IACS Institutional Repository

Nayak, P K

Ravi, S

Study of Critical Current Density from ac Susceptibility Measurements in (La 1-xY x)2Ba2CaCu5Oz Superconductors

Changmai, R

Baruah, G D

Indian J Phys. 82 (6), 679-683 (2008) 
Cavity-modified laser induced fluorescence of a blended 
system of polymer film 
R Changmai1* and G D Baruah2 
'Department of Physics DHSK College, Dibrugath-786 001, Assam Indw 
department of Physics, Dibrugarh University, Dibrogarh-786 004 Assam Inciui 
E-mail ranjan_changmai(f/ redjffmail com 
Abstract Laser induced fluorescence spectrum in the range 5600-6900A of a block of neat film of polymer 
blend system of Poly (1,4-Phenylene Vinylene) (PPV) and p-Benzoqumone (PBQ) has been obtained with the 
help of a 500 mW A r (5145 A) laser Drastic fluorescence quenching has been observed when this block of 
polymer blend system with a thickness of 2 5 mm, breadth 15 mm and length 15mm, is placed in an external cavity 
consisting of two dielectric mirrors separated by a distance of 20mm We have also observed substantial 
narrowing down of fluorescence in the higher wavelength side of the spectrum Our measurement also allows 
us to shed light on the fundamental question of energy transfer and gain mechanism 
Keywords LIF, Quenching, Polymer film 
PACS Nos. 33 50, Dq, 33 50 Hv, 42 70 Jk 
1. Introduction 
A polymer is a very large molecule comprising hundreds of thousands of atoms formed 
by successive linking of one or two, occasionally more, types of small molecules into a 
chain or network structure. The repeating units are usually obtained from low molecular 
weight, simple compounds, referred to as monomers. The conversion process, monomer 
to polymer is known as "polymerization". The formation of polyethylene is a simple example 
of this process. Frequently the end groups of polymers are unknown. They may arise 
from impurities in the reaction mixture. In some cases, the end group of polymers can by 
controlled. However the properties of a polymer are governed entirely by the bulk of the 
polymer molecule rather than the end groups. Polymer and plastics possess distinctive 
features that have motivated their study as laser gain media, and high gain materials for 
solid state polymer lasers [1-10]. Particularly, conjugated polymers are believed to carry 
Corresponding Author 
©2008IACS 
680 R Changmat and G D Baruah 
great promise for application as gain material in organic solid-state lasers They cover the 
entire visible spectral range and offer the prospect of being electrically driven The first 
electrically driven organic laser has been realized using a crystalline material [11] but 
lasing from conjugated polymers has so far been achieved in an optically pumped device 
In the first optically pumped thin film polymer laser a planar microcavity was employed 
[3] In a planar microcavity a polymer layer is sandwiched between two highly reflecting 
mirrors The laser light then propagates perpendicular to the active layer, just as in a 
vertical-cavity surface emitting semiconduting laser Since then various other optically driven 
polymer lasers have been reported, using a variety of cavity configurations [12,13,14] It is 
worthwhile to note that even non-conjugated polymers those with one carbon atom with a 
double bond followed by an array of atoms with single bonds can behave as conjugated 
polymer after they were doped with suitable materials This is contrary to the common 
believe that for a polymer to be able to conduct electric current, it must consist alternately 
of single and double bonds between the carbon atoms Poly (1 4-phenylene vinylene) hac 
been the subject of great interest in recent years It has potential applications in light 
emitting diodes, light emitting electrochemical cell and plastic lasers [15,16] To improve 
its solubility and optical properties substituents are introduced to the main chain Tc 
study the luminescence properties in detail it is very important to explore the factors that 
can influence the luminescence In the present work we are concerned with the laser 
induced fluorescence of a neat polymer film consisting of a blended system of PPV and 
a luminescent organic molecule PBQ We report here that the fluorescence spectrum 
undergoes drastic changes in its intensity and narrows down considerably when placed in 
an external cavity 
2. Experimental 
In the present experiment we have used poly (p -phylenevinylene) (PPV) ad p-benzoquinone 
(PBQ) of reagent grade PBQ was blended into PPV matrix in the ratio of 20% by weight 
The material prepared is dissolved in an organic solvent acetone and kept in a cell for 48 
hours. Acetone evaporates and leaves behind a jellylike neat film of the blended mixture 
It is possible to prepare neat polymer films of different dimensions and in this experiment 
the film of dimension of thickness 2 5 mm, breadth 15 mm and length 15mm has been 
used. An argon ion laser (500mW all lines) is used for exciting the fluorescence of the 
sample and the exciting radiation is 514 nm The salient feature of the experiment is that 
the laser radiation is allowed to fall on the sample and the fluorescence is observed on a 
spectrograph in the same direction. The fluorescence is recorded on a commercially 
available film. A bright fluorescence is observed in the red-yellow sector of the spectrum 
In this work we also compared this fluorescence with that observed when the blended 
system of polymer is placed inside a cavity constructed between two dielectric mirrors 
The intensity of the fluorescence spectra is measured with the help of a computer, a 
scanner, a printer and a software designed specifically for the purpose 
Cavity-modified laser induced fluorescence of a blended system of polymer film 681 
3. Results and discussion 
Figure 1(A) shows the fluorescence spectrum of the PBQ/PPV system in the red-yellow 
sector ranging from 5500 to 7000 A. This fluorescence system undergoes dramatic change 
in shape when the sample is placed inside the cavity. As may be seen is Figure 1 (B) a 
discrete band appears at 6750 A and the entire spectrum narrows down to diffuse band 
within the wavelength range of 5800 A to 6000 A This indicates conclusively that the 
fluorescence system is affected by the cavity and the mechanism is analogues to gain 
narrowing process. It is also seen that when the PBQ and PPV ratio is changed (PBQ is 
50% by weight) fluorescence spectrum undergoes further modification in shape as shown 
in Figure 1 (C), when the system is placed in the cavity The discrete band at 6750 A 
Figure 1. Laser induced fluorescence spectra of the blended system of PPV/PBQ 
disappears. Figure 2 shows the nature of the measured intensity of the fluorescence 
spectra of the blended systems. It may be noted that PBQ is a well known organic 
material exhibiting fluorescence and emission in the visible region [17-18]. The present 
experimental observations is an example of modification of fluorescence in the presence 
of cavity. Wegmann et a/. [19] presented measurement which indicated laser emission 
from n conjugated polymer blend system when placed into an external cavity. From the 
present work it is difficult to say whether the modified spectra are due to stimulated 
emission or it is simply a narrowed down fluorescence spectrum. In either case, our 
682 R Changmai and G D Baruah 
observations allow us to throw light on the fundamental question of gam mechanism m 
this class of material. Energy transfer form PBQ to PPV and vice versa is imponant 
parameter to be investigated. In conclusion, we may indicate that blending an optically 
active material in a polymer matrix and placing the system in external resonator affects 
the fluorescence A system of this type may serve as an excellent material for polymei 
laser 
21811 
242 00 
252 
67 00" 50 38 
Figure 2. The fluorescence spectra of the blended system (A) Without cavity (B) With cavity PBQ r 
20% by weight and (C) PBQ is 50% by weight 
Acknowledgment 
The authors gratefully acknowledge AICTE, Govt, of India for their financial assistance 
through a research grant. No* 1-51/FD/EF (24)7 2005-2006 
References 
[1] F Hide, M Diza-Ganca, B Schwartz, M Andersson, Q Pei and A Heeger Science 273 1833 
(1996) 
[2] M Diza-Ganca, F Hide, B Schwartz, M D McGehee, M Andersson, O Pei and A Heeger Appl Phys 
Lett 70 3191 (1997) 
[3] N Tessler, G J Denton and R H Friend Nature (London) 382 695 (1996) 
[4] G J Denton, T Tessler, M J Stevens and R H Friend Adv Mater 9 547 (1997) 
[5] S Frolov, M Ozaki, W Gellerman and K Yoshmo Jpn J Appl Phys, Part 2 35 L1371 (1996) 
[6] S Frolov, W Gellerman, M Ozaki, K Yoshmo and Z V Vardeny Phys Rev Lett 78 729 (1997) 
[7] S V Frolov, M Shkunov, Z V Vardeny and K Yoshmo Phys Rev B 56 R4363 (1997) 
[8] H Broumer, V Krasnikov, A Hilberer and G Hadzuoannou >4oV Mat 8 935 (1996) 
[9] G Gelinek, J Warman, M Remners and D Neher Chem Phys Lett 265 320 (1997) 
Cavity-modified laser induced fluorescence of a blended system of polymer film 683 
[10] C Zenz, W Graupner, S Taseh, G Leismg and U Scherf Appl Phys Lett 71 2566 (1997) 
[11] J H Schon, Ch Klor, A Dodabalapur and B Batlog Science 289 599 (2000) 
[12] Y Kawabe, Ch Speigelberg, A Schiiltgen, M F Nobor B Kippelen, E A Mash, P M Allemand M 
Kuwata-Gonokami, K Takeda and N Peyghambanan Appl Phys Lett 72 141 (1998) 
[13] M D Mc Gehee, M-A Diaz-Garcia, F Hide, R Gupta, E K Miller, D Noses and A J Heeger Appl Phys 
Lett 72 1536 (1998) 
[14] C Zenz, W Graupner, S Taseh, G Leismg J Appl Phys 84 5445 (1998) 
[15] M Granstroem, K Petristch, A C Ans, A Lux MR Andersson and RH Friend Nature 395 257 (1998) 
[16] Sha-Yu Li, Fie Xiong, Hai-Rong Zhang, Xue-Fang Lu, Yi Li and Guo-Qiang Yang Ghin J Chem 22 
80 (2004) 
[17] M G Jayswal and R S Singh Indian J Pure Appl Phys 2 358 (1964) 
[18] M G Jayswal and R S Singh Spectochim Acta 27 1597 (1965) 
[19] G Wegmann, H Giesea, A Greiner and R F Mahrt Phys Rev B57 R 4218 (1998) 


English

Cavity-Modified Laser Induced Fluorescence of a Blended System of Polymer Film

Singh, K Kunjabali

Das, H L

Indian J. Pliys. 82 (6), 685-693 (2008) 
Microstructures, stress, strain and surface characterization 
of TD polycrystalline CdS thin films 
K Kunjabali Singh1* and H L Das2 
department of Physics, TM College. Oinam-795 134, Manipur, India 
''Department of Physics, Gauhati University, Guwahati-781 014, Assam, India 
E-mail kunjabahoti'yahoo com 
Abstract : We report tn this paper the effects of substrate temperature (Ts) and film thickness (T,) on the 
microstructures, stress, strain and the surface morphology in thermally deposited (TD) polycrystalline CdS thin 
films using XRD and SEM techniques The XRD and the SEM results reveal that the quality of polycrystalline 
nature of CdS films grown under identical film growth conditions were observed significantly improved with Ts 
and Tr The structural phases of the films were observed to possess two phase structures Films grown at Ts 
£ 423K were found f e e . cubic ZnS structure with (111), (220) and (311) planes, and hexagonal phase 
structure at Ts > 443K with (002) and (112) planes The evaluated lattice parameters in cubic and hexagonal 
phases were found with close agreements with their standard values. The films were observed to contain a 
large regions of localized micro-strains, the sizes of which were observed to decrease as Ts increases from 
373K to 473K in films of 2500A thickness, and Tf increases from 1500A to 4000 A at 473K (Ts) The films grown 
from 300K to 423K (Ts) were found to be unstressed, and fairly stressed at elevated Ts The stresses were 
remarkably increased with increasing Tt 
Keywords • CdS thin films, XRD, SEM, substrate temperature, film thickness, stress, strain 
PACS Nos. : 61.10 Nz, 68.37 Hk, 68.35 Gy. 
1. Introduction 
The physical properties of a thin solid film are characterized by its structure sensitive 
properties like crystalinity, structural phase, lattice constants, grain-size etc. which influence 
remarkably on the mechanical, optical and photo-electronic properties of the film. Attention 
is, therefore, given to structural analysis of the films, particularly the CdS thin fims by 
different deposition techniques [1-8] correlating electrical, optical and photo-electronic 
properties of the films. A vacuum evaporated thin film may be either amorphous or 
polycrystalline in nature. A polycrystalline thin film generally contains a large number of 
Corresponding Author 
©2008IACS 
686 K Kunjabali Singh and H L Das 
crystallites of various sizes oriented quite haphazardly. These are known as grains, the 
sizes of which are closely related to the transport properties of the films. The grain-sizes 
increase with Ts, particularly beyond the re-crystallization temperature of the film and Tf 
XRD technique was used for structural analysis. It was observed that CdS thin films 
possess f.c.c. cubic zinc blend structure below Ts < 423K with most preferential orientation 
along (111) plane at 20 = 26.45°. The values of sin201/sin202 corresponding to (111) 
and (220) planes could confirm the f.c.c. cubic structure of the film with orientation along 
(111) plane parallel to the substrate, while hexagonal ZnS (wurtzite) structure was obtained 
at Ts > 443K with strong reflection plane along (002) at 20 = 26.450 parallel to the 
substrate and c axis perpendicular to the substrate as confirmed by JCPDS - X-ray 
powder file data [9,10]. A two phase structures sphelerite and wurtzite structures were, 
therefore, expected between 423K and 443K. Similar results were also obtained by other 
workers [11-14]. The lattice parameters in both cubic and hexagonal structures were 
evaluated, and the systematic errors in measurements of 20 in XRD data corresponding 
to each plane of reflections in the films, were eliminated from Nelson - Riley (N-R) plots 
The internal stress and micro-strains in the films developed owing to the effects of Ts and 
Tf were also systematically evaluated. 
2. Experimental details 
CdS bulk powder (Koch Light Laboratory, UK with purity 99.999%) was used as the 
source material for the present work. The material has density of 4.8 x 103 Kg m3 and 
m.p.~ 1750°C under normal atmospheric pressure. Some amount of sample was taken 
Prior to the deposition of the films, the material was dried and then grinded into fine 
powder form. Tantalum (Ta) boat (m.p. ~ 2990°C and density ~ 16.6 x 103 Kgnr3) was 
prepared from Ta -sheet (M/S Hind High Vacuum Co. Pvt. Ltd., Bangalore) and was used 
as the heater for deposition of the sample. CdS thin films of thickness 2500A at different 
Ts in the range (300 - 473)K, and Tf in the range (15 - 40) x n where n =100A at Ts = 
473K were suitably deposited on chemically and ultrasonically cleaned glass substrates 
by means of HINDHIVAC (12A4) coating unit. The substrate temperatures of the films 
were measured with the help of a copper - constantant thermocouple coupled with a 
digital micro-voltmeter, and the thickness of the films were measured with the help of a 
suitably designed Multiple Beam Interferometer with an accuracy of ± 15A. The vacuum 
maintained was ~ 1.33 x 10"4 Pa. The rate of deposition of the films was controlled 
properly by controlling the heater current, and was maintained between (2.14 - 2.50) 
AS"1. The as grown CdS thin films had a dimension of 10 x 10 x 1.35 mm3. The structural 
characterizations of the films were determined by means of Phillips X* pert Pro - Automated 
Powder X-ray (model APD 1700) diffractometer with CuKa - radiations U = 1.572A). The 
X-ray tube was operated at 40KV - 20mA for all the present observations. The surface 
morphology of the films was studied with the help of JEOL JSM - 6360 Scanning Electron 
Microscope (SEM) computerized with EIZO FLEXSCAN T566. For SEM studies appropriate 
film sizes 7.0 x 7.0mm2 were suitably cut with the help of a fine diamond head cutter. 
Microstructures, stress, strain and surface characterization of TD etc 687 
3. Results and discussions 
3.1. Effect of substrate temperature on structural properties • 
The XRD patterns of CdS thin films of Tr 2500A grown at 7s, 300K, 423K and 473K are 
shown in Figures 1 (a, b and c ), which reveal the polycrystailine growth of the films The 
diffraction patterns clearly reveal peaks corresponding to cubic f.c.c. zinc blend structure 
at 20 = 26.50°C at Ts in the range (373 - 423)K with most preferential reflections along 
700 -, 
(a)Ts = 300K
 1 0 0o ~ 
il 
800 500 
f 400 
c 300 
200 
100 
10 
>• 600 4 
E 400 A 
*«**!*•# 
20 30 40 50 60 70 
20 
200 
(b)Ts - 423K 
*+' . *M<**V*¥* 
10 20 30 40 
20 
50 60 70 
2500 
2000 
~ 1500 
| 1000 
500 
0 
10 
^ ******* ' *"+"+. 
20 " 30 
(c)Ts = 473K 
40 
20 
50 60 70 
Figure 1. (a, b and c) XRD patterns of CdS thin trims, 2500A at different Ts 
(111) plane together with some other small peaks at (220) and (311) planes. The films 
grown at elevated Ts (473K) show sharp peak at 20 = 26.45° with most preferential 
reflection along (002) plane with other reflection (112) plane, which shows the hexagonal 
wurtzite structure [9,10]. It is observed that the peak heights increase with increasing Ts. 
The lattice parameters in cubic phase were evaluated using the relations 
aca/=AV"(rt2 + /c2-h/2)/2sin0 
dcai=acal/f(ri> + l<* + l*) 
and in hexagonal phase, using Vegerd's law 
ahex=^V2acal' and cca/=^~4 /3 a< 
0) 
(2) 
(3) 
688 K Kunjabali Singh and H L Das 
1/cf2 = 4/3 [h2 + hk + k2 ] / a 2 + / 2 / c 2 . (4) 
The systematic errors in measurements of 20 corresponding to each plane of reflections 
were eliminated with the help of N-R plots. 
3.2. Effect of substrate temperature on grain-sizes, average stress and strains: 
The grown CdS thin films consist of various grains of different sizes oriented haphazardly, 
which depend on Ts. The grain or crystallite sizes were calculated using the Scherrer 
formula [16] 
Dhkl = kXIP20cosd (5) 
where the value of shape factor k is taken as 0.94, /320, the width of the peak at half 
of the maximum peak intensity and 6 the Bragg angle. 
Figure 2 shows the variation of grain - size with Ts of the films. It is observed that 
grain-sizes increase with increasing Ts from 180A (Ts = 373K) to 229A (Ts = 423K) 
beyond which it increases rapidly to 305A (Ts = 473K) as also been observed in the 
SEM micrographs from Figure 3(a and b). 
3161 
296 
-< 276 
8 236 
i 216 
o 196 
176 
156 
373 
Substrate Temperature (Ts) (K) 
Figure 2. Grain Size vs. Ts plot. 
Figure 3 (a and b). SEM micrographs of CdS films, 2500A (t) at different Ts. 
Microstructures, stress, strain and surface characterization of TD etc 689 
The films were observed under various stresses on account of some thermal expansion 
co-efficient of the films and the substrates at different Ts, which depended on various 
factors like lattice parameters, grain-sizes, orientation of grains [17,18]. The stresses 
were calculated using the relation[19,20] 
S = E/2y(ao-a)/ao (6) 
where a0 and a are the lattice parameters of CdS bulk and thin film materials, E and y 
the Young's modulus and Poisson's ratio of the material of the films respectively Standard 
values of a0 [21], E and y [22] for CdS bulk material were used for calculation of stress 
The peak width f320 used to obtain the grain - size can be assumed to be an exclusive 
contribution for obtaining micro-strains. The micro-strains were calculated using the relation 
[23] 
e = 028 [Cot 0/4]. (7) 
111 3.362 
220 2.069 
311 1.756 
002 3.367 
112 1.761 
100 
43 
30 
100 
60 
Table 1. dhkl- values at different Ts of the CdS thin films 
hkl XPDS I/I0 f c.c cub ZnS structure at Ts 
d(A ) 300K 373K 
20 d (A) 20 d (A) 
423K 
21) d (A) 
h e p structure 
at Ts =• 473K 
21) d ( A) 
26 455 3 366 
52 450 1 746 
26 50 
43 85 
52 00 
3 361 
2 066 
1 760 
26 45 
47 53 
3 894 
1 789 
Table 2. Lattice parameters, micro-strains and av stress in CdS films at different Ts 
Ts 
(K) 
hkl JCPDS- standard 
(A) (A) 
a ca l 
(A) 
a0 from NR-plots micro-strains av stress 
L (x 104) (GPa) 
373 
423 
111 
200 
311 
111 
220 
311 
5.83 
5.83 
acub 
(A) 
5 830 
5.537 
5 790 
5.830 
5.843 
5 837 
5 83 
5 83 
83.51 
65 71 
h.c.p. -structure 
>m N-
( A) (A) (A) 
< W a o f r o m N ' R P,otche*-
( A) 
e (x 10*4) a v. Stress 
(GPa) 
002 6.745 4 769 
473 112 a = 4.136 A 4.689 3.316 
c = 6.713 A 
4.172 
7.787 
5.414 
3.394 48.94 
1 789 0.45 
690 K Kunjabali Singh and H L Das 
Table 3. d-values from XRD data of the films at different Tf grown at 473K (Ts). 
hkl XPDS 1500 A 3500A 4000 A 
20° d (A) 26° d (A) 20° d (A) 
002 3.367 26.55 3.864 
103 1.898 
112 1.761 
Table 4. Lattice parameters, micro-strains and av. stress in Tf dependent CdS films. 
Tf hkl JCPDS ahex. chex a0 from N-R plot micro-strain av. stress 
(A) (A) (A) (A) e (x 10"4) (GPa) 
1500 002 4.7515 7.728 .... 52.0 
002 a = 4.136 4.760 7.773 
3500 103 c = 6.713 4.233 6.911 4.190 44.91 0.71 
26.50 
48.10 
52.30 
3.887 
1.951 
1.293 
28.50 
48.55 
52.30 
3.887 
2.284 
1.293 
4000 
112 
002 
103 
112 
.
C .
3.618 
4.760 
4.326 
3.032 
 
 
4.951 
7.773 
6.852 
4.951 
4.235 38 38 1 292 
Table 5. Values of grain-sizes measured at different Ts and T, of the films 
Values 
ft/2 (degree) -+ 
20 (degree) -» 
D
 hkl (A) -+ 
300 
Ts (K), Tf 
373 
0.45 
26.46 
180 
= 2500A 
423 
0.355 
26.50 
229 
473 
0.264 
26 45 
305 
1500 
0.282 
26.50 
287.2 
T, (A), (Ts 
2500 
0.243 
26.49 
333.64 
= 473K) 
3500 
0.242 
26.50 
333.56 
4000 
0.207 
26.49 
390 
where p20 and e were evaluated from the corresponding XRD spectra. The crystal has 4 
atoms per unit cell with unit cell dimension a = 5.83A, 5.843A and 5.837A corresponding 
to (111), (220) and (311) planes respectively supporting the f.c.c. cubic zinc sulphide 
structure. Further, the sin201/sin202 value corresponding to the first two reflection planes 
(111) and (220) which yielded 0.75 confirmed the cubic structure [24] with reflection along 
(111) plane parallel to the substrate, while the crystal exhibits hexagonal wurtzite structure 
at Ts > 443K with strong orientation along (002) plane parallel to the substrate and c axis 
perpendicular to the substrate [9,10]. A two phase structures-spheralite and wurtzite are 
expected to exist between Ts, 423K - 443K. Similar results were also reported by other 
workers [11,12,13,14]. Table 1-shows a comparative study of observed d-values with JCPDS 
standard values in both structural phases in the reflection planes, and were in close 
Microstructures, stress, strain and surface characterization of TD etc 691 
agreements with standard values. The lattice parameters aca(-values and a0 from N-R plots 
as shown in Figures 4 (a and b) in the films are shown in Table 2. 
6.5 
6 
5.5 (a) 
5
 V
 ' Ts = 373K 
•< 
8 
45 
4 
3 5 Ts=473K 
3 ( b) 
25 
0 05 1 15 2 25 3 35 4 45 
1/2 (cos2tf/smfM cos2 O/o) 
Figure 4 (a and b). Nelson-Riley plots of the films at different Ts 
3.3. Effect of film thickness on micro-structural properties 
Figures 5 (a and b ) depict the XRD patterns of the films grown at 473K (Ts) with Tf 35n 
and 40n (with n assigned value as above) respectively. The diffraction patterns reveal the 
polycrystalline h.c.p. (wurtzite) structure with preferential reflections along (002), (103) and 
(112) planes which was improved with increasing Tr 
(b) t = 4000A 
1400 
1200-1 
1000 
800 
600 
4ooi y ^ V -100CH 
0 
10 20 30 40 50 60 70 10 20 30 40 50 60 70 
(a) t - 3500A 
3000 
2500 
2000 \ 
' 1500 
c 
200 
20 20 
Figure 5 (a and b). XRD patterns of CdS films at different thickness 
The lattice parameters evaluated with the help of eqs. (1), (2), (3) and (4) along with 
micro-strains and average stress in the films have been shown in Tables (3 and 4 ). The 
values of a0 determined from N-R plots of the films of Tf 35n and 40n (Figure not shown) 
as shown in Table 4, were observed in close agreement with their standard values. Table-
5 represents the values of grain-sizes at different Ts at (111) plane and T, at (002) plane 
of the films, and the variation with Tf at Ts at 473K has been shown in Figure 6.

Microstructures, Stress, Strain and Surface Characterization of Td Polycrystalline CdS Thin Films

Bordoloi, Namita Sharma

Choudhury, D K

Indian J Phys 82 (6), 779-787 (2008) 
P J P | 
Form factors and charge radii of light and heavy flavoured 
mesons in a QCD inspired quark model with two 
loop static potential 
Namita Sharma Bordoloi1 and D K Choudhun/* 
department of Physics Cotton College Guwahati 781 002 Assam India 
2
 Department of Physics Gauhati University Guwahati 781 014 Assam India 
Abstract We incorporate the effects of two loop static potential in the results of form factors and charge 
radii of both light and heavy flavoured pseudo scalar mesons in a QCD inspired quark model pursued by us For 
heavy light mesons we also report the results in the infinite heavy quark mass limit 
Keywords Two loop static potential form factors charge radii 
PACS Nos. 12 3 9 J h , 1 2 3 9 P n 1 2 3 9 X 
1. Introduction 
The study of meson wave functions and their phenomenology is an important topic in 
QCD [1] as one of the basic ingredients of the static and dynamical properties of a 
hadron is its wave function 
Meson wave functions are in principle calculable from simple QCD potentials within 
Schr&dinger equation approach if they are less singular than 1/r2 [2, 3] Furthermore, the 
application of QCD potentials to light quark systems is handicapped due to their ad-hoc 
relativistic modifications [4] Any QCD potential based approach must take into account 
the above limitations 
Some time back [5, 6], such a specific QCD Inspired quark model has been proposed 
by us The model uses the two body Schrodinger equation and obtains the first order 
perturbed wave function of the QCD potential using Dalgarno's method [7] In order to test 
the model wave function, we have calculated [5] the properties of light and heavy flavoured 
mesons such as form factors, charge radii, masses and decay constants both in finite 
Corresponding Author 
© 2008 IACS 
780 Namita Sharma Bordoloi and D K Choudhury 
and infinite heavy quark mass limit [6], as well as the Isgur-Wise function of heavy light 
mesons [8] 
Recent years have however seen unprecedented progress in the study of QCD potentials 
Explicit calculations have been performed to obtain two loop static QCD potentials pioneered 
by Peter [9] and Schr6der [10, 11]. 
In this paper, we reinvestigate the results of [5, 6] incorporating the two loop effects on 
the form factors and charge radii of mesons both in the finite and infinite heavy quark 
mass limit In section 2, we discuss the formalism, while in section 3, we summarise the 
results Section 4 contains the summary and conclusion 
2. Formalism 
2 1 The Model: 
The essential features of the model have been reported in reference [5] We summarise 
them for completeness With the spin independent Fermi-Breit Hamiltonian with three 
colour degree of freedom 
H = -(4asl3r) + br + c (1) 
and taking H1 =br + c (2) 
as perturbation, one obtains the relativised normalized wave function with confinement 
parameter b to be 
r 
Vreucon, ( ' ) = ( " ' I f ^ ) ^ ' { ^ ^ { r I ^ e* (3) 
with the normalization constant 
2 i 
N1 = 2 2 / ll22l\r(3-2e)-±nb4r(S-2e)
 + ±n
zb24r(7-2e)U (4) 
where M = mqmQ /{mq + mQ) (5) 
is the reduced mass, and ^ =(4/*as/3)~1 (6) 
and e = 1-(1-4as/3)~1 (7) 
2.2. Two Loop Static Potential: 
The static two loop potential is conveniently defined in V scheme as 
l/(r) = -Cpay(l/r2)/r (8) 
Form factors and charge radii of light and heavy flavoured mesons etc 781 
Here av represents the effective coupling constant which incorporates the entire radiative 
corrections into its definition and 
CF=(A£-1)/2A/C (9) 
Traditionally, the strength of the quark gluon interaction is characterized by the coupling 
constant cx-^(q2) defined in MS regulanzation scheme V scheme is a physically based 
alternative to this usual scheme. The relationship of avi^lr2\ to the conventional 
coupling constant in MS scheme at the two loop level has been reported by Peter [9] 
and later Schroder [11] has corrected the error in the gluon sector Incorporating these 
corrections, the relationship between av and « ^ at the NNLO level is 
av(yr2)^a^s^(aMS/4n)(a,^/30L)^(aMS/Anf 
(a2 + ft2)(L2 + ^ 2 /3 ) + (A+2ft )a I)L) + 0(a2rs)} (10) 
where L = 2ln(£rexpy) (11) 
and p0=ncA-lTFnf (12) 
and A = ~34 C5 - 4CFTFn, - 2£ CATFnF (13) 
are the first two terms of the beta function, fi being the scale parameter, a1 and a2 
being the one loop and two loop constants. The scale parameter are judiciously chosen 
in order to reduce large logarithmic corrections. There are several choices such as : 
i) The first choice fi = 1/r which eliminates r dependence in L altogether, 
ii) the second choice fi = exp(-yE - a1 /2/30)/r eliminates the one loop coefficient 
completely ; and the 
iii) and the third choice fi^exp(-yE)lr2 removes all terms involving the Euler 
constant YE from the coefficients 
2.3. Form Factor and Charge Radii: 
The elastic charge form factor for a charged system of point quarks has the form [12,13] 
eF((^)=^(el/Ql)lr\F{rfsm(Qlr)dr ( 14 ) 
/ - 1 
782 Namita Sharma Bordoloi and D K Choudhury 
where e, is the charge of the ith quark/antiquark and 
2 i Q=Em/0/Sm' (15) 
Q describes how the virtuality Q is shared between the quark antiquark pair of the 
meson. Then, using eq. (3) in eq. (14) with improved approximations in the series expansion 
of (Q/) and integrating over r, 
e F (
°
2 ) = i^ i : ? e 'x v 
v £ - 1 (l + # f / 4 ) " (2-2£)X,-l(2-2e)3C?fx/3 + -^(2-2£)50fxf r(2-2e) 
\ £ - 2 
^ ^ ( l
 + ^ 0f/4) ~ (4-2£)X,-|(4-2£)3QfX/3+J-(4-2£)504xf r(4-2f) 
\E-3 
614^^6(1 + ^ ° f / 4 ) " (6-2e)X|-l(6-2ej3Gf^ + ?l6(6-2e)5Gfxf r ( 6 - 2 f ) 
(16) 
where x = * ' * • « 120 l« 
5
 of 
* • " ' * 
(17) 
and A/1 is the normalization constant. 
In the lAscheme, a0 and e as defined in eqs. (6) and (7) should read as 
< = ( 4 A « V 3 ) - 1 
and 
ev = ( l - 4 a y / 3 ) ~ 1 . 
The average charge radii square for the mesons are obtained from the relation 
V)-+sr*V\ /tf»o 
(18) 
(19) 
(20) 
Using eq. (16) in eq. (20), we obtain the average charge radii square in 
V scheme for the mesons having quark masses m, and mJ and charges e, and eJ 
respectively, 
Form factors and charge radii of light and heavy flavoured mesons etc 783 
('2M2 u m, + e, m. 
| 2 4 2 
+ ±Sb2aZ«r(6~2t„)\{^^ 64 ^ v/ 2 4 128 I 
(2 -2r v ) r (2~2^) - -^^ 3 (4 -2^)r (4 -2^) + 
J . j.2*,2„V6 
64 i u
z
^ X ( 6 - 2 ^ ) r ( 6 - 2 ^ v ) 
(21) 
2.4. /nffn/te Heavy Guarfc Mass Limit: 
In the infinite heavy quark mass limit, mQ - * * and the reduced mass n reduces to m 
and consequently $ change to $" , 
^-=(4mQa, /3)~1 (22) 
These changes are incorporated in eqs. (16) and (21). 
3. Results 
3.1. Relationship between am and av : 
Taking nf = 4 and nf = 5 and the corresponding value of « ^ at the scale of the c-
and b- quark mass am(mc) = 0.39,nf = 4 and ^ W ^ = 022>nt = 5 . l14 l respectively, the 
value of av(Ur2) for the three choices of /2 have been evaluated earlier [15]. It shows 
that the two loop potential invariably scales up the effective coupling constant. 
3.2. Form factors: 
We have calculated form factors and charge radii of light and heavy flavoured mesons 
with two loop potential within a QCD Inspired quark model using the well known 
QCD factors, Nc = 3, CF = 4/3, TF = 1/2,C^ = 3 along with the numerical value p(3) = 1.202 
and Ye = 0.5772 .We have also improved our earlier calculation at leading order level 
[5]. 
784 Namita Sharma Bordoloi and D K Choudhury 
Table 1 presents the numerical values of form factors of pions with two loop effects 
and compares with experimental data. Table 2 presents the data comparison for kaons. 
The tables for both pions and kaons show that data seems to favour b = 0. 
Even though there is no experimental information available for form factors of mesons 
having charm and bottom quarks, we plot the results for form factors of D mesons in 
Figures 1 and for B mesons in Figures 2 for b = 0 and b = 0.01 GeV2 with 
am(mc) = 0.391n, =4 for D mesons and with a^(mb)^0.22,nf =5 for B mesons 
respectively (LO lines); with av = 0.65 for D mesons and with av = 0.26 for B mesons 
(NLO lines). These figures show that two loop effects (NLO) and those without such 
effects (LO) are widely different for b = 0.01 GeV2, but the change is insignificant for b = 
0 for both D and B mesons. Figures 1 and 2 also display the results of infinite heavy 
quark mass limit ( NLO(inf) and LO(inf) lines). Effect of such limit is sensitive to both O2 
and the confinement parameter b. 
Table 1. Form Factors of pions with o v = 0 . 6 5 . 
&(GeV/c)2 
0. 015 
0.027 
0.037 
0.050 
0.101 
0.125 
0.203 
0.253 
|F„f[14] 
0.944 ±0.007 
0.898± 0.008 
0.876± 0.011 
0.830± 0.010 
0.680± 0.017 
0.665± 0 023 
0.529±0.040 
0.336±0.073 
|FJ2(*>=0) 
0.934 
0.886 
0.848 
0.803 
0.658 
0.603 
0.466 
0.402 
|F„|2(b = 0.0lGe\/2) 
0.866 
0.792 
0.745 
0.699 
0.605 
0.583 
0.532 
0.502 
Table 2. Form Factors of Kaons with av = 0.65 . 
Q*(GeV/c)2 
0.0175 
0.0275 
0.0375 
0.0475 
0.0575 
0.0675 
0.0775 
0.0850 
0.0950 
|FK|2 US) 
0.965± .024 
0.973± .031 
0.918± .043 
0.877± .055 
0.880±0.071 
0.742±0 088 
0.720± 0.120 
0.730± 0.010 
0.800±0.140 
l^|2(fc=o) 
0.936 
0.903 
0.871 
0.841 
0.813 
0.786 
0.760 
0.742 
0.719 
|F„|2(fc = 0.0lGeV2) 
0.930 
0.899 
0.872 
0.850 
0.829 
0.811 
0.795 
0.784 
0.770 
Form factors and charge radii of light and heavy flavoured mesons etc 785 
4\ 
LN 
LOflM* 
U Oi4 0 1 l i 1 0 
tf(lHK).01 OeV 
Figure 1 FD(Q2) vs Q2 for D mesons 
ou i 3 o « iib i)t 1 u 
Figure 2 FB(Q2) vs Q2 for B mesons 
U V * 0 4 O • t f 
Q* (b-0.01 G^V1) 
3 3 Charge Radii 
We present the results obtained for the charge radii for pions and kaons in Table 3 and 
compare with the data as well as with the predictions of the light front quark model [16] 
using b = 0 and b = 0 01GeV2 Data favours 6 = 0 The mean square charge radii of the 
heavy pseudo scalar mesons have not been measured yet We present the results of 
our calculations of their charge radii square both for finite heavy quark masses as well in 
the infinite mass limit in Table 4 (with 6 = 0) and compare our results with those of 
Table 3. The Mean square charge radii of pions and kaons with av = 0 65 
This work, b=0 
This work, b=0 01GeV2 
Hwang [16] 
Burden et a/ [19] 
Expt 
^ 
0 370 
1 025 
0 443 
0 314 
0 439±0 008[17] 
K+ 
0 309 
0 440 
0 349 
0 240 
0 34±0 050[18] 
K° 
-0 064 
-0 092 
-0 0676 
-0 020 
-0 054± 0 026[20] 
786 Namita Sharma Bordoloi and D K Choudhury 
Hwang [16]. The results for the D mesons are in good agreement, but for B mesons, they 
differ significantly. 
Table 4. The mean square charge radii of D (otv = 0.65) & B mesons (av - 0.26). 
Meson 
& 
D° 
PS 
8* 
8° 
fl8 
* 
(r*)F,(b = 0) 
0.134 
-0.234 
0.126 
2.960 
-1.471 
•1.373 
2.050 
w. ,(b = 0.01Gel/2) 
0.182 
-0.363 
0.182 
3.371 
-1.680 
-1.681 
3.372 
('% [16] 
0.184 
-0.304 
0.124 
0.378 
-0.187 
-0.119 
0.043 
(r% (161 
0.248 
-0.496 
0.181 
0 496 
-0.248 
-0.181 
-
4. Conclusion 
We have calculated form factors and charge radii of light and heavy flavoured mesons 
with two loop potential within a QCD inspired quark model .We have improved our earlier 
calculation at the leading order level. The predictions are consistent with current experimental 
data as well as with those of other models except for charge radii of B mesons. 
The earlier analyses [5, 6] revealed two limitations of the QCD Inspired quark model: 
Firstly, in the earlier work, calculated results were in good agreement with other 
theoretical predictions and experimental data when the running coupling constant was 
frozen at around av = 0.6 to 0.65, in contrast to a^ = 0.39 for D mesons and aMS = 
0.22 for B mesons. Two loop calculations raise the value of the running coupling constant 
from a ^ = 0.39 for D mesons to <xv = 0.65, and justify setting a = 0.65 to obtain 
agreement with other theoretical work [16] . For B mesons however, two loop calculations 
raise a ^ = 0.22 to av = 0.26 which falls short of the desired value of 0.6. 
Secondly, whereas spectroscopic analysis places the value of the confinement 
parameter b to be equal to 0.183GeV2 , the earlier work shows that model predictions fit 
best with data for b = O.ln the present work too, the same feature remains. A larger value 
of b cannot be incorporated in the model. 
The model presumably needs reformulation so that significant confinement effects 
can be incorporated and also so as to be able to justify obtaining good results for B 
mesons with av = 0.6. 
References 
[1] S J Brodsky and G P Lepage in Perturbative QCD, (ed.) A H Mueller (Singapore : World Scientific) 
(1989) 
Form factors and charge radii of light and heavy flavoured mesons etc 787 
[2] A De Rujula et al, Phys. Rev. D12 147 (1975) 
[3] F E Close An Introduction to Quarks & Partons p374 (London, U K • Academic Press) (1979) 
[4] S Godfrey and N Isgur Phys. Rev. D32 189 (1985) 
[5] D K Choudhury and P Das Pramana 44 519 (1995) 
[6] D K Choudhury and P Das Pramana 46 349 (1996) 
[7] A Dalgarno in Quantum Theory I Elements, (New York : Academic Press) (1961) 
[8] D K Choudhury and N S Bordoloi IJMPA 15 3667 (2000) 
[9] M Peter Phys. Rev Lett. 78 602 (1997) 
[10] Y Schroder hep-ph/9812205 
[11] Y Schroder hep-ph/9909520 
[12] D P Stanley and D Robson Phys. Rev D21 3180 (1980), bid 26 223 (1982) 
[13] S Flugge in Practical Quantum. Mechanics (New York Springer Verlag) (1964) 
[14] D E Groom et al, Eur. Phys J C15 1 (2000) 
[15] D K Choudhury and N S Bordoloi Modem Physics Letters A17 1909 (2002) 
[16] C W Hwang, hep-ph/0112237 
[17] S R Amendolia et a/, Nucl. Phys. B277 168 (1986) 
[181 
[19: 
[20; 
S R Amendolia et al, Phys. Lett B178 435 (1986) 
C J Burden et al, Phys. Lett. B371 163 (1996) 
W R Molzon et al, Phys. Rev Lett. 41 1213 (1978)

Form Factors and Charge Radii of Light and Heavy Flavoured Mesons in a QCD Inspired Quark Model with Two Loop Static Potential

Das, Abhijeet

Saikia, A

Indian J. Rhys. 82 (6), 789-794 (2008) 
Quark contribution in the evolution equation 
of the scaling violation of f
 2 
Abhijeet Das1* and A Saikia-' 
depa r tmen t of Phi/sics Dibru College, Dibrugarh - 786 UO'A Assam Indui 
-Dispur College. Guwaruiti 7S1 006 Assam Inriw 
E mail abh])eet_dds(r/ yahoo com 
Abstract : In a number of analyses in the literature, relations connecting the gluon density with the scaling 
violation of the structure function are reported In all these approaches, the DGLAP equation for the gluon is 
considered after neglecting the quark term In this work, we calculate the contribution coming from the quark term 
and estimate the error that enters when the quark contribution is neglecte?d 
Keywords DGLAP evolution equations, low x, Taylor expansion 
PACS Nos. 12 38 Bx, 12 38 Aw, 12 38 -t 
1. Introduction 
In the literature, there are a number of approximate relations connecting the gluon density 
with the scaling violation of the structure function F2 [1-3] These relations are obtained 
by evaluating the convolution integral involving the gluon density in the DGLAP equation 
[4] after neglecting the convolution integral involving the singlet structure function i.e. the 
quarks. In the above methods the gluon convolution integral is evaluated by Taylor 
expanding the gluon distribution about a chosen point. 
In this work, we estimate the quark contribution in the DGLAP evolution equation for 
the singlet structure function using the framework of [5]. We then perform a numerical 
analysis in which we compare the analytically obtained value of the quark contribution 
using a test input distribution with the numerically obtained ones. We then determine the 
amount of error involved in neglecting the quark contribution for a number of standard non-
perturbative parametrisations available in the literature [7-12]. 
Corresponding Author 
© 2008 IACS 
790 Abhijeet Das and A Saikia 
In Section 2 we describe the formalism and obtain the analytical expression for the 
quark contribution. In Section 3 we perform the numerical analysis to compare the 
analytically and numerically obtained values of the quark contribution using a test 
distribution. In Section 4 we evaluate the amount of error involved in neglecting the quark 
contribution for various input distributions and in Section 5 we conclude with some 
discussions. 
2. Formalism 
The DGLAP integrodifferential equation for the singlet structure function is given by 
a^l).M)[{3 + 4h(1-x)}/j(^) 
•(1) 
We consider the following term in the RHS of eq. (1) as the quark contribution 
QC = [{3
 + 4ln(1-x)}F/(x,tf) + 2Ji^{(l + 2 2 )F 2 s ^ > Q 2 l -2F 2 s (x ,0 2 ) (2) 
Writing z - 1 - u, the integral in (2) becomes 
,.j*{w(1-»f}*(£.*)-2j>(**) 
0 v J 0 
= ' 1 - ' 2 - (3) 
We evaluate the ^ in (3) as follows. The function F2s(x/1- u% O2) is expanded about 
a point z = a where 0 < a < 1 - x such that \u-a\<R, R being the radius of 
convergence. The nth term in the expanded integrand will be 
Tn^u-a) n_, F"-
1
 1 + (1-u)2 
(n-1)! u (4) 
where F""1 = ^ - M 
du 
. We suppress the O2 dependence of F2! 
w=o 
Quark contribution in the evolution equation of the scaling violation ofF„ 791 
The series can be integrated term by term and the resulting series converge. Integrating 
Tn we get at low x 
1 - * n-1 
4i = }Tndu = ^ar 
O r=0 
M ) ^ 1 2(1-xf' 2(1-* )^ 
n- r + 1 n- r n - r - 1 
n-1 
-2> 
r=0 
1_ _ 2 2 
n - r + 1 n - r n - r - 1 (5) 
- _J"ft_L jr""1 where ar=^vF> 
The completed integrated series can be obtained by evaluating the terms Jv J2, J3 . 
The second Integral l2 in (3) can be trivially evaluated. The integral I thus being evaluated, 
the quark contribution becomes, 
AF*tf-*—a)+LF**) qC
~3'* M - « 12 
+ 1 ^ 45 1-a -a 
VI 
150 
tf5) 
1-a 
x 
1-a 
a 
-a 
+ H F 2 S ( 3 ) 
90 ^ 1-a 
-« 
29 
60480 
£?(6» 
1-a 
• a 
r«0) 
p-a 
-a + a + fl(a) (6) 
where a is a parameter and fl(a) is given by 
fl(a) = f-7--aW*> 32 1-a - a 
^120 3! 
F*4) 
1-a - a 
{ 11l__a2 
240 2! 
f 11 _ a 4 ^ 
8400 4! 
rf3)f-X -a I 
1-a 
F S ( 5 ) 
1-a •a 
29 a_ 
161280 5! 
5 "\ 
.5(6) 
1-a -a + •• (7) 
Let a0 be the value of a for which R(a) = 0. This gives 
. = * * ? < 1 > qc~-.2 1-a -CC + SQ 
792 Abhijeet Das and A Saikia 
The function F2S(1) will be sensitive to the variation of x only when x/1-a = a-ai) 
Hence for x —> 0, a -* SQ and we get 
qc
 3i-2 1-a a (8) 
where a satisfies the condition (from (7)), 
[32 -a 
7«(2) 
1 - a 
11 _ a2 \ 
240 2! 
-s(3) 
1-a 
3 A 1 a 
120 3! 
F 
r2 
<4)( X ) 
1-a 
A 4 > 1_1_ _ a 
8400 4! 
-S(5) 
1-a 
29 _ a ^ 
161280 5! 
-S(6) 
1-a 
+ ••• = 0 0) 
The value of a is determined by solving (9) with a paramethsation for F2S. 
3. Numerical analysis 
As a test distribution we take 
£ ( x ) - V ( 1 - A ) ' ( 1 + yx) 
which at low x 
F2s(x) = A ^ - (10) 
Inserting (10) in (9) and considering the first three terms of the series, we get the 
cubic 
( (A + 2)(A+3) A + 2 l
 3 (71 A + 2'i 2 
1 + - — - a a 
[23 11(A + 2)' 
T6 + 240~~ a -
(32 
7 11(A + 2) (A + 2)(A + 3) , | 
= 0 
32 240 120 
For the representative values A = - 0.5, - 0.3, and - 0.08, this gives 
(11) 
7 3 47
 2 241 51 _ 
- a a + a =0 
8 32 160 160 
(12) 
180 3 219
 2 3137 4019 _ 
5 4 60 300 (13) 
Quark contribution in the evolution equation of the scaling violation of F 793 
1218
 3 1007 2 3051 35347 a — a + a 
125 80 200 10000 (14) 
respectively 
The above cubics are solved numerically by Horner's method [6] to find the value of 
a Using the value of a the quark contribution from (8) is calculated The quark 
contribution is also calculated numerically using Simpson's rule from (2) The results are 
shown in Table 1 
Table 1. Comparison of analytical and numerically evaluated values of qc 
A 
- 0 5 
- 0 3 
- 0 08 
a 
0 27211738583 
0 27774612055 
0 28345901264 
Error in 
evaluating (/ 
-5 0 x 1010 
60 x 108 
4 0 x 107 
Value of qc 
(anal) x x* 
-1 562816773 
-1 0046321418 
-0 2898924899 
Value of qc 
(num) x x* 
-1 574607 
-1 026599 
-0 2975729 
The results clearly show that the analytically derived expression for the quark contribution 
is in accurate agreement to the numerically computed values 
4. Results 
The quark contribution (8) derived from the DGLAP evolution equation for singlet structure 
function (1) will be compared to the gluon contribution given by the following term in (1) 
gc^dz^+^-zf^tf (15) 
which will allow us to determine the amount of error that enters when the quark contribution 
is neglected. We consider a number of standard non-perturbative parametnsations available 
in the literature [7 -12] and calculate the percentage error that enters when the quark 
contribution is neglected The results, for four quark flavours are shown in Table 2 
Table 2. Estimation of error that enters when quark contribution is neglected for various inputs 
A a qc & gc Actual value Approx value error 
(x x*) (x x* ) (%) 
- 03 Aq = 0 411 0 277746 qc = - 1004632 1 96324619 2 37615 17 38 
- 0 3 Ag = 0 775 0 58778 gc = 3 066 
- 0 04 Aq = 0 92 0 284445 qc = 0 147086 7 74825 7 88357 1 72 
- 0 04 Ag = 2 07 0 706696 gc = 3 808488 
- 0 26 Aq = 0 222 0 29135 qc = - 1 018093 5 799305 6 0253218 3 75 
- 0 33Ag = 1 9 0 50518 gc = 3 171222 
Paramete-
nsations 
MRS(A) 
MRS(R) 
MRST(01) 
A 
Xq = 
Xg = 
Xq = 
Ag = 
Aq = 
Ag = 
794 Abhijeet Das and A Saikia 
In the above analysis we have observed that for inputs with positive values of A [8, 
10, 12], the values of q.c. are diminishingly small but for the inputs with negative values 
of A [7, 9, 11], the values of q.c. are considerable. Neglecting the quark contribution thus 
brings in error particularly when the parton distributions have an x* dependence with 
negative values of A. 
5. Conclusions and discussions 
We evaluated the convolution integral for the quark sector in the DGLAP equation (1) 
analytically using the framework of [5] and calculated the quark contribution q c defined 
in eq. (2). We found that the analytical result for q c is in excellent agreement with the 
numerically computed value of the same. The fact that the formalism allows us to obtain 
the point of expansion in terms of the parameter A characterising various parton distributions 
available in the literature, was taken advantage of to calculate the amount of error entering 
in the relations like that in Ref. [1 - 3] where the quark contribution is neglected compared 
to the gluon contribution. We analysed various standard non-perturbative parameterizations 
in vogue indicating that the error for inputs having positive values of A is diminishingly 
small while those having negative values of A, the error is considerable. 
References 
[1] K Prytz Phys. Lett. B311 286 (1993); Phys. Lett. B332 393 (1994) 
[2] K Bora and D K Choudhury Phys. Lett. B354 401 (1995) 
[3] M B Gay Ducati and V P B Goncalves Phys. Lett. B390 401 (1997) 
[4] Yu L Dokshitzer Sov. Phys. JETP 46 641 (1977); V N Gribov and L N Lipatov Sov. J. Nucl. Phys. 
28 822 (1978); G Altarelli and G Parisi Nucl. Phys. B126 298 (1077) 
[5] A Saikia Pramana - J. Phys. 52 257 (1999) 
[6] E A Volkov Numerical Methods (Moscow : Mir Publishers) (1982) 
[7] A D Martin, W J Stirling and R G Roberts Phys. Lett. B306 145 (1993) 
[8] A D Martin, W J Stirling and R G Roberts Phys. Rev. D50 6734 (1994) 
[9] A D Martin, W J Stirling and R G Roberts Phys. Lett. B387 419 (1996) 
[10] A D Martin, R G Roberts, W J Stirling and R S Thorne Eur. Phys. J. C4 463 (1998) 
[11) A D Martin, R G Roberts, W J Stirling and R S Thorne Eur. Phys. J. C23 73 (2002) 
[12] A D Martin, R G Roberts, W J Stirling and R S Thorne Eur. Phys. J. C28 455 (2003)

Quark Contribution in the Evolution Equation of the Scaling Violation of F2

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|7 | I Atsumi I Isihara M Koyama and M Matsu/awa Pins Rt\ A42 0WI ( I W ) 
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[ 19| W Futsth (private communication) (1988) 
|211 K B Mac Adam (to be published) (1991) 
[2^| 1 Roy PhD J In sis (University of ( alcutta India) (1992) 
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Study of Critical Current Density from ac Susceptibility Measurements in (La <SUB>1-x</SUB>Y<SUB> x</SUB>)<SUB>2</SUB>Ba<SUB>2</SUB>CaCu<SUB>5</SUB>O<SUB>z</SUB> Superconductors

Cavity-Modified Laser Induced Fluorescence of a Blended System of Polymer Film

Microstructures, Stress, Strain and Surface Characterization of Td Polycrystalline CdS Thin Films

Form Factors and Charge Radii of Light and Heavy Flavoured Mesons in a QCD Inspired Quark Model with Two Loop Static Potential

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Notes for Contributors

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