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Foreword

RTHCOM ING PUBLICATIONS
Volume 80, (2006)
Indian J. Phys. 80 (5), 555-556 (2006)
Special Session o f Condensed Matter Days-2005^on the Majumdar -  Ghosh Model, 
held at Berhampur University, Orissa, India, dufdng 29-31, August, 2005
The M G Hamiltonian -  a pedagogic review 
D ipan Kumar Ghosh
Resonating valence bond states in 2 and 3D-brief histcky and recent examples 
G B askaran
Majumdar-Ghosh-Iike spin models in low dimensions 
Indrani Bose
Non-linear sigma model approach to quantum spin chains 
SuMATHi R ao
A solvable quantum antiferromagnet model
B ik a s  K  C h a k r a b a r t i  a n d  J u n - i c h i  I n o u e
Regular Articles
Atomic & M olecu la r Physics
The effect o f high magnetic field on the scattering o f electron with atomic hydrogen 
A joy C hakraborty
Condensed M atter &  M aterials Physics
Temperature and wavelength dependence o f transient photoconductivity in ZnTe 
S Johnston, R K A hrenkiel, S Bhunia and D N  Bose
General & interdisciplinary Physics
Monte-Carlo simlation o f  electron transport in solids and biological materials 
M  Farhad Rahimi and N ima Ghal-eh
N u c le a r  P h ysics
Measurement of cross sections for the reactions *^ C^u(n,t3r)“ ®Co, ^^Cu(n,p)^Ni,
®^Cu(n,2n)<^Cu, 5«Ni(n,2n)57Ni and 58Ni(n,p)^«"'^»Co at 14.8 M eV  neutron energy
A  K  M  Harun-A r-Rashid, M U Khandakbr, M N  Islam. A  K M M  H Meaze,
Sk a  Latif, M  a  Halim, K  N aher, M  N C howdhury, M S U ddin, M A  Hafiz and 
M  M  Rahman
(  Cont*d. on next page ]
556 Forthcoming Publications
An investigation of the **0 + '*0 elastic scattering by phenomenological and double­
folding potentials in optical model formalism at the energies between ~
145 MeV
M E K u r k c u o g l u  a n d  H A y t e k in
Notes
On the acoustic deformation potentials of some II-IV-V2 ternary chalcopyrih;s and their 
mixed versions
L  K  S a m a n t a , S  C h atterjee  a n d  P  C h a n d r a
Some comments on fractality of proton at small x  
D  K  C h o u d h u r y  a n d  R u pjy o t i G o g o i
I  ACS  Proceedings
Ripon Professorship Endowment Lecture
Orbital ice and its melting phenomenon 
T a p a n  C h atterji


Forthcoming Publications

Chatterjee, Tapan

Orbital Ice and its Melting Phenomenon

From Editor's Desk

Johnston, S

Ahrenkiel, R K

Bhunia, S

Bose, D N

Indian J Phys 80 (7), 703-706 (2006)
Temperature and wavelength dependence of transient photoconductivity
inZnTe
S Johnslon', R K Ahrenkiel', S Dhuniu  ^and D N Bose^*
'National Renewable Encigy Luboialory, Cioldcn, rolorado-8()4()l USA 
 ^Saha Institute of Nuclear Physics, 1/AF Bidhiinnagar, Kulkala-700 064, India 
’University Science Instrumentation Centre. Culcuttii University, Kolkata'700 00‘J. India
E-mail lulubiiset^yahou.co in
Rettnvecl 8 Scpiemhei 2005, fucepted 4 May 2006
A bstract Transient pholuconduciivity studies have been carried out between 125-293 K on Zn-unnculcd p-ZnTc singi-c crystals 
using a contaLiless r l  lechniquc with 532 nm laser pulse excitation Ihc  samples had resistivity ot 13 7 ohm-cm with hole concentration 
8 42 X lU'Vcm’ at 300 K The decay was Lharuclerized by 2 decay times The laster decay was found to increase from 1 x 10^ s ui 125 K 
to reach u maximum ol 23 x 10 s at 250 K and then decrease to 20 x lO*^ ' s at 293 K The decay tunes observed uie attributed to comer 
cKutaiion Irom and trapping due to a shallow doiuii level at -0  04 cV When the excituiion wavelength was varied fioin 532 nm to 
592 nm, both the signal amplitude and the decay limes increased, with surface effects being dominant at shorter wavelengths The results 
aiL compared with steady-state phoioconduciiviiy measurements
Keywords ’ Zinc iclluiidc, transient photoconductivity, trapping
I'ACS Nos ’ 72 20 Jv, 72 40 +w
1. Introduction
ZnTe i.s a IT-VI compound scmiconduclor whose properties 
have nol been studied as extensively as those of ZnSe 
and CdTe. It has a direct band-gap of 2.26 eV at 300 K 
with cubic zmc-blcnde structure (lattice constant of 6 1037 
A) [1] and. unlike other II-VI compounds, is generally 
p-type Alloyed with other Il-Vl semiconductors, it is a 
promising material for a variety of optoelectronic devices 
in the visible and near ultraviolet optical ranges Although 
the importance of ZnTe-based materials for these 
applications has been recognized, the difficulty in preparing 
n-type material with good conductivity has hampered 
progress in the past. Recently, breakthroughs have occuned 
in realizing low resistivity ii-lype doped ZnTe epitaxial 
layers [2]. Pure green light emitting diodes (LEDs) based 
on ZnTe have also been realized by various techniques
13].
Minority carriers play an important role in all such 
devices and hence their thorough understanding is crucial. 
‘Corresponding Author
Since the properties of the 11-VI compounds are mostly 
dominated by point defects, knowledge of the behavior of 
carrier lifetimes with different defect modification processes, 
such as annealing under different conditions, are important.
Timc-resolved photoconductivity measurements on 3 
types of ZnTe were reported by Bose el al [4]. From the 
fast decay, the minority carrier lifetimes were found to be 
4.6 X 10"® s for as-grown, 3.2 x 10-^  s for Zn-annealed and
3.1 X 10"’ s for In-doped semi-insulating (SI) ZnTe. Here, 
we report the study of temperature dependence of transient 
photoconductivity observed on a longer time scale on Zn- 
annealed p-ZnTe This decay is determined by carrier 
excitation from and trapping due to a .shallow donor level 
at -0.04 eV as discussed below
2. ExperimenlJil
ZnTe was prepared by microwave synthesis from 7Ns 
purity elements followed by single crystal growth from a 
4% Te-rich melt which resulted in the lowering of growth
(D2006 lACS
7()4 S Johnston, R K Ahrcnkiel, S Bhunia and D N Bose
tempcralure from ihe mclling point of ZiiTe which is 
J295°C to IISO '^C Crystals were grown by very slow 
lowering of the quart/ crucible by the Bridgman technique 
as described by Bhunia and Bose [51 icsulling in growth 
along the <1I1> direction as determined by Lauc 
photographs Samples were cut peipcndicular to the growth 
axis and pieparcd by lapping and etching Resistivity and 
Hall cftect studies were earned out between 120-300 K to 
determine the cairiei *.onccntration and mobility which 
were found to be I 6 x 10'^ ’ enr^ and 46 cm^/Vs in as- 
giown samples at 3(K) K Since the ^-type nature of ZiHe 
IS attributed to the predominant defects being Zn vacancies, 
annealing was earned out in a Zn atmosphere at 650“C for 
12 hours which resulted m decrease m hole concentration 
to 8 42 X l()'Vcni\ increase in hole mobility to 54 cmW  
s and increase in resistivity from 8 5 ohm-cm to 13 7 ohm- 
cm The hole activation energy was found to be 149 meV 
for the Zn-annealed samples Cu/n is known to give an 
acceptor level at this eneigy and such a level was also 
detected in photolumincscencc studies [6)
From optical absorption expeiiments, a plot of (or/ii')  ^
VA' hv  showed a direct band-gap of 2 26 eV at 300 K 
However, theie was a long tail in the absorption edge 
extending to -2  I-eV which is attributed to defects giving 
rise to shallow levels above the valence band This is 
important in explaining the wavelength dependence of 
photoconductive decay
'I’lansient photoconductivity was studied using a 
contactless radio-frequency photoconductive decay 
technique as reported by Ahrcnkiel et al [7] A pulsed 
light source derived Irom the second harmonic of a YAG 
laser at 532 nm was used to excite the sample The light 
pulse was strongly attenuated to provide energy of 
////pulse The experimental technique uses an ultra-high 
frequency source (424 MHz) to induce phase-locked 
currents in the semiconducting sample These sample 
currents produce, in turn, an electromagnetic field that is 
proportional to the sample conductivity. The source also 
acts as a detector and measures the electromagnetic field 
produced by the sample. The distance between the source 
and the sample could be varied by an electrically driven 
micro-positioner so that the signal from the sample-dark- 
conduclivity remained constant The coupling was adjusted 
so thai the detector produced a fixed signal irrespective of 
sample si/e and dark conductivity This signal was directed 
to a high-lrequcncy budge circuit and the system coupling 
varied until the dark response was balanced by a bridge 
ciicuil When operated over a frequency range between 
420-430 MHz, the resonant detection system had a Q
factor of over 200. The system rise-time is about 4 ns 
which is the minimum detectable lifetime.
3. Results
The transient photoconductive decays at 293 K were 
characterized by fast and slow components as shown m 
Figure I with excitation using a 532 nm laser pulse When 
the sample tempcratuie was varied Irom 125 K to 293 K the 
slower transient in the Zn-anncaled sample decay varied 
as shown in Figure 2 While the fast decay is attributed 
to a direct radiative recombination process, the slower 
decay is due to trap emission and captuic Irom shallow 
trap.s. It IS observed that the amplitude of the response 
increased with temperature as seen m Figure 2 whi^ le the 
decay time varied as shown in Figure 3 It increased 
slowly from 125 K to 210 K, rapidly till 250 K where it 
reached a maximum value of 22 5 ps and then decreased 
to 20 ps at 293 K
Figure 1. TianMcnl ptu)lucandii(.livity of (a) as-grown ZnTc, (b) Zii- 
unncalcd Zn'l'c and (c) Sl-Znic with laser exciiaiion al <^ = 532 nm
0 2 4 6 6 10
Time (ps)
Figure 2. Transient pholueonductivity in ZnTe between T = 125 
-2 ‘J3 K (^  = 532 nm).
Temperature and wavelength dependence of transient photoconductivity in ZnTe 705
Temperature (K)
I'lgurc 3. Vaiiation of tranMcnl decay limes ■ iih temperature (>l 
nno
When ihe incidcnl wavelength was varied, the decay 
lime increased with increasing wavelength Since the band- 
gap ol ZnTe is 2 26 eV corresponding to A = 549 nm, laser 
pulses at 532 nm wavelength resulted in strong absorption 
near the surface whereas at longer wavelengths, the incident 
ladialion was able to penetrate into the bulk This resulted 
in surface recombination being important at shortei 
wavelengths while bulk recombination was dominant at 
longer wavelengths The decays are characterized by 2 
lime constants T| and whose values are given in Table 1
I'ulilc 1 Vatijiion 
wjvclcni;ih
or iransiciit decay times with incident lase>
WjvLlcngih A Max response Decay time Decay lime
nm) aiul pholon 
cneigy (cV)
(mV) h  (MS) r: (MS)
-‘i.'iS (2 234) 1 .5 X 10^ 5 543 -
5H5 (2 119) 1 85 X 10^ 24 42 98 27
592 (? 095) 3.0 X tO^ 36 04 177 3
592 (low uilensity) 2 0 X 10^ a 485 76 97
4. Discussion
While the fast decay in photoconductivity as observed in 
: Figure 1 IS due to direct electron-hole recombination, the 
slower decays with larger time constants are due to trapping 
and release of earners from shallow traps. The earlier paper
[4] focused on the fast decay times due to direct 
recombination from which the values of the recombination 
coiTstam B -  1.4 X IQ-’ cm  ^ s '‘ and 4 x 10-'" cm  ^ s"', 
jrespeciively were obtained for the two types of samples.
The present results on the other hand, are concerned with 
the slower decay.s determined by traps This carrier trapping 
time constant is given by
where a  = capture cross section, v,|, = thermal velocity of 
carriers = HF cm/s and N, -  trap concentration For a 
decay lime of 5 543 ps, if N, is assumed to be lO'Vcm' 
then <T= 1 68 X 10-"'/cm*, a value characteristic of neutral 
traps. This assumes all the traps to be empty and available 
lor capture of carriers At high injection levels, the 
concentration of empty traps becomes (A/, -  n) and hence, 
the capture rale decreases giving longer decay times This 
IS observed from Table 1 when the pulse energy is varied 
at 592 nm
If the trap depth is E„ the rate of release from traps is 
then obviously proportional to exp {-EJkT) Thus, the 
rapid increase in frce-camer life-timc between 210 K and 
250 K IS due to de-trapping or release of earners from 
traps. Assuming that
T = To exp {-EJkT),
where T = free carrier life time and To is a constant, the 
exponential increase can thus be altnbuted to a trap depth 
of 0.049 eV
Steady-stale photoconductivity and.photolumine.sccnce 
studies by Bhunia ei al [61 had revealed the presence of 
an In donor with an activation energy of 0 040 eV which 
controlled electron lifetime at T < 220 K and a deep 
acceptor at -an energy 0 404 cV above the valence 
band which determined the lifetime at T > 220 K. I ’he 
steady-state photoconductivity showed a sharp maximum 
al 220 K compared with a maximum at 250 K for the 
transient studies It is thus evident that the presence of 
the In donor level accounts for the sharp increase in 
photoconductive deetty time with increase of temperature 
between 210 K and 250 K, since it results in increase in 
effective free carrier lifetime. The quenching of 
photoconductivity above 220 K is due to the deep acceptor 
Otc as explained earlier which could be passivated by H 
plasma treatment. It may be mentioned that ICP analysis 
had shown the concentration of In in ZnTe as 0-.28 ppm 
and that of Cu at 0.042 ppm
It IS observed from Table 1 that the decay time increased 
significantly with incident wavelength and the response 
am.plitude also increased. This may be attributed to several 
rea.sons At photon energies at and above the band gap 
of 2.26 eV (548 nm) electron-hole pairs are created which 
can, on the cessation of the laser pulse, exhibit fast direct 
recombination as seen from Figure 1. At these energies
706 S Johnston. R K Ahrenkiel, S Bliunia and D N Bose
where a  is reasonably high, absorption occurs near the 
suriaec Wilh increasing wavelength a  decreases and so 
the light pcnelralcs deeper into the sample and hence, the 
role of surhiu* recombination is reduced However, now 
absuiplion at lovvei-lhan-bandgap energies occurs due to 
excitation ol electrons from filled acceptor levels above the 
valence h.ind and hence, free holes are not created. This 
Lontiibiites to the increase of the free carrier lifetime of 
elections in the tondiiclion hand Thus, the transient 
lesoonse at 585 nm (Figure 4) and 592 nm does not exhibit
the veiy last decay due to direct recombination Comparing 
the transients at these two wavelengths, it is lound that 
the amplitude ul the response mcicases by a factor ot I 6 
while the decay time increases by a factor of 1.5 Since the 
magnitude ol photoconductivity depends on the / if  
pioiluci, the incieasc m response amplitude can be 
attributed to the inciease in iree carrier life-time. The 
longer decay with time constant Ti is also seen to depend 
on the excitation energy and may be due to capture of 
eleuions m shallow acccplois to restore equilibrium.
Ahrenkiel et al [8] have also shown by model 
calculations that the exact nature of the decay depends on 
the trap density as compared with the injection level Thus 
from Table 1, it is seen that the decay time depends on the 
incident light intensity, both Ti and T2 decreasing 
considerably al lower incident intensities This is because 
al higher injection levels, the occupancy of the traps is 
higher, leading to a lower rate of capture [9J
5. Conclu.sions
Direct determination of transient photoconductivity m Zn- 
anncaled single crystal ZnTe was carried out by a 
contactless r.l technique using laser pulses al wavelengths 
of 532-592 nm The temperature dependence of the transient 
decay was explained on the basis of carrier trapping and 
excitation from a shallow In donor at a trap depth of - 0 .^  
cV The wavelength and intensity dependence of iF^ e 
photoconduclivc decay have also been studied
Kererences
l^ l Laiidoll Bom stcin N u inen ta l Uiifa and F iuutim uil 
Reldlioinliipi III S iiciue and lechnolofiv (cd.s) O Madelung, 
II Weiss and M Schulz (Berlin Springer) (1982)
[2] 11 Ogawa, G S Irhin, H Nakuyama, Ml Ni<^ hio and A Yoshida Jpn 
j. Appl riiys Pari 2 33 L 980 (1994)
|3 j K Salo, I' Asahi, M Hanalusa, A Noda, A Aiakawa, M Udiida, 
O Oda, Y Yamada and T lagULlii P/iw Siai Sol /\ 180 267
( 2000)
|41 D N Dose, R K Ahrenkiel and S Dhunia J Appl Fliy\ 86 6.S99 
(1999)
[5 1 S Bhunia and D N Bose J Cr\u Uuwth 186 .S.3.S (1998)
[6 1 S Bhunia, D Pal and D N Dose Senutond- Sn  ledi 13 I4J4 
(1998)
[7J R K Ahrenkiel. R Ellingson, S Johnslon and M Wanlass App! 
Hhys U lt 72 3470 (1998)
[8J R K Ahrenkiel, D M Keyes and D L Levi Hrot J3ih European 
PV Solar Energy Conf (Nice, France) (1995)
19] R K Ahrenkiel in SemiLonduaors and Semimeiali Vol. 39 
(eds) R K Willurdson and A C Beer (New York Academic) 
(1993)


English

Temperature and Wavelength Dependence of Transient Photoconductivity in ZnTe

Bhatt, N K

Vyas, P R

Jani, A R

Gohel, V B

Thermodynamic Properties of Aluminium (Al) at High Temperatures and High Pressures using Local Pseudopotential with Mean-Field Potential Approach: A Comparative Study

li,mJ PIm 80(7), 763-766(2006)
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Notes for Contributors

Majumdar, Parihasarathi

Thermal Stability of Radiant Black Holes

Raychaudhuri, Amalkumar

80  (9 ), 9 4 3 -9 4 6  (2 0 0 6 )
 ^ S I C \ K R E V I E W V O L U M F i  9 8 ,  N U M B E R  4 M A Y  1 5 .  1 9 5 5
Relativistic Cosmology. I
Aualkuuak RAVciiAnDirusi
Theoretical Physics Department, Indian Assoctahan far the Cidtevation of Science, Jadavpur, Calcutta, India 
(Received December 28, 1953)
The paper presents some general relations obtaining in relativistic cosmology. It appears from these 
that a simple cliangc over to anisotropy without the introduction of spin docs not solve any of the out­
standing difficulties of isotropic cosmological models
I. INTRODUCTION
Pe r h a p s  the only poin t in which all the current theories of cosmology are found to be in agreement 
is the time-dependent nature of the spatial geometry 
It therefore seems of considerable interest to investigate 
Ihc temporal behavior of a gravitating system as 
observed by a member of the system itself in its 
neighborhood I t  is true th a t there exists a fairly large 
umouiit of literature where a study has been made on 
similar problems, however they all depend on some 
culditional assumptions of which homogeneity and (or) 
some sym m etry postulates seem to be very common 
ones While there m ay indeed be some great aesthetic 
appeal in favor of such a.ssumptions, y e t they seem 
nevertheless open to serious doubts even on a smoothed- 
out scale a n d  very  definitely do no t provide an exact 
picture of the universe when one considers the finer 
details. Further, the introduction of such assumptions 
lead to rather ambiguous situations when one runs into 
some difTicultics, e.g., the well-known difficulties re­
garding the time-scale* and the original singularity (a 
creation m the finite past?) of the isotropic cosmologic 
models of general relativity  have been variously a t­
tributed to the assumption of homogeneity and isotropy 
on the one hand*"* and to a failure of the general theory 
of relativity on the other.*
In this paper, an a ttem p t is therefore m ade to study 
the temporal behavior of a gravitating cloud on the 
basis of the E instein gravitational equations under 
very general conditions. This would presumably give 
one an idea about the potentialities and limitations of 
the general theory of relativ ity  in providing a satis­
factory solution to  the cosmological problem.
‘ Recent researches have led to a doubling of the nebular 
distances and thus the "age” of the universe has been corre­
spondingly Increased. However, it seems doubtful whether even 
this revised time scale would be consbtent with the estimates of 
the age of the earth by A Holmes, Nature 163, 453 (1949) and 
some of the astronhysical estimates fF  Hoyle, Nature 163, 196 
(1949)]. -
■ A S. Eddington, Science Progr 34, 225 (1939).
■R C Tolman, Revs Modern Phys 21, 374 (1949); G Omer, 
Jr , Astrophys J. 109, 164 (1949).
* R. C. Tolman, Rdatimly, Thermodynamics and Cosmology 
(Clarendon Press, Oxford, 1934), pp 438-39.
' For a concise review of these theories, see H Bondi. Cosmology 
(Cambridge University Press, Cambridge, 1952).
II. THE DIFFERENTIAL EQUATION GOVERNING THE 
SPATIAL EXPANSION (OR CONTRACTION)
We shall assume th a t there is no interaction except 
through the Einstein gravitational equations
- S irr" ,»  R > ^ , - Afi",, ( 1)
where the symbols have their usual significance. Under 
this circumstance the world lines of m a tte r will be 
time-like geodesies. Further, we shall consider th a t 
there is no chaotic motion. We m ay now take the 
world lines of m atte r as our /-lines and if the coordinate 
along these lines measures the proper interval, the line 
element can be w ritten in the form*
dP-{-2geidtdx*-\- gtkdx^dx^, (2)
where the three-space m etric d a ^=  g ,kdx*dx‘‘ is  negative 
definite. W ithout loss of generality we can take 
a t a particular point (say /=ac**ex*«»ai:*™0), i.e., the 
(x*x*x*) space is orthogonal to the world line a t  this 
point. The condition tha t the /-lines are geodesics 
gives
^.*.4-0 , (3)
so tha t g u ’s vanish everywhere on the /-axis. We shall 
take this /-axis as the world line of our observer.
I t  m ay be noted th a t we are no t taking g4t*'0 
everywhere, i.e., we are not assuming th a t the geodesic 
congruence of the world lines is normal. This, according 
to Gddel,* corresponds to the existence of a spin relative 
to the compass of inertia. O ur considerations therefore 
include the spinning cosmological models as well.
The energy m om entum tensor is given by
dx*' i x ,  dx" dx"
d s  d s  d j dj
(4)
so th a t for our cosmic fluid whose world lines are the 
Mines (i.e., d x f / d s ^ ^ l t  d x '/d j-O ) , r»*, has only one 
nonvanishing component a t points on the /-axis (where 
g ,* -0 ) , namely
r*4 -p .
I t  is now easy to deduce the following relation from  
the field equations (1):
________ .R*4-A-4»p. (5)
* la  thu paper, G reek indicee run from 1 to 4 while Latin 
indices run from 1 to 3
* K. G6del, Revs. Modem Phyi. 21. 447 (1949).
1123 (S> 2006 lACS
9 4 4
A M A L K U M A R  R A Y C H A U D H U R I
A d irect calculation of R * t gives a t any po in t on the 
/-axis,
at*
+  * -  g tt. l) (?.«, « -  gml, i), (6)
where dots indicate differentiation with respect to  lim e 
and g tk.i stands for (3/c)x‘)ff,*.
We can show th a t i i f ^  L ( d / d t )  lo g v / ( -  g)  T  
vanishes in case of isotropic expansion (or contraction) 
and IS positive otherwise. For, a t  the particu lar poin t 
we can diagonalize the (3X 3) m atrix  g,* by a transfor­
m ation  of the form
(7 )
w here / ‘ is a  function of its  argum ents, analy tic a t 
poin ts on the /-axis Such a transform ation  does not 
how ever d istu rb  the value of any of the term s in (6). 
H ence, w ithou t loss of generality  we can assum e g.h to 
be diagonalized, so th a t
lo g \ / ( - g ) ^
-  iCg “ ff* »g'" / / « . - .*g‘" t / i - ]
*  4Cg"g” gi.*+g"g**(7i»*+g” g"^«*]
 ^  ^ F / ^ “  y « Y   ^ /gg»
I z L V g i i  F j j /  Vgaa g i i /  ' ‘gM g i i '  j
- ^ ( s a y ) , (8)
where ^  vanishes if and only if g i t" " a g th  a t  the poin t 
under consideration, a  being independent of the pairs of 
indices i  and  ft, i e., <t> vanishes if the expansion (or 
contraction) a t the poin t be isotropic. W e note, fu rther, 
th a t the relation g ,k ’“ ag^k is invarian t under transfor­
m ation  (7).
If  be the velocity vecto r of m a tte r , then w ith our 
choice of coordinate system  it  is sim ply the u n it vector 
along the /-line a t  the poin t and  hence w‘« 0  and  v**= 1 
80 th a t
■  4 (Wi; fc—®*. ^ ) *  4 (®<. *— 0 “  4 (g t<  It '-g k 4 . t) . (9)
T he vanishing of the tensor onk is the necessary and  
sufficient condition for the geodesic congruence of the 
world lines to be no rm a l.' W e shall identify  the an ti­
sym m etric tensor ci>,k w ith spin, as seems natu ra l from 
the classical relation u « 4 c u r l v . '  U sing (9), we can
* L. P Eiwnhu-t, Riemannian Geometry (Princeton University 
Press. Princeton, 1949), p 115
■ Tnis definition of spin b slightly different from that of Gddel. 
While we consider an antuymmetric tensor, Gddel defines the 
spin as a vector constructed from thb tensor, the velocity vector 
and the Levl-Civita tensor However, so long as the field u
purely gravitational, so that the velocity field forms a geodesic 
con^uence, physical conclusions from either definition are very 
similar.
w rite,
4g‘*g'"(gt*. m —g-i*. <) (gll. h — gkt. l) =
and this expression is easily seen to be negative definite. 
W riting this as — 2w® as is consistent w ith the classical 
analogy, we get from (6) and (8),
a* i r a  v
g ) ] - f - - l— lo g \ / ( —g)J + 0*—2cj*. (10) 
so th a t substitu tion  in (5) gives
(l/G )(a»C /a /*)=  (A -4Tp-0»-j-2u,’')/3 , (11)
w here we have p u t C*~ —
In the ord inary  isotropic case, the corresponding 
equation is
(l/C 0(a»G /a/>)=  (A -4 x p )/3 . (12)
F u rther, one can deduce an equation formally exactly 
sim ilar to (12) on the basis of N ew tonian mechanics,“* 
where A corresponds to a repulsive force proportional 
to  distance and the term  involving density  arises from 
N ew tonian g ravitational a ttrac tio n  Considering Eq 
(11), therefore, one m ay say th a t on the classkal 
analogy the spin gives rise to a repulsive force (the 
centrifugal repulsion) while any aniso tropy in  the local 
expansion efiectively increases the g ravitational a ttrac­
tion
From  the divergence relation we get
pG*=C (const). (Ivl)
H owever, the dependence of 0* and  cj* on G  remains in 
general a rb itra ry , and  thus i t  does n o t seem possible to 
in teg rate  (11) w ithou t in troducing fu rther assumptions.
m . THB CASE OF A NONSPINNINO  
GRAVITATING SYSTEM
In  case the spin vanishes, ai*=*0 and  E q. (11) becomes 
( l /G )0 » G /a /» ) -  (A - 4 t p - 0 * ) /3 .  (14)
I t  follows a t  once th a t a  necessary condition for a 
com pletely sta tic  behavior in  the neighborhood is
P = A /4 t ,
i.e., the local density  m ust satisfy the  relation  obtaining 
in the E instein  s ta tic  universe. I t  is to  b e  noted tha t 
th is condition is arrived  a t  w ithou t any  assum ption 
regarding sym m etry  or conditions obtain ing  in the 
d is tan t piarts of the universe. I t  is now cosy to see tha t 
if the whole universe is s ta tic  the  relation m ust be 
satisfied everyw here. I t  therefore follows th a t no non­
spinning nonhomogeneous universe can be s ta tic  and 
the only static  nonspinning universe is the E instein 
universe."
** See reference 5, Chap IX
“ The empty de Sitter universe b  also sometimes regarded as 
static However, In the coordinate system in which thb univerw 
:s static, the /-lines are not geodesics and thus the static nature b 
only apparent due to the absence of matter.
1 1 2 4
945
R E L A T I V I S T I C  C O S M O L O G Y .  I
We note fu rth e r th a t  E q . (14) shows th a t, in the 
absence of spin, w ith  a  given value of p an d  0 / G  
(determinable in principle from  local observations) a t  
a certain in s tan t, the tim e behavior in the locality is 
identical for all locally isotropically  expanding system s 
and is given by  E q . (12).^ F u rth er, in general, we have 
AS the first integral of (14):
A 8 » C /1  1 \
---- (G>-Ga*)-|------ ( --------- )
3 3 VG G a /
- f3 ^, ^*GdG, (15)
where we have used E q , (13) and  the subscrip t A  
indicates the values corresponding to  an  assigned state . 
Hence, finally, the tim e in terval between two states A  
and B  is given by
■ i
d Q , (16)
OiilA 8vpA / ^ \ ‘  1*
where we have w ritten  Q  for G / G a  and 
hence, if G a > G b , then  over the whole range of in te­
gration C ?^ l an d  therefore 0 ^ 0 , the equality  sign 
occurring only if vanishes (i.e , isotropic expansion 
or contraction). W e m ay  hence enunciate the theorem  
that in the absence of spin, th e  tim e in terval between 
a state A  of given p an d  0 / G  an d  ano ther s ta te  B  of 
specified volum e ratio  (i.e., G b / G a  given) is a m axim um  
m case of isotropic expansion (or contraction) if G a > G b  
and there is no zero of 0  in the  in terval considered. 
In particu lar, in the  cosmological problem  in the absence 
uf spin, am ong m odels w hich s ta r t  from  the singular 
state G « 0 , the tune scale to the present s ta te  is a 
maximum for the isotropic models.
I t m ay appear therefore th a t so far as the difficulty 
regarding the sho rt tim e scale of relativ istic  isotropic 
models is concerned, noth ing  w ould be gained by  sim ply 
changing over to  nonisotropic models w ithou t in tro ­
ducing spin. H ow ever, the ac tual situa tion  is slightly  
different. T he cosmological constan t A is n o t inde­
pendently know n. F o r the  isotropic models, the neces­
sity of a good fit w ith  d a ta  of second order (e.g., the 
departure of th e  velocity-d istance relation  from  line­
arity  and the plausible bounds to  the value of pressure) 
sets an upper bound to  A which proves insufficient to  
give a long tim e scale. W hen, how ever, one gives up 
the assum ption  of iso tropy, th e  second-order d a ta  do
" EquRtion 12 has been shown to be valid by R. C Tolmui 
[Proc Natl. Acad. 5d. 20, 169 (1934)1 at the center of symmetry 
iji a spherically symmetric system. J L. Synge CProc Natl Acad 
Sci zO, 635 (1934)3 has obtained the same equation on the 
usumpLion of “symmetry” about the world line at the point.
not se t any  precise bound on A as some o ther arb itra ry  
param eters come into the picture. One can thus  allow 
m uch higher values of A and obtain  correspondingly 
longer tim e scales. T hus the longer tim e scales are due 
to an  increased freedom in  the choice of A ra th er than  
to  aniso tropy itself.^ H owever, it should be noted th a t 
the in troduction  of such an  arb itra ry  param ete r robs 
the theory  of m uch of its  appeal and  indeed if one sets 
A = 0 , then the theorem  we have ju s t proved shows 
th a t a simpile change-over to anisotropy would only 
decrease the tim e scale.
F u rther, E q. (14) shows th a t if A =  0, then  C cannot 
have any m inim um , so th a t one has to s ta r t from a 
singularity  a t  a finite tunc in the p as t as in isotropic 
models. T hus a  simple change-over to  aniso tropy does 
not solve any of the difficulties.
rv. SOME IMPORTANT RELATIONS 
In  this section, we shall prove some in teresting  
relations B y direct calculations we have, a t  any  po in t 
on the /-axis, for the contracted  Riem ann-Christoffcl 
tensor com ponents,
(17)
where J?*** are the corresponding tensor com ponents for 
the three-space g,kdx*dx*‘, and  the T 's  are the 
Christoffel 3-index symbols. C ontracting  E q . (17), we 
get, a fter some simplifications (the contraction  here is 
from  1 to 3),
=  [ l o g v / ( - g ) ] + f — l o g > / ( - f ) ldfi La/ J
+  ,g k m - U ^ ^ g l4 .  (18)
A very  great simplification in the above relation can 
be a tta in ed  by  m aking a  fu rther specification of the 
coordinate system . So far, the three-space has been 
taken to  be only orthogonal to the /-axis. I f  now we 
m ake a  transform ation ,
f - / - b « ( * ') ,
where the function ^  satisfies the following conditions,
(0.Oo=O» (0. tk)o=i(g4lb.H-g4<.fc),
the suffix 0  indicating th e  value a t  T hen  in th e
new coordinate system  (^4<,a)o will be an tisym m etric , 
i.e ,
( tf4 < ,fc )o —  —  ( 0 4 b , <)o-
Such a coordinate transform ation , how ever, does n o t 
affect any  of the results so far obta ined  and  (18) now
See, in thia connection, the popen cited in reference 3.
1125
946
becomes (on droppmg bars)
R*i' D o g \/( - «)D
and comparing with 
relation
dt
AMALKUMAR.  R A Y C H A U D H U R I
(11), we get the interesting
.(60»-
h [ ^ i o g V ( - e ) ] -
0
G
(2 1 )
2a^\ (19)
so th a t using (1), (4), and (10), we get
l r 3  1*
-  - l o g v ' ( - g )  -A .  (20)
3ld/ J
Differentiating (20) with respect to / and using (13),
If the spin vanishes and the expansion also be locally 
isotropic, then from (17), (4), and (1), we find that 
R**k is of the form K 6*t I t  is easy to see tha t a threc- 
spacc whose contracted Riemann-Christoffcl tensor is 
of this form is locally isotropic. Hence we have the 
theorem tliat if spin be absent and if the expansion be 
locally isotropic, then the space is locally isotropic 
Thus local spatial isotropy follows from the restricted 
assumption of locally isotropic expansion (or contrac­
tion) in case of vanishing spin.
1126


Relativistic Cosmology I

Acharya, M

Singh, L P

Dash, G C

Non-parallel Vortex Instability of Natural Convection Flow Over a Non-isothermal Inclined Flat Plate with Simultaneous Thermal and Mass Diffusion

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