# Signatures of room-temperature superconductivity emerging in two-dimensional
domains within the new $\rm{Bi/Pb}$-based ceramic cuprate superconductors at
ambient pressure

S. Dzhumanov1<EMAIL_ADDRESS>D.D. Gulamova2 Sh.S. Djumanov3 1Institute of
Nuclear Physics, Uzbek Academy of Sciences, 100214, Ulugbek, Tashkent,
Uzbekistan  2Institute of Material Science, Uzbekistan Physics-Sun Scientific
Production Association, Uzbek Academy of Sciences, Tashkent, 100084,
Uzbekistan  3Tashkent State Technical University, Tashkent 100095, Uzbekistan

###### Abstract

We predict the possibility of realizing room-temperature superconductivity in
different two-dimensional (2D) domains within the ceramic high-$T_{c}$ cuprate
superconductors at ambient pressure and experimentally confirm this prediction
of 2D room-temperature superconductivity in the newly derived
$\rm{Bi/Pb}$-based ceramic cuprate superconductors containing many grain
boundaries, interfaces and multiplate blocks. We argue that, in these
high-$T_{c}$ materials, besides bulk superconductivity in three-dimensional
(3D) domains there is also strongly enhanced 2D superconductivity emerging in
the 3D-2D crossover region well above the superconducting transition
temperature $T_{c}$. We study the possibility of the existence of distinct 3D
and 2D superconducting phases in high-$T_{c}$ ceramic cuprates, in which the
unconventional (tightly-bound polaronic) Cooper pairs behave like bosons and
condense below certain critical temperatures into 3D and 2D Bose superfluids
in 3D and 2D domains. We show that the superconducting transition temperature
in 2D domains is much higher than in 3D domains and can reach up to room
temperature. We report signatures of room-temperature superconductivity
occurring at different grain boundaries and 3D/2D interfaces and in multiplate
blocks within the ceramic superconductors
$\rm{Bi_{1.7}Pb_{0.3}Sr_{2}Ca_{n-1}Cu_{n}O_{y}}$ (where $n=2-30$), synthesized
by using the new melt technology in a large solar furnace (in Parkent). The
samples of these materials synthesized under the influence of concentrated
solar energy have the bulk $T_{c}$ values ranging from 100 K to about 140 K
and the more higher superconducting transition temperatures, possibly even as
high as room temperature in the 3D-2D crossover region. The remnant 2D
superconductivity in newly derived $\rm{Bi/Pb}$-based ceramic cuprate
superconductors is observed at temperatures $T\gtrsim 200-300$ K well above
the bulk $T_{c}$ and the onset of room-temperature superconductivity is
evidenced by the observations of a sharp step-like drop in the resistance and
a well-detectable partial Meissner effect at around 300 K and ambient
pressure.

###### keywords:

High-$T_{c}$ cuprate superconductors , New Bi/Pb based ceramic cuprate
superconductors , Grain boundaries and interfaces , Multiplate blocks , Bose-
liquid superconductivity , Three- and two-dimensional superconducting phases ,
Room-temperature superconductivity

††journal: arXiv

##  1 Introduction

Remarkable progress in condensed matter physics is often driven by discoveries
of new superconducting materials and superfluid (Fermi or Bose) liquids. In
particular, the discovery of $\rm{La}$-based high-$T_{c}$ cuprate
superconductors [1, 2, 3] and then the dramatic increase of the critical
temperature $T_{c}$ of the superconducting transition in other discovered
$\rm{Y}$-, $\rm{Bi}$\- and $\rm{Hg}$-based cuprate superconductors to above 90
K [4], 110 K [5] and 130 [6] ushered in a new era of physics and technology.
These important discoveries stimulated efforts to find new materials with even
higher superconducting transition temperatures, possibly even close to room
temperature. Various experiments showed [3, 7, 8] that the superconductivity
is occurred in the bulk three-dimensional (3D) cuprate material rather than in
the $\rm{CuO_{2}}$ plane. Therefore, the above values of $T_{c}$ observed in
different high-$T_{c}$ cuprates correspond to the bulk superconducting
transition temperatures.

Starting from 1993, there has been considerable interest in the increase of
$T_{c}$ in some families of cuprate superconductors to a maximum value at
different applied pressures [9, 10, 11] and the record values of $T_{c}\simeq
153-164$ K reported for $\rm{Hg}$-based cuprate superconductors are remained
still unchanged. For a long time, despite intense research effors (see
Refs.[12, 13, 14, 15, 16]), the increasing of $T_{c}$ up to room temperature
in various high-$T_{c}$ cuprates was remained a difficult problem. So far,
room-temperature superconductivity was not reported for high-$T_{c}$ cuprate
materials even under pressure.

Recently, there has been growing interest in the discovery of room-temperature
superconductivity in other classes of materials and the new so-called
high-$T_{c}$ hydrides have been synthesized at very high pressures [17, 18,
19]. However, room-temperature superconductors discovered under high pressures
will not have a wide-range practical applications. Therefore, in the last
decade, new attempts were made to synthesize the promising ceramic cuprate
superconductors with the onset temperature of the superconducting transition,
$T_{c}^{onset}\gtrsim 150-180$ K at ambient (atmospheric) pressure, by using
more advanced technology [20, 21]. Prominent among high-$T_{c}$ cuprate
superconductors are the so-called $\rm{Bi/Pb}$-based ceramic cuprate
compounds, such as $\rm{Bi_{1.7}Pb_{0.3}Sr_{2}Ca_{n-1}Cu_{n}O_{y}}$ (with
$n=2-30$), which may present an alternative path to realizing room-temperature
superconductivity at atmospheric pressure.

For many decades, there has been speculation (see, e.g., Refs.[14, 18]) that
Little’s model of high-$T_{c}$ superconductivity in one-dimensional organic
polymers with polarizable side chains [22] and Ginzbur’s model of two-
dimensional (2D) alternating conducting/insulating sandwich layers [23] may be
possible routes to room-temperature superconductivity. Such early theoretical
predictions are based on the Bardeen-Cooper-Schrieffer (BCS)-like theory of
Fermi-liquid superconductivity. Still, most researchers trying to find the new
high-$T_{c}$ superconductors make predictions on the basis of the BCS-like and
Migdal-Eliashberg theories of Fermi-liquid superconductivity regarding the
possibility of increasing the critical temperature $T_{c}$ up to room
temperature (see Refs.[15, 16, 17]). However, the high-$T_{c}$ cuprates
undergoing a $\lambda$\- superconducting transition at $T_{c}$ [24, 25] just
like $\lambda$-transition in liquid 4He can be in the regime of Bose-liquid
superconductivity [26, 27] and the BCS-like theories are incapable of
predicting the actual $T_{c}$ in such unconventional superconductors.

In this work, we present theoretical and experimental results of room-
temperature superconductivity emerging in different 2D domains within the
newly derived $\rm{Bi/Pb}$-based ceramic cuprate superconductors containing
many grain boundaries, interfaces and mutiplate blocks. We show that, in these
ceramic high-$T_{c}$ materials, besides bulk 3D superconductivity there is
also strongly enhanced 2D superconductivity occurring well above the bulk
$T_{c}$ in the 3D-2D crossover region. We argue that the unconventional Cooper
pairs in high-$T_{c}$ cuprates behave like bosons and condense below certain
critical temperatures into 3D and 2D Bose superfluids in 3D and 2D domains. We
examine the possibility of the existence of 3D and 2D superconducting phases
in ceramic cuprate superconductors predicted by the theory of 3D and 2D Bose
superfluids. We find that superconducting transition temperature in 2D domains
is much higher than that in 3D domains and the highest critical temperature
$T_{c}$ for some high-$T_{c}$ cuprates, e.g., $\rm{Bi/Pb}$-based ceramic
cuprate superconductors can reach up to room temperature in 2D domains. We
report experimental signatures of room-temperature superconductivity emerging
at different grain boundaries and interfaces and in multiplate blocks within
the ceramic cuprate superconductors
$\rm{Bi_{1.7}Pb_{0.3}Sr_{2}Ca_{n-1}Cu_{n}O_{y}}$ (with $n=2-30$), synthesized
by using the new melt technology under the influence of concentrated solar
energy. We claim that the occurrence of such a room-temperature
superconductivity in these high-$T_{c}$ materials is evidenced by the
observations of a sharp drop in the resistance and a pronounced partial
Meissner effect at around 300 K and ambient pressure.

##  2 Theoretical blackground

The cuprate compounds are typical polar materials and charge-transfer-type
Mott-Hubbard insulators [28]. Therefore, in these materials the charge
carriers (holes or electrons) introduced by doping in a polar crystal strongly
interact with optical phonons [29] and their self-trapping is favorable just
like the self-trapping of holes in ionic crystal of alkali halides [30, 31].
In doped high-$T_{c}$ cuprates, the Cooper pairing of self-trapped carriers
(polarons) results in the formation of tightly-bound (bosonic) Cooer pairs
[32]. These unconventional Cooper pairs in 3D high-$T_{c}$ cuprates condense
into a 3D Bose superfluid below the bulk $T_{c}$ [32, 33]. We now consider the
ceramic cuprate superconductors consisting of 3D and 2D domains. Then the
actual $T_{c}$ in 3D domains is determined from the self-consisting solutions
of the integral equations of a 3D Bose superfluid [34]

$\displaystyle\frac{1}{\gamma_{B}}=\int^{\xi_{BA}}_{0}\sqrt{\varepsilon/\xi_{BA}}\frac{coth\left[\sqrt{(\varepsilon+\tilde{\mu}_{B})^{2}-\Delta^{2}_{B}}/2k_{B}T\right]}{\sqrt{(\varepsilon+\tilde{\mu}_{B})^{2}-\Delta^{2}_{B}}}d\varepsilon,$
(1)
$\displaystyle\frac{2\rho_{B}}{D_{B}}=\int^{\infty}_{0}\sqrt{\varepsilon}\times$
$\displaystyle\times\left\\{\frac{\varepsilon+\tilde{\mu}_{B}}{\sqrt{(\varepsilon+\tilde{\mu}_{B})^{2}-\Delta^{2}_{B}}}coth\left[\frac{\sqrt{(\varepsilon+\tilde{\mu}_{B})^{2}-\Delta^{2}_{B}}}{2k_{B}T}\right]-1\right\\}d\varepsilon,$
(2)

where $\gamma_{B}=D_{B}\tilde{V}_{B}\sqrt{\xi_{BA}}$ is the interboson
coupling constant in a 3D Bose superfluid, $\rho_{B}$ is the density of
attracting (superfluid) bosons, $D_{B}=m_{B}^{3/2}/\sqrt{2}\pi^{2}\hbar^{3}$
is the density of states, $m_{B}$ is the mass of free bosons,
$\tilde{\mu}_{B}$ is the renormalized chemical potential of an interacting
Bose gas,
$\tilde{V}_{B}=V_{BA}-V_{BR}/[1+D_{B}V_{BR}(\sqrt{\xi_{BR}}-\sqrt{\xi_{BA}})]$
is the effective interaction potential between bosons, $\xi_{BA}$ and
$\xi_{BR}$ are the cutoff parameters for the attractive $V_{BA}$ and repulsive
$V_{BR}$ parts of the interboson interaction potential, $\Delta_{B}$ is the
coherence parameter (i.e. superfluid order parameter) of condensed bosons.

The critical temperature $T_{c}^{2D}$ of the superconducting transition in 2D
domains is determined from the self-consistent solutions of the integral
equations of a 2D Bose superfluid [34]

$\displaystyle\frac{1}{\gamma_{B}}=\int^{\xi_{BA}}_{0}\frac{coth\left[\sqrt{(\varepsilon+\tilde{\mu}_{B})^{2}-\Delta^{2}_{B}}/2k_{B}T\right]}{\sqrt{(\varepsilon+\tilde{\mu}_{B})^{2}-\Delta^{2}_{B}}}d\varepsilon,$
(3) $\displaystyle\frac{2\rho_{B}}{D_{B}}=\int^{\infty}_{0}\times$
$\displaystyle\times\left\\{\frac{\varepsilon+\tilde{\mu}_{B}}{\sqrt{(\varepsilon+\tilde{\mu}_{B})^{2}-\Delta^{2}_{B}}}coth\left[\frac{\sqrt{(\varepsilon+\tilde{\mu}_{B})^{2}-\Delta^{2}_{B}}}{2k_{B}T}\right]-1\right\\}d\varepsilon,$
(4)

where $\gamma_{B}=D_{B}\tilde{V}_{B}$ is the interboson coupling constant in a
2D Bose superfluid, $\rho_{B}$ is the density of 2D superfluid bosons,
$D_{B}=m_{B}/2\pi\hbar^{2}$.

Numerical solutions of Eqs. (1) and (2) determining the bulk
$T_{c}=T^{3D}_{c}$ in 3D superconducting domains can be obtained for arbitrary
$\gamma_{B}$. While the approximate analytical solutions of these equations
are obtained for $\gamma_{B}<1$ [34]. In particular, the bulk $T_{c}$ at
$\gamma_{B}>0.3$ can be determined approximately from the expression

$\displaystyle
T_{c}=T^{*}_{BEC}[1+c\gamma_{B}\sqrt{\sqrt{2}k_{B}T^{*}_{BEC}/\xi_{BA}}],$ (5)

where $T^{*}_{BEC}=3.31\hbar^{2}\rho^{2/3}_{B}/k_{B}m^{*}_{B}$ is the
renormalized Bose-Einstein condensation temperature in a 3D Bose-liquid, in
which $m_{B}$ is replaced by the renormalized mass $m^{*}_{B}$ of interacting
bosons defined similarly in Ref.[35], $c=\pi^{3/2}/3.918$.

The solutions of Eqs. (3) and (4) determining the critical temperature
$T^{2D}_{c}$ in 2D superconducting domains are obtained analytically [34] and
the superconducting transition temperature at grain boundaries and interfaces
and in multiplate blocks within the ceramic high-$T_{c}$ cuprate materials for
arbitrary $\gamma_{B}$ is determined from the following relation:

$\displaystyle
T_{c}^{2D}=-\frac{T_{0}^{*}}{ln[1-exp(-2\gamma_{B}/(\gamma_{B}+2))]},$ (6)

where $T^{*}_{0}=2\pi\hbar^{2}\rho_{B}/k_{B}m^{*}_{B}$.

We use the analytical expressions (5) and (6) to examine the possibility of
the existence of distinct 3D and 2D superconducting phases in ceramic
high-$T_{c}$ materials and to predict the existence of two distinctly
different regimes of high-$T_{c}$ superconductivity in them and the possible
route to room-temperature superconductivity. In order to determine the
characteristic superconducting transition temperatures in the bulk and the
3D-2D crossover region of high-$T_{c}$ cuprates, we estimate $T_{c}$ and
$T^{2D}_{c}$ by assuming that the mass $m_{p}$ of polaronic carriers in 3D
domains is of the order of $2m_{e}$ [36] (where $m_{e}$ is the mass of free
electrons) and in 2D domains is somewhat larger than $2m_{e}$. It is natural
to assume that, in high-$T_{c}$ cuprates, the preformed bosonic (polaronic)
Cooper pairs have the mass $m_{B}=2m_{p}$ and only the interacting bosons with
the renormalized mass $m^{*}_{B}>m_{B}$ can condense into Bose superfluids at
$T\lesssim T_{c}$ (in 3D domains) and $T\lesssim T^{2D}_{c}$ (in 2D domains).
By taking $m_{p}\simeq 2m_{e}$, $m_{B}=2m_{p}$, $m^{*}_{B}=1.05m_{B}$ and
$\rho_{B}\simeq 4\cdot 10^{19}$ $cm^{-3}$ for 3D superconducting domains in
ceramic cuprate materials, we find $T^{*}_{BEC}\simeq 81.7$ K. Using Eq.(5) we
further estimate the bulk $T_{c}$ by assuming that two bosonic Cooper pairs
interact with each other by means of exchange optical phonons having
relatively low energy $\hbar\omega_{0}\simeq 0.03$ eV and the cutoff energy
$\xi_{BA}$ for the attractive part of the interboson interaction potential can
be replaced by $\hbar\omega_{0}$. Then, at $\gamma_{B}=0.7$ the analytical
expression (5) for the bulk $T_{c}$ predicts the value of $T_{c}\simeq
1.508T^{*}_{BEC}\simeq 123$ K. We can now estimate the superconducting
transition temperature in 2D domains by assuming that $m_{p}\simeq 3m_{e}$,
$m_{B}=2m_{p}$, $m^{*}_{B}=1.05m_{B}$ and $\rho_{B}\simeq 2\cdot 10^{13}$
$cm^{-2}$. Under these assumptions, we obtain $T^{*}_{0}\simeq 176$ K. If we
take the same value of $\gamma_{B}=0.7$ for a 2D Bose superfluid in the
analytical expression (6), we find $T^{2D}_{c}\simeq 1.105T^{*}_{0}\simeq 195$
K. The above predictions of the theory of Bose-liquid superconductivity in
high-$T_{c}$ cuprates indicate that the expected value of $T^{2D}_{c}$ in the
3D-2D crossover region will be much higher than the expected value of $T_{c}$
in the bulk. Therefore, the 3D domains in ceramic high-$T_{c}$ cuprates become
non-superconducting above the bulk $T_{c}(=T^{3D}_{c})$, but high-$T_{c}$
superconductivity is still maintained in 2D domains in a wide temperature
range above $T_{c}$. Such a remnant 2D superconductivity persisting at grain
boundaries and interfaces and in multiplate blocks within the ceramic
high-$T_{c}$ cuprate materials might be encouraging in achieving room-
temperature superconductivity at atmospheric pressure. In order to make this
argument more quantitative, the results of our numerical calculations of the
critical superconducting transition temperature $T^{2D}_{c}$ as a function of
$\gamma_{B}$ are presented in Fig.1. As may be seen in Fig.1, at some values
of $\rho_{B}$ and $m_{B}$, the critical temperature $T^{2D}_{c}$ strongly
depending on $\gamma_{B}$ eventually reaches to room-temperature at a certain
value of $\gamma_{B}\gtrsim 0.8$.

Figure 1:  The calculated critical temperature $T^{2D}_{c}$ of the
superconducting transition in 2D systems as a function of interboson coupling
constant $\gamma_{B}$. The parameters used are $\rho_{B}\simeq 2.8\cdot
10^{13}cm^{-2}$, $m_{p}=3m_{e}$, $m_{B}=2m_{p}$ and $m^{*}_{B}\simeq
1.05m_{B}$. Figure 2:  Alternating 3D non-superconducting/2D superconducting
sandwich layers in ceramic cuprate superconductors above the bulk $T_{c}$.
Bold-type arrows indicate a crossover from 3D non-superconducting (metallic)
state to 2D superconducting state in these sytems.

Here we point out a possibility that many grain boundaries parallel to each
other, interfaces and parallel multiplate blocks in specially grown ceramic
cuprate superconductors (see, e.g., experimental results presented in Sec.3)
will be involved in alternating 3D non-superconducting/2D superconducting
sandwich layers above the bulk $T_{c}$. As can be seen in Fig.2, the crossover
from the bulk 3D regime to surface 2D regime of superconductivity in these
alternating 3D/2D sandwich layers might be possible path to realizing room-
temperature superconductivity in such distinctive ceramic high-$T_{c}$
materials. Apparently, there is some experimental evidence for the signatures
of superconducting transition well above the bulk $T_{c}$ in different
families of high-$T_{c}$ cuprates. In particular, experimental results on the
anomalous resistive transitions between 125 and 260 K in the cuprate
superconductors $\rm{Y_{3}Ba_{4}Cu_{7}O_{x}}$ (with the bulk $T_{c}\simeq 90$
K) [12] indicate that superconductivity above the bulk $T_{c}$ emerges at
grain boundaries in accordance with the prediction of the above expression (6)
for $T^{2D}_{c}$. Further, there is also experimental evidence for the
signatures of superconductivity at temperatures much higher than the bulk
$T_{c}$ in other high-$T_{c}$ cuprates, where resistive transitions were also
observed at a temperature of about 260 K [13]. Concurrently, the possibility
of traces of superconductivity at temperatures $T>>T_{c}$ in the systems
$\rm{HgBa_{2}Ca_{n-1}Cu_{n}O_{2n+2+\delta}}$ and
$\rm{Bi_{2}Sr_{2}CaCu_{2}O_{8}}$ was reported [13]. Thus, the above key
experimental findings provided the confirmation of the predictions of the
theory of 3D and 2D Bose superfluids regarding the possibility of
superconductivity well above the bulk $T_{c}$ in $\rm{Y}$-, $\rm{Bi}$-and
$\rm{Hg}$-based cuprate superconductors. Among these high-$T_{c}$ materials,
some $\rm{Bi}$-based ceramic cuprate superconductors may be the promising
systems to investigate the 3D-2D crossover regime of superconductivity, which
might be relevant to room-temperature superconductivity. In the following, we
report experimental evidence of 2D room-temperature superconductivity
occurring in the 3D-2D crossover region within the newly derived
$\rm{Bi/Pb}$-based ceramic cuprate superconductors, which contain many grain
boundaries, interfaces and multiplate blocks.

##  3 New melt technology for synthesizing of $\rm{Bi/Pb}$-based ceramic
cuprate superconductors under the influence of solar energy

The possibility of producing superconducting materials with the highest
critical temperature to a considerable extent depends on the methods of their
synthesizing. The solid-state reaction method is often used to synthesize the
samples of high-$T_{c}$ cuprate superconductors. However, this method has the
following shortcomings: the synthesis of samples is the multi-stage process
and the preparation of well-textured samples is difficult. In this regard, the
melt technologies of sample synthesis are promising in obtaining high-$T_{c}$
cuprate superconductors with higher critical temperature and have certain
advantages over other method of synthesizing high-$T_{c}$ superconducting
samples. For the synthesis of the samples of high-$T_{c}$ cuprate
superconductors, the resistance and induction furnaces, optical furnace, laser
and other energy sources were used for melting a mixture of starting
materials. In the last years, the new melt technology for synthesizing
$\rm{Bi/Pb}$-based ceramic cuprate superconductors were developed by using the
concentrated solar energy for melting a mixture of source materials [37]. The
advantages of this method in the synthesis of the samples of high-$T_{c}$
ceramic materials have been described elsewhere [20, 21]. The new melting
technology based on the use of solar energy allows us to obtain the well-
textured ceramic cuprate superconductors
$\rm{Bi_{1.7}Pb_{0.3}Sr_{2}Ca_{n-1}Cu_{n}O_{y}}$ (with $n=2-30$), in which the
onset of superconductivity occurs at critical temperatures close to room
temperature. The samples of these ceramic cuprate superconductors were
prepared by a melt process in a large solar furnace (in Parkent).

Figure 3:  Melting process of a mixture of source materials and synthesis of
ceramic cuprate superconductors $Bi_{1.7}Pb_{0.3}Sr_{2}Ca_{n-1}Cu_{n}O_{y}$ in
a large solar furnace. Figure 4:  Microstructures of ceramic cuprate
superconductors (a) $Bi/Pb$ (2.2. 29.30 ) and (b) $Bi/Pb$ (2.2.19.20) [21]
obtained by using the electron microscopy and the microscopy Solver Next and
SEM.

Nominal compositions, $\rm{Bi_{1.7}Pb_{0.3}Sr_{2}Ca_{n-1}Cu_{n}O_{y}}$ were
prepared from adequate mixtures of precursors $\rm{Bi_{2}O_{3}}$, $\rm{PbO}$,
$\rm{SrSO_{3}}$, $\rm{CaO}$ and $\rm{CuO}$. Mixtures of these starting source
materials were then put in a solar furnace and melted by concentrated solar
radiation with a radiant flux of about 420-480 $W/cm^{2}$ in this furnace,
which is shown in Fig.3. The appropriate temperature gradient in the melt
process is of order 1480 0C. The samples were prepared in the form of bars or
plates with the size $5\times 5\times 45mm$ and in the form of disks with a
diameter of $14-26mm$ and a thickness of $2.3-5.4mm$. The microstructure of
the prepared samples were studied using the microscopy solver Next, SEM and
electron microscopy Zeiss. Phase analysis of the samples was performed by
$X$-ray diffraction (DRON UM-1) technique (using $\rm{CuK_{\alpha}}$ radiation
and $\rm{Ni}$ filler). Samples of the new $\rm{Bi/Pb}$-based cuprate
superconductors contain many grain boundaries parallel to each other, twin
boundaries, interfaces and multiplate blocks (see Fig.4).

##  4 Experimental evidence of room-temperature superconductivity in
$\rm{Bi/Pb}$-based ceramic cuprate superconductors

According to the $X$-ray diffraction data and observed resistive transitions,
the synthesized samples of $\rm{Bi/Pb}$-based cuprate superconductors are
multiphase materials, where different superconducting phases are formed at
different critical temperatures. In particular, 3D superconducting phases with
different bulk $T_{c}$ values are observed in the temperature range
$100K\lesssim T_{c}\lesssim 130K$. Analyses of the microstructure of ceramic
samples using electron microscopies showed that the new superconducting
systems $\rm{Bi_{1.7}Pb_{0.3}Sr_{2}Ca_{n-1}Cu_{n}O_{y}}$ consist of many small
and large grains (i.e. nanocrystals), twin domains and closely packed parallel
multiplate blocks, as shown in Fig.4. The microstructure of the chip of
ceramic samples obtained from glass-crystalline plates and annealed at
temperatures $T\simeq 843-850$ 0C for 3-94 h represent closely packed
multilamellar blocks, which contain coupled stacks of many quasi-2D parallel
layers with nano-sized thickness.

Figure 5:  Temperature dependence of the resistance for the sample of
$\rm{Bi/Pb}-2245$.

The samples of $\rm{Bi/Pb}$-based ceramic cuprate superconductors were used to
study the superconducting properties in a wide temperature range from 77 to
300 K. The critical temperatures of the superconducting transitions were
determined by the resistance-temperature measurements carried out using the
standard four-probe technique with silver past contacts. The contact
resistance was far less than $1\Omega$ in each case. The superconducting
transition temperature in a series of the studied samples is defined by the
maximum slope of the resistive transition in the resistance-temperature
curves. Fig.5 shows the resistance-temperature dependence of the synthesized
sample of $\rm{Bi_{1.7}Pb_{0.3}Sr_{2}Ca_{n-1}Cu_{n}O_{y}}$
($\rm{Bi/Pb}-2245$), where the onset of superconductivity in 2D domains
seemingly occurs at the temperature $T^{onset}_{c}\simeq 210$ K well above the
bulk $T_{c}\simeq 120$ K. In Fig.6, we show the resistance-temperature
dependence for the synthesized sample of $\rm{Bi/Pb}-2267$. For this sample,
the onset of superconductivity is observed at room temperature
$T^{onset}_{c}\simeq 395$ K and a sharp step-like drop in the resistance is
observed at around this temperature. The onset of superconductivity in the
studied sample of $\rm{Bi_{1.7}Pb_{0.3}Sr_{2}Ca_{n-1}Cu_{n}O_{y}}$
($\rm{Bi/Pb}-221920$)occurs at $T^{onset}_{c}\simeq 310$ and a sharp drop of
the resistance is observed at around this temperature (see Fig.7). Fig.8 shows
the resistance-temperature curve for the synthesized sample of
$\rm{Bi_{1.7}Pb_{0.3}Sr_{2}Ca_{n-1}Cu_{n}O_{y}}$ ($\rm{Bi/Pb}-222930$). First,
the resistance of this sample increases slightly from 350 K down to 295 K and
then a sharp drop in the resistance is observed around the temperature
$T^{onset}_{c}\simeq 295$. The superconducting transition exhibits a step at
about 295 K.

Upon lowering the temperature, other superconducting transitions in the
studied samples of $\rm{Bi/Pb}-2245$, $\rm{Bi/Pb}-2267$, $\rm{Bi/Pb}-221920$
and $\rm{Bi/Pb}-222930$ are observed at $T_{c}\simeq 100-130$ K. Our
experimental results indicate that the bulk 3D superconductivity in these
$\rm{Bi/Pb}$-based ceramic cuprate materials occurs only at more lower
temperature than the onset temperature of 2D superconductivity emerging at
different grain boundaries and 3D/2D interfaces and in multiplate blocks. The
superconducting transition in the multiphase superconductors
$\rm{Bi_{1.7}Pb_{0.3}Sr_{2}Ca_{n-1}Cu_{n}O_{y}}$ occurs first at about room
temperature and is manifested as a step-like resistive transition in the
resistance-temperature curve. Such a resistive transition is indicative of the
presence of more than one superconducting phase and the persistence of remnant
2D superconducting phase in these ceramic materials above the bulk $T_{c}$ up
to room temperature. Here we notify that the 3D and 2D superconducting phases
coexist below the bulk $T_{c}$ and the ceramic samples of the
$\rm{Bi/Pb}$-based cuprate superconductors below this critical temperature
predominantly consist of 3D superconducting phase.

Figure 6:  The resistance-temperature dependence for the sample of
$\rm{Bi/Pb}-2267$ . Figure 7:  The resistance-temperature dependence for the
sample of $\rm{Bi/Pb}-221920$.

Another key criterion for emerging of room-temperature superconductivity in
the synthesized samples of $\rm{Bi_{1.7}Pb_{0.3}Sr_{2}Ca_{n-1}Cu_{n}O_{y}}$ is
the detection of the presence of a pronounced Meissner effect below a
characteristic temperature of the step-like resistive transition occurring at
around 300 K. We detected an incomplete Meissner effect in the most samples of
these ceramic superconducting materials at room temperature.

In Fig.9, we illustrate the detection of such a well-defined Meissner effect,
that is, the expulsion of magnetic flux from the $\rm{Bi/Pb}$-based ceramic
cuprate superconductors at room temperature. Upon lowering the temperature,
fractional Meissner effect in these materials increases gradually and remained
always smaller than 100% even below the bulk $T_{c}$, very likely due to the
presence of the insulating phase in underdoped and even in optimally doped
high-$T_{c}$ cuprates [38, 39, 40]. In insulating domains, the Cooper pairs do
not exist in $k$-space any longer and become localized pairs (i.e. bipolarons)
in real space [41], as observed in $\rm{Bi}-2212$ [42]. The variations of the
Meissner effect in ceramic high-$T_{c}$ cuprate superconductors reflect
variations of the insulating volume and superconducting volumes of 3D and 2D
domains. One can assume that a partial superconducting volume of 2D domains in
the studied samples of $\rm{Bi/Pb}$-based cuprate superconductors and related
partial Meissner effect detected at room temperature varies from 5% to 25%.

Figure 8:  The resistance-temperature dependence for the sample of
$\rm{Bi/Pb}-222930$. Figure 9:  The partial Meissner effect detected in the
$\rm{Bi/Pb}$-based ceramic cuprate superconductor
$\rm{Bi_{1.7}Pb_{0.3}Sr_{2}Ca_{n-1}Cu_{n}O_{y}}$ (n=30) at room temperature.

##  5 Conclusion

We have presented the theoretical and experimental results concerning the
possibility of the existence of 3D and 2D superconducting states in specially
synthesized multiphase ceramic cuprate superconductors
$\rm{Bi_{1.7}Pb_{0.3}Sr_{2}Ca_{n-1}Cu_{n}O_{y}}$ (with $n=2-30$). These newly
derived high-$T_{c}$ materials consist of 3D and 2D superconducting domains
and are particularly interesting from the viewpoint of novel superconducting
properties and room-temperature superconductivity of such systems. We have
argued that the high-$T_{c}$ cuprates are unconventional (bosonic)
superconductors, in which the tightly-bound (poloronic) Cooper pairs behave
like bosons and condense into 3D and 2D Bose superfluids in 3D and 2D domains.
We have shown that, in ceramic high-$T_{c}$ cuprate superconductors, besides
bulk 3D superconductivity there is also strongly enhanced 2D superconductivity
emerging in the 3D-2D crossover region and persisting well above the bulk
$T_{c}$. We have examined the possibility of the existence of distinctly
different 3D and 2D superconducting phases in these materials predicted by the
theory of Bose-liquid superconductivity. We have determined the critical
superconducting transition temperatures in 3D and 2D systems, which depend on
the effective mass of interacting bosons, the density of superfluid bosons and
the interboson coupling constant. We have found that the superconducting
transition temperatures in bosonic cuprate superconductors is much higher in
2D domains than in 3D domains and the bulk superconductivity is destroyed
above $T_{c}(=T^{3D}_{c})$, but 2D superconductivity persists up to the onset
temperature $T^{onset}_{c}=T^{2D}_{c}>>T^{3D}_{c}$. We have predicted the
possibility of realizing room-temperature superconductivity at different grain
boundaries and 3D/2D interfaces and in multiplate blocks within the new
ceramic cuprate superconductors. Apparently, some families of such cuprate
superconductors synthesized by using advanced technologies might be
particularly promising in the experimental search for room-temperature
superconductivity at ambient pressure. We have used the new technology for
synthesizing the ceramic cuprate superconductors with very high critical
temperatures, ever observed previously, and finally discovered room-
temperature superconductivity in the $\rm{Bi/Pb}$-based ceramic high-$T_{c}$
materials $\rm{Bi_{1.7}Pb_{0.3}Sr_{2}Ca_{n-1}Cu_{n}O_{y}}$, synthesized by
using the new melt technology in a large solar furnace, with the highest
critical temperatures $T^{onset}_{c}\simeq 295-310$ K at ambient pressure. The
samples of $\rm{Bi/Pb}$-based cuprate superconductors contain grain boundaries
parallel to each other, twin boundaries and interfaces and multilamellar
blocks represening the alternating 3D non-superconducting/2D superconducting
sandwich layers above the bulk $T_{c}$, which are favorable for realizing
room-temperature superconductivity. The remnant 2D superconductivity
persisting at different grain boundaries and 3D/2D interfaces and in
multilamellar blocks up to high temperatures $T=T^{onset}_{c}\simeq 295-310$ K
and the onset of room-temperature superconductivity in the synthesized samples
of $\rm{Bi/Pb}$-based ceramic cuprate materials are evidenced by the
observations of a sharp step-like drop in the resistance and a pronounced
partial Meissner effect at around 300 K. Thus, by the use of the series of
$\rm{Bi/Pb}$-based ceramic cuprate materials, we have succeeded in finding the
new class of room-temperature superconductors at ambient pressure.

## Acknowledgments

We would first like to thank B.S. Yuldashev for his attention and
encouragement in the process of carrying-out this work. We would also like to
thank our colleagues I. Khidirov, B.L. Oksengendler, U.T. Kurbanov and E.Kh.
Karimbaev for valuable discussions and suggestions. This work was support by
the Foundation of the Fundamental Research, Grant No $\Phi$-$\Phi$A-2021-433.

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# Radiation from relativistic jets

,a Y. Mizunob, P. Hardeec, H. Sold, M. Medvedeve, B. Zhangf, Å. Nordlundg, J.
T. Frederikseng, G. J. Fishmanh, R. Preecea,
a National Space Science and Technology Center, 320 Sparkman Drive, VP 62,
Huntsville, AL 35805, USA
b National Space Science and Technology Center, 320 Sparkman Drive, VP 62,
Huntsville, AL 35805, USA, (visiting Univ. of Nevada, Las Vegas)
c Department of Physics and Astronomy, The University of Alabama Tuscaloosa,
AL 35487, USA
d LUTH, Observatore de Paris-Meudon, 5 place Jules Jansen, 92195 Meudon Cedex,
France
e Department of Physics and Astronomy, University of Kansas, KS 66045, USA
f Department of Physics, University of Nevada, Las Vegas, NV 89154, USA
g Niels Bohr Institute, University of Copenhagen, Juliane Maries Vej 30, 2100
Copenhagen Ø, Denmark
h NASA-Marshall Space Flight Center, National Space Science and Technology
Center 320 Sparkman Drive, VP 62, Huntsville, AL 35805, USA
E-mail

###### Abstract:

Nonthermal radiation observed from astrophysical systems containing
relativistic jets and shocks, e.g., gamma-ray bursts (GRBs), active galactic
nuclei (AGNs), and Galactic microquasar systems usually have power-law
emission spectra. Recent PIC simulations of relativistic electron-ion
(electron-positron) jets injected into a stationary medium show that particle
acceleration occurs within the downstream jet. In the presence of relativistic
jets, instabilities such as the Buneman instability, other two-streaming
instability, and the Weibel (filamentation) instability create collisionless
shocks, which are responsible for particle (electron, positron, and ion)
acceleration. The simulation results show that the Weibel instability is
responsible for generating and amplifying highly nonuniform, small-scale
magnetic fields. These magnetic fields contribute to the electron’s transverse
deflection behind the jet head. The “jitter” radiation from deflected
electrons in small-scale magnetic fields has different properties than
synchrotron radiation which is calculated in a uniform magnetic field. This
jitter radiation, a case of diffusive synchrotron radiation, may be important
to understand the complex time evolution and/or spectral structure in gamma-
ray bursts, relativistic jets, and supernova remnants.

##  1 Introduction

Shocks are believed to be responsible for prompt emission from gamma-ray
bursts (GRBs) and their afterglows, for variable emission from blazars, and
for particle acceleration processes in jets from active galactic nuclei (AGN)
and supernova remnants (SNRs). The predominant contribution to the observed
emission spectra is often assumed to be synchrotron- and inverse Compton
radiation from these accelerated particles for gamma-ray bursts [1-7] and for
AGN jets [8-13]. It is assumed that turbulent magnetic fields in the shock
region lead to Fermi acceleration, producing higher energy particles [14, 15].
To make progress in understanding emission from these object classes, it is
essential to place modeling efforts on a firm physical basis. This requires
studies of the microphysics of the shock process in a self-consistent manner
[16, 17].

##  2 Method of calculation

Three-dimensional relativistic particle-in-cell (RPIC) simulations have been
used to study the microphysical processes in relativistic shocks. Such PIC
simulations show that rapid acceleration takes place in situ in the downstream
jet [18-33]. Three independent simulation studies confirm that relativistic
counter-streaming jets do excite the Weibel instability [34], which generates
current filaments and associated magnetic fields [35], and accelerates
electrons [18-22].

In order to determine the luminosity and spectral energy density (SED) of
synchrotron radiation, it is general practise to simply assume that a certain
fraction $\epsilon_{\rm B}$ of the post-shock thermal energy density is
carried by the magnetic field, that a fraction $\epsilon_{\rm e}$ is carried
by electrons, and that the energy distribution of the electrons is a power-
law, $d\log n_{\rm e}/d\log\varepsilon=p$ (above some minimum energy $E_{\rm
m}$ which is determined by $\epsilon_{\rm e},\epsilon_{\rm B}$ and $p$). In
this approach, $\epsilon_{\rm B}$, $\epsilon_{\rm e}$, and $p$ are treated as
free parameters, to be determined by observations. However, more constraining
data now require additional free parameters such as the introduction of broken
power-law to reproduce the spectral energy distributions of TeV blazars for
instance [11]. Due to the lack of a first principle theory of collisionless
shocks, a purely phenomenological approach to modeling radiation is applied,
but one must recognize that emission is then calculated without a full
understanding of the processes responsible for particle acceleration and
magnetic field generation [17]. It is important to clarify that the
constraints imposed on these parameters by the observations are independent of
any assumptions regarding the nature of the shocks and the processes
responsible for particle acceleration or magnetic field generation. Any model
proposed for the actual shock micro-physics must be consistent with these
phenomenological constraints.

Since magnetic fields are generated by the current structures produced in the
filamentation (Weibel) instability, it is possible that “jitter” radiation
[36-44] is an important emission process in GRB and AGN jets. It should be
noted that synchrotron- and ‘jitter’-radiation are fundamentally the same
physical processes (emission of accelerated charges in a magnetic field), but
the relative importance of the two regimes depends on the comparison of the
deflection angle and the emission angle of the charges [36]. Emission via
synchrotron- or “jitter”-radiation from relativistic shocks is determined by
the magnetic field strength and structure and the electron energy distribution
behind the shock, which can be computed self-consistently with RPIC
simulations. The full RPIC simulations may actually help to determine whether
the emission is more synchrotron-like or jitter-like.

The characteristic differences between Synchrotron- and jitter radiation are
relevant for a more fundamental understanding of the complex time evolution
and/or spectral propertis of GRBs (prompt and afterglows) [45]. For example,
jitter radiation has been proposed as a solution of the puzzle that below
their peak frequency GRB spectra are sometimes steeper than the “line of
death” spectral index associated with synchrotron emission [35-38], i.e., the
observed SED scales as $F_{\nu}\propto\nu^{2/3}$, whereas synchrotron SEDs
should follow $F_{\nu}\propto\nu^{1/3}$, or even more shallow (i.e.
$F_{\nu}\propto\nu^{\alpha}$ where $\alpha\leq 1/3$, e.g., [37]). Thus, it is
crucial to calculate the emerging radiation by tracing electrons (positrons)
in self-consistently evolved electromagnetic fields. This highly complex
analytical and computational task requires sophisticated tools, such as multi-
dimensional, relativistic, PIC methods.

###  2.1 New Computing Method of Calculating Synchrotron and Jitter Emission
from Electron Trajectories in Self-consistently Generated Magnetic Fields

Consider a particle at position ${\bf{r}_{0}}(t)$ at time $t$ (Fig. 1). At the
same time, we observe the associated electric field from position $\bf{r}$.
Because of the finite propagation velocity of light, we actually observe the
particle at an earlier position $\bf{r}_{0}(\rm{t}^{{}^{\prime}})$ along its
trajectory, labeled with the retarded time $t^{{}^{\prime}}=t-\delta
t^{{}^{\prime}}=t-\bf{R}(\rm{t}^{{}^{\prime}})/c$. Here
$\bf{R}(\rm{t}^{{}^{\prime}})=|\bf{r}-\bf{r}_{0}(\rm{t}^{{}^{\prime}})|$ is
the distance from the charge (at the retarded time $t^{{}^{\prime}}$) to the
observer’s position.

Figure 1: Definition of the retardation effect. From an observers point, r,
one sees the particle at position $\bf{r}_{0}(\rm{t}^{{}^{\prime}})$ where it
was at retarded time t’ (from Figure 2.2 in [46]).

The retarded electric field from a charged particle moving with instantaneous
velocity $\boldsymbol{\beta}$ under acceleration $\boldsymbol{\dot{\beta}}$ is
expressed as [48],

$\displaystyle\bf{E}$ $\displaystyle=$
$\displaystyle\frac{q}{4\pi\epsilon_{0}}\left[\frac{\bf{n}-\boldsymbol{\beta}}{\gamma^{2}(1-\bf{n}\cdot\boldsymbol{\beta})^{\rm{3}}\rm{R}^{2}}\right]_{\rm
ret}+\frac{q}{4\pi\epsilon_{0}c}\left[\frac{\bf{n}\times\\{(\bf{n}-\boldsymbol{\beta})\times\boldsymbol{\dot{\beta}}\\}}{(1-\bf{n}\cdot\boldsymbol{\beta})^{\rm{3}}\rm{R}}\right]_{\rm
ret}$ (1)

Here,
$\bf{n}\equiv\bf{R}(\rm{t}^{{}^{\prime}})/|\bf{R}(\rm{t}^{{}^{\prime}})|$ is a
unit vector that points from the particle’s retarded position towards the
observer. The first term on the right hand side, containing the velocity
field, is the Coulomb field from a charge moving without influence from
external forces. The second term is a correction term that arises when the
charge is subject to acceleration. Since the velocity-dependent field falls
off in distance as $R^{-2}$, while the acceleration-dependent field scales as
$R^{-1}$, the latter becomes dominant when observing the charge at large
distances ($R\gg 1$).

The choice of unit vector $\bf{n}$ along the direction of propagation of the
jet (hereafter taken to be the $Z$-axis) corresponds to head-on emission. For
any other choice of $\bf{n}$ (e.g., $\theta\lesssim 1/\gamma$), off-axis
emission is seen by the observer. The observer’s viewing angle is set by the
choice of $\bf{n}$ ($n_{\rm x}^{2}+n_{\rm y}^{2}+n_{\rm z}^{2}=1$). After some
calculation and simplifying assumptions (for detailed derivation see [46]) the
total energy $W$ radiated per unit solid angle per unit frequency can be
expressed as

$\displaystyle\frac{d^{2}W}{d\Omega d\omega}$ $\displaystyle=$
$\displaystyle\frac{\mu_{0}cq^{2}}{16\pi^{3}}\left|\int^{\infty}_{-\infty}\frac{\bf{n}\times[(\bf{n}-\boldsymbol{\beta})\times\boldsymbol{\dot{\beta}}]}{(1-\boldsymbol{\beta}\cdot\bf{n})^{2}}e^{i\omega(t^{{}^{\prime}}-\bf{n}\cdot\bf{r}_{0}({\rm
t}^{{}^{\prime}})/{\rm c})}dt^{{}^{\prime}}\right|^{2}$ (2)

This equation contains the retarded electric field from a charged particle
moving with instantaneous velocity $\boldsymbol{\beta}$ under acceleration
$\boldsymbol{\dot{\beta}}$, and only the acceleration field is kept since the
velocity field decreases rapidly as $1/R^{2}$. The distribution over
frequencies of the emitted radiation depends on the particle energy, radius of
curvature, and acceleration. These quantities are readily obtained from the
trajectory of each charged particle.

Since the jet plasma has a large velocity $Z$-component in the simulation
frame, the radiation from the particles (electrons and positrons) is heavily
beamed along the $Z$-axis as jitter radiation [36, 37, 38, 44].

##  3 Radiation from relativistic electrons: a simple case to test computing
method

Here we have calculated the radiation from two electrons with Lorentz factor
($\gamma=15.8,40.8$) [29, 30]. The electrons gyrate in the $x-z$ plane with
the uniform magnetic field ($B_{\rm y}$) and the results are shown in Figures
2 & 3.

Figure 2: The paths of two charged particles moving in a fixed homogenous
magnetic field (left panel) ($\gamma=15.8,40.8$). The particles produce a time
dependent electric field. An observer situated at great distance along the
n-vector sees the retarded electric field from the gyrating particles (right
panel). As a result of relativistic beaming, the field is seen as pulses
peaking when the particles move directly towards the observer.

Figure 3: The observed power spectrum from two charged particles, gyrating in
a magnetic field at different viewing angles. The viewing angles are 0∘, 1∘,
2∘, 3∘, 4∘, 5∘, and 6∘ ($n_{\rm y}\neq 0$). With larger angles the frequencies
above the Nyquist frequency should be strongly damped, however they increase
due to aliasing [46]. The units on both axes are arbitrary. The theoretical
synchrotron spectrum for a viewing angle equal to 0∘ is plotted for comparison
as a red curve for the electron with $\gamma=40.8$ (multiplied by 2 for
clarity).

The spectra observed far from the electron at angles with respect to the $z$
direction are shown in Fig. 3. The higher frequencies ($>f_{\rm c}$) are
strongly damped with increasing angles as $e^{(-f/f_{\rm c})}$, see [48].
Since the critical frequency $f_{\rm
c}=\frac{3}{2}\gamma^{3}\left(\frac{c}{\rho}\right)=2309$, where $\rho=11.03$
for the electron with $\gamma=40.8$ is larger than that for $\gamma=15.8$, the
radiation from the electron $\gamma=40.8$ is dominant [30]. The electron with
$\gamma=15.8$ gyrates about three times in this period, the ripples in the
spectrum shows the electron cyclotron frequency. However, in order to resolve
it much longer time is required [46]. We have very good agreement between the
spectrum obtained from the simulation and the theoretical synchrotron spectrum
expectation (red curve) from eq. 3 (eq. 7.10 [46]).

Synchrotron radiation with the full angular dependency for the parallel
polarization component is given by [48],

$\displaystyle\frac{d^{2}W_{||}}{d\omega d\Omega}$ $\displaystyle=$
$\displaystyle\frac{\mu_{0}cq^{2}\omega^{2}}{12\pi}\left(\frac{r_{\rm
L}\theta^{2}_{\beta}\beta^{2}}{c}\right)^{2}\frac{|K_{\frac{2}{3}}(\chi/\sqrt{\cos\theta\beta^{3}})|^{2}}{(\cos\theta\beta^{3})^{2}},$
(3)

where $\theta$ is the angle between n and the orbital plane
$\theta^{2}_{\beta}\equiv 2(1-\beta\cos\theta)$, $\chi=\omega r_{\rm
L}\theta^{3}_{\beta}/3c$ and $r_{\rm L}$ the gyro-radius $\gamma mv/(qB)$. For
$\beta\rightarrow 1$ and $\theta\rightarrow 0$ this expression converges
toward the solution one normally finds in text books [48, 47].

It should be noted that the method based on the integration of the retarded
electric fields calculated by tracing many electrons described in this section
can provide a proper spectrum in turbulent electromagnetic fields. On the
other hand, if the formula for the frequency spectrum of radiation emitted by
a relativistic charged particle in instantaneous circular motion is used [47,
48], the complex particle accelerations and trajectories are not properly
accounted for and the jitter radiation spectrum is not properly obtained (for
details see [46, 49]). The results described above validate the technique used
in our code as described previous section [29, 30, 46, 49].

##  4 Discussion

We have started to calculate emission directly from our simulations using the
same method described in the previous section. In order to calculate the
(jitter-like) synchrotron radiation from the particles in the electromagnetic
fields generated by the filamentation instability, the retarded electric field
from a single particle is Fourier-transformed to give the individual particle
spectrum as described in the previous section. The individual particle spectra
are added together to produce a total spectrum over a particular simulation
time span [46, 49]. It should be noted that for this calculation very large
simulations over a long time ($t_{\rm s}$) are required using a small time
step ($\Delta t$) in order to increase the upper frequency limit to the
spectrum (Nyquist frequency $\omega_{\rm N}=1/2\Delta t$). Frequency
resolution is limited by the time span ($\Delta\omega=1/t_{\rm s}$) [46, 49].
For a case with the time step $\Delta t=0.01/\omega_{\rm pe}$ and the time
span $t_{\rm s}=50/\omega_{\rm pe}$, a calculated spectrum will have the
highest frequency, $50\omega_{\rm pe}$ and the frequency resolution (the
lowest frequency), $0.02\omega_{\rm pe}$. $\omega_{\rm pe}$ is calculated with
an appropriate plasma density. Simulations over a long time allow us to obtain
multiple spectra at sequential time spans so the spectral evolution can be
calculated. Synthetic spectra obtained in the way we have described above
should be compared with GRB prompt and afterglow observations.

In the case of AGN jets, diffusive synchrotron radiation has already been
invoked by several works [39, 41, 43] to reproduce spectra of 3C273, M87 and
Cen A knots from radio to X-rays. For TeV blazars, taking into account the
relative importance of the energy densities contained in the small-scale and
large-scale magnetic fields may be an elegant alternative to the choice of a
broken power-law for the energy distribution of radiating electrons.

###### Acknowledgments.

K.I.N., Y.M., and P.H. are supported by NSF-AST-0506719, AST-0506666, NASA-
NNG05GK73G, NNX07AJ88G, and NNX08AG83G. M.M has been supported by NSF-
AST-0708213, NASA-NX07AJ50G, and NNX08AL39G, and DoE-DE-FG02-07ER54940.
Simulations were performed at the Columbia facility at the NASA Advanced
Supercomputing (NAS) and IBM p690 (Copper) at the National Center for
Supercomputing Applications (NCSA) which is supported by the NSF. Part of this
work was done while K.-I. N. was visiting the Niels Bohr Institute. He thanks
the director of the institution for generous hospitality. K.-I. N. prepared
this proceeding during his visit at Meudon Observatory, and thanks the
director of the institution for generous hospitality.

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