Physics Letters A 361 (2007) 167–172
www.elsevier.com/locate/pla
Self-organization of intense light within erosive gas discharges
V.P. Torchigin ∗ , A.V. Torchigin
Institute of Informatics Problems, Russian Academy of Sciences, Nakhimovsky prospect 36/1, 119278 Moscow, Russia
Received 21 April 2006; received in revised form 31 July 2006; accepted 8 September 2006
Available online 20 September 2006
Communicated by F. Porcelli
Abstract
Process of appearance of fire balls at gas discharges is considered. It is shown that the intense white light radiated by atoms excited at gas
discharge is subject to self-organization in such a way that miniature ball lightnings appear.
© 2006 Published by Elsevier B.V.
PACS: 42.65.Jx
Keywords: Self-organization; Optical solitons; Space solitons; Optical quadratic nonlinearity; Whispering gallery waves; Propagation
1. Introduction
Studying a phenomenon of ball lightning (BL), we put forward a hypothesis that BL is a light bubble comprising of a
thin spherical layer of conventional compressed air where an
intense light is circulating in all possible directions [1]. Since
the refraction index n of the compressed air is greater than that
of surrounding air, the layer shows itself as a planar lightguide
which curvature is different from zero. Such lightguide can confine the light launched into it. In turn, the intense light produces
the electrostriction pressure which tends to near air molecules
and provides the air compression. Looking for relations between the light bubble and other nonlinear optical objects, we
concluded that the light bubble can be considered as an optical
incoherent space soliton (OISS) [2]. Unlike known plane OISS
which curvature is equal zero, the curvature of the light bubble
is different from zero.
Studying theoretically a behavior of light bubbles in the
air atmosphere, we showed that their behavior coincides completely with the mysterious and puzzling behavior of BLs observed by many eyewitnesses. It has been explained how a light
bubble can bypass obstacles, find out holes in walls and chimneys at roofs and penetrate through them in rooms changing its
* Corresponding author.
E-mail address: [email protected] (V.P. Torchigin).
0375-9601/$ – see front matter © 2006 Published by Elsevier B.V.
doi:10.1016/j.physleta.2006.09.016
shape. Penetration of light bubbles through window panes has
been explained also. At last, it has been shown how a light bubble can catch up a flying airplane and penetrate in its salons.
The explanations are presented in [3,4] and there is no necessity to repeat them once more. Explanation of these puzzles is
unthinkable for other known theories.
It turned out that BLs are not single representatives of light
bubbles. As was shown, so-called autonomous objects (AOs)
produced in a laboratory in last two centuries at attempts to obtain artificial BLs are light bubbles too [5]. As a rule, AOs are
produced by means of so-called erosive gas discharges at which
an evaporation of electrodes in a discharge gap takes place. Reasons of a favorable action of the erosive discharge on a process
of AOs appearance were explained and a mechanism of AO formation was analyzed [6]. It was shown that AOs appear due
to instability of the homogeneous intense light produced at gas
discharges. A further analysis of this problem enables us to conclude that a term “self-organization” is more suitable for this
case. It is more correct to tell, that AOs appear from the homogeneous intense light generated at gas discharges owning to
“self-organization” of such light.
The purpose of the Letter is to show that the self-organization
of intense light takes place also in numerous experiments where
balls of fire were observed. Unlike autonomous objects which
can exist independently after ceasing a gas discharge during a
fraction of a second, fire balls exist only during process of a gas
discharge and disappear immediately after its termination. In
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Fig. 1. Dependence of the refractive index n within ball on the distance r from its center. (a) Initial state; (b) after self-organization.
the same time fire balls can be observed within several minutes
when a gas discharge takes place. Numerous investigations of
fire balls were carried out in last two centuries but at present
there is no general accepted explanation of their physical nature.
In the first section we analyze a model of processes responsible for appearance of a fire ball in a discharge gap. We consider
a process of self-organization of intense light where an occasional fluctuation of the gas refraction index is transformed
gradually into a shell of light bubble. We show that a fire ball
can be considered as a light bubble which life time is small
because of an insufficiently great increase in the refraction index in the shell of the light bubble. As a result, radiation losses
in such light bubble are great and its life time is small. After
that we analyze possible mechanisms responsible for appearance of fire balls of relatively great diameter. In the last section
an explanation of available experimental data on the base of
theoretical results obtained in previous sections is presented.
2. Model of appearance of light bubbles within
homogeneous intense light
Consider a fluctuation of gas density appeared as a result
of a chaotic motion of gas molecules. Such fluctuations appear
in any gas constantly and they are responsible for the known
optical effect of molecular light scattering. Let for the sake of
simplicity a form of the fluctuation is a ball which radius r0
is essentially greater than light wavelengths are. Since the refraction index n of a gas is proportional to the gas density, the
dependence of n on the distance r from the ball center can be
presented by the curve shown in Fig. 1(a). It is supposed that
the spherical layer of thickness r is a transitive layer where n
decreases linearly from n0 + n within the ball to n0 outside it.
Let the intense light radiated by excited atoms is homogeneous
in the region where the fluctuation is located. This means that
there are no preferable directions and the densities of the light
energy in all points are identical. These assumptions simplify
considerations significantly because the light intensity does not
depend on coordinates.
Trajectories of light beams propagating in a cross-section of
the ball by plane z = 0 are shown in Fig. 2. Similar picture takes
place in any cross-section because of central symmetry of the
considered configuration. As is known any light beam propagating in an inhomogeneous optical medium bends in the direction
where the refraction index increases. The longer the light beam
propagates in an inhomogeneous medium the greater its bend-
Fig. 2. Trajectories of light beams in the transitive layer.
ing is. In Fig. 2 this concerns the beams which propagate in the
transitive layer of thickness r perpendicular to the ball radius.
We will consider such beams only.
The radius R of curvature of the beam is determined as follows
R −1 = grad(n) sin θ,
(1)
where θ is an angle between directions of grad(n) and the beam.
In the considered case
R −1 ≈ n/r.
(2)
Clearly, that the bend of beams to the center of the ball entails an increase in the light intensity inside the transitive layer.
Indeed, in this case any cross-section of the transitive layer is
crossed by the greater number of light beams entered the transitive layer as compared with the case where R −1 = 0. One can
see that an increase in the light intensity is proportional to the
length of the trajectories of beams which are in the transitive
layer.
An increase in the light intensity in the transitive layer can
cause an increase in its refraction index n if the optical medium
is nonlinear. In turn, the increase in n entails an increase in the
light intensity and so on. As a result a light bubble can appear
where a character of dependence n(r) is shown in Fig. 1(b).
There are several mechanisms which provide a nonlinearity
of optical medium in the case. These are the well-known optical
Kerr-effect and the optical electrostriction effect in gases. These
are new nonlinear optical effects discovered at analysis of experimental results obtained at investigations of AOs properties.
First of them takes place in gas mixtures and is connected with
the fact that molecules of gas mixture component with maximal
V.P. Torchigin, A.V. Torchigin / Physics Letters A 361 (2007) 167–172
169
Fig. 4. Dependence of the intensity within transitive layer IT on the surrounding
intensity I0 (r0 = 10−5 m, r = 10−6 m, n = 10−4 , n2 = 10−5 cm2 /W).
Fig. 3. Schematic diagram for calculation of length of a light beam in the transitive layer.
n are involved in the area where an intense light propagates. The
second one is connected with the fact that electrons of low temperature plasma are pushed out from the area where an intense
light propagates. As a result, n in the area increases. The electrons attract positive ions and, therefore, plasma is pushed out
from the area [7].
We will suppose that an action of all these mechanisms results in an increase in n by δn at an increase in the light intensity
by I and the following relation takes place
δn = n2 I,
(3)
where n2 is the index of the total nonlinearity when a joint action of the mechanisms takes place.
Let us determine an increase in the light intensity in the transitive layer. We have from the drawing shown in Fig. 3,
Cos ϕ = 1 − r(r/R)/(R − r0 + r).
(4)
≈ 1 − ϕ 2 /2
Under assumption that ϕ < 1 and Cos ϕ
from (4) and (2)
−1/2
.
ϕ ≈ 21/2 (n) 1 − n(r0 − r)/r
we have
(5)
Taking into account that an increase C in the light intensity
in the transitive layer is proportional to ϕ, parameter C can be
expressed as follows
1/2
C = (n + δn)/n 1 − (r0 − r)n/r
−1/2
× 1 − (r0 − r)(n + δn)/r
.
(6)
As is seen, C = 1 at δn = 0 and C tends to infinite as the
term in the last square brackets tends to zero. This takes place
if the term (n + δn)/r tends to (r0 − r)−1 . Taking into
account (2), we can conclude that C tends to infinity when the
radius of curvature R = (n + δn)/r for light beams in the
transitive layer tends to the radius of the transitive layer r0 .
Denoting the light intensity in the transitive layer by IT , we
have IT = CI0 where I0 is the light intensity in surrounding
space. Then from (6) and (3) we obtain
1/2
IT = I0 (1 + n2 IT )/n 1 − (r0 − r)n/r
−1/2
× 1 − (r0 − r)(n + n2 IT )/r
.
(7)
This expression determines dependence between I0 and IT .
For the sake of illustration a particular case of the dependence is
presented in Fig. 4. An increase in IT occurs more quickly than
increase in I0 . This is explained by the fact that the increase in
I0 is accompanied by an increase in the length of the light trajectory resided in the transitive layer. When the length becomes
equal to the length of the transitive layer the light beam trajectory becomes closed and the light beam can circulate repeatedly
around the transitive layer. The length of the beam trajectory
resided within the transitive layer increases many times over in
this case. The transitive layer shows itself as a planar lightguide
which curvature is different from zero. As is known, similar
lightguides can confine the light launched within them. The
light can circulate in the lightguide during some time after termination of gas discharge when the intensity I0 becomes equal
to zero. Such situation takes place for autonomous objects considered in [8].
A wave approach is more appropriate for analysis of light
circulation than the beam approach applied above. In this case
ought to consider light waves of whispering gallery type circulating in the transitive layer in all possible directions. Such
waves are used in optical resonators of whispering gallery
type [9]. A glass balls of several ten micrometers in diameter
are used as such resonator. The refraction index of glass is about
n ≈ 1.45. In this case the difference of the refraction indexes
of glass and surrounding space n ≈ 0.45. This is very great
difference which provides a safe confinement of light waves
within glass balls. The difference n in a fire ball is essentially
smaller than that in a glass ball. The more the difference the
better confinement is. Since the difference is proportional to the
light intensity in the transitive layer, the more light intensity the
more confinement is. This conclusion corresponds to the conclusion derived from the analysis based on the beam approach.
Ought to add a single specification that a confinement increases
continuously with an increase in background light intensity I0
and there is no jump at the moment when a light beam in the
transitive layer becomes closed one.
Since fire balls disappear immediately after ceasing gas discharge, the light intensity IT in the transitive layer is insufficient
to form a lightguide with small radiation losses. Nevertheless,
the self-organization of intense light takes place in fire balls also
because occasional fluctuation of gas density is transformed in
the light bubble. The light intensity IT in the shell of the light
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Fig. 5. Instability of two identical light bubbles (dotted lines show boundaries
between light beams propagating in different bubbles. (a) Two identical bubbles; (b) light intensity in the right bubble is greater than that in the left one; (c)
light intensities are similar in the bubbles.
bubble can be essentially greater than the background light intensity I0 .
3. Merge of light bubbles
The cross-section size of fluctuations of gas density which
are germs of future fire balls are small as compared with diameters of fire balls observed by experimenters. The same situation
is for AOs. Results of investigation of interaction between AOs
and liquid nitrogen show that AO diameter can be smaller than
ten micrometers [10]. In the same time experiments on AOs
show that there are AOs which sizes are comparable with the
cross-section of the set-up used for AOs production. Mechanism of appearance of great AOs was considered in [6]. It was
shown that small AOs can merge at their contact and, as a result,
AO of greater diameter can appear. Since the physical natures
of both AOs and fire balls are the same and all these objects can
be considered as light bubbles, the same effect can take place
for fire balls too.
Let us show that a steady-state of two identical contacting
light bubbles (Fig. 5(a)) is instable. Suppose that there is some
disturbance at which the light intensity in the right bubble is
increased. In this case the air density in its transitive layer increases too and the boundary between the beams propagating to
the right and to the left shifts to the left as is shown in Fig. 5(b)
by dotted line. A part of the light circulating in the left bubble passes to the right one and the initial misbalance increases.
As a result, all light of the left bubble passes to the right one.
The compressed air in the left bubble remains because transient
processes connected with the light are essentially faster than
that connected with compression of the air. The first ones are
determined by the light speed c whereas the second ones are
determined by the sound speed u. As is known, c/u ≈ 106 .
Having lost the electrostriction pressure exerted by the light,
the compressed air in the left bubble begins to expand. As a
result, a crash can be heard. As for the right bubble, that the increase in the light intensity entails an increase in the air pressure
in the transitive layer by means of attraction in the bubble shell
additional air molecules from surrounding space. As a result the
bubble diameter increases.
It is easy to check, that two contacting bubbles of different diameters with identical light intensity are instable too. The
light passes to the bubble with the greatest diameter. The same
is valid for bubbles shown in Fig. 5(c). Thus, a relatively great
final light bubble which is essentially greater than initial bubbles can be formed in a process of gas discharge.
There are the following evidences in favor of merging small
light bubbles. These bubbles are characterized various radiation losses for various wavelengths. It is connected with the fact
that a ratio d/λ is relatively small for them. Here d is the bubble diameter, λ is the wavelength of the light circulating in the
bubble shell. Radiation losses in such bubbles increase with a
decrease in the ratio d/λ [11]. As a result, light waves from
the red region of the spectrum leave the bubble shell faster than
light waves from the blue region and bubble color shifts in blue
side of the spectrum. What is why liquid nitrogen with small
light bubbles penetrated within it is blue–green in color [10].
Since a source of light waves circulating in a shell of great final
bubble is waves of small light bubbles, a color of the final bubble should be moved in the blue side of spectrum too. Indeed,
the shell of great bubble in experiments with ultrasonic streams
is violet in color [12].
4. Analysis of experimental results
Luminous balls at gas discharges were observed by experimentalists over last two centuries. Surveys of these experiments
are presented by Singer [13] and Barry [14]. Before an explanation of experiments on the base of the presented model it
is expedient to remind properties of light bubbles concerned
a specificity of their movement in space. They move always in
the direction of the gradient of the refraction index in the space
where they are located. There is the following contradiction for
fire balls. On the one hand, a fire ball should be located in the
region of intense light to accumulate it in its shell. On the other
hand, the temperature in the region is greater than that in surrounding space. As a result, the refraction index in the region
is smaller than that in surrounding space with other things being equal. In this case the fire ball should leave the region. The
contradiction is authorized as follows. An erosive gas discharge
is used. In this case additional portions of substance evaporated
from electrodes are introduced in the region. In this case the refraction index in the region increases and can be maximal at
a corresponding regime of evaporation. In this case a fire ball
resides in the region. This reception is used in all experiments
known to the authors.
For example, very pictorial experiments were carried out by
Plante in 1875–1890 [14]. He made observations with two par-
V.P. Torchigin, A.V. Torchigin / Physics Letters A 361 (2007) 167–172
allel flat dump surfaces separated by an air gap. The dump
surfaces were formed by pads or filter paper disks moistened
with distilled water. When the pads were connected across the
capacitor and battery poles, a small ball of fire discharge formed
between the two surfaces. The ball of fire occurred between
moist areas and would not occur between dry areas. The discharge moved about the surfaces randomly but remained between damp areas. The ball-of-fire discharge would continue
between the damp surfaces until the voltage source could no
longer support the discharge. The ball disappears immediately
after ceasing the discharge.
The phenomenon can be explained as follows. A final light
bubble is forming in the discharge gap due to merging initial
small light bubbles which appear due to the self-organization
of an intense light. The light intensity I0 produced by atoms
excited at the discharge within light bubbles is insufficient to
provide a significant increase in the refraction index within
bubble shells. In this case both the initial light bubbles and final light bubble disappear as soon as the discharge disappears.
A behavior of the resulting bubble in process of gas discharge
is identical to that of the light bubble which can exist for itself. Both bubbles move in the direction where the air density
is maximal. If the filter paper becomes dry owning the high
temperature within gas discharge, the balls move in the region
where a conversion of water into vapor takes place. The evaporation increases the air density not only because of entering
water vapor in the discharge region but also due to a decrease
in the air temperature. This fact explains a random motion of a
fire ball between plates covered by damp filter paper. The ball
cannot stay on the same place because the filter paper in the
region where the ball is located becomes dry and the air density decreases. On the contrary, the air density in the adjacent
region where the filter paper is damp increases owning an evaporation of water, and the ball is forced to move to the region. A
characteristic crackling noise produced by gas discharges can
be explained by sharp expansion of the air compressed in the
shells of initial small light bubbles at their merging. When intense light passes from one bubble to others, the compressed
air in the first bubble begins to expand. The expansion causes
crackling.
Similar experiments were repeated many times by other investigators. For example, Hesehus [15] extended the experiment technique of Plante and used a transformer to produce
a 104 -V alternating current source. A copper plate and water
surface with a separation of 2–4 cm were used as electrodes.
Rays, ball of fire, flames, conical, oval and spherical forms and
images were produced, similar to those produced by Plante.
Leduc [16] used a photographic plate to trace the path and
discharge characteristics of the discharge. Luminous spheres
appeared on a negative electrode if two electrodes in a form
of thin metal edges were placed on photosensitive layer of photographic plate. After separation of the sphere from the negative
electrode, the edge of the electrode became dark and a small luminous ball slowly moved to the positive electrode. From 1 to
4 minutes are required for the ball to pass the distance about
5–10 cm. A trajectory of the ball was extremely complex and
unpredictable.
171
This phenomenon can be explained as follows. The temperature of the photosensitive layer in the region where the fire
ball is located is maximal because the region heats up light radiation from the ball. This entails evaporation of the substance
from the layer and the gas density in the region is maximal.
When the evaporating substance will be exhausted, evaporation
takes place from some adjacent region and the ball shifts to the
region. The ball never returns to a former region because there
can be no evaporation after the ball leaved the region. What is
why crossed lines are absent at the trajectory of the ball. When
the fire ball achieved the positive pole, the ball disappeared and
the source of a current started to behave as if its poles would be
connected by a conductor. It testifies that the fire ball changed
properties of a photosensitive plate. The regions through which
it had passed became conductors.
Toepler [17] showed that a fire ball tends to preserve its integrity. Moving between electrodes, it bypasses plates located
at its way and penetrates though small holes in these plates. It
is not surprising because a fire ball as any light bubble tends
to preserve its form and integrity. Mechanisms responsible for
penetration of light bubbles through holes and bypassing obstacles are considered in [3,4].
Exhaustive experiments were conducted in the mid-1950s
by Nauer [14] in order to duplicate in a generic fashion most of
the earlier experiments and evaluate the results with respect to
electric discharge and plasma theory. Fire balls were obtained
in the manner of Plante when both electrodes were damp paper
pads. The ball was highly mobile and bright orange in color.
Hesehus experiments were also repeated and confirmed. Nauer
showed that the electrical and thermal conductivity of the flat
plate apparently influenced the discharge form and appearance
to some degree.
A little bit other situation takes place in numerous experiments carried out in 1990th. Erosive gas discharges of several
milliseconds duration produced by means of discharge of a battery of capacitors were used for production of relatively long
lived light bubbles. Electrodes in the discharge gap were covered by the substance which was evaporated in process of the
discharge. A drop of water [18], wax, polymers, cotton, shaving
of a tree [19], various metals [20] were used as the evaporated
substance. AOs flying from the discharge gap were observed.
Their life-time after ceasing the discharge is about a fraction of
a second. In this case the light intensity in the discharge gap I0
is great enough and the lifetime of AOs is greater by two orders
of magnitude than that of white light in the conventional air atmosphere. It is explained by a specificity of the molecular light
scattering in AO shell. Unlike conventional light scattering in
3D space where scattering losses are irreplaceable, scattering
losses in AO shell are replaceable because a great part of the
scattering light scatters in the same shell and continues to circulate within it. The light lifetime increases by a factor of about
two orders of magnitude in this case.
Very pictorial are experiments where a ditch with liquid nitrogen is placed near a discharge gap [10]. In this case the air
density gradient is directed towards the surface of the liquid nitrogen and the gradient value is significantly greater than that in
other experiments. AOs produced at gas discharge move to the
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V.P. Torchigin, A.V. Torchigin / Physics Letters A 361 (2007) 167–172
surface and penetrate into the liquid nitrogen which begins to
shine by blue–green color. The phenomenon is explained in [5].
Ought to note that shining of the liquid nitrogen was observed
not only at a discharge of a battery of capacitors but also at a
flash of conventional power flash lamp used at photographing
(the energy of the battery of capacitors was 300–600 J) [10].
A further increase in the lifetime within AO shell can be obtained by a significant increase in the light intensity within a
discharge gap. Such physical conditions take place at strikes
of conventional natural linear lightning. In this case the air
pressure in the shell increases in such a degree that air molecules occur packed close together. This entails a significant
increase in the light lifetime because fluctuations of the air density which are responsible for light scattering disappear almost
completely [1].
There are strong reasons to suspect that features of vacuum
discharges can be explained by appearance of AOs and fire
balls. Vacuum in a discharge gap takes place at initial phase
only. At a steady-state there are vapors of metals from a cathode. As is known, a vacuum discharge can be accompanied by
appearance of avalanches of 109 –1011 electrons called by ectons [21]. An appearance of the avalanches can be explained as
follows. Vapors of metals and intense light are favorable for an
appearance of AOs. An interaction of AOs with a metal sheet
has been investigated experimentally [16]. It was shown that
AOs are attracted to a metal sheet. They can burn through the
sheet if its thickness is small or they can burn out a crater on the
sheet surface if AOs energy is insufficient for burning through.
This phenomenon can be explained easily [2]. Indeed, nearing
the sheet surface, AO evaporates a metal because AO heats the
surface due to AO light radiation. Setting down in the region
with maximal gas density, AO evaporates the metal until AO
energy exhausts. Heating the cathode is accompanied by the
conventional phenomenon of thermo electronic emission and
electrons emit from the cathode. When AO energy exhausts, AO
disappears and thermo electronic emission ceases. As a result,
an avalanche of electrons is formed. It is unknown another reasonable explanation of ceasing the electron emission. Ought to
note, that characters of craters formed by AOs at erosive gas discharges on the surface of a metal sheet [22] and craters formed
on the cathode surface at vacuum discharge [21] are identical.
A description of a random motion of fire-ball in Plante’s
experiments reminds a description of a random motion of a
cathode spot at vacuum discharges [23]. Possibly, light bubbles
are responsible for appearance of not only fire-balls but also
cathode spots.
5. Conclusion
Recognition of the fact that there are light bubbles in the
nature enables us to explain many phenomena which were considered as anomalous ones till now. These are not only puzzles
of natural ball lightnings but also mysterious properties of autonomous objects. In the Letter an attention was focused on
properties of nonautonomous objects which exist in time of a
gas discharge only. It was shown that physical nature of these
objects as well as both autonomous objects and natural ball
lightnings are the same. A difference is in their sizes and stored
energy. A phenomenon of a self-organization of intense light
leads to the conclusion which changes radically Kirchhoff’s notions about properties of equilibrium light radiation. At great
enough light intensity in a gas a state of equilibrium is impossible because of a self-organization of the intense light. Possibly,
these processes take place within sun and stars.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
[23]
V.P. Torchigin, Phys. Dokl. 48 (3) (2003) 108.
V.P. Torchigin, A.V. Torchigin, Opt. Commun. 240 (4–6) (2004) 449.
V.P. Torchigin, A.V. Torchigin, Phys. Scr. 68 (2003) 388.
V.P. Torchigin, A.V. Torchigin, Phys. Lett. A 328 (2–3) (2004) 189.
V.P. Torchigin, A.V. Torchigin, Phys. Dokl. 49 (10) (2004) 553.
V.P. Torchigin, A.V. Torchigin, Phys. Lett. A 337 (2005) 112.
V.P. Torchigin, A.V. Torchigin, Europhys. Lett. D 32 (2005) 383.
V.P. Torchigin, A.V. Torchigin, Europhys. Lett. D 36 (2005) 319.
S.M. Spillane, T.J. Kippenberg, K.J. Vahala, Nature 415 (2002) 621.
C.K. Dimitrov, et al., in: R.F. Avramenko (Ed.), Ball Lightning in a Laboratory, Himiya, Moscow, 1994, pp. 79–86.
A.N. Oraevskiy, Quantum Electron. 32 (5) (2002) 377.
A.I. Klimov, G.I. Mishin, Pis’ma Zh. Tekh. Fiz. 19 (13) (1993) 19.
S. Singer, The Nature of the Ball Lightning, Plenum, New York, 1971.
J.D. Barry, Ball Lightning and Bead Lightning, Plenum, New York, 1980.
N.A. Hesehus, Zh. Rus. Chem.–Phys. Soc. 8 (1900) 311.
S. Leduc, Comptes Rendus 129 (1899) 37.
M. Toepler, Ann. Phys. IV 6 (1901) 339.
A.I. Egorov, S.I. Stepanov, Zh. Tekh. Fiz. 72 (12) (2002) 102.
V.L. Bychkov, A.V. Bychkov, I.B. Timofeev, Zh. Tekh. Fiz. 74 (1) (2004)
128.
S.E. Emelin, V.S. Semenov, V.L. Bychkov, N.K. Belisheva, A.P. Kovshil,
Zh. Tekh. Fiz. 67 (3) (1997) 19.
G.A. Mesyats, Usp. Fiz. Nauk 165 (6) (1995) 601.
R.F. Avramenko, V.I. Nikolaevs, L.P. Poskacheva, in: R.F. Avramenko
(Ed.), Ball Lightning in a Laboratory, Himiya, Moscow, 1994, pp. 95–112.
V.A. Nemchinskiy, Cathode spot, in: A.M. Prokhorov (Ed.), Physical Encyclopedia, vol. 2, Sovetskaya Encyclopedia, Moscow, 1990, p. 246.