IACS Institutional Repository

Sinha, Santanu

Santra, S B

Finite-Size-Scaling Theory for Anisotropic Percolation Models

Janarthanan, S

Kumar, T Kishore

Selvakumar, S

Pandi, S

Sagayaraj, P

Anand, D Prem

IndianJ.Phys. 82(10) 1287-1292(2008) 
Investigation on the mechanical, dielectric and 
photoconductivity properties of N-Methyl Urea NLO single 
crystals 
S Janarthanan1, T Kishore Kumar1, S Selvakumar2, S Randi1, P Sagayaraj3 and 
DPremAnand4* 
'Department of Physics, Presidency College, Chennai-600 005, India 
2Department of Physics, L N Government College, Ponneri-601 204, Tamilnadu, India 
3Department of Physics, Loyola College, Chennai-600 034, India 
4Department of Physics, St. Xavier's College, Palayamkottai-627 002, Tamilnadu, India 
E-mail : dpremanand@yahoo.cain 
Received 30 November 2007, accepted 29 May 2008 
Abstract : Single crystal of N-Methyl Urea (NMU), an organic nonlinear optical material was grown by slow 
evaporation technique from methanol as solvent. The grown crystals were characterized by single crystal XRD, 
microhardness, dielectric and photoconductivity studies. The non-centrosymmetric space group of the crystal 
was elucidated by single crystal XRD analysis. The Vickers Hardness Number (VHN) values of NMU suggest 
that the mechanical strength of (110) plane is harder than (100) and (011) planes. It was observed from the 
dielectric studies that the dielectric constant and dielectric loss of NMU sample has high values at low frequency 
and it remains almost constant at high frequency range. Photoconductivity studies of NMU crystal revealed 
positive photoconducting nature of the sample. 
Keywords : Growth from solutions, single crystal growth, nonlinear optical materials, Vickers hardness 
number (VHNV 
PACS Not. : 42.70.Mp. 81.10.Dn, 77.84,-s 
1. Introduction 
Nonlinear optics (NLO) has emerged as one of the most attractive fields of current 
research in view of its vital applications in areas like optical modulation, optical 
switching, optical logic, frequency shifting, optical data storage, telecommunication and 
signal processing [1,2], In this communication, attempts were made to synthesize 
organic compounds with non localized AT electron systems to realize nonlinear 
susceptibilities better than inorganic counterparts [3], Due to the technological importance 
© 
'Corresponding Author ©2008 I ACS 
1288 SJanarthanan, TKIshore Kumar, SSefvakumar, SPancH, P Sagayaraj and D Prem Anand 
of these applications NMU is identified to be an excellent crystal with promising NLO 
properties. Recently, the morphological instability and defect formation in NMU-methanoi 
growth system are reported [4]. The NLO property of the grown crystal is confirmed 
by Kurtz powder second harmonic generation (SHG) test. Satisfactory quality and 
reasonable size of NMU crystals were grown by the temperature reduction method from 
methanol and ethanol solvents [5]. Bulk crystal of NMU of dimensions 1.5 * 0.3 x 0 2 
cm3 has also been obtained from methanol solution by constant volume decreasing 
temperature method [6]. The crystallization of NMU both from solution and vapour 
techniques were successfully achieved by Ardoino et al [7], In addition to that the 
structural characterization of NMU crystals has been performed by high resolution X-
ray diffraction analyses and X-ray topography techniques. The results indicate that the 
NMU crystals exhibit inclusions at the seed-crystal interface and the dislocation starts 
from the seed [8]. A thorough scan of literature reveals that only limited reports are 
available on this material. Therefore we have taken initiatives to investigate the 
mechanical, dielectric and photoconductivity studies, so as to ascertain the suitability 
of this crystal for photonics device fabrication. 
2. Experimental procedure 
NMU was purified by azeotropic distillation with toluene using Dean and Stark 
apparatus. In order to remove water, the purified salt was re-crystallized thrice from 
ethanol. The crystals obtained were stored under dry conditions in a dessicator. The 
solution was saturated at 40VC and the solubility of NMU was maintained as 9 g/100 
ml at 40"C in methanol solvent. Seed crystals were formed due to spontaneous 
nucleation. Single crystals of NMU of dimension 1 .5 * 0.3 * 0.2 cm3 were harvested 
in a period of 20 to 30 days. The photograph of as grown crystals of NMU is shown 
in Figure 1. 
Figure 1 . Photograph of as grown NMU single crystals. 
Investigation on the mechanical, dielectric and photoconductivity properties etc. 1289 
3, Results and discussion 
3.1. Single crystal XRD : 
The grown crystal was subjected to X-ray diffraction studies using ENRAF NONIUS 
CAD-4 single crystal X-ray diffractometer, with MQK* (A = 0.71073 A) radiation. The cell 
parameters were obtained from least squares refinement employing SHELXL-97 program. 
The lattice parameters are a = 8.478 A, b = 6.986 A, c = 6.926 A and the crystal 
belongs to orthorhombic system with space group P212121. These values agree well 
with the reported values [9]. 
3.2. Microhardness studies : 
NMU single crystal was subjected to Vickers microhardness test with the applied loads 
varying from 10 to 50 g for an indentation time of 10 seconds. Indentation was done 
on the well-defined faces (110), (100) and (011) planes of the crystal. The Vickers 
microhardness values were calculated using the equation Hv = 1.8544 (P/d2) kg/mm2. 
Vickers microhardness profile as a function of applied test loads is illustrated in Figure 
20 30 40 50 
Load p(gm) 
Figure 2, Variation of Vickers hardness number with applied toad on [110], [100] and [011] planes. 
The value of the work hardening coefficient was estimated from the plot of log 
P versus log d, drawn by the least square fit technique. It was observed that the 
Vickers hardness number increases with load. The work hardening coefficients of NMU 
were found to 2.022, 2.032 and 3.105 along (110), (100) and (011) planes respectively. 
According to Onitsch, 1.0 < n < 1.6 for hard materials and n > 1.6 for soft materials 
[10-12]. Hence, it is suggested that NMU can be categorized as a soft materials. 
3
-3. Dielectric studies : 
A
 rectangular sample of NMU crystal with dimension 5 x 3 x 2 mm3 was subjected 
1290 SJanarthanan, TKishore Kumar, SSelvakumar, SPandi, PSagayarajandDPremAnana 
to dielectric measurements. The sample coated with silver paint was placed between 
the copper electrodes of the sample holder. 
The variation of dielectric constant and dielectric loss of NMU crystal as a 
function of frequency at room temperature are shown in Figures 3 and 4 respectively. 
From Figure 3, it is observed that the dielectric constant has high values at low 
frequencies and then it decreases with increasing frequency, the trend continues up to 
10 KHz and gets saturated beyond this range. Above this value, er is independent of 
frequency with further increase in frequency. The high dielectric constant of NMU single 
crystal at low frequencies may be attributed to the dependence on the electronic, ionic, 
orientation and space charge polarizations. 
80 
60 
i 
840 
I 20 
0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 
log frequency 
Figure 3. Variation of dielectric constant of NMU crystal as a function of frequency. 
1.5 
1 °'5 
0.0 
2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 
log frequency 
Figure 4. Variation of dielectric loss of NMU crystal as a function of frequency. 
The space charge contribution will depend on the purity and perfection of the 
material. It has noticeable influence in the low frequency region. The orientation effect 
Investigation on the mechanical, dielectric and ptwtocondtHMtyprw etc. 1291 
may be active even upto 1010 Hz. The ionic and electronic polarizations always exist 
below 1013Hz [13]. Hence, the larger values of dielectric constant exhibited by NMU 
at low frequencies may be attributed to space charge polarization arising due to crystal 
defects. Similarly in Figure 4, the dielectric loss decreases with increase in frequency 
and at high frequency range it remains almost a constant. In addition, the low values 
of dielectric loss (Figure 4) indicate that the grown crystals possess lesser number of 
electrically active defects [14]. 
3.4. Photoconductivity studies : 
The photoconductivity studies of the NMU single crystals were carried out using 
Keithley 480 picoammeter. In the absence of any radiation on the sample, and by 
varying the applied field from 20 to 240 V/cm, the corresponding dark current values 
were recorded by the picoammeter. To measure the photo current, the sample was 
illuminated with a halogen lamp (100 W) containing iodine vapour, by focusing a spot 
of light on the sample with the help of a convex lens. The applied voltage was 
increased from 20 to 240 V and the corresponding photo current was recorded. Photo 
current and dark current are plotted as a function of the applied field as shown in 
Figure 5. It is observed from the plot that the dark current is less than the photo 
current, thus suggesting that NMU exhibits positive photoconductivity. This phenomenon 
is attributed to the generation of mobile charge carriers caused by the absorption of 
photons [15]. 
50-
8. 40-
I30; 
i 10-
01 
I — • — Dark currant™| 
1 | — • — Photo currant I ^ 
1
 ———• *-
1 •• « 1 • 1 1 •> 1 1 1 
500 1000 1500 
Applied field (voWem) 
2000 
Figure S. Field dependent conductivity of NMU single crystal. 
3.5. NLO tost : 
For comprehensive analysis of the second order non linearity, Kurtz powder technique 
was used. The sample was illuminated using a Q-switched Nd:YAG laser emitting 1064 
nm, 40 ns laser pulses with spot radius of 1 mm. The second harmonic generation 
developed in pure NMU crystal was confirmed from the emission of green radiation. 
1292 SJanarthanan, TKishore Kumar, SSetvakumar, SPandi, PSagayarajandDPremAnand 
4. Conclusions 
NMU single crystals were successfully grown by slow evaporation technique using 
methanol as a solvent. The structure was confirmed by XRD. The dielectric study 
reveals that dielectric constant and dielectric loss of the crystal decreases with 
increasing frequency. The anisotropy of the hardness of NMU crystal was confirmed by 
the micohardness measurement. The photoconductivity studies ascertained that NMU 
exhibited positive photoconductivity. Studies on metastable zone width, induction period, 
ESCA/AUGER near the phase transition temperature are in progress. 
Acknowledgements 
We sincerely thank Dr. L Joseph Milton Gaspar, Research Associate, Polymat, The 
University of the Basque Country, San Sebastian, Spain and Dr. John Maria Xavier, 
Department of Chemistry, Loyola College, Chennai for their constant help, support and 
encouragement 
References 
[1] H S Natwa and S Miyata (eds.) Nonlinear Optics of Organic Molecules and Polymers (New York : CRC 
Press) (1997) 
[2] P N Prasad and D J Williams Introduction to Nonlinear Optical Effects in Molecules and Polymers (New 
York: Wiley) (1991) 
[3] H S Nagaraja, V Uphadhayaya, P Mohan Rao, P Sreeramana Aithal and A P Bhat J. Crystal Growth 193 
641 (1998) 
[4] A McEwan, R I Ristic, B Yu Shekunov and J N Sherwood J. Crystal Growth 167 701 (1996) 
[5] L Zeng, M Zha. M Ardoino, P Franzosi, L Zanotti, 6 Zuccalli and C Paorici J. Crystal Growth 166 528 
(1996) 
[6] L Zeng, M Zha, L Zanotti, C Razzetti and C Paorici J. Crystal Growth 147 74 (1995) 
[7] M Ardoino, L Zeng, C Razzetti, M Zha, L Zanotti and M Curti Mat Chem. Phys. 66 299 (2000) 
[8] C Ferrari, W J Liu, S S Jiang, M Zha and L Zanotti Mat Chem. Phys. 66155 (2000) 
[9] D Prem Anand, S Selvakumar, M Gulam Mohamed, K Rajarajan, K Ambujam and P Sagayaraj Convergence 
7 52(2005) 
[10] E E A Shepherd, J N Sherwood and 6 S Simpson J. Crystal Growth 167 709 (1996) 
[11] RI Ristic, B Yu Shekunov and J N Sherwood J. Crystal Growth 167 693 (1996) 
[12] R T Bailey, F R Cruickshank, D Pugh, J N Sherwood, G S Simpson and S Wilkie J. AppL Phys. 78 1388 
(1995) 
[13] B Narasimha, R N P Choudhary and K V Rao J. Mater. Sd. 231416 (1988) 
[14] C Balarew and RDuhlew J. Solid Sate Chem. 65 1(1984) 
[15] V N Joshi Photoconductivity (New York : Marcel Dekker) (1990) 


Investigation on the Mechanical, Dielectric and Photoconductivity Properties of N-Methyl Urea NLO Single Crystals

Notes for Contributors

Karmakar, Mahua

Sarkar, Bikash Kumar

Barman, Supriya

Mazumdar, P S

Singh, S Dorendrajit

Singh, W Sambhunath

Bhattacharya, Manabesh

Indian J. Phys. 82 (11), 1495-1502 (2008) 
Evaluation of kinetic parameters from thermoluminescence 
glow curves 
Mahua Karmakar1*, Bikash Kumar Sarkar2, Supriya Barman3, P S Mazumdar1, 
S Dorendrajit Singh4, W Sambhunath Singh5 and Manabesh Bhattacharya5 
1
 Department of Physics, A PC College, New Barrackpore, Kolkdta 700 131, West Bengal, India 
department of Physics, M D Mahavidyapith, Birati, Kolkata-700 051, West Bengal, India 
3
 Bhangar Balika Vidyalaya, Bhangar, 24 Parganas (South), West Bengal, India 
^Department of Physics, Manipur University, Canchipur, Imphal 795 003, Manipur, India 
5
 Department of Physics, Manipur College, Singjamei Imphal-795 008, Manipur, India 
0
 Department of Physics, K N College, Berhampur 742 101, West Bengal, India 
L mail mahua, apc^yahoo com 
Received 12 Mav 2008, accepted 4 Julv 2008 
Abstract In the present paper two methods of determination of trapping parameters from 
thermoluminescence glow curves are considered One is due to Kirsh and the other is suggested by Vejnovic et 
af By taking into account both the computer generated and the experimental TL peaks, It is established that the 
Kirsh method is superior to that of Vejnovic et at The possible application of Kirsh method to the case of 
temperature dependent frequency factor is also studied 
Keywords : Thermoluminescence, trapping parameters, temperature dependent frequency factor 
PACSNo. . 78.60 Kn 
1. Introduction 
Thermoluminescence (TL) is often used for studies of trapping levels is solids, in particular 
for the evaluation of trap depth or the activation energy [1]. The study remains active 
because of its applications in the field of dosimetry and dating [1,2]. A number of methods 
[1] are currently been applied for the evaluation of the trapping parameters from a TL glow 
peak i.e. activation energy (E), order of kinetics (b) and frequency factor (s). Vejnovic et 
al [3] proposed a method of determination of TL trapping parameters. In the present paper 
Corresponding Author 
©2008IACS 
1496 Mahua Karmakar et al 
the method of Vejnovic et al is denoted by the acronym VPRD VPRD method makes 
use of a single expression for activation energy The main difficulty of using such expression 
is requirement of prior knowledge of order of kinetics It seems that Vejnovic et al is not 
aware of this fact pointed out by Gartia et al [4] Moreover they have not taken note of 
the existence of a number of expressions of activation energy [5,6] 
Kirsh [7] suggested a method of evaluation of TL trapping parameters The method is 
simple and straight forward in the sense that unlike other conventional methods of 
determination of trapping parameters such as peak shape, initial rise and various heating 
rates [8], it permits simultaneous evaluation of order of kinetics b and activation energy 
E In the present paper a critical comparison of VPRD method and Kirsh method has 
been done Finally an attempt has been made to adjudge the validity of Kirsh method for 
the case of temperature dependent frequency factor [1,8] 
2. Theory 
The expression for TL intensity I can be expressed as [7] 
1- dn_ 
dt (D 
with dt 
( n \ 
\no J 
snQ exp 
- £ 
kT (2) 
where T is the temperature at time f, n and n0 are respectively, the number of trapped 
carriers at time t and initial number of trapped carriers, k is Boltzmann constant, b is the 
order of kinetics For a linear heating profile given by the eq [8] 
T^T^pt (3) 
where p is the linear heating rate and 7"0 is the initial temperature Equation (2) can be 
recast as [4] 
•L = exp[(um-u) + F(u,um)] (/> = 1) (4) 
and 
— = exp(um-u) 1 - ^ F M j ] " " 1 ( 6 ,1 ) (5) 
lm is intensity corresponding to the peak temperature Tm 
F(^um) = u2mexp{um) E2(um) E2(u) 
u u„ 
(6) 
Evaluation of kinetic parameters from thermolummescence glow curves 1497 
With u = E/kT and um = E/kTm 
E (u) is the second exponential integral [9] 
Following (2) one can write [7] 
n _ A(T) 
(7) 
A(T) is the area of the peak from temperature T upto its end point temperature Te and 
is given by 
A(T)^ji(r)dr 
T 
From eqs (2) and (3) for any two points of the TL curve [7] 
(8) 
Jln(/) 
A\n(n / nQ) 
E J ln ( / /7 ) 
k A\r\(n I n0) (9) 
where J represents the difference between any two points of the TL curve n/n0 is 
calculated from eq (7) From eq (9) it follows that a plot of A\n(l)/A\n(nin0) against 
Jln(1 T)/Aln(n/n0) will give a straight line of slope - (Elk) and intercept b Thus Kirsh 
method allows simultaneous evaluation of b and E Knowing b and E the frequency factor 
s can be obtained from the equation [8] 
-A*£ exp kT. m / •ib-*£°»{w)dT 
Now according to VPRD method expressions for trapping parameters are given by [3] 
(10) E = ^ H 
Jaff 
and fo = 
-VVA2 -4V*2 
2A, (11) 
with 4>=-H 'eff - 0 2 w. eff . (12) 
1498 Mahua Karmakar et a/ 
A, = -0 592 In "eft 
LWeffJ 
- 1 3 7 8 + 1 5 2 ^ f f (13) 
and A, = 0 9 'eft (14) 
where Se„ and weff are respectively, the areas of the falling side and the whole peak s 
has been found out by using the approximate equation 
S =
^re x p kT„ U(b-1)^IsL (15) 
Normally in the analysis of the TL curves, the frequency factor is assumed to be 
temperature independent But this assumption needs modification in view of temperature 
dependence of capture cross-sections [1,7,8,10,11] As a results s varies as Ta ( -2 < a < 2) 
i e we can write s{T) = s0 Ta where s0 is a constant For the case of temperature 
dependent frequency factor the analogues of equations (4)~(6) are given by [12] 
i;=[ir) eM("™-")+F^M (b=:1> (16) 
t /„ 
exp{um-u) A & - 1 _y , 1
— r - ^ . f / n ) 
b 
b-1 (6*1) (17) 
With F(u,um) = (u2m+aum)uamexp(um) Ea+2(Um) Ea+l(u) - a + 2 l u m 
,a+1 
a+2^ 
(a+1 
Where En(u) is the n-th order exponential integral [9] 
3. Results and discussion 
It is clear from eq (7) that in order to find nJn0 one has to compute the area of the peak 
which requires the computation of the integral (Te being the end point 
T 
temperature) Now considering the case of non first order kinetics (b * 1 ) [sqs (5) and 
(6)] the area of the portions of the peak from temperature T to the end point temperature 
T is given by 
Evaluation of kinetic parameters from thermoluminescence glow curves 1499 
r b--\ F(u',um) dT' (19) 
Te is obtained by solving the equation l(Te) = 0 (Te > Tm) numerically by using Newton, 
McAuley method [13] Changing the independent variable from T to u (= ElkT) eq (19) 
can be expressed as 
A{U) =
 J JexPKi~a') 1 ~ W « J to 1 u'~2du' (20) 
Integral occurring in eq (20) has been evaluated by using Gauss Legendre quadrature 
method [14] with range (u, ue) being split into a number of subranges The integral over 
each subranges has been carried out by using 16-pomt Gauss Legendre quadrature The 
second exponential integral has been evaluated [15] by using its continued fraction 
representation [16] For the first order kinetics similar numerical technique has been 
followed 
As a check of our computer code we have reproduced the values of Seff and weff of 
numerically computed TL peaks reported by Vejnovic et al [3] The order of kinetics (b) 
and activation energy (E) have been found out by using the technique of standard least 
square regression [17] 
In Table 1 we present the trapping parameters of a number of computer generated TL 
peaks considered by Vejnovic et al 
Table 1. 
rate {[\ = 
E„(eV> 
16 
1 6 
04 
1 6 
04 
01 
01 
1 6 
04 
01 
01 
Trapping parameters of 
0 5K/sec) 
K 
2 5 
1 9 
1 5 
1 5 
1 5 
1 5 
0 7 
0 7 
1 9 
1 9 
2 5 
Sin (sec"1) 
1013 
1013 
1013 
109 
109 
109 
108 
10s 
105 
105 
105 
[3] Em, Ek and Ev denote respectively the input 
some numencally computed TL peaks corresponding to the 
Ek (eV) 
1 608 
1 601 
0 399 
1 601 
0 400 
0 100 
0 090 
1 586 
0 396 
0 099 
0 098 
\ 
2 524 
1 908 
1 501 
1 510 
1 513 
1 511 
0 691 
0 686 
1 893 
1 885 
2 450 
sK(sec~1) 
1 166*1013 
1 017*1013 
9 380x1012 
1 022x109 
1 032x109 
1 0 1 7 X 1 0 9 
8 904x107 
8524x104 
8679x104 
8259x104 
7 269x10 
Ev (eV) 
1 599 
1 600 
0 400 
1 600 
0 400 
0 100 
0 100 
1 600 
0 400 
0 100 
0 100 
bv 
2 493 
1 896 
1 498 
1 498 
1 498 
1 498 
0 700 
0 700 
1 894 
1 894 
2 485 
linear heating 
sv (sec1) 
9 86x1012 
9 90X1012 
9 93x1012 
9 95x108 
9 95x108 
9 95x108 
1 00x106 
1 00x105 
9 95x104 
9 95x104 
9 91X104 
1500 Mahua Karmakaretal 
values of activation energy, activation energy as calculated by Kirsh method and activation 
energy as calculated by VPRD method bm, bk and bv% sm, sk and sv are the corresponding 
values of order of kinetics and frequency factor respectively A close inspection of Table 1 
reveals that in most of the cases the values of the trapping parameters as calculated by 
the Kirsh method are more closer to the input values than those of VPRD method 
Encouraged by our findings in the case of numerically computed peaks we consider 
well analyzed experimental first order 165 5°C TL peak (/? = 25°C/min) [5] of Ca doped 
KCI, X-irradiated at room temperature followed by thermal cleaning upto 150°C and 320°C 
second order TL peak of bluish green microhne ( /?= 40°C/min) [18] In Table 2 we depict 
the results of application of Kirsh method and VPRD method to the aforesaid experimental 
TL peaks EcV bcf and Scf are the values of trapping parameter obtained by using the 
rigorous method of curve fitting It is evident from Table 2 that also in the case of 
experimental TL peaks the values of the trapping parameters as calculated by using Kirsh 
method show a better overall agreement with the curve fitted values than those by using 
VPRD method 
Table 2. Trapping parameters of some experimental TL peaks 
Ecf(eV) bcf S t f (sec-1) Ek (eV) bk sk(sec"1) Ev (eV) bv sv (sec"1) 
1 36 
1 42 
1 0 
2 0 
1 46x1014 
3 44x1010 
1 36 
1 43 
1 00 
2 01 
1 53*1014 
3 84X1010 
1 36 
1 42 
1 00 
2 00 
1 46*1014 
3 41x1010 
Finally we apply Kirsh method to some computer generated TL peaks with the 
temperature dependent frequency factor as reported by Flemming [19] A perusal of Table 
3 shows that in this case the error in the determination of the activation energy is of the 
order of 10% as in the case of other methods of analysis such as peak shape, various 
heating rates and initial rise [8] We get good agreement in the values of b This is 
because of the fact that the symmetry factor ng of a TL peak with temperature dependent 
frequency factor is not much sensitive to value of temperature exponent a [12, 19] According 
to Chen and Kirsh [8] the activation energy obtained in the case of temperature dependent 
frequency factor should be corrected by subtracting akTm from its calculated value In 
Table 3. Trapping parameters of some numerically computed TL peaks ith temperature dependent frequency 
factor All peaks correspond to a linear heating rate of 0 42 K/s 
a 
2 
2 
2 
2 
T„(K) 
595 
595 
580 
580 
E«(eV) 
1 303 
1 497 
0 409 
0 591 
K 
2 0 
2 0 
1 0 
1 0 
(sec-1) 
1 27*1010 
1 27*1010 
158 63 
158 63 
Ek (eV) 
1 395 
1 404 
0 489 
0 510 
\ 
1 981 
2 011 
0 963 
1 037 
(s0Tma)k 
(sec*1) 
1 27x1010 
1 27x1010 
151 82 
158 97 
ECK <eV) 
1 292 
1 507 
0 389 
0610 
Evaluation of kinetic parameters from thermolummescence glow curves 1501 
table 3 we also present the corrected value of Ek given by Eck = Ek - akTm as obtained 
by following the prescription of Chen and Kirsh They are in good agreement with input 
value of E 
4. Conclusion 
In the present paper a comparative study of Kirsh method [7] and VPRD method [3] for 
the determination of TL trapping parameters have been made by considering both computer 
generated and experimental TL peaks In general, the prediction of Kirsh method is found 
to be better than those of VPRD method As mentioned by Vejnovic et al [3] themselves 
VPRD method does not give good results for b - 0 7 but Kirsh method gives good results 
even for b = 0 7 This is because of the fact that Kirsh method has a sound mathematical 
basis resulting directly from basic TL equations whereas the development of VPRD method 
requires some approximation Moreover as we have already noted that unlike other 
conventional method of determination of trapping parameters such as initial rise, various 
heating rates and peak shape Kirsh method admits of simultaneous evaluation of activation 
energy (E) and order of kinetics (b) It is gratifying to note that Kirsh method can also be 
applied to the case of temperature dependent frequency factor 
Acknowledgment 
Two of us (WSS and MB) are grateful to University Grants Commission, New Delhi for 
partial financial assistance 
References 
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World Scientific) (1997) 
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[4] R K Gartia, S J Singh and P S Majumdar Phys Stat Sol (a) 106 291 (1988) 
[5] P S Mazumdar, S J Singh and R K Gartia J Phys D21 815 (1988) 
[6] R K Gartia, S J Singh and P S Mazumdar Phys Stat Sol(a) 114 p407 (1989) 
[7] Y Kirsh Phys Sat Sol (a) 129 15 (1992) 
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(1965) 
[10] S G Elkomoss, M Samimi, M H All and P Sifferet J Appl Phys 57 5331 (1985) 
[11] P N Keating Proc Phys Soc 78 p1408 (1961) 
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[13] E V Knshnamurty and S K Sen Numencal Algonthms (New Delhi East West) (1994) 
[14] W H Press, S A Tenkolssky, W T Vellenng and B P Flannery Numencal Receipes m Fortran 
(Cambridge Cambridge University Press) (1992) 
[15] W G Devi Vidyasagar University Journal of Physical Sciences 3 18 (1997) 
1502 Mahua Karmakar et a/ 
[16] A Erdeyli Higher Transcendental Functions Vol 2 Ch 9 (New York : McGraw Hill) (1988) 
[17] E J Dudewicz and S N Misra Modem Mathematical Statistics (New York : Wiley) (1988) 
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[19] R J Flemming J. Phys. D23 950 (1990) 


Evaluation of Kinetic Parameters from Thermoluminescence Glow Curves

Forthcoming Publications

Raghuwanshi, Sanjeev Kumar

Talabattula, Srinivas

Indian J. Phys. 82(12) 1681-1687(2008) 
mp 
Asymmetric group velocity dispersion and pulse distortion in 
a uniform fiber Bragg grating 
Sanjeev Kumar Raghuwanshi* and Snnivas Talabattula 
Applied Photonics Laboratory, Department of Electrical Communication Engineering, 
Indian Institute of Science, Bangalore-560 012, India 
E-mail sanjeevkrfa ece use ernet in 
Received 2 March 2008, accepted 4 Ju/y 2008 
Abstract : In a uniform fiber Bragg grating, if the input signal is a Gaussian pulse the dispersion is zero near 
center wavelength and becomes appreciable only near the band edges and side lobes of the reflection spectrum 
However for chirped Gaussian pulses, group velocity dispersion and the reflected light must become asymmetric 
Here the chirped Gaussian pulses can be treated as a symmetric but nonuniform input signal The present paper 
describes that for the case of symmetric Gaussian pulse, the group velocity dispersion and pulse distortion 
remain symmetric however strong the grating may be On the other hand both tend to be more asymmetric for the 
case of strong grating while the input signal is symmetric with nonuniform shape 
Keywords : Coupled mode theory of fiber Bragg gratings, group velocity dispersion effect, chirped Gaussian 
pulses 
PACS Nos. : 42.79.Dj, 42 79 -e 
1. Introduction 
Recently, there has been growing interest in the dispersive properties of fiber Bragg 
gratings for applications such as dispersion compensation, pulse shaping and fiber and 
semiconductor laser components [1]. Although many of these rely on the ability to 
tailor the dispersion in the nonuniform gratings, here we introduce the basis for 
determining delay and dispersion from the known (complex) reflectivity of a uniform 
Bragg grating. The amplitude and power reflection coefficients in a uniform fiber Bragg 
grating for the case of non-phase-matched contradirectional modes coupling are given 
by [2-14] : 
-K s\nh(f2L) 
^sinh(/?L) + /rtcosh(rtL) v ' 
#n .. A ,. © 2008 IACS 
Corresponding Author 
1682 Sanjeev Kumar Raghuwanshi and SrinivasTalabattula 
\pf~ 
n2 + 
2 sinh
z(f?L) 
Q2cost\2(f2L) + lEi) 
2 
sinh2(flL) (2) 
Here the parameters ru k and Q represent the mismatch between the momentum of 
the mode with the refractive index perturbation, mode coupling coefficient and the phase 
detuning parameter, respectively. Turan Erdogan [2] presents a derivation of the above 
equations. The group delay and dispersion of the reflected light can be determined from 
the phase of the amplitude reflection coefficient [2]. If we denote 0p = phase{p) then, 
the delay time rv for light reflected off a grating is 
0„ 
T
" du 2nC dX (3) 
1MB S 1M0 14606 1881 
Mnm) 
Figure 1 . Calculated reflection spectra and 
corresponding group delay for an uniform Bragg 
grating with (a), (b) KL = 2 and (c), (d) KL = 8. 
I 
xAA/\i" 
1&82 " 0 6 18488 18S0 18502 15804 18808 lV . t f 
X(nm\ 
4rdM IJ —^Vr i v . — i 
•TO 2 18*04 18488 184*8 1M 18802 18804 18808 1»0« 
Figure 2. Calculated group velocity dispersion for 
the same case as Figure 1 with (a) KL = 2 and (b) 
*L = 8. 
Figure 1 shows the delay rp calculated for the two examples of the grating. We see 
that for unchirped uniform grating both the reflectivity and delay are symmetric about 
the center wavelength. Since the dispersion dp (in ps/km) is the rate of change of 
delay with wavelength, we find 
" d\ 
2T„ A2 d2e 
2nC d\2 
Asymmetric group velocity dispersion and pulse distortion in a uniform fiber Bragg grating 1683 
~2nC\ 2 ^ 
dcv2 (4) 
In a uniform grating, the dispersion is zero near center wavelength as shown in Figure 
2 and becomes appreciable only near the band edges and side lobes of the reflection 
spectrum, where it tends to vary rapidly with wavelength. 
2. Asymmetric pulse distortion 
Gaussian pulses can be considered as a symmetric signal, on the other hand chirped 
Gaussian pulses can be presumed to be nonuniform but symmetric signal. Figure 3 
demonstrates various kinds of Gaussian pulses with different values of chirp parameter 
[3]. Now if we feed these Gaussian pulses to a uniform Bragg grating, we can visualize 
the pulse distortion as well as reflected mode field due to these chirp factors. We 
consider only the positive chirped Gaussian pulses into analysis because the negative 
chirp factor will only reverse the whole effect. First we present the results of chirp free 
pulses. Figure 4 shows the fundamental reflected mode field and corresponding peak 
power reflection spectrum. 
5 o 
°[ ~-\/\}^J 
"S r* 
C l l X 
» 
.• T5S0 
we* 
1S«' 1»'0 1650 5 DM 
Figure 3. Gaussian pulses (left side) and 
corresponding phase spectrum (right side) for various 
values of chirp factor. 
Figure 4. Fundamental reflected mode field and 
corresponding peak power reflection spectrum while 
C=0and /,L = 2 
Figures 5 and 6, show the reflected light for the two cases, one for weak 
grating (KL = 2) and another for strong grating (*L = 6), while chirping factor is kept 
at C = 0 and C = 2, respectively. One can see from Figures 5 and 6 that for chirp 
free pulses the reflected power is symmetric in both the cases, while for C = 2, they 
are considerably asymmetric. Reflected pulse would be more asymmetric for strong 
grating. 
1684 Sanjeev Kumar Raghuwanshi and SnnivasTalabattula 
r 
I M S 154115 1550 15501 1351 1549 1549 5 1550 1350 5 1551 
»(nm) x (nm) 
Figure 5. Reflected light at z = 0, while (a) C = 0, 
(b) C = 0 a n c U i . = 2 
1649 5 1610 1550 1 
x<nm) 
1549 1S48 5 ia<j0 l&fc)-* 
JL(nm) 
Figure 6. Reflected light at z = 0, while (a) C = 0 (b) 
C = 2and hL = 8 
3. Asymmetric group velocity dispersion 
Dispersion plots and corresponding contour maps for various values of chirp factors are 
shown in Figures 7, 8 and 9, respectively for KL = 2. For the case of chirp free pulse 
1560 0 
15604 
1550? 
1550 
1549 B 
1549 4 
— 
™ _ 
. = - , -
j n y — u i ^
 m 
*-"
 11
"
LM
' 
<- r~~*- r -*» 
~ - -
-
1550 41 
I 1S8° 
Ccrtotr Map 0-(paftm) 
ft48 154J 6 5 1550 1550 5 1561 
U - ^ - i 
ColoirMaplXpftAvn) 
Figure 7. Dispersion and corresponding contour 
map for C = 0 and ^ L = 2) 
Figure 8. Dispersion and corresponding contour 
map for C = 2 (/vL = 2) 
the dispersion is zero at center wavelength where the peak reflection occurs; on the 
other hand, for considerably chirped pulses the dispersion curve no longer remains 
symmetric but picks up a considerable curvature. For highly chirped Gaussian pulses 
(C = 4), the dispersion curve picks up larger curvature at all the wavelength of interest 
This feature can be considered as an asymmetric response of uniform fiber Bragg 
grating even for weak grating case. Figures 10 and 14 show the fundamental reflected 
mode field and corresponding peak power reflection spectrum for the case of moderately 
strong {KL = 4) and strong grating {KL = 6). The reflection bandwidth of the spectrums 
are 0.4 nm (Figure 10) and 0.6 nm (Figure 14), respectively. Dispersion and contonr 
maps for different values of C and KL are displayed in Figures 11-13. Comparison of 
Asymmetric group velocity dispersion and pulse distortion in a uniform fiber Bragg grating 1685 
2 1 0 1 ? 
Figure 9. Dispersion and corresponding contour 
map for C = 4 (*X = 2) 
2 1 0 
Figure 11. Dispersion and corresponding contour 
map for C = 0 (KL = 4) 
M 
1649 6 I S M 15803 1661 2 1 0 1 2 
Figure 13. Dispersion and corresponding contour 
map for C = 4(«£. = 4) 
| 0 6 
{., 
1M9 1M9 1 V S 1 *< 
Figure 10. Fundamental reflected mode field and 
corresponding peak power reflection spectrum while 
C = 0 and KL = 4 
r 
t 
1550 8^ 
1550 4 L 
) 
1550 2 
?«60l 
i54«ef 
1546 5 15A0 1550 5 1561 
X<nm> Contot* Map tXo»*mm 
Figure 12. Dispersion and corresponding contour map 
for C«2( /vL = 4) 
Joe 
1&4SS 1560 15605 1511 
Figure 14. Fundamental reflected mode field and 
corresponding peak power reflection spectrum while 
C=0andKL = 6 
1686 SanjeevKumarRaghuwanshiand Snnivas Talabattula 
4 
Ajjl Tnv** -J t 1 
| 1549 8 
1 19496| 
1^49 15496 1550 15605 1451 2 1 0 1 2 
Cortour Map CMpMnn) 
Figure 15. Dispersion and corresponding contour 
map for C = 0 (* L = 6) 
80 
60 
40 
| 20 
r 
° 20 
-40 
-ao 
r _r
" 
"4" 
^549 1549 5 1540 
-
•1 
\ 
jri 
1550 5 15* 
15506 
1550 4 
1580 2 
1590 
1549 8 
1S4»6 
1549 4 
f '*" 
* 
' / 
/ 
-
r -
 T 
___ _ 
_ 
_ _ - ™ _ 
/ ' 
( | 
\ / 
- - ~ r 
„. 
•-
^ 
I 
I 
I 
2 1 0 1 2 
Coriow Map O^psmn) 
Figure 16. Dispersion and corresponding contour map 
forC = 2 ( * L = 6) 
-4 ^ 
_x 
^ " i 
_
 A 
1550 6 
1550 4 
1550 2 
1 1550 
1549 fit 
1549 6 V 
1549 4 f 
1549 1549 5 1550 15505 1551 
X(rwn) 
2 1 0 1 2 
Contour Map CHps/nm) 
Figure 17. Dispersion and corresponding contour map for C = 4 (*L = 6) 
Figures 7, 11 and 15 suggests that for the case of symmetric Gaussian pulse the zero 
dispersion bandwidth is more pronounced for the case of strong grating (B.W. « 0.6 
nm) [15] while for chirp free pulses dispersion accumulates more for some singular 
points in the case of a strong grating (±80 ps/nm). The dispersion graph becomes 
significantly asymmetric for strong grating for considerably chirped Gaussian pulses. It 
is quite symmetric for weak grating (±5 ps/nm) case even for the same simulation 
parameters. 
4. Conclusion 
We have found that for the case of strong grating the dispersion is large at some 
particular point of bands which may be due to rapid phase variations. Infact, the weak 
Asymmetric group velocity dispersion and pulse distortion in a uniform fiber Bragg grating 
grating shows lesser dispersion but the dispersion is more spread out over the band 
as compared to strong grating The dispersion graph tends to be more asymmetric for 
the strongly chirped pulses. Simulation also shows that the dispersion is high due to 
strong grating at some singular point of wavelength There is no effect of chirp factor 
on absolute value of dispersion as such The only advantage of using strong grating 
for chirp free pulse is that we can achieve a large band of zero dispersion at some 
center wavelength where the peak reflection occurs 
References 
[1] F Ouellette Optics Letters 16 847 (1987) 
[2] T Erdogan IEEE J Lightwave Technol 15 1277 (1997) 
[3] G P Agarwal Nonlinear Fiber Optics (New York Academic) (2001) 
[4] A Othonos and K Kalli Fiber Bragg Gratings Fundamentals and Applications in Telecommunications 
and Sensing (London Artech House) (1999) 
[5] K 0 Hill IEEE J Lightwave Technol 15 1263 (1997) 
[6] L R Chen, S D Benjamin, P W E Smith and J E Sipe IEEE J Lightwave Technol 15 1503 (1997) 
[7] C R Giles IEEE J Lightwave Technol 15 1391 (1997) 
[8] J E Sipe, B J Eggleton and T A Strasser Opt Commun 15 269 (1998) 
[9] A Carballar, M A Muriel and J Azana IEEE Photon Technol Lett 11 694 (1999) 
[10] S K Liaw, K P Ho, Y K Chen and C C Lee Optical and Quantum Electronics 31 77 (1999) 
[11] Stefano Longhi Phys Rev E64 037601 (2001) 
[12] S Longhi, M Marano, P Laporta, O Svelto and M Belmonte J Opt Soc Am B19 2742 (2002) 
[13] Jaepong and Byoungho Lee IEEE Photon Technol Lett 17 408 (2005) 
[14] Stefano Longhi Phys Rev E72 056614 (2005) 
[15] S K Raghuwanshi and S Talabattula J Instrum Soc India 37 297 (2007) 


Asymmetric Group Velocity Dispersion and Pulse Distortion in a Uniform Fiber Bragg Grating

Dutta, Tapati

Tarafdar, Sujata

Indian J. Phys. 82 (2), 201-207 (2008) 
Chaotic behaviour of population on a square lattice 
Tapati Dutta2 and Sujata Tarafdar1* 
'Condensed Matter Physics Research Centre, Department of Physics, 
Jadavpur University, Kolkata-700 032, India 
^Department of Physics, St Xavier's College, Kolkata-700 016, India 
E-mail sujatatarafdarrt/'rediffmail com 
Abstract : Coverage of occupied sites on a square lattice is allowed to evolve according to a set of rules 
The rules imply an attractive interaction for growth of new members, the original members 'die', and the new 
population' multiplies and redistributes randomly over the lattice We show that this scenario leads to a steady 
coverage, cycles with a finite number of points and ultimately chaos as model parameters vary. The calculated 
results are verified by computer simulation. An immobile situation, where migration or redistribution over the 
lattice is restricted is also simulated. 
Keywords Bifurcation, chaos, square lattice, computer simulation 
PACS Nos. • 82.40.B], 47.20 Ky, 02 50 -r 
1. Introduction 
Simple cellular automata algorithms are known to generate a variety of spatio-temporal 
patterns [1-3]. We present here an algorithm for evolution of population on a square lattice, 
which gives rise to chaotic behaviour. The term 'population' represents here, a collection 
of some kind of entity, physical, biological or otherwise, distributed on a square lattice. 
The probabilities for growth or death of these entities are site specific, i.e. dependent on 
the number of occupied nearest neighbours. We call the fractional number of occupied 
sites the 'coverage' and study evolution of the coverage under certain rules. 
Several versions of this model are possible, we study one version in detail, since it 
gives particularly interesting results. On varying model parameters the population distribution 
or coverage evolves to a steady state upto a certain parameter value, then shows repeated 
bifurcations and period doubling leading to chaotic behaviour interspersed with periodic 
windows. The bifurcation diagram looks remarkably like that of the celebrated logistic 
map [4]. 
* Corresponding Author 
© 2 0 0 8 I ACS 
202 Tapati Dutta and Sujata Tarafdar 
Many non-linear equations and iterated maps are known to generate chaos [5, 6], but 
we have not come across such reports for the type of system discussed here. 
2. Model 
Our model is as follows. We consider a square lattice, with unit spacing. The sites may 
be occupied or vacant. To start with - a fraction p0 of sites are randomly occupied. Next 
- the vacant sites can be filled up with a certain probability. We assume this growth 
probability to be dependent on the nearest neighbourhood environment. Then the original 
sites are evacuated and the newly added population multiplies by a certain constant 
factor A. This next generation population with concentration p1 are now randomly 
redistributed over the square lattice. This is the 'migration' process. These steps are 
repeated over and over to see how the population pn at the nth stage evolves. 
The neighbourhood dependent growth probability is B*, where k is the number of vacant 
nearest neighbour (n.n.) sites surrounding a vacant site, and B is a constant such that 0 
< B < 1. The situation is illustrated in Figure 1. 
rn M 
Prob - 1 
o o I 
1 0 I I I *. I 1 1 1 
I ! I Prob-B P | 
and so on as no. of vacant n.n. increase 
0 
0 
1 
0 
0 
Prob«B 4 
Figure 1. Schematic illustration for the growth rule on the square lattice. 
The evolution process can be represented by the following iterative map 
pn+1 = / \ (1-pJ[pn + B(1-pn)]4. (1) 
Here, the term (1 - pn) on the right represents the probability of a site being vacant. 
The fourth power binomial expression, when expanded, gives the probabilites of this vacant 
site having 1 to 4 vacant n.n. with the correct weightage. If all neighbours are occupied, 
the growth probability is 1, and it decreases with the number of vacant n.n. This in effect 
represents an attractive interaction between the entities occupying the sites. The factor 
Chaotic behaviour of population on a square lattice 203 
A multiplies the new population, the original generation dies out and the right hand side 
becomes the {n + 1)th generation population p ^ r If we now assume these new generation 
members to randomly redistribute themselves over the whole lattice, we are ready to 
repeat the process to get pn+2 and so on. 
The probability parameter B can take values between zero to one The multiplying 
parameter A can take any positive value but obviously, if it becomes so large that the 
product on the right exceeds 1, the situation becomes unphysical As we vary A and B 
within the permitted range, very interesting behaviour for the coverage (or population 
concentration) is observed. For B = 0.5, when we vary A, a steady coverage is observed 
upto A= 4.82 approximately At A = 4.824 (we vary A in steps of 0.001) the steady state 
bifurcates to a 2-point cycle. Here the coverage oscillates continuously between two values 
As A is increased further there are repeated bifurcations and finally 'chaos' where the 
coverage has an infinite period never repeating itself. There are intermittent periods where 
finite cycles return and penod-3 cycles can also be seen. Finally at A = 6 102 the range 
of p covers the whole range of physically meaningful values Further increase in A leads 
to unphysical negative values or values higher than 1. All these results are independent of 
the initial coverage pQ. Figure 2 shows the bifurcation diagram with 6 = 0 . 
Figure 2. The variation of coverage with A shows repeated bifurcations and chaos 
The fine structure of the figure is extremely intricate, as in the bifurcation diagram for 
the logistic map [4]. Figure 3 shows the details under higher magnification Emergence of 
the period three region is clearly visible here. So the occurrence of chaos is to be expected 
[7], according to the well known rule 'period three implies chaos'. A Misiurewicz point [8] 
where chaotic bands cross, is also visible. To quantify the chaotic behaviour, we have 
calculated the Lyapunov exponent according to the prescription in [8] around A - 6.02 
and obtained a positive value of 0.568. 
To verify these remarkable results, we have simulated the model on a computer. A 
square lattice of size 300 x 300 is generated. Sites on it are randomly occupied with an 
204 Tapati Dutta andSujata Tarafdar 
initial concentration pt The vacant sites are then scanned to note k, the number of 
vacant n n and they are filled up with a probability Bk The new sites are counted and 
multiplied by the factor A and old occupied sites are evacuated The existing sites are 
now redistributed over the whole lattice, and the process is repeated, several thousands 
of times. The behaviour of the coverage agrees with the calculated results shown in figure 
(2) It takes typically 60 - 70 steps to reach the characteristic evolution pattern which is 
independent of the initial coverage. For B < 6.102, p takes values from 0 to 1. Obviously 
the unphysical values p > 1 or p < 0 are not possible in the simulation 
Figure 3. Details from a blow-up of Figure 2 show periodic windows with cycles of period 3 within the 
chaotic region 
It is interesting to note that when we remain confined to physically acceptable values 
of p in the range 0 to 1, full blown chaos can be observed only for B close to 0 5 For B 
= 0.2, a small steady state value of p is obtained upto A ~ 11, and after that the value 
oscillates between ±~>. On the other hand for larger S, eg with 6 = 0 8 , steady state 
is obtained for A ~ 2, but no bifurcation is observed with 0 < p < 1. 
3. Applications 
We discuss in brief some possible physical situations where this model may be 
applied 
3.1. Ferro-magnetic particles in magnetic field: 
Consider a 2-dimensional arrangement of spin-half particles, with a ferro-magnetic interaction 
between neighbours. When placed in an external magnetic field H, each particle can be 
in the ground state, the spin aligned in the direction of the field, or in the higher level with 
anti-parallel alignment. The difference in energy between the two levels is 2juH, where V 
is the magnetic moment. At a finite temperature more spins are in lower level. Spins can 
flip, depending on the temperature, keeping the total energy constant. This causes a 
spatial rearrangement, preserving the number of up and down spins. Let us now introduce 
Chaotic behaviour of population on a square lattice 205 
a radio-frequency field, which acts as a 'pumping' source of energy, similar to that in a 
laser The frequency of the r.f. field is v, where 
hv = 2nH. (2) 
The present model may now be applied to this system as follows. We assign the state 1 
/ e 'occupied' to the spins in the upper energy level and 0 or Vacant' to those in the 
iower. Assuming ferromagnetic interaction between nearest neighbours only, particles in 
the upper level, draw up nearest neighbours according to the rule illustrated in figure (1). 
The number of particles already in the upper level is reduced by a factor C, as they drop 
to the lower level, with emission of a photon. The number pushed up to the upper level 
are multiplied by factor A, due to the pumping. The temperature of the system is such as 
to redistribute or randomise the spatial arrangement of the spins with the total energy 
held constant. This sequence of events is represented by the equation below 
pn^=Cpn + AV-pn)\pn+B0-pn)]\ (3) 
This equation, with suitable choice of parameters A, B, C, leads to cycles with periods 
2, 4, 8 and so on. We found that for C = 0.6 and A = 4, B = 0.4 gives a period-2 cycle, 
B = 0.43 a penod-4 cycle and B = 0.44 a period-8 cycle. Values of p become unphysical 
for higher B. For C = 0.4, we get 2, 4 and 8 period cycles for B having values 0.53, 0.55 
and 0.57 respectively. The full phase behaviour is yet to be explored. The percolation 
behaviour of a magnetic system with 'bootstrapping' was studied in [9]. 
3.2 Condensation and evaporation: 
A somewhat similar model was suggested for condensation and evaporation of particles 
on a plane by Dutta et al [10]. The iterative map for this model can be represented in the 
present scheme as follows 
pm, = pn -Cpn +0-pn)[P-pn) + Dpn? (4) 
Here C represents evaporation probability at an occupied site and Eh is the adsorption 
probability at a vacant site with k occupied n.n. The probability of growth is less for sites 
with more occupied neighbours, so this situation mimics a repulsion between the particles. 
C and D are related to temperature and pressure, and it was found that under certain 
conditions the steady coverage bifurcates to a 2-point cycle. 
For the condensation / evaporation case Dutta et al [10] also studied an immobile 
particle model, through computer simulation, where the lateral movement of the particles 
is not allowed, so there is no randomisation after deposition. We also study an analogous 
version of the present model. 
Dutta et a/[11] previously studied a system with attractive interaction between particles 
condensing on a square lattice. Here the algorithm was different from the algorithm used 
in the present paper, and the results showed a phase transition and hysteresis, rather 
than bifurcation. 
206 Tapati Dutta andSujata Tarafdar 
Equation (1) is not valid unless the pn entities are randomly distributed on the square 
lattice So only computer simulation is applicable to study the immobile case, with no 
migration of the new generation members We start as before, with a random distribution 
of a concentration p of occupied sited, and occupy new sites according to the rule described 
already But, here we forbid lateral motion of the newly deposited members. The additional 
(A - 1) pn daugther members produced by the multiplying factor A are distributed randomly 
over vacant sites 
The results tor the immobile case are almost identical to the mobile case This is 
rather surprising considering that in the 'immobile' case the generation which grows 
according the environment-specific rule have to remain confined to their original sites 
whereas in the 'mobile' case the second generation particles are randomly redistributed 
over the lattice before the next deposition step starts. 
4. Discussion 
Nonlinear systems described by iterative maps are well known [5, 6], they give rise to 
fascinatingly complex diagrams However, these systems are of special interest when 
they can be associated with real situations, - physical, chemical or biological The present 
model though hypothetical, may be applicable to various problems such as adsorption/ 
desorption or reaction-diffusion [12] problems in chemistry, magnetic systems in physics 
[9] or population dynamics [13] studies 
The number of neighbours considered for each site determines the order of the polynomial 
expression occurring in our map One can consider variations such as a linear chain, with 
two neighbours, or include second neighbours, to get a total of eight neighbouring sites 
on a square lattice 
Different modifications are possible, to represent attractive or repulsive interactions 
between the participating entities The growth or evacuation probability can be made site 
specific according to the demands of the problem Allowing the parameters to take 
unphysical values also gives results which may be interesting from the point of view of 
mathematics, though not meaningful in a practical sense. We hope to report some such 
results in future. In conclusion we may say that we have presented an interesting iterative 
map, associated with growth/death or adsorption / desorption on a regular lattice 
Acknowledgment 
The authors gratefully acknowledge stimulating discussion on non-linear systems with 
Prof. Jyoti Das and Prof. Bandana Barua. 
References 
[1] T Vicsek Fractal Growth Phenomena (Singapore World Sci) (1992) 
[2] S Wolfram Physica D10 1 (1984) 
[3] J A C Gallas and H J Herrmann Physica A356 78 (2005) 
[4] R May Nature 261 459 (1976) 
Chaotic behaviour of population on a square lattice 207 
[5] H G Schuster Deterministic Chaos (3rd revised ed) (New York . Wiley) (1995) 
[6] Hao Bai-Lin Chaos Vol I (Singapore : World Sci.) (1984) 
[7] T-Y Li and J A Yorke Amer. Math. Monthly 82 985 (1975) 
[8] R C Hilborn Chaos and Nonlinear Dynamics (First edition India * Oxford Univ Press) (2004) 
[9] D Stauffer Int. J. Mod. Phys. CB 1141 (1997) 
[10] T Dutta, N I Lebovka and S Tarafdar Phys. Rev. E70 061609 (2004) 
[11] T Dutta, N I Lebovka and S Tarafdar Phys. Rev. E66 046104 (2002) 
[12] Y Hayase, O Descalzi and H R Brand Physica A356 19 (2005) 
[13] Yu M Szirezhav and D O Logofet Stability of Biological Communities (Moscow . Mir Publishers) 
(1983) 


English

Chaotic Behaviour of Population on a Square Lattice

Banerjee, B

Roy, S K

Indian J Pfos 82 (3), 333-339 (2008) 
A plausible explanation of non-zero isotope shifts in 
superconductors 
B Banerjee and S K Roy* 
Department of Physics Visva-Bharati University Santiniketan-731 235 West Bengal India 
E-mail binayakbanerjee(" rediffmail com 
Abstract We propose to consider a charge Bose liquid model as a phenomenological explanation of three 
pjzzling features of non-zero isotope shifts ( a ) in superconductors These features are (i) large value of a 
0 5) in normal superconductors, (ii) an overall trend of lower values of a with increasing 7 and (in) a very 
striking feature of negative value of a (= - 0 013 in Bi 2 2 2 3 with 7 = 108K) [6] in high 7t superconductors This 
phpnomenon of non-zero isotope shifts observed in high Tc oxides are explained on the baiss of (bi) polaronic 
theory 
Keywords Isotope-shift exponent Polaron, nearest neighbour (n n) electron-phonon interaction (e~ph) 
PACS No. 7 20 + f 
1 Introduction 
A charge Bose liquid, which was proposed by Schafrots [1] as a phenomenological 
explanation of superconductivity and later derived microscopically from Frohlich [2] electron-
phonon (e-ph) interaction (bipolaronic superconductivity) is an intriguing possibility for high 
Tc metal oxides From the research on superconductivity of strong electron-phonon (e-ph) 
interacting systems, we know that, the strong-coupling between electrons located at lattice 
sites with phonons leading to the generation of polarons/ bipolarons [3]. 
We propose to explain the phenomenon on the basis of small polaronic theory and 
suggest several plausible explanations. As for the apex oxygen anharmonicity, it is due 
to the polaronic effects as demonstrated by some works [4]. We propose to make an 
attempt in this direction to explain the observation of non-zero and negative a on the 
basis of small polaronic cloud. It is argued that some of these striking variation in a do 
not necessarily imply a large phonon contribution to Tc and do not invalidate the electronic 
mechanism with a small phonon contribution Moreover, in BCS theory, there is no reason 
* Corresponding Author 
© 2 0 0 8 I A C S 
334 B Banerjee and S K R0y 
why a can not be negative, when the coulomb pseudo potential \i is comparable with 
the electron-phonon coupling constant 
2. The model 
In this work, we consider a single band Hubbard Model in the presence of a strong local 
(on-site) electron-phonon interaction Thus we start with the model Hamiltonian, 
H - Hhub + He ph + Hph (1) 
where Hhub is the one band Hubbard Hamiltonian and is given by 
"/,uft ^^crCja + U^n^nd -»^nlo 
I JO 
where t denotes the hopping energy of an electron between the lattice sites / and ; 
c*m (cl(r) represents the electron creation (annihilation) operator of spin a(- +1 for both 
up and down spins, respectively) at lattice site /, nirr + c*acia is the number operator U 
is the on-site Hubbard interaction term v is the chemical potential introduced due to 
doping 
He-ph ,s t h e o n " s l t e electron-phonon term, 
« ^ = 2 f t ( Q )K + ^ ) (3) 
Here gf(q) is the strength of the on-site electron-phonon interaction term and is 
expressed by, 
7,(9) = Y dV i)exp(-iqRA (4) 
Also H
 h is the free phonon term in eqn (1) 
Hph = S w ^ ^ (5) 
Now we will perform a canonical transformation exp(S) H exp (-S) with respect to the 
local e-ph interaction generated through the operator, 
q Vq 
Thus one eliminates the term He_p/? from the resulting Hamiltonian H for narrow band 
system Thus one obtains the Hamiltonian, having the site energies renormalised and 
A plausible explanation of non-zero isotope shifts in superconductors 335 
also repulsive on-site interaction reduced by the polaronic self-energy Thus, the model 
polaronic Hamiltonian for narrow band system is obtained as, 
W =
 -'p£C^CAT +UeY,n*n<l - " ] £ " ' n! " E ^ "' 
1,1 cr 
(7) 
This Hamiltonian is just like the Hamiltonian [5] with only an extra term of v , chemical 
potential where tp = t exp (-g2) is the polaronic hopping, t is the bare hopping and g 
denotes the electron-phonon (e-ph) coupling strength The effective on-site Coulomb repulsion 
•s Up given by, 
Ue = U-2Ep, 
where Ep -_ ^i£^!t (8) 
V in eq (7) is the inter-site attraction induced by the e-ph coupling 
>n strong-coupling narrow band systems, the Migdal theorem breaks down in the so-
called anti-adiabatic limit [7], / e, when <w ^ Wpt co being the characteristic phonon 
freuency and W is a renormalised half-bandwidth. 
For moderate values of g, Wp>V t the small polarons form spatially overlapping Copper 
pairs, the regime of BCS-type polaronic superconductivity For large values of gr, t is very 
small and for W « V, bipolarons may form 
3 Calculation of T 
c 
Now the second and third terms on the right hand side of the eq (7) is simplified by 
Hartree-Fock approximation with introduction of the order parameters 
4>=tei*,T> (9) 
Then the Hamiltonian, in the BCS form, may be written as, 
H = J ^ i?to + £4c K T c-a + hc) 
ko 
(11) 
where, £k =ek -fi, ek = -tp Zyk and 7/c = 1 / Z ^ e x p (/lr.f?/y); f?;/ being the nearest 
v 
neighbour lattice vector. For a square lattice yk =cos /cx + cos/cy and Z = 4 Then the 
336 B Banerjee and S K Roy 
superconducting gap parameter is given by, 
Ak-UeAQ-ZYkVA,. (12) 
Now using the Green's function technique, the superconducting correlation is obtained 
as for BCS superconductors 
(13) 
where, Ek = ^ 1 ^ a n d P ^ r T • 
Kill 
Combining eqs (12) and (13) one obtains at T= Tc , 
: A \ 
1-- ZViA ( A ) 
I A) Ue'o 
and 
where 
1 = ^ / 2 - 1 4 / , ( A ] A>J 
(14) 
(15) 
, „ - / • ; * . , w **,»[,><'.;'>,} (16) 
For, n = 0, 1, 2 
Where the subscript r denotes a reduced variable expressed in units of the polaronic 
half band width Wp = Wbexp(-g2), where Wb ( = zt) is the half-bandwidth of the bare 
electronic band. Here N(er) is the electron density of states. 
Combining the set of coupled eqs (14) and (15) we get for Tc, 
2 "\ 
1 = ZV /„ 
i + 4,/0 (17) 
Following Alexandrov [7], the eq. (17) may be written as, 
1 = ZVJ2 + 2ZVf*rJ, + (zvtf -Ue)-J0+ ZVUe (J0J2 - 4 ) 
where J„-Jefer J V ( e f > 1 ( € r - ^ . t a n / i ^ 6 ' ^ ) 
(18) 
(19) 
with n = 0, 1, 2. 
A plausible explanation of non-zero isotope shifts in superconductors 337 
It should be noted that in the weak coupling limit the values of the integrals J, and J2 
are much smaller than JQ. Also in the weak coupling limit (W » ZV) we assume, J1 = 
j = 0 and then J0 takes the form as 
^~ln(l.14/^VT^
 (20) 
then we get the approximate expression for Tc as, 
2W 
T?»™-<l.UWp^-rf.exp\ - 2 - p f | (21) 
4. Calculation of a 
The isotope-shift exponent (ex) is defined as, 
cx=--('d\nTc/d\nM). (22) 
Now, the factors which depend on the isotope mass (M) are as follows : 
g2 ocy[MtWp~ exp ( g2), (23) 
dWp 1 
dM ~2jM m 
It indicates that W decreases with increasing isotopic mass which has been supported 
by Guo-meng Zhao et al [8] and Bussmann-Holder et al [9]. 
Now to derive an expression for a we have to differentiate eq. (7) with respect to M. 
Thus, 
a-* 2 
( H / O / I , / 2 , 
1- 0 
21 1 / 2 - 2 / - / , + /*/, 
ft 'S2-2/S1 + /2St) (25) 
where,
 / =
 UA (26) 
1 + <Vt> 
and Sn = jder N(er)-^-sec h*l ft e ' " " ' (27) 
-1 ^ 
with n = 0, 1, 2. 
338 B Banerjee and S K Roy 
In the absence of effective coulomb repulsion {Uc = 0) the expression for a takes the 
form, 
, 2 ( 
a 
1 
( ZVftSz j (28) 
The expression for a corresponding to T*pprox eq. (21) is 
a 
approx _QC 2Wn 
( ZVnf-Uej (29) 
5. Results and discussion 
We have studied Tc and a within the framework of a polaronic model as a function of the 
electron-phonon interaction strength (p2) and doping (through chemical potential jnr) We 
also compared the results obtained from the approximated formulas (21) and (29). Naturally 
a suitable choice of parameters like V, Wb, \in Wp are essential. Our choice of parameters 
is guided by the references [8] and [10]. 
In Figure 1 we have compared the variation of Tc with p2 from the exact and approximate 
eqs, in the absence of on-site repulsion, for two values of /ur (0.6 and 0.4). Tc obtained 
from the exact calculations increases with increasing g2 i.e., with decreasing polaronic 
bandwidth, and saturates to a value of V/3, independent of j u r , for large values of cf. 
700 -i 
Figure 1. Variation of Tc with g2 for V = 2000 K, Wb = 12000 K and V9 = 0. 
For Curves A and C, iir = 0.6 and for Curves B and D, itr = 0.4. Solid curves are for exact value of f 
and dotted C curves are for approximate Tc. 
In Figure 2 the variation of exact values of a (obtained from eq. 28) with gr2 is shown 
for ^ r - ° - 6 a n d °-4 a i s negative for the entire range of g values and becomes zero as 
A plausible explanation of non-zero isotope shifts in superconductors 339 
j -> 0 as well as for large values of g where Tc saturates, /.a, dTc/dg2 ^ 0 . For 
intermediate values of g, the region of interest for polaronic superconductivity, a is negative 
a*d decreases (becoming more negative) with decreasing values of nr For BCS-type 
oolaronic superconductivity a is negative even for large g. 
oo 
-05 
- 1 0 
-1 5 
- 2 0 
-2 5 
-3 0 
- 3 5 
0 1 2 3 4 5 6 
Figure 2 Variation of exact a with tf for Wb = 12000K and V = 2000K Curves A and B are for / i r = 
f> and 0 4 Ut = 0 for all curves 
6. Conclusion 
Our study shows that a is negative for polaronic BCS-type superconductivity for the 
entire range of doping and become more negative with the introduction of Coulomb 
repulsion Positive values of a may be obtained within the polaronic theory for large 
values of g, where superconductivity is due to the Bose condensation of biopolarons 
References 
|1] MR Schafroth Phys Rev 100 463 (1955) 
[2] A S Alexandrov and J Ranninger Phys Rev B23 1796 (1981), ibid 24 1164 (1981) 
[3] A S Alexandrov and N F Mott Dep Prog Phys 57 1197 (1994) 
[4] J Mustre de Leon Phys Rev Lett 68 3236 (1992) 
[5] Xin-Zhong Van Phys Rev B73 052501 (2006) 
[6] A G Migdal Sow Phys JETP 7 996 (1958) 
[7] A S Alexandrov Phys Rev B46 14932 (1992) 
[8] Guo-meng Zhao and Donald E Morris Phys Rev 1351 16487 (1995) 
[9] Busmann-Holder et al, Eur Phys J B44 4 (2005) 
MO] G A Thomas et al, Phys Rev Lett 61 1313 (1988) 
V A 
- 7 — 1 1 » 1 ' 1 


A Plausible Explanation of Non-Zero Isotope Shifts in Superconductors

Baruah, Rulee

Duorah, Kalpana

Duorah, H L

Indian J. Phys. 82 (5), 523-529 (2008) 
The rapid neutron capture process in an explosive 
astrophysical environment 
Rulee Baruah1*, Kalpana Duorah2 and H L Duorah2 
1
 Department of Physics, HRH The Prince of Wales Institute of Engineering and Technology, 
Jorhat-785 001, Assam, India 
1
 Department of Physics, Gauhati University, Guwahati-781 014, Assam, India 
E-mail ruleebaruahC" yahoo co in 
Abstract The rapid neutron capture process (r-process) is one of the major nucleosynthesis processes 
responsible for the production of heavy elements beyond iron Recent models of r-process nucleosynthesis rely 
on a neutrino-heated bubble developing at late times, which provides both the necessary conditions and the 
requisite amount of ejected mass for the r-process Meyer et al [1] showed that the hot bubble that forms outside 
the protoneutron star during a supemovae (SN) explosion may be a viable site for the r-process as long as the 
entropy per baryon can be made sufficiently high But in a very neutron rich environment such as a neutron star, 
the r-process could occur even at low entropy The high entropy wind is not the correct r-process site, owing 
to the inherent deficiencies in the abundance pattern below A=110 as well as the problems in obtaining the high 
entropies in SN II explosions required for producing the massive r-process nuclei up to A ~ 195 and beyond We 
have tried to associate the explosion entropies with the site-independent classical approach (nn and T()) and 
thereby compare the results of the two approaches from the abundances at different entropy conditions We 
find that an entropy of ~ 300 with Y0 ~ 0 45 can lead to a successful r-process This is in agreement with the r-
process abundance peaks at nn ^  1032 cm3 and 7"9^15 
Keywords . Nucleosynthesis, entropy, abundances, supemovae 
PACS Nos. 97 10 Cv, 25 40 Lw 
1. Introduction 
The existence and distribution of chemical elements and their isotopes are a consequence 
of nuclear processes. To attempt to understand the sequence leading to the formation of 
elements, it is necessary to study the universal abundance curve. The first attempt to 
construct such an abundance curve was made by Goldschmidt in 1937. An improved 
curve was given by Brown in 1949 and then by Suess and Urey [2]. These curves are 
derived mainly from terrestrial, meteoritic and solar data and sometimes from mass 
Corresponding Author 
©2008IACS 
524 Rulee Baruah, Kalpana Duorah andHL Duorah 
spectroscopic data. All these tables along with the more recent one by Wapstra et al [3 
form the basic data for our analysis 
If we look at the abundance curve we see that H is the most abundant element 
followed by He; Li, Be, B are very rare. From C to O the abundance increases - then to 
Sc, there is a fairly regular downward trend which again increases up to Fe. Beyond Fe 
there is an almost uniform abundance of all other elements. Thus it is material to attempt 
to explain the interactions of nuclei of elements and their origin by a synthesis or build-
up process starting with one or other or both of the building blocks. The synthesis from h 
to Fe can be explained by charged particle interactions. But this does not lead to significant 
nucleosynthesis beyond the iron group as the coulomb barrier becomes very high. So we 
attempt to study and present here the nucleosynthesis process chiefly the r-process 
responsible for the production of heavy elements beyond iron. We choose Supernovae as 
the site for the r-process because the Supernovae light curve shows the presence of 
98Cf254 found in nature. 
Stars in the mass range 10-30 M 0 evolve to form iron cores of 1.3 to 1.6 M 0 . These 
iron cores collapse according to well known instabilities, photodisintegration and electron 
capture. As the central density exceeds ~ 1010 gm cm"3, electron capture on nucleons 
and nuclei becomes so fast that the collapse approaches free fall. The collapse is halted 
as the central density reaches about twice the nuclear -matter density. An outward bound 
shock wave then forms in the matter that is continuing to fall onto the nearly stationary 
core. After the shock wave has propagated out to several hundred kilometers, conditions 
behind the shock at 100 to 200 kms are suitable for neutrino heating. Matter that has 
fallen into 200 km has just internal energy to dissociate into free nucleons and so is 
relatively cool after being dissociated. This material can absorb the high energy neutrinos 
from the accreting core but does not efficiently reemit it since its temperature is low The 
neutrino heating blows a hot bubble above the proto-neutron star [4]. 
Meyer et al. [1] first calculated the r-process under conditions appropriate to a neutrino 
heated bubble and found that the solar r-process abundances could be replicated. They 
showed that the hot bubble that forms outside the protoneutron star during a SN explosion 
may be a viable site for the r-process as long as the entropy per baryon can be made 
sufficiently high. A high entropy per baryon means that few nucleons reassemble beyond 
He4. Those few a-particle that do reassemble into heavier nuclei, can capture many free 
neutrons, so an r-process can occur. 
2. Nuclear physics considerations in the R-process path 
Our method is aimed at comparing the net results of r-processing under the different 
astrophysical conditions of high and low entropies, but with necessarily the same nuclear 
data. For our purposes, such evolution occurs as the temperature falls from ~ 1010 to 10y 
K. Beginning at about 1010 K, nuclear statistical equilibrium favors the assemblage of 
nucleons into a-particles and heavy nuclei. As the temperature drops below about 5x109 
K, the reactions responsible for converting a -particles back into heavy nuclei begin fall 
The rapid neutron capture process in an explosive astrophysical environment 525 
out of equilibrium. By 3x109 K, charged particle reactions freeze out. Below this temperature, 
the r-process occurs until the temperature reaches ~ (1-2) x109 K, where the neutron 
reactions also cease as the neutrons are depleted [4]. From the considerations of neutron 
capture on neutron magic nucleus which is quite difficult, we find that the Qn value is 
most likely somewhere between 2 to 4 MeV. 
The condition for dynamical equilibrium between (n, y) and (y, n) reactions is expressed 
as 
0) X (A, Z) + n « X ( A + 1, Z ) +
 Y + Q„ ( A Z) 
where Qn ( A Z) is the neutron binding to nucleus X(A, Z). 
Writing n{ A, Z ) and nn for the number densities of nuclei {A, 2) and neutrons 
respectively, the statistical balance in this reaction is expressed by 
log (n(A + ^,Z)/n( A,Z)) = log nn- 34.07 - (3/2)\ogT9 + (5.04/T9)Q„ (2) 
Using the condition that in equilibrium n ( ^ + 1 f Z ) « n ( A f Z ) w e obtain Qn as 
Q„ = ( 175.04) (34.07 + (3/2) log 7, - log nn ). 
The temperature and density conditions considered here range from 
7 = 1.0 - 2.0 x 109 K and nn = 1020 - 1030ci7T3 
Variation of Qn values with respect to temperature and density 
(3) 
"n 
10?4 
102 4 
10^4 
102 4 
102 4 
10^4 
7-9 
1.0 
1.2 
1.4 
1.6 
1 8 
2 0 
On 
1 99 
2 40 
2 85 
3.29 
3.73 
4 17 
"n 
102 6 
102 6 
102 6 
102 6 
102 6 
102 6 
^9 
1 2 
1 4 
1 6 
1 8 
2.0 
2.2 
On 
1 95 
2.28 
2 66 
2.99 
3.38 
3.74 
"n 
10?fl 
102B 
102 8 
1 0 3 0 
1028 
1028 
T
» 
1 6 
1 8 
2 0 
2 4 
2 2 
24 
On 
2.02 
2 29 
2 58 
2 2 
2 87 
4 11 
"n 
1 0 3 0 
10' * 
1 0 * 
103 0 
1 0 a i 
T
, 
1 8 
2 0 
2 2 
2 6 
2 8 
On 
1 58 
1 79 
2 0 
2 42 
2 63 
In terms of temperature T9 in units of 109 K and density p5 in units of 105 gm cm-3 
we introduce the entropy at this stage as 
S = 3.34 x Ti/p5 . 
We consider the determination of Qn ( A. Z ) on the basis of smooth Weizsacker 
atomic mass formula neglecting shell, pairing and quadruple deformation effects : 
M{A,Z) = (A-Z)Mn+ZMh-(Mc2)[aA-p(A-2Z)2!A-yA2'3-eZ(Z-1)JAV3] (A) 
526 Ftulee Baruah, Kalpana Duorah and H L Duorah 
Mn and Mh are masses of neutron and proton, a, /J, y and e are constants m 
energy units which represent volume, isotopic, surface and coulomb energy parameters 
respectively. 
With this we calculate the neutron binding energy as 
Qn ( A Z ) = 6 n ( M 1 , Z ) = c 2 ( M ( A Z ) + Mn - /W(/ l + 1, Z ) ) (5, 
Based on the treatment of Burbidge et al. [5] we modify the above expression fo' 
M ( A, Z ) as 
M(AtZ) = Mw(A,Z)- (1/c2) ((f(N) + g(N)) (6) 
MW(A, Z) represents the Weizsacker expression given by equation (4). 
We now obtain 
Qn = f (AZ) + f'{A-Z), using 
f\A-Z) = f'{N) = df(N)/dN = f (A + \ Z) - f (AZ) 
On simplification and on putting Z = A - N we rewrite this equation as 
Qn (AN) = f (A N) + ff(Nl 
f'(N) is the excess neutron binding energy to nuclei with a specified N over that given 
by the smooth Weizsacker mass formula normalized to zero at the beginning of the 
shell in which N lies. Using mass tables of Wapstra et al. [3] for various /V, the resulting 
average values have been plotted against A/. Another form of averaging is then affected by 
300 . 
Proton number (Z) 
Figure 1. Average r-process path used in the calculation. 
The rapid neutron capture process in an explosive astrophysical environment 527 
drawing a smooth curve through the points obtained in this way. By substituting Q for 
different values of T9 and nn and reading f'(N) values from N and f'(N) plot, we get 
equations relating to A and N. 
3. Abundance calculation 
The charge of the most stable isobar ZA is given by 
ZA - (4/2) [1 + (e/4fi)A2,*]\ (7) 
Then we determine a factor BA using expression 
BA=(0.58A2/3-078)/[(A/2)-ZA]% for * < 1 5 0 
BA = (80M)[1+ (c /4fi)A2/3] for A > 150. (8) 
The fermi energy term given by Fermi [6] is 
SA = 6 7 / ^ . (9) 
The effective /T decay energy ^ expressed in MeV is then determined as 
Wfi =BA(ZA-Z-2.5) + 0.S \+0oddA 
- SA 2 even Z, even A . 
+ < V 2 o d d z> ev^n A (10) 
Then the mean lifetime of a nucleus at the waiting point is calculated as 
V=io4/%5- (H) 
under the assumption that in a steady state the abundances of elements in the neutron 
capture path are proportional to (j decay lifetimes. The abundance of an element is 
then given by 
n(A) = const (rp. /AAJ (12) 
where AA is the number of neutrons added to a nucleus. 
The computed abundances are then calculated with observed universal abundances 
taken from Suess and Urey [2] for different values of Qn, and is shown in Figure 2. 
528 Rulee Baruah, Kalpana Duorah and H L Duorah 
t 
[Vcsamed • Otecrved] 
V t * 1 9 9 M f \ (T ,= l.« • l « l » 2 4 l 
I normalized \u \ = 12"> 
55 M 75 12 « 1QS 11) 121 12t 140 147 1% 16$ 174 IIS IK 24* 
A • 
>~C*ailaipfl • Observed1 
Of = 4' i3\ l t \ i l , = 3li B | = l ^ 
tiornili/rtl to \ - 123» 
• • • » 
• . ' • • * • M 
M t) 7f 77 M 93 W ) 112 120 121 134 14* Itt 1M 1411* 11* 191202 
Q ^ S U A (T, = 15.1,-1., 
mormjli/ed to \ = Wm 
-fT-
•.:*:.• - \ •..y:::v.••••'• 
M (7 74 It 90 101 110 119 127 IX 14S IS) 111 170 17» lift <r I 
\» CaKulatefl y Observes | 
0 ^ 2 3! \ h \ i k ^
 BrMi« 
tnnrmjih/fd t<> \ = 12* 
o 
' •»* 
SS 6? 7«, 81 91 10<T 111 120 121 1J6 t*fc 1S4 1 « 171 '80 M ^ 
A • A-
Figure 2. Abundances of nuclei produced in the rapid neutron capture process 
4. Results and conclusion 
We have studied the r-process isotopic abundances at various temperatures ranging from 
1.0 to 2.2 x 109 K and neutron number densities ranging from 1020 to 1032 cm*3. At all 
these ranges we get appreciable abundance curves comparable to the observed ones 
We mostly concentrate on our analysis at energies greater than 2 Mev as this is the 
condition prevailing in the Supernovae envelopes and neutron capture occurs during the 
later expanding stage. We start our calculation at A - 40 which is the possible product 
of the p-process. Most of our abundance curves were found to be significant by a 
The rapid neutron capture process in an explosive astrophysical environment 529 
normalization at A=126. We tried to analyze the abundance peaks at all the conditions 
considered. The calculated peaks were found to correspond the observed ones in most of 
the curves. 
We see that at extreme densities during core collapse we get elements which are not 
seen in observation. At later expansion/cooling and low density state the elements are 
produced in the r-process chain which is also observed experimentally. We propose that, 
during iron cores' collapse according to well known instabilities, the elements that are 
formed instantly undergo photodisintegration and electron capture. Only in the later stage, 
they undergo neutron capture and correspondingly beta decay to produce the observable 
elements. For example at nn=1024and 7g = 2.0, the chain shows all the isotopes that are 
observed by Wapstra et al. [3]. We conclude that this is one process by which all the 
heavy elements have been produced. These elements were created during SN explosion 
and in the later expansion stages they were distributed all over the universe. 
References 
[1] B S Meyer, G J Mathews, W M Howard, S E Woosley and R D Hoffman Astrophys J 399 656 
(1992) 
\2\ H E Suess and H C Urey Rev Mod Phys 28 53 (1956) 
[3] A H Wapstra and G Audi Nucl Phys A 729 337 (2003) 
[4] S E Woosley, J R Wilson, G J Mathews, R D Hoffman and B S Meyer Astrophys J 433 229 (1994) 
|5| E M Burbidge. G R Burbidge, W A Fowler and F Hoyle Rev Mod Phys 29 547 (1957) 
[6] E Fermi Nucl Phys (Chicago University of Chicago) p142 (1950) 
[7] I N Borzov and S Goriely Phys Rev C 62 03550 (2000) 
[8] A Chetia and H L Duorah // Nuovo Cimento 94B 93 (1986) 
[9) Y Z Qian, P Voge! and G J Wasserburg Astrophys J 494 285 (1998) 
(10J M Tamamura, S Wanajo, N Itoh, K Nomoto and S Nozawa Nucl Phys A 718 638c (2003) 


The Rapid Neutron Capture Process in an Explosive Astrophysical Environment

Krori, K D

Dutta, Sabita

Das, Kanika

Mahanta, Chandra Rekha

Indian J Phvs 82 (5), 531 -537 (2008) 
Raychaudhuri equation, big bang and accelerating universe* 
K D Kron1*, Sabita Dutta2, Kanika Das* and Chandra Rekha Mahanta4 
•Theoretical Physics Forum, Lachitnagar Bye lane 7 Sub lane 2 Guwahati 781 007 Assam India 
^Department of Physics Shnmanta Sankar Academy Panbazar Guwahati 781 001 Assam India 
^Department of Mathematics Gauhati University Guwahati 781 014 Assam India 
Mathematical Physics Forum Cotton College Guwahati 781 001 Assam India 
E mail kdkron(« rediffmail com 
Abstract In this paper, we show that with appropriate inputs the Raychaudhuri equation can provide 
physically rational clues to certain cosmological phenomena 
Keywords Raychaudhuri equation, big bang acceleration 
PACSNo 98 80 Hw 
1. Introduction 
Relativistic cosmology has been plagued by the problem of singularity (big bang) 
corresponding to scale factor tending to zero at a finite time in the past Tolman and 
Eddmgton had held the view that the singularity might have resulted from the simplifying 
assumptions of homogeneity and isotropy Later investigations have, however, shown that 
this view is not valid It has, rather, been observed that a singularity in classical cosmology 
is inevitable 
A theoretical confirmation of the inevitability of a singularity has been provided by the 
Raychaudhuri equation [1, 2] 
v£ + 2(o2 -2cr2 - ~ 0 2 -0ava -4TTG(P + 3p) = 0 (1) 
o 
where, 
v" = unit velocity vector 
* Corresponding Author 
f
 Dedicated in respectful memory of Professor A K Raychaudhuri 
© 2008 IACS 
532 K D Krori, Sabita Dutta, Kanika Das and Chandra Rekha Mahanta 
^ = acceleration (departure of v*1 from geodesicity). It arises from the 
pressure gradient in the case of a perfect fluid. 
0 = expansion 
o)2 = vorticity 
a
2
 = shear 
p = matter (energy) density 
p = fluid pressure 
The last term in eq. (1) represents the gravity effect and is abetted by shear. 
With ouy* = dQ/ds, where s parametrises a geodesic, eq. (1) can be written in the 
form 
d0
 + 1 ()2 = v^ + 2u)2 - 2o2 - 4TIG (p + 3p).
 (2} 
For a general metric, defining a scale function, ft, such that 0 = 3 ( f t / f t ) , Raychaudhun 
et al [3] have written eq. (2) in the alternative form 
ft 1 .,, 2
 2 2 2 AnG t 0 x 
In the absence of v^ and to2, from eqs. (2) and (3) 
respectively we may write 
f
 + V < 0 ,4, 
ds 3 
and 
ft< 0. (5) 
Wald [4] has shown from eq.(4) and Raychaudhuri et al from eq. (5) that a collapse to 
a singularity is inevitable. 
Therefore, as is well-known, on the basis of the Raychaudhun equation, a number of 
singularity theorems have been derived [5]. The purpose of this paper is, however, to 
uncover some new facets of this equation. We show here that with appropriate inputs, 
this equation may provide clues to certain important cosmological developments. 
Raychaudhun equation, big bang and accelerating universe 533 
2 (a) Time-dependent A 
It we accept the view that the universe was born with a big bang from a singular state, 
then we have to answer the question What triggered the big bang ? Guth [6] recently 
advanced a plausible answer Dark energy had done it ' Indeed it is estimated [7] that 
the energy of the universe is 65% dark energy, 30% dark matter and 5% known matter 
(visible and invisible) showing that the dark energy is the most dominant form of energy in 
the universe 
When Raychaudhun formulated his equation, dark energy was unknown and the cause 
of the big bang was an unsolved mystery It would therefore be reasonable, according to 
the above statistics, to incorporate dark energy into the equation But how to do it 9 
Turner [7] emphasizes the role of A , introduced by Einstein as a repulsive " cosmological 
constant", in the context of the present-day accelerating universe One may therefore 
regard \(t) as a candidate for dark energy It may be noted that Sussman et al [8] and 
Fukui [9] have also considered a time dependent A as representing dark energy (see 
also [19]) We can then incorporate dark energy into the Raychaudhun equation through 
the following enlarged law of gravitation [2, 10, 11] 
^ v " f l ^ f l + ff,,! \ = -BnGT„x (6) 
with a more general conservation equation 
(BnGT^ +gr'" A(t)) - 0 (7) 
Using eq (6), the Raychaudhun equation (3) is generalized into the form 
= vMu+ to - cr - (p + 3p)+ 4(f) (8) 
R 3 * 3 3 3 v ; 3 v ; 
Obviously A(t) opposes the gravity effect due to 4/rG(p + 3p) in eq (8) It may be 
mentioned in passing that this interpretation of \(t) as representing dark energy 
distinguishes it from A introduced with an opposite sign by some other authors [3] and 
[12] in the enlarged law of gravitation 
Now, if -A(t) is so large at f? -> 0 that it makes the right-hand side of eq (8) equal 
o 
lR Ft - » 0 , a big bang results and the universe is born Further, so long as the right-
hand side remains positive, the universe continues to accelerate.Thus, A(t) causes both 
a big bang as well as the acceleration of the universe 
We may check the possibility of a big bang in the simplest cosmological theory, viz, 
m Friedmann cosmology (with SnG = 1) The equations to be considered are 
534 K D Krori, Sabita Dutta, Kanika Das and Chandra Rekha Mahanta 
(i) Raychaudhuri equation (from (8)) 
Conservation equation (from (7)) 
(9) 
P + 3—(f> + p) = -A. 
The solution is 
R = R0tn,p = p0/t2, A = AQ/t2 
(10) 
(11/ 
where 
and 
A> =
 4 P o O + r)2-Po 
Y comes from the assumed relation 
P = YP. 
:'12) 
(13) 
Assuming a sufficiently large value for p0, AQ becomes larger still (see(12)). Then, we 
find that 
R 
R 
Po0+y) 
2 
( P b ( i + r ) _ / | 
2 
1 (14) 
is positive and remains positive even at f -> 0 resulting in a big bang. 
2. (b) Bulk Viscosity: 
The bulk viscosity of a kind of viscous cosmic fluid (dark energy ?) is considered by 
some authors ([13], [14] and other references therein) to be responsible for the observed 
acceleration of the universe. It modifies pressure, p, into a pressure, p*, given by 
p* = p + n (15) 
where n represents the viscous process. According to Isreal and Stewart ([15], [16]), n 
is given by the equation 
ft + T 71 = -3C X 
R 2 { R r S T ) (16) 
Ftaychaudhuri equation, big bang and accelerating universe 535 
where T is the temperature and r is the relaxation time If we neglect the second term 
on the right-hand side of eq. (16) and also r, we obtain the Eckart result [17] 
* = -
3
^ (17) 
leading to 
p* = p - 3 £ - = p-C6 (18) 
where 0 is the expansion. The viscosity coefficient, £, must be positive so that the 
entropy increases according to the second law of thermodynamics 
Now, substituting p* for p in the Raychaudhun equation (3) 
R 1 „ 2
 2 2 2 AnG 
— = - V* + - CD - - CT* 
R 3 '" 3 3 3 
0 + 3(p-f0)] 
/e 
fl 1 „ 2
 2 2 2 4TTG, 0 . A _ _ 
~R^3^3(° 3 a 2 - - ^ - ( P + 3p) + 4/rG£0 (19) 
Obviously, the bulk viscosity term in eq.(19) opposes the gravity effect due to 
(4/rG/3) (p + 3p). Now, if the viscosity term is so large that it makes the right-hand side 
in eq (19) positive, then it may cause the acceleration of the universe as envisaged by 
some authors referred to above 
2. (c) Negative Pressure: 
Let us replace p by - p in eq. (3). Then, we have 
R 1
 u 2 2 2 2 4TTG 
R 3 '" 3 3 3 
i e , 
" 1 •* 2
 2 2 2 47rG . -
— = - v* + - o r - - a* p + 4;rGp (20) 
R 3 '" 3 3 3 
In eq.(20), the pressure term opposes the gravity effect due to (4JIG/3)P . Now, if the 
pressure term is large enough to make the right-hand side in eq. (20) positive, then it 
may cause rapid expansion of the universe. That negative pressure causes rapid expansion 
has been much discussed in the literature. 
536 K D Kron, Sabita Dutta, Kamka Das and Chandra Rekha Mahanta 
2. (d) Scalar Potential: 
Banerjee and Das [18] have recently considered the universe as matter-dominated as a' 
present, and as embedded in a scalar field, 0, with a scalar potential, 1/(0) They have 
shown that the acceleration of the universe may be caused by a simple tngonometnca 
potential Now, it can be shown from their field equations that 
£
 = -i^[p + 2 0 2 - 2V«t»] (21 
Taking note of eq (21), the Raychaudhun equation (3) can be written in the form 
R 1
 u 2 2 2 2 47rGr _ i 2 nt, ,^,1 
le 
R 1
 u 2 2 2 2 4nGr rt,2"l 8wGl//A. 
Rs3< + 3^-3^-—lp+2^r—V{$) (22' 
Eq (22) shows that if 1/(0) is sufficiently large so as to make the right-hand side positive 
the acceleration of the universe should occur 
Thus, from the above four examples, it is evident that with appropriate inputs, the 
Raychaudhun equation indeed reveals interesting cosmological possibilities 
The authors express their profound gratitude to the Government of Assam for providing 
all facilities at Cotton College, Guwahati-781001, to work out this paper 
References 
[1] A K Raychaudhun Phys Rev 98 1123 (1955) 
[2] A K Raychaudhun Theoretical Cosmology (Oxford Clarendon Press) p80 (1979) 
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[4] R M Wald General Relativity (Chicago The University of Chicago Press) p230 (1984) 
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[6) A H Guth Cosmic Questions (New York The New York Academy of Sciences) (ed) J B Millc 
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[10] R C Tolman Relativity Thermodynamics and Cosmology (Oxford Clarendon Press) p189 (1934) 
[11] P A M Dirac General Theory of Relativity (New York John Wiley & Sons) p68 (1975) 
[12] S Weinberg Gravitation and Cosmology (New York John Wiley & Sons) p155 (1972) 
ptaychaudhuri equation, big bang and accelerating universe 537 
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[18] N Banerjee and S Das Gen. Rel. Grav. 37 1695 (2005) 
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Raychaudhuri Equation, Big Bang and Accelerating Universe

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