IACS Institutional Repository

Sarkar, Utpal

Chattaraj, Pratim Kumar

Dynamic Reactivity Profiles of Helium and Neon Atoms Interacting with an Intense Laser Field

Bose, Indrani

Majumdar-Ghosh-Like Spin Models in Low Dimensions

Chakraborty, Ajoy

Iniliim J. Phys. 80 (6 ), 615-619 (2006)
The effect of high magnetic field on the spattering of electron with atomic hydrogen
Ajoy Ciakraborty 
Laban H rld  V idyap ith .
A D -3 6 9 , Salt Lake KoIkata-700 064, India
c-m ail: tp asg@ iiah en d ra .ia cs .res in 
Received 15 September 2004, accepted 20 March 2006
A bstract In ves tiga tion  has been c a n ie d  out to  show  the e ffe c t  o f  h igh m agnetic fie ld  o ve r  the scattering param eters fo i e lectron -hydrogen
scattering. The m agnitudes o f  the m agnetic f ie ld  vary from  0.2 to 5 a.u. Th ree first-order m odels have been em p loyed  to study the system These 
models reveal the relative e ffect o f  high magnetic fie ld  on the projectile and the target both separately and simultaneously. The scattering cross sections 
arc compared amongst them selves and a lso w ith  the corresponding fie ld -free  results. It has been found that the e ffec t o f  fie ld  distortion o f  the target 
IS neglig ib le on the scattering parameters. But the dressing o f  the incom ing electron fo r the m agnetic fie ld  bring sign ifican t qualitative change.
K ey w o rd s  Electron, hydrogen, high m agnetic fie ld , scattering cross section
1»ACS N o s . . 34.50.-s. 34.80 .-i, 34 90.-»-q
1. Introduction
It is well known that the external field can change the flavour of 
basic physics. The Stark effect and the Zeeman effect are two 
famous examples. In particular, magnetic field has been employed 
to study various aspects of physical system. While exploring 
the various properties of magnetized plasma IJ | or during the 
study of energy spectrum of hydrogen like excitons [2J, existence 
of magnetic field alters the situation dramatically. Moreover, 
magnetic cooling, magnetic bottling, and magnetic trapping are 
frequently used in Bose Einstein Condensation as well as in the 
antiparticle physics [3,4]. The large numbers of theoretical 
investigations on hydrogen atom wave functions in presence 
of high magnetic field have opened up a new domain of quantum 
chaos [5,6]. Therefore, the scattering process in presence of 
external magnetic field can provide a large number of information 
about the structure and dynamics of the different atomic 
systems. Moreover, this kind of study has important application 
in revealing various properties of astrophysical system, e ,g ,, 
surface property of neutron star.
Motivated by the above facts, we plan to study the simplest 
scattering system Le. scattering of an electron by a hydrogen 
atom in a strong magnetic field. This problem is much more 
complicated from the corresponding field-free case. First of all.
the incident electron in presence of high magnetic field moves 
one dimensionally along the direction of the magnetic field. The 
electron looses its plane wave character and moves in cylindrical 
wave fashion, characterized by Landau levels having two 
quantum numbers n and .v. To get the wave function i)f an electron 
in a particular /i>th state, an infinite sum over the other quantum 
number a, has to be performed. Secondly, the exact wave function 
of the Hydrogen au>m in presence of magnetic field is not known 
and is much more different from the corresponding field-free 
counterpart. An a b  in itio  calculation for this system is rather 
difficult to perform.
Here, we investigate electron-hydrogen atom scattering in 
the presence of a strong magnetic field using first-order theory. 
We are interested to judge the qualitative change of the 
scattering parameters when the magnetic field is on, with respect 
to the field-free electron-hydrogen scattering. We take the field- 
free scattering parameters as a base line and employ three 
different models! In model (a), we assume the modified wave 
function of the incident electron in the magnetic field while 
hydrogen atom remain unperturbed during the interaction time. 
In the second model (b), we take one of the most accurate 
available variational wave function of the atomic hydrogen under 
the magnetic field [7], and the incident electron is considered to 
be free from the external magnetic field. To take the necessary
©2006 lACS
616 A joy Chak rahorty
account of the effect of the dressing of both colliding systems, 
we introduce the third picture, model (c) where the electron’s 
wave function as well as the hydrogen’s wave function are 
dressed by the magnetic field. These three models will help us 
to reveal the relative effect of strong magnetic field in the 
electron-atom scattering. To the best of our knowledge^ this is 
the first quantum mechanical attempt to explore the magnetic- 
field-assisted electron-atom scattering. Previously, potential 
scattering of electron in the back ground of strong magnetic- 
field has been studied by several workers [8-12].
2. Theory
We brielly de.scribe our theoretical model in this .section. Atomic 
units are used through out the paper. Wc take the magnetic field 
along z axis. The Schrodinger wave equation in presence of 
magnetic field can be written as
(1)
where is the interaction hamiltonian between electron and 
hydrogen ; and H j are the hamiltonian of free electron and 
the electron of the hydrogen atom in presence of the magnetic 
field, respectively.
and
/f, = 1 / 2 (P , -A r
//2 = 1/2(P,,
(2)
(3)
where and Pj^ is the momentum of the free electron and the 
bounded electron, respectively and A is the vector potential 
representing the field and is given by
A = (0.0, (l/ 2 )«  r  sin 0 )  . (4)
The total wave function of the system is expressed as
!^ = :t/ (r ,) i2 (r2 ) , (5)
where V  (r,) and i2(r2) are the wave functions of the incoming 
electron and bounded electron, respectively. When electron is 
dressed by the magnetic field, the wave function U  (rj) satisfies 
the equation
with energy eigen value e  given by
2 I 2.
(6)
(7)
Here, K  is the one-dimensional momentum of the electron 
along the direction of the magnetic field. The magnetic field is 
given in atomic unit (2.35 x 10^  tesla equals to 1 a.u.). However,
in cylindrical coordinate, (/(r,) is nothing but well known 
Landau level and is given by
(8)
where is the cylindrical wave function (here we adopi
the convention of Ventura] 8], for other choice see [9]). k is the 
wave number along Z axis, n is the principle quantum no. and  ^
is another quantum no. which for a particular /i, runs from zeiu 
to infinity. The first order transition amplitude is given by
By = { u ,  (/•, )i2, (r. )|e. |(/, ( r, )42, ( r, )) (')!
Here, V is the interaction potential between target and 
projectile. The subscript / and / refer to the initial and final 
channel wave functions, respectively.
In model (a) and (c), U (Tj ) is represented by eq. (8 )  when, 
as in model (b), it is plane wave. For model (a) and (c), involves 
sum over and . We use ordinary field-free hydrogen wavt 
function in model (a), whereas in model (b), we lake tlu 
magnetized hydrogen atom wave function due to Gallas |7| In 
parabolic coordinates, the wave function is given by :
S 2 { ^ , r j ) -  N n  ' e x p ---- (10)
where N  is the magnetic field-dependent normalization constant 
and c/, c are the variational parameters. For the ground state, 
the identity a  -  b always holds. This variational wave function 
is reliable and valid for a large span of magnetic field. Finally m 
model (c), as wave functions of both the target and the projectile 
are dressed, we have used simple wave function due to Galkin
[13]. In this wave function, the radial part of the magnetized 
hydrogen atom is taken as
(11)
whereas the angular part of the wave function remains as usual 
This wave function is very similar to field-free hydrogen atom 
However, the range parameter /3  of the radial part is used as a 
variational parameter. This basis set is used to obtain the energs 
eigen-value of the hamiltonian described in eq. (3). The energy 
minimization procedure is now applied to get the value of // 
and the optimum value of energy.
Using that sorts of wave function, we obtain the first Born 
cross section for all the models. To-evaluate the first-ordei 
transition amplitude (B j), we have used Gauss-Legendre 
quadrature technique to get the numerical results. The 
convergence of the results are tested. Moreover, by augmenting 
our model (a) and model (c), we can reproduce the results ol 
Ventura [8} and in the zero field situation, our model (b) gives 
the field-free scattering results. For model (a) and (c), the
The effect o f  high magnetic f ie ld  on the scattering o f  electron w ith atom ic hydrogen 617
definition of the cross section of a particular transition from n. 
to fif states is given by
2^ / n \~*
(12),
In the model (b), the definition of the cross section is samei 
as the field-free case.
3. Results and discussion I
We consider here the direct elastic scattering of an electron off| 
an atomic Hydrogen in presence of high magnetic field. In modelf 
(a) and (c) where we have taken the dressing of the incident 
electron in magnetic field, the scattering process is one-' 
Jimensional; whereas in the other two models the scattering is 
ihree-dimensional. In model (a) and (c), the forward (backward) 
scattering corresponds to the scattering of the electron along 
against) the direction of the applied field. In models (b) and the 
icld-frec case, by forward cross section we mean the probability 
>f the electron to scatter to the forward hemisphere 
[ 0 < 6 <  7 1/2 ) and backward cross section corresponds to the 
.cattering to the backward hemisphere. The forward, backward 
ind the total scattering cross sections are evaluated and 
•ompared using different models. Here, we report the variation 
)f the cross section with the energy of the incoming electron 
aking magnetic field as a parameter. The magnetic field values 
lave been varied from 0.2 a.u. to 5 a.u. The magnetic field value 
vhich affects the scattering procedure appreciably is of present 
nterest and we report these results.
At 0.2 a.u. magnetic field (not shown), the results of model 
a) and model(c) coalesce up to the incident energy of nearly 1 
i.u. Afterwards, marginal difference in results between the two 
ppears, the results of model (a) being higher. Similarly, the
results of model (b) and field-free results coincide with each 
other for whole the energy region. The same characteristic nature 
is found for the magnetic field value of 1 a.u. It implies that 
magnetic field value cannot influence the forward scattering 
cro.ss section below the magnetic field value of 1 a.u. At 1 a.u. 
magnetic field, the behaviour of the forward scattering cross 
section are shown in the Figure 1. It is noticed from the figure 
that the results of four different models are well .separated. On 
the other hand, high magnetic field ( // > 1 a.u.) affects the 
elastic forward cross section appreciably which is evident from 
Figure 2. The results of all the models differ significantly at 5 
a.u. of magnetic field for the whole energy range.
1CP
^  10' r
o 10’
10’
H = 5 a u
\
■\.
2 4 6
Energy (in a u )
F ig u re  2. Forw ard cross section at H =5  a u C^’iirvcs 
m odel (b )  ; ____m cKlcl(c) . zero  fie ld  restill.
iiiodcl(a)
In the case of backward scattering up to I a.u. of magnetic 
field, the predictions of model (a) and model (c) coalesce each 
other as the model (b) does with the field-free model for the 
whole energy region (Figure 3). However, as in the case of 
forward scattering, all these results become separated at higher 
magnetic field (Figure 4). We conclude from Figures 3 and 4 that
igure 1, Forward cross section at H=1 a.u. Curves: .... model(a) 
Kxlcl (b) ; ___model(c) ; zero field result.
Figure 3. Backward cross section at Hk I a.u. Curves: .... model(a) 
model (b) ; ___model(c) ; zero field result.
618 A joy  C hakraborty
the differences of the backward scattering cross sections of 
model (a) with model(c) and that of model (b) with the field-free 
case occur in the low energy regime. Also the results predicted 
by the one-dimensional models ((a) and (c)) and that predicted 
by three-dimensional models ((b) and field-free) become close 
to each other in the high energy region as magnetic field 
increases. These two features are different from the forward 
cross section behaviour. It is noticed from Figure 4 and also 
from Figures 2 and 3 that at high enrgy region, the backward 
cross sections of two different models are going to meet while 
the forward cross sections are well separated. Thus, we hasten 
to add that the distortion of the target due to the magnetic field 
is prominent in the forward channel than in the backward 
channel.
Figure 4. Backward cross section at H =5  a.u. C-urves 
m odel (b )  ; ____ m o d e i(c ) ; zero  fie ld  result.
. m ode l(a ) ;
From these four figures, another interesting facts can be 
noted. For the whole energy range, the forward cross sections 
dominate over the corresponding backward cross sections 
except at the near zero energy domain where these two are nearly 
same. This feature is universal and independent of magnetic
Figure 5. Total cross section at Hsl a.u. Curves: 
model (b) ; ___model(c) ; zero field result.
... mcxlel(a)
field and model. The behaviour of the total cross section 
(consists of forward and backward cross section) is shown m 
Figure 5. The qualitative feature of the total cross section of all 
the models are like the forward cross section as they aie 
dominating.
4. Conclusion
We have investigated the elastic electron-hydrogen scattering 
at high magnetic field. The results are presented to show ihc 
qualitative difference from the field-free scattering parametci 
for a limited region of magnetic field strength. However, usinj: 
the same methodology, one can extend it to the other field value 
of interest. The effect of high magnetic field are considered on 
the projectile and target, separately and simultaneously on both 
of them. Results are compared among them also with the field 
free scattering parameter. In the models (a) and (c), the dressin‘j 
of the incoming electron is taken. These two results change 
both qualitatively and quantitatively from the field-free results 
Therefore, the dressing of the projectile is more impoitani 
phenomenon. Also, in these two models, the near zero encig\ 
elastic forward and backward scattering cross sections diverge 
Therefore, we hasten to add that this divergence is a featiiic ui 
the electron's motion in magnetic field. On the other hand, in 
model (b), only the target is dressed and for all the concernin'^  
magnetic field, this results are similar to tho.se of field-free results 
Also below I a.u, of magnetic field, the forward as well as 
backward elastic cross section of model (a) and model (c) aic 
almost identical. Therefore, the effect of target distortion due to 
magnetic field is nominal and not very significant below 1 a u 
magnetic field so far as the elastic scattering is concerned. The 
incoming electron being free, is largely affected by the applied 
magnetic field. In fact, the plane wave geometry is changed inU' 
cylindrical geometry and the three-dimensional motion is reduced 
to one dimension. This brings large change in scattering process 
Hydrogen atom ground state on the other hand, being 
spherically symmetric, is least affected by the external magnetic 
field. Classically, it can be justified that the hydrogen atom is a 
diamagnetic substance [14] and so magnetic field could not 
bring large change in it. Difference of the qualitative behavioin 
of forward and backward cross section, shows that the effect ot 
high magnetic field is significant in the forward channel. Due to 
the unavailability of any theoretical or experimental data, we 
could not judge the merits of the different models. Still we hope 
that our predictions could serve as a platform for an unexplored 
domain of quantum scattering. Also, it would be interesting to 
employ higher order approximation to study the problem.
Ackfiowledgnient
Author is very much grateful to Dr P. K. Sinha, Dr P. Chaudhun 
and Prof. A. S. Ghosh for their helpful suggestions and fruitful 
discussions during the progress of the work.
The effect o f  high magnetic f ie ld  on the scattering o f  electron w ith atom ic hydrogen 619
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The Effect of High Magnetic Field on the Scattering of Electron with Atomic Hydrogen

l ,u U a n J .P f> y s .flO (2 ),2 n -2 l4  (2006)
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Notes for Contributors

Advisory Board of Editors

Karan, S

Sen Gupta, S P

Microhardness Studies on Urea-Doped Solution Grown Single Crystals of Epsomite

Ghosh, A

Mitra, P

l„dm) rhvs. 80(9). 867-871 (2006)
Counting of black hole microstates
A Ghosh and P Mitra*
Theory Division, Saha Instilule of Nuclear Physics, 1/AF Bidhannogar, Kolkatu 700 064, India 
E-mail panhasarathi miirii@saha ac i
Ahslrait The cniropy ot a black hole can be obtained by couiUing stales in loop quantum gravity The dominant term depends t 
ilii’ Iinniir/i parameter involved in the quantization and is proportional to the area of the horizon, while there is a logarithmic correctit
uiih to rllic ien t -1 /2
Keywords ; Black hole, entropy, loop quantum gravity 
P\CS Nos ■ 04 70 Dy, 04 60 Pp
11 IS an honour and a pleasure to write in the volume 
ilulicalctl to Professor Amal Kumar Raychaudhun, eminent 
iliLoreiital physicist and revered teacher of generations of 
I’hysics students The theory of gravitation, with which he 
pu'otcupied himself, is progressing steadily, and although 
d full quantum theory is not yet at hand, a lot of interesting 
icsiilts arc available
A framework for the description of quantum gravity 
dsinji holonomy variables has become popular us loop 
t|iianium gravity (IJ A start was made in this work in the 
(Jireciion ol counting of black hole microstates. Further 
progress was made in |21, [3] and in [4], In the present 
•irtiUe wc shall try to tie up some loose ends left there 
(Jther discussions of the subject can be found in [5,6]
In this approach, there is a classical isolated horizon 
quantum states are sought to be built up by 
‘^''''Ociatmg spin variables with punctures on the horizon 
riie entropy is obtained by counting the possible slates 
il'di arc consistent with a particular area, or more precisely 
^^ iih a particular eigenvalue of the area operator [IJ
set units such that 47Ty/’5,= 1, where y is the 
Inimirzi parameter and ff, the Planck length. Equating the 
classical area A of the horizon to the eigenvalue of the 
operator we find 
 ^nrrcsptindmg Author
0 )
where the p-th puncture carries a spin j^ j, more accurately 
an irreducible representation labelled by j^, and contributes 
a quantum o f  a r e a  2 ^  +1) to the total area spectrum
For mathematical convenience let us replace the half-odd 
integer spins by integers which makes the area
equation Henceforth, will be
referred to as the "spin' earned by the p-lh puncture. A 
puncture carrying zero spin contributes nothing to the 
spectrum, hence such punctures are irrelevant. Since the 
minimum ‘spin’ each puncture should carry is unity the 
total number of punctures cannot exceed A/y/2 . At the 
same time, the largest ‘spin’ a puncture can carry is also 
bounded, n ^  N, where + 2) = A.
A sequence of ‘spins’ np, each I < np < N, will be 
called permissible if it obeys (1) The p-th puncture gives 
(np + 1) number of quantum states In this way each 
permissible sequence gives rise to a certain number of 
quantum states. The task is to find the total number of 
slates for all permissible sequences. Let it be d(A). One 
can subdivide the problem as follows ; Fix any puncture, 
say p = 1 Consider the subset of all pennissible sequences
© 2006 lACS
S 6 S A (}fu)\h (iml I* Mitro
siu.h lhal puiiLliiri' I ciiiiK's Xpin' I F^ or siith sfquciiLL's 
llu‘ ijica cquuluiM (1) loads
(2)
S o  iho lolal mimhor ol t|iMi)lum s laks  o.ivon by all 
so(|tjoiioos ohoyiiq? (2) is <H/\ v 1 ) Ilul iho piiiKluio I 
iisoll ijiivi-s iv\() si.ilos Ihoioloio, Iho lolal lUimboi t|uaiilLim 
slalos j:i\ori by llio sLibsol ol pL'imissiblo sot|uonLCs in 
which puiiULiu I i-anios 'spin I is 2(HA ) In Iho
iioM slop Lonsidoi Iho siibsi‘1 ol all poiiiiissiblo sot|uoiiLOs 
such lhal inmUiiii ' I caiiios ‘spin’ 2 Aigumonls siinilai lo 
ilu above loads lo iho lolal iiunibu  ol slalos lor such 
siibsol ol soc|uoncos lIs ^</(/\ v2  ) ( ’oiilimim^ ihis 
piociss,  w'o Olid up wilh a rouiiroiico rolalion
(HA) - ^ l / M  1 >(/(/\ -yjnin 1 2) j I V i 1 (^ 1
Hus IS similar lo iho lolalion in |H bin dillois Ironi il m 
having' all values ol /;i - /, / + I i / allovvod
In solvino, ( )^ wo oinplo> a Inal soluiion J(A) -  ovplA/l) 
riion (^) puls a coiidiiioii on A
D o - I
Ihoiotoio, .1 soluiion loi A oboyiii” iho above eijualion 
iniplios a soluiion ol Iho loeunoiico iclalion (3) boi largo 
aioa A >> I, wo have N »  I Moieovoi, lor /I = o( I ) Iho 
summand lalls oil oxpononlially lor laigo n So loimally wo 
oan oxloiid Iho sum U[) lo mrinily 'Hus numciically >ields 
A -  OSbl riio orroi we make m eslimaling A In evlouding 
Iho sum all Iho wa> lo inrimly is f)(c ') Iho lolal 
dogoneiacy (HA) Ihon givos use lo a Moll/mami oniiopv 
A(d) = In (/(/I) -  A A 111 physical uniis
A lA )-
a A
d/iyr;,
which yields AlAi], it we choose Lhe parametei y -  A in  
this IS lhe basic idea behind lhe counling and ihereby, 
making a piedielion lot the j^-parameloi in ordei lhal an 
eniiioly quanium geomelric calciilalion malchcs a 
somielassiLal loimula Thus wo eannol derive the 
semiclassical vvoild bul can adpisl oui paiameteis m the 
iheoiv such that lhe semiclassical world emerges
In lhe above counling process we completely miss 
which conUguralion of spins dommales the counting, m 
olliei words coriiiibuies lhe largest numbci ol quanium
(4)
slates A common misconcepcion is lhal the smallest 
n -  I al every punelure gives rise to the largest 
ol (|uantum stales It aiiscs From the mluitiori ihai sun 
conliguiatioii maximr/es the luimbcr ol punctures anj , 
Ihorelore seiuiclassieally lavouicd The Following arnh^ , 
will show that such an mluilion is meorrecl We locus m 
puncUiics canying idcnlieal spins This is soiiicvsiui i 
analogy vviih slalislical mechanic's where w'e look |(,r 
particles canying lhe same energy. Let lhe number h 
pimelures caiiymg ‘spin’ n be So m the aiea ec|Udini|, 
(1) lhe Slim over |)unctures can he leplaeed by ihc sm, 
ovei spins
/I -  ^  vA'Di t 2)
hqLi.ilion (h) liiilliei symboh/es the lael lhal spins ,ir, 
mine lundamenlal in llus |iroblem than punctures 
eoiiriguralioii ol ‘ s|uns’ v„ will be called permissible i! n 
obeys (b) raeh conliguialion yields [ I,,(/i + I)''quaiiiiip. 
slates bul each ot the coiiligurations can be- chusen p.
V )'/Il ' ways (punetuics are considered disimjniisli
able) I heic-lore, lhe total number ol quanium stales cue. 
by such a eonliguration is
Howevei. ilie conliguMiion m (7) niLiy iiol be peimissihii 
lo obtain a permissible eoiiliguralion, we maxmu/e In J 
by varying stib)ecl Ur the coiislrainl (6). In the vaii.iiioii 
we assume lhal s„ »  1 for each n (or onlv vlil! 
coniiguialums dominate lhe counling) Such an assiiinpiii''i 
clearly breaks clow'u i1 A ~  o(l) The vanaliorial c-qu.itii'ii 
r!)ln</^  -  /irVA. where A is a Lagrange mulhplici, gnc"
S'.. = (/M-l)e (Si
Clearly, loi eonsisteney, A eiheys (4) -with N = U f  HI' 
As already observed this hardly makes a difference, mnu' 
precisely the difference.s are exponentially suppiew^ 'i! 
(j{(’ for large areas. Moreover, although each v„ »  * 
the sum X IS convergeni, since large n terms 
exponentially suppressed This can be explicitly seen 
plugging in (S) into (6), which yields
S  V. !)> (// ' =0342/1
Let us denolc the eonliguration (8) dominating the counHir 
by V The total number of quantum stales is obvion'il' 
= X  where the sum extends over all pcrmisMi’’'^
Countiufi of black hole murostates 869
However, the largest number of stales come 
,,pni sonic dominant configuration \  . So we can expand 
, iiu^ rc accurately the entropy \nd, around this dominant 
,^,ll^ Hlrallon and the result should be expiessibic in the
In il -- In (I- ~ -  ^  \  ^ , where Ss„
ihe nrca equation 2) = 0 , which
b\ requiring that the displaced configuration + 
p, jlso obevs the area equation (6) One may wonder at 
ii,|ioini whether such a condition can ever be met since 
,1, ,I,i' niiegeis whereas y[n[n-\-2) arc irrational Strictly 
.ikiiie in the aica equation (b) we requnc that the sum 
N ', /^mn + 2) should be close to A. In other words 
MiiL'c - \ A  ^ J , wheie A «  A, must exist such that 
i(u Mini lies 111 the range This amounts to saying that 
V.i^ ^nin + 2) be a numbci e - o(1), wheic e may 
uiih Lonliguralions but the variation is slow The 
iiKiiii'. A, which depends on V, is symmetne A simple 
,,m\LiUuii gives / s__-l/(S',„) ■ The total number
>1 !,iU's van be expressed as
( 10)
Ihe above steps illustrate the basic points of the 
calculation which now can be adapted to the actual 
counting The actual counting problem involves another 
crucial condition Each puncture carrying a representation 
labelled by \pm ' n must be associated with a slate |n,ni) 
wlicie m IS halt-odd integer valued spin projections, -n/2 
^ ni < n/2 The condition is that ~ where the sum
extends ovei all punctures I'herelore. a sequence of ‘spins’ 
M,. IS [lermissihlc il it obeys the area equation and the spin 
projection equations simultaneously The task is to count 
the numbei ol states lor all such permissible sequences A 
iccLirrenee relation, similai to (3), can be found also in this 
case following (31, we relax the spin-projection equation 
to X , ,”'/' = *' wheie v is a half-odd integer that can lake 
any sign Let the total numbei ol stales be (l,(A) As 
bcfoie fix a puiictine, say 1, and let it carry ‘spin’ I For 
such sequences, the aiea and the spin pirojcclion equations 
become ^ ~  ‘‘tid 
respectively Theiclore, the number of quantum stales for 
all iieimissible configurations such that the puncture 1 
caiiics ‘spin' 1 is (l^^^^JA-yf^) + r/ ,,j(A--s/3) 
Continuing this process as before, we end up with the 
recuircnce relation
wli.ie ilu' sum extends over all Huctuations The large 
iliUiuiiitiis die out exponentially The Gaussian sum over 
ilikiu.iimns would have pioduccd a factor I/ ^det(K ) if
111. Jeli.i kmc lion were not there It is easy to sec that K
II.;. .1 z e r o  eigenvalue so this hypothetical
'zLini would be divergent Hut the della function makes 
I'u '.uin o\ei the zero mode ol K finite Note that each 
eigenvalue ol K scales like IM, so the fluctuations 
ik, \Nhich have to be rewritten in terms ol normal modes 
■>' nl K, have to be converted to S s 'J y fA , producing 
L\lia I acini s ol J a foi each summation. As one summation 
I. lenioved by the delta function.
d ^Cd, J J V a /V a , (ID
d  - Jxp (A A) [ ^  a;  ]' ^  y p l  ( 27t:i„ )'' ‘
J ( A ) - ^  c/, + 2)^4 1, (13)
wheie the largest ‘spin’ N contributes only one stJite to 
the above sum, provided \> belongs to the set of allowed 
values of m = ( N/2, .N/2] In order to solve (13), wc
consider the Fouiier iranslorm ol c/,,(A)
'vfure ( does not involve A. Plugging (8) into (7) and
'ii^ l'leciing 0( 1 ) factors, we find
dJA ) = f  — d jA ) e “-  (14)j 2^  4;r
and le-express the recurrence lelation in terms of d,„(A)
(^) ^ + cos(wm) (p5 j
In an attempt to solve (15), we again employ a Inal 
solution c/^(A) = exp which on being plugged into
the recurrence lelalion yields u condition on :
1 = 2 .  ^^  cos (/mo) ( 16)
( 12)
that the factors of Va cancel, wc get d = exp (AA) 
to factors of o(l) which will anyway be of o(1) m the 
Liiiropy and therefore have been neglected throughout in
the- L , l l . ' l l l - i l . /« ,
The above equation (16) clearly shows that is a periodic 
function of co. It is also multi-valued. However, it has a 
local maximum at ft; = 0 and in a small neighbourhood of 
this maximum it can be approximated by a power senes 
= A 4 tijftjr 4 ci4ft;‘‘ 4 Values of A, a2, a ,^ etc can 
then be obtained from (16) by comparing various powers
87 0 A Ghosh and P Mitra
ol (o ll can be easily shown that obeys the same 
iccurrcnce relation as (3), therefore the same as before,
finally, wc aic interested in 
" 47T
r/„(/n ^ J = r  , where a  -  oO), (18)
•'■’^ 4 r J a
which yields an entropy ,V(/\) = A / \ - - I n o r
S {A )--
XA
4 7 t y (
-In A. (19)
in physical units Thus, incorpoiation ol the projection 
equation ^ / n - 0  does not alter the leading expression of 
entiopy, hence does not give a different requirement on 
the )^-paramclcr to make the leading entropy agree with the 
semiclassical formula, but gives a universal log-correction 
to the semiclassical formula with a factor of 1/2
I'he counting using the dominant configuration is 
cleaner when the projection equation is incorporated Here 
we firesenl the detailed calculation As before, let 
denotes the number ol punctures carrying ‘spin’ n and 
pio|eciion m The area and spin projection equations take 
the form
A -  ^  V. m + 2), 0 = 5^ni.s„ „
..)'
n „ ' / n „ v j
varying .v„„, subject to the two conditions (20) 
calculation is identical as before and the result can be 
expressed in terms of two Lagrange multipliers X, a
E c ,
- A 2) ,
(22)
Consistency requires that X and or be related to each other 
as (22) satisfy
the spin projection equation wc must require the sum 
^  = 0  This IS possible if and only
if which essentially implies a~
0 (The value 2i7T is excluded by positivity requiremeiusi 
Therefore, the condition on X becomes the same as before 
The sum = also the same as before
The total number of quantum states for all permissible 
configurations is clearly d(A) = 2^  To estimati;
d{A) we again expand Inr/ around the dominani 
configuration (22), denoted by . As before, it givc<> 
In d = In  ^ ,„’S + o(Ss^ ^ ) whcie K
IS the symmetric matrix „v  =
All variations must satisfy the
two conditions (20) which give the two conditions 
„,i/n(n+Y) = 0 and ~ (^■ liking into
account these equations the total number of states cun be 
expressed as
(20)
A configuration .v„„, will be called permissible if it satisfies 
both of these equations (20) Since now the m-quantum 
numbers arc also specified, each puncture is in a definite 
quantum state specified by two quantum numbers n, m. 
The total number of quantum states for all configurations 
IS the number of ways a configuration can be chosen This 
can be done in two steps Note that ■ So
fir-it, the configuration can be chosen in (X  '" ) ' / ! !  ' ' ' ' 
ways Then out of the configuration s„„, can be chosen 
in .v„ ' ways and finally a has to be taken
'riuis wc gel
x5 y]n{n + 2)
= C'd, (23)
(21)
To obtain permissible configurations which contnbute the 
largest number of quantum slates we maximize Ind^ by
where C' is again independent of A. Inserting (22) into (21) 
and dropping o(l) factors, we get
=exp(AA)(5^T„„,y'YP[(27rT„,„)"^ (24)
Plugging thc.se expressions into d we finally get
d = where a  -  o(l), (25)
Va
leading once again to the formula (19) for the entropy 
origin of an extra ^  can be easily traced in this approach, 
which IS the additional condition =0. Thus the
Countinfi o f black hole microstates 871
,iK'fticit‘nt o f  the lo g -c o r r e c t io n  is  a b s o lu te ly  ro b u st and  
not depend  o n  the  d e ta ils  o f  th e  c o n f ig u r a t io n s  at all 
II IS d irectly  l in k e d  w ith  the  b o u n d a ry  c o n d it io n s ,  the  
m ust s a tis fy
Keri-rentes
' 1 ] A Ashtekar, J Baez and K Krasnov Adv Thcnr Math Ph\'\ 4 I
(:noO)
[21 A Ghosh and P Mura Pfm Rev 1)71 027S02 (2005) 
ni K A Meissner C/ou Quant Grav 21 5245 (2004)
|4| A Ghosh and P Milru H m  Lett H616 114 (2005)
|5| FTamaki and H Nomura PIm Rev D72 107501 (2(K)5)
[6] C -H Chon et al gr-qc/05110H4
[7| K V Krasnov Plm Rev D55 1505 (1907), A P Polychronakos 
1)69 044010 (2004)


English

Counting of Black Hole Microstates

Selvanayagam, S

Velmurugan, D

Yamane, T

Suzuki, A

PHENIX, a High Throughput Structure Determination Tool for Single-wavelength Anomalous Diffraction Data

Bhattacharya, Manabesh

Singh, W Shambhunath

Singh, S Dorendrajit

Ghosh, Satrajit

Mazumdai, P S

Computerised Glow Curve Deconvolution of Pure NaCl, NaCl : I and Common Salt of Manipur using Mixed Order Kinetics Model

Singh, R K

Pandey, K K

Ultrasonic Behaviour of Nanocrystalline Ni<SUB>3</SUB>Fe (15nm)

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