text: suppressed, (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) δm2 sin2 2θ (cid:28) sin2 2θ , when |V | (cid:29) cos 2θ , (3.12) eﬀ 0 eﬀ 0 2E which can eﬃciently block the production of sterile neutrinos. The value of T to block νs (cid:18) (cid:19) production at BBN time can be roughly estimated as follows: G δm2 T δm2 M2 1/4 V ∼ X ET 4 > ⇒ νs > X . eﬀ M2 νs 2E MeV 2E2 G X X − Take G ∼ 104G and M ∼ 1 GeV, we get T ∼ 10 3MeV around BBN time. This X F X νs is what wee see the smallness of T all the way to BBN time from the dot-dashed line in νs (cid:12) (cid:12) Fig. 2. (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) As the Universe cools down, the matter potential V gets smaller very quickly. When eﬀ δm2 |V | < cos 2θ , matter eﬀect can be neglected and θ (cid:39) θ . Again, we discuss in the eﬀ 0 eﬀ 0 2E simpliﬁed framework of two-ﬂavor case, ν -ν or ν -ν in the mass eigenstates, a s 1 2 ν = cos θ ν − sin θ ν , ν = sin θ ν + cos θ ν . a eﬀ 1 eﬀ 2 s eﬀ 1 eﬀ 2 Before BBN, because of the highly suppressed θ , there are mostly ν or ν neutrinos in eﬀ a 1 the thermal bath, and a small amount of ν or ν states. After θ is not suppressed any s 2 eﬀ more, ν has the scatter(cid:114)ing process to produce ν through ν + ν → ν + ν with rate 1 2 1 2 2 2 Γ ∼ G2 T 5 sin2 2θ , and ν + ν → ν + ν with rate Γ ∼ G2 T 5 sin4 2θ . If Γ is larger than X νs 0 1 1 2 2 X νa 0 8πG Hubble parameter H = ρ , then ν and ν , or ν and ν , will reach equal numbers R 1 2 a s 3 quickly. Since the sterile neutrinos’ self-interaction will induce rapidly elastic scattering, ν ν → ν ν , it can redistribute momenta among sterile neutrinos. As long as the scattering s s s s rate Γ ∼ G2 T 5 is larger than the Hubble parameter H (G is the Newton’s constant), sterile X X νs neutrinos will soon reach the Fermi-Dirac distribution, leading to the ﬂavor equilibrium, n = n . For G = 108G and M = 1.2MeV it was estimated that the equilibrium νs νa X F X would be approached around T ∼ 40KeV [15], although numerically being a formidable γ challenge for such a large interaction. Before a detailed discussion on N at or after CMB time, we should note that the eﬀ above investigation only took into account the sterile neutrinos produced from oscillations. However, oscillation is not the only contributing process. Generally, we expect there is a 9 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: suppressed,

text: 2θ

text: 2θ

text: when

text: |V

text: cos

text: 2θ

text: eﬀ

text: eﬀ

text: which

text: can

text: eﬃciently

text: block

text: the

text: production

text: of

text: sterile

text: neutrinos.

text: The

text: value

text: of

text: T

text: to

text: block

text: νs

text: production

text: at

text: BBN

text: time

text: can

text: be

text: roughly

text: estimated

text: as

text: follows:

text: G

text: T

text: V

text: X

text: ET

text: νs

text: X

text: eﬀ

text: νs

text: MeV

text: G

text: X

text: X

text: Take

text: G

text: and

text: M

text: GeV,

text: we

text: get

text: T

text: around

text: BBN

text: time.

text: This

text: X

text: F

text: X

text: νs

text: is

text: what

text: wee

text: see

text: the

text: smallness

text: of

text: T

text: all

text: the

text: way

text: to

text: BBN

text: time

text: from

text: the

text: dot-dashed

text: line

text: in

text: νs

text: Fig.

text: As

text: the

text: Universe

text: cools

text: down,

text: the

text: matter

text: potential

text: V

text: gets

text: smaller

text: very

text: quickly.

text: When

text: eﬀ

text: |V

text: cos

text: 2θ

text: matter

text: eﬀect

text: can

text: be

text: neglected

text: and

text: θ

text: θ

text: Again,

text: we

text: discuss

text: in

text: the

text: eﬀ

text: eﬀ

text: simpliﬁed

text: framework

text: of

text: two-ﬂavor

text: case,

text: ν

text: -ν

text: or

text: ν

text: -ν

text: in

text: the

text: mass

text: eigenstates,

text: a

text: s

text: ν

text: cos

text: θ

text: ν

text: sin

text: θ

text: ν

text: ν

text: sin

text: θ

text: ν

text: cos

text: θ

text: ν

text: a

text: eﬀ

text: eﬀ

text: s

text: eﬀ

text: eﬀ

text: Before

text: BBN,

text: because

text: of

text: the

text: highly

text: suppressed

text: θ

text: there

text: are

text: mostly

text: ν

text: or

text: ν

text: neutrinos

text: in

text: eﬀ

text: a

text: the

text: thermal

text: bath,

text: and

text: a

text: small

text: amount

text: of

text: ν

text: or

text: ν

text: states.

text: After

text: θ

text: is

text: not

text: suppressed

text: any

text: s

text: eﬀ

text: more,

text: ν

text: has

text: the

text: process

text: to

text: produce

text: ν

text: through

text: ν

text: ν

text: ν

text: ν

text: with

text: rate

text: Γ

text: T

text: 2θ

text: and

text: ν

text: ν

text: ν

text: ν

text: with

text: rate

text: Γ

text: T

text: 2θ

text: If

text: Γ

text: is

text: larger

text: than

text: X

text: νs

text: X

text: νa

text: Hubble

text: parameter

text: H

text: ρ

text: then

text: ν

text: and

text: ν

text: or

text: ν

text: and

text: ν

text: will

text: reach

text: equal

text: numbers

text: R

text: a

text: s

text: quickly.

text: Since

text: the

text: sterile

text: neutrinos’

text: self-interaction

text: will

text: induce

text: rapidly

text: elastic

text: scattering,

text: ν

text: ν

text: ν

text: ν

text: it

text: can

text: redistribute

text: momenta

text: among

text: sterile

text: neutrinos.

text: As

text: long

text: as

text: the

text: scattering

text: s

text: s

text: s

text: s

text: rate

text: Γ

text: T

text: is

text: larger

text: than

text: the

text: Hubble

text: parameter

text: H

text: (G

text: is

text: the

text: Newton’s

text: constant),

text: sterile

text: X

text: X

text: νs

text: neutrinos

text: will

text: soon

text: reach

text: the

text: Fermi-Dirac

text: distribution,

text: leading

text: to

text: the

text: ﬂavor

text: equilibrium,

text: n

text: n

text: For

text: G

text: and

text: M

text: it

text: was

text: estimated

text: that

text: the

text: equilibrium

text: νs

text: νa

text: X

text: F

text: X

text: would

text: be

text: approached

text: around

text: T

text: although

text: numerically

text: being

text: a

text: formidable

text: γ

text: challenge

text: for

text: such

text: a

text: large

text: interaction.

text: Before

text: a

text: detailed

text: discussion

text: on

text: N

text: at

text: or

text: after

text: CMB

text: time,

text: we

text: should

text: note

text: that

text: the

text: eﬀ

text: above

text: investigation

text: only

text: took

text: into

text: account

text: the

text: sterile

text: neutrinos

text: produced

text: from

text: oscillations.

text: However,

text: oscillation

text: is

text: not

text: the

text: only

text: contributing

text: process.

text: Generally,

text: we

text: expect

text: there

text: is

text: a

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: A graph-theoretic interpretation of the Lucas Sequence is now established by Theorem 1.2, and from its proof emerges a graph-theoretic interpretation of the terms of the Dickson poly- nomial. 3. Acknowledgments We would like to thank Professor Robert Coulter for bringing our attention to the connection between Lucas Sequences and Dickson polynomials, which led to a much shorter proof of our main result. We also thank both him and Professor Felix Lazebnik for their suggestions on the preliminary draft of this note. Finally, we thank the referee for carefully reviewing this note, and for catching a crucial indexing error. References [1] G. Hopkins and W. Staton, Some identities arising from the Fibonacci numbers of certain graphs, The Fibonacci Quarterly, 22 (1984), 255–258. [2] C. Kimberling, Divisibility properties of recurrent sequences, The Fibonacci Quarterly, 14 (1976), 369–376. [3] C. Kimberling, Greatest common divisors of sums and diﬀerences of Fibonacci, Lucas, and Chebyshev polynomials, The Fibonacci Quarterly, 17 (1979), 18–22. [4] R. Lidl, G. L. Mullen and R. Turnwald, Dickson Polynomials, Pitman Monographs and Surveys in Pure and Applied Mathematics, 1965 [5] H. Prodinger and R. Tichy, Fibonacci numbers of graphs, The Fibonacci Quarterly, 20 (1982), 16–21. James Alexander, Department of Mathematical Sciences, University of Delaware, Newark, Delaware, United States Email address: alex@math.udel.edu Paul Hearding, Department of Mathematical Sciences, University of Delaware, Newark, Delaware, United States Email address: hearding@math.udel.edu 4

text: A

label: paragraph

text: graph-theoretic

text: interpretation

text: of

text: the

text: Lucas

text: Sequence

text: is

text: now

text: established

text: by

text: Theorem

text: and

text: from

text: its

text: proof

text: emerges

text: a

text: graph-theoretic

text: interpretation

text: of

text: the

text: terms

text: of

text: the

text: Dickson

text: poly-

text: nomial.

label: section

text: Acknowledgments

text: We

text: would

text: like

text: to

text: thank

text: Professor

text: Robert

text: Coulter

text: for

text: bringing

text: our

text: attention

text: to

text: the

text: connection

text: between

text: Lucas

text: Sequences

text: and

text: Dickson

text: polynomials,

text: which

text: led

text: to

text: a

text: much

text: shorter

text: proof

text: of

text: our

text: main

text: result.

text: We

text: also

text: thank

text: both

text: him

text: and

text: Professor

text: Felix

text: Lazebnik

text: for

text: their

text: suggestions

text: on

text: the

text: preliminary

text: draft

text: of

text: this

text: note.

text: Finally,

text: we

text: thank

text: the

text: referee

text: for

text: carefully

text: reviewing

text: this

text: note,

text: and

text: for

text: catching

text: a

text: crucial

text: indexing

text: error.

text: References

text: G.

text: Hopkins

text: and

text: W.

text: Staton,

text: Some

text: identities

text: arising

text: from

text: the

text: Fibonacci

text: numbers

text: of

text: certain

text: graphs,

text: The

text: Fibonacci

text: Quarterly,

text: C.

text: Kimberling,

text: Divisibility

text: properties

text: of

text: recurrent

text: sequences,

text: The

text: Fibonacci

text: Quarterly,

text: C.

text: Kimberling,

text: Greatest

text: common

text: divisors

text: of

text: sums

text: and

text: diﬀerences

text: of

text: Fibonacci,

text: Lucas,

text: and

text: Chebyshev

text: polynomials,

text: The

text: Fibonacci

text: Quarterly,

text: R.

text: Lidl,

text: G.

text: L.

text: Mullen

text: and

text: R.

text: Turnwald,

text: Dickson

text: Polynomials,

text: Pitman

text: Monographs

text: and

text: Surveys

text: in

text: Pure

text: and

text: Applied

text: Mathematics,

text: H.

text: Prodinger

text: and

text: R.

text: Tichy,

text: Fibonacci

text: numbers

text: of

text: graphs,

text: The

text: Fibonacci

text: Quarterly,

text: James

text: Alexander,

text: Department

text: of

text: Mathematical

text: Sciences,

text: University

text: of

text: Delaware,

text: Newark,

text: Delaware,

text: United

text: States

text: Email

text: address:

text: alex@math.udel.edu

text: Paul

text: Hearding,

text: Department

text: of

text: Mathematical

text: Sciences,

text: University

text: of

text: Delaware,

text: Newark,

text: Delaware,

text: United

text: States

text: Email

text: address:

text: hearding@math.udel.edu


text: 2 P. F. Wang, C. Wang, and J. L. Han et al. 1998; Crawford et al. 2001; Weltevrede & Johnston 2008). To understand various polarization features, numerous theoretical researches on pulsar polarization have been de- veloped on the emission processes (e.g. Xu et al. 2000; Dyks et al. 2010; Wang et al. 2012; Kumar & Gangadhara 2012) or the propagation eﬀects (e.g. Wang et al. 2010; Beskin & Philippov 2012), both of which can lead to depolariza- tion. The depolarization of pulsar linear polarization is gen- erally attributed to simultaneous interaction of two modes of orthogonally polarized radiation (McKinnon & Stinebring 1998; Karastergiou et al. 2005) or owing to randomization of position angles in weaker magnetic ﬁelds in outer mag- netosphere (Manchester et al. 1975; Morris et al. 1981). For example, McKinnon (1997) attributed the depolarization to birefringence of the X mode and O mode above pulsar polar caps. von Hoensbroech et al. (1998) qualitatively explained depolarization at high frequencies by considering the evo- lution of one natural plasma mode within pulsar magneto- sphere. However, the formation and distribution of the two modes within a pulsar beam were not mentioned; the re- fraction eﬀect was simply discussed by analogy with the cal- cite crystal; the polarization limiting eﬀect was simply anal- Figure 1. Emission regions on a bundle of magnetic ﬁeld lines (cid:39) ysed; and the evolution of linear polarization across ﬁeld line with θ 2/3θnm for a given sight line. The magnetic moment planes was not investigated. To get a better understanding m is inclined by an angle of α with respect to the rotation axis of the frequency dependence of pulsar linear polarization, Ω. The thick line outlines all tangential emission points of ﬁeld emission processes together with propagation eﬀects need lines. θnm represents the angle between the wave vector n and the magnetic momentum m. The grey area represents the region to be investigated systematically. for outﬂowing relativistic particles along a bundle of magnetic Recently, we jointly studied the polarized curvature ra- ﬁeld lines. s = 0.0 denotes the magnetic axis, s = 1.0 denotes the diation together with propagation processes within pulsar last open ﬁeld line (LOF). magnetosphere numerically (Wang et al. 2014). We suc- ceeded in demonstrating the distributions of the X-mode and O-mode within a pulsar magnetosphere and explained the Pulsar magnetic ﬁelds are assumed to be of the static dipole depolarization across an entire pulsar beam. We found that form in the radio emission region, the depolarization is serious if the corotation of relativis- B = B (R(cid:63) )3[3rˆ(rˆ · mˆ ) − mˆ ], (1) tic particles is not considered. When the rotation is taken (cid:63) r into account, signiﬁcant linear polarization will be produced. here B is magnetic ﬁeld strength at the magnetic equa- (cid:63) However, the frequency dependence behaviour of linear po- tor of a neutron star, R is the neutron star radius, rˆ is the (cid:63) larization was not incorporated in the analysis. unit vector along r, and mˆ represents the unit vector of the Stimulated by the observational and theoretical inves- magnetic dipole moment. Within the static dipole magne- tigations, we hereby develop the joint research of emission tosphere, emission generated from the regions of the same and propagation processes to understand frequency depen- polar angle θ within a given magnetic ﬁeld line plane will dence of pulsar linear polarization by considering the polar- point towards the same direction, as shown in Fig. 1. ized curvature radiation process together with propagation Along a ﬁeld line, emission radiated at diﬀerent heights eﬀects in a pulsar magnetosphere. In Section 2, we present has diﬀerent frequencies. The characteristic frequency for theoretical basics for our calculations. Linearly polarized ra- curvature radiation reads, diation within an entire pulsar beam, polarized pulse proﬁle 3γ3c and their evolution with frequency are described in Section ν = . (2) 3. Comparisons with observations are elaborated in Section CR 4πρ 4. Discussions and conclusions are presented in Section 5. Here, γ is the Lorentz factor of a relativistic particle, ρ 2 THEORETICAL BASICS represents the curvature radius for instaneous particle tra- jectory. The larger curvature radius ρ for particle trajec- 2.1 Geometry and emission tory in the high magnetosphere naturally leads to lower fre- Pulsar radio emission is generally believed to be gener- quency emission. This is the physical basis for the radius to ated by relativistic particles streaming out along the curved frequency mapping. magnetic ﬁeld lines within the open magnetosphere. Due to For an inclined dipole magnetosphere, the pulsar emis- the bending of ﬁeld lines, relativistic particles will experience sion beam is generally compressed in the meridional plane perpendicular acceleration and produce curvature radiation. deﬁned by the rotation axis and magnetic axis (Biggs 1990). Coherent curvature radiation from bunches of relativistic For simplicity, our calculations assume that pulsar beams particles serves as one of the most probable mechanisms for are circular, i.e., the last open ﬁeld lines which deﬁne the pulsar radio emission. The coherent particle bunch can be boundary of a pulsar beam have the same ﬁeld line constant treated as a point-like huge charge for the sake of simplicity. of r , as deﬁned by equation (15) of Gangadhara (2004). e,lof (cid:13) c 0000 RAS, MNRAS 000, 1–10 ##LTLine## ##LTLine##

text: P.

text: F.

text: Wang,

text: C.

text: Wang,

text: and

text: J.

text: L.

text: Han

text: et

text: al.

text: Crawford

text: et

text: al.

text: Weltevrede

text: Johnston

text: To

text: understand

text: various

text: polarization

text: features,

text: numerous

text: theoretical

text: researches

text: on

text: pulsar

text: polarization

text: have

text: been

text: de-

text: veloped

text: on

text: the

text: emission

text: processes

text: (e.g.

text: Xu

text: et

text: al.

text: Dyks

text: et

text: al.

text: Wang

text: et

text: al.

text: Kumar

text: Gangadhara

text: or

text: the

text: propagation

text: eﬀects

text: (e.g.

text: Wang

text: et

text: al.

text: Beskin

text: Philippov

text: both

text: of

text: which

text: can

text: lead

text: to

text: depolariza-

text: tion.

text: The

text: depolarization

text: of

text: pulsar

text: linear

text: polarization

text: is

text: gen-

text: erally

text: attributed

text: to

text: simultaneous

text: interaction

text: of

text: two

text: modes

text: of

text: orthogonally

text: polarized

text: radiation

text: (McKinnon

text: Stinebring

text: Karastergiou

text: et

text: al.

text: or

text: owing

text: to

text: randomization

text: of

text: position

text: angles

text: in

text: weaker

text: magnetic

text: ﬁelds

text: in

text: outer

text: mag-

text: netosphere

text: (Manchester

text: et

text: al.

text: Morris

text: et

text: al.

text: For

text: example,

text: McKinnon

text: attributed

text: the

text: depolarization

text: to

text: birefringence

text: of

text: the

text: X

text: mode

text: and

text: O

text: mode

text: above

text: pulsar

text: polar

text: caps.

text: von

text: Hoensbroech

text: et

text: al.

text: qualitatively

text: explained

text: depolarization

text: at

text: high

text: frequencies

text: by

text: considering

text: the

text: evo-

text: lution

text: of

text: one

text: natural

text: plasma

text: mode

text: within

text: pulsar

text: magneto-

text: sphere.

text: However,

text: the

text: formation

text: and

text: distribution

text: of

text: the

text: two

text: modes

text: within

text: a

text: pulsar

text: beam

text: were

text: not

text: mentioned;

text: the

text: re-

text: fraction

text: eﬀect

text: was

text: simply

text: discussed

text: by

text: analogy

text: with

text: the

text: cal-

text: cite

text: crystal;

text: the

text: polarization

text: limiting

text: eﬀect

text: was

text: simply

text: anal-

text: Figure

text: Emission

text: regions

text: on

text: a

text: bundle

text: of

text: magnetic

text: ﬁeld

text: lines

text: ysed;

text: and

text: the

text: evolution

text: of

text: linear

text: polarization

text: across

text: ﬁeld

text: line

text: with

text: θ

text: for

text: a

text: given

text: sight

text: line.

text: The

text: magnetic

text: moment

text: planes

text: was

text: not

text: investigated.

text: To

text: get

text: a

text: better

text: understanding

text: m

text: is

text: inclined

text: by

text: an

text: angle

text: of

text: α

text: with

text: respect

text: to

text: the

text: rotation

text: axis

text: of

text: the

text: frequency

text: dependence

text: of

text: pulsar

text: linear

text: polarization,

text: Ω.

text: The

text: thick

text: line

text: outlines

text: all

text: tangential

text: emission

text: points

text: of

text: ﬁeld

text: emission

text: processes

text: together

text: with

text: propagation

text: eﬀects

text: need

text: lines.

text: θnm

text: represents

text: the

text: angle

text: between

text: the

text: wave

text: vector

text: n

text: and

text: the

text: magnetic

text: momentum

text: m.

text: The

text: grey

text: area

text: represents

text: the

text: region

text: to

text: be

text: investigated

text: systematically.

text: for

text: outﬂowing

text: relativistic

text: particles

text: along

text: a

text: bundle

text: of

text: magnetic

text: Recently,

text: we

text: jointly

text: studied

text: the

text: polarized

text: curvature

text: ra-

text: ﬁeld

text: lines.

text: s

text: denotes

text: the

text: magnetic

text: axis,

text: s

text: denotes

text: the

text: diation

text: together

text: with

text: propagation

text: processes

text: within

text: pulsar

text: last

text: open

text: ﬁeld

text: line

text: (LOF).

text: magnetosphere

text: numerically

text: (Wang

text: et

text: al.

text: We

text: suc-

text: ceeded

text: in

text: demonstrating

text: the

text: distributions

text: of

text: the

text: X-mode

text: and

text: O-mode

text: within

text: a

text: pulsar

text: magnetosphere

text: and

text: explained

text: the

text: Pulsar

text: magnetic

text: ﬁelds

text: are

text: assumed

text: to

text: be

text: of

text: the

text: static

text: dipole

text: depolarization

text: across

text: an

text: entire

text: pulsar

text: beam.

text: We

text: found

text: that

text: form

text: in

text: the

text: radio

text: emission

text: region,

text: the

text: depolarization

text: is

text: serious

text: if

text: the

text: corotation

text: of

text: relativis-

text: B

text: B

text: mˆ

text: mˆ

text: tic

text: particles

text: is

text: not

text: considered.

text: When

text: the

text: rotation

text: is

text: taken

text: r

text: into

text: account,

text: signiﬁcant

text: linear

text: polarization

text: will

text: be

text: produced.

text: here

text: B

text: is

text: magnetic

text: ﬁeld

text: strength

text: at

text: the

text: magnetic

text: equa-

text: However,

text: the

text: frequency

text: dependence

text: behaviour

text: of

text: linear

text: po-

text: tor

text: of

text: a

text: neutron

text: star,

text: R

text: is

text: the

text: neutron

text: star

text: radius,

text: rˆ

text: is

text: the

text: larization

text: was

text: not

text: incorporated

text: in

text: the

text: analysis.

text: unit

text: vector

text: along

text: r,

text: and

text: mˆ

text: represents

text: the

text: unit

text: vector

text: of

text: the

text: Stimulated

text: by

text: the

text: observational

text: and

text: theoretical

text: inves-

text: magnetic

text: dipole

text: moment.

text: Within

text: the

text: static

text: dipole

text: magne-

text: tigations,

text: we

text: hereby

text: develop

text: the

text: joint

text: research

text: of

text: emission

text: tosphere,

text: emission

text: generated

text: from

text: the

text: regions

text: of

text: the

text: same

text: and

text: propagation

text: processes

text: to

text: understand

text: frequency

text: depen-

text: polar

text: angle

text: θ

text: within

text: a

text: given

text: magnetic

text: ﬁeld

text: line

text: plane

text: will

text: dence

text: of

text: pulsar

text: linear

text: polarization

text: by

text: considering

text: the

text: polar-

text: point

text: towards

text: the

text: same

text: direction,

text: as

text: shown

text: in

text: Fig.

text: ized

text: curvature

text: radiation

text: process

text: together

text: with

text: propagation

text: Along

text: a

text: ﬁeld

text: line,

text: emission

text: radiated

text: at

text: diﬀerent

text: heights

text: eﬀects

text: in

text: a

text: pulsar

text: magnetosphere.

text: In

text: Section

text: we

text: present

text: has

text: diﬀerent

text: frequencies.

text: The

text: characteristic

text: frequency

text: for

text: theoretical

text: basics

text: for

text: our

text: calculations.

text: Linearly

text: polarized

text: ra-

text: curvature

text: radiation

text: reads,

text: diation

text: within

text: an

text: entire

text: pulsar

text: beam,

text: polarized

text: pulse

text: proﬁle

text: and

text: their

text: evolution

text: with

text: frequency

text: are

text: described

text: in

text: Section

text: ν

text: Comparisons

text: with

text: observations

text: are

text: elaborated

text: in

text: Section

text: CR

text: 4πρ

text: Discussions

text: and

text: conclusions

text: are

text: presented

text: in

text: Section

text: Here,

text: γ

text: is

text: the

text: Lorentz

text: factor

text: of

text: a

text: relativistic

text: particle,

text: ρ

text: THEORETICAL

text: BASICS

text: represents

text: the

text: curvature

text: radius

text: for

text: instaneous

text: particle

text: tra-

text: jectory.

text: The

text: larger

text: curvature

text: radius

text: ρ

text: for

text: particle

text: trajec-

text: Geometry

text: and

text: emission

text: tory

text: in

text: the

text: high

text: magnetosphere

text: naturally

text: leads

text: to

text: lower

text: fre-

text: Pulsar

text: radio

text: emission

text: is

text: generally

text: believed

text: to

text: be

text: gener-

text: quency

text: emission.

text: This

text: is

text: the

text: physical

text: basis

text: for

text: the

text: radius

text: to

text: ated

text: by

text: relativistic

text: particles

text: streaming

text: out

text: along

text: the

text: curved

text: frequency

text: mapping.

text: magnetic

text: ﬁeld

text: lines

text: within

text: the

text: open

text: magnetosphere.

text: Due

text: to

text: For

text: an

text: inclined

text: dipole

text: magnetosphere,

text: the

text: pulsar

text: emis-

text: the

text: bending

text: of

text: ﬁeld

text: lines,

text: relativistic

text: particles

text: will

text: experience

text: sion

text: beam

text: is

text: generally

text: compressed

text: in

text: the

text: meridional

text: plane

text: perpendicular

text: acceleration

text: and

text: produce

text: curvature

text: radiation.

text: deﬁned

text: by

text: the

text: rotation

text: axis

text: and

text: magnetic

text: axis

text: (Biggs

text: Coherent

text: curvature

text: radiation

text: from

text: bunches

text: of

text: relativistic

text: For

text: simplicity,

text: our

text: calculations

text: assume

text: that

text: pulsar

text: beams

text: particles

text: serves

text: as

text: one

text: of

text: the

text: most

text: probable

text: mechanisms

text: for

text: are

text: circular,

text: i.e.,

text: the

text: last

text: open

text: ﬁeld

text: lines

text: which

text: deﬁne

text: the

text: pulsar

text: radio

text: emission.

text: The

text: coherent

text: particle

text: bunch

text: can

text: be

text: boundary

text: of

text: a

text: pulsar

text: beam

text: have

text: the

text: same

text: ﬁeld

text: line

text: constant

text: treated

text: as

text: a

text: point-like

text: huge

text: charge

text: for

text: the

text: sake

text: of

text: simplicity.

text: of

text: r

text: as

text: deﬁned

text: by

text: equation

text: of

text: Gangadhara

text: e,lof

text: c

text: RAS,

text: MNRAS

text: ##LTLine##

text: ##LTLine##


text: On frequency dependence of pulsar linear polarization 3 asymmetry, with heights for the leading part of the beam getting larger than those for the trailing part, as shown in the right panels of Fig. 2. For emission of a single particle bunch at a given height in the pulsar magnetosphere, the radiation ﬁeld E(t) and the corresponding Fourier components E(ω) can be calcu- lated by using the circular path approximation, as described by Wang et al. (2012). The emission from a relativistic par- ticle bunch is beamed in a 1/γ cone around the velocity direction. Therefore, the detectable emission from a given ν = 0.8GHz without rotation ν = 0.8GHz with rotation height and rotation phase is contributed by relativistic par- ticles not only at the tangential emission point of a ﬁeld line but also on nearby ﬁeld lines within the 1/γ cone. By inte- grating the emission within an entire emission cone from all ﬁeld lines, we will obtain the total polarized radiations in the given direction. In general, curvature radiation is highly linearly polarized, and the rotation has great inﬂuences on polarized curvature radiation process (Wang et al. 2014). 2.2 Propagation ν = 1.4GHz without rotation ν = 1.4GHz with rotation Once polarized emission is generated through the curva- ture radiation process, they will be coupled to local plasma modes to propagate out in a pulsar magnetosphere, which is ﬁlled with relativistic plasmas that can be simply assumed to be cold (with a single Lorentz factor γ) and with a den- sity of N = ηN . Here, η is the multiplicity factor and p GJ N represents the Goldreich-Julian density (Goldreich & GJ Julian 1969). Two eigen transverse modes, the X-mode and O-mode, of the plasma have been carefully investigated by Wang et al. (2010) and Beskin & Philippov (2012). The X- mode waves have a refraction index of n = 1 and propagate X ν = 3.0GHz without rotation ν = 3.0GHz with rotation rectilinearly; while the O-mode waves have a refraction in- dex of n < 1 and suﬀer refraction (Barnard & Arons 1986). Figure 2. Emission heights (in the unit of R(cid:63)) for diﬀerent fre- O The polarization states of the X-mode and O-mode change quencies in a pulsar beam, plotted in the rotation phase ϕ and the sight line angle ζ. Corotation of relativistic particles is not during their propagation. In the inner magnetosphere near considered for the left panels, but considered for the right panels. the emission region, the polarization states of both modes The parameters used for model calculations are an inclination an- evolve adiabatically, i.e., with the polarization of the X- − gle of α = 30o, a Lorentz factor of γ = 500, and a rotation period mode wave orthogonal to the local k B plane and the − of a pulsar P = 1s. polarization of the O-mode wave within the k B plane (Cheng & Ruderman 1979). As the waves propagate to the higher magnetosphere, the adiabatic condition is not satis- In general, the beam size in the meridional direction does ﬁed and wave-mode coupling happens, i.e., one mode leaks not aﬀect our conclusions in this paper. to the other and vice versa. The altitude where the mode The emission regions at three diﬀerent frequencies of coupling happens is named as being the polarization limiting ν are calculated as shown in Fig. 2. The height ranges are radius (Cheng & Ruderman 1979; Wang et al. 2010). Above CR quasi symmetric about the magnetic axis for emission at a the polarization limiting radius, the polarization states of given frequency generated from the static dipole magneto- both modes are frozen and will not be further aﬀected by sphere. Emission near the beam centre comes from a lower the plasma, except for possible cyclotron absorption in the altitude, as shown in the left panels of Fig. 2, because the outer magnetosphere. curvature radii of the inner ﬁeld lines at a lower altitude In addition to evolution of a single wave mode, the dis- are comparable to those of the outer ﬁeld lines at a higher tributions of polarized waves within each 1/γ emission cone altitude. Further, the entire emission regions get deeper in also change during propagation. As shown by Wang et al. the magnetosphere for a higher frequency, which is caused (2014), the O mode refraction will lead detectable X and O by the smaller curvature radii of ﬁeld lines in the deeper modes at a given position to be incoherent, as they origi- magnetosphere. It should be noted that relativistic particles nate from diﬀerent parts of a pulsar beam. Adiabatic walk- within a pulsar magnetosphere not only stream along the ing will result in the ordering of polarized waves within the curved magnetic ﬁeld lines, but also corotate with magneto- 1/γ cone, i.e., with the O-mode wave vectors following the sphere. When the corotation is considered, the trajectories polarization plane of initial emission, but wave vectors for and velocities of relativistic particles will be bent towards the X-mode rotate by 90o. Therefore, incoherent superposi- the rotation direction compared with those for the static tion of the orthogonal X-mode and O-mode emission causes dipole magnetosphere. Hence, the emission regions become depolarization for total radiation power. If the corotation of (cid:13) c 0000 RAS, MNRAS 000, 1–10

text: On

label: title

text: frequency

text: dependence

text: of

text: pulsar

text: linear

text: polarization

text: asymmetry,

text: with

text: heights

text: for

text: the

text: leading

text: part

text: of

text: the

text: beam

text: getting

text: larger

text: than

text: those

text: for

text: the

text: trailing

text: part,

text: as

text: shown

text: in

text: the

text: right

text: panels

text: of

text: Fig.

text: For

text: emission

text: of

text: a

text: single

text: particle

text: bunch

text: at

text: a

text: given

text: height

text: in

text: the

text: pulsar

text: magnetosphere,

text: the

text: radiation

text: ﬁeld

text: E(t)

text: and

text: the

text: corresponding

text: Fourier

text: components

text: E(ω)

text: can

text: be

text: calcu-

text: lated

text: by

text: using

text: the

text: circular

text: path

text: approximation,

text: as

text: described

text: by

text: Wang

text: et

text: al.

text: The

text: emission

text: from

text: a

text: relativistic

text: par-

text: ticle

text: bunch

text: is

text: beamed

text: in

text: a

text: cone

text: around

text: the

text: velocity

text: direction.

text: Therefore,

text: the

text: detectable

text: emission

text: from

text: a

text: given

text: ν

text: without

text: rotation

text: ν

text: with

text: rotation

text: height

text: and

text: rotation

text: phase

text: is

text: contributed

text: by

text: relativistic

text: par-

text: ticles

text: not

text: only

text: at

text: the

text: tangential

text: emission

text: point

text: of

text: a

text: ﬁeld

text: line

text: but

text: also

text: on

text: nearby

text: ﬁeld

text: lines

text: within

text: the

text: cone.

text: By

text: inte-

text: grating

text: the

text: emission

text: within

text: an

text: entire

text: emission

text: cone

text: from

text: all

text: ﬁeld

text: lines,

text: we

text: will

text: obtain

text: the

text: total

text: polarized

text: radiations

text: in

text: the

text: given

text: direction.

text: In

text: general,

text: curvature

text: radiation

text: is

text: highly

text: linearly

text: polarized,

text: and

text: the

text: rotation

text: has

text: great

text: inﬂuences

text: on

text: polarized

text: curvature

text: radiation

text: process

text: (Wang

text: et

text: al.

text: Propagation

text: ν

text: without

text: rotation

text: ν

text: with

text: rotation

text: Once

text: polarized

text: emission

text: is

text: generated

text: through

text: the

text: curva-

text: ture

text: radiation

text: process,

text: they

text: will

text: be

text: coupled

text: to

text: local

text: plasma

text: modes

text: to

text: propagate

text: out

text: in

text: a

text: pulsar

text: magnetosphere,

text: which

text: is

text: ﬁlled

text: with

text: relativistic

text: plasmas

text: that

text: can

text: be

text: simply

text: assumed

text: to

text: be

text: cold

text: (with

text: a

text: single

text: Lorentz

text: factor

text: γ)

text: and

text: with

text: a

text: den-

text: sity

text: of

text: N

text: ηN

text: Here,

text: η

text: is

text: the

text: multiplicity

text: factor

text: and

text: p

text: GJ

text: N

text: represents

text: the

text: Goldreich-Julian

text: density

text: (Goldreich

text: GJ

text: Julian

text: Two

text: eigen

text: transverse

text: modes,

text: the

text: X-mode

text: and

text: O-mode,

text: of

text: the

text: plasma

text: have

text: been

text: carefully

text: investigated

text: by

text: Wang

text: et

text: al.

text: and

text: Beskin

text: Philippov

text: The

text: X-

text: mode

text: waves

text: have

text: a

text: refraction

text: index

text: of

text: n

text: and

text: propagate

text: X

text: ν

text: without

text: rotation

text: ν

text: with

text: rotation

text: rectilinearly;

text: while

text: the

text: O-mode

text: waves

text: have

text: a

text: refraction

text: in-

text: dex

text: of

text: n

text: and

text: suﬀer

text: refraction

text: (Barnard

text: Arons

text: Figure

text: Emission

text: heights

text: (in

text: the

text: unit

text: of

text: for

text: diﬀerent

text: fre-

text: O

text: The

text: polarization

text: states

text: of

text: the

text: X-mode

text: and

text: O-mode

text: change

text: quencies

text: in

text: a

text: pulsar

text: beam,

text: plotted

text: in

text: the

text: rotation

text: phase

text: ϕ

text: and

text: the

text: sight

text: line

text: angle

text: ζ.

text: Corotation

text: of

text: relativistic

text: particles

text: is

text: not

text: during

text: their

text: propagation.

text: In

text: the

text: inner

text: magnetosphere

text: near

text: considered

text: for

text: the

text: left

text: panels,

text: but

text: considered

text: for

text: the

text: right

text: panels.

text: the

text: emission

text: region,

text: the

text: polarization

text: states

text: of

text: both

text: modes

text: The

text: parameters

text: used

text: for

text: model

text: calculations

text: are

text: an

text: inclination

text: an-

text: evolve

text: adiabatically,

text: i.e.,

text: with

text: the

text: polarization

text: of

text: the

text: X-

text: gle

text: of

text: α

text: a

text: Lorentz

text: factor

text: of

text: γ

text: and

text: a

text: rotation

text: period

text: mode

text: wave

text: orthogonal

text: to

text: the

text: local

text: k

text: B

text: plane

text: and

text: the

text: of

text: a

text: pulsar

text: P

text: polarization

text: of

text: the

text: O-mode

text: wave

text: within

text: the

text: k

text: B

text: plane

text: (Cheng

text: Ruderman

text: As

text: the

text: waves

text: propagate

text: to

text: the

text: higher

text: magnetosphere,

text: the

text: adiabatic

text: condition

text: is

text: not

text: satis-

text: In

text: general,

text: the

text: beam

text: size

text: in

text: the

text: meridional

text: direction

text: does

text: ﬁed

text: and

text: wave-mode

text: coupling

text: happens,

text: i.e.,

text: one

text: mode

text: leaks

text: not

text: aﬀect

text: our

text: conclusions

text: in

text: this

text: paper.

text: to

text: the

text: other

text: and

text: vice

text: versa.

text: The

text: altitude

text: where

text: the

text: mode

text: The

text: emission

text: regions

text: at

text: three

text: diﬀerent

text: frequencies

text: of

text: coupling

text: happens

text: is

text: named

text: as

text: being

text: the

text: polarization

text: limiting

text: ν

text: are

text: calculated

text: as

text: shown

text: in

text: Fig.

text: The

text: height

text: ranges

text: are

text: radius

text: (Cheng

text: Ruderman

text: Wang

text: et

text: al.

text: Above

text: CR

text: quasi

text: symmetric

text: about

text: the

text: magnetic

text: axis

text: for

text: emission

text: at

text: a

text: the

text: polarization

text: limiting

text: radius,

text: the

text: polarization

text: states

text: of

text: given

text: frequency

text: generated

text: from

text: the

text: static

text: dipole

text: magneto-

text: both

text: modes

text: are

text: frozen

text: and

text: will

text: not

text: be

text: further

text: aﬀected

text: by

text: sphere.

text: Emission

text: near

text: the

text: beam

text: centre

text: comes

text: from

text: a

text: lower

text: the

text: plasma,

text: except

text: for

text: possible

text: cyclotron

text: absorption

text: in

text: the

text: altitude,

text: as

text: shown

text: in

text: the

text: left

text: panels

text: of

text: Fig.

text: because

text: the

text: outer

text: magnetosphere.

text: curvature

text: radii

text: of

text: the

text: inner

text: ﬁeld

text: lines

text: at

text: a

text: lower

text: altitude

text: In

text: addition

text: to

text: evolution

text: of

text: a

text: single

text: wave

text: mode,

text: the

text: dis-

text: are

text: comparable

text: to

text: those

text: of

text: the

text: outer

text: ﬁeld

text: lines

text: at

text: a

text: higher

text: tributions

text: of

text: polarized

text: waves

text: within

text: each

text: emission

text: cone

text: altitude.

text: Further,

text: the

text: entire

text: emission

text: regions

text: get

text: deeper

text: in

text: also

text: change

text: during

text: propagation.

text: As

text: shown

text: by

text: Wang

text: et

text: al.

text: the

text: magnetosphere

text: for

text: a

text: higher

text: frequency,

text: which

text: is

text: caused

text: the

text: O

text: mode

text: refraction

text: will

text: lead

text: detectable

text: X

text: and

text: O

text: by

text: the

text: smaller

text: curvature

text: radii

text: of

text: ﬁeld

text: lines

text: in

text: the

text: deeper

text: modes

text: at

text: a

text: given

text: position

text: to

text: be

text: incoherent,

text: as

text: they

text: origi-

text: magnetosphere.

text: It

text: should

text: be

text: noted

text: that

text: relativistic

text: particles

text: nate

text: from

text: diﬀerent

text: parts

text: of

text: a

text: pulsar

text: beam.

text: Adiabatic

text: walk-

text: within

text: a

text: pulsar

text: magnetosphere

text: not

text: only

text: stream

text: along

text: the

text: ing

text: will

text: result

text: in

text: the

text: ordering

text: of

text: polarized

text: waves

text: within

text: the

text: curved

text: magnetic

text: ﬁeld

text: lines,

text: but

text: also

text: corotate

text: with

text: magneto-

text: cone,

text: i.e.,

text: with

text: the

text: O-mode

text: wave

text: vectors

text: following

text: the

text: sphere.

text: When

text: the

text: corotation

text: is

text: considered,

text: the

text: trajectories

text: polarization

text: plane

text: of

text: initial

text: emission,

text: but

text: wave

text: vectors

text: for

text: and

text: velocities

text: of

text: relativistic

text: particles

text: will

text: be

text: bent

text: towards

text: the

text: X-mode

text: rotate

text: by

text: Therefore,

text: incoherent

text: superposi-

text: the

text: rotation

text: direction

text: compared

text: with

text: those

text: for

text: the

text: static

text: tion

text: of

text: the

text: orthogonal

text: X-mode

text: and

text: O-mode

text: emission

text: causes

text: dipole

text: magnetosphere.

text: Hence,

text: the

text: emission

text: regions

text: become

text: depolarization

text: for

text: total

text: radiation

text: power.

text: If

text: the

text: corotation

text: of

text: c

text: RAS,

text: MNRAS


text: ∗ (Dated: May 27, 2020) Abstract The neutron-drip-line nucleus 23N is investigated in a three-body model consisting of a 21N core and two valence neutrons. Using the Faddeev formalism with the realistic neutron-neutron potential and the neutron-core potential, we calculate the ground-state properties of 23N including its two- neutron separation energy, and obtain good agreement with the experiments. We also ﬁnd an excited 23N state with a shallow two-neutron separation energy at about 0.18 MeV. By evaluating the root-mean-square matter radii, the average distances between the two valence neutrons, and the average distances from the core to the center-of-mass of the valence-neutron pair, we show that the excited state of 23N has a distinct halo structure. Through calculating the correlation density distributions of the 23N three-body system in conﬁguration space, we ﬁnd that the excited state of 23N has a triangular shape, which is similar to but much more extended than the ground state. This scaling symmetry between the ground and excited states indicate the existence of an Eﬁmov state in 23N. PACS numbers: 21.10.Gv, 21.60.-n, 27.30.+t 1

text: (Dated:

text: May

label: date

text: Abstract

text: The

label: abstract

text: neutron-drip-line

text: nucleus

text: is

text: investigated

text: in

text: a

text: three-body

text: model

text: consisting

text: of

text: a

text: core

text: and

text: two

text: valence

text: neutrons.

text: Using

text: the

text: Faddeev

text: formalism

text: with

text: the

text: realistic

text: neutron-neutron

text: potential

text: and

text: the

text: neutron-core

text: potential,

text: we

text: calculate

text: the

text: ground-state

text: properties

text: of

text: including

text: its

text: two-

text: neutron

text: separation

text: energy,

text: and

text: obtain

text: good

text: agreement

text: with

text: the

text: experiments.

text: We

text: also

text: ﬁnd

text: an

text: excited

text: state

text: with

text: a

text: shallow

text: two-neutron

text: separation

text: energy

text: at

text: about

text: MeV.

text: By

text: evaluating

text: the

text: root-mean-square

text: matter

text: radii,

text: the

text: average

text: distances

text: between

text: the

text: two

text: valence

text: neutrons,

text: and

text: the

text: average

text: distances

text: from

text: the

text: core

text: to

text: the

text: center-of-mass

text: of

text: the

text: valence-neutron

text: pair,

text: we

text: show

text: that

text: the

text: excited

text: state

text: of

text: has

text: a

text: distinct

text: halo

text: structure.

text: Through

text: calculating

text: the

text: correlation

text: density

text: distributions

text: of

text: the

text: three-body

text: system

text: in

text: conﬁguration

text: space,

text: we

text: ﬁnd

text: that

text: the

text: excited

text: state

text: of

text: has

text: a

text: triangular

text: shape,

text: which

text: is

text: similar

text: to

text: but

text: much

text: more

text: extended

text: than

text: the

text: ground

text: state.

text: This

text: scaling

text: symmetry

text: between

text: the

text: ground

text: and

text: excited

text: states

text: indicate

text: the

text: existence

text: of

text: an

text: Eﬁmov

text: state

text: in

text: PACS

text: numbers:


text: ± well with the experimental result, i.e., S = 3.07 0.31 MeV [20]. The r.m.s. matter radius 2n of the ground-state 23N is about 3.0 fm, which corresponds to the size of a stable nucleus ≈ with mass number A 27. This radius is much smaller than 22C, whose r.m.s. radius is ± about 5.4 0.9 fm [44]. Therefore, although loosely-bound compared with stable nuclei, the halo structure of the 23N ground state may not be as clear as its isotone halo nucleus 22C. TABLE III. The two-neutron separation energies S and r.m.s. matter radii r of the ground 2n m and excited states of 23N, calculated with three sets of parametrization and from the experiments. ∗ The superscript denotes the excited state. ∗ ∗ S r S r 2n m 2n m (MeV) (fm) (MeV) (fm) Set A 4.13 2.969 0.315 4.272 Set B 3.64 2.985 0.185 4.358 Set C 3.13 3.004 0.069 4.476 ± Exp [20] 3.07 0.31 - - - − The 1 excited state of 23N, which has not been discovered in experiments yet, is predicted 2 ∗ in this work to have an extremely shallow two-neutron separation energy S . With diﬀerent 2n ∗ parameterization, we obtain S to be in the region of 0.069–0.315 MeV, with a mean value 2n ∗ of 0.185 MeV. The r.m.s. radii of this excited state r is about 4.3 fm, which corresponds m ≈ to the size of a stable nucleus with A 80. Therefore, the excited state of 23N can be unambiguously described as a giant halo state. To illustrate the structure of 23N, we calculate the average distances between the two valence neutrons r and the average distances from the core to the center-of-mass of the nn valence-neutron pair r in the 21N+n+n system. The results in Table IV indicate that C(nn) both ground and excited states of 23N have triangular shapes. By calculating the ratios of r and r respectively in the ground and excited states, we ﬁnd that nn C(nn) ∗ r ≈ r ≈ nn ∗ nn 1.9 . (7) r r C(nn) C(nn) The same ratio obtained in both states suggests that these two states have similar geo- 8 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: well

text: with

text: the

text: experimental

text: result,

text: i.e.,

text: S

text: MeV

text: The

text: r.m.s.

text: matter

text: radius

text: of

text: the

text: ground-state

text: is

text: about

text: fm,

text: which

text: corresponds

text: to

text: the

text: size

text: of

text: a

text: stable

text: nucleus

text: with

text: mass

text: number

text: A

text: This

text: radius

text: is

text: much

text: smaller

text: than

text: whose

text: r.m.s.

text: radius

text: is

text: about

text: fm

text: Therefore,

text: although

text: loosely-bound

text: compared

text: with

text: stable

text: nuclei,

text: the

text: halo

text: structure

text: of

text: the

text: ground

text: state

text: may

text: not

text: be

text: as

text: clear

text: as

text: its

text: isotone

text: halo

text: nucleus

text: TABLE

text: III.

text: The

label: caption

text: two-neutron

text: separation

text: energies

text: S

text: and

text: r.m.s.

text: matter

text: radii

text: r

text: of

text: the

text: ground

text: m

text: and

text: excited

text: states

text: of

text: calculated

text: with

text: three

text: sets

text: of

text: parametrization

text: and

text: from

text: the

text: experiments.

text: The

text: superscript

text: denotes

text: the

text: excited

text: state.

text: S

text: r

text: S

text: r

text: m

text: m

text: (MeV)

text: (fm)

text: (MeV)

text: (fm)

text: Set

text: A

text: Set

text: B

text: Set

text: C

text: Exp

text: The

text: excited

text: state

text: of

text: which

text: has

text: not

text: been

text: discovered

text: in

text: experiments

text: yet,

text: is

text: predicted

text: in

text: this

text: work

text: to

text: have

text: an

text: extremely

text: shallow

text: two-neutron

text: separation

text: energy

text: S

text: With

text: diﬀerent

text: parameterization,

text: we

text: obtain

text: S

text: to

text: be

text: in

text: the

text: region

text: of

text: MeV,

text: with

text: a

text: mean

text: value

text: of

text: MeV.

text: The

text: r.m.s.

text: radii

text: of

text: this

text: excited

text: state

text: r

text: is

text: about

text: fm,

text: which

text: corresponds

text: m

text: to

text: the

text: size

text: of

text: a

text: stable

text: nucleus

text: with

text: A

text: Therefore,

text: the

text: excited

text: state

text: of

text: can

text: be

text: unambiguously

text: described

text: as

text: a

text: giant

text: halo

text: state.

text: To

text: illustrate

text: the

text: structure

text: of

text: we

text: calculate

text: the

text: average

text: distances

text: between

text: the

text: two

text: valence

text: neutrons

text: r

text: and

text: the

text: average

text: distances

text: from

text: the

text: core

text: to

text: the

text: center-of-mass

text: of

text: the

text: nn

text: valence-neutron

text: pair

text: r

text: in

text: the

text: system.

text: The

text: results

text: in

text: Table

text: IV

text: indicate

text: that

text: C(nn)

text: both

text: ground

text: and

text: excited

text: states

text: of

text: have

text: triangular

text: shapes.

text: By

text: calculating

text: the

text: ratios

text: of

text: r

text: and

text: r

text: respectively

text: in

text: the

text: ground

text: and

text: excited

text: states,

text: we

text: ﬁnd

text: that

text: nn

text: C(nn)

text: r

text: r

text: nn

text: nn

text: r

text: r

text: C(nn)

text: C(nn)

text: The

text: same

text: ratio

text: obtained

text: in

text: both

text: states

text: suggests

text: that

text: these

text: two

text: states

text: have

text: similar

text: geo-

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: A+B! A 1 1 B! User A Relay User B Multiple access phase! Self-interference cancellation! A+B! A B! 1 1 A+B! A+B! User A Relay User B Broadcast phase! 6 CA! AB! BA+AB! BA! BC! BA! Relay CB! CA+AC! Relay User A User B User A User B Self-interference cancellation! BA+AB! AB! AC! CA+AC! CB+BC! BC! AC! CA! CB! User C User C Multiple access phase! Broadcast phase! Fig. 5. A diagram illustrating signal space alignment for network coding in a MIMO Y channel. channel paths among all transmit and receive pairs includ- layer (analog) network coding [4], [5] was introduced. As ing relays in the channel. Under this fully-connected multi- illustrated in Fig. 1-(b), physical-layer network coding allows hop channel model, unlike the layered multi-hop interference the exchange of packets A and B within two communication channel, it was demonstrated that the use of relays cannot phases by canceling the known self- interference signal a prior. increase sum-DoF of the channel [18]. A major beneﬁcial Speciﬁcally, in the ﬁrst phase, the two users send packet A aspect of exploiting relays in the non-layered multi-hop in- and B to the relay simultaneously. In the second phase, the terference channel, however, was found by recent work [19]. relay broadcast the linear combination of A and B to the two The beneﬁt is the considerable reduction of CSI feedback users. Since each user has knowledge about self-interference amount required for interference alignment compared to the signal a prior, it can cancel the self-interference effect from method in [18]. By proposing a relay-aided blind interference the received signal. This leads the performance improvement alignment technique, it was shown that the optimal sum- double compared to the case of the resource orthogonalization DoF of the channel is achievable without CSIT, provided that method. Drawing upon this idea, [4] showed that the rates the relay with global CSI forwards the received signals to of information exchange are double of these achieved by receivers in such a way that aligns interference. Similar to traditional interference management strategies. blind interference alignment [1, Section 4.9], this interference The idea of linear index coding was initially introduced in management technique exhibits a possibility to mitigate in- [7] for a noiseless broadcasting channel, in which a transmitter terference successfully with limited CSIT in the non-layered sends packets for multiple users, each with other users’ multi-hop interference channel. packet in prior as side-information. The transmitter desires IV. MULTI-WAY COMMUNICATION NETWORKS WITH to deliver multiple packets to their respective users over a RELAYS shared noiseless link. The principal goal of index coding is the design of transmit codes that minimize the number of Multi-way communication networks with relays models a required channel uses, while making all users decode the general communication scenario where multiple users ex- desired packets with the received signals from the transmitter change information with the help of multiple relay nodes. This and their own side-information. As depicted in Fig. 1-(c), wireless network architecture has received attention because of its broad applications to cellular networks, sensor networks, for instance, a transmitter desires to deliver packet A, B, and D2D communication. The most distinctive feature of and C to user A, B, and C, respectively, and each user has knowledge of the others’ desired packets. In this case, multi-way relay networks are that source and destination the optimal transmission strategy is to broadcast one XORed nodes are inseparable. This feature introduces a new potential of exploiting interference signals as an efﬁcient interference packet A + B + C to all three users by spanning a channel use, as each user decodes the desired packet using the received management approach. Network coding and index coding are representative tech- A + B + C and stored side-information {B, C}, {A, C}, and niques for exploiting interference. For multi-hop wired net- {A, B}. works, [6] originally introduced the idea of network coding The concept of interference exploitation was extended by to boost network throughput. The crucial principle underlying combining the ideas of interference alignment and network network coding is that intermediate nodes in networks forward coding; this is referred to as signal space alignment for functions (a linear combination) of their received packets, network coding [22]. Consider a three user multi-way com- instead of independent packets. The same spirit of exploiting munication scenario where a user with two antennas wants interference was extended into wireless networks. In the two- to exchange two independent messages each via a relay with way relay channel in which two nodes exchange information three antennas; this is referred to as a MIMO Y channel [22]. with each other via a shared relay node, the concept of physical As illustrated in Fig. 5, when each user sends two independent ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: A+B!

font: default

text: A

text: B!

text: User

text: A

text: Relay

text: User

text: B

text: Multiple

text: access

text: phase!

text: Self-interference

text: cancellation!

text: A+B!

text: A

text: B!

text: A+B!

text: A+B!

text: User

text: A

text: Relay

text: User

text: B

text: Broadcast

text: phase!

text: CA!

text: AB!

text: BA+AB!

text: BA!

text: BC!

text: BA!

text: Relay

text: CB!

text: CA+AC!

text: Relay

text: User

text: A

text: User

text: B

text: User

text: A

text: User

text: B

text: Self-interference

text: cancellation!

text: BA+AB!

text: AB!

text: AC!

text: CA+AC!

text: CB+BC!

text: BC!

text: AC!

text: CA!

text: CB!

text: User

text: C

text: User

text: C

text: Multiple

text: access

text: phase!

text: Broadcast

text: phase!

text: Fig.

text: A

text: diagram

text: illustrating

text: signal

text: space

text: alignment

text: for

text: network

text: coding

text: in

text: a

text: MIMO

text: Y

text: channel.

text: channel

text: paths

text: among

text: all

text: transmit

text: and

text: receive

text: pairs

text: includ-

text: layer

text: (analog)

text: network

text: coding

text: was

text: introduced.

text: As

text: ing

text: relays

text: in

text: the

text: channel.

text: Under

text: this

text: fully-connected

text: multi-

text: illustrated

text: in

text: Fig.

text: physical-layer

text: network

text: coding

text: allows

text: hop

text: channel

text: model,

text: unlike

text: the

text: layered

text: multi-hop

text: interference

text: the

text: exchange

text: of

text: packets

text: A

text: and

text: B

text: within

text: two

text: communication

text: channel,

text: it

text: was

text: demonstrated

text: that

text: the

text: use

text: of

text: relays

text: cannot

text: phases

text: by

text: canceling

text: the

text: known

text: self-

text: interference

text: signal

text: a

text: prior.

text: increase

text: sum-DoF

text: of

text: the

text: channel

text: A

text: major

text: beneﬁcial

text: Speciﬁcally,

text: in

text: the

text: ﬁrst

text: phase,

text: the

text: two

text: users

text: send

text: packet

text: A

text: aspect

text: of

text: exploiting

text: relays

text: in

text: the

text: non-layered

text: multi-hop

text: in-

text: and

text: B

text: to

text: the

text: relay

text: simultaneously.

text: In

text: the

text: second

text: phase,

text: the

text: terference

text: channel,

text: however,

text: was

text: found

text: by

text: recent

text: work

text: relay

text: broadcast

text: the

text: linear

text: combination

text: of

text: A

text: and

text: B

text: to

text: the

text: two

text: The

text: beneﬁt

text: is

text: the

text: considerable

text: reduction

text: of

text: CSI

text: feedback

text: users.

text: Since

text: each

text: user

text: has

text: knowledge

text: about

text: self-interference

text: amount

text: required

text: for

text: interference

text: alignment

text: compared

text: to

text: the

text: signal

text: a

text: prior,

text: it

text: can

text: cancel

text: the

text: self-interference

text: effect

text: from

text: method

text: in

text: By

text: proposing

text: a

text: relay-aided

text: blind

text: interference

text: the

text: received

text: signal.

text: This

text: leads

text: the

text: performance

text: improvement

text: alignment

text: technique,

text: it

text: was

text: shown

text: that

text: the

text: optimal

text: sum-

text: double

text: compared

text: to

text: the

text: case

text: of

text: the

text: resource

text: orthogonalization

text: DoF

text: of

text: the

text: channel

text: is

text: achievable

text: without

text: CSIT,

text: provided

text: that

text: method.

text: Drawing

text: upon

text: this

text: idea,

text: showed

text: that

text: the

text: rates

text: the

text: relay

text: with

text: global

text: CSI

text: forwards

text: the

text: received

text: signals

text: to

text: of

text: information

text: exchange

text: are

text: double

text: of

text: these

text: achieved

text: by

text: receivers

text: in

text: such

text: a

text: way

text: that

text: aligns

text: interference.

text: Similar

text: to

text: traditional

text: interference

text: management

text: strategies.

text: blind

text: interference

text: alignment

text: Section

text: this

text: interference

text: The

text: idea

text: of

text: linear

text: index

text: coding

text: was

text: initially

text: introduced

text: in

text: management

text: technique

text: exhibits

text: a

text: possibility

text: to

text: mitigate

text: in-

text: for

text: a

text: noiseless

text: broadcasting

text: channel,

text: in

text: which

text: a

text: transmitter

text: terference

text: successfully

text: with

text: limited

text: CSIT

text: in

text: the

text: non-layered

text: sends

text: packets

text: for

text: multiple

text: users,

text: each

text: with

text: other

text: users’

text: multi-hop

text: interference

text: channel.

text: packet

text: in

text: prior

text: as

text: side-information.

text: The

text: transmitter

text: desires

text: IV.

text: MULTI-WAY

text: COMMUNICATION

text: NETWORKS

text: WITH

text: to

text: deliver

text: multiple

text: packets

text: to

text: their

text: respective

text: users

text: over

text: a

text: RELAYS

text: shared

text: noiseless

text: link.

text: The

text: principal

text: goal

text: of

text: index

text: coding

text: is

text: the

text: design

text: of

text: transmit

text: codes

text: that

text: minimize

text: the

text: number

text: of

text: Multi-way

text: communication

text: networks

text: with

text: relays

text: models

text: a

text: required

text: channel

text: uses,

text: while

text: making

text: all

text: users

text: decode

text: the

text: general

text: communication

text: scenario

text: where

text: multiple

text: users

text: ex-

text: desired

text: packets

text: with

text: the

text: received

text: signals

text: from

text: the

text: transmitter

text: change

text: information

text: with

text: the

text: help

text: of

text: multiple

text: relay

text: nodes.

text: This

text: and

text: their

text: own

text: side-information.

text: As

text: depicted

text: in

text: Fig.

text: wireless

text: network

text: architecture

text: has

text: received

text: attention

text: because

text: of

text: its

text: broad

text: applications

text: to

text: cellular

text: networks,

text: sensor

text: networks,

text: for

text: instance,

text: a

text: transmitter

text: desires

text: to

text: deliver

text: packet

text: A,

text: B,

text: and

text: communication.

text: The

text: most

text: distinctive

text: feature

text: of

text: and

text: C

text: to

text: user

text: A,

text: B,

text: and

text: C,

text: respectively,

text: and

text: each

text: user

text: has

text: knowledge

text: of

text: the

text: others’

text: desired

text: packets.

text: In

text: this

text: case,

text: multi-way

text: relay

text: networks

text: are

text: that

text: source

text: and

text: destination

text: the

text: optimal

text: transmission

text: strategy

text: is

text: to

text: broadcast

text: one

text: XORed

text: nodes

text: are

text: inseparable.

text: This

text: feature

text: introduces

text: a

text: new

text: potential

text: of

text: exploiting

text: interference

text: signals

text: as

text: an

text: efﬁcient

text: interference

text: packet

text: A

text: B

text: C

text: to

text: all

text: three

text: users

text: by

text: spanning

text: a

text: channel

text: use,

text: as

text: each

text: user

text: decodes

text: the

text: desired

text: packet

text: using

text: the

text: received

text: management

text: approach.

text: Network

text: coding

text: and

text: index

text: coding

text: are

text: representative

text: tech-

text: A

text: B

text: C

text: and

text: stored

text: side-information

text: {B,

text: C},

text: {A,

text: C},

text: and

text: niques

text: for

text: exploiting

text: interference.

text: For

text: multi-hop

text: wired

text: net-

text: {A,

text: B}.

text: works,

text: originally

text: introduced

text: the

text: idea

text: of

text: network

text: coding

text: The

text: concept

text: of

text: interference

text: exploitation

text: was

text: extended

text: by

text: to

text: boost

text: network

text: throughput.

text: The

text: crucial

text: principle

text: underlying

text: combining

text: the

text: ideas

text: of

text: interference

text: alignment

text: and

text: network

text: network

text: coding

text: is

text: that

text: intermediate

text: nodes

text: in

text: networks

text: forward

text: coding;

text: this

text: is

text: referred

text: to

text: as

text: signal

text: space

text: alignment

text: for

text: functions

text: (a

text: linear

text: combination)

text: of

text: their

text: received

text: packets,

text: network

text: coding

text: Consider

text: a

text: three

text: user

text: multi-way

text: com-

text: instead

text: of

text: independent

text: packets.

text: The

text: same

text: spirit

text: of

text: exploiting

text: munication

text: scenario

text: where

text: a

text: user

text: with

text: two

text: antennas

text: wants

text: interference

text: was

text: extended

text: into

text: wireless

text: networks.

text: In

text: the

text: two-

text: to

text: exchange

text: two

text: independent

text: messages

text: each

text: via

text: a

text: relay

text: with

text: way

text: relay

text: channel

text: in

text: which

text: two

text: nodes

text: exchange

text: information

text: three

text: antennas;

text: this

text: is

text: referred

text: to

text: as

text: a

text: MIMO

text: Y

text: channel

text: with

text: each

text: other

text: via

text: a

text: shared

text: relay

text: node,

text: the

text: concept

text: of

text: physical

text: As

text: illustrated

text: in

text: Fig.

text: when

text: each

text: user

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text: 7 messages to the relay in the multiple access phase, the relay Time slot 1 Time slot 2 observes a total of six incoming signal streams. Since a relay is able to use a three dimensional signal space, the users BS or (WiFi AP)! Decode A and B ! Decode C and D ! cnInoeottwphoeerrakbtriovcaoedldycinasgsetncpdahntahseeea,scitghhnebarleselasaolyigtanhpeapdtlaiaepssataihresoifmgntuwalltoi-dsuiismgeneranblsseioafomnr. At=!1User 1h1[1] h31[1h]32h[11][1]AU+sehr 23[1]BLh3[311t=][=1(A]1A, B+)h32[1]B L1[2t]=(C2, D) User 1 User 3t=C!2! forming technique that cancels unmanageable interference to h2[1] tohwenustrearns.smAittthsiegunsael,r tehned,soea-ccahlluesderseislf-aibnlteertfoerﬁernsctes,uabntrdactthietns B! User 2 h42[1] User 4 hL441[[11=]]A(A+, Bh)42[1]B L2[2](C, D) User 2 User 4 D! extract the desired signal from its partner, in a process that Time slot 3 is analogous to network coding. This approach achieves the Precoding optimal sum DoF in the MIMO Y channel and in a broad class Cotrdaends mmiussltiiocna st vAA+vBB +vCC +vDD vA ? h⇤2[3] vC ? h⇤4[3] of multi-way communication networks with relays. What BS has! vB ? h⇤1[3] vD ? h⇤3[3] A! B! C!! D! Recently, the interference exploitation idea also was applied What u sAe!r 1 has! h⇤1[3] h⇤3[3] What user 3 has! to fully-connected multi-way relay channels. Unlike the prior L1[2](C, D) C! physical-layer network coding approaches that only exploit the User 1 User 3 L3[1](A, B) Side-information! self-interference signal as the main source of side-information, What user 2 has! h⇤2[3] h⇤4[3] What user 4 has! a new physical-layer network coding strategy called space- B! D! time physical layer network coding was proposed in [23]. The L2[2](C, D) User 2 User 4 L4[1](A, B) essence is the exploitation of overheard interference signals Fig. 6. A diagram illustrating space-time physical-layer network coding in a 12/ 42 as side-information in addition to self-interference signals two-pair two-way interference channel with a two-antenna relay. for fully-connected multi-way relay networks. As illustrated in Fig. 6, consider a fully-connected multi-way information exchange scenario where two pairs (user 1-3 and user 2-4) time relay transmission. In the ﬁrst phase, using the fact that exchange packets with their partners via a relay with two the relay is able to decode M independent packets per channel antennas. This scenario can model the case where two D2D use thanks to multiple antennas, M users among K users are user pairs cooperatively exchange video ﬁles by sharing a scheduled to send packets into the network at each channel WiFi access point. In time slot 1, user 1 and user 3 send use until the relay acquires full knowledge of packets in two packet A and C. Then, since user 2 and user 4 have the network. During this phase, the remaining K − M users a single antenna, the both cannot decode the desired packet overhear the linear combinations of transmitted packets per reliably due to mutual interference. Instead, they just store channel use, thereby learning the interference patterns. In the the overheard linear combination of two packets A and C to second phase, the relay starts to control information ﬂows by exploit it later as side information. Whereas, the access point broadcasting the superposition of packets so that all the users with two antennas is able to decode the two packets reliably harness their side-information in the form of: 1) what they by applying a zero-forcing decoder. In time slot 2, user 2 and sent and 2) what they overheard in decoding. See the details 4 send packets B and D. Then, similarly, user 1 and user for this generalization in [23]. 3 store the linear combination of B and D, while the access V. RESEARCH CHALLENGES point decodes the two packets thanks to the multiple antennas. Recall that during the previous two time slots, the access point Despite the theoretical performance gains of the advanced has decoded all the packets, A, B, C, and D. Furthermore, interference management techniques described in the previous each user has obtained two different types of side-information: sections, there are many research challenges that remain to 1) what it sent and 2) what it overheard. In time slot 3, transform theory to practice. A. Out-of-Cell Interference the access point sends out a linear combination of the four packets using space-time precoding so that every users exploits Most of the interference management techniques reviewed both different types of side-information simultaneously. For in this article were developed in idealized interference net- example, the precoding vector carrying packet B is selected work settings. These settings consider a particular number of so that the packet B does not propagate to user 1 who does not cooperative transmit-and-receiver pairs, ignoring the potential have knowledge of packet B as side information. By carefully interference from outside of the cooperative set and often choosing precoding vectors with the same principle, every deemphasizing noise effects. There exists apparent limita- users is able to decode the desired packets successfully from tions in translating the performance gains obtained from the the received signal in time slot 3 using the two types of side advanced interference management techniques into practical information obtained in the prior time slots. wireless systems due to their simpliﬁed natures. Space-time physical layer network coding can be explained The practical challenge is to reevaluate the gains obtained through the lens of linear index coding in a general fully- from the interference management algorithms using models connected multi-way relay network conﬁguration comprised that accurately capture the impact of the irregular spatial of a single antenna K users and a relay equipped with M structure of wireless node locations and channel propagation antennas. The information exchange among users takes place characteristics depending on operating frequency bands (e.g., using two phases: i) side-information learning and ii) space- low frequency or mmWave bands). One possible direction is ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine##

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