IACS Institutional Repository

Santra, S B

Chanu, S Ranjita

Indian J. Phys. 82 (5), 615-620 (2(X)8) 
A numerical model for studying glass dissolution in water 
S B Santra* and S Ranjita Chanu 
Department of Physics, Indian Institute of Technology Guwahati. 
Guwahati-781 039, Assam, India 
E mail santra(</ ntg ernet m 
Abstract Glass, a multicomponent vitreous system, can be considered as a random mixture of the constituent 
oxides Due to random local environment, the binding energy of silica in the solid is expected to be randomly 
distributed Dissolution of this random solid in water is modeled via a Monte Carlo (MC) algorithm Time evolution 
o\ such a system under normal conditions shows that the system becomes rough to porous with the formation 
of dry silica gel layer It is found that the chemical reaction of corrosion occurs at a constant speed in the steady 
state As an effect, a self-established potential gradient is established and the system evolves into a critical 
state 
Keywords Glass, random solid, invasion percolation, Monte C'ailo 
PACS Nos. 05 10.Ln, 05.65 +b, 64.60.Ht 
1. Introduction 
Since multicomponent silicate glasses are used as the disposal media for radioactive and 
toxic wastes [1], it is important to understand the dynamics of corrosion or dissolution of 
glass in an aqueous solution. Glass, a multicomponent vitreous system, can be considered 
as a random mixture of the constituent oxides with random binding energy of silica in the 
solid. Such a solid could be called as a random solid. A random solid is then a 
multicomponent vitreous system in which, due to random local chemical environment, the 
binding energies of the constituent molecules of the solid are randomly distributed. If 
such a solid is in contact with an aqueous solution, the solid element dissolves at a slow 
rate and forms a complex with the solution. The complex breaks almost instantaneously 
and the solid element in the solution is available for redeposition on the rough or irregular 
solid surface exposed to the solution. As the process continues, the solution penetrates 
into the solid and the solid becomes porous. A self-established potential gradient drives 
the solution molecules inside the solid with a constant drift velocity. The solution molecule 
Corresponding Author 
©2008IACS 
616 S B Santra and S Ranjtta Chanu 
forms cluster of different sizes inside the solid At the non-equilibrium steady state
 5 
power law distribution of the cluster size is found This is then a self-organized cntica 
state between a random solid and an aqueous solution driven by a self-established potentia 
gradient Below, we will present the model and discuss the results obtained 
2. The model 
In order to study the time evolution of a random solid in contact with an aqueous solution 
a numerical model of corrosion or dissolution is developed in 2 dimensions (2D) In th»<; 
model, it is assumed that the random binding energy is uniformly distributed between c 
and 1 The solid shown in Figure 1(a) is a dense structure of elements with randon 
binding energy r( The solid of width L is infinitely long in the y-direction The aqueojr 
solution, infinite in volume, is placed at the bottom of the semi infinite solid The white 
space in Figure 1 represents the solution The initial solid-solution interface is marked b/ 
a thick line Random solid element R dissolves slowly in the solution S, makes a compound 
RS and the compound breaks instantaneously into R and S The chemical reaction o* 
dissolution could be represented as 
R + S-+RS->R + S '1 
The solid element R in the solution is now available for redeposition on the interface 
Assuming diffusion of the solid element R is very fast in the solution, redeposition ^ 
made at a randomly chosen site on the externally accessible perimeter with unit probability 
Generally, dissolution is a slow process and redeposition is faster It is mimicked here by 
considering no further dissolution during redeposition The slowest possible dynamics o* 
the system then involves dissolution and redeposition processes together with reconstruction 
of the rough interface at the single particle level This is usually the case of qlass 
dissolution in aqueous solutions in the macroscopic scale [2] Different physical processes 
involved in a single Monte Carlo (MC) step are explained with the help of Figure 1 The 
process are (i) extraction of externally accessible perimeter of the solid, (H) dissolution of 
the site with minimum random number (minimum binding energy) on the perimeter with 
unit probability (in Figure 1(b), it is r3), (in) modification of the external perimeter, (ivi 
t t t 
Random Solid Random Solid Random Solid 
Hi r l 2 r l 3 r ,4 F l , rM r l 2 r n r u r^ r u rI2 r n f l 4 r^ 
it *7 r8 r9 rio r6 r7 r8 r9 ri0 r6 r7 r8 r9 r,0 
H r2 rt r4 r< r, r2 u r5 r, r2 r4 r, 
) • (0.0) * tO 0) 
x x=L 
Water Water Water 
Figure 1. A single MC step is represented here (a) The arrangement of random numbers rt represents a 
random solid, (b) r3 is identified as minimum random number on the solid-solution interface and it is 
dissolved, (c) the solid element is redeposited at a randomly chosen site r, of the modified solid surface 
A new random number rci is assigned with the redeposited site The process is then repeated 
A numerical model for studying glass dissolution in water 617 
redeposition of the solid element present in the solution on a randomly chosen site of the 
modified external perimeter (r, in Figure 1(c)), (v) assignment of a new random number ru 
to the redeposited site The whole process is then repeated and time is increased from t 
to f+1 Note that the total number of particles is conserved and the system evolves at 
equal solid to liquid and liquid to solid flux rate (one particle per time step) throughout the 
simulation 
In Figure 3, the system morphology is shown for a system of width L = 64 at time 
steps t = 211 The solution is represented by light gray scale and the random solid is 
represented by dark gray scale The black dots represent the perimeter sites It can be 
seen that the interface becomes rough and the solution has penetrated into the solid 
leaving behind a restructured solid like a dry silica gel as observed in glass alteration 
experiments [2] It should be noted here that in the study of self-stabilized etching of 
random solids by finite etching solution [3] (infinite solid and finite solution) final stable 
morphology obtained was fractal as it was observed by Balazs in the corrosion of thin 
metal films [4] 
f=2 1 1 
Figure 2. Morphology of the random solid system at f=211 Dark gray scale represents the random solid 
and the light gray scale represents the solution Black dots are the externally accessible perimeter sites in 
contact with the solution The solid line represents the lattice boundary at r=0 
3. Results and discussion 
Simulations have been performed on the square lattice of widths ranging from L = 64, 128 
and 256 Data are averaged over maximum 1000 samples It is seen from the system 
morphology that the solution molecules enter into the system The solution molecules 
618 S B Santra and S Ranjtta Chanu 
inside the solid form clusters and move collectively In order to characterize the motion o' 
the solution molecules inside the solid, the profile of solution molecules are calculated as 
a function of the penetration depth along y The solution profile is defined as the average 
number of solution molecules per row A/W(y) and plotted in Figure 3(a) against y the 
coordinate of rows It can be seen that the solution molecules move inside the solid as a 
Gaussian profile leaving behind a restructured solid To distinguish the restructured sold 
from the original random solid, the average energy per row < r(y) > has been calculates 
as a function of y In Figure 3(b) < r(y) > is plotted against y It can be seen that ? 
gradient in the average energy has developed spontaneously The region of higher potentia1 
corresponds to the restructured solid whereas lower potential region corresponds to the 
unexplored random solid with average energy 0 5 Note that the solution profile is just n 
front of the potential gradient It seems that the spontaneously developed potential gradient 
pushing the solution profile deeper and deeper into the solid with time 
8 | . 
£ 4 
2 
0 ' 
0 7 
(b) 
A 
¥0 6 
0 200 400 600 800 1000 
y 
Figure 3 (a) Water profile number of water molecules per row as a function of y the row index inside 
the solid (b) Plot of average energy <r(y)> versus y A self-established potential gradient is observed n 
the system 
The motion of solution profile can be studied measuring the average peak position <yN 
of the profile with time Peak positions of the solution profile is the distance of the peak 
from the initial interface at t = 0 It is measured as a function of time t for L = 64,128 and 
256 In Figure 4, <y > is plotted against scaled time tIL Different symbols correspond to 
the system sizes circle for L = 64, squares for L = 128, and triangles for L = 256 It 
should be noticed that the data for different width of the systems collapse onto a single 
curve when f is scaled as tIL The time scale of the system is thus proportional to the 
width L of the system The solid line represents a straight line with slope one It can be 
seen that the data in double logarithmic scale is parallel to the solid line Thus, <yp> the 
mean displacement is proportional to time t This means that the solution profile is drifted 
by the potential gradient into the solid with a constant velocity v = d < yp >/dt [5] 
Therefore in the t -»oo limit, the whole random solid will dissolve completely leaving a 
pure silica gel 
A numerical model for studying glass dissolution in water 619 
During the motion of solution molecules through the random solid, the solution molecules 
occupy nearest neighbour sites and form clusters The clustering of solution molecules 
occur on its own due the dynamics of the system without fine tuning of any parameter of 
Figure 4. Plot of average peak position of the solution profile <yp> inside the solid as a function scaled 
time tIL Different symbols corresponds to different system sizes circle for L=64, squares for /.=128, and 
triangles for L=256 The solid line represents a straight line with slope one It is found the profile is drifted 
with a constant velocity into the solid 
the system from outside The size of a cluster $s$ is defined as the number of solution 
molecules connected by nearest neighbour bonds The probability to have a cluster of 
$s$ number of solution molecules (size s) at time t for a given system size L is given by 
Ps(t)-ns(t)INtot(t) (2) 
where ns(t) is the number of clusters of size s out of Ntot(t) clusters at time t The 
probability distribution P$(t) of cluster size is plotted in Figure 5 for a system size 
L = 1024 Different symbols correspond to different time steps circle for f = 218, square 
9 
log2s 
Figure 5. Plot of cluster size distribution P8(t) against s Different symbols correspond to different time 
steps circle for t = 218, square for t = 219, and triangle for t = 220 The solid line represents a straight line 
with a slope - 2 A power law distribution of cluster size is observed 
620 S 6 Santra and S Ranjita Chanu 
for t = 219, and triangle for t = 220. The solid line represents a straight line with a slope 
-2. It can be seen that the cluster size has a power law distribution Ps{t)-ssr with an 
exponent rs * - 2 . It should be mentioned here that the exponent obtained here is close 
to that of the cluster size distribution in the case of percolation [6]. The system driven by 
a self-established potential gradient thus evolves into a critical state through the self 
clustering of solution molecules This demonstrates self-organized criticality [7] in the 
process of dissolution of a random solid in a solution driven by a self-established potential 
gradient. This is a new observation. 
4. Conclusion 
The process of glass dissolution in a solution is modeled considering the glass as a 
random solid The molecules in solution penetrate the solid at a constant speed A self 
established potential gradient develops in the system which drives the solution profile 
deep into the solid. During the drift of the solution profile it is found that there is self-
clustering of solution molecules The cluster size distribution is found to be a power law 
with an exponent approximately -2. The non-equilibrium steady state is then a self 
organized critical state of a random solid and an aqueous solution driven by a self-
established potential gradient 
Acknowledgment 
This work is financially supported by BRNS, DAE, India, grant no.2005/37/5/BRNS 
References 
[1] R H Doremus Glass Sciences (New York John Wiley & Sons) (1973), R K Her The Chemistry of 
Silicon (New York John Wiley & Sons) (1979), A Paul Chemistry of Glasses (London Chapman 
and Hall) (2nd ed) (1990) 
[2] M Nikoshita, M Harada, Y Sato and Y Hariguchi J Am Ceram Soc 74 783 (1991), 0 Deruelle, 0 
Spalla, Ph Barboux and J Lambard J Non-Cryst Solids 241 237 (2000) 
[3] B Sapoval, S B Santra and Ph Barboux Europhys Lett. 41 297 (1998), S B Santra, B Sapoval, Ph 
Barboux and F Devereux C R Acad Sci 326 129 (1998), S B Santra and B Sapoval Physica A266 
160 (1999) 
[4] L Balazs Phys Rev E54 1183 (1996) 
[5] S B Santra and W A Seitz Int J Mod Phys C11 1357 (2000) 
[6] K Christensen and N R Moloney Complexity and Cnticality (London Imperial College Press) (2005) 
[7] P Bak How Nature Works the Science of Self-Organized Criticality (New York Copernicus) 
(1996) 


English

A Numerical Model For Studying Glass Dissolution in Water

Debnath, B

Talukdar, R

Bhattacharjee, B

Julian J Phvs 82 (5), 633-638 (2008) 
Emission characteristics of intermediate mass fragments on 
the residual part of the projectile nucleus 
B Debnath, R Talukdar and B Bhattacharjee* 
Department of Physics Gauhati University 
Guwahati 781 014 Assam India 
E mail bb_22(" rediffmail com 
Abstract In the study of projectile multifragmentation a number of properties such as multiplicities energy 
of fragments etc of the emitted intermediate mass fragments (IMFs) are found to vary significantly with Zh 
where Zb is a measure of the mass of the fragmenting system In this work we report the variation of <N(MF> with 
/, for 950 MeV/A MKr interactions with different targets of photonuclear emulsion The maximum value of <Nim> 
has been found to vary systematically with the target mass Further from this study it has been observed that 
Z^ is linearly correlated with the number of emitted projectile protons (A/p) for the studied interactions 
Keywords Nucleus nucleus collision multifragmentation intermediate mass fragments 
PACS Nos 25 75 -q 25 70 Pq , 13 85 Hd 
1. Introduction 
In projectile multifragmentation process, a projectile spectator, on excitation, splits into 
several pieces of intermediate mass fragments (IMFs) which span the mass-range between 
alpha particle and fission fragments It is believed that studies on the decay of such 
excited nuclear systems may provide information about the nuclear collision dynamics 
The sum of all projectile fragments (PFs) with charge Z = 2, which is also known as 
bound charge Zb, gives the measure of the mass of the fragmenting system [1] Correlation 
between average number of IMFs and the mass of the fragmenting system is one of the 
most interesting aspects of studying projectile multifragmentation For a given collision 
system, the magnitude of Zb is independent of the beam energy and is also taken as a 
measure of the degree of centrahty of the collision [2,3] On the other hand, when the 
variation of <NIMF> is studied with Zb for a given projectile, in reactions with the lighter 
targets, the maximum value of the mean multiplicities of IMFs depend on the bombarding 
Corresponding Author 
©2008IACS 
634 B Debnath, R Talukdarand B Bhattacharjee 
energy if the later is less than a specific minimum value. As reported by ALADIN grour 
[4], specific minimum bombarding energies are about 400, 800 and 1000 MeV per nucieor 
for collisions of gold projectiles with aluminium, carbon and beryllium targets respectively 
From their report it is therefore readily evident that for the same projectile, as the target 
mass increases, the specific minimum value of the beam energy for attaining the maximum 
mean multiplicity of IMFs, when plotted against Zb> decreases. 
An advantage of studying high energy nucleus-nucleus collisions with nuclear emulsion 
is that the technique provides an opportunity to study nuclear reactions with variable 
targets (such as H, C/N/O and Ag/Br). As in most of the emulsion works, the bombarding 
energy is generally a few GeV/A or above, no such work has so far been reported foi 
which the incident beam energy is found to be below the critical minimum for the studied 
interactions. As a result, no variation in the maximum value of <NIMF> could be seen m^> 
the variation of target mass for the studied emulsion reactions. 
As the incident beam energy for the emulsion pellicles of the present investigation i$, 
significantly low which is 950 MeV/A, in this work an attempt has therefore been made to 
study the variation of <NIMF> with Zb for ^Kr interactions with various targets of photonuclear 
emulsion to conclude whether the beam energy of the present investigation is above or 
below the specific minimum value for the studied interactions. 
Further, a number of workers have taken the number of projectile protons (AM as r. 
good measure of the impact parameter of a collision [5,6]. The correlation between <A//Mr-
and N is another important aspect of projectile multifragmentation that has been studied 
in details by different workers [6]. As both Zb and N are taken as a measure of degree 
of centrality of the collision, an attempt has also been made to find the correlation between 
these two parameters for interactions of same Kr projectile with various targets of nucleai 
emulsion. 
2. Experimentals 
The technique of nuclear emulsion allows for a complete detection of all PFs with charge 
Z > 1 and is the most suitable method to study all the possible projectile fragmentation 
channel. The capability of emulsion to record all the PFs irrespective of their charge and 
emission angle makes emulsion experiments superior or at least competitive to other 
experimental technique for the study of projectile multifragmentation [2,7]. For the present 
investigation, NIKFI BR-2 nuclear emulsion pellicles irradiated horizontally with 84Kr beam 
having energy 950 MeV/A at SIS synchrotron at GSI, Darmstadt were used. These emulsion 
plates were line scanned and various tracks were then classified according to the standard 
emulsion terminology on the basis of their ionization (I) in the emulsion [8]. The identification 
of PFs is unique due to their confinement in the extreme forward angle, 0PF = 0.2/pL 
radian, where pL is the beam momentum. The charge of different PFs were measured by 
estimating various track parameters such as blob density (B), gap density (H), the Gap 
Length Coefficient (G), 8 -rays density and track width [9]. For this study, only those 
non-interacting singly charged projectile fragments (protons, deuterons and tritons) are 
Emission characteristics of intermediate mass fragments etc 635 
considered as Np which are emitted within 0-7 degree opening angle and are free from 
comparatively large scattering. Charge of Z = 1 PFs are estimated by blob and gap 
counting method. 
3. Results and discussion 
Frequency distribution of multiply charged projectile fragments with Z > 2 emitted from 
H4Kr-H, 84Kr-C/N/0, 84Kr-Em and 84Kr-Ag/Br interactions are plotted in Figure 1. From this 
figure it can be readily seen that, irrespective of target mass, the emission of light projectile 
fragments are most abundant. In case of heavier target, the remnant part of the projectile 
nucleus splits into lighter fragments more preferably. On the other hand, heavy PFs are 
emitted more frequently from the lower target mass system. These results are in good 
agreement with the results reported by Cherry et al. [10]. 
0.20 
015 
% 0.10 
0.05-1 
0.00 
Kr-H 
llti,«»|..i.|.ii.T,ii,ti...|ii 
.1 , Kr-CNO . 
HiU.f.ylT..jiT.«.>T.iJ.;H)l|l 
. I . Kr-Em 
Hull Kr-AgBr 
10 15 20 25 30 35 40 
Figure 1. Frequency distribution of multiply charged projectile fragments with Z - 2 for the collisions of 
krypton beam with different targets of nuclear emulsion at 950 MeV/A The distributions for various targets 
are plotted on the same scale. 
It has been reported by ALADIN group [3] that in case of collisions with lighter targets, 
incident beam energy should be greater than a specific minimum value in order to achieve 
a complete disassembly of the projectile spectator thereby attaining the maximum fragment 
multiplicity. Keeping this result in our mind, we have investigated the variation of <NIMf> 
with Zb for collisions of 950 MeV/A 84Kr with different targets of nuclear emulsion and the 
results are plotted in Figure 2(a)-(c). 
From these plots one can easily observe the increase of maximum value of mean 
multiplicity of IMFs (<NIMF>max) with the increasing target mass for the present beam 
energy of E/A = 950 MeV for 84Kr. Such variation of <NIMF>wax with the variation of target 
636 B Debnath, R Talukdar and B Bhattachaqee 
mass suggests that the available beam energy of present investigation is less than the 
critical minimum value of bombarding energy for the interactions of 84Kr with lighter mass 
targets of nuclear emulsion. It may be noted that, due to low statistics, we have not 
reported here the plot for <NIMF> vs Zb for 84Kr-H interactions. Nevertheless, when it was 
V 
3.5 
3.0 
2.5 
2.0 
1.5 J 
1.0 
0.5 ^  
0.0 
3.5
 n 
(a) Kr-C/N/O 
10 15 20 25 30 35 40 
3.0J (b)Kr-Em 
2.5 
2.0 
J* 1-5H 
i.o4 
0.5 J 
0.0 
10 15 20 25 30 35 
z 
40 
3.5-, 
3.0 
2.5 
2.0 
A 
V 
1.0 
0.5 
0.0 
(c) Kr-Ag/Br 
10 15 20 
z_ 
25 
— i — 
30 
r 
35 40 
Figure 2. Average multiplicity distribution ot IMFs as a function of Zb for the interactions of (aJ^Kr-C/N/O 
(b^Kr-Em and (c^Kr-Ag/Br. 
tried to fit the experimental data points obtained from 84Kr-H interactions, the value of 
<A//MF>max was found, as expected, to be less than the all other values of <A//MF>max. The 
mean and maximum values of <NlMF> for various interactions of present investigation are 
listed in Table 1. Jain et al [11] have reported the average and maximum values 
of <NIMF> for 197Au-Em interactions at 10.6 A GeV to be 2.68±0.10 and 3.65±0.82 
respectively. 
Emission characteristics of intermediate mass fragments etc 637 
Table 1. Values of average <NIMF>, maximum mean multiplicities of IMFs and R? of the best fit of simple 
Gaussian distribution of the experimental data 
Target A v 9 - <N,Mf> Max <AA IMF* R
?(%) 
Ag/Br 
C/N/O 
H 
1.55±0 10 
1.36±0 10 
1.2±0.14 
0.72±0.13 
2 82±0 23 
2 51±0 12 
2.27±0 18 
1.09±0 31 
93 1 
94 8 
93 1 
As mentioned in section 1, both the number of projectile protons (N ) and the mass of 
the fragmenting system (Zb) are related with the geometry of the collision. To find the 
correlation between these two parameters, the variation of <Zb> with N1 for the interactions 
of 84Kr with different emulsion targets are studied and the results are plotted in Figure 3. 
A 
N 
Kr-Ag/Br 
Kr-Em 
Kr-C/N/O 
Kr-H 
N 
p 
Figure 3. Correlation between average Zb and Np for the reactions of 84Kr with Ag/Br, Em, C/N/O and H 
at 950 MeV/A Solid (Ag/Br), dot (Em), dash (C/N/O) and dash dot (H) lines are the best fitted lines for the 
respective experimental data points 
As expected, a negative correlation of <Zb> with Np is readily evident from this plot. 
The errors shown in this figure are the maximum and minimum errors for various data 
points and are estimated as independent statistical errors only. Though the errors for the 
above plots are relatively large due to poor statistics of studied interaction of each type, 
nevertheless one can not ignore a systematic increase in the values of <Zb> for a particular 
value of Np with the decrease of target mass. However, no such regular pattern could be 
observed from this plot on the probable dependence of the strength of the correlation with 
the target mass. The values of the slopes for the best fitted lines of the data points for 
the reactions with H, C/N/O, Em and Ag/Br are respectively found to be -1.62 ± 0.20, 
-1.68 ± 0.14, -1.27 ± 0.25 and -1.36 ± 0.27. 
638 B Debnath, R Talukdar and B Bhattacharjee 
4. Summary 
From the result obtained from this investigation it has been observed that the mechanise 
of fragmentation of spectator part of the projectile nucleus depends on the target mass 
Emission of heavy PFs is more frequent in lighter target mass system. For the interactions 
studied under the present investigation with the available beam energy, the complete 
disassembly of the projectile spectator with the light targets have not been achieve? 
resulting an increase in the values of <A//MF>max with the increase of target mass As the 
mass of the target decreases for the studied krypton interactions, <Zb> values are founa 
to increase for a specific value of Np. 
Acknowledgment 
Authors thankfully acknowledge the financial assistance of Department of Atomic Energv 
Board of Research in Nuclear Sciences, Mumbai to carry out this piece of research work 
in the form of a research project. 
References 
[1] J Hubele et al, Z Phys A340 263 (1991) 
[2] A Dabrowska et al Acta Physica Polonica B31(3) 725 (2000) 
[3] M L Cherry et al, Phys Rev C52(5) 2652 (1995) 
[4] A Schuttauf et al, Nucl Phys A607 457 (1996) 
[5] C Cavata et al, Phys Rev C42 1760 (1990) 
[6] P L Jain and G Singh Phys. Rev C46(1) R10 (1992), Phys Rev C47(5) 2382 (1993) 
[7] B Bhattacharjee Nucl Phys A748 641 (2005), B Bhattacharjee and S Sengupta Int J Mod Ph\^ 
E14(8) 1223 (2005) 
[8] B Bhattacharjee et al, Radiant Meas 36 291 (2003) 
[9] P H Fowler and D H Perkins Phil. Mag 46 587 (1955), C F Powell et al, The Study of Elementdt\ 
Particles by Photographic Method (London Pergamonn) p 093 (1959) W H Barkas Nuckw 
Research Emulsion (New York, London . Academic Press) Vol 1 (1963) 
[10] M L Cherry et al, Eur Phys. J C5 641 (1998) 
[11] P L Jam et al, Phys Rev C50(2) 1085 (1994) 


Emission Characteristics of Intermediate Mass Fragments on the Residual Part of the Projectile Nucleus

Hazarika, Farzana B

Baruah, G D

Indian J. Phys. 82 (6), 76 ] -766 (2008) 
Vector model of quantum interference laser 
Farzana B Hazarika* and G D Baruah 
Department of Physics, Dibrugarh University, 
Dibrugarh - 786 004, Assam, India 
E-mail gdbaruah(u rediffmail com 
Abstract The present work reports a vector model of quantum interference laser in terms of phasor 
diagrams which involve the motion of organized vectors in terms of detuning The time evolutions of these 
vectors have been worked out using different decay parameters in the usual expression for transition probabilities 
versus time in arbitrary units The vector model of density matrix is used to describe inversion with reference to 
Pomcare sphere 
Keywords Vector model, Quantum Interference Laser 
PACS Nos. 42 50-p, 42 55-f 
1. Introduction 
Quantum interference is a challenging principle of quantum theory and the essentials of 
quantum mechanics could be grasped from an exploration of the double slit experiment 
According to Feynman[1], each photon not only goes through both slits simultaneously 
but traverses every possible trajectory on the way to the target, not just in theory, but in 
fact. In order to see how this might possibly occur, experiments have focused on tracking 
the paths of individual photons. What happens in this case is that the measurement in 
some way disrupts the photon trajectories (in accordance with uncertainty principle), and 
somehow, the results of the experiment become what would be predicted by classical 
physics: two bright lines on photographic plate, aligned with the slits in the barrier. If we 
cease the attempt to measure, however, the pattern will become multiple lines in varying 
degrees of lightness and darkness. 
From what has been described above it is apparent that uncertainty principle is involved 
in explaining the phenomenon of interference and therefore the phenomenon is more 
appropriately termed as Quantum interference. Quantum interference research is being 
Corresponding Author 
© 2 0 0 8 I ACS 
762 Farzana B Hazarika and G D Baruah 
applied to a growing number of fields such as superconductivity quantum interference 
device (SQUID), quantum cryptography, quantum computing, quantum beat, the Hanle 
effect, Self induced transparency and Lasing without inversion or Quantum interference 
laser[2-9]. 
Under special conditions coherent atomic transitions can cancel absorption. The 
mam idea of LWI is that absorption cancellation provides the possibilities to obtain light 
amplification even if the population of the upper level is less than the population of the 
lower level. Such a situation can be realized, for instance, in a three level system, when 
two coherent atomic transitions destructively interfere and, hence, cancel absorption. 
To present the basic physics of LWI it is best to consider the theory of this effect in 
a three level /1-configuration and then to demonstrate how the concept of lasing without 
inversion can be realized experimentally. The configuration of /t-three levels atomic system 
is presented in Figure 1. It is formed by upper levels \a> and | o through interaction with 
electromagnetic fields £1 and E2% respectively, in such way that only atomic transitions 
\a> - | o and \a> - |b> are allowed. The physical reason for canceling absorption in this 
system is the uncertainty in atomic transitions | o - \a> and \b> - \a> which results in 
destructive interference between them. This situation is similar to Young's double slit 
problem, where interference is a consequence of uncertainty in determining through which 
of two slits the photon passed. 
w 
V V L1 
Figure 1. Three level atom in the A-configuration interacting with two fields of frequencies WL1 and W^ 
In the present work we report a vector model to represent quantum interference laser 
in terms of phasor diagrams which have been worked out from the concept of transition 
probabilities and their time evolutions with suitable 
decay parameters being introduced. We have also * — 
the occasion to use the formalism of vector model 
of density matrix and Poincare sphere to indicate 
their possible correlation with quantum interference 
lasers.
 h 
J*L 
2. Atom field interaction 
Yb The usual representation of the energy level diagram 
for two level atom indicating decay rates ya and yb F ,9 u r e 2- Ener9y ,evel d , a 9 r a m for *" 
. _.. _ level atom indicating decay rates ya is shown in Figure 2. ., 
* and Yb 
Vector model of quantum interference laser 763 
It may be noted that the atom decays to the lower level as a result of its interact.on 
with vacuum. The theory of spontaneous emission of Weisskopf - Wigner justified the 
inclusion of phenomenolog.cal decay rates
 Ya and yb in the Schrodinger equation for 
the atomic probability amplitudes [10] The probability of stimulated absorption including 
damping is given by 
exp(-y(f) s\r\(w~v)t/2 
L (w \ ) / 2 
3. Moving vectors 
The transition probability for arbitrary value of time may be plotted for different values of 
detuning (co-v) Using the eq (1) we have worked out the graph which represents the 
time evolution of population in the upper state for different values of detuning at 90 100 
110, 120 and 130 MHz This is shown in Figure 3 
140 n
 1 
120 Jt 
20 40 60 80 
Time(jiS) 
Figure 3 Transition probability vs time tor different values of detuning and decay CL >t<<
 ; 05 
As shown in Figure 3 we may join all the five maxima in a group of detunings as an 
arrow and a number of these arrows can be drawn in this way We observe that these 
vectors evolve in time. These vectors can be used to represent transition probabilities for 
the changes of population in the upper state The zero transition probabilities at arbitrary 
time of 62n (n = 1, 2, 3 ..) indicate some type of saturation A phasor diagram has been 
constructed with the help of the moving vectors The graphs obviously represent some 
kind of resonance. 
As may be seen in Figure 3, these vectors change directions as well as magnitudes, 
a situation which is analogous to the process of lasing without inversion or quantum 
interference laser. 
Let us now follow a wave as it bounces back and forth between two mirrors inside a 
laser cavity as shown in Figure 4. 
764 Farzana B Hazarika and G D Baruah 
Consider the initial field at the plane just to the right of Mv labeled by EQ. It propagates 
to M2 and back to the starting plane and undergoes a change in amplitude a,a2 and a 
phase factor d'lk2d^ as it travels that round trip and thus generates the field labeled £ 
This field experiences the same amplitude change and phase factor as E0 and in turn 
generates E2* and so on. 
o> 
Figure 4. Resonating cavity and phasor diagram 
The phasor diagram does not occur if 0 = 0, but 0 should be sufficiently small. The 
total field propagating to the right may be represented by 
£? =^E;=E0 = EQa,a2e-™ + ^ [ a ^ r * 2 * ] 2 +. 
l - a ^ e 120 (2) 
The field returning from M2 is just a2 times the round-trip phase factor &,k2d\ multiplying 
the wave going to the right. 
We now proceed to analyze the Phasor diagrams as shown in Figure 5 which are 
drawn with the help of the moving vectors. 
As may be inferred from Figure 5 the vectors in the phasor diagram first closes up as 
in the case of resonating cavity but uncoiled again and expands. It is also observed that 
when decay is small the expansion is slow but for bigger decay the expansion is more. 
This is a general nature of all the phasor diagrams we have constructed with different 
decay constants and with the same set of detuning values. Thus we find that there is a 
distinct difference between the phasor diagram for a resonating cavity and the phasor 
diagrams of moving vectors. The former is contracting while the later is initially contracting 
but expanding afterwards. The phasor diagrams also indicate that as the decay y increases 
the area decreases exponentially which may be illustrated by plotting decay versus area. 
As regards its physical significance it may be noted that the decay parameter y is related 
Vector model of quantum interference laser 
765 
to the lifet.me as r = 1/y Th.s follows from the theory of spontaneous em.ss.on In the 
present case the area approximately behaves like lifetime 
i:-' 
V 
\ 
•J»l 
, > 
• s 
/v. 
• \ • « t ; 
Figure 5 Phaser diagrams for moving vectors 
4. Density matrix in motion 
In this section we introduce the concept of the vector model of density matrix in motion 
[11]. This model has value not only in solving the equation of density matrix in motion but 
also in providing a physical picture of a particular system The equation are equivalent to 
the Bloch equation [12] appearing in nuclear magnetic resonance In this connection we 
may note that the phenomenon of the effect of an adiabatic rapid passage in inverting the 
population distribution of an assembly of nuclear spins has been recognized and predates 
the maser itself by approximately a decade. The technique is straightforward In a system 
in which a radiation field far from resonance with a transition frequency in a material 
sample interacts with a sample, the field frequency is swept through the material resonance 
until it is again far off resonance but with the frequency deviation reversed. If the sweep 
rate has been adjusted to lie within relatively easily attainable limits in certain materials, 
it is found that the population of the material sample after the passage is inverted relative 
to its value prior to the passage. For laser media when decays ya and yb are small 
compared to y in the Bloch model which is accurate and may be easier to use. We 
have in the geometrical model of density matrix a moving vector defined as 
fl=-yf? + R><8 (3) 
766 Farzana B Hazarika and G D Baruah 
The R vector precesses clockwise about the effective field B with diminishing 
magnitude. It is possible to follow the R vectors for some nonzero value of detuning as 
in the case of moving vectors and observe the condition of interference. 
It is appropriate to conclude our discussion on moving vectors and phasor diagrams 
with reference to an elegant concept known as Poincare representation of describing the 
state of polarization of light. Conceptually Poincare representation is simple but it remained 
neglected for a long time until its utility was pointed out [13].ln a so-called Poincare 
sphere there is one to one correspondence between all the points on the surface of a 
sphere (of unit radius called the Poincare sphere) and all possible forms of elliptic vibrations 
that can be conceived, may be represented by this sphere. One also notes that all 
surface points represent states of complete polarization. One could extend the concept 
and represent partially polarized states by points inside the sphere such that distance 
from the centre is a measure of the degree of polarization. Given the Poincare sphere, 
the change in the state of polarization of light as it passes through any medium can be 
visualized as a trajectory or moving vectors on the surface of the sphere. In fact, 
Pancharatnam [14] exploits this idea to actually design an achromatic quarter waveplate. 
5. Conclusion 
The present work is concerned with a vector model of transition probability which may be 
used to visualize quantum interference laser. The density matrix in motion and the concept 
of Poincare sphere has been introduced and their possible use in vector model is indicated. 
Acknowledgment 
The authors gratefully acknowledge financial support from AICTE, New Delhi. 
References 
[1] R P Feynman, R B Leighton and M. Sands The Feynman Lectures on Physics Vol 1 
(Massachusetts : Addison-Wesley Publishing Co. Reading) Chapter 37 (1965) 
[2] C Cohen Tannoudji Quantum Mechanics Vol. 1 (New York : John Wiley & Sons) (1997) 
[3] M O Scully and M S Zubairy Quantum Optics (London : Cambridge University Press) (1997) 
[4] N Fleischauer and C H Keitel Opt. Comm. 94 599 (1992) 
[5] J Mompart and R Corbatan Opt. Comm. 147 229 (1998) 
[6] A S Zebrov Phys. Rev. Lett. 75 1499 (1999) 
[7] O Kocharovskaya Opt. Comm. 77 215 (1999) 
[8] L M Narducci and M O Scully Opt. Comm. 81 379 (1999) 
[9] Xiang Ming and Jin-sheng Pen Opt. Comm. 154 205 (1998) 
[10] V F Welsskopf and E Wigner Z. Phys. 63 54 (1930) 
[11] M Sergent (III), M O Scully and W E Lamb (Jr) Laser Physics (Massachusetts : Addisson-Wesley 
Publishing Co. Reading) p18 (1974) 
[12] F Bloch Phys Rev. 70 460 (1946) 
[13] G N Ramachandran and S Ramaseshan Handbuch der Physik Vol 25 Part 1 (1961) 
[14] Collected works of S Pancharatnam (India : Oxford University Press) Raman Research Institute, 
Bangalore, (1975) 


Vector Model of Quantum Inteference Laser

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Notes for Contributors

Castillo, G F Torres Del

Indian J. Phys. 82 (9), i 105-1112 (2008) 
Hidden symmetries of the motion of a charged particle in an 
uniform magnetic field 
G F Torres del Castillo 
Departamcnto de Fisica Matematica, Institute de Ciencias, 
Universidad Aut6norna dc Puebla, 72570 Puebla, 
Puebla, Mexico 
Received 31 January 2008, accepted 19 July 2008 
Abstract : It is shown that, by means of an appropriate canonical transformation, the constants of motion 
for a charged particle in a uniform magnetic field can be associated with ignorable variables in the Hamiltonian. 
The canonical coordinates where the hidden symmetries of the Hamiltonian become obvious are related with the 
generating functions of the rigid translations. 
Keywords : Hidden symmetries, Hamiltonian formalism. 
PACS Nos. : 45.20.Jj, 02.20.Sv 
1. Introduction 
As is well known, the existence of ignorable coordinates in the Hamiltonian of a mechanical 
system allows one to find constants of motion in a straightforward manner. Each ignorable 
coordinate, as well as each constant of motion that does not depend explicitly on the 
time, is associated with a one-parameter group of canonical transformations that leaves 
the Hamiltonian invariant. However, in some cases, the existence of several one-parameter 
groups of symmetries of the Hamiltonian cannot be associated with the simultaneous 
existence of ignorable coordinates. A simple example of this situation is given by the 
problem of a particle in a central force field; the Hamiltonian is invariant under the rotations 
about all axes passing through the center of force, but the number of ignorable coordinates 
is 1 at most. Nevertheless, all components of the angular momentum are conserved as a 
consequence of the mentioned invariance. 
In some other cases, there may be constants of motion that are not associated with 
transformations on the configuration space. For instance, one can readily verify that in the 
case of a particle in a uniform gravitational field, with Hamiltonian 
© 2008 IACS 
1106 G F Torres del Castillo 
where g is the acceleration of gravity, px is a constant of motion (associated with the 
ignorable coordinate x) and A = pKpy +m2gx is also conserved. However, the conservation 
of A is not related with the invariance of H under transformations on the phase space 
induced by transformation of the configuration space. In fact, for any mechanical system, 
the constants of motion associated with the invariance of the Hamiltonian under 
transformations induced by groups of transformations on the configuration space are 
homogeneous functions of degree 1 of the canonical momenta. 
The problem of a classical charged particle in a uniform magnetic field also involves 
constants of motion not related to transformations on the configuration space (see Section 
2). Following the procedure employed in Ref. [1], we find that expressing the Hamiltonian 
in terms of an appropriate set of canonical coordinates, the existence of these constants 
of motion follows from the existence of a pair of ignorable canonically conjugated variables 
Apart from constant factors, the new canonical coordinates are the components of the 
kinematical momentum and the generators of translations, which would coincide with the 
former, if the magnetic field is absent. 
2. Hidden and obvious symmetries 
The vector potential for a uniform magnetic field B-Bz can be chosen as 
* = ~8xr = ~B(--yx + xy). (2) 
Therefore, the Hamiltonian for a charged particle of mass m and electric charge q in this 
uniform magnetic field can be taken as 
(see, e.g., Refs. [2,3]), making use of the standard symplectic structure of the phase 
space, where px, pyl and p2 are the canonically conjugate momenta of the Cartesian 
coordinates x, y, and z, respectively. In this manner, only the coordinate z is ignorable 
(and its conjugate momentum, pv is conserved). 
According to Hamilton's equations, from eq. (3) it follows that p = mf+(q/c)A. Since 
the vector potential is not uniquely determined by a given magnetic field, the canonical 
momenta turn out to be gauge-dependent; this minor drawback is a consequence of 
employing the canonical formalism. (However, it is also possible to give a gauge-independent 
Hamiltonian formulation of the interaction with a magnetic field, see, e.g., Ref. [4] and the 
Hidden symmetries of the motion of a charged particle in a uniform magnetic field 1107 
references cited therein.) In what follows we will make use of the standard Hamiltonian 
formalism in order to show, in this usual context, how to exhibit the symmetries of the 
Hamiltonian (3), employing canonical coordinates. In Section 4 we will discuss the effect 
of the gauge transformations. 
One can readily verify (making use of the Hamilton equations or of the Poisson brackets) 
that 
are constants of motion. Since K1 and K2 are not homogeneous functions of degree 1 of 
the canonical momenta, they are not related with transformations on the configuration 
space, such as translations or rotations; they correspond to hidden symmetries of the 
Hamiltonian (3). As pointed out above, being constants of motion that do not depend 
explicitly on the time, JC, and K2 generate one-parameter groups of canonical transformations 
that leave the Hamiltonian invariant. The infinitesimal generators of these one-parameter 
groups are the Hamiltonian vector fields XK and XK , where for any differentiate 
function f, 
x =y| df d 8f d l 
f
~
L{dp, dq, dq, dp, I' (5 ) 
and qf, p, is an arbitrary set of canonical coordinates. (Then, the infinitesimal generator of 
the group of canonical transformations generated by pz is d/dz.) Thus, 
*' dx 2c dpy *2 dy 2c clpx ® 
The fact that these vector fields involve d/dpx and d/dpy means that the canonical 
transformations generated by /C, and K2 do not come from transformations on the 
configuration space. 
A straightforward computation shows that XKl and XKi commute, which also follows 
from the fact that the Poisson bracket {Kv K2) is a constant 
'
 1 2)
 V i a * , dp, dp, 3q,) K' c K() 
(For any pair of functions, f, g, the commutator of the Hamiltonian vector fields Xf and Xg 
is related with the Hamiltonian vector field corresponding to their Poisson bracket by 
[*r.Xg]--XM.) 
1108 G F Torres del Castillo 
Since the velocity of the particle along the z-axis is constant, we can restrict ourselves 
to the motion on the xy-plane. The Hamiltonian now will be 
fc 1 qB )2 ( qB x2 
2m * 2c J Pv+7T-
X 
> 2c (fi) 
and since XK and XK do commute, according to the Frobenius theorem, they are tangent 
to a family of two-dimensional surfaces that foliate the phase space. By inspection, or 
making use of the method of characteristics, one finds that the functions that simultaneously 
satisfy the conditions XKf = 0 and XKf^0 (which are equivalent to \Kvf}-0 and 
{K2, f) = 0 are arbitrary functions of 
F l 5 E X
~ ^ 3 P y a n d F 2 s y + ^ P * ( 9 ) 
and, therefore, the two-dimensional surfaces mentioned above are given by F1 = const, 
F2 = const. 
The Hamiltonian h obeys XK h = 0 (as a consequence of the conservation of the 
Ku)t which implies that it must be a function of F1 and F2. In fact, one readily finds that 
On the other hand, one finds that the Poisson bracket {Fv F2) is also a constant, 
(cf. eq. (7)). Since {F v F2) is different from zero, F1 and F2 cannot be the momenta of a 
system of canonical coordinates, but, absorbing the constant factor appearing on the 
right-hand side of eq. (11) into Fy and F2, we can identify a pair of canonically conjugated 
variables. For example, we can take u - (1/2) F1 and pu = (qB/2c) F2, that is 
x c qB 
Us2-JBpy> p » s p * + 2^ y <12> 
In terms of these canonically conjugated variables, the Hamiltonian h is given by 
Hidden symmetries of the motion of a charged particle in a uniform magnetic field 1109 
which has the standard form of the Hamiltonian of a one-dimensional harmonic oscillator 
(As is well known, the problem of a charged particle is a uniform magnetic field in 
quantum mechanics can be related with the problem of a one-dimensional harmonic 
oscillator) 
Since we are considering now a four-dimensional phase space, in addition to u and 
Pj we have to choose another pair of canonically conjugated variables, v and pr say, 
which will not appear in the Hamiltonian h and, therefore, are constants of motion According 
to the derivations above, a simple choice is essentially given by K1 and K2, 
x c qB 
(see eqs (4) and (7)) 
In summary, (u, v, pu, pv) is a system of canonical coordinates in terms of which the 
Hamiltonian ht given by eq (13), possesses two ignorable coordinates, by contrast with 
eq (8), then explaining the existence of the constants of motion /C, and K2 
We end this section showing that the existence of the constants of motion K^ and K2, 
together with the fact that h itself is conserved (since it does not depend explicitly on the 
time), allows us to find easily the orbit of the particle and the hodograph in terms of the 
original variables Indeed, combining eqs (4) and (8) we obtain 
~ 1 (qB c „ , v, „ 
£ = — — x K- + y +—AC, 
2m { c qB 4 qB 1 
2 i 1 
- P x - ^ 1 P y - ^ (15) 
where E is the value of h Thus, in the xy plane and in the pj> plane, the orbit is a circle 
and the values of the constants of motion KA and K2 determine the centers of both 
circles 
3. Relationship with the generators of translations 
The elementary problem considered above is also interesting because of the difference 
between the canonical momentum and the kinematical momentum, mv In the Lagrangian 
or the Hamiltonian formalism, the interaction with a magnetic field makes it necessary to 
introduce the vector potential and, in Cartesian components, the canonical momentum 
turns out to be given by 
pt =mv, +^-,4,, (16) 
c 
where v is the velocity of the particle 
1110 G F Torres del Castillo 
As pointed out in Ref [5], under rigid translations in the configuration space, the 
Cartesian components of the velocity of the particle must be invariant; therefore, if <p 
denotes the generating function of rigid translations along the Cartesian axis x;l we must 
have 
Kf;) = 0, {*,.<M=<=V (17) 
However, from eq (16) it follows that {mvn pj} = {p, -(q/c) Ah ps} = -{q/c)dAf/dx}, which 
may be different from zero, thus showing that, when there is a nonvanishing magnetic 
field, the canonical momentum is not the generating function of rigid translations 
In terms of the notation and conventions employed in Section 2, one find that the x-
and y-component of the kmematical momentum are 
qB 
mvx=pUf mvy=-?—u (18) c 
and that the Poisson bracket relations (17) are satisfied taking 
QB 
<PI-PV. ^ 2 = ^ (19) 
Thus, the canonical coordinates (u, v, pu, pv) introduced at the end of the preceding 
section are, essentially, the Cartesian components of the kmematical momentum and the 
generating functions of translations along the x- and y-axis 
It may be noticed that, while the kmematical or the canonical momentum is not 
conserved <p, and <P2 are constants of motion (which follows from the fact that the magnetic 
field itslef is invariant under rigid translations [5] or from their relation with the ignorable 
coordinates v and pv given by eqs (19)) A somewhat unexpected result, since the group 
of translations of the plane is Abelian, is that the Poisson bracket between the generating 
functions of translations given by eqs (19) is different from zero, {<Pv<P2} = ~qB/c 
Nevertheless, as pointed out in Section 2, the infinitesimal generators of the transformations 
generated by them, as well as the corresponding one-parameter groups of canonical 
transformations, do commute In the quantum version of this problem, the nonvanishing of 
the commutator of the operators corresponding to <R, and <P2 leads to a ray-representation 
of the group of translations of the plane [6] 
4. Gauge transformations 
Under a gauge transformation, the vector potential A is replaced by 
4 ' = A + V<f, (20) 
Hidden symmetries of the motion of a charged particle in a uniform magnetic field 1111 
where £, is an arbitrary (differentiate) function of the coordinates, so that the magnetic 
field, B, is left invariant. As pointed out above, the canonical momentum is given by 
p = mr+(q/c)A; therefore, the gauge transformation (20) is accompanied by the 
transformation 
P ' - P + J V £ .
 (21) 
Then 
p'xdx + p'ydy + p'zdz = pxdx + pydy + pzdz + d( 2 A 
which implies that the transformation (21), together with 
is a canonical transformation. Hence, the Poisson brackets calculated in terms of x, y, z, 
px, pr pr coincide with those calculated using x, y, z, p'x,p'yl p'z as canonical variables. 
In other words, the underlying symplectic structure of the phase space is gauge-independent 
This means that we only have to make the replacement of the canonical momenta p, by 
p,\ according to (21), and all the Poisson bracket relations given above remain valid, as 
well as all the relations that do not contain explicitly the canonical momenta, such as 
eqs. (10), (13), (18), and (19). 
For instance, in terms of p', the constants of motion K, and K2 are given by 
qdf qB qd$ qB 
K ,
-
p
* ~ F c * 2 c y ' K*-p"-6dy-2c*' 
and, as one can readily verify, the Poisson bracket {Kv K2)f calculated using the new 
canonical variables, is equal to - qB I c (cf. Eq. (7)). 
S. Final remarks 
In the example considered in this paper, two constants of motion, /C, and K2, not related 
with obvious symmetries of the original Hamiltonian, have been identified. Since the Poisson 
bracket {Kv K2} is a constant different from zero, the existence of K, and K2 can be 
seen as a consequence of a pair of ignorable canonically conjugated variables. A set of 
canonical coordinates where the symmetries become obvious is formed, essentially, by 
the Cartesian components of the kinematical momentum and the generating functions of 
translations, which are independent only if the magnetic field does not vanish. 
In the example of a particle in a central force field, mentioned in the Introduction, 
not all components of the angular momentum can be associated with ignorable 
1112 GF Torres del Castillo 
coordinates because the Poisson bracket of any two of these components is neither zero 
nor a constant. 
Acknowledgments 
The author wishes to thank Drs. A Ramfrez-Palacios, B Mielnik and M Montesinos for 
useful discussions, and to the referee for helpful remarks. 
References 
[1] G F Torres del Castillo and J L Calvario-Acocal Rev. Mex. Fis. 54 127 (2008) 
[2] H Goldstein Classical Mechanics (2nd ed) (USA Reading MA : Addison-Wesley) (1980) 
[3] M G Calkin Lagrangian and Hamiltonian Mechanics (Singapore : World Scientific) (1996) 
[4] G F Torres del Castillo Rev. Mex, FIs. 50 88 (2004) 
[5) G F Torres del Castillo in Group Theoretical Methods in Physics - Proceedings of the XXV 
International Colloquium on Group Theoretical Methods in Physics, Cocoyoc, Mixico, 2-6 August 
2004, Institute of Physics Conference Series Number 185, (eds) G C Pogosyan, L E Vicent and K 
B Wolf (Bristol : Institute of Physics), p527 (2005) 
[6] L C Cortes-Cuautli and G F Torres del Castillo Int. J. Theor. Phys. 45 1799 (2006) 


Hidden Symmetries of the Motion of a Charged Particle in a Uniform Magnetic Field

Parga, G Ares De

Ortiz-Dominguez, M

Mares, R

A Deduction of the Landau-Lifshitz Equation of Motion Without Mass Renormalization in Classical Electrodynamics

Advisory Board and Editors

Kumar, Santosh

Pal, Janki

SEPs Associated CMEs Affecting Near Earth Environment

Khare, P K

Bangre, Gautama

Mishra, Pankaj

Mishra, Jyoti

Indian J. Phys. 82(10) 1301-1308(2008) 
Exploration of space charge behaviour of polyvinyl 
carbazole using measurement of thermally stimulated 
discharge current 
P K Khare*, Gautama Bangre, Pankaj Mishra1 and Jyoti Mishra 
Department of Postgraduate Studies and Research in Physics and Electronics, Rani Durgavati University, 
Jabalpur-482 001, Madhya Pradesh, India 
Hindustan College of Science and Technology, rarah (Mathura)-281 122, 
Uttar Pradesh, India 
Recdued 3 December 2007, accepted 11 June 2008 
Abstract : Measurements of thermally stimulated depolarization currents (TSDC) have been employed to 
investigate the space charge behaviour and molecular motion in polyvinyl carbazole (PVK) foil samples at 
different poling fields and temperatures. One relaxation peak was observed around 70-110°C. it was found that 
with increasing applied electric field the order of peak current increases and peak shifts towards lower temperature 
side. The order of current for samples poled at 50°C was higher than the samples poled at 60°C. The activation 
energy decreases with the increase in poling field and temperature. The current maxima occuring above the 
polarizing temperature, have distributed relaxation times. The analysis of the results suggest that intense charge 
trapping in surface and bulk traps and of the injection of carriers from electrodes under strong field at near 
surface region created by heterocharge. 
Keywords : Polyvinyl carbazole, thermally stimulated discharge currents, charge trapping, heterocharge. 
PACS Nos. : 77.22.Ej, 77.22.Cmf 77.22Jp 
1. Introduction 
Polymer films which are polarized in a strong electric field at an elevated temperature 
and are cooled with the field on, show semipermanent electric polarization which 
persists for years at room temperature; they are called polymer electrets [1]. Electrets 
formed by thermal methods are referred to thermoelectrets. The measurement of 
thermally stimulated current (TSC) which is caused by the thermal release of trapped 
charge carriers into either the conduction or valence band giving rise to TSC maximum, 
have been widely used to determine the density and energy distribution of charge 
carrier trapping centers in solids. A TSC peak arises in fundamentally different ways 
'Corresponding Author ° 2 0 0 8 I A C S 
CD 
1302 P KKhare, Gautama Bangre, PankajMishra andJyotiMishra 
for band transport in the presence of traps and for thermally activated hopping, in the 
first case classical theories [2] assumes temperature controlled detrapping of charge 
carriers and either fast transport in the conduction band or recmobinations in the bulk. 
The peak parameters allow one to obtain information about the depths, densities and 
possibly the distribution of traps which released the charge carriers. The second 
approach assumes that carriers hop from one localized state to another without ever 
moving in an extended state. These carriers are transported through the sample until 
they reach the electrode and the decrease of TSC is due to absorption of the carriers 
by the electrode. This approach allows one to derive information on the transport 
process [3]. The TSDC technique is a basic tool to identify and evaluate the dipole 
reorientation process, trapping and recombination levels in electrets and widely used to 
investigate the nature of various mechanisms in polymeric dielectrics. 
Polyvinyl carbazole (PVK) displays a wide range of physical and chemical 
properties which are interesting as far as basic research and applications are 
concerned. The present paper reports the results of thermally stimulated discharge 
current measurement on solution grown polyvinyl carbazole samples to understand the 
mechanism of transport of the charge carriers during polarization. 
2. Experimental details 
The isothermal immersion technique was utilized for preparing polyvinyl carbazole (PVK) 
foil samples. The solution was prepared in a glass beaker by dissolving 5 gms PVK 
in 100 ml of chemically pure dimethyl formamide (DMF) at 40°C. The solution was kept 
for 24 hours to get a homogeneous and transparent solution. The solution thus 
prepared was then poured onto an optically plane cleaned glass plate floating on 
mercury. The solvent was allowed to evaporate in an oven at 50°C for 24 hours to yield 
the desired samples. This was followed by room temperature outgassing at I0~5torr for 
a further period of 12 hours to remove any residual solvent. The plate was then drawn 
slowly out of the solution at a rate of 0.2 mm s~\ leaving a uniform film on the plate. 
Samples thus obtained were uniformly smooth and could be easily peeled from the 
glass surface. The preconditioned samples were sandwiched between aluminized 
electrodes over a central circular area of 36 mm diameter. The thickness of the sample 
was of the order of 50 fxm, which was estimated by measuring the capacitance of the 
fabricated sandwiches. By using aluminium electrodes of 1.6, 2.6 and 3.6 cm in 
diameter it was observed that even the smallest currents measured were proportional 
to the electrode area. This indicates that the current flows through the bulk of the 
samples and not around the edges. Different steps for the preparation of a thermoelectret 
are as follows : (i) the sample is heated to the desired polarizing temperature Tp\ it 
is kept at Tp for some time (in the present case, 1 hour) to reach thermal equilibrium; 
(ii) then an electric field (Ep) is applied at Tp and kept on for a period of polarizing 
time (fp), it is then cooled slowly under the field application to room temperature. The 
Exploration of space charge behaviour of polyvinyl carbazote etc. 1303 
field is then removed. The current was recorded with an electrometer (600B, Keithley) 
which was carefully shielded and grounded to avoid ground loops or extraneous 
electrical noise, as a function of applied field and temperature. The polarized samples 
were subsequently short-circuited for an arbitrary time of 10 min, so as to remove any 
frictional and stray charges present. The short circuit TSCs were then recorded by 
reheating the samples at a linear rate of 3°C/min [4-6]. 
3. Results 
The experimental conditions under which the thermally stimulated discharge currents 
were measured are summarised below. 
Polarizing field (Ep) - 100-300 volts 
Polarizing temperature (Tp) - 40°, 50° and 60°C 
Heating rate - 3°C/min 
Electrodes - Aluminium (both sides) 
The thermally stimulated discharge current (TSDC) spectra for polyvinyl carbazole 
(PVK) films polarized with polarizing fields 100, 200 and 300 volts at temperatures 40°, 
50° and 60°C are illustrated in Figures 1-6 for similar electrodes (i.e. AI-AI). The initial 
rise plots (i.e. log / vs. 1000/7) for these figures are shown in Figures 7-12. The value 
of activation energy (Table 1) has been calculated from these initial rise plots. The 
points taken for the calculation of activation energy are in the initial rising part of the 
peak. 
Table 1. Values of activation energy (eV) corresponding to polarizing fields. 
Temp(°CH 
Field (volts)i, 
100 
200 
300 
rm°o 
100 
95 
90 
40 
eV 
0.27 
0.21 
0.16 
rm°c 
100 
90 
70 
50 
eV 
0.26 
0.20 
0.16 
60 
rm°c ev 
105 0.25 
95 0.19 
75 0.15 
3.1. Field dependence : 
The polarizing field dependence of TSDC thermograms was investigated by polarizing 
the PVK films with different fields 100, 200 and 300 volts at various fixed temperatures 
(40, 50 and 60°C). 
The field dependent TSDC thermograms for PVK films are shown in Figures 1-3. 
The curves are generally characterized by one broad peak located in between 
70-105°C. In general, the magnitude of TSD current increases with increase in the 
Polarizing field, however the position of the peak maxima is found to be independent 
°f £p and shifts, towards lower temperature. 
32. Temperture dependence : 
The temperature dependence of TSDC thermograms was investigated by polarizing PVK 
1304 PKKhare, Gautama Bangre, PankajM/shraandJyottMishi 
0 7 i 
06-
£ 05-
© 
X 04-
tf 03-
O 02H 
01 
0 50 100 
TEMPERATURE (°C) 
150 
I S ] 08 
0 
06 
05 
04 
03H 
02 
01 
0 
50 100 
TEMPERATURE (°C) 150 
Figure 1. Thermally stimulated depolarization current 
of polyvinyl carbazole (PVK) sample poled at 
temperature 40°C with various fields : • -100, • - 200 
and • - 300 volts 
Figure 2. Thermally stimulated depolarization current 
of polyvinyl carbazole (PVK) sample poled at 
temperature 50°C with various fields • -100, • - 200 
and • - 300 volts 
film at different temperatures 40°, 50° and 60°C with various fixed polarizing field 100 
200 and 300 volts. 
The temperature dependence TSDC thermograms for PVK films are shown in 
Figures 4-6. The curves are characterized by a peak located between 70-100°C. The 
magnitude of the current increases with polarizing temperature. The TSDC peak shifts 
towards lower temperature side as the polarizing temperature increases. 
08-j 
0 7 1 
06-| 
< 
X 04-j 
01 
05 
< 04 
b 
x<)3 
LU 0 2 
§0.1 H 
50 100 
TEMPERATURE (°C) 
150 
i i — 
50 100 
TEMPERATURE (PC) 
150 
Figure 3. Thermally stimulated depolarization current Figure 4. Thermally stimulated depolarization current 
of polyvinyl carbazole (PVK) sample poled at of polyvinyl carbazole (PVK) sample with a field of 
tomnarBtura fiO°n with vArimift fiaiH* * * . inn • . om 100 volts at various temoerature : • - 40. • - 50 and te perature 60°C ith various fields : • -100, • - 200 
and • - 300 volts 
100 volts at various te perature : • - 40, • - 50 and 
60*0. 
It is evident from all the observed TSDC thermograms (Figures 1-6) initially the 
current starts at higher order and gradually it decays upto a certain temperature and 
after that with the increase in polarizing field or temperature it increases and a broad 
peak appears. Current saturated at the end of thermograms. In aii the thermograms the 
Exploration of space charge behaviour of polyvinyl carbazole etc. 1305 
0.8 
08-j 
- 0 5 H 
X 
g04 
oH 
o 
U -] 
0.7-j 
< 0.6 
b 
-0.5H 
fe0.4 
UJ 
gJ0.3 
S0.2H 
0.1 
50 100 150 
TEMPERATURE (°C) 
Figure 5. Thermally stimulated depolarization current 
of polyvinyl carbazole (PVK) sample with a field of 
200 volts at various temperature : • - 40, • - 50 and 
A-60°C. 
0 50 100 150 
TEMPERATURE (°C) 
Figure 6. Thermally stimulated depolarization current 
of polyvinyl carbazole (PVK) sample with a field of 
300 volts at vanous temperature : • - 40, • - 50 and 
A-60°C. 
order of current for the samples poled at 60°C found lower than the samples poled at 
50°C. 
In order to clearify the nature of TSDC peak, the series of observations were 
taken at different field and temperature conditions. In the present paper only representative 
results are shown. 
4. Discussion 
When the polymer film is heated up to a temperature close to its glass transition 
temperature, the dipoles and the free charges present in the polymer are mobilized. 
Application of a field causes an alignment of dipoles and a drift of the free charges. 
After some time, when the sample is cooled to room temperature, the main chains of 
the polymer are immobilized and the free charges are frozen-in. When heating of the 
thermoelectret is started space charge of opposite polarity combined to give rise to the 
depolarization current. Then the current is contributed by the disorientation of dipoles 
and the release of ionic charges from the defects or dislocation. As the thermal 
treatment is continued the charge carriers are injected on the dielectric surface from 
the electrodes and due to the Maxwell-Wagner effect, also disappear and become the 
TSD current. Thermal activations at a constant rate causes the release of charges due 
to their mobilization giving a peak at the site of maximum release of charges [7]. 
The origin of TSD current can be attributed to the following mechanisms. 
(i) Polarization due to polar side group/dipole alignments; 
00 Ions trapped in defects on dislocation in crystalline region; 
(i«) Space charge build up due to the migration of Ions over microscopic distances 
and 
W The space charge Injected from the electrode. 
Polyvinyl carbazole (PVK) is a polar polymer, the major contribution to the polarization 
1306 PKKhare, Gautama Bangm, PankajMJshraandjyotiMishra 
may be due to alignment of the dipoles under the effect of the electric field at hiqh 
temperature. In the present investigation, TSDC thermogram exhibits a broad peak 
(70°-105°C) and it represent presence of multiplicity of relaxation mechanism. The 
multiplicity of relaxation in PVK may be because of the presence of trapping levels of 
different depths. The appearence of the peak implies that the injection of ions may be 
significant in this polymer. It is also possible that PVK contains a high number of 
impurity molecules prior to field treatment and these molecules are dissociated in to 
various ionic species [8]. 
The discharge current availaible at the external circuit may be due to the 
rearrangement of ordered dipoles. From Table 1, it is evident that the activation energy 
varied between 0.16 to 0.27 eV i.e. charge carriers are trapped in levels distributed over 
a range of 0.11 eV, characterized by a continious relaxation distribution. Further more. 
as the polarizing field increases the barrier height of the traps is lowered, accounting 
for the observed decrease of the activation energy with increasing polarizing field. The 
increasing trend of current is in conformity with increase of conductivity of PVK with 
field and this can be attributed to the increase in local conduction in the thermally 
generated defects in PVK. In the present case, various results obtained may be 
explained by stepwise superposition of various phenomena responsible for persistant 
polarization in electrets [9]. 
The broadness of the peak may be explained by assuming a distributed, or 
multiple, dielectric relaxation caused by a variation in activation energy, when the 
rotation of the dipoles does not proceed in the same environment. Alternatively, it may 
be due to a distribution in relaxation times, when the rotational masses of the dipoles 
are not equal. The broadness of the peak in the present case is, however, most likely 
to be a function of distribution in relaxation times, because the peak occurs near the 
Tg of the polymer, where the side groups move in unison with the main chain. Because 
the distribution of dipoles in the amorphous phase is most likely to be random, it may 
be expected that this is a complex process involving both distribution of the activation 
energy and relaxation times [10]. 
As the temperature is increased more charge is trapped in shallow traps. This 
larger amount of trapped charge on release gives a higher current and a higher sharp 
peak. With a further increase of temperature, due to increased mobility, carriers are 
shifted to deeper traps. Release from such deeper traps being difficults, the current 
show a decrease trend [11]. 
Assuming that the trap sructure is not ifluenced by heating, energetically different 
traps will be filled at elevated temperature rather than at room temperature. Assuming 
a discrete set of trapping levels, each represented by an activation energy a l / / \ and an 
attempt to escape frequency V", the detrapping time constant for each trapping level 
can be assumed to be of the from 
Exploration of space charge behaviour of polyvinyl carbazole etc. 1307 
r , .=hr 1exp( i ; , / /c7) (D 
/ = 1, ,/V0 
where N0 is the total number of trapping levels, r, the mean detrapping time for Ath 
trapping level, k is the Boltzmann constant and T is the temperature. 
For a trapping level to contribute to the charge motion in the sample during 
charging, r, has to be smaller than the charging time Tc. This means that with 
increasing temperature, the number of "active" (contributing to charge motion) trapping 
levels N(T) increases whereas the number of inactive (not contributing to the charge 
motion) levels [A/0-A/(7)] decreases. These inactive levels store the charge delivered to 
the sample during charging process. In view of the above, the higher temperature peak 
is expected to shift toward lower temperature with increase in poling temperature which 
is experimentally observed in the present case [12]. 
1.5 
H 
0.5 H 
<0.8 
£o.6i 
8 0.4 
'0.2 
2.6 2.7 2.8 2.9 
1000/T(K-1) 
Figure 7. Log / vs. 1000/7"(K"1) for polyvinyl carbazole 
(PVK) sample poled at temperatue 40°C with various 
field • -100, • - 200 and • - 300 volts. 
2.6 2.7 2.8 2.9 3 3.1 
1000/T(K-1) 
Figure 8. Log / vs. 1000/7 (K~1) for polyvinyl 
carbazole (PVK) sample poled at temperatue 50°C 
with various field • -100, • - 200 and A - 300 volts. 
2.6 2.7 2.8 2.9 
iooan"oc1) 
Hgure 9. Log / vs. 1000/7 (K"1) for polyvinyl 
carbazole (PVK) sample poled at temperatue 60°C 
with various field • -100, • - 200 and A - 300 volts. 
2.8 2.9 3.1 
1000^00 
Figure 10. Log / vs. 1000/7 (K~1) for polyvinyl 
carbazole (PVK) sample poled with a field of 100 volts 
at different temperature: • - 40, • - 50 and A - 60°C. 
The magnitude of peak increases with the increase in the field of polarization and 
this could be explained on the basis that under the influence of electric field detrapped 
excess charge carriers increases resulting in the corresponding increase in peak 
1308 PKKhare, Gautama Bangre, PankajMishraandJyotiMishn 
14n 
12 
b 
^0 8 X 
goeH 
0 4-
0 2-
/ 09 
A 08 
/ ^°7" / < * * 
^ - O * " *04~ 
- ^ ^ ^ 80 3-
*"* 3 ^ 02^ 
01 
0 
26 27 28 29 3 31 26 27 28 29 31 32 
1000TT(K-1) 100<VT(K-') 
Figure 11. Log Ivs 1000/ T(K~*) for polyvinyl carbazole Figure 12. Log / vs. 1000/7" (K~1) for polyvinyl 
(PVK) sample poled with a field of 200 volts at different carbazole (PVK) sample poled with a field of 300 volts 
temperature • • - 40, • - 50 and A - 60°C at different temperature: • - 40, • - 50 and A - 60°C 
current. The shift of the current maxima occurs because the excess charges are 
generally depleted or enhanced, creating a lower or higher deriving field and charges 
may get swept less or more quickly towards the electrodes. 
References 
[1] J Van Turnhout Thermally Stimulated Discharge of Polymer Electrets (Amsterdam Elsevier) (1975) 
[2] H J Wintle J Polym Sci Polym Phys 12 2135 (1974) 
[3] D M Taylor and T J Levis J. Phys D Appl. Phys 41346 (1971) 
[4] M S Gour, P K Khare and Ranjeet Singh Indian J Phys 80 89 (2006) 
[5] P K Khare, J M Keller, M S Gaur, Ranjeet Singh and S C Datt Polymer International 39 303 (1996) 
[6] M S Gaur, Reeta Singh, P K Khare and Ranjeet Singh Polymer International 37 33 (1995) 
[7] Y K Kulshrestha and A P Srivastava Polymer J 11 515 (1979) 
[8] P K Khare and R S Chandok Phys Stat Soldi (a) 147 509 (1995) 
[9] H C Sinha, I M Talwar and A P Srivastava Thin Solid Films 82 229 (1981) 
[10] P K Khare Indian J. Pure Appl Phys. 32 160 (1994) 
[11] P K Khare and A P Srivastava Indian J Pure Appl Phys 31 126 (1993) 
[12] H Von Seggern and J E West J Appl. Phys 55 2754 (1984) 


Exploration of Space Charge Behaviour of Polyvinyl Carbazole Using Measurement of Thermally Stimulated Discharge Current

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