We derive and solve by the spectral method the equations for a neutral system
of ultra-relativistic electrons that are compressed to the radius of the
nucleus and subject to a driving force. This driving force can be thought of as
originating from a nuclear breathing mode, a possibility we discuss in detail

Ludwig, H.

Ruffini, R.

Xue, S. -S.

English

arXiv

By the spectral method we derive and solve the equations for a neutral system of ultra-relativistic electrons that are compressed to the radius of the nucleus and subjected to a driving force. This driving force can be thought of as originating from a nuclear breathing mode, a possibility we discuss in detail

Xue, S.-S.

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Nucleus Driven Electronic Pulsation

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Nonlinear Phenomena in Complex Systems, vol. 17, no. 4 (2014), pp. 384 - 386
Nucleus Driven Electronic Pulsation
H. Ludwig,∗ R. Ruﬃni,† and S.-S. Xue‡
Dipartimento di Fisica and ICRA, Sapienza Universita` di Roma, Rome, ITALY
(Received 5 September, 2014)
By the spectral method we derive and solve the equations for a neutral system of ultra-
relativistic electrons that are compressed to the radius of the nucleus and subjected to a
driving force. This driving force can be thought of as originating from a nuclear breathing
mode, a possibility we discuss in detail.
PACS numbers: 52.27.Ny; 52.30.Cv; 95.30.Qd
Keywords: giant nucleus oscillation, strong electric field
1. Introduction
The creation of electron-positron pairs
through strong ﬁelds, which has been predicted
theoretically by F. Sauter as early as 1931
[1] but not yet been observed, is nowadays
of high interest for both laboratory scientists
and astrophysicists [2, 3]. The obstacle to be
solved is the creation of a suﬃciently strong
electric ﬁeld. In this note we will present an
approach to investigate the dynamic response
of an ultra-relativistic compressed electron gas
subjected to the breathing mode of the underlying
nucleus, a system that promises to generate
strong electric ﬁelds, depending on the mutual
dynamics of negative and positive charges. This
model can, with slight modiﬁcation like an
adjustment of proton density, be extrapolated
to astrophysical scales. In [4] we presented the
stationary solutions of pulsating electron gas
in the framework of the Thomas-Fermi model,
and we will now recapitulate the main results
that are of importance for the present work. In
this model the nucleus is modeled as a sphere
of homogenous positive charge distribution. The
∗E-mail: hendrik.ludwig@icranet.org; Also at
ICRANet, P.zza della Republica 10, I-65122 Pescara,
ITALY; Also ICRANet, Nice, FRANCE
†E-mail: ruffini@icra.it; Also at
ICRANet, P.zza della Republica 10, I-65122 Pescara,
ITALY; Also at ICRANet, Nice, FRANCE
‡E-mail: xue@icra.it; Also at ICRANet, P.zza della
Republica 10, I-65122 Pescara, ITALY
electrons are considered as a charged perfect ﬂuid
with the equation of state of a Fermi gas at zero
temperature.
2. Spectral method formalism
Since we want to treat the regime around
nuclear density we can assume the ultra-
relativistic limit
ρ =
3
4
(3pi2n4)1/3, p =
1
4
(3pi2n4)1/3 (1)
for the equation of state, where ρ, p, and n are
the energy density, pressure, and number density
of the electron gas respectively. Throughout the
text, natural units c = 1, ~ = 1 are used. When we
assume the nucleus of proton number Z to have a
radius rnuc = ∆Z
1/3 in units of the pion mass, and
provide suﬃcient external pressure to restrain the
electrons to the volume of the nucleus, we obtain
the very simple equilibrium conﬁguration
ρ =
9
16∆4
(
3
2pi
)2/3
, p =
3
16∆4
(
3
2pi
)2/3
,
n = nnuc =
3
4pi∆3
.


(2)
The parameter ∆ measures the departure
of proton density from nuclear densities. Positive
and negative charges completely cancel, so there
is no electric ﬁeld in equilibrium. Introducing
a time-dependent displacement ﬁeld ξ(t, r) and
treating Euler’s and Maxwell’s equations to ﬁrst
384
Nucleus Driven Electronic Pulsation 385
order, one obtains for stationary modes the
second order ordinary diﬀerential equation
ξ′′(r) +
2
r
ξ′(r) +
[
3ω2 −
(
12
pi
)1/3 3α
∆2
− 2
r2
]
× ξ(r) = 0. (3)
Here α is the ﬁne structure constant, and ω is
the frequency of the mode that has to be chosen
according to appropriate boundary conditions.
This leads to
ωn =
[(
12
pi
)1/3 α
∆2
+
pi2n2
3 r2nuc
]1/2
and
ξn(x) = C j1 (pin r/rnuc) , (4)
where C is an arbitrary factor and j1(z) is the ﬁrst-
order spherical Bessel functions of the ﬁrst kind.
The last allows to construct an orthonormal basis
(see Figure 1)
ξn(r) = npi
√
2 r−3/2nuc j1(npi r/rnuc), n = 1, 2, 3...
(5)
on the interval [0, rnuc] with respect to the scalar
product
〈ξn(r), ξm(r)〉 =
∫ rnuc
0
r2ξn(r)ξm(r)dr = δnm.
(6)
To model a driving force acting on the system
we can expand the Dirac delta function on the
surface
δ(r − rnuc) = −(2 rnuc)
1
2
∞∑
n=1
(−1)nξn(r) (7)
and the identity function
id(r) = −3(2 r
5
nuc)
1
2
pi2
∞∑
n=1
(−1)n
n2
ξn(r) (8)
in terms of this basis. The spatial delta function
at the surface can be used in combination
with a temporal delta function to model an
instantaneous impulse applied to the surface,
2 4 6 8 10
r ( ! )
-0.10
-0.05
0.05
0.10
0.15
0.20
0.25
" ( ! )
FIG. 1. The basis functions ξn for n = 1, 2, 3, 4 are
shown for rnuc = 10, corresponding to Z = 1000 and
∆ = 1. For ascending n, the lines are solid, dashed,
dotted and dash-dotted, respectively.
or in combination with a harmonic temporal
function to drive the surface periodically. The
identity function in combination with a harmonic
temporal function models the breathing mode of
a homogenously charged sphere. To ﬁrst order,
Gauss’ law predicts an electric force proportional
to the radius resulting from any contraction or
expansion of the sphere.
3. Solving the equation of motion
According to Eqs. (3) and (4), the basis
functions fulﬁl the equation
ω2nξn(r)−⊠ξn(r) = 0 (9)
Nonlinear Phenomena in Complex Systems Vol. 17, no. 4, 2014
386 H. Ludwig, R. Ruﬃni, and S.-S. Xue
where ⊠ is a diﬀerential operator in r. Making the
ansatz
ξ(t, r) =
∞∑
n=1
Tn(t)ξn(r) (10)
and plugging it into the equation of motion with
an external force term
∂2ξ(t, r)
∂t2
−⊠ξn(t, r) = f(t, r) (11)
we can expand in the basis (ξn) to obtain the
independent equations
T¨n(t)− ω2nTn(t) = fn(t), (12)
where
f(t, r) =
∞∑
n=1
fn(t)ξn(r). (13)
If the force is applied just for an instant at t = 0,
we can write fn(t) = fnδ(t), and assuming the
system initially unperturbed (Tn(t) = 0 for t < 0)
the solution to (12) is
Tn(t) =
fn
ωn
θ(t) sin(ωnt). (14)
If the force is applied at a frequency Ω starting
at t = 0 we can write fn(t) = fnθ(t) sin(Ω t),
and assuming the system initially unperturbed
the solution to (12) is
Tn(t) = fnθ(t)
ωn sin(Ω t)− Ω sinωnt
ω3n − ωnΩ2
. (15)
Using for fn now the coeﬃcients of (7) or (8), we
can construct various dynamical situations in the
compressed atom.
4. Conclusions
Using the formalism developed above, it
is now possible to investigate the response of
the electron gas to the nuclear breathing mode
for diﬀerent system sizes from atoms and giant
atoms to neutron stars. The proton density should
be determined from beta-equilibrium, while the
frequency of the breathing mode has to be inferred
from studies of nuclear incompressibility such as
[5, 6]. An investigation has to take into account
the energy budget of the initial breathing mode,
comparing it to the energy transferred to electron
gas and electric ﬁeld, to determine if dampening
of the former occurs ﬁrst.
Acknowledgement
We thank the National Academy of Sciences
of Belarus for their hospitality. H. Ludwig
is supported by the Erasmus Mundus Joint
Doctorate Program by Grant Number 2012-1710
from the EACEA of the European Commission.
References
[1] F. Sauter. Z. Phys. 82, 742 (1931).
[2] G. Preparata, R. Ruffini, and S.-S. Xue.
Astron. Astrophys. 338, L87 (1998).
[3] A. Ipp, J. Evers, C. Keitel, and
K. Hatsagortsyan. Phys. Lett. B 702, 383
(2011).
[4] H. Ludwig, R. Ruffini, and S.-S. Xue.
ArXiv:1402.3468 (2014).
[5] J. Blaizot, J. Berger, J. Decharge´, and M. Girod.
Nuc. Phys. A 591, 435 (1995).
[6] C. Wang, K. Chung, and A. Santiago.
Phys. Rev. C 55, 2844 (1997).
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