A THEORETICAL STUDY OF THE EFFECT OF STRONG ELECTRIC ANDMAGNETIC FIELDS ON COMPACT STELLAR OBJECTS - AN APPLICATION TO

TINY LABORATORY SAMPLES

Arpita Ghosh

Department of Physics, Visva-Bharati

For

Pre-Ph.D. Seminar, Department of Physics

Visva-Bharati

List of Publication

1. Eur. Phys. J. A (2011) 47, 56.

2. J. Astrophysics. Astr. (2011) 32, 377.

3. Monthly Notice of Royal Astronomical Society (2012) 425, 1239.

4. Phys. Letts A (in press).

CONTENTS

• Introduction.

• A Theoretical Study of the Magnetically Deformed Inner Crust Matter of Magne-tars.

• Work Function Associated with Ultra-relativistic Electron Emission from StronglyMagnetized Neutron Star Surface.

• The Effect of Strong Quantizing Magnetic Field on the Cold Emission of Electronsfrom Magnetars.

• Suppression of Relativistic Field Emission of Electrons in 1+1 - Dimension.

• Conclusion.

Brief Overview of Magnetic Field Strength in Natural System

Effect of strong magnetic field on crustal matter of magnetar s

• Capable of distorting and magnifying images of stellar objects (magnetic lens-ing).

• Change of self-energy of electrons.

• Cause photons to rapidly split and merge.

• Distort atoms into long thin cylindrical shape .

• Affects equation of state for dense stellar matter.

• Affects elementary process (electromagnetic and weak).

• Affects electron emission process from polar region of magn etars.

• Work function of crustal matter associated with electron em ission be-comes anisotropic.

Schematic Diagram for the Internal Structure of a Typical Ne utronStar/Magnetar

Typical Structure of a Rotating Neutron Star/Magnetar with Magnetic Lines ofForces

Inner crust matter of Magnetars

• Surface magnetic field of magnetars: ≥ 1014 G

• Inner crust region: Width ∼ 0.5 − 0.7 km, Density ∼ 1011 − 1014 gm/cc

• Structure of inner crust :

Array of cylindrically deformed (cigar shaped) metallic ions (mainly metallic iron).=⇒

Cylindrical type distribution of electron gas around each nucleus −→

axis of each cylinder along the direction of magnetic field −→

we assume azimuthal symmetry for electron distribution.

Quantum Nature of Electron Gas

• Cylindrical distribution of electrons around the nucleus −→ distorted Wigner-Seitz (WS) cells.

• Landau levels of these electrons are populated.

• Proton are massive enough =⇒ no quantum mechanical effect of magnetic field.

Formalism in brief

• Relativistic version of Thomas-Fermi method in presence of strong magnetic field.• Magnetic field B is constant and along z-axis (Aµ = 0,0, xB,0).

• We start with Poisson equation:

∇2φ = 4πene(r − rn) −4πZe

Vnθ(rn − r) (1)

• In cylindrical coordinate the contribution of protons −→ θ(rn − r)θ(zn − z)

• Density of degenerate electron gas :

ne =eB

2π2

νmax∑

ν=0

(2 − δν0)pF , ν → Landau QN, νmax → upper limit (2)

• To solve the Poisson equation in cylindrical coordinate we use Thomas-Fermicondition :

µe = (p2F +m2e + 2νeB)1/2 − eφ = const. (3)

• In strong magnetic field we get a magnetic constriction in the direction perpen-dicular to the field, and the electron distribution around the nucleus becomescigar shaped.

• Here for the sake of simplicity, we consider cylindrical type shape.

• Radial contraction factor:

γ ∝ 4

(

a0ρ

)2

∝B

B0, (4)

where B is external magnetic field, B0 ≈ 108G, a0 is the Bohr radius and ρ isthe radial parameter.

• We start with cylindrical form of Poisson equation:

∂2ψ

∂r2+

1

r

∂ψ

∂r+∂2ψ

∂z2= λ2ψ (5)

Where λ = 2e3B/π.

• With the separation of variables ψ(r, z) = R(r)L(z):

d2R

dr2+

1

r

dR

dr+ ξ2R = 0, (6)

d2L

dz2− (ξ2 + λ2)L = 0, (7)

• The combined solution:

ψ(r, z) = CJ0(ξr) exp[

±(ξ2 + λ2)1/2z]

(8)

where ξ(B) is a real constant and C(B) is normalization constant.

• Unlike spherical distribution of electrons around the nucleus; in the cylindricallydistorted case surface electric field can not be zero.

• Electric field on the curved surface E1

E1 = −2C exp(−Λ(s2 − r2max)1/2)(J0(ξrmax)Λez + J1(ξrmax)ξer) (9)

• similarly at the plane faces E2

E2 = −2C exp(−Λzmax)(J0(ξ(s2 − z2max)

1/2)Λez

+ J1(ξ(s2 − z2max)

1/2)ξer) (10)

where Λ = (λ2 + ξ2)1/2

• Due to electromagnetic induction there will be charge polarization on the sur-faces of cylindrically deformed WS cells.

• As a result ≈ 100 charged WS cells are observed to form a bundle of chargeneutral combination, instead of a single cylindrical cell.

• The electron Fermi momentum, pF (r, z):

pF =[

ψ2(r, z) −m2ν

]1/2, with mν = (m2 + 2νeB)1/2 (11)

• Electron density:

ne(r, z) =eB

2π2

νmax∑

ν=0

(2 − δν0)[

ψ2(r, z) −m2ν

]1/2(12)

• The upper limit of Landau quantum ν:

νmax =ψ2(r, z) −m2

e

2eB= νmax(r, z) (13)

Variation of ne with r and z

Variation of νmax with z and r

Variation of zmax and rmax with B :

• From νmax(rmax, zmax) = 0

Different kinds of energies associated with the electron gas within WS cells –

a© Cell-averaged kinetic energy density :

Ek =1

V

∫

ǫkd3r =

2

V

eB

2π2

∫

d3rνmax∑

ν=0

(2 − δν0)

×∫ pF(r,z)

0dpz

[

(p2z +m2ν)

1/2 −me

]

(14)

b© Cell-averaged electron nucleus interaction energy :

Een = −2

VZe2

∫

d3rne(r, z)

(r2 + z2)1/2(15)

Kinetic energy

e− n Interaction energy

Variation of Ee−n and Ek with B

c© Electron-electron direct interaction :

Ediree =1

Ve2∫

d3rne(r, z)∫

d3r′ne(r′, z′)

×1

[

(r − r′)2 + (z − z′)2]1/2

(16)

d© Electron-electron exchange interaction :

E(ex)ee = −

e2

2

∑

j

∫

d3rd3r′1

[

(r − r′)2 + (z − z′)2]1/2

×ψi(r, z)ψj(r′, z′)ψj(r, z)ψi(r

′, z′) (17)

e© Kinetic pressure within the cell:

P(r, z) =eB

4π2

νmax(r,z)∑

ν=0

pF (p2F +m2ν)

1/2 −m2ν ln

pF + (p2F +m2ν)

1/2

mν

(18)f© Cell averaged P :

P =1

V

∫

P(r, z)d3r (19)

Variation of kinetic pressure within WS-cell

Variation direct energy, exchange energy (e− e field type and Coulomb type) andcell averaged pressure with magnetic field

Variation e− e and e− n Coulomb interaction with magnetic field

Ratio of electron kinetic pressures in cylindrically deformed and spehricallysymmetric cases.

Conclusions:

• Assumed cylindrical shape WS cells contract in presence of strong magneticfield.

• The Landau QN changes with both r and z coordinates and also with B.

• The inner crust matter is observed to be more stable if the electron distributionis cylindrical in nature.

Variation of Work Function for Neutron Star Crustal Matterwith Magnetic field

• To investigate the formation of magneto-sphere for a strongly magnetized neu-tron star or a magnetar, we need the variation of work function with strong mag-netic field.

• Since the electrons are emitted mainly by cold or field emission process from thepolar region by the action of strong electric field, created by rotating magneticfield, the work function plays a vital role in electron field emission from neutronstar polar region.

• Since the emission occurs in strong magnetic field environment, we need to knowthe variation of work function with magnetic field.

Schematic diagram for the electron emission near the polar region of a neutron star.Here the magnetic field is along z-axis.

• We have investigated electron emission along z-direction and also perpendiculardirection to the magnetic field.

• Here the cylindrical region indicates the deformed electron distribution in pres-ence of strong magnetic field.

• We have followed the mechanism of cold cathode emission.

• We assume that the frequency of incident electromagnetic wave is greater thanthe threshold frequency.

A Longitudinal Emission

• Here the electrons are emitted along +z-direction.

• The electrostatic potential at Q produced by the emitted electron:

φ(p)(r, z) = e∫ ∞

ξ=0J0(ξr) exp[∓(z − h)ξ]dξ,

for z > or < h (20)

Where AP = h, PD = r and CQ = z.

• For charge distribution within the distorted WS cell (cylindrical region), we startwith the cylindrical form of Poisson’s equation:

∂2ψ

∂r2+

1

r

∂ψ

∂r+∂2ψ

∂2z= λ2ψ (21)

• The solution:

ψ(r, z) =

∫ ∞

ξ=0J0(ξr)a(ξ) exp[(ξ2 + λ2)1/2z]dξ

with z < 0 (22)

a(ξ) is some unknown spectral function.

• The secondary field in coherence with φ(p)(r, z) and ψ(r, z):

φ(s)(r, z) = e∫ ∞

ξ=0J0(ξr)f(ξ) exp(−ξz)dξ

with z > 0 (23)

where f(ξ) is again some unknown spectral function.

• This fictitious field pulls out electrons.

• Using the continuity conditions for tangential and transverse components of elec-tric field and the displacement vector respectively:

E(t) = E(p)t + E(s)t and D⊥ = D(p)⊥ +D(s)⊥ (24)

• We get

f(ξ) =

1 −

(

1 + λ2

ξ2

)1/2

1 +

(

1 + λ2

ξ2

)1/2

exp(−ξh) (25)

• Then we have the secondary potential

φ(s)(r, z) = e∫ ∞

ξ=0J0(ξr)

1 −

(

1 + λ2

ξ2

)1/2

1 +

(

1 + λ2

ξ2

)1/2

exp[−ξ(h+ z)]dξ for z > 0

(26)

• Hence the secondary field along the axial direction at r = 0:

E(s)z = −

∂φ(s)

∂z(27)

= e∫ ∞

ξ=0

1 −

(

1 + λ2

ξ2

)1/2

1 +

(

1 + λ2

ξ2

)1/2

exp[−ξ(h+ z)]ξdξ (28)

• The force acting on an electron at z = h :

F(s)z = eE

(s)z = e2

∫ ∞

ξ=0

1 −

(

1 + λ2

ξ2

)1/2

1 +

(

1 + λ2

ξ2

)1/2

exp(−2hξ)ξdξ (29)

• The work done in pulling out an electron from just inside the metal surface toinfinity:

Wf = −∫ ∞

0F

(s)z dh (30)

= −e2

2

∫ ∞

ξ=0

1 −

(

1 + λ2

ξ2

)1/2

1 +

(

1 + λ2

ξ2

)1/2dξ (31)

• After evaluation we get :

Wf =λ

3e2 =

1

3

2B

πB(e)c

1/2

mee3 (32)

• Where B(e)c ≈ 4.43 × 1013G, the typical field strength to start populating the

Landau levels for electrons in the relativistic scenario.

• This equation gives the variation of work function with the strength of magneticfield (∼ B1/2) associated with the emission of electrons along the direction ofthe magnetic field.

The variation of longitudinal part of work function with magnetic field.

• From our investigation it is obvious that the work function increases with theincrease in magnetic field strength.

• Physically it means that with the increase in magnetic field strength, electronsbecome more strongly bound inside the crustal matter.

• As a consequence the electron emission current becomes extremely small in thecase of neutron stars with ultra-strong magnetic field.

• =⇒ charge density in the magneto-sphere will be low enough =⇒ intensity ofradio wave emission will be extremely low.

B Transverse Emission

• We now consider the emission of electrons in the direction transverse to the mag-netic field direction.•Here p is the position of an emitted electron.•The electronic potential at q:

φ(p)(r, z) = e∫ ∞

ξ=0J0[ξ(r − r0)] exp(∓zξ)dξ (33)

for z > 0 or z < 0

Where ap = r0, cq = r and pd = z.

• Whereas the form of ψ(r, z) remains unchanged.

• Using some of the properties of J0 we have

ψ(r, z) =+∞∑

k=−∞

Jk(ξr0)∫ ∞

ξ=0Jk[ξ(r+ r0)]a(ξ) exp[−(ξ2 + λ2)1/2 | z |]

dξ (34)

• Similarly we have

φ(p)(r, z) = e+∞∑

k=−∞

Jk(2ξr0)∫ ∞

ξ=0Jk[ξ(r+ r0)] exp(−ξ | z |)dξ (35)

and

φ(s)(r, z) = e+∞∑

k=−∞

Jk(ξr0)∫ ∞

ξ=0Jk[ξ(r+ r0)]f(ξ) exp(−ξ | z |)dξ (36)

• Then from the continuity conditions on the curved surface along r direction,putting z = 0, r = R and redefining the spectral function a(ξ) = a(ξ)/e:

∫ ∞

ξ=0ξdξ

+∞∑

k=−∞

J ′k[(R+ r0)ξ]{a(ξ)JR(ξr0)

− Jk(2ξr0) − Jk(ξr0)f(ξ)} = 0 (37)

• Similarly the continuity condition along z-direction on the curved surface at z = 0and r = R:

∫ ∞

ξ=0

+∞∑

k=−∞

Jk(ξr0)Jk[(R+ r0)ξ]a(ξ)(ξ2 + λ2)1/2dξ

=

∫ ∞

ξ=0

+∞∑

k=−∞

Jk(2ξr0)Jk[ξ(R+ r0)]ξdξ

+∫ ∞

ξ=0

+∞∑

k=−∞

Jk(ξr0)Jk[ξ(R+ r0)]f(ξ)ξdξ (38)

• Hence after some simple algebraic manipulation we have:

φ(s)(r, z) = e∫ ∞

ξ=0J0(ξr)

F(1 + λ2

ξ2)1/2 −G

1 −

(

1 + λ2

ξ2

)1/2

exp(−ξz)dξ (39)

Where

F =J1[(R− r0)ξ]

J1(Rξ)and G =

J0[(R− r0)ξ]

J0(Rξ)(40)

• we have used some well-known relations of Jk and J ′k.

• Hence the corresponding electro-static field at (R, z = 0):

E(s)(R,0) = −∂φ(s)(r, z)

∂r

= e∫ ∞

ξ=0

J1[(R− r0)ξ]

(

1 + λ2

ξ2

)1/2− J0[(R−r0)ξ]J1(Rξ)

J0(Rξ)

1 −

(

1 + λ2

ξ2

)1/2ξdξ (41)

• The work function associated with the emission:

Wf = −e∫ ∞

RE(s)(R,0)dr0 (42)

• Now using some well-known integrals for J0 and J1 we finally get

Wf = −e2∫ ∞

ξ=0

(

1 + λ2

ξ2

)1/2−[

J1(Rξ)J0(Rξ)

]

1 −

(

1 + λ2

ξ2

)1/2dξ (43)

• This integral for work function has been obtained numerically and found to bediverging in nature.

Conclusions

• Work function is showing anisotropic nature in presence of strong magnetic field.

• Along the field direction, work functionWf ∝ B1/2, =⇒ the electron will be morestrongly bound inside the crustal region if the magnetic field is strong enough.

• In the transverse direction the work function is found to be large enough com-pared to the longitudinal part.

• In the extreme case, transverse part of the work function diverge for ultra strongmagnetic field strength.

• We expect that such anisotropic behavior and identical type variation of workfunction in the strong magnetic field environment should also be there in thecase of laboratory samples.

Fowler-Nordheim (F-N) Electron Cold Emission Formalismin Presence of Strong Magnetic Field

• Unlike other electron emission process, in cold emission, electrons are pulledout from metal surface by strong electric field.

• This is a purely quantum mechanical tunneling process through surface barrier.

• Formalisms for both non-relativistic and relativistic versions of field emission ofelectrons in presence of a strong quantizing magnetic field, relevant for stronglymagnetized neutron stars or magnetars, are developed.

• In the non-relativistic scenario, where electrons obey the Schrodinger equation,we have noticed that when Landau levels are populated for electrons in presenceof strong quantizing magnetic field the transmission probability exactly vanishesunless the electrons are spin polarized.

• On the other hand, the cold electron emission under the influence of strong elec-trostatic field at the poles is totally forbidden from the surface of those compactobjects for which(i) electrons are unpolarized or spin is neglected and(ii) the surface magnetic field strength is ≫ 1015G.

• Whereas in the relativistic case, where the electrons obey Dirac equation, thepresence of strong quantizing magnetic field completely forbids the emission ofelectrons from the surface of compact objects if B > 1013G, i.e., when theLandau levels are populated.

A Non-relativistic scenario

• In presence of strong magnetic field the electron distribution at the crustal regiongets deform.

• Instead of spherical, the distribution is assumed to be ellipsoidal in nature.

• In this investigation we have assumed, for the sake of simplicity, cylindrical typedistribution of electrons around positively charged nuclei.

• We have Followed Fowler-Nordheim’s epoch making paper in the proceeding ofthe Royal Society of London, to investigate the electron emission from the polesby cold emission mechanism.

• The strength of electric field at the poles of a strongly magnetized neutron star isF ∼ 2 × 108p−1B12V cm

−1, B12−→ magnetic field strength in units of 1012

at the poles and P−→ time period in second.

• Strong magnetic field is also indirectly responsible for magnetosphere formation:

• High energy emitted electrons produce curvature γ-photons.

• This photons can create e− − e+-pairs if energy > 2 ×me.

• To solve the problem analytically, following F-N we take triangular shape surfacepotential V (z) = C−Fz, C−→ is the constant surface barrier and F−→ drivingforce for electron emission.

• The electrons before emission satisfy the Schrodinger equation in cylindrical co-ordinate (We assume (h = c = 1)):

−1

2m

[

1

ρ

∂

∂ρ

(

ρ∂ψ

∂ρ

)

+1

ρ2∂2ψ

∂θ2+∂2ψ

∂z2

]

−ieB

2m

∂ψ

∂θ+

(

e2B2ρ2

8me−E

)

ψ = 0 (44)

• Electrons just liberated, satisfy the equation:

−1

2m

[

1

ρ

∂

∂ρ

(

ρ∂ψ

∂ρ

)

+1

ρ2∂2ψ

∂θ2+∂2ψ

∂z2

]

−ieB

2m

∂ψ

∂θ+

(

e2B2ρ2

8me− E + V (z)

)

ψ = 0

(44a)

• With separable solution:

ψ(ρ, θ, z) = φnρ,m(ρ, θ)fν(z) (45)

• The radial part along with θ dependency is given by

φnρ,m(ρ, θ) =exp(imθ)

(2π)1/2ρ−1−|m|0

[

(| m | +nρ)!

2|m|nρ! | m |!

]

× ρ|m| exp

(

−ρ2

4ρ20

)

Lnρ,m

(

ρ2

2ρ20

)

(46)

ρ0−→ is the Larmor radius.

• Energy without spin:

E =p2z

2me+ µBB(2nρ + ν ± ν + 1) (47)

• Energy with spin:

E =p2z

2me+ µBB(2nρ + ν ± ν + 1) ∓ µBB (48)

µB−→ Bohr magneton.

• Electron just emitted, equation along z-axis:

−1

2m

d2f0dz2

+ V (z)f0 = (E − µBB) f0 = wf0 (49)

• Electrons within the crustal matter satisfy:

d2f0dz2

+ w2kf0 = 0 (50)

• The solution of this equation:

f0 =1

w1/2k

[

a exp(iwkz) + a′ exp(−iwkz)]

(51)

• For electrons just tunneled out, the solution is

f0(y) = y1/2H(2)1/3

(

2

3y3/2

)

(52)

• We define the quantity:

Q =2

3(2meF)1/2

(

C − w

F

)3/2

(52a)

• The transmission probability for electrons:

D(w) =| a |2 − | a′ |2

| a |2(53)

• The field emission current for electrons:

R =eB

2π2

∫ ∞

0f(w)D(w)

pz

medpz (53a)

• From the continuity of wave function and their derivatives at the interface:

D(w) ≈[w(C − w)]1/2

Cexp

(

−4

3(2meF)1/2

(

C − w

F

)3/2)

=[w(C − w)1/2]

Cexp (−2Q) (54)

• Obviously magnitude of D strongly depends on the magnitude of Q.

• This factor ≈ ∞ in absence of electron spin −→ makes transmission probabilityfor electrons →0.

• To study the effect of magnetic field on cold electron current from the poles, weassume following F-N:C = µe + wf , µe→ electron chemical potential and

wf = wc × (B/B(e)c )1/2eV, wc ≈ 82.93.

• Then one can express Q in the following form:

Q ≈ h1/2f (β × 0.5 + wch

−1/2f × 10−6)3/2 × 107 = QB +Qwf (55)

• β = 0−→ conventional electron spin polarization, β = 1−→ with no electronspin and β = 2−→ opposite spin polarization.

• For β = 1 or 2−→QB extremely large =⇒ Q also large enough (−→ ∞) =⇒

transmission current R vanishingly small.

• For β = 0, QB = 0=⇒Q = Qwf is finite =⇒ transmission current is finite.

• In this case the variation of transmission current with electric field (also withmagnetic field):

R = 0.26F1/224 exp(−9.8 × F14) (56)

The solid curve is for the variation of electron field current with electric field andthe dashed one is for the variation of that with magnetic field.

BRelativistic scenario

• The potential is introduced in field theoretic manner.

• The radial part of Dirac equation satisfied by the upper component:[

βλ +∂2

∂ρ2+

1

ρ

∂

∂ρ− k2ρ2

]

R(ρ) = 0 (57)

β2λ = E2 −m∗2 − p2z + 2λk with λ = ±1=⇒ up and down spin states.k = eB/2 and m∗ = m+C − Fz with m∗ = m for electrons when V (z) = 0.

• The solution of the above equation:

R(ρ) = N exp

(

−t

2

)

Ln(t) (58)

• Ln(t)−→ Legendre Polynomial, t = k2ρ2 andenergy eigen value Eν = (P2

z +m2 + 2νeB)1/2.

• Motion along z-axis:

d2fν

dz2+ (E2 −m∗2 − 2νeB)fν = 0 (59)

• Using the transformation:

z =m+ C − uF1/2

F(60)

• We have

d2fν

du2+ (α2 − u2)fν = 0 −→ 1 − D QM HO (61)

Where α = (E2 − 2νeB)1/2/F1/2 with α2 = 2l

we have E2 = 2(νeB + lF) and

fν,l = N exp

(

−u2

2

)

Hl(u) = (fν)I (say) (62)

• Electrons within the crustal matter (V (z) = 0):

d2fν

dz2+ α2fν = 0 (63)

Here α = (E2 −m2 − 2νeB)1/2

• Solution:

fν =1

α1/2

[

a exp(iαz) + a′ exp(−iαz)]

= (fν)II (say) (64)

• Continuity at z = 0:

(fν(0))II = (fν(0))I and(

f ′ν(0))

II=(

f ′ν(0))

I(65)

• Hence:

a+ a′ = Nα1/2 exp

(

−u20

2

)

Hl(u0) (66)

• and

iα1/2(a− a′) = N exp

(

−u20

2

)

[

u0Hl(u0) − 2lHl−1(u0)]

(67)

Where u0 = u(z = 0) = m∗/F1/2

• Defining

a+ a′ = X and a− a′ = iY , −→ X, Y Real Quantities (68)

• Hence the transmission coefficient

T =| a |2 − | a′ |2

| a |2= 0 (69)

Conclusions :

• Electron emission is allowed if electron spin is considered.

• To have non-zero transmission current, in addition, electrons must have conven-tional spin polarization.

• For extremely high magnetic field, wf→ ∞, transmission current →0.

• In the strong magnetic field case, charge density in the magneto-sphere will beextremely low.

• For the relativistic scenario electrons can not be extracted whatever be thestrength of surface electric field at the poles if B > 1013G, i.e., the Landaulevels for the electrons are populated.

• Our theoretical results do not contradict the known observational data on electronemission from magnetars.

Relativistic Field Emission of Electrons in1 + 1-Dimension

• In this part we have presented a formalism for electron emission in 1 + 1-dimension with relativistic scenario.

• We assume 1-Dimensional Dirac equation satisfied by electrons at the polar re-gion of a strongly magnetized neutron star.

• We have included a triangular type potential barrier at the surface near polarregion.

• Further, to avoid Klein paradox, the potential is introduced in a field theoreticmanner.

Relativistic Emission Model in Presence of ScalarPotential Barrier

• We use the following form of Dirac equation in presence of scalar potential V (x):

[~α.~p+ β(m+ V (x))]ψ = Eψ (70)

• Where as before V (x) = C − eFx −→ C is the surface barrier andF (absorbing the electron charge, we express eF by F ) is the constant electricfield −→ acts as the driving force for electron emission.

• We use the following representation:

α = σy =

(

0 −ii 0

)

and β = σx =

(

0 11 0

)

(71)

• We define: m∗ = m+ C −→ effective electron mass.

• Spinor solution:

ψ =

(

uv

)

(72)

• Hence

−dv

dx+ (m∗ − Fx)v = Eu (73)

du

dx+ (m∗ − Fx)u = Ev (74)

• After eliminating v and with the transformation:

ξ = F1/2

(

m∗

F− x

)

(75)

• We have

−d2u

dξ2+ ξ2u =

(

E2

F− 1

)

u (76)

• This equation may be re-arranged in the following form:

d2u

dξ2+ (λ− ξ2)u = 0 (77)

=⇒ equation for 1-D HO with λ = 1 + 2ν,

• we have

E2 = 2(ν + 1)F

and

u = NE

F1/2exp

(

−ξ2

2

)

Hν(ξ) = uI (say) (78)

• Within the crustal matter, Dirac equation in 1 + 1-dimension:

(~α.~p+ βm)ψ = Eψ (79)

• This equation can be transformed as

d2u

dx2+ w2

ku = 0 (80)

where wk = (E2 −m2)1/2 is the free electron momentum.

• Following Fowler-Nordheim:

u =1

w1/2k

[

a exp(iwkx) + a′ exp(−iwkx)]

= uII (say) (81)

• Assuming the interface at x = 0:

uII(0) = uI(0) and u′II(0) = u′I(0) (82)

• Hence

a+ a′ = Nw1/2k

E

F1/2exp

(

−ξ202

)

Hν(ξ0) (83)

and

iw1/2k (a− a′) = NE exp

(

−ξ202

)

[

ξ0Hν(ξ0) − 2νHν−1(ξ0)]

(84)

• These two equations may be arranged in the following form:

a+ a′ = X and a− a′ = iY (85)

where X and Y are two real quantities.

• Hence

T = 1 −|a′|2

|a|2, (86)

vanishes exactly.

Conclusion:

• This investigation shows that in 1 + 1-dimension the transmission coefficient offield electron emission vanishes exactly in the relativistic scenario, in presenceof scalar type surface potential at the poles.

• We have also notice that transmission coefficient is also zero for vector typesurface potential, of course it has no relevance in astrophysics