside and the discharge layer, it bleaches the solarized region, causing the solarized layer to gradually become thinner and thinner, eventually disappearing altogether. Most older Lichtenberg figures are completely bleached. Although they no longer show any solarization, some specimens may show slight residual "fogging" caused by irradiation damage to the acrylic. Some specimens exhibit little initial solarization, while a small number of specimens appear to permanently retain their amber color. Curiously, it has been found that fully charged specimens will retain their green color and stored charge indefinitely when kept at dry ice temperatures. These specimens then change to an amber color after being discharged days, or weeks, later. There are apparently several different processes that are associated with solarization.

Recently, it has also been discovered that the solarization layer may be fluorescent. An amateur scientist from Australia, Daniel Rutter, discovered that monochromatic light from a green laser pointer changes color when passed through the solarized layer of a Lichtenberg figure. More recently, we discovered that the light from a near-ultraviolet source (such as a Blu-ray laser or even bright blue LED's) may cause the solarized region to glow with a brilliant yellow-green fluorescence. This occurs on some specimens but not others, and as the solarization fades over time, so does the mysterious fluorescence.

Natural Lichtenberg figures - fulgurites and lightning discharges

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, keraunographic markings or simply ferning patterns. Although the exact causes are subject to some debate, they 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 within two days. A small Lichtenberg figure has also been observed at the entry 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 the similarity between the figure above and the Lichtenberg figure below (illuminated 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 quite similar. So it should not be surprising that the branching forms of lightning also have fractal characteristics. This similarity can clearly be observed during "anvil crawler" and horizontal "spider lightning". Spider lightning follows a thin, positively charged cloud layer, and the slowly propagating discharges can crawl across the sky for 30-40 miles - literally spanning from horizon to horizon. On a much smaller scale, transient Lichtenberg figures (often mistakenly called St. Elmo's Fire) often appear on the outer surface of cockpit windows of airplanes as they fly within thunderstorms.

Similar branching fractal patterns also occur when thunderstorms generate electrically conductive leaders that propagate downward from a charged cloud to the ground below. When one of these leaders connects with an unfortunate object on the ground, a high current surge (called the return stroke) back upward through the completed path, resulting in Cloud-to-Ground (CG) lightning strike. Exceptional examples of downward propagating positive leaders have been 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 positive lightning bolt. Positive lightning is a significantly rarer (and considerably more dangerous!) form of lightning than 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 brightly glowing tips of the positive leaders smoothly propagate, unlike negative leaders which propagate in a series of discrete jumps (called stepped leaders). The first descending leader to finally connect with the Earth below 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), and the actual elapsed time for the clip was only a little longer than three thousandths of a second. The speed of the propagating leaders was between 3 x 10⁴ and 6.5 x 10⁵ meters/second. This clip even contains a single frame which captures the beginning of the return stroke from the Earth going back up one of the leader channels. Even at the majestic scale of natural lightning, you can clearly 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 typically lasts for several hundred 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 his page on YouTube.

Lichtenberg Figures can also be seen at some high energy pulsed power facilities, where deionized water is sometimes 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 huge electrical stress, becoming an electrical conductor that safely dissipates unwanted residual energy from the system. The filamentary breakdown paths form Lichtenberg figures that dance across 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)

Are There Practical Uses for Lichtenberg Figures?

Analysis of Lichtenberg figures is a powerful tool for diagnosing high voltage breakdown phenomena within solids and along dielectric interfaces. By examining these figures, experts can diagnose and prevent future electrical faults within a variety of high voltage devices, such as the transformers and capacitors used by electrical utilities. Historically, they have been a powerful tool in measuring the polarity and magnitude of transient overvoltages on power lines during direct and indirect lightning strikes. These measurements were critical for the development of modern electrical power transmission and distribution systems. Lichtenberg Figures are still used as a forensic clue for identifying the cause of injury or death of human and animal lightning victims.

In 2009, researchers at Texas A&M University have proposed using 3D Lichtenberg Figures created within various polymeric materials as "templates" for growing vascular tissue. There are significant similarities between branching Lichtenberg Figures and animal circulatory systems - a fact not lost on many medical researchers. The hope is that, by creating branching 3D Lichtenberg Figures inside biodegradable polymers, such as polylactic acid (PLA), scientists could these as "molds" for vascular tissue to develop and grow within. Vascularization is essential for growing functional replacement tissues and organs. The 18th century technology of Lichtenberg Figures may play a profound role in organ replacement therapy during the 21st century!


Like holding lightning in your hand

Holding a Lichtenberg Figure is about the closest you can come to holding fossilized lightning in your hand. Captured Lightning® is indeed an accurate description for our sculptures. Most of our Lichtenberg figures were produced by irradiating various acrylic shapes using a 5 MeV research 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-rays. And, as with snowflakes, every Lichtenberg Figure is a one-of-a kind treasure.

Following are a pair of 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. (Caution: you'll need a high speed Internet connection to view these since they are each ~6 MB files and will take a little while 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 had the opportunity to own ones as spectacular as these. Stoneridge Engineering is proud to be the world's most experienced provider of these beautiful and rare treasures. We offer a wide selection of 2D and 3D figures ranging in size from affordable 2 x 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:

Gallery 1    Gallery 2    Special Eye Candy Page

Everyone is a genius at least once a year. The real geniuses simply have their bright ideas closer together. – G.C. Lichtenberg Thunder is good; thunder is impressive. But it is the lightning that does the work. - Mark Twain

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 Review: 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)
20. Yu.N. Vershinin, S.V. Barakhvostov, "Electron Processes in the Pulse Breakdown of Solid Dielectrics", 3rd International Conference on “Technical and Physical Problems in Power Engineering”, (TPE-2006), May 29-31, 2006 - Gazi University, Ankara, Turkey (detonation theory of high field breakdown in solid dielectrics)
21. Vershinin, Yu. N., "Parameters of Electronic Detonation in Solid Dielectrics", Technical Physics, Vol. 47, No. 12, 2002, pp. 1524–1528. Translated from Zhurnal TekhnicheskoÏ Fiziki, Vol. 72, No. 12, 2002, pp. 39–43, ISSN: 10637842
22. N, I, Kuskova, "Spark Discharges in Condensed Media", Technical Physics, Volume 46, Number 2, 2001, pp182-185
23. Theodore Gray, "Theo Gray's Mad Science: Experiments You Can Do At Home - But Probably Shouldn't", Black Dog & Leventhal Publishers, 2009, ISBN 978-1579127916
24. Jen-Huang Huang, Jeongyun Kim, et al, "Rapid Fabrication of Bio-inspired 3D Microfluidic Vascular Networks", Journal of Advanced Materials, Volume 21,Issue 35, 1-5, DOI: 10.1002/adma.200900584

Download a one-page condensed explanation (377 kB PDF file)
Download the above explanation in German (translated by Harry Meier)