INFINITELY MAGNETIZED HYDROGEN ATOM AND PULSAR CRUST

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INFINITELY MAGNETIZED HYDROGEN ATOM AND PULSAR CRUST

C. Angelie, C. Deutsch, M. Signore

To cite this version:

C. Angelie, C. Deutsch, M. Signore. INFINITELY MAGNETIZED HYDROGEN ATOM AND PULSAR CRUST. Journal de Physique Colloques, 1980, 41 (C2), pp.C2-133-C2-138.

�10.1051/jphyscol:1980221�. �jpa-00219813�

INFINITELY MAGNETIZED HYDROGEN ATOM AND PULSAR CRUST

Abstract .- A semi-classical approach is used to compute the very strongly magneti- zed H atom . Relativistic corrections to the ground state are shown to be negligi- ble, as well as the finite proton mass effects .

1 . Introduction .- The standard models for pulsars and the commonly accepted scenarios 1 2

'' describing their formation point to ma- gnetic field intensities B of the order of 1 01 2- 1 01 3 G at their surface.

The.properties of matter in these conditions are very different from those of ordinary matter, due to the fact that the energies associated with the magnetic motion of elec- trons become much larger than their Coulomb energies .

Of the oscillatorlike energy levels corres- ponding to the motion of an electronJ_B, practically only the lowest-lying one plays a significant role in the description of the properties matter : as the spacing of the magnetic levels is of the order of the elec- tron rest mass and temperatures are about lO -10 °K (for not too young pulsars), the excitation of higher transverse levels is negligibly small .

T h e states corresponding to this ground le- vel are conveniently chosen to be the ei- genstates of the angular momentum along B, with non positive eigenvalues -M (M = 0,1, 2, . .. .) . r hey have spin antiparal,lel to B, and zero excitation of the radial motjlon . The density of probability in such a M state

is sharply peaked at a value of the radial

coordinate equal to the cyclotron radius PM <\, [J2MH)/eB] 1 / 2.

A relatively simple model for an N-electron atom of atomic number Z can be built if one separates the transverse and longitudinal, with respect to B, motions of the electron . In this scheme, the transverse motion is determined by the magnetic field alone, the electrons having at their disposal the abo- ve M states .

.The Coulomb field of the nucleus, averaged over the transverse motion, determine only the longitudinal motion. The corresponding available states consist of a very deep ground level and of excited parity doublets situated very close to the Rydberg terms

- 2

•\> -n , with n denoting the number of no- des of the longitudinal wave functions . This separation is a good assumption only if the cyclotron radius of the outermost electron is much smaller than its Bohr ra- dius in an atom of atomic number

Z , i ja . pM << a /Z . It then represents the well-known Adiabatic Approximation (AA) first introduced in atomic physics by

Schiff and Snyder . 3

In order to put the AA on a sound basis, we shall pay attention to the much larger ran- ge of strong field intensities with

Present adress :Service de Physique•du Laser, CEN-Saclay Departement Chimie- Physique.

1"* t -4-

C. Angelie, C . Deutsch and M. Signore

, Laboratoire de Physique des Plasmas, Université Paris XI, 91405 ORSAY Groupe d'Astrophysique Relativiste, Observatoire de Paris, 92 MEUDON

Résumé .- On examine quelques propriétés, importantes pour la structure des pulsars, de l'atome d'hydrogène arbitrairement magnétisé. On montre que l'hypothèse de cy- lindrification forcée sous-jacente à l'Approximation Adiabatique (AA) demeure va- lable pour toute valeur de l'intensité magnétique B. On peut ainsi obtenir d'une manière unifiée, le spectre AA (B -*•<») , celui du spectre diamagnétique (B faible) , l'échelle dilatée de Landau à la limite d'ionisation (en l'absence d'effet Stark cinétique), et le spectre de Bohr (B = 0) . On démontre que les corrections relati- vistes sont négligeables dans le fondamental . Ce qui permet de se limiter à un traitement non relativiste des propriétés statistiques de la surface des pulsars . Les effets de masse finie du proton sont également très faibles (- m /m ) .

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1980221

JOURNAL DE PHYSIQUE

and simplify the discussion of the basic atom properties by restricting to Z = 1, i.e. to hydrogen. Such a restriction is not dramatic because our results can be easily extrapolated to other species.

We shall focus our attention on three chal- lenging aspects of the AA, which recently received a lot of attention.

2. Semi-classical approach to the nonrelati- vistic spectrum.- T o get a more general perspective which includes less extreme si- tuations (white dwarfs, atomic physics ex- periments, etc

...

^{) ,} one could wonder how to interpolate in between the two extreme cases, i .e. B = ^m (cylindrical geometry) and B = 0

(spherical Bohr atom) .,To reach this goal, we shal'l point out the a priori surprizing result that for any B values the nonrelati- vistic spectrum can be quantized in a cylin- drical coordinate system

' .

So, the cylindrification procedure3 which overcomes the lack of coordinate separabili- ty through a factorization of a Z+d harmonic oscillator with a 1-d Coulomb appears as a specialization of the above statement to the B + m limit.

rhe extension to any B value is obtained through the Bohr-Sommerfeld quantization rules, which are a natural by-product of the virial theorem respectively applied to each variable u in the form

-

wU denotes the frequency of the correspon- ding .motion averaged over many periods. We claim the existence of the quantization ru- les, even in the absence of separability of the variables.

T O support this point of view, let us consi- der the 3-d Kepler problem V(p,Z) =

-a (P

+

^{Z 2,} -l/ in cylindrical geometry.

The periods TZ and T remain well defined, P

as well as the average pulsations and 'GS P and the virial quantities.

< m ~ > = <-FZZ> and <mv 2 > = <-F p>

,

₍₂₎

P P

Restricting to closed tra jectories6, one gets

- -

- -

1,(3/2~3/2

= %

^{= w} P - a K e

2 Virtotal

with <=>

-

<g> =

2 r 2

'The rules (3) provide

@23/21E1 3/2

Vir total = { )inp +

3 + I M ~ }

a e&

Whence

I E I

^{= (mea} 2

/ ~ v S

^(np +

2 +

^{l~1)-2, (4)}

yields back the exact Bohr spectrum in the B = 0 limit.

Let us apply the same procedure to the ar- bitrarily magnetized case .The nucleus is supposed infinitely heavy.

The total virial ( Q g = 2~ eB Vir total = <% r

+

^{<rn e Q B} 2 2 ^p >

p,' p2

altogether with <-z = nz@wz and

<iie>

⁼

me e

np@Ed, provides a quantization procedure for any B, through the quantum numbers

%,

n and M _P

.

In the B + 0 limit, one recovers m = M and n = r+,

+

ⁿ

+ 1 ~ 1

+ ^1, with the plausible

P

correspondence

3

= !L

- 1 MI. %

is the number of nodes of the wave function along + B//&

.

^{When B -+ m}

^{, $}

introduces the AA

.

The p-average is performed first while % = 2V br 2V

-

¹ according to the parity of the longitudinal wave-function

I .

The spectrum may be worked out quantitati- vely within the BKW approximation. The exact conservation of M permits the intro- duction of

The quantization is then obtained,through

u

az2

+ r

P ^I(~%)+ a p

₊

_{2 : ~ (2 ~t}

₁

_{MI+ 1)}

.

^{When B -+ the}

h

^{usual 1-d}

spectrum is recovered.Taking the longitu-

C1 dinal spectrum in the Balmer-like form,

( 5 ' one gets the global result (n S

2

⁾

Where

rn

B is the effective magnetic momentum. The

where N, I M ~ , n = 0 , 1, 2,

...

excited (Rydberg) states are characterized

For

IZ I

<< p1 (0) and M = 0 , in the vicini-

by their frequency Z mee

- while

ty of the ionization limit, one has n - x3n3 1

HeB 1 )1/2 the transverse motion is monitored by D ^E = (2~+1.1)=

-

^const{-.

B ' he

P he AA valid when - = -

%

m2e22 ^e (10)

3 3 < < l '

B P B n so that

allows the factorization

xNn

^{(p ,Z ) =}

RN(pIZ)WNn(Z) with N

-

^{n and P n =}

^{3 .}

Eq

.

⁽⁵⁾ is then decoupled into

and

rhe corresponding BKW spectrum is deduced from

C1

-

(p2+Z 2,

where p (Z) denote the roots of the inte- 1, 2

grand. Restrincting first to

I z I

^{>> p (0)}

,

we obtain 2

in the absence of motional Stark effect.

,These hydrogenic results could be extended to the hydrogenic part of the spectrum of several elements (He, Cs, etc

. .

^{.) which}

have an easily polarizable core in the presence of an outer valence electron 7

.

It should be kept in mind that the above results are valid for the excited states only. Hopefully, the corresponding infor- mation for the ground state has been 'sup- plied very recently by Simon et a1 819.

These authors have proposed, for a fixed M, and large B, the exact expression

with

&I-1 1

Q~ = Q~

- ³ iio

^{i!(M-i) M > 1}

B is in number of Bo and EM(B) in number a of e 2/ 2a0

.

^Eq

.

(12) yields back the stan-

In B' dard AA estimate EM(B) =

- -

₄

+

(") B(IMI+M+ 1 ) when B -+ 0 4 .

JOURNAL DE PHYSIQUE

2. _Relativistic corrections to the ground state (AA).- Recently, several authors lolll

I

have speculated about the existence of non- negligible relativistic corrections down to the bottom of the spectrum.

This is a topic of importance in view of the obvious dependence of the bulk (statistical) properties of the pulsar crust on the atomic ground state1' 2.

In contradistinction to these claims, 'we present a simple derivation 5112 which show, that if these corrections could exist, they are indeed quantitatively negligible, within the AA framework.

Let us consider the standard Dirac Hamilto- nian

+ + 2

H = ca.a

+

^{Bm c} _e

-

^e$

,

(13)

+ + e g + +

where a = P

+

^{-xr and @ =} 2. ^{a and} f3 are

2 r

the standard Dirac matrices. Again, the pro- ton mass is taken infinite.

The free transverse motion arises from (H = Ho

+

^h)

+ +

Ho = caL al+ Bmec 2

while h = caZPZ

-

e@ may be taken as a per- turbation (h << Ho)

.

Restricting to the positive energy eigenfuc- tion space (Eo > O), one considers the four- component spinors I-,P> and I+,P> which are simultaneous eigenfunctions of H and JZ. So,

0

we lcok for solutions in the form

The perturbation expansion parameter

introduces the decomposition

e@ is of rank 1, and caZPZ of rank 2. 1 The condition

+

^qo) = 0 gives

while +1,,2 = H1/2J,o/2Eo '

The free longitudinal motion does not con- tribute to the spectrum. The conditions

(J,,,J,,) = (J,1,J,1/2) = 4 yield

El is obtained from

f+ (2) may then be deduced from the matrix eiements of Eq. (17) taken between I-,P>

and I+,P>. The only non zero's are the diagonal ones. If we denote by IN',M> with N = N'

+

^SZ

+

1 the nonrelativistic ei-

-,

genstates of

-

TI pertaining to N', one

gets 2me

and

In addition to the NR transverse average

<NI@IN>, we get a new <N-lI$IN-l>. So, we expect that the excited part (N f 0) expe- riences sensible relativistic corrections.

However, it should be kept in mind that in the superstrong case, only the N = 0 sta-

practice to the observation. So, we retain 2 1.r

Mc only the I-,P> state with aZ = -1 and Eo =

m c2. Eq. (18) thus simplifies to +

e where 1.r =

-

_{m +m} m1m2

,

^{M =} ml+m2. a is the ef- 1 2

fective potential for the effective magne-

with the spinor eigenstate Choosing the gauge as

without small components. The spin is a cons- tant of the motion, and nothing has changed from the NR treatment as far as the N = 0 states are concerned.

These results have been confirmed by a more

-

computational approach13, which produces a 2 2 very small ground state correction s { e l _Hc

.

On the other hand, the excited states would experience a relative increase of their bin-

N W , ding energy s << 1.

mec

3. Separation of the center of mass.- In the previous discussions, we took for grantedthe one-body reduction of the e-P system bymeans of an infinite proton mass. Many people have questioned this step which is so obvious in the B 4 0 limit, but very difficult to justi- fy, in the opposite B -t m situation. Simon et a1.' have shown that this procedure is abso- lutely correct in the present situation, as well as the separation of the c-m. motion.

0 ' ~ o n n e l l ~ has produced a simple and transpa- rent version of the tricky mathematical de- rivationsused by Simon et al.

. '

He makes use of the equivalence of

a _X =

- y,

^{ay = T ,} bx ^{az =}

^o

Eq. (21) simplifies to

where LZ is the Z-component of the angular momentum. Eq.(22) makes clear that the ef- fect of the finite proton mass can be ta- ken into account simply by replacing ml by 1.r and inserting g in the linear B term.

The numerical corrections can only be

References.

-

/1/ Ruderman, M., in Physics Dense Matter, Ed. C.J. Jansen, p. 117 (1974)

/2/ Banerjee, B., Constantinescu, D.H., and Rehak, P., Phys. Rev. (1974) 2384

/3/ Schiff, L.I., and Snyder, H., Phys.

Rev.

55

^{(1939) 59}

/4/ Angelie, C. and Deutsch, C., Phys.

Lett.

67A

(1978) 357

/5/ Angelie, C., These 3eme Cycle, Orsay, June 1978

/6/ This is the key assumption

/7/ Deutsch, C., Phys..Rev. A 2 (1976) 2311

/8/ Avron, J., Herbst, I. and Simon, B., Phys. Lett.

62A

(1977) 214

/9/ Avron, J., Herbst, I. and Simon, B., Ann. Phys. (N.Y.)

114

(1978) 431 /lo/ Garstang, R.H., Rep. Prog. Phys.

40

(1977) 105

/11/ Glasser, M.L. and Kaplan, J., Phys.

Lett.

53A

(1975) 373

C2-138 JOURNAL DE PHYSIQUE

/12/ Angelie, C., and Deutsch, C., Phys.

Lett. (1978) 353

/13/ Virtamo J.T., and Lindgren, K.A.U., Phys. Lett. (1979) 329

/14/ O.Conne1, R.F., Phys. Lett. (1979) 389

/15/ Carter, B.J., J. Math. Phys.

10

(1965) 192.