IACS Institutional Repository

Rout, S K

Panigrahi, S

BaTi0.4Zr0.6O3 Solid Solution : Phase Formation Kinetics and Reaction Mechanism Through Solid Oxide Reaction

Advisory Board and Editors

From Editor's Desk

Rajendran, S

Babu, Ravishanker

Kanimozhi, K

Indian J. Phys. 81 (5 & 6), 539-545 (2007) 
Studies on solid polymer Electrolyte based 
on PEO/PVC fiends 
S Rajendran*, Ravishanker Babu and K Kanimozhi 
Department of Physics, Alagappa University, Karaikudi-630 003, Tamilnadu, India 
E-mail sraj54(r/>yahoo com 
Received 30 December 2006, accepted 19 June 2007 
Abstract Novel polymer blend electrolyte was prepared using poly (ethylene oxide), polyvinyl chloride) 
and lithium perchlorate (LiCI04) as the complexing salt by employing solvent casting technique The prepared 
films were subjected to XRD, FTIR, a c impedance spectroscopy and thermogravimetry / differential thermal 
analysis (TQ/OTA) Quantitative analysis of the FTIR spectra provides the specific interactions between the 
constituents The thermal stability of the film is found using TG/DTA studies The maximum conductivity value of 
PVC (25)-PEO(75)~LiCI04{8) film is found as 1 32 x 105 S/cm at room temperature As the PVC concentration 
increases in the electrolyte, the conductivity is found to decrease The temperature dependent ionic conductivity 
is also carried out in the temperature range 303-333K and the results are discussed 
Keywords Polymer blend, FTIR, ac impedance spectroscopy, thermal studies 
PACS Nos. 80 Te, 72 80 Le, 82 45 Gj, 81 70 Pg 
1. Introduction 
In the last couple of decades, solid polymer electrolytes (SPEs) have attracted considerable 
attention due to their interesting properties such as ease of preparation of thin film and 
other forms, wide range of composition and hence control of properties, good electrode-
electrolyte interfacial contacts, high value of ionic conductivity as well as their wide 
application potentials in high energy density battenes, electrochromic displays, sensors 
and fuel cells [1,2]. Poly (ethylene oxide), in particular, is an exceptional polymer which 
makes complexes with high concentrations of a wide variety of salts to form polymeric 
electrolytes [3]. The complexes of PEO with a number of alkali salts such as LiBF4, 
* Corresponding Author 
© 2 0 0 7 1 A C S 
540 S Rajendran, Ravishanker Babu and K Kanimozhi 
UPF6, LiB(C6H5)4 [4], LiSCN [5], LiS03CF3 and LiCI04 [6] are reported. Even though 
these electrolytes give good conductivity above their melting temperature i.e. above 60°C 
appreciable conductivity could not be attained at room temperature due to the crystalline 
nature of PEO, which is a major obstacle in the context of ionic conductivity. To minimize 
the crystallization, works have been done by incorporating ceramic fillers into the PEO 
matrix. 
The main thrust of the current research in SPE is to develop the electrolytes that 
show high conductivity and give better mechanical properties. Enhancement in ionic 
conductivity could be attained by blending of polymers [7], cross linking [8], insertion of 
ceramic fillers [9], plasticization [10], etc. Among the various methods polymer blending 
is a feasible approach in which one conducting component acts as a plasticizer to reduce 
the non conductive crystalline phase of the other leading to an increase in the ionic 
conduction. 
Proof of the blend concept was first demonstrated by Rhoo et al [11] on PVC/PMMA 
blend polymer electrolytes, further the characteristics of a Lithium-ion polymer battery 
that employ a PVC/PMMA polymer electrolyte was reported by Kim et al [12]. Many 
workers conceived this idea and worked on different blends such as PVC/PMMA, PVdF/ 
PMMA, PAN/ PEO [13-17] to find an appropriate blend which would be suitable for lithium 
battery applications. 
In the present work we have prepared blend electrolytes consisting of polymers PVC, 
PEO and LiCI04 as the salt in different weight ratios and investigations on conductivity, 
complexation and thermal stability have been carried out. PVC which acts as a mechanical 
stiffener, is used in order to increase the thermal stability and mechanical stability of the 
polymer electrolyte. It is well known that the thermal stability of PEO based electrolytes 
bereft of ceramic oxides is not good. Hence to increase the thermal stability of PEO 
based polymer electrolyte PVC has been incorporated. 
2. Experimental Studies 
PVC (Mw = 1.5 x 105), PEO ( M w s 1 x 105) and LiCI04 purchased from Aldrich were 
used in this study. All the electrolytes were prepared by the solvent casting technique. 
The solutions for film cast were prepared by dissolving appropriate amounts of PVC, PEO 
and LiCI04 together in anhydrous THF. The solution was stirred continuously for 10 hours 
until a homogeneous suspension was obtained. The resulting homogeneous mixture was 
cast on the Teflon-bushes, allowing THF to evaporate slowly in an argon atmosphere at 
room temperature and then at 40°C for an hour for complete evaporation of residual solvent 
if any. The resulting films were visually examined for its dryness and free standing nature. 
Chemical storage, film casting and cell assemblies were performed in a vacuum 
atmosphere. 
Studies on solid polymer electrolyte based on PEO/PVC blends 541 
3. Results and discussion 
3.1 X-ray diffraction studies : 
t 
Figure 1 shows the XRD patterns of PVC-PEO-Li<|04 complexes and those of pure PVC, 
PEO and UCI04. It is seen that PVC exhibits amoibhous phase. The XRD pattern of pure 
PEO (Figure 1b) indicates two crystalline peaki at 19.7° and 23.9° [18]. The peaks 
c 
•e 
3-
f 
26 (degrees) 
Figure 1. XRD plots for Pure (a) PVC ; (b) PEO ; (c) L iCI04 ; (d) PVC:PEO:LiCI04 (0:100:8); 
(e) PVC:PEO:LICI04 (25:75:8) ; (f) PVC:PEO:UCI04 (50:50:8) ; (g) PVC:PEO:LiCI04 (75:25:8) and 
(h) PVC:PEO:UCI04 (100:0:8). 
542 SRajendran, Ravishanker Babu and K Kanimozhi 
appearing in pure PEO are observed in the complexes S1 and S2 with lesser intensities 
and are found to be absent in other complexes. This is in accordance with the concentration 
of PEO in the complex. Most of the peaks pertaining to LiCI04 disappear in the polymer 
complex, which indicates the complexation of lithium salt with the polymer electrolyte. 
3.2 FTIR studies : 
FTIR studies on blends are useful to identify miscibility of the polymers used for blending. 
Figure 2(a-h) depicts the FTIR spectra 
of pure PVC, PEO, LiCI04 and 
complexes respectively. Figure 2(d) gives 
the spectrum of pure PVC+LiCI04 
complex. 
The y (C-CI) stretching vibration of 
pure PVC appearing at 610 cm1 is 
shifted to 692 cm*1 in the complex. The 
other CH2 bending vibrations like CH2 
wagging, CH2 twisting and 5$ym 
deformation which appear at 1254, 1333 
and 1428 cm-1 in pure PVC are shifted 
to 1258,1328 and 1419 cm1 respectively. 
Some peaks present in pure PVC (2360, 
1734, 1559, 966, 494 cm1) are found to 
be absent in the complexes. 
Figure 2(h) elucidates the spectrum 
of PEO+LiCI04. The peaks at 930, 820 
cm"1 are assigned to CH2 rocking 
vibrations and may be attributed to the 
gauche form. The broad peak at 1130 
cm"1 in the pure PEO shifts to 1145 
cm1 in the complex. The peak at 1105 
cm"1 is shifted to 1110 cm'1 
corresponding to C-O-C stretching 
vibration. Vibrational peaks of PEO 
appearing at 2900, 1600, 1325, 1225 
cm"1 are absent in the polymer 
complexes. 
The band at 470 cm*1 in pure LiCI04 
is assumed to be replaced by two bands 
at 479 and 434 cm1 indicating two kinds 
of environment for the CIO"4 ion in the 
4000 400 
Wavenumber (cnr*) 
Figure 2. FTIR plots for Pure (a) PVC; (b) PEO; (c) 
UCI04; (d) PVC:PEO:UCI04 (0:100:8); (e) PVC:PEO:LiCI04 
(25:75:8); (f) PVC;PEO:LiCI04 (50:50:8) ; (g) 
PVC:PEO:LiCI04 (75:25:8) and (h) PVC:PEO:UCI04 
(100:0:8) 
Studies on solid polymer electrolyte based on PEO/PVC blends 543 
complex. The vibrational peaks 1610, 1300, 107$, 750 cm*1 of LiCI04 are found to be 
absent in the polymer complexes. * 
In addition, some new peaks at 2237, 2162,! 1965, 1800, 1725, 1620, 1469, 1031 
cm"1 are observed in the polymer complexes otl|er than those of the starting materials 
viz. PVC, PEO, LiCI04 which proves the complication of the system. 
3.3 Conductivity studies: [ 
The polymer electrolyte films were sandwiched between two stainless steel (SS) blocking 
electrodes. The ionic conductivities of the polymer electrolytes were calculated from the 
bulk resistance obtained from the isotherm. Measurements were made by complex 
impedance analyzer in the frequency range 40 Hz-100 KHz. The ionic conductivity values 
of PVC-PEO-LiCI04 electrolyte films were obtained in the temperature range 303-333K 
and are tabulated in Table. The maximum conductivity has been obtained for the film S1 
(Table 1). But it has poor mechanical stability. Hence, the film S2 having good mechanical 
Table 1. Conductivity values of PVC: PEO:LiCI04 
Composition Conductivity values x 10*5 S/cm 
Rm PVC:PEO:LiCI04 303 K 318 K 333 K 
s 1 . 
s2 
s3 
s4 
s5 
0 : 
25 
50 
75 
100 
100 
75 
50 
25 
: 0 
8 
8 
8 
8 
8 
4.587 
1.322 
0.409 
0.115 
0.002 
22.290 
2.401 
0.676 
0.123 
0.011 
89.170 
10.440 
1.529 
0.127 
0.013 
stability may be used for the battery application. The room temperature conductivity of 
S2[PVC(25)-PEO(75)-LiCI04)] is estimated as 1.32 x 10"5 S/cm. Figure 3 represents the 
variation of the logarithm of electrical conductivity (logo) with inverse absolute temperature 
for various polymer electrolytes with fixed concentration of LiCI04. It is evident from the 
table that as the temperature increases the conductivity also increases for the polymer 
complexes. This can be explained by free volume model [19]. As the temperature increases 
the free volume increases which leads to increase in ion mobility and segmental mobility 
that will assist ion transport and virtually compensate for the retarding effect of the ion 
clouds. The graphs in Figure 3 show an unique nature suggesting no linear dependence 
which suggest that ion cloud follow Williams-Landel-Ferry (WLF) mechanism [20], in 
otherwords ion transport in polymer electrolyte is dependant on segmental motion [21]. 
3.4 Thermal analysis: 
Thermal analysis of PVC-PEO-LiCI04 (film S2) is shown in Figure 4. The DTA curve shows 
a small endothermic peak at 64.4°C which indicates the melting of the polymer. It can be 
544 S Rajendran, Ravishanker Babu and K Kanimozhi 
seen from TG curve that the film looses its weight gradually on increasing the temperature. 
A well defined exothermic peak is seen around 302°C indicating the decomposition of 
polymer electrolytes. Consistent with this thermal event, the weight decreased rapidly 
-2 , 
-3 
-4 j 
CD 
•7 
-8 
2.99 3.04 3.09 3.14 3.19 
1000/T 
3.24 3.29 3.34 
Figure 3. Arrhenius plot of log a against reciprocal temperature of PVC:PEO:LiCI04 . 
from 265°C. The first and second decomposition of the film take place between 50-80 and 
250-280°C respectively. Weight losses of the polymer electrolyte at 100, 200 and 300°C 
are found as 7, 11 and 28% respectively. The weight losses may be due to the evaporation 
of the moisture and solvent. 
200 250 300 
Temperature (°C) 
Flgur* 4. TG/DTA curve for PVC: PEO: LiCI04 (25:75:8). 
4. Conclusion 
Five different compositions of PVC-PEO-LiCI04 polymer electrolyte systems have been 
prepared and the polymer electrolyte (film S2, i.e. PVC/PEO, 25/75) is found to be the 
Studies on solid polymer electrolyte based on PEO/PVC blends 545 
best film on the basis of both conductivity and mechanical stability. The film is found 
stable upto 250°C and then the film losses its weight gradually till 265°C beyond which 
the polymer electrolyte loses weight drastically because of onset of degradation. As the 
film S2 possesses desirable properties, it can be uapd for Li battery applications. 
References 
M B Armand, J M Chabagno and M Duclot in Second Intern Meeting on Solid Electrolytes, St 
Andrews, Scotland, p2022 (1978) 
F M Gray Solid Polymer Electrolyte-Fundamentals and Technological Applications (Germany VCH, 
Weinheim) (1991) 
D W Kim, J R Park and H W Rhee Solid State Ionics 83 49 (1996) 
W Wieckzorek and J R Stevens J Phys Chem. B 101 1529 (1997) 
J Przyluski and W Wieczorek Solid State Ionics 36 165 (1989) 
J Y Cherng, M Z A Munshi, B B Owens and W H Smyrl Solid State Ionics 28 857 (1988) 
H J Rhoo, H T Kim, J K Park and T S Hwang Electrochim Acta 42 571 (1997) 
H T Kim, K B Kim, S W Kim and J K Park Electrochim Acta 45 4001 (2000) 
H Hong, C Liquan, H Xuejie and X Rongjian Electrochim Acta 37 1671 (1992) 
D Peramunage, D M Pasquanello and K M Abraham J Electrochim Soc 142 1789 (1995) 
O Bohnke, G Frand, M Rezrazi, C Rousselot and C Truche Solid State Ionics 66 97 (1993) 
M Alamgir and K M Abraham J Electrochem Soc 140 196 (1993) 
T Mam and J R Stevens Polymer 33 834 (1992) 
M B Armand in J R Mac Callum, C A Vincent (Eds) Polymer Electrolyte Reviews (Amsterdam 
Elsevier) p1 (1987) 
E A Reitman, M L Kaplan and R J Cava Solid State Ionics 17 67 (1985) 
P R Sorenson and T Jacobson Eletrochim Acta 27 1675 (1982) 
D Fauteux and C Robitaille J Electrochem Soc 133 307 (1986) 
K Such, Z Flonanczyk, W Wieczorek and J Przytuski Second Int Symposium on Polymer Electrolytes 
(1989) 
M Rathner and D F Shnver Chem Rev 88 109 (1988) 
M L Williams, R F Landell and J D Ferry J Am Chem Soc 77 3701 (1955) 
Y Okamoto, T F Ych, H S Lee and T A Skotheimk J Polym. Sci Part A Polym. Chem. 31 2573 
(1993)

English

Studies on Solid Polymer Electrolyte Based on PEO/PVC Blends

Choudhury, D K

Gogoi, Rupjyoti

Indian 7. Phys. 81 (5 & 6), 607-614 (2007) 
Deep inelastic neutrino scattering and fractal models of 
nucleoli structure functions at small x 
D K Choudhury* and Rupjyoti Gogoi 
Department of Physics, Gauhati University, Guwahati - 781 014, Assam, India 
E-mail dkcj?hys(«)yahoo.co in 
Received 5 October 2006, accepted 16 May 2007 
Abstract : The concept of self-similarity in the structure of the proton at small x used successfully in the 
analysis of recent HERA data, is now extended to deep inelastic neutrino scattering. The fractal models are then 
tested with the available CCFR neutrino data 
Keywords * Self-similanty, fractal dimension, deep inelastic scattering, structure function, low x 
PACS Nos. . 05.45.Df, 12 38.-t, 13 60.Hb 
1. Introduction 
While notion of self-similarity and fractals has been extensively studied in hadron multiparticle 
production processes during last decades [1-7], its application in deep inelastic scattering 
is rather more recent when Lastovicka [8,9] proposed a functional form of the structure 
function F2(x,&) reflecting self-similarity at small x and could provide an excellent description 
of HERA data [10,11] with the fitted parameters Dv D2 and D3 identified as fractal or self-
similar dimensions. Later [12], the monofractal limit of Lastovicka's model was studied in 
an analogy with classical monofractals [13,14] as well as with a variant of statistical 
quark model [15]. More recently [16], a new parameterization of proton structure function 
at small x has been proposed to overcome the problem of negativity of one of the fractal 
dimension D3 of the proposed fractal model. 
The present paper makes a similar analysis of the nucleon structure functions in deep 
inelastic neutrino scattering. Unfortunately, the deep inelastic neutrino scattering experiments 
Corresponding Author 
©2007IACS 
608 D K Choudhury and Rupjyoti Gogoi 
have not yet explored the very small x values unlike the corresponding electron-proton 
scattering experiments. The CCFR neutrino scattering data are available [17,18] in the 
kinematic range 0.0075 < x < 0.75 and 1.3 < Q2 < 125.9 GeV2. This is to be compared 
with HERA data [10,11] which cover a region of four momentum transferred squared 0.045 
< Q2 < 150 GeV2 and of Bjorken x, 6.2x10'7 < x < 0.2. In spite of this kinematical 
constraint on DIS neutrino data, in the present paper, we will test the validity or otherwise 
of the fractal models pursued in electron-proton scattering and see if any new insight can 
be obtained from this rather novel approach of looking at the structure of the proton. 
In Section 2, we develop the formalism while Section 3 is devoted to the results 
and discussions. 
2. Formalism 
As noted in Ref. [16], two alternative forms of the unintegrated parton densities ft(x,cf)are 
available: 
log/;(x,q2) = A l o g % l o g f ^ + % ] + ^ l o g ^ Z + C^log + &o (1) 
and 
<*, \ogf,(x,(f) = Di\ogyx\og l % J + £>>log;/+ O,log I % + ( ? 2 1+3,(2) 
where Dv D? D3 are the flavor independent self-similarity dimension and D0' are the flavor 
dependent normalization constants. 
Integrating over q2, one obtains the integrated parton densities qjLx.Q2) as 
o8 
q,(x.<?)=lf,(x.<?)dfm (3) 
The structure function F2 and xF3 are given as [19]: 
^
w
 = x (u+d) + x(<y+u) [ | V ^ | 2 + | V c d | 2 ] + 2 x s [ n / 0 S | 2 + | \ / c s | 2 ] + 2xs (4) 
and 
x f f = - x ( u + d ) + x ( d + u ) [ |V o d | 2 + |V c d | 2 ]+ 2x8[\Vmf+\V„f]-2xs,® 
Deep inelastic neutrino scattering and fractal models etc 
where u=u(x,&) etc and Vlf are the CKM matrix elements [20]. 
609 
Here, 
and 
pvN _ r 2 + r2 
3 ss 
(6) 
(7) 
Using (1) and (2) in eqs. (4) and (5), we get two sets of structure functions. 
Set I: 
Ar t z v-^ + 1 e*>(%x F%N(x,di)= e U°X 
V ;
 1 + 0,+D, log 
[,-H'*%)f1+tf \<V1 
'tf (8) 
and 
Set II: 
v
 ' 1 + 0,+O.logV 
(x-A^ i +%)fi+%r-i (9) 
and 
V
 ' 1-D,-qk>9% 
v
 ' 1-0,-D.log)/ 
| > o t ( 1 + % x *• 1 + Q
2
, 
X-Dj+1 
' « 
-1 (10) 
(xH^)(i
 + %p- ,(11) 
where the normalization constants D / and D / are appropriate combinations of flavor 
dependent constants Dj occurred in (1) and (2): 
e ^ [ e « + e * ] + [V*+e<S] [\vj+\vj] + 2e^[\vj+\Vcsf] + 2B^ , ( 12 ) 
eA>" = _p* + 0 * ] + [e* + e* ] [ |V^|2+|^|2] + 2e*[ | ^ | 2 +Ks|2]-2e*.(13> 
610 D K Choudhury and Rupjyoti Gogoi 
3. Results and discussion 
The numerical values of the self-similarity dimensions Dv D2 and D3 as well as Q02 are 
already determined from DIS HERA data and reported in Refs. [8] and [16]. The flavor 
dependent normalization constants 0/and D/are determined from CCFR data and are 
recorded in Tables 1, 2. 
Table 1. Results of fit for D^O and D, * 0 for 1.3 < Q2 < 125 9 GeV2 and 0.0075 < x < 0.35 for Ref.[ 8] 
and Ref. [16]. 
xF, 
Fit <V 
Of fixed 
„2 
r / / d o f D0' 
D1 fitted 
-,2 
D1 fixed 
r
2
 /dof 0 / *2 *2/dof Df
 x
2
 ,
2/dof 
D, fitted 
2 
Ref.[8] 2.1205 12.143 0.136 1.9078 11.653 0.131 4.1574 217.766 2.447 4.3462 365 525 4 107 
±0.263 ±0.262 ±0.698 ±0.560 
Ref.[16] 2.0803 4.159 0.047 2.2628 5.934 0.067 4.1057 78.452 0.881 4.1755 92.001 1.034 
±0.220 ±0.215 ±0.456 ±0.470 
1.5 
1 . 
Q^LyGeV1 
i T i i mif t i i i m i j 
?+»•» 
• CCFR 
— Ms work 
— LMtoviclu 
0.5 
10 
,1.11 Hill I I t I Hill 
Q/«*3.2GeV* 
—i—t i i n i y r -T r M I I I ; ! 
Q'-SGeV* Q2-7.9 GeV2 
10 
z 
10' 
1J5 
rf 1.55 
Q2-12.6 GeV* 
1.251 
10 
1.90 
Q^OGeV 2 Q2«31.6GeV* Q2-50,l GeV2 
10* 
rr 
r 
[ 
L 
L 
r 
1 Hi I 1—i l l rvr 
f
* ^ : * " • - — - ] 
• ^v^,^^ J 
• \ 
. J 
• CCFR 
— thbwork
 a 
— Lutovick* 
• 1 
L4,lt„... , ,,.»,..—»,—. > m n 
10" 10 
1.4 
A 
0^79,4 GeV1 
. , ' J 
• J 
• 1 
• CCFR • 1 
— thUwork * I 
— Unovicka
 m 1 
. 1—i i i f i 11 
w »• 
0.8 
30.4 
fcT 
n 
Q2«125.9GeV* 
. . . . . . ; , 
^ \ 
• 1 
• J 
• CCFR J 
thbwoifc . j 
— LMtovWu 1 
1 ii 1 11 1 ifi 1 1 
10' 10v 
Figure 1. F2(x,&) versus x In bins of Q2 with D1 « 0 (eqs. 8 &10). 
Deep inelastic neutrino scattering and fractal models etc 611 
Table 2. Results of the fit for D^O and D1 * 0 for 1 3 < Q 2 £ 20 GeV2 and 0.0075 < x < 0.35 for Ref. [8] 
and Ref. [16]. 
xF, 
Fit 
D1 fixed 
x2 
D1 fitted \ D1 fixed D1 fitted 
Oo' X2 Z2'*01 D0< X2 /dof r
2
 ^ / d o f 0 / • /dof 
Ref.[8] 2.1205 3.766 0.057 1.9078 3 456 0 052 4 1B74 75.905 1.150 4 3462 130 864 1983 
±0.263 ±0.262 ±0*98 ±0 560 
Ref.[16] 2.0803 2.437 0 0369 2.2628 2 440 0 037 4.1057 29.095 0.441 4.1755 35.461 0.537 
±0.220 ±0.215 ±0*456 ±0.470 
In Figures 1 and 2, we plot F2(x,Q2) vs x in bins of Q2 with D1 = 0 and D1 * 0, 
respectively. A study of Figure 1 for F2(x,Q2) vs x (with D1 = 0) shows that for lower 
values of Q2 (Q2 < 5 GeV2), model of Ref. [8] fits better than that of Ref. [16]; but for 
intermediate Q2 range (Q2= 12.6 - 20 GeV2), the model of Ref. [16] is preferred by data. 
The predictions of both the models for each Q2 bin are found to have a cross over point 
Q'-OGeV* Q*»3.2GeV* Q2«5GeV* 0**7.9 GeV2 
• CCFR (Mtwwk I 
— Luto»lck* 
• CCfR 
— thkwo* « 
10' 10" 10" 10" 
Figure 2. FJx.Q2) versus x in bins of Q2 with D1 * 0 (eqs. 8 &10). 
612 D K Choudhury and Rupjyoti Gogoi 
at a particular point. As Q2 is increased, the cross over point shifts towards large x. As 
an illustration while at Q2 = 1.3 GeV2, the cross over point is at x = 0.01, at Q2 = 125.9 
GeV2, it shifts to x = 0.298. For higher Q2 (Q2 > 20 GeV2), however predictions of both 
the models overshot the data. 
A similar pattern persists even for F2(x,Q2) vs Q2 with D1 * 0 in Figure 2. The x2 for 
1.3 < Q2 < 125.9 GeV2 and the lower Q2 range 1.3 < Q2 < 20 GeV2 are shown in Table 
1 and 2. From x2 analysis, the present model (Ref. [16]) fits better than that of Lastovicka 
(Ref. [8]). As expected, if confined to low Q2 range (Q2< 20 GeV2), ;tr2/dof is improved. 
In Figures 3 and 4, we plot xF3 vs x in bins of Q2 with D1 = 0 and D1 * 0, 
respectively. Corresponding x2 are again shown in Tables 1 and 2, respectively. Except 
for low Q2 range (which basically explores the smallest available x of CCFR data), both 
the models fail to reproduce the experimental pattern; rather they invariably overshot the 
data. This feature is presumably due to the predominant valence nature of non-singlet 
Q'-UGeV 2 Q2-0.2G«V* Q2-5GeV* Q*«7.9 GeV2 
Q'-SO.l G«V* 
? 
•sT 
i t i m i l l i i, i Mini 
4j0.6^ •§ it 0.7$ 
, A i, t m i l l i i I.I i l l 
r—i 
V 
V 
I 
I 
L 
I 
h 
L 
r r - tTTTT-^ - f 1 
It 
y / / 
i 
/ • • " • • 
n 
i 'i i i 
-
1 
-
-t 
j 
-
• 
• cent 
— thbwmk 
— LMlevtuka 
.. 
m 
iii i i m 
^Je.9 
10m 
x 
10w 
i i i i m i t ; / i i i i i i 
• CCFR 
I I ,1 I 11 Mi ,i 1 , 1 I I I 
10" 10 
Figure 3. xF3(xtQp) versus x in bins of Q2 with O t • 0 (eqs. 9 &11). 
Deep inelastic neutrino scattering and fractal models etc 613 
structure function F2(x,Q2) where the probability of gluon radiation creating self-similar 
pattern is very small. 
To conclude, while the fractal models of Refe. [8] and [16] provide an excellent 
description for the HERA data at small x, for (JCFR neutrino data, the models have 
limited validity. This is most probably due to the lact that CCFR experiment has not yet 
reached the small x regime where the notion of djelf-similarity may be more appropriate. 
Future DIS neutrino experiment in such small x fegime can alone provide fresh insight 
into the structure of the proton by interpreting th4 data based on fractality. 
Q^-l^GeV* 
| I I n i | i • i i t 
Q ^ G e V * 
10' 10' 
Q ^ G e V 2 Q2»7.9 GeV* 
Q^H^GeV 1 Q^OGeV* 0^31.6 GeV Q2«50.1 GeV2 
Figure 4. xF3(x,Q*) versus x in bins of Q2 with D1 * 0 (eqs. 9 &11). 
iow 
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[12] D K Choudhury and Rupjyoti Gogoi Indian J. Phys. 80 659 (2006) 
[13] BB Mandelbrot Fractal Geometry of Nature (New York: W H Freeman) (1983) 
[14] Michael F Barnsley Fractal Everywhere (New York: Academic) (1993) 
[15] S N Banerjee, A Bhattachaiya, B Chakraborti and S Banerjee Int. J. Mod. Phys. A17 4939 (2002) 
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[18] CCFR Collaboration: WJ Seligman ef al, Phys. Rev. Lett. 79 1213 (1997) 
[19] M Ghick, S Kretzer and E Reya; astro-ph / 9809273 
[20] Particle Oata Group: C Caso et al, Euro. Phys. J. C3 (1998)

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Misra, Anirban

Datta, Sambhu N

Indian J Phys. 81(9) 881-888(2007) 
$ 
Relativistic Hamiltonian for neutral atom of spin-1/2 
nucleus in homogeneous magnetic field 
Anirban Misra 
Department of Chemistry, North Bengal University, Siligun Darjeeling-734 430 West Bengal India 
and 
Sambhu N Datta* 
Department of Chemistry, Indian Institute of Technology Bombay, Powai, Mumbai 400 076 India 
Email sndatta<®chcm ntb ac in 
Abstract : A relativistic formalism to construct the Hamiltonian for a general N-electron neutral atom with a 
spin 1/2 nucleus and placed in a homogeneous magnetic field [J. Chem Phys 123 064101 (2005)] is discussed 
here Projected Coulomb and Breit interactions are taken to describe the inter-particle interaction through second 
order in fine structure constant The center of mass motion is separated from the movement of the neutral (/V+1)-
fermion system Using a series of suitable unitary transformations, the Hamiltonian can be brought into a 
separable form The correct atomic Hamiltonian in the non-relativistic limit is also discussed The treatment can 
be extended to the case of a molecule containing atoms of only spin-Vfe nuclei 
Keywords : Relativistic Hamiltonian, projected interaction, non-relativistic limit 
PACS Nos. : 31 30 Jv, 31.30 jc, 31 30 jy, 12.20 Ds 
1. Introduction 
In most of the state of art quantum theoretical calculations on atoms and molecules, 
one usually considers a faceless nucleus that provides only a Coulombic potential that 
can be legitimately viewed as an external one. Sometimes the effects of nuclear and 
electron spins are treated in a perturbative manner. However, for a multi-electron atom 
with a different nuclear spin, one can expect that there is a different form for the 
Hamiltonian operator. 
In our earlier work, the relativistic dynamics of one and two fermion systems 
placed in a homogeneous magnetic field were discussed in detail [1-3]. A generalization 
of this treatment has also been carried out explicitly [4]. In this work we describe the 
•Corresponding Author © 2007 IACS 
882 Anirban Misra and Sambhu N Datta 
formulation of a relativistic Hamiltonian for a neutral N-electron atom with a spin-y2 
nucleus in simpler terms. 
In Section 2 of this article, the phenomenological relativistic Hamiltonian is 
written for a (N+1) particle system with proper inter-particle electro-dynamical interaction 
[5]. In Section 3, the separation of the center of mass motion is carried out and the 
dynamics of each one-particle part is brought to a separable form. The non-relativistic 
Hamiltonian and a few general points of observation are discussed in Section 4. 
2. Phenomenological Hamiltonian 
The total relativistic Hamiltonian for an atom with a spin-Vfe nucleus and N electrons, 
and placed in a homogeneous magnetic field, can be written from a generalized version 
[6] of the celebrated Bethe-Salpeter equation [7]. We write 
A/+1 
X 
/-2 
" = ^(nM+£M'U)+u d) 
where the first particle is the nucleus and rest N particles are electrons. The operator 
hD(i\r) is the Dirac Hamiltonian for the /-th particle, 
hD(i I r,) = ca, • /r, + j3,m,c2 - fiar)0(i)Pfa]2B. (2) 
In the above, m„ n, and juano(i) are the corresponding rest mass, operator for the 
mechanical momentum and the anomalous magnetic moment; B is magnetic field 
strength, and or„ /?, and a, are the respective Dirac matrices in standard representation 
The last term in the above equation arises from the interaction of the anomalous 
magnetic moment with the external magnetic field. We adopt the symmetric gauge 
here for convenience, that is, A(r) = Vz B x r,. The mechanical momentum is written 
as 7r, = p, — L A ( r l ) 1 where q, is the charge of the /-th particle. 
c 
The interaction U is taken as projected so that it can be used safely to all 
orders in perturbation theory, without fear from infinite degeneracy or continuum 
dissolution [5], 
U = A.(UB + UM)A+. (3) 
N 
In the above, / l + = f | A,+, k^ being the generalized Casimir projection operator, 
A„=5(l (+i»o(/") /£, ) (4) 
where E, is the diagonal form of h^i). The generalized Breit term UB is 
u = y q,q' i a,a> fc•*"«)•(*',r») 
Relativistic Hamiltonian for neutral atom of spin-1/2 nucleus in homogeneous magnetic field 883 
where r„ = I /} - r, |, contains the retardation effect through order v2/c2. The magnetic 
interaction UM is the Barker-Glover generalization [8] of the interaction of the intrinsic 
magnetic moment with the external electromagnetic field, 
U» Mi V, 
1£/</£A/+1 2/77,C r r 
», 
2mjC riJ hi 
(6) 
2/77 Q 
where/', =—r~AW)('') for / = 1, ..., A/+1. The quantity UM is not included in the 
treatment in Ref. [6]. 
3. Simplification of dynamics 
The total relativistic Hamiltonian is expressed in a new set of coordinates, (f?CM> {r/}). 
The center of mass coordinate is defined by 
/V+1 
" C M - Z-r * j '* ' 
and the relative coordinates are 
(7a) 
(7b) 
where MT is the total rest mass MT -~Zjm, , and r,' are defined for / = 2,..., /V+1. 
This allows one to write the Dirac Hamiltonian operators as 
[M; 
+/3,m1c2-/iaro(1)B/31CT1z, 
and for / = (2, N + 1), (8) 
(i) = cal-\-±K + pl--\ql(Acu + Al)-Z ' 4 
The center of mass motion is also embedded in the relativistic Hamiltonian described 
in the previous section. The movement of the center of mass is separated from a 
884 Anirban Misra and Sambhu N Datta 
neutral atom following the techniques put forward by Avron et al [9]. It is found that 
the total pseudo-momentum K = ]£ k, is a constant of motion for the Hamiltonian H, 
though the individual mechanical momentum n{ and individual pseudo-momentum 
K = P, + — A a r e not constants of motion. The components of K are found to satisfy 
c
 on 
the commutation relations KxK = -i—B where Q is the total charge written as 
A/+1 C 
° = 2 9/ . These components are distinct only for a neutral atom. As JC commutes 
with H, one may replace the total pseudo-momentum K by its eigenvalue K [9]. This 
changes the Hamiltonian H(K) into the effective Hamiltonian H(K) and the projectors 
yl^(K) and A+(K) into the reduced projectors A,+(K) and A+{K). 
Using suitable unitary transformations one can transform the effective Hamiltonian 
into an exceedingly simple form. The translational motion for a neutral N-electron atom 
is discarded by considering the reduced Hamiltonian at zero total pseudo-momentum 
H(0) that represents the relative movements of the particles [4], 
«(0) = /6(1)+2/6(') + W(0). (9) 
1=2 
In the above 
/?o(1) = -ccc,g% k\ + P,m,c2 - /iano(1)e/Vx1z % (10) 
tfD(i) = ca,gn; + /5 , /n .c 2 - / v , (W, tv (for / = 2 A/), (11) 
and 
l/(0) = /UO)(l/B+tfMH<0) (12) 
with 4(0) = r K and A° =i|l+-3TT for '= 1 M-
To gain certain advantages we bring first /fc(1) and then each /&(/) into a form that 
is separable for positive- and negative-energy states. In order to bring the Hamiltonian 
into a separable form, the odd Dirac matrices for the particles are replaced by even 
terms through the successive use of a set of Tsai [10] and Weaver [11] transformations, 
and one obtains the transformed one particle Hamiltonians as 
T1 /2 
^1/2 
i*2 /«2 
peiativistic Hamiltonian for neutral atom of spin-1/2 nucleus in homogeneous magnetic field 885 
and 
M'') = A 
|1/2 
n£cA + c2\ n* ~fe*\ -/W')/3r /z + c2p'* (13) 
where the r's are Pauli spin matrices. The transformed Dirac operators hT(f) are 
manifestly separable. To obtain their eigenvalues, one needs to substitute the eigenvalues 
of the z components individual mechanical momentum and of r1z in the above [2]. 
The above process also transforms the interaction operator. The odd terms in the 
transformed interaction can be removed by a series of Foldy-Wouthuysen type of 
transformations. The final form of the interaction operator becomes 
*^res = *^Bee "*" ^Darwin "** ^Darwin "*" ^ spin-orbit + ^spin-orbit 
+
^correction "*" ^ hyperfine ^contact interaction + **hyperfme ( « 4 ) 
The components of the above interaction operator are explicitly given by 
t/Bee = - ^ E V - 1 + e 2 X V 1 .
 { Q i = _ 
KJ 
Ze) 
,2fc2/ 
uDarwin Oir?C? 
ne2h2V + 2ne)-
~ 2 3 2J ZJ5 (r«). 
mecr i >(>/) 
_ ^ 6 ^ ( 1 + 2 ^ )
 V ' 3 
"Darwin - « . J 2 „ 2 ZJ ° \'l >• 
_Ze2fr1 + 2 ^ ) y ' r,' 
"sptn-orbtt 7 1 X 5 2 ^ 2 * a <
 r>3Xpl 
+
 , - i J 2 - 2 - ai ^3xPi< 
886 Anirban Misra and Sambhu N Datta 
^correct™ = ~ „ . , * » X £ 17' t f" Pi + ^ 3 ' / C r P',)-Pl\ 2Mmec , i [r, v 
s ' l 7";'p;4w'p;)'p; • 2/n|c2 T /~> I r„ 
U, hyperfine 4me<f , ,(>,)[ rn rv 
U, 2ne
2h*(\ + ne)2 
contact Interaction —j3, L Lar°,° Ci). 
3mec , /<>,) 
Ze2fif 1 + 2/1, 2(1 + ^ ) 1 
4cz M' Mma vE'4 xp, + 
Ze2ft 
4M2c2 (15) 
The prime on each summation sign indicates that summations are taken for / = 2, ., 
N +1. The genesis of the above terms is very clear from this treatment that is 
discussed in detail in Ref. [4]. Most of these terms are well known. However, the third 
and the last term in the above, U'Darwin and l/'Hyperf.ne. are not commonly encountered 
in the literature, are lower than the respective unprimed terms by factors of m%IM2 and 
mJM respectively, and will need attention as more powerful experimental techniques 
develop. 
4. Discussion 
Most of the systems of interest in Chemistry are weakly relativistic. With a few 
exceptions like nuclear spins that control NMR, and X-ray spectroscopy, most of the 
chemical effects are in the non-relativistic limits. Therefore, one would like to know the 
non-relativistic limit of the transformed Hamiltonian. 
The Hamiltonian in the non-relativistic limit is obtained by expanding the one-
particle operators hT(iYs through order v2/c2, and replacing the transformed Casimir 
projection operators by unit operators. We get 
'nonrel = W U + AW~ + WEDr + 4WEDr + WNmr + W^ + U' . Epr rNmr 'rel (16) 
In the non-relativistic Hamiltonian, W ^ is the Schrodinger free-particle Hamiltonian, and 
Relativistic Hamiltonian for neutral atom of spin-1/2 nucleus in homogeneous magnetic field 887 
AWSCh is a corrective term due to the nuclear recoil, IVEpr is the interaction of the 
magnetic field with the total orbital angular momentum and the total spin angular 
momentum of the electrons and the diamagnetic interactions, the term AWEpr represents 
the interaction of the orbital angular momentum of the recoiling nucleus with the 
magnetic field. The term WNmr is for the energy of interaction of the nuclear spin 
magnetic moment and the magnetic field, and IV^, is the kinetic energy corrections. 
These contributions, are explicitly written as 
Here, L and S are the total orbital angular momentum and the total spin angular 
momentum operators for the electrons, and / is the spin angular momentum operator 
for the nucleus. 
This theory has the following accomplishments : (1) The nuclear magnetic 
resonance and electron paramagnetic resonance interaction terms have been derived 
here from the appropriate relativistic Hamiltonian for an all-fermion atom placed in a 
homogeneous magnetic field. (2) The correction term to electron paramagnetic resonance 
is generally not considered in literature. However, its effect would be of the order of 
hyperfine interactions and hence must be accounted for in an accurate analysis of the 
EPR spectrum. (3) Its effect should also be detected in NMR spectroscopy for a L * 0 
atom. 
As a whole, we observe that Tsai and Weaver transformations can be 
systematically applied to derive the N-electron Hamiltonian for an atom with a spin-12 
nucleus. It is possible to rigorously derive the transformed Hamiltonian which is 
separable for individual positive- and negative-energy states for all the particles, 
specifically here through order \P/c2 in interaction. The non-relativistic limit of the 
888 AnirbanMisra andSambhuN Datta 
transformed Hamiltonian is easily calculated. From this, the origin of the electronic and 
magnetic resonance spectroscopies can be explained. 
Although the treatment here is for an all-fermion atom, it can easily be extended 
to the treatment of a molecule containing only the spin-te nuclei. In the latter case, 
particles 2, 3, etc. will be considered as nuclei, and the rest as electrons. The center 
of mass needs to be redefined. The operator W^ has to be modified, and one will 
have to apply a careful accounting of the dynamics while taking the Bom-Oppenheimer 
approximation and the non-adiabatic coupling into considerations. This part of the work 
remains to be done. 
We have recently derived the expression for the retarded interaction between a 
spin-0 boson and a spin-1/2 fermion [12]. This derivation opens up the possibility of 
writing the Hamiltonian operator for a general molecule that consists of spin-0 and spin-
1/2 nuclei, like carbon, oxygen, and of course, hydrogen. 
Acknowledgment 
AM and SND are grateful for financial support from DST and CSIR respectively. 
References 
[1] S N Datta and A Misra J. Chem. Phys. 114 1478 (2001); ibid. 114 9209 (2001) 
[2] S N Datta and A Misra Int. J. Quantum Chem. 82 209 (2001) 
[3] S N Datta and A Misra Int. J. Quantum Chem. 90 812 (2002) 
[4] A Misra and S N Datta J. Chem. Phys. 123 064101 (2005) 
[5] J Sucher Phys. Rev. A22 348 (1980) 
[6] I Lindgren, S Salomonson and B As6n Phys. Reports 389161 (2004); I Lindgren, S Salomonson and 
D Hedendahl Phys. Rev. A73 056501 (2006) 
(71 E E Salpeter and H A Bethe Phys. Rev. 841232 (1951) 
[8] W A Barker and F N Glover Phys. Rev. 99 317 (1955) 
[9] J E Avron, I W Herbst and B Simon Ann. Phys. (New York) 114 431 (1978) 
[10] W Tsai Phys. Rev. D71945 (1973) 
[11] D L Weaver J. Math. Phys. 18 306 (1977) 
[12] S N Datta and A Misra J. Chem. Phys. 125 084111 (2006)

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