On the rich eight branch spectrum of the oblique propagating longitudinal waves in partially spin polarized electron-positron-ion plasmas

0.31 MB·

by  Pavel A. Andreev

On the rich eight branch spectrum of the oblique propagating longitudinal waves in
partially spin polarized electron-positron-ion plasmas
Pavel A. Andreev∗
Faculty of physics, Lomonosov Moscow State University, Moscow, Russian Federation.
Z. Iqbal†
Department of Physics, G. C. University Lahore,
Katchery Road, Lahore 54000, Pakistan and Faculty of physics,
6 Lomonosov Moscow State University, Moscow, Russian Federation.
1 (Dated: January 6, 2016)
0
We consider the separate spin evolution of electrons and positrons in electron-positron and
2
electron-positron-ion plasmas. Weconsiderobliquepropagating longitudinalwavesin thissystems.
n Workinginaregimeofhighdensityn0 ∼1027 cm−3 andhighmagneticfieldB0 =1010 Gwereport
a
presenceofthespin-electronacousticwavesandtheirdispersiondependencies. Inelectron-positron
J
plasmas, similarly to the electron-ion plasmas, we find one spin-electron acoustic wave (SEAW) at

5 propagation parallel or perpendicular to the external field and two spin-electron acoustic waves at
the oblique propagation. At the parallel or perpendicular propagation of the longitudinal waves in
]
electron-positron-ionplasmaswefindfourbranches: theLangmuirwave,thepositron-acousticwave
h
p andpairofwaveshavingspinnature,theyaretheSEAWand,aswecalledit,spin-electron-positron
- acousticwave(SEPAW).Attheobliquepropagationwefindeightlongitudinalwaves: theLangmuir
m wave,Trivelpiece-Gouldwave,pairofpositron-acousticwaves,pairofSEAWs,andpairofSEPAWs.
Thus, for the first time, we report existence of the second positron-acoustic wave existing at the
s
a obliquepropagation and existence of SEPAWs.
l
p
.
s I. INTRODUCTION spin waves with a linear spectrum is also obtained in
c
Ref. [7]. Further applications of the method of many
i
s particle QHDs have been given in Refs. [9, 10] which
y The field of spin quantum plasmas has been rapidly
predicts a mechanism of instabilities which arises due to
h growingoverthe lastdecade. Takabayasi[1]derivedand
theinteractionofneutronbeamwithelectron-ionmagne-
p analyzedthequantumhydrodynamicequationsforasin-
[ tized spin-1/2 quantum plasma, including instability of
gle spin-1/2 particle. The effects of electron spin on the
the spin-plasma waves. Further analysis of spin-plasma
1 plasma dynamics were first studied by Kuz’menkov at.
waves was presented in Refs. [11, 12]. Considering ki-
v al,[2,3]in2001. Theseauthorshavedevelopedamethod
netics in the extended phase space suggested by Kagan
1 ofexplicitderivationofmany-particlequantumhydrody-
in 1961 [13]-[15], where the spin or magnetic moment is
6 namic(QHD)equations. Theseequationsweretruncated
7 consideredalongwiththecoordinateandmomentum,the
to consist of the continuity equation, the Euler equa-
0 fine structure of the Berstein modes is found [16]. The
tion, the energy balance equation,the magnetic moment
0 finestructureisdemonstratedonthe exampleofthe sec-
1. evolution equation for spin-1/2 quantum plasmas. The ondmode. Itarisesduetothepresenceoftheanomalous
starting point of these derivation was the many-particle
0 magnetic moment of electrons.
Pauli equation. These set of equations contain the ef-
6
fectsofthespin-spinexchangeinteractionsandCoulomb Recently, a set of QHD equations for charged spin-
1
: exchange interactions. Another form of derivation was 1/2 particles is derived from the Pauli equation in Ref.
v recentlysuggestedin Ref. [4]. A simplifiedformofQHD [17]. ItformsseparatespinevolutionQHDs(SSE-QHDs)
Xi equations were considered in Refs. [5, 6]. Whichtreatsspin-upandspin-downelectronsastwodif-
ferent fluids. It is essential if the populations of spin-
r The method of many particle QHDs was applied to
a up electrons and spin-down electrons in the presence of
study the eigenwave problem for spin-1/2 electron-ion
external magnetic field is different (n 6= n ). This dif-
↑ ↓
plasmas with an account of the ion motion [7]. The
ference of populations of quantum states is responsible
dispersion relations of electrostatic and electromagnetic
for difference of Fermi pressuresof the spin-up andspin-
waves has been studied and found that the spin-plasma
down electrons. Revealing in a new type of a soundlike
waves(foundforelectronsinRef. [8])existinthevicinity
solution called the spin-electron acoustic wave (SEAW)
ofelectronandioncyclotronfrequenciesinthespectrum
[17], see some additional discussion below, after formula
ofwavespropagatingperpendicularto the externalmag-
(2). Spin current evolution in terms of SSE-QHD was
netic field, and the dispersion relation for self-consistent
considered in Ref. [18].
In the framework of the hydrodynamic model and
linear-response-functionformalismthe effects ofspinpo-
∗Electronicaddress: [email protected] larizationontheLangmuirandzerosoundwavesisinves-
†Electronicaddress: [email protected] tigated in Ref. [19]. It was found that spin polarization
2
increasesthecoefficientofspatialdispersionofLangmuir spin corrections on the spectrum of magnetosonic waves
waves. Ithasalsoshownthatphasevelocityofzerosound in non- relativistic and relativistic degenerate electron-
increases with increase of the degree of polarization. positron-ion(e-p-i)plasmas,wheretherelativisticeffects
Electron-positron (e-p) plasma is distinct because it areincludedintheFermipressureonly. Ahydrodynamic
consists of particles which havemass symmetry and anti and kinetic models for spin-1/2 electron-positron quan-
charge symmetry. Naturally electron-positron plasmas tum plasmas has been developed in Ref. [37] which in-
are found in many astrophysical environments like early corporates the Coulomb, spin-spin, Darwin and annihi-
universe [21], in neutron star magnetosphere [22], [23]. lationinteractions. Therewasconcludedthatthecontri-
Naturally existence of electron-positron plasma in com- butions of the annihilation interactions shifts the eigen-
pact starshas been investigatedby applying a simplified frequencies of the transverse electromagnetic plane po-
modelofagravitationallycollapsingorpulsating baryon larized waves and transverse spin-plasma waves.
core. It has shown that possible electric processes that New longitudinal wave in the degenerate e-p-i spinless
lead to the production of electron-positron pairs in the quantum plasma has been reported in Refs. [38]–[42]. It
boundaryofabaryoncoreandcalculatethenumberden- was called positron acoustic wave (PAW). For ultrarel-
sity of electron-positron pairs n = 1028 cm−3 [24]. ativistic electrons and non-relativistic positrons the dis-
pair
The degenerate electron-positron plasmas with ions are persionrelationofPAWs in the intermediate waverange
believedtobefoundincompactastrophysicalbodieslike wereobtained[40]. Thenonlinearwavestructureoflarge
neutron stars and the inner layers of white dwarfs [25– amplitude PAWs in e-p plasma with electron beam has
27]. been discussed in Ref. [38].
Inthepresentwork,weemploytheseparatedspinevo-
Physicists also trying to generate e–p plasmas in lab-
lution QHD for the e-p and e-p-i magnetized degenerate
oratories. In this context different schemes have been
plasmas. At consideration of separate evolution of spin-
proposed for the laboratory generation. For exam-
up andspin-downelectronsand positronswe discuss the
ple, in large-scale conventional accelerators, the possi-
oblique propagation of longitudinal waves. So, in the
bility of recombining high-quality electron and positron
presentworkwecalculatethespectrumofSEAWs,PAWs
beams via magnetic chicanes [28]. Pederson et. al, [29]
and predict spin-electron-positronacoustic waves.
have been presented plan for the creation and diagno-
sis of electron-positron plasmas in a stellarator, based
on extrapolation of the results from the Columbia Non-
II. ANALYTICAL MODEL
neutral Torus stellarator,as well as recent developments
inpositronsources. Interactionofultrashortlaserpulses
The SSE-QHDs developed in Refs. [17], [43] can be
with gaseous or solid targets could lead to the genera-
applied to the electron-positron plasmas and electron-
tion of the optically thin e-p plasma with above solid
statedensitiesintherangeof(1023−1028)cm−3[30]. Re- positron-ionplasmas. Therefore,wepresentthe continu-
ity and Euler equations for each spin projection of each
cently,ithasshownthat,byusingacompactlaser-driven
species.
setup, ion-free electron–positron plasmas can be gener-
In this paper we consider the evolution of the longitu-
atedinthe laboratory[31]. Their chargeneutrality,den-
sity about 1016cm−3 and small divergence finally open dinal waves in non-relativistic plasmas. Hence the spin
evolution does not affects our results. The Fermi spin
up the possibility of studying electron–positron plasmas
current (the thermal part of the spin current for the de-
in controlled laboratory experiments.
generate spin-1/2 fermions) obtained in Ref. [44] is not
The wave propagation phenomenon in electron-
considered here. Explicit spin contribution in the spec-
positron plasma is different as in usual electron-ion
trum of the Langmuir waves arising via the spin-orbit
plasma. Using a two-fluid model and a kinetic model, it
interactionis found in Ref. [45] (see formula (34) for de-
hasbeenobservedthatmanywavephenomenalikeacous-
scribing propagation perpendicular to the external mag-
ticwaves,whistlerwaves,Faradayrotation,lowerhybrid
neticfield). Inourapproximationweneedthecontinuity
waves and shear Alfven waves are absent in the nonrela-
and Euler equations for each subspecies.
tivistic e-pplasmas[32,33]. Mostofapplicationsrelated
The continuityequationinthe SSE-QHDarisesas fol-
totheplasmawavephenomenonpresentedinabovemen-
lows [17]
tionedstudieshasfocusedontheelectron–ionspinquan-
tumplasmas. However,someapplicationswerepresented ∂ n +∇(n v )=(−1)isT , (1)
t as as as az
for electron-positron plasmas and electron-positron-ion
plasmas for both spin-1/2 quantum plasma and spin- where a = e,p for electrons and positrons correspond-
less quantum plasma. For instance, the set of spin-1/2 ingly, s = u,d for the spin-up and spin-down conditions
QHD equations developed for electron-ion plasmas was ofparticles,n andv aretheconcentrationandveloc-
as as
applied for e-p plasmas by Brodin and Marklund [34]. ity field of particles of species a being in the spin state
They found new spin depended Alfv´enic solitary struc- s, T = γ(B S −B S ) is the z-projection of spin
az h¯ x ay y ax
tures obeying the Kortewegde Vries equation, where the torque, i : i = 2, i = 1, with the spin density projec-
s u d
nonlinearityiscausedbyspineffects. Mushtaqet. al[35], tions S and S , each of them simultaneously describe
ax ay
[36] studied the effects of quantum Bohm potential and evolution of the spin-up and spin-down particles of each
3
FIG. 2: (Color online) The figure shows the low frequency
FIG. 1: (Color online) The figure shows the dispersion of part of spectrum of oblique propagating longitudinal waves
the oblique propagating longitudinal waves in the electron- in e-p plasmas, where Σ=0.1.
positron plasmas. It shows four waves. Upper branch de-
scribes the Langmuir wave. Second line from the bottom,
which looks almost horizontal and depicted by the dashed
line, presents the Trivelpiece-Gould wave. Two other waves
arethelowerandupperbranchesofthespin-electronacoustic
waves. Detailsofthelowfrequencybranchesareshowninthe
next figure.
species. Therefore, functions S and S do not bear
ax ay
subindexesuandd. Inthismodelthez-projectionofthe
spin density S is not an independent variable, it is a
az
combination of concentrations S =n −n .
az au ad
The time evolution of the velocity fields of all species
of particles for each projection of spin v and v is
au ad
governedby the Euler equations [17]
mnas(∂t+vas∇)vas+∇Pas FIG. 3: (Color online) The figure shows the longitudinal
waves in e-p-i plasmas propagating parallel to the external
magnetic field. Ratio between theelectron andpositron con-
=q n E+ 1[v ,B] +(−1)isγ n ∇B centrations ischosen to beβ=0.1. Theupperbranchshows
a as c as a as z the Langmuir wave dispersion. Three linear dependencies
(cid:18) (cid:19) present SEAW, PAW and SEPAW.
γ
+ a(S ∇B +S ∇B )+(−1)ism(T −v T ), (2)
ax x ay y az as az
2
with Pas = (6π2)32na35s¯h2/5m, Taz =e γh¯a(J(M)axBy −
J B ), which is the torque current, where J =
(M)ay x (M)ax
(vau +vad)Sax/2, and J(M)ay =e (vau +vad)Say/2 are
the convective parts of the spin current tensor. All
species affect each other via the electric field: ∇E =
4π q n and ∇×E=0.
a,s as as
The SSE-QHDs was applied to two-dimensional elec-
P
tron gas in plane samples and nanotubes located in ex-
ternal magnetic fields [46]. It was found that in two-
dimensional electrongas Langmuir wavereplacedby the
couple of hybrid waves by considering separate spin-up
electrons and spin- down electrons evolution. One of
them is the modified Langmuir wave and the other is
FIG. 4: (Color online) The figure shows the longitudinal
SEAW.SurfaceSEAWswasconsideredinRef. [47]. Lin- waves in e-p-i plasmas propagating parallel to the external
earinteractionbetweensurfaceSEAWandsurfaceLang- magnetic field for β =0.5 and Σ=0.1.
muir wave (surface plasmons) is found in Ref. [47]. In
4
theSEAW).Moreover,SSE-QHDmodelappliedtostudy
the oblique propagation of longitudinal waves in mag-
netized spin-1/2 plasmas and found that instead of two
wellknownwaves(LangmuirwaveandTrivelpiece–Gould
wave) four wave solutions appeared in separate spin-up
and spin-down degenerate magnetized plasma [43].
We deal with e-p plasmas and e-p-i plasmas located
in an external magnetic field B = B e and study
ext 0 z
the dispersion of waves in these systems. We assume
that electrons, positrons, and ions have non-zero uni-
form equilibrium concentrations. The equilibrium veloc-
ity fields ofall species areequalto zero. For e-pplasmas
concentrationsofallelectronsandallpositronsareequal
n =n . Theirspinpolarizationareequaltoeachother
0e 0p
FIG.5: (Coloronline)ThefigureshowstheTrivelpieceGould as well ηe =ηp, where ηa =3µBB0/2εFa, with the Bohr
waveinthee-p-iplasmasforthedifferentratiosbetweencon- magneton µB and Fermi energy εFa = (3π2n0a)32¯h2/2m
centrations of electrons and positrons. Upper (lower) line is ofspeciesa. Thus,wehaven =n andn =n ,
0eu 0pd 0ed 0pu
constructed for β=0.5 (β =0.1) and Σ=0.1.
where n = n (1 − η )/2, n = n (1 + η )/2,
0eu 0e e 0ed 0e e
n = n (1 + η )/2, n = n (1 − η )/2. For the
0pu 0p p 0pd 0p p
e-p-i plasmas we have n = n +n . Consequently,
0e 0p 0i
theequilibriumconcentrationsofelectronsandpositrons
are not equal each other. Therefore, their spin polariza-
tion η ∼ n−2/3 are not equal each other either. Hence,
a 0a
parameters n , n , n and n are four indepen-
0eu 0ed 0pu 0pd
dent parameters (one can use another set of parameters
n , n , η , η ). Relations between parameters depend
0e 0p e p
on the external magnetic field. Next, we consider linear
evolution of perturbations. For the oblique propagating
plane waves k={k ,0,k } we find the following disper-
x z
sion equation:
sin2θ cos2θ ω2
+ Lau
a=e,p(cid:18)ω2−Ω2 ω2 (cid:19)"1−(ωs2in−2Ωθ2 + coωs22θ)Ua2uk2
X
FIG. 6: (Color online) The figure shows the longitudinal
waves in e-p-i plasmas propagating perpendicular to the ex-
ternal magnetic field for β = 0.1. As in Fig. 3 we have ω2
four branches: Langmuir wave(in other words theupperhy- + Lad =1, (3)
bridwave),SEAW,PAWandSEPAW.Itshowsthefrequency 1−(ωs2in−2Ωθ2 + coωs22θ)Ua2dk2#
squareshiftforallbranchesonΩ2,withthedimensionlesscy-
clotron frequency Σ=0.1. where ω2 = 4πe2n /m, and ω2 = 4πe2n /m
Lau 0au Lad 0ad
are the Langmuir frequencies for the spin-up and spin-
downparticlesofspeciesa, andω2 =ω2 +ω2 , Ω=
La Lau Lad
order to discuss the in-depth analysis of SEAW, which eB0/mc is the cyclotron frequency, k2 =kx2+kz2, Ua2s =
waspredictedbymethodofSSE-QHDs,methodofsepa- (6π2n0as)32¯h2/3m2, θ is the angle between direction of
rate spin evolution quantum kinetics, which separately wavepropagationkandtheexternalmagneticfieldB =
0
describes spin-up and spin-down electrons, was devel- B e .
0 z
oped in Ref. [48]. By applying this method, the effects For the electron-positron plasmas we have n = n ,
0e 0p
of SSE on the real dispersion and Landau damping of n = n , n = n . Hence, equation (3) in this
0eu 0pd 0ed 0pu
SEAW were addressed and real and imaginary parts of regime is an equation of the fourth degree relatively ω2
spectrums of ion-acoustic waves and zeroth sound have for the oblique propagation. At the propagationparallel
also been found. Nonlinear SEAWs in presence of the or perpendicular to the external magnetic field equation
exchange interaction are considered in Ref. [49], where (3) simplifies to equation of the second degree.
the existence of spin-electron acoustic soliton is demon- Intheregimeofe-p-iplasmasallfourparametersn ,
0eu
strated. Subsequently,intheRef. [50]ithasbeendemon- n , n andn aredifferentfromeachother. Asthe
0ed 0pu 0pd
strated that the existence of SEAW leads to an explana- result equation (3) is an equation of eight degree rel-
tionofthemechanismoftheelectronCooperpairforma- atively ω2 for the oblique propagating waves. In the
tion in the high temperature superconductors as result regimes of the parallel or perpendicular propagation it
of electron-spelnon interaction (spelnon is the quanta of simplifies to the equation of fourth degree relatively ω2.
5
III. NUMERICAL ANALYSIS B. Electron-positron-ion plasmas
In our numerical analysis we use a single value of the Describing longitudinal waves in e-p-i plasmas, we
electron concentration n ≡ n = 1027 cm−3. We have start with propagation of waves parallel to the external
0e 0
differentregimesofthepositronconcentrations. Tomea- magnetic field. Fig. 3 (Fig. 4) shows results for β =0.1
sure the concentrations of positrons in units of the equi- (β =0.5).
librium concentration of electrons we introduce parame- In the beginning of Sect. III we have described the
ter β = n /n . For the electron-positron plasmas we mechanism of shift of the Langmuir wave dispersion de-
0p 0e
have β = 1. For the electron-positron-ion plasmas we pendence. Same effect reveals itself in the spectrum of
haveβ ∈(0,1). Weconsidertwovaluesofβ forelectron- Trivelpiece-Gould wave,as it is depicted in Fig. 5. First
positron-ion plasmas. They are β =0.5 and β =0.1. of all we see presence of four branches in both cases.
Changing β at the fixed number of electrons n we Comparingthese resultswiththewell-knownresultsand
0e
change the full concentration of light particles in the results found in Refs. [17], [43] we make the following
system. It changes the effective Langmuir frequency, conclusions. The upper dispersionbranchbelongsto the
which is the frequency of Langmuir wave at k → 0, Langmuir wave. One of these linear branches describes
ω2 = 4πe2(n + n )/m = 4πe2n (1 + β)/m. It theSEAWasitfollowsfromtheprevioussubsectionand
L,eff 0e 0p 0e
creates difference in the behavior of the Langmuir wave Ref. [17]. One of two other branches is the PAW found
spectrum on different figures. It is well known result in Refs. [38], [39], which exists due to different concen-
which is not affect the effects caused by the spin polar- trations of electrons and positrons. In addition to three
ization considered in this paper. earlierfound wavesolutions we obtainan extra solution.
For presentation of numerical results we use the fol- Inthee-p-iplasmaswefindfoursubspecieswithdiffer-
lowing dimensionless variables: the dimensionless fre- entspinpolarizations(1+η )/2,(1−η )/2,(1−η )/2=
e e p
quency ξ = ω/ωLe, the dimensionless cyclotron fre- (1+ηeβ−23)/2, and (1+ηp)/2 = (1−ηeβ−32)/2 instead
quency Σ = Ωe/ωLe, and the dimensionless wave vector of two existing in e-i or e-p plasmas. Therefore, we have
κ=vFek/3ωLe. found reacher spectrum of the spin-electron acoustic ex-
citations.
Since new wave arises in the e-p-i plasmas due to the
A. Electron-positron plasmas account of the SSE we call it the spin-electron-positron
acoustic wave (SEPAW). The spectrum changes at the
We start our analysis with relatively simple case of e- change of positron number. As we mention above, it
p plasmas (β = 1). Due to the equal concentrations of changes the full number of the light particles. However,
electronsandpositronstheyhavesamespinpolarization. it changes the spin polarization of positrons depending
Due to difference of the sign of their electric charges the on the Fermi energy of positrons. Therefore, the change
numbersofspin-upelectronsandspin-downpositronsare of β affects the SEPAW.
equalto eachotherandthesesubspecies movesinphase. We calculate spectrum of longitudinal waves propa-
Same picture we have for spin-down electrons and spin- gating perpendicular to the external field as well. We
up positrons. see that it increases square of all waves on Ω2 as it fol-
At the propagation of waves parallel or perpendicu- lowsfromFig. 6. The frequenciesofacousticwavestend
lar to the external field we find two wave solutions: the to zero at k → 0 in the regime of parallel propagation.
Langmuir wave and SEAW, with the properties similar Being shifted in the regime of the perpendicular propa-
to the properties of these waves in e-i plasmas described gation we have ω → Ω at k → 0. It is different for the
in [17]. Langmuirwave. Intheregimeofparallelpropagationwe
Considering the oblique propagation of longitudinal findω2 →(1+β)ω2 atk →0,whilefortheperpendicu-
Le
waves in e-p plasmas we obtain Fig. 1 showing four larpropagationweobtainω2 →(1+β)ω2 +Ω2atk→0.
Le
waves. This regime demonstrates existence of two Therefore,theshiftoffrequencyω(k →0)issmallerthan
SEAWs, similarly to the e-i plasmas considered in Ref. Ω. In our case, for Ω=Σω ≈0.1ω . It is hardly visi-
Le Le
[43]. Hence, Fig. 1 shows the Langmuir wave, the up- bleinFig. 6,sinceω ≃ω [ (1+β)+Σ2/2 (1+β)]=
Le
per SEAW, the Trivelpiece-Gould wave, and the lower ω [1.05+0.01].
Le
p p
SEAW correspondingly, in order of the decrease of their The upper linear branch in Figs. (3), (4), (6) presents
frequency. Relative behavior of the lower SEAW and the SEAW similar to wave in the e-i plasmas. The mid-
the Trivelpiece-Gould wave is shown in Fig. 2. We see dle linear branch presents the PAWs, where spin effects
that they do not have any overlapping. The SEAW has increases frequency of the PAW. The lower branch is
smaller frequencies for all physically possible wave vec- the SEPAW, which is the SEAW in the subsystem of
tors. positrons. It is locatedlowerthan SEAW in the electron
Tosummarizethissubsectionwereportofexistenceof subsystem due to the smallar concentration of positrons
two SEAWs in the e-p plasmas. We also report increase in compare with the concentration of electrons β <1.
ofthe Langmuirwavefrequency due to spin polarization As it is directly follows from the dispersion equation
entering spectrum via the Fermi pressure. (3) number of the dispersion branches doubles at the
6
oblique propagation. Thus, we have eight longitudinal
waves, which reduces to four branches at the parallel
and perpendicular propagations. It is well-known that
in e-i plasmas at the oblique wave propagation we find
Trivelpiece-Gouldand Langmuir waves. The e-i plasmas
with the SSE picture is more interesting. Instead of the
Langmuir wave and the SEAW one can find: the Lang-
muir wave,the Trivelpiece-Gouldwave,the lowerSEAW
and the upper SEAW. If we forget about spin separa-
tion and consider e-p-i plasmas one can find the PAW
[38], [39] at the parallel or perpendicular propagation.
Atthe transitionto the oblique regimewe expectto find
theLangmuirwave,theTrivelpiece-Gouldwave,andtwo
PAWs. TothebestofourknowledgethesecondPAWex-
istingattheobliquepropagationhasnotbeenreportedin
literature. Hence, in this paper, we report the existence FIG. 7: (Color online) The figure shows the oblique propa-
gatinglongitudinalwavesinthee-p-iplasmas. Inthisregime
of this wave.
we find eight branches described in the text. Figure is con-
Main subject of this paper is the e-p-i plasmas with
structed for β=0.1 and Σ=0.1.
the account of the SSE. Therefore,we obtain: the Lang-
muirwave,theTrivelpiece-Gouldwave,thepairofPAWs,
lowerandupper SEAWs, andreportedfor the firsttime,
a pair of SEPAWs. The spectrum of all these waves is
shown in Fig. 7 for β = 0.1. We see that the decrease
of the positron concentration causes shifts of dispersion
dependencies of PAWs and SEPAWs into area of larger
frequencies,whilethedispersiondependenciesofSEAWs
do not show any visible changes. Consequently the dis-
persion dependencies of SEAW, PAW, and SEPAW be-
comes closer. It is hard to distinguish the low frequency
partofthespectruminFig. 7. Hence,wepresentFig. 8,
wherethelowfrequencypartofthespectrumisdepicted.
Fig. 8 contains the Trivelpiece-Gould wave and lower
branchesof the spin-electronacoustic,positronacoustic,
and spin-electron-positron acoustic waves.
FIG. 8: (Color online) The figure shows details of the low
frequency part of spectrum of the oblique propagating longi-
tudinalwaves for β=0.1 and Σ=0.1.
C. Area of applicability of obtained results
T , spin-up electrons T , spin-down electrons T ,
Let us consider the following thought experiment in Fe Feu Fed
and positrons T . In this regime, our model presented
context of analysis of propagation of waves parallel to Fp
by equation (1) and (2) does not work. A strong col-
the external magnetic field.
lision damping in the system of positrons might destroy
Keepingafixednumberofelectronswecanaddanum-
thePAWandSEPAW.Increasingnumberofpositronsto
ber of positrons and take away same number of ions
reachconditionsforthedegeneratepositrongasweenter
to keep the quasi-neutrality of the system. This imag-
area of applicability of our results.
inary quasicontinuous process we could find changes in
For n = 1027 cm−3 and n = 0.1n = 1026 cm−3,
the spectrum of the SEAW, along with the discussed 0e 0p 0e
at B = 1010 G, we have η = 0.025 and η = 0.116.
above changes of the Langmuir wave spectrum. In addi- 0 e p
tion to these changes we expect to find two other waves: It gives TFes(1 ± ηe)32TFe ≈ TFe = 3.47× 107 K and
the PAW and the SEAW. However, these two waves are TFps = {0.23,0.21}TFe ≈ 0.2TFe = 0.7×107 K. Hence,
predicted for plasmas with the degenerate electrons and ourresultscanbeusedfortheplasmaswithtemperatures
positrons. below106 K.Atlargerconcentrationssimilarresultscan
If we consider a temperature regime with the degen- be found for larger temperatures.
erate electrons T ≪ T , and with the both subspecies
Fe
of electrons are degenerate either T ≪T ,T (it re-
Feu Fed
quiresrelativelysmallspinpolarizationforthefinitetem- IV. CONCLUSION
peratures T), we find that positrons, for a small number
of them, a non-degenerate T ∼ T or T ≫ T . Here Separatespinevolutioninsystemswiththepartialspin
Fp Fp
we have used the Fermi temperatures for the electrons polarizationhasrevealeditselfinexistenceofnewwaves.
7
Thereby, we have found the SEAWs in the e-p plasmas. has been reported for the first time.
OneSEAWexistsattheparallelandperpendicularprop-
agation. Two branches exist at the oblique propagation.
Their appearance is related to different Fermi pressure
for the spin-up and spin-down electrons and positrons.
V. ACKNOWLEDGEMENTS
Fore-p-iplasmaswehavedemonstratedexistenceofpair
of SEAWs, pair of PAWs, and pair of SEPAWs along
with the Langmuir and Trivelpiece–Gould waves, at the The workofP.A.wassupportedbythe RussianFoun-
oblique propagation. Three of them have been found for dation for Basic Research (grant no. 16-32-00886) and
thefirsttime: pairofSEPAWsandsecond(upper)PAW. the Dynasty foundation. One of us Z. Iqbal gratefully
Theseeightbranchesreducestofourbranchesatthepar- acknowledge the higher education commission Pakistan
alleland perpendicular propagation. These branchesare for the financial support under IRSIP Award No. 1-
Langmuirwave,SEAW,PAWandSEPAW.TheSEPAW 8/HEC/HRD/2015/4027.
[1] T. Takabayasi, Prog. Theor. Phys. 14, 283 (1955). [27] S. L. Shapiro and S. A. Teukolsky, Black Holes, White
[2] L.S.Kuz’menkov,S.G.Maksimov,andV.V.Fedoseev, DwarfsandNeutronStars: ThePhysicsofCompactOb-
Theor. Math. Fiz. 126, 136 (2001. jects, WILEY-VCH Verlag GmbH & Co. KGaA, Wein-
[3] L.S.Kuz’menkov,S.G.Maksimov,andV.V.Fedoseev, heim, Germany (2004).
Theor. Math. Fiz. 126, 258 (2001). [28] R. G. Greaves, M. D. Tinkle and C. M. Surko, Phys.
[4] T. Koide, Phys. Rev.C 87, 034902 (2013). Plasmas 1, 1439 (1994).
[5] M.MarklundandG.Brodin,Phys.Rev.Lett.98,025001 [29] T. S. Pedersen , J. R. Danielson , C. Hugenschmidt, G.
(2007). Marx, X. Sarasola, F. Schauer, L. Schweikhard, C. M.
[6] G.BrodinandM.Marklund,NewJ.Phys.9,277(2007). Surkoand E. Winkler, NewJ. Phys.14, 035010 (2012).
[7] P. A. Andreev and L. S. Kuz’menkov, Moscow Univ. [30] Y. Wang, P. K. Shukla, B. Eliasson, Phys. Plasmas 20,
Phys.Bull. 62, 271 (2007). 013103 (2013).
[8] D. V. Vagin, N. E. Kim, P. A. Polyakov, and A. E. [31] G. Sarri et.al, NatureCommunications 6, 6747 (2015).
Rusakov,Izv.RANSer.Fiz.70(3),443(2006)[Bull.Rus. [32] G.P.ZankandR.G.GreavesPhys.Rev.E51,9(1995).
Acad.Sci.: Phys. 70(3), 505 (2006)]. [33] N. Iwamoto, Phys. Rev.E 47, 1 (1993).
[9] P.A.AndreevandL.S.Kuz’menkov,Phys.AtomicNu- [34] G. Brodin and M. Marklund 14, 112107 (2007).
clei 71, 1724 (2008). [35] A. Mushtaq, R. Maroof, Z. Ahmad, A. Qamar, Phys.
[10] P.A.AndreevandL.S.Kuz’menkov,Int.J.Mod.Phys. Plasmas 19, 052101 (2012).
B 26, 1250186 (2012). [36] R. Maroof, S. Ali, A. Mushtaq, and A. Qamar, Phys.
[11] G.Brodin,M.Marklund,J. Zamanian,A.Ericsson, and Plasmas 22, 112102 (2015).
P.L. Mana, Phys. Rev.Lett. 101, 245002 (2008). [37] P. A. Andreev,Phys. Plasmas 22, 062113 (2015).
[12] A. P. Misra, G. Brodin, M. Marklund, and P.K. Shukla, [38] Y. N.Nejoh, Aust.J. Phys. 49, 967 (1996).
J. Plasma Phys. 76, 857 (2010). [39] N. L. Tsintsadze, L. N. Tsintsadze, A. Hussain, and G.
[13] Yu.Kagan, Zh.Eksp. Teor. Fiz. 41, 844 (1961). Murtaza, Eur. Phys. J. D 64, 447 (2011).
[14] Yu.Kagan, Zh.Eksp. Teor. Fiz. 41, 1536 (1961). [40] N. L. Tsintsadze, R. Chaudhary, A. Rasheed, J. Plasma
[15] Yu.Kagan, Zh.Eksp. Teor. Fiz. 51, 1893 (1966). Physics 79, 587 (2013).
[16] A.Hussain,M.StefanandG.Brodin,Phys.Plasmas21, [41] H. Metref and M. Tribeche, Phys. Plasmas 21, 122117
032104 (2014). (2014).
[17] P.A. Andreev,Phys.Rev.E 91, 033111 (2015). [42] M.Tribeche,K.Aoutou,S.YounsiandR.Amour,Phys.
[18] M. Iv.Trukhanova,Phys. Lett.A 379, 2777 (2015). Plasmas 16, 072103 (2009).
[19] A.D.Margulis and VI.A.Marguli, Zh.Eksp.Teor.Fiz. [43] P.A.Andreev,L.S.Kuz’menkov,AnnalsofPhysics361,
93, 1800 (1987). 278 (2015).
[20] J. C. Ryan,Phys. Rev.B 43, 4499 (1991). [44] P. A. Andreev,L. S. Kuz’menkov,arXiv:1510.03468.
[21] Rees,M.J.1983Whattheastrophysicistwantsfromthe [45] A. Yu. Ivanov, P. A. Andreev, L. S. Kuz’menkov, Prog.
very early universe. In: The Very Early Universe: UK Theor. Exp.Phys. 2015, 063I02 (2015).
Cambridge University Press, pp.29–58. [46] P.A.Andreev,L.S.Kuz’menkov,Eur.Phys.Lett.113,
[22] N.P.ArendtJr.,andA.J.Eilek,Astrophys.J.581,469 17001 (2016).
(2002). [47] P. A. Andreev,L. S. Kuz’menkov,arXiv:1512.07940.
[23] P.A. Sturrock,Astrophys.J.164, 529 (1971). [48] P. A. Andreev,arXiv:1409.7885.
[24] W. B. Han, R. Ruffiniand S. S. Xue, Phys. Rev.D 86, [49] P.A.Andreev,arXiv:1504.08234(acceptedinPhys.Plas-
084004 (2012). mas).
[25] D.Lai, Rev.Mod. Phys.73, 629 (2001). [50] P. A. Andreev, P. A. Polyakov, and L. S. Kuz’menkov,
[26] A. K. Harding and D. Lai, Rep. Prog. Phys. 69, 2631 arXiv:1507.03295.
(2006).