IACS Institutional Repository

Borah, Rashmi Rekha

Bhuyan, K

Bhuyan, P K

Imtum J Phvs. 82 (5), 545-550 (2008) 
Correlation distance of ionospheric total electron content in 
the Indian low latitude region 
Rashmi Rekha Borah*, K Bhuyan and P K Bhuyan 
Department of Physics Dibrugarh University Dibrugarh-786 004, Assam India 
E mail rashmirekha bc« rediffmail torn 
Abstract The Global Positioning System Satellite (GPS) provides an opportunity for studying Ionospheric 
tjtal electron content (TEC) We used GPS TEC data collected at 16 stations across India since 2003 to obtain 
correlation distance of ionospheric total electron content in the Indian low latitude ionosphere The correlation 
coefficients at each local time for 118 pairs of stations are calculated for every month It has been found that the 
r
- 0 7 correlation distance in longitude is approximately 1500 km in winter months 1100 km for summer months 
and 1200 km for equinoxes The r= 0 7 correlation distance in latitude is approximately 900 km in winter months 
and 700 km in summer and equinox months The nighttime correlation distance is found to be greater than the 
daytime distance 
Keywords TEC, GPS correlation 
PACS No. 94 20 cf 
1. Introduction 
The Global Positioning System (GPS) is a satellite based positioning system widely used 
for navigation, relative positioning and time transfer GPS also offers opportunities for 
Ionospheric research [1, 2, 3] The dispersive ionosphere introduces a time delay in the 
1 57542 GHz (L1) and 1.22760 GHz (L2) simultaneous transmissions from GPS satellites 
orbiting at 20,000 km The relative ionospheric delay of the two signals is proportional to 
the total amount of electrons along the ray path, which is also known as the total electron 
content (TEC). Time delay measurements of L1 and L2 frequencies can, therefore, be 
converted to TEC along the ray path from the receiver to the satellite With the increasing 
positional accuracy requirements, the ionospheric parameter TEC is becoming a parameter 
of greater importance because it is directly related to the group delay that the ionosphere 
imposes upon the RF signal propagation time. Global distributions of TEC variations as 
Corresponding Author 
© 2008 IACS 
546 Rashmi Rekha Borah, K Bhuyan and P K Bhuyan 
well as its characteristics at equatorial, middle and high latitudes have been investigated 
by a number of workers. TEC measurements from GPS supplements TEC obtained from 
earlier beacon satellites. 
The study of correlation of total electron content at pairs of stations is important not 
only from the scientific view point but also from practical considerations such as prediction 
of TEC at a location where no observational facility exists. Correlation distance is a measure 
of the variability of the ionosphere at the concerned locations. As the solar activity 
increases, the variability increases at each location reducing the correlation distance 
Pioneering works on TEC correlation distance were done by Rao et al [4] and daRosa 
[5]. Klobuchar and Johanson [6] had observed that at midlatitudes the correlation of TEC 
decreases with increasing longitudinal or latitudinal separation of the monitoring stations 
Bhuyan and Tyagi [7] had performed a correlation analysis of TEC data over India for a 
period of low solar activity and found that the correlation coefficient decreases from about 
0.9 at a distance of about 300km to 0.5 at a distance of 2000 km. Because of the 
complexity of the Indian zone ionosphere, the correlation of TEC measured at a given set 
of stations decreases rapidly as the latitudinal/ longitudinal separation increases. In this 
study, the correlation coefficients at each local time for 118 pairs of stations are calculated 
for every month. We have chosen the period from 10 to 16 hours local time to perform 
correlations of the day-to-day variability at pairs of stations. We refrained from using TEC 
values over longer intervals of a day, in order to avoid inadvertently correlating diurnal 
shape changes at pairs of stations. For nighttime correlation we have chosen the period 
from 23 to 03 hours local time. 
2. Data Base 
The total electron content of the ionosphere along the line of sight from the receiver to the 
GPS satellite (slant TEC) is given by the expression 
At
 = (40.36/ 'cf2)xSTEC (1) 
where At is the time delay introduced during the passage of the signal through the 
ionosphere, c is the velocity of e.m. radiation and f is the transmission frequency. GPS 
slant TEC data were collected at 18 stations: Trivandrum (8.5°N, 77°E), Agatti (10.75 N. 
72.5°E), Port Blair (11.75"N, 92.5"E), Bangalore (13°N, 77.5°E), Mumbai (18.5°N, 72.5E). 
Hyderabad (17.5UN, 78.5"E), Vishakhapatnam (17.5 "N, 83'E), Ahmedabad (23 N, 72.5 E). 
Bhopal (23°N, 77.25UE), Raipur (21WN, 81.5°E), Jodhpur (26.25X 73°E), Delhi (28.75 N. 
77.25°E), Aizwal (23.5°N, 93°E), Guwahati (26°N, 92°E), Bagdogra (27°N, 88.5'E), Kolkata 
(22.5K 88.25°E), Shimla (31.08'N, 77.06°E) and Lucknow (27.25"N, 80.88°E) across India 
since 2003. The slant TEC measurements made are the sum of the real slant TEC, the 
GPS satellite differential delay bs (satellite bias) and the receiver differential delay, bP 
(receiver bias). Therefore, the vertical TEC can be expresses as, 
VTEC = (STEC-[bR + bs])/S(E) (2) 
Correlation distance of ionospheric total electron content in the Indian low latitude 547 
where STEC is the slant TEC measured, E is the elevation angle of the satellite in 
degrees, S(E) is the obliquity factor with zenith angle at the ionospheric pierce point 
flPP) and VTEC is the vertical TEC at the IPP. The obliquity factor, S(E) (or the mapping 
function) is defined as 
S(E) = 1 
cos(z) = < i -
( RExcos(E) 
RE + hs 
N2 0 5 
(3) 
where RE is the mean radius of the earth in km, hs is the ionosphere (effective) height 
above the earth's surface, z is the zenith angle and E is the elevation angle in degrees 
(PV.S. Rama Rao et al. [8]). 
3. Significance of correlation coefficient 
The percentage improvement, PI, in the prediction of TEC is related to the correlation 
coefficient r by 
P/ = 100 1 - ( W * } 
It was pointed out first by Gautier et al [9] in the prediction of foF2 and after that 
Klobuchar [10] successfully used it for TEC prediction. This relationship shows that a 50 
percent reduction of TEC uncertainty requires a correlation coefficient of 0.87. A correlation 
coefficient of 0.7 between TEC values taken at the same local time at pairs of stations 
will result in an improvement of only 29% in prediction of TEC at one station using data 
from the second station. In this study this value of r = 0.7, corresponding to a 29 percent 
uncertainty reduction, is defined as the 'correlation distance'. 
4. Results and discussion 
4.1. Longitude separation: 
The correlation coefficients for the period of 10 to 16 hour local time for 118 pairs of 
stations were calculated for every month and these monthly values were averaged, in 
order to obtain a seasonal mean value to see linear dependence of r upon ionospheric 
longitude separation. Figure 1 shows daytime and nighttime correlation coefficient with 
longitude separation for each season. We found that the r = 0.7 correlation distance in 
longitude is approximately 1500 km in winter months, 1100 km for summer months and 
for equinoxes correlation distance is approximately 1200 km. The nighttime correlation 
distance is greater than the daytime distance. Klobuchar and Johanson [6] had observed 
that at midlatitudes the correlation of TEC decreases with increasing longitudinal or latitudinal 
separation of the monitoring stations. They found that for a 29 % TEC prediction improvement 
at a given geographic location, a TEC monitoring station must be located within 
548 Rashmi Rekha Borah, K Bhuyan and P K Bhuyan 
approximately 2900 km longitudinal separation and within 1800 km latitudinal separation 
Because of the complexity of the Indian zone ionosphere, the correlation of TEC measured 
at a given set of stations decreases rapidly as the latitudinal/ longitudinal separation 
increases. Bhuyan and Tyagi [7] had performed a correlation analysis of TEC data over 
India and found that the correlation coefficient decreases from about 0.9 at a distance o< 
about 300km to 0.5 at a distance of 2000 km during a period of low solar activity Tc 
study the nighttime behavior, we average the correlation coefficient values for the period of 
22 to 04 hours local time. The correlation coefficients of 118 pairs of stations were calculated 
for each month and these monthly values were averaged, in order to obtain a seasonal 
mean value. Nighttime correlation distance for r = 0.6 are 612 km during winter months 
1615km for summer and 950km for equinox. Nighttime correlation distance during summer 
is very high compared with other seasons. 
Daytime 
o o 
o 
0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 
1 Nighttime 
500 1000 1500 2000 2500 50 0 1000 1 5 0 0 2 0 0 0 2 G 0 0 0 5 00 1 0 0 0 1 S00 ?0 t i f 
Longitudinal Separation, km 
Figure 1. Daytime and nighttime correlation coefficient with longitude separation for each season 
4.2 Latitude Separation : 
Figure 2 shows latitude separation of station pairs with correlation coefficient. The correlation 
coefficients for the 10 to 16 hour local time period for 118 pairs of stations are calculated 
for each month and these monthly values are averaged, in order to obtain a seasonal 
mean value. The seasonal mean values of correlation coefficients are plotted as a function 
of station spacing in latitude. We found that the r = 0.5 correlation distance in latitude is 
approximately 2000 km in winter months, 1600 km for summer months and for equinoxes 
correlation distance is approximately 1500 km. If we consider r = 0.7 correlation distance 
in latitude is approximately 900 km in winter months and 700 km in summer and equinox 
months. The nighttime correlation distance is shorter than the daytime distance. 
The spatial and temporal variability of the F region remains of great research interest 
It is unlikely that short-term solar EUV changes can produce the observed spatial TEC 
variability. The lower correlation distance for stations at nearly the same longitude, but 
Correlation distance of ionospheric total electron content in the Indian low latitude 549 
separated in latitude, could likely be due to the different day to day strength of the 
neutral wind in the F region as a function of latitude. The neutral wind, normally blowing 
away from the sub solar point, can drive ionization up or down in altitude along magnetic 
0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 
Nif&ihittUOT e 
0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 
Longitudinal Separation, km 
Figure 2. Daytime and nighttime correlation coefficient with latitude separation for each season 
field lines to regions of lower or increased loss, which would significantly change TEC. 
Equatorial electrojet strength which transport ionization from the magnetic equator to the 
low midlatitudes can also change the TEC as a function of latitude. P.V.S. Rama Rao et 
al [8] have observed that the day-to-day variations in TEC are correlated to the day-to-
day variability in the corresponding EEJ strengths. Also, the seasonal variation of TEC 
shows a feature similar to that observed in the integrated equatorial electrojet (IEEJ). 
Simultaneous observation of ionization density bubbles and amplitude scintillations from 
multi-satellite measurements in the Indian low and equatorial latitudes.has been reported 
by Paul et al. [11]. 
Mar*04 May04 June'04 -**— July'04 
Aug'04 Sep'04 OcT04 NovAM 
-^ -Dec '04 Jan*05 Feb'05 
0 2 4 6 8 10 12 14 16 18 20 22 
Local Time 
Figure 3. Temporal variation of coefficient of correlation for Ahmedabad-Aizwal during the period March 
2004 to February 2005. 
550 Rashmi Rekha Borah, K Bhuyan and P K Bhuyan 
Figure 3 shows temporal variation of coefficient of correlation for Ahmedabad - Aizwai 
pair of stations during the period March'04 to Feb'05. It is observed that during the middav 
hours TEC correlation is high and in early morning and nighttime hours TEC correlation is 
low for all months. Similar features could be seen for station pair Delhi-Raipur. In Figure A 
the month vs correlation coefficient plot for station pair Ahmedabad-Aizwal shows seasonal 
dependence of r. Daytime TEC correlation is low during the months of April, May and 
June and high in late winter, the month of July and the autumnal equinox months The 
nighttime TEC correlation is lowest for the equinox months April and October and highest 
for February, March. During June, July, September and November nighttime TEC correlation 
is higher than that of daytime. 
090 
0 80 
0?0 
OtO 
0 50 
0 40 
0^0 
0<?0 
0 10 
0 00 
F M A M J J A b O N D J l 
Months <P«b 2004 to Fer> ?00S) 
Figure 4. Seasonal variation of coefficient of correlation for Ahmedabad-Aizwal during the period Februarv 
2004 to February 2005 
Acknowledgment 
This work was carried out with partial support from the Space Application Centre 
Ahmedabad. The authors would like to thank Dr. K. Bandyopadhyay, Dr. M.R. Sivaraman 
and all others associated with the 'GAGAN' project. 
References 
[1] G E Lanyi and T Roth Radio Science 23(4) 483 (1988) 
[2] D Coco GPS World 47 (1991) 
[3] K Davies and G K Hartman Radio Sci 32 1695 (1997) 
[4] N Narayana Rao, M Y Youakim and K C Yeh SAMSO TR71-163 (1971) 
[5] A V daRosa Space Research XIV 209 (1974) 
[6] J A Klobuchar and J M Johanson Air Force Geophys Lab Report GL -TR-77-0185 (1977) 
[7] P K Bhuyan and T R Tyagi Indian J Radio & Space Phys. 18 134 (1989) 
[8] P V S Rama Rao at al., Ann Geophys 24 3279 (2006) 
[9] Gautier et al, Conference Record, First Annual IEEE Communications Convention (1965) 
[10) J A Klobuchar Progress in Astronautics and Aeronautics 164 485 (1996) 
[11] A Paul et al, Radio Sci. 37 1087 (2002) 
pDay(10-16LT) 


English

Correlation Distance of Ionospheric Total Electron Content in the Indian Low Latitude Region

Krori, K D

Dutta, Sabita

Das, Kanika

Indian J. Phys. 82 (6), 665-672 (2008) 
Direct derivation of Sagnac effect 
K D Krori!\ Sabita Dutta2and Kanika Das* 
theoretical Physics Forum, Lachitnagar, Bye Lane ~7, Sub Lane 2 Guwahati-781 007 Ass.im India 
'Department of Physics, Shrimanta Shankar Academy Panba/at fiuwahati 781 001 Assam India 
^Department of Mathematics Gauhati University Guwahati 781 014, Assam India 
E mail kdkroritt/ icdiffmail com 
Abstract RIZZI and Ruggiero and later Ruggiero independently have shown that it is possible to calculate 
Sagnac effect in flat, Schwarzschild, Lense-Thirring (slowly spinning sphere), Kerr and Godel metrics in 
analogy with Aharonov-Bohm effect 
One may reasonably wonder Is it possible to derive this effect independently, / e by some direct method ,} 
In this paper, we show that the answer to this question is indeed in the affirmative 
Keywords Sagnac effect, general relativity 
PACS No. 04 20 q 
1. Introduction 
M.G. Sagnac predicted the so-called Sagnac effect [1] in 1905 and experimentally verified 
it in 1913 [2] for light. Since then, a lot of experimental and theoretical studies have been 
carried out on this effect [see Ref [3] and Ref. [4] and other references therein] It has 
been found that this effect holds for both luxons (light-like particles) and tardyons (material 
particles). 
The subject of derivation of the Sagnac effect has attracted much attention. Recently 
Rizzi and Ruggiero [3] and later Ruggiero [4] have derived the effect in analogy with 
Aharonov-Bohm effect. One may reasonably wonder : Is it not possible to calculate the 
effect independently, i.e. by some direct method ? We show in this paper that it is 
indeed possible to derive this effect by a direct, simple method in different types of metrics 
viz., flat, Schwarzschild, Lense-Thirring, Kerr and Godel metrics. 
Corresponding Author 
© 2008 IACS 
666 K D Krori, Sabita Dutta and Kanika Das 
2. Direct derivation of Sagnac effect in some metrics 
We first demonstrate the method of derivation in a flat metric and then apply it to some 
other metrics. 
2.1. Flat metric: 
We start with the following flat metric : 
ds2 - c2dt2 - dr2 - r2dd2 - r2 sine2 eft2 
For r= R (constant) and 6 =n/2 (constant), eq.(1) becomes 
ctf^ctdf-tfdj2 
We apply the transformation 
0 =0 + 12 t 
where Q is an angular speed. 
Using eq. (3) in eq. (2), 
ds2 = 1- FPQ
1 
c2di* - fl2^2 - 2R2Q d<t> dt. 
(1) 
(2) 
(3) 
(4) 
From eq. (4), the proper time-interval is 
dx = R
2£22 / 2 dt 
d$ 
*%* ^ * % *
 i 2t f*f l* 
r tfa2^
 2/\ tfQ2\ 
1 • 
1- c2 1--
R2Q2 > 
(5) 
From eq. (5), for a beam of particles co-propagating along a semicircular path, we 
have, 
n-}* 
Direct derivation ofSagnac effect 667 
2R2Q d0 
c*h-£rf vd-^!f vfi-^L2' 
= /4 (1s t integral) + B (2nd integral) + - 2nR
2tt 
(6) 
fl2^2) 
Again, from eq. (5), for a similar beam of particles counterpropagating along a 
semicircular path, we have, 
•I-r2 = I dx 
ds ds 
.j_*Il*ll_J.+ J 
(-0» . f l 2 ^ ^ ) » 
dt 
c2 c2 
i 2R2Q(-d<t>) 
v[i-*£r % - * £ ? vfi-^fl8^ 
c2 
= A (1st integral) + B (2nd integral) - - 2nR
2Q 
\_f?a?\ 
(7) 
The proper time-difference between the two beams therefore is 
An R2® 
4 T = T 1 - T 2 = J 
° ' tfQ2)2 1 - -
c2 
(8) 
This is the same as eq.(33) in [3]. Obviously eq. (8) is true for all types of particles — 
luxons as well as tardyons. 
668 K D Krori, Sabita Dutta and Kanika Das 
2.2. Schwarzschild metric: 
The Schwarzschild metric for a spherical object is 
ds* 1-
2GM 
re2 
c2dt2 ' 2GM^ 
~ re2 
dr2 -r2 de2-r2s\n26 d <t>2 (9) 
where M is the mass of the object. 
For r = R (constant) and 0 - — (constant), 
ds2 = 1-
2GM\ 
Re2 J c
2dt2-R2dd>2 
Using the transformation (3) in eq. (10) 
2GM tfQ2 ds< = 1 - -
Re2 c2 
c2dt2 - R2d <p2 - 2R2nd$ dt 
(10) 
(11) 
From eq. (11), the proper time-interval is 
dr = 
2GM tfQ 2 \ 
Rc2 c2 
dt 
ds ds , t 
— dd> 
dt (d<t>) v 
2/-»2 \ 
1- 2GM _ R'Q 
Rc2 c2 
R2«±d<t, 
dt v 
2GM R2n2^ 
Rc2 c2 
2R2® d<t> 
L 2GM R2Q2 )2 
1-
Rc2 c2 
(12) 
From eq. (12), for a beam of particles co-propagating along a semicircular path, we 
have 
r,=\dr = \-
ds ds 
dt d<t> 
d<t> 
c< 1-
2GM tfQ 2 \ 
Rc2 c2 
Direct derivation of Sagnac effect 669 
,2 d$ 
* Ff — d<l> * _ 
f 0?
 | T 2R2Qd0 
c21'., _ 2GM _ fl2£>2 ^ o ^ 2G/W R2r22 )2 
Rc2 c2 Re2" c2 
= A (1st integral) + B (2nd integral) 
2nR2Q 
o\ 2GM tfn2)* 
° ^ RcT~~cT 
(13) 
Similarly, from eq.(12), for a similar beam of particles counterpropagating in a semicircular 
path, we have, 
. „ 2nR2Q 
r2 = A+B-- (14) 
{ 2GM R2Q2 
Re2 ~~'c2 
Hence, the proper time-difference between the two beams is 
An PpQ 
2GM R2®2 
Rc2 c2 
(15) 
This is the same as eq. (27) in [4]. 
2.3. Lense-Thirring metric (slowly spinning sphere) 
The Lense-Thirring metric is [5] 
ds2 =11 J l - ^ c 2 dt2 -(: + ^}\dr2 + r2d62 +r2s\n26d02] 
rc2 ) { rc2 jL 
^ sin2 0oty edt (16) 
670 K D Krori, Sabita Dutta and Kanika Das 
For r = R (constant) and 6 =/r/2 (constant), eq. (16) becomes 
ds2 = 
' 2GM\ 
Re2 J C
2
 dt2 - 1 + 2GM 
Rcz 
R2 d(t> -o 4GJ _,-
o~3 dQcdt, 
Using the transformation (3) in eq. (17), 
(17) 
ds2 = 1 -
2GM 
Re2 
1 + 
2GM 
Re2 
Rz£22 AGJQ 
RcA 
C2dt2 
1 + 2GM \„2 ,1*2 
Re2 
R< d<S>d - „^EWQ_^J 
Rcd RcJ 
dQ dt (18) 
Now, proceeding as before, we can find that the proper time-difference between two 
identical oppositely circulating beams of particles is 
1 + 2GM 
AT = An LI Ac
2
 j R
2Q- 2GJ 
Re2 
C2 r 
1 2GM (^ t 2GM\R2Q2 | AGJQ 
Re' Re2 c2 Rcq 
(19) 
This is the same as eq. (11) in [4]. 
2.4. Kerr metric: 
The usual form of Kerr metric is 
PC2 A 
?o\ 
2 2 2GMafrs\rf6 
r +ar + T-T, 
P2c2 
. 2 n - / i 2 4G/War sin2 6 , r .. sinz0 d 0 z + — c/0 cdt 
P cr (20) 
where 
p2 = r2 + a 2 cos 2 0 , 
A = r * - 2 ^ + *9Ma = j/<*t 
cr 
J being the absolute value of angular momentum. 
Direct derivation ofSagnac effect 671 
For r= R (constant) and 6 = - (constant), (20) becomes 
<*? = 1 - 2GMr fdi2-
Re2 ) 
Using the transformation (3) in (21), 
tf+a2 2GMaP 
Re2 
.« 4GMa^ ^ 
FfcP (21) 
ds2 = 1 - 2GM 
Re2 
R2
 + a
2
 + 
2GMa2 
Re2 
Q2 AGMaQ 
c2 Re2 
c2 dt2 
„ 2 2 2GMad Fr+ar + - „ 
Re2 
d<t>2-2 ,2 2 2GM3
2
 ) R< + ad + ' 
fte2 
r2- 2G/Wa 
flc 
d0 dt (22) 
Now, proceeding as usual, we can find that the proper time-difference between two 
indentical oppositely circulating beams of particles is 
AT = x 
,2 2 2GMa2 * Ff + af* 
Re2 
Q- 2GMa 
Re 
1 - 2GM 
Re2 
R2 + a2 + 2GMa 
Re2 
2
 ^ AGMaQ 
c2 Re2 
(23) 
This is the same as eq. (17) in [4]. 
2.5. Gddel metric: 
The Godel metric is 
ds2 = c2 df2 dr< 
1 + l 2a J 
I2aj d <!>
2
 - dz2 + 2r2-^d 0 cdt (24) 
V2a 
where a is a constant > 0. 
Now, for r = R (constant) and z = constant, 
ds2 = c 2 dt2 - R2 1 l 2 a J d02 + ^ d<p cdt •42a (25) 
672 K D Krori, Sabita Dutta and Kanika Das 
Using the transformation (3) in eq. (25) 
ds2 = 1 - R 2 R) 
2\ 
1
-'ii 2
 2Rl Q'-
•ht 
Qc <? 
xtftf-R2 1 2 a J 
2" 
d<t>2- 2R2£2 
< 
1-
2aJ 
2\ 
) 
_2R2c 
yfeai
 j 
d0dt (26) 
One can see that in this case the proper time-difference between two identical oppositely 
circulating beams of particles is 
Ar= x 
cr 
R*Q 
2a 
L 
•J2~B 
1 - R2 2a 
n2-*™n 
& a 
<? 
(27) 
This is the same as eq. (21) in [4]. 
3. Concluding remarks 
Thus we find that it is possible, without having recourse to any analogy or any external 
aid, to derive Sagnac effect in a number of metrics in a straight-forward manner. 
Acknowledgment 
The authors express their profound gratitude to the Government of Assam for all facilities 
provided at Cotton College, Guwahati-781001 (India) to work out this paper. 
References 
[1] M G Sagnac C. ft Acad Sci. Paris 141 1220 (1905) 
[2] M G Sagnac C. ft Acad Sci. Pahs 157 708 (1913) 
[3] G Rizzi and M L Ruggiero Gen. Rel. Grav. 35 1745 (2003) 
[4] M L Ruggiero Gen. Rel. Grav. 37 1845 (2005) 
[5] R Adler, M Bazin and M Schiffer Introduction to General Relativity (New York : McGraw-Hill Book 
Company), p257 (1975) 


Direct Derivation of Sagnac Effect

Zoliana, B

Thapa, R K

Indian J. Phys. 82 (6), 729-734 (2008) 
111 J Pi 
Application of projection operator method to define basis 
functions for use in photoemission calculations 
B Zoliana1* and R K Thapa2 
•Department of Electronics, Govt Zirtin Residential Science College, Aizawl, M,zoram-796 001, India 
department of Physics, Mizoram University, Aizawl, Mizoram- 796 001, India 
E-mail bzohana(« yahoo com 
Abstract : We have applied projection operator method of group theory in deriving the symmetry adapted 
wavefunctions defined by
 ¥l which represents the surface electronic states in the case of a strong periodic 
potential. The method involves the defining of basis function pertaining to a particular symmetry point of Cu(110), 
from which v, is formulated by the LCAO method This has been applied to calculate photocurrent from Cu (110) 
surface state. We have also used this method of calculation for a metal like Al 
Keywords : Periodic potential, projection operator, wavefunctions, photocunent 
PACSNo. : 79.60.Bm 
1. Introduction 
For the use of Projection Operator for the case of strongly bonded metals, we define the 
crystal potential by a Kronig-Penney 8 -potential which is periodic with the periodicity of 
i 10 . 
..AJ 
mcfal 
1 
1 
I' 
L 
,1 
i-i i va< 
r 
V(y) 
• - » 
FJgurt 1. Schematic representation of Kronig-Penney 3 -potential model. 
* Corresponding Author 
©2008IACS 
730 B Zoliana and R K Thapa 
the lattice as shown in Figure 1. In our calculation, the Schrodinger equation will then be 
solved to obtain the solution in terms of Green function [1]. 
The method involves the projection operator operating on linear combination of atomic 
orbitals (LCAO) representation of the surface states giving the basis function and this 
basis function has been incorporated in defining y/t. This had been applied to calculate 
photocurrent from the Cu (110) surface state. The calculation has been carried out in one-
dimension and it was found that the basis function derived in one dimensional form can 
be extended to a metal like Al. 
2. Formalism 
2.1. Derivation of basis function by projection operator method: 
For Cu (110), surface state occurs [2] in L? - L, band gap. The LCAO representation for 
L2- point is -r=r(x+ y + z) . The projection operator formula [3] is given by 
v3 
y T 
Here lp is the dimension of the unitary irreducible representation of the group g, g is 
order of g and 2 ^ extends over all the transformation T of g. 
T 
Eq.(1) operating on the LCAO representation for L2, gives a set of basis functions 
—f=\x + y + z). This basis function in one dimensional form is incorporated in the derivation 
of initial state wave functions. 
2.2. Derivation of initial state wavefunction: 
The solution of Schrodinger equation in terms of Green function, in one dimensional form 
may be written by the integral equation 
V{y)\i/(y)G(z,y)dy
 m (2) 
Let the Kronig-Penney potential be represented by a linear Dirac 5 -function i.e 
V{y)>*[2£] X*(y-na)
 (3) 
where P = Km ^:Xt * b with Xo : = 2 ( v i ~ Ei). Here, b is the width of Kronig-Penny potential 
b-~*0 £ 
moad and V1 is the height of the potential, p is the strength of the s -function barrier 
Application of projection operator method to define basis functions etc 731 
and it is assumed to be positive. The final electronic wavefunction after substitution of the 
above value (eq.3) in eq. (2) gives 
^ (z )= -^Lre / / c a s in /c / z~s in /c / ( z -a ) l iM 
sin/c,aL ' /V 'J (4) 
2C/psin2fc,a 
where r(0) = —— - —_ - . (5) 
kla(e,ka-e~,H,a)(ooska-coskta) 
Introducing the atomic orbital <j>(z) which includes the basis function derived by P.O 
method as discussed above the initial state wavefunction can be represented by 
, . IvWQWe*'^ k, cos k,z- xs\n k,z), z<0 (bulk& surface) 
[if/l(0)klexz, z>0( vacuum) v ' 
2/sin/ca r(0) 
kt cos k,a - k,e~lk a - x sin k,a 
where * , « » - , --_,z _._,;.' ..._.._• < 7 > 
2.3. Photocurrent calculations: 
The photocurrent density formula from golden rule approximation can be written as 
ffi.^ry^^iH'I^H2 5(E-Ef)S(Ef - E, -hw)10(E -»<«>)[1- f0{E)] (8) d£2 ti ^ 
where ^ ( fF , ) refer to the initial (final) state wavefunctions and perturbation H ' can be 
written as 
mc "*.«5+i!*M (9) 
where > \ , ( z ) = ^ d ^ , / ^ ( z ) is the component of vector potential along z-axis. The 
final state wavefunction |V,) used is the scattering state of the step potential[4] which is 
encountered by the electron. The matrix element / - (v, | H' |y/,) in eq. (8) can be written 
in one dimensional form as : 
The integrals in Eq. (10) are evaluated numerically by writing FORTRAN programmes. 
732 B Zoliana and R K Thapa 
3. Results and discussions 
3 1 Copper 
The plot of photocurrent against the photon energy ho) for Cu is shown in Figure 2 For 
the same value of surface state [5] energy (2.72 eV below Fermi level) and potential 
barrier height (11.4 eV) , photocurrent was calculated for two values of surface widths 
We find that for d= 10 a.u., a maximum in the value of photocurrent occurs at fico = 14 
eV. With further increase of photon energy, photocurrent decreases to a minimum value 
at h(D = 19 eV and the next hump occurs again at ttw = 23 eV But for a narrow 
surface width (of = 0), the behaviour of photocurrent is quite different as shown in Figure 
2. We do not find any peak for values of photon energy below and above 19 eV The 
behaviour of photocurrent shows a qualitative agreement with those shown by other 
metalslike W, Si, etc. in which Kronig-Penney potential model was also used [5] 
its
) 
«£, 
| 
R 
710E+00 
7 00E+00 
6 90E+00 
6 80E+00 
6 70E+00 
6 60E+00 
6 50E+00 
6 40E+00 
6 30E+00 
6 20E+00 
10 15 20 25 
Photon Energy (eV) 
30 
Figure 2. Plot of variation of photocurrent with photon energy for surface widths d = 10 a u and d = 0 
in copper using Kronig-Penney 5 -potential where y/t is derived by projection operator method of Group 
Theory 
3.2. Aluminium: 
Figure 3 shows the plot of photocurrent versus photon energy (hco ) for the surface width 
d= 10 a.u and narrow surface width d= 0 a.u. We have chosen the initial state energy 
£, for a high lying surface state occurring at 0.41 eV below Fermi level, the location of 
the Fermi level and the work function 0 for AI are taken as 11.7 eV and 4.25 eV, 
Application of projection operator method to define basis functions etc 733 
respectively. The photocurrent profile for d = 10 a.u. showed a strong photoemission at 
photon energy too = 9 eV, which was followed by a decrease in photocurrent and the 
minimum occurring at too =15 eV (the plasmon energy of Al is lnop = 15.3 eV). There 
is another hump in the photocurrent data at hco = 20 eV. The behaviour of photocurrent 
in the case of narrow surface width is quite different. For example, almost a flat peak in 
a photon energy range 10 eV-14 eV was obtained. Although the minima in photocurrent 
was seen at hco = 15 eV, contribution to photoemission was far less important. However 
the photoemission results for d = 10 a.u. seems to be in qualitative agreement with the 
experimental results of Levison et al [7] (see Figure 3 (inset)). For example, the experimental 
data of Levinson etal [7] showed a maxima in photocurrent at hw = 13 eV with occurrence 
of a minima at the plasmon energy (^ 15.3 eV). Also, it showed similar behaviour with 
the calculated results of Thapa et al [8,9]. 
5.00E+01 
4.50E + 01 -
4.00E+01 
— 3.50E + 01 
to 
E 
£ 3.00E + 01 
S 2.50E + 01 
8 
o 2.00E + 01 
CL 
1.50E+01 
1.00E + 01 
5.00E+00 
0.00E + 00 
10 15 20 25 30 
Photon Energy (eV) 
Figure 3. Plot of photocurrent variation against photon energy In the case of aluminium, tor surface width 
d = 10 a.u. and d = 0 by employing Kronig-Penney tf-potential where v, is derived by projection 
operator method of Group Theory. 
Inset • Experimental results for aluminium and beryllium as obtained by Levinson et al [7]. 
4. Conclusion 
In our calculations of photocurrent, we have seen that photoemission was considered to 
take place in metal, for location of surface state below the Fermi level. Symmetry state 
or direction to the existence of surface state in a particular band gap was ignored. 
Therefore, to include band structure effect, we have applied projection operator method of 
T — i 1 — i 1 r 
\^^Jb^o 
06 0? 0J 0V tO M 12 
* * • > V -7 N 
\ 
k
»-x 
734 B Zoliana and R K Thapa 
group theory to derive initial state wavefunction ^, pertaining to a particular band gap 
When compared to experimental data and other previous theoretical results like Kronig* 
Penney potential model of Thapa et al [6], we find that calculated data with this approach 
of calculations seemed to agree in a better way. This is quite evident from the photoemission 
result from Al especially for which also experimental result is available. However, there 
are still certain drawbacks remaining for further corrections. For example, we have not 
considered the nature and type of potentials which defines the bulk and surface regions 
This plays equally important role in the definition of dielectric response function too. From 
all the above facts, the model employed in our calculations seems to be highly simplified 
However, as evidenced from the comparison with other approach of calculations, as well 
as with experimental results [7], our method seems to be quite appropriate. 
Acknowledgment 
B Z acknowledges Minor Research Project from UGC and RKT a grant from CSIR (Delhi) 
References 
[1] S G Davison and M Steslicka Basic Theory of Surface State (Oxford : Clarendon) (1992) 
[2] E Bertel Phys. Rev. B50 4925 (1995) 
[3] J F Cornwall Group Theory and Electronic Energy Band in Solids (Amsterdam, Holland North 
Holland Publishing Company) (1969) 
[4] B Zoliana A Theoretical Formulation of initial Statewave functions for photoemission calculations, 
Ph.D. Thesis, NEHU (2004) 
[5] E W Plummer and E Eberherdt Advance in Chemical Physics 49 533 (1982) 
[6] R K Thapa and N Kar Phys. Rev B51 17980 (1995) 
[7] H T Levlnson, E W Plummer and P J Feibelman Phys. Rev. Lett 43 952 (1979) 
[8] R K Thapa, P Das and N Kar Mod Phys. Letts. B8 361 (1994) 
[9] Zaithanzauva Pachuau, B Zoliana, P K Patra, D T Khathing and R K Thapa Phys. Letts A294 52 
(2002). 


Application of Projection Operator Method to Define Basis Functions for Use in Photoemission Calculations

Borah, Sankar Moni

Pal, Arup Ratan

Bailung, Heremba

Chutia, Joyanti

Indian J Pliys 82 (6), 741 -746 (2008) 
4^ 
Hardness study of titanium nitride thin films deposited on 
bell-metal by cylindrical magnetron sputtering 
Sankar Mom Borah*, Arup Ratan Pal, Heremba Bailung and Joyanti Chutia 
Plasma Physics Laboratory Material Sc it nets Division 
Institute of Advanced Study in Science & Technology (IASS1) 
V.gyan Path Paschim Boragaon Gorchuk Guwohati 781 03r> Assam Incln 
F mail sankarmonif< gmail com 
Abstract Deposition of thin films by reactive magnetron sputtering is a very suitable method for the 
surface protection of materials Titanium nitride thin film coatings provido very good protection to materials from 
wear and tear, corrosion oxidation etc Thin films of titanium nitride deposited by reactive magnetron sputtering 
in argon/nitrogen gaseous mixture acts as hard protective coating and also prevent the bell metal substrate from 
reacting with the environment and getting oxidized An investigation on the hardness of titanium nitride films 
deposited by reactive magnetron sputtering has been carried out by the method of nano indentation and an 
analysis of the films for different deposition conditions is presented The TiN film hardness is found to increase 
13 times at 50% nitrogen in the gaseous mixture 
Keywords Titanium nitride cylindrical magnetron sputtering hardness 
PACS Nos 52 77 Dq 52 80 Sm 
1. Introduction 
Sputtering deposition has become a generic name for a variety of sputtering processes 
Among the various sputtering processes, the one, which is most widely used today, 
involves the use of magnetic field perpendicular to the electric field and is known as 
magnetron sputtering Magnetron sputtering uses confinement of excited plasma in crossed 
magnetic and electric fields that results in high deposition rates and good reproducibility 
[1,2]. Titanium nitride is a rocksalt structure (NaCI), the compound consisting of Ti atoms 
filled in fee based lattice with all octahedral sites filled with nitrogen atoms Some of its 
physical properties are as follows Lattice parameter 0 424 nm, Melting point 3173 K, 
Thermal stability (Aff298) 336 6 kJmol \ Young's modulus 450 GPa TiN thin films have 
found applications in hard, decorative, wear and corrosion resistant coatings and also 
Corresponding Author 
© 2 0 0 8 I A C S 
742 SankarMom Borah, Arup Ratan Pal, Heremba Baifung andJyanti Chutia 
diffusion barriers in IC technology [3] Further, it has good electrical, thermal, mechanical 
and chemical properties. In micro-electronics instrumentation, it finds application for its 
electrical characteristics and for its diffusion barrier properties [4] Study of TiN deposition 
has generated a lot of interest among the scientific community because of its lustrous 
golden yellow coloured film which has been used for decorative applications. The golden 
colour of the film is due to the high reflectance of TiN at the red end of the visible 
spectrum with low reflectance near the ultraviolet region [5]. The formation process of TiN 
thin films by reactive sputtering has been studied by many researchers over the years In 
the case of reactively sputtered TiN film coatings, nitrogen partial pressure plays an 
important role in determining the mechanical properties [6]. Deposition of thin films by 
reactive magnetron sputtering is a very suitable method for the surface protection of 
materials Reactive magnetron sputtering is used for the elaborate study of titanium nitride 
thin films [7] Here, we have presented an analysis of the hardness study of TiN thin film 
deposited on bell-metal in reactive Ar/N2 gas environment by direct current magnetron 
sputtering method. 
2. Details 
A. Experimental set up and procedure: 
The experimental cylindrical magnetron device is a stainless steel chamber having 
dimensions of 30 cm in diameter and 100 cm in length A small titanium cylinder of 
length 25 cm and outer diameter 3.25 cm is placed co-axially inside the chamber which 
acts as the cathode For deposition of TiN, the bell metal substrates of size 0.8 cm x 0 8 
cm are placed at a distance of 8 cm below the cathode A schematic diagram of the 
experimental set up is shown in Figure 1. For generation of a steady axial magnetic field, 
Pirani Gauge 
Penning Gauge 
N2|Ar ^as 
if 
ER EA 
Cathode 
p| Substrate holder 
bate vaivi 
si X YRecordar 
Rotary Pump 3 
Diffusion Pump 
Figure 1. Schematic diagram of experimental set-up E-electrlc field, B-Magnetic field, ER-End reflectors, 
Lp-Langmuir probs, MM-Magnetic coils, PS-Power supply 
Hardness study of titanium mtnde thin films deposited on bell-metal etc 743 
two coils are placed around the body of the chamber
 8.m.lar to Helmholtz co.ls type 
configuration Low pressure is created ins.de the chamber us.ng a combination of rotary 
and diffusion pumps The base pressure of the chamber is of the order of 10* Torr and 
working gas pressure is of the order of 10 3 Torr A piran, gauge and an ionization gauge 
are used for measurement of pressure ms.de the chamber The discharge power ,s supplied 
from a stabilized DC power supply (1500 V, 5 A) working ,n the voltage-regulated mode 
Gas is injected to the chamber to raise the neutral pressure up to 10 * Torr by using a 
double valve system consisting of a stop valve and a needle valve (fine control) The 
deposition condition of the TiN films is as follows - discharge voltage is 600 V magnetic 
field is 0.01 Tesla and total gas pressure is 2 x 103 Torr The deposition time is 75 
minutes. 
B. Results and discussions: 
X-ray diffraction study of the deposited titanium nitride thin films at different nitrogen 
percentages [(a)-25% N2. (b) 50% N2, (c) -75% N2and (d)-100% NJ in the gas mixture 
shows that single phase TiN thin films have been deposited over the bell-metal substrates 
for each of the different deposition conditions A clear sharp and intense peak of TiN 
corresponding to (200) lattice texture has been observed as shown in Figure 2 
{200] 
(*J 
3 fc) 
id) 
2D 30 40 50 
26 (degrees) 
6 0 
Figure 2. XRD plots of TiN at (a) - 25% N2 (b) 50% N2, (c) 75% N, and (d) 100% N^ 
The surface morphology of the TiN thin films has also been studied using the atomic 
force microscope. Careful observation of these films shows that the film growth over the 
bell-metal substrate maintains a uniform pattern throughout and that they have good 
adherence to it The average roughness of the films is found to be of the order of 20 nm 
All the films exhibit dense columnar structure. 
Mechanical properties of the deposited thin film coatings, like hardness and Young's 
modulus has been studied using the nano-indentation technique For obtaining maximum 
hardness, the mdentor's depth of penetration was about one tenth of the deposited film's 
thickness. This was done so that the calculated value was a true measurement of the 
744 SankarMoniBorah, Arup Ratan Pal, Heremba BailungandJyantiChutia 
coating hardness. A typical indentation depth profile done by the nano-indenter on the 
film coating is shown in Figure. 3. 
Sc»n r*tt 1.489 wz 
Figure 3. Image showing the indent profile on TiN sample 
Figure 4 shows the applied load versus the corresponding indentation depth profile for 
the deposition of TiN thin film coating at 50% N2 in the gas mixture. It is clearly seen 
that with increasing load there is a gradual deviation of the loading curve away from its 
initial position. This explains the property of plastic deformation wherein the material loses 
its elastic property. The deviation is found to be minimum at this deposition condition. 
z 
•o 
4000-
3000 -
2000-
1000-
04 
—•— 500 nN f 
* - 1000MN f 
—•- 2D00nN Jw*l 
-•-3000nN ^ r I 
• 4000nN J t I i 
x SOOO^iN ^ p \ I J 
Wmm%% «C'vai **r \a • *—\W &—i— 
100 200 300 
Indertation depth (nm) 
400 500 
Figure 4. Load vs indentation depth curves at different loads. 
The hardness, H of the deposited TiN thin films at different nitrogen percentages (25%, 
50%, 75%) in the gas mixture has been calculated using the method of Oliver and Pharr 
Hardness study of titanium nitride thin films deposited on bell-metal etc 745 
(1992), according to the relation 
where Pmgx is the peak indentation load and A is the indentation contact area, determined 
from the mdenter shape function Generally, loading data are influenced more by the 
material's plastic properties while the unloading data by the elastic properties Hardness 
is calculated from the loading data of the curves Generally the hardness values determined 
by nanoindentation method are functions of surface roughness chemical state of the 
surface layer and mdenter size effect Proper care is taken to minimize the influence of 
these effects on the hardness value 
In Figure 5, the variation of hardness and the Young's modulus with different IM 
percentages in the gas mixture is shown It is found that the maximum hardness of the 
film is at 50% N2 in the gas mixture The TiN film coating at this condition increases the 
hardness of bell-metal by nearly 13 times The increase in the film's hardness is an 
important property as it increases the durability of the material The Young's modulus 
value is also highest at 50% N2 in the plasma gaseous environment 
^ £ 
cs 
</> 
8 C 
Ha
r 
30-
20-
10-
0-
- • -
—O-
- Young s modulus 
-Haidness 
V / ± 
cr 
Y+- uncoated bell-metal 
1 r 
A" \ \ 
1 ' 
H 
A 
. \ I 
\ 
\ j 
j 
0 
- f 
 200 
< 
o 
150 to 
3 
a 
100 5 
H50 
25 50 75 
N2 gas percentage 
Figure 5. Plot of Hardness and Youngs modulus at different N//o in the gas mixture 
3. Summary 
Titanium nitride thin films have been deposited on bell-metal by reactive sputtering in 
cylindrical magnetron device in argon and nitrogen gas mixtures Nitrogen gas percentage 
m the mixture plays a significant role in the quality of the deposited film XRD analysis 
shows that single phase TiN films corresponding to (200) lattice texture has been formed 
under the above deposition condition The optimum deposition condition for the best quality 
titanium nitride thin film is found to be at 50% nitrogen in the gas mixture Hardness 
study shows that maximum hardness occurs for the film deposited at 50% nitrogen in the 
gas mixture. Apart from its hardness property, the radiant golden colour of these films 
makes it suitable to be used commercially for decorative applications 
746 Sankar Moni Borah, Arup Ratan Pal, Heremba Bailung and Jyanti Chutia 
Acknowledgments 
This work was supported by a grant from the Department of Science and Technology, 
Government of India. The authors acknowledge the facilities provided by the Centre for 
Nanotechnology, the Central Instrumentation Facility and Department of Physics of IIT 
Guwahati for XRD, SEM and AFM analyses of the samples. Acknowledgment is also due 
to Prof. D. Ghosh of SINP, Kolkata for providing the nano-indentation facility. 
References 
[1] K Wasa and S Hayakawa Handbook of Sputter Deposition Technology Principles, Technology 
and Applications ( Westwood New Jersey : Noyes Publications) (1992) 
[2] R Behrisck and K Wittmaack Sputtering by Particle Bombardment HI: Characteristics of Sputtered 
Particles, Technical Applications (Berlin : Heidelberg: Springer-Verlag) (1991) 
[3] Xuan-qian Xu, H Ye and T Zou J Zhejiang Univ Science A 7(3) 472 (2006) 
[4] L Xiao, D Yan, J He, L Zhu, Y Dong, J Zhang and X Li Appl. Surf. So. 253 7535 (2007) 
[5] S Niyomsoan, W Grant, D L Olson and B Mishra Thin Solid Films 415 187 (2002) 
[6] S Yang, D B Lewis, I Wadsworth, J Cawley, J S Brooks and W D Munz Surf. Coat. Technol 131 
228 (2000) 
[7] S E Rodil and J J Olaya J. Phys.: Condens. Matter 18 S1703 (2006) 


Hardness Study of Titanium Nitride Thin Films Deposited on Bell-Metal by Cylindrical Magnetron Sputtering

Kalita, K

Indian J. Phys. 82 (6), 801-805 (2008) 
It 
Interaction of loosely bound radioactive and stable nuclei 
via elastic scattering and fusion cross sections 
K Kalita 
Department of Physics, Gauhati University 
Guwahati-781 014, Assam, India 
E-mail ku_kalita(" yahoo com 
Abstract Elastic scattering angular distributions have been measured for 7Be+ ?7AI system at E
 h = 17, 19, 
21 MeV in the angular range ocm = 1 2 - 4 3 ° An optical mode! (OM) analysis of these data have been carried 
out in order to extract OM potential parameters and reaction cross sections One proton stripping cross sections 
were also measured at these energies The fusion cross sections were deduced by subtracting the integrated 
transfer cross-sections from the reaction cross sections obtained from elastic scattering The 7Li + 27AI elastic 
scattering angular distributions were also measured at EL=10,13,16, 19 and 24 MeV in the angular range 
°cm 1 2 ~ 7 2 ° leading to the OM potential parameters The cx-evaporated spectra were also measured at 0La^ 
= 52 - 132°, compared with reproduced data with statistical model calculations and fusion cross sections were 
extracted from them The Coupled channel fusion cross section including static nuclear deformations (CCDEF) 
calculations describe these data quite well These data were compared with data on similar loosely bound 
systems and found to be consistent 
Keywords Radioactive beam, heavy ion scattering process, statistical model calculations 
PACS Nos. 25 70 Be, 24 10 Ht, 25 70Ef 
1. Introduction 
Considerable amount of experimental and theoretical efforts have been devoted over past 
few decades to understand the nuclear reactions induced by loosely bound radioactive 
and stable nuclei. These nuclei, being rather loosely bound are susceptible to break-up 
and can substantially populate the transfer channel (with +ve Q values) when inducing 
nuclear reactions. There has been a focused interest in this field with the aim of studying 
the effect of break-up of such projectiles on the fusion cross sections in the context of 
enhancement and suppression [1,6]. In addition, the data on reactions induced by 
radioactive nuclei provide valuable input for understanding the problems of astrophysical 
interest. For instance, we may mention the measurement of S17(0) factor through 
investigations of the reaction d(7Be, 8B)n reaction [7,8]. The break-up threshold of 7Be, 
©20081 ACS 
802 KKalita 
7Li, 6Li, 9Be are low and they all lie between 1.6 MeV to 2.45 MeV. With the intension of 
making a comparison among the loosely bound projectiles we have carried out the elastic 
scattering angular measurements at ELab=17,19, 21 MeV (for 7Be + 27AI) and at ELab=10, 
13, 16, 19, and 24 MeV (for 7Li + 27AI ). Also we measured the fusion cross section by 
detecting the a-evaporated particles from the compound nucleus formation. A complete 
data set of this type should provide answers to the role of shape, size and other nuclear 
structure related degrees of freedom in influencing both fusion and elastic scattering. 
2. Experimental 
The details of the experimental setup has been reported in Ref. [9,10]. We report here 
the quasi-elastic scattering and transfer reaction cross section measurements made for 
the 7Be + 27AI system at EL = 17, 19, 21 MeV in the angular range 0cm= 12 - 43° using 
inter-University Accelerator Centre (IUAC), New Delhi and hence, the fusion cross-sections. 
Here 7Be (radioactive) in secondary beam (-10 kHz intensity), which is mirror nuclei of 
7Li (~1010 Hz) was bombarded on 27AI target (~ 1 mg/cm2 thickness) separately. An 
optical model (OM) analysis of quasi-elastic scattering data was carried out and the OM 
parameters determined and, the reaction cross sections were extracted. The fusion cross 
sections were derived at these energies by subtracting the experimental integrated transfer 
cross sections obtained from the one proton transfer reaction (7Be + 27AI —> 6Li + 28Si, 
Q = + 6MeV) data. These fusion cross sections were found to be consistent with those 
obtained from the coupled channel calculations using CCDEF [11]. For 7Li + 27AI system 
the elastic scattering angular distributions were carried out at EL= 10,13, 16, 19 and 24 
MeV in the angular range 0Lab= 12 - 72° and the a-evaporation spectra were measured 
at BLab = 52 - 132° from the compound nucleus formation at TIFR Pelletron accelerator 
facilities, Mumbai and was extracted the fusion cross section following the procedure 
done by Kailas et al [12]. The a-evaporation spectra were also reproduced with the 
statistical model calculation using PACE code [13] and when compared with the measured 
one it is found to be consistent. The CCDEF [11] calculations describe these data quite 
well. 
3. Result 
The details of the experimental setup and part of the results are reported in Ref. [10]. The 
elastic scattering data have been analysed in the optical model framework using Optical 
Model code SNOOPY [14] in order to extract reaction cross sections at the measured 
energies. Following the procedure of Ref. [12], the measured alpha spectra integrated 
over 1 MeV energy bins, were compared with the energy spectra expected for a statistically 
equilibrated compound nucleus as predicted by Monte Carlo statistical code PACE [13] 
as shown in Figure 1. The comparison of the fusion cross sections of the two systems 
as well as with the 9Be + 27AI system [15] along with the corresponding CCDEF [11] 
calculations is shown in Figure 2. In this calculation the ground-state deformation parameter 
Interaction of loosely bound radioactive and stable nuclei etc 803 
fo of the target nucleus (27AI) was taken to be 0.31 [16] and that of the first excited 
state at 0.842 /32 was also taken to be 0.31. 
10' 
10° 
10-2 
T 
L L _ _ 
_1 , 1 
- J 1 
1 • 
1 1 .. 
U>fcW\ 
l i m . _ 
-3 
10 15 20 
Ea(MeV) 
25 30 30 5 10 15 20 
Ea (MeV) 
Figure 1. a -energy spectra from 7Li + 27AI—>34S* CN at various angles corresponding to various lab 
energies 
E 
I000 
100 
1 
, -. —.,-.,
 7, 
iii 
I 
1 
, , , 1 r-
*-<*.* - ^ 
/' 
i 
i 
9 
J. . I i 1_ 
p — 
* * * 
© 
• 
_ 
• 
-
k 
1 *- 1 
7U + 27AI, this work 
7Be + 27AI, this work 
CCDEF on 7Li + 27AI 
9Be • 2 7 AI . ret {12] 
CCDEF on 9Be + 27Al 
CCDEF on 7Be + 27Al 
1 . 1 
H 
-j 
3 
H 
—i 1 
10 15 
Ecm(MeV) 
20 25 30 
Figure 2. Comparison of the experimental fusion cross sections with CCDEF calculations for 7Be, 7Li, 9Be 
+ 27AI systems. 
804 KKalita 
4. Conclusions 
In conclusion, a new measurement of quasi-elastic scattering and one-proton stripping 
reaction cross sections have been carried out on 7Be + 27AI system and the optical 
model (OM) parameters deduced The fusion cross sections were extracted by subtracting 
the experimental integrated transfer cross section from the total reaction cross sections 
obtained from OM fitting to quasi-elastic data The fusion cross sections agree well with 
the simplified coupled channel calculations Also we measured the elastic scattering angular 
distributions and a-evaporated spectra for 7Li + 27AI system From the latter we obtained 
the fusion cross sections by reproducing the a-spectra by statistical model calculation 
using the PACE code The OM analysis of the elastic scattering data results in good 
description with the optical potential parameters for 6Li + 28Si and 9Be + 28Si (loosely 
bound) systems [17] The present value of fusion cross section at 24 MeV agrees very 
well with the value reported by Padron et al [18] The comparison of the fusion cross 
section for 7Be , 7Li + 27AI systems with recently measured fusion cross sections for 9Be 
+ 27AI system [15] shows that the magnitudes of the latter are somewhat higher in the 
entire energy range This may be attributed to the presence of valence neutron in 9Be 
Finally, the fusion cross sections for these mirror nuclei (7Be, 7Li) along with 9Be on 27AI 
are found to be about the same within limits of en or 
Acknowledgments 
The help provided by various person (R Singh, S Verma, J J Das, A Jhingan, N Madhavan, 
S Nath, T Varughese, P Sugathan, V V Parkar, K Mahata, K Ramachandran, A Snvastava, 
A Chatterjee, S Kailas, S Barua, P Basu, H Majumdar, and A K Sinha) during the 
experiments is gratefully acknowledged 
The cooperation and help provided by pelletron crews both at IUAC, New Delhi as well 
as at TIFR, Mumbai during the experiments is also gratefully acknowledged 
References 
[1] N Takigawa, M Kuratani and H Sagawa Phys Rev C47 R2470 (1993) 
[2] C H Dasso and A Vittun Phys Rev C50 R12 (1994) 
[3] C H Dasso and A Vittun Nucl Phys A597 473 (1996) 
[4] M S Hussein and A F R Detoledopiza Phys Rev Lett 72 2694 (1994) 
[5] J Takahashi et al, Phys Rev Lett 78 30 (1997) 
[6] A Szanto de Toledo et al, Nucl Phys A679 175 (2000) 
[7] Weipmg Lie et al Phys Rev Lett 77 661 (1996) 
[8] J J Das et al, Phys Rev C73 015808 (2006) 
[9J K Kalrta, A Jhingan, S Barua, J J Das et al, Indian J Pure & Appl Phys 43 567 (2005) 
[10] K Kalita, S Verma, R Singh, J J Das et al, Phys Rev C73 024609 (2006) 
[11] C H Dasso and S Landowne Comput Phys Commun 46 187 (1987), J Fernandez-Niello C H 
Dasso and S Landowne Comput Phys Commun 54 409 (1989) 
Interaction of loosely bound radioactive and stable nuclei etc 805 
[12] S Kailas, R Vandenbosch, A Charlop, A Garcia et al, Pramana J Phys 35 439 (1990) 
[13] A Gavron Phys. Rev. C21 230 (1980) 
[14] P Schwant, Indiana University Cyclotron Facility Report, SNOOPY (1984) (unpublished) 
[15] G V Marti, P R S Gomes, M D Rodrguez, J 0 Fernandez et al, Phys Rev C71 027602 (2005) 
[16] D Sundholm and J Olsen Phys Rev. Lett 68 927 (1992) 
[17] M Hugi, J Lang, R Muller, E Ungncht et al, Nucl Phys A368 173 (1981) 
[18] I Padron, P R S Gomes, R M Anjos, J Lubian et al, Phys Rev C66 044608 (2002) 


Interaction of Loosely Bound Radioactive and Stable Nuclei Via Elastic Scattering and Fusion Cross Sections

Ray, Saibal

Usmani, A A

Rahaman, F

Kaum, M

Chakraborty, K

Electromagnetic Mass Model Admitting Conformal Motion

lndianlPhys.%l{% 1219-1220(2008) 
QARTHCOMING PUBLICATIONS 
Volume 82, (2008) 
IMF and geomagnetic field: Anew feature 
R G RASTOGI 
Applications of degenerate / non-degenerate modes coupling 
in an optical waveguide 
SANJEEV KUMAR RAGHUWANSHI AND SRINIVAS TALABATTULA 
Effect of firing temperature on electrical and structural characteristics 
of Sn02 thick films 
R Y BORSE AND A S GARDE 
Fluctuation and fractal characteristics of ring like and jet like 
events produced at SPS energies 
DIPAK GHOSH, ARGHA DEB, PRABIR KUMAR HALDAR AND SIMA GUPTAROY 
Optical and texture property studies in a nematic liquid crystal mixture 
JITENDRA KR KUSHWAHA, V P ARORA, K K RAINA AND V K AGARWAL 
Surface and hardness studies on as-grown (100) faces of zinc (tris) 
thiourea sulphate crystals 
V GANESH AND K KISHAN RAO 
Exploration of space charge behaviour of polyvinyl carbazole using 
measurement of thermally stimulated discharge current 
P K KHARE, GAUTAM BANGRE, PANKAJ MISHRA AND JYOTI MTSHRA 
Structural characterization and electrical conductivity of naphthalene 
doped polyblend films of polystyrene (PS) and polymethyl methacrylate 
(PMMA) 
R J DHOKNE, V S SANGAWAR, P S CHIKHAUKAR, V S THOOL, 
A U UBALE AND A R JUNGHARE 
Universal second order switched-capacitor filter for different value 
G N SHINDE AND S R BHAGAT 
1220 Forthcoming Publications 
Doublephotoionizationof Ar(1s22s22p6 3s23p6) 'S*, 
R KUMAR, R P K RAY AND S N TIWARY 
Elastic cross section for electron scattering by N20 molecule 
SUNITA SINGH AND DEO RAJ 
SEPs associated CMEs affecting near earth environment 
SANTOSH KUMAR AND JANKI PAL 
Electric dipole transitions for Mg XI (z = 12) 
GULDEM URER AND LEYLA OZDEMIR 
Investigation on the mechanical, dielectric and photoconductivity 
properties of N-methyl urea NLO single crystals 
S JANARTHANAN, T KISHORE KUMAR, S SEIVAKUMAR, S PANDI, 
P SAGAYARAJ AND D PREM ANAND 
Asymmetic group velocity dispersion and pulse distortion in an uniform 
fiber Bragg gratings 
SANJEEV KUMAR RAGHUWANSHI AND SRINIVAS TALABATTULA 


Forthcoming Publications

Rastogi, R G

Chandra, H

Indian J. Phys. 82(10) 1247-1254(2008) 
IMF and geomagnetic field: A new feature 
R G Rastogi and H Chandra* 
Physical Research Laboratory, Ahmedabad-380 009, Gujarat, India 
E-mail: hchandra@prl.res.in 
Received 12 September 2006, accepted 11 June 2008 
Abstract : It is shown that during magnetic storms, besides the decrease of AH at low latitudes synchronous 
with the decrease of ring current index Dst, there are significant changes in H associated with the change of 
IMF-Bz. Large initial phase of the storms is associated with persistent positive Bz continuing after the IP shock. 
Large positive AH are observed even during the peak phase of the storms due to sudden large northward 
turning of IMF-Bz. This seems to be a new feature detected during the nighttime with reduced amplitude at 
equatorial latitudes. 
Keywords : Equatorial electrojet, magnetic storms, space weather. 
PACSNo. : 91.25.-r 
1. Introduction 
The association between the Interplanetary magnetic field and geomagnetic activity is 
well known. Largest Kp or Ap values were found to correspond with the largest 
southward IMF component by Wilcox et al [1] and Patel et el [2J. Increasing southward 
component of IMF increases the reconnection rate at the southward side of the 
Magnetosphere. Nishida [3] showed good correlation between the geomagnetic horizontal 
component at Huancayo and the Bz component of IMF (measured by IMP 1 
satellite). 
Rastogi and Chandra [4] first reported the association between the equatorial 
electrojet and solar wind, based on the spaced receiver ionospheric drift measurements 
from Thumba near the dip equator. They showed an increase in the westward drift 
speed during daytime or in the eastward drift speed during nighttime with the increase 
°f the component of Interplanetary magnetic field normal to the ecliptic /.a. IMF-Bz 
(positive when northward). Rastogi and Patel [5] noted that large and quick changes 
^m 
"Conwpondtag Author O2008IACS 
1248 R G Rastogi and H Chandra 
of the IMF-Bz from southward to northward direction are associated with the reversal 
of ionospheric electric field over Jicamarca. Later Rastogi and Kroehl [6] showed that 
the decrease of AH during DP2 fluctuations at Huancayo and the decrease of 
ionospheric eastward electric field were associated with similar increases of IMF-Bz 
component. 
Moos [7] identified a well-defined pattern in the so called "X disturbance" in the 
horizontal component of the geomagnetic field (H) at Colaba, Bombay. He found 
occasional sudden rise of H followed by a rapid decrease lasting for few hours and a 
slow recovery lasting for 2-3 days. Chapman [8] defined these as "magnetic storms" 
and named various phases of the magnetic storms as sudden commencement (SC), 
initial phase (IP), main phase (MP) and recovery phase. Chapman and Ferraro [9-12] 
suggested that the geomagnetic storms are initiated by the impact on the earth of the 
ionized solar plasma bubble ejected from the Sun. The decrease of H field was 
suggested due to the generation of current around the earth due to the polarization of 
solar plasma. Singer [13] suggested the trapping of charged particles from the Sun 
after penetrating the earth's geomagnetic field and the creation of the radiation belt and 
the equatorial ring current. 
Suguira and Chapman [14] described the results of 346 magnetic storms at 24 
observatories covering the period 1902-1945. They found that no abnormal features of 
the geomagnetic storms were seen at the equatorial electrojet stations unlike the 
abnormal intensification of quiet time solar daily variations of H at other stations. 
Rastogi [15] found that the maximum depression of H during the main phase of the 
magnetic storm showed a maximum over the magnetic equator larger than expected of 
the Dst index. Recently Rastogi [16] showed that during magnetic storms, associated 
with southward IMF-Bz, there are definite enhancements in the decrease of H field at 
the dayside equatorial stations. Sometimes during the storm period, abnormally large 
changes in H field are observed associated with large sudden changes of IMF-Bz. 
A strong interplanetary shock, with a pressure increase of the order of 30 nPa, 
reached the earth's magnetosphere at about 0630 UT on 10 November 2000 resulting 
in a strong sudden commencement of magnetic storm of duration of a day [17]. The 
solar wind parameters (1) solar wind flow speed, (2) solar wind ion density, (3) 
Interplanetary magnetic field component Bz as recorded by satellite WIND are reproduced 
in Figure 1 from the SPIDR (Space Physics Interactive Data Resource) website. The 
time delay between the arrival of the shock at the satellite and at the earth was on 
the average of 12 minutes and the variation of this time delay with time is also shown 
in Figure 1. The time delay is taken into account on the temporal variations of the 
solar wind parameters. The sudden increase of the solar wind speed and ion density 
and an abnormally large positive impulse of IMF-Bz are clearly shown in the figure 
associated with SC. The abnormal features of this storm had been the wild fluctuations 
of IMF-Bz from about +15 to - 1 5 nT during the period of large wind density. Around 
IMF and geomagnetic field: A new feature 1249 
15 
10 
5 
o 
-5 
-10 
14 
12 
10 
20 
15 
10 j *yj 
Interplanetary magnetic field Bz, nT 
j/ V lv---Vv 
r ~ i 
Ion density per cm3 
X Flow Speed x 102 km/sec 
2000-11-10 12:00 12:00 2000-11-12 2000-11-11 
urc 
Figure 1. Temporal variations of solar wind parameters recorded at satellite WIND on 10 and 11 November 2000. 
1100 UT when the solar wind density and speed returned to normal a very large 
change of IMF-Bz from a value of -12 nT at 1150 UT to a value of +9 nT at 1315 
UT was recorded. 
SC of 110 nT was recorded at Tirunelveli in India at 0640 UT The Dst value 
reached a minimum of -96 nT at 1230 UT. The effects of large geomagnetic storm of 
November 10, 2000 at low and equatorial stations are presented in the present 
communication. 
2. Results 
In Figure 2 are shown the variations of hourly mean values on 10 November, 2000 of 
the parameters (i) density of solar wind, 4)\) solar wind velocity, (iii) IMF-Bz, (iv) Dst 
index and (v) the storm time variations of H (H-SqH) at Indian Stations Tirunelveli (TIR), 
Kodaikanal (KOD), Pondicherry (PON), Alibag (ABG), Ujjain (UJJ) and Sabhawala 
(SAB). The solar wind density showed a sharp increase from 0530 UT to 0730 UT, 
thereafter a rapid decrease till 1230 UT and practically remained constant later. The 
solar wind velocity increased from 500 km/s at 0530 UT to 700 km/s at 0730 UT and 
1250 Ft GRastogi and H Chant* 
STORM-TIME VARIATION - H, nT 
10-NOVEMBER, 2000 
Figure 2. Variation of hourly mean values of (i) solar wind density, (ii) solar wind velocity, (iii) IMF-Bz, (iv) Dst 
index and (v) storm time variations of H (H~ SqH) at Indian stations Tirunerveli (TIR), Kodaikanal (KOD), Pondicherry 
(PON), AKbag (ABG), Ujjain (UJJ) and Sabhawala (SAB) and at Huancayo (HUA) and M'Bour (MBO) on 10 
November 2000. 
remained constant thereafter. IMF-Bz increased from 0 nT to 10 nT between 0430 and 
0630 UT, decreased to - 5 nT at 0830 UT. After 1130 UT it showed gradual increase 
up to 1330 UT. It is to be noted that the storm-time variations of H at any of the 
stations showed large positive initial phase associated with an increase of Dst index. 
This was due to the continued compression of the magnetosphere due to the large 
values of solar wind velocity and density coupled with the positive value of IMF-Bz 
Storm time variations of H and Dst index started decreasing after 0830 UT with the 
IMF-Bz turning negative and reached the minimum value at 1300 UT followed by the 
recovery phase of the storm. 
Figure 3 compares the variation of the quarter hourly mean values of IMF-Bz on 
10 November, 2000 with the magnetograms at some of the low latitude stations around 
the world. It can be seen that IMF-Bz had Ijeen steadily increasing since 0430 UT to 
0630 UT even before the occurrence of SC. The amplitude of SC at Tinmelveli was 110 
nT at 0640 UT. During the period 0430 UT to 0630 UT there was a flat minimum of 
H at these stations. It was only the sudden decrease of IMF at 0630 UT that caused 
the SC. Another event of importance was a large slow increase of H at Huancayo of 
about 70 nT at 1130 UT which seems to be associated with the large and steady 
IMF and geomagnetic field: A new feature 1251 
H, nT 
26050 
26000 
27600 
27550 
i i i i i i i 
I I I I I I I I I I I I I M I I I I | I 1 ' H 
0 3 6 9 12 15 18 21 24 
M m TIRUNEL VELI 
HT4BnT 
00 02 04 06 08 10 12 14 16 18 20 22 UT 
06 09 12 15 18 21 l/T 
10-NOVEMBER, 2000 
Figure 3. Qurter hourly values of IMF-Bz along with the magnetograms of some low latitude stations for 10 
November 2000. 
increase of IMF-Bz between 1130 and 1330 UT. The local time corresponding to 1130 
UT at these stations are 0630 LT at Huancayo (HUA), 1022 LT at MBO, 1618 LT at 
Alibag (ABG), 1834 LT at PHU, 2106 LT at Guam (GUA) and 2002 LT at Honolulu 
(HON). This impulse can be seen at other stations but had been very weak at Indian 
stations Tirunelveli and Alibag, which were in the local evening sector (1618 LT at 
Alibag). There were no suitable longitudinal distribution of equatorial electrojet stations 
and so the local time effect of such events cannot be investigated. 
Prof. K Yumuto of the Kyushu University, Japan had organized geomagnetic 
observatories under the Circum-pan Pacific Magnetometer Network. The daily maps of 
X and Y fields at these stations are available in their website. Figure 4 shows the 
recast stack plots of the X component magnetograms at some of the stations on a 
uniform sensitivity scale. The stations are distributed in the geographic longitude 
between 114° E to 150° E cover dip of 59° N to 67° S. The SC at 0630 UT and a 
positive pulse around 1200 UT on 10 November 2000 are clearly recorded at all 
stations (Table 1). The first impulse at 0630 UT was by the IP shock when the solar 
1252 AG Rastogi and H Chandra 
0 4 8 12 16 20 24 
HOURS UT 
10-NOVEMBER, 2000 
Figure 4. The X component magnetograms at a number of stations covenng dip of 59° N to 67° S for 10 
November 2000 in the longitude region of 114°E-150°E . 
Table 1 . Dip and geographic longitudes of the stations shown in Figure 4 
Station name 
Popov Island 
Moshin 
Rikubetsu 
Onagawa 
Kagoshima 
Chichijima 
Lunpmg 
Muntmlupa 
Yap 
Cebu 
Oavao 
Weipa 
Learmonth 
Birdsville 
Katanning 
Adelaide 
Canberra 
Station code 
ppi 
MSR 
RIK 
ONW 
KAG 
CBI 
LNP 
MUT 
YAP 
CEB 
DAV 
WEP 
LMT 
BSV 
KAT 
ADL 
CAN 
Dip°N 
59 
58 
57 
52 
46 
36 
36 
16 
6 
3 
- 1 
-39 
-56 
-57 
-68 
-66 
-67 
Geog. longitude °E 
131.7 
142 3 
143.8 
141.5 
130.7 
142.2 
121.2 
121.0 
138.5 
123.9 
125.6 
141.9 
114.1 
139.2 
117.6 
138.4 
149.4 
IMF and geomagnetic field: A new feature 1253 
wind shock struck the magnetosphere while the second impulse at 1130 UT was due 
to the abnormal large change of IMF-Bz from a value of - 1 2 nT at 1115 UT to a value 
0f +9 nT at 1350 UT without any significant change of solar wind density or speed. 
The local time corresponding to the second impulse around 1130 UT for these stations 
was between 1900 LT and 2130 LT. It is interesting to note that the amplitude of the 
pulse was minimum at equatorial stations and steadily increased with increasing dip 
latitude north or south. 
3. Conclusion 
There have been numerous studies related to the geomagnetic storm effects in the 
density, composition and dynamics of the sub-auroral, middle and low latitude ionosphere 
and thermosphere e.g. Kelley [18], Fuller-Rowell et al [19], Basu et al [20], Fejer [21]. 
Prompt penetration and disturbance dynamo electric fields, though of smaller magnitude, 
are the important sources of low latitude ionospheric electrodynamic disturbances. 
Sharp electric field perturbations with time scales typically shorter than about an hour 
are due mostly to the prompt penetration of solar wind/magnetospheric electric fields 
to middle, low and equatorial latitudes [21 and references therein]. Quasi-period (DP2) 
magnetic field fluctuations with time scale of about half an hour to several hours at 
high latitudes and in the dayside of magnetic equator are signatures of convection 
electric fields controlled by IMF-Bz [3]. Slower varying electric field disturbances with 
time scales of few hours to tens of hours are identified as ionospheric disturbance 
dynamo electric fields caused by enhanced energy deposition in to the auroral 
ionosphere e.g. Blanc and Richmond [22]. 
The effects of magnetic storm at low latitude stations are not entirely due to the 
overhead equatorial ring current but are modified by the change of solar wind 
parameters (velocity, density and IMF-Bz). A new feature of large positive impulse in 
H during the nighttime associated with large steady change of IMF-Bz with reduced 
amplitude at lower latitudes is detected. 
References 
[1] J M Wilcox, K H Schatten and N F Ness J. Geophys. Res. 72 19 (1967) 
[2] V L Patel. L J Cahill and A J Dessler J. Geophys. Res. 72 426 (1967) 
[3] A Nishida J. Geophys. Res. 73 5549 (1968) 
[4] R G Rastogi and H Chandra J. Atmos. Terr. Phys. 36 377 (1974) 
[51 R G Rastogi and V L Patel Proc. Indian Acad. Sd. A82 121 (1975) 
[6] R G Rastogi and H Kroehl Indian J. Radio & Space Phys. 7 84 (1978) 
U] N A F Moos Magnetic observations made at the Government Observatory, Bombay 1d47-1905 and their 
discussions Part II, The phenomenon and its discussion (1910) 
[8] S Chapman Proc. Roy. Soc. Lond. A95 61(1918) 
[9] S Chapman and V C A Ferraro Trans. Magn. Atmos. Bectr. 36 77 (1931a) 
RG Rastogi and H Chandra 
[10] S Chapman and V C A Ferraro Trans. Magn. Atmos. Electr. 36 171 (1931b) 
[11] S Chapman and V C A Ferraro Trans. Magn. Atmos. Electr. 37147 (1932a) 
[12] S Chapman and V C A Ferraro Trans. Magn. Atmos. Electr. 421(1932b) 
[13] S F Singer EOS Trans. American Geophys. Union 38 175 (1957) 
[14] M Suguira and S Chapman Abh. Akad. Wiss. Gottingen Maths. Phys. Klasse Sonderhefti 1 (1960) 
[15] R 6 Rastogi Ann. Geophysicae 17 438 (1999) 
[16] R G Rastogi Earth Planets Space 58 1 (2006) 
[17] E Boralv et al Proc. AGU Chapman conference on storm-sub storm relationships, March 12-16, Lonavala 
India p22 (2001) 
[18] M C Kelley The Earth's Ionosphere, Plasma Physics and Electrodynamics (San Diego, California: 
Academic) (1989) 
[19] T J Fuller-Rowel, M V Codrescu, R J Moffet and S J Quegan J. Geophys. Res. 99 3893 (1994) 
[20] B G Fejer. in Auroralphenomena and Solar-Terresstrial relations CAWSES handbook - 1 , Boulder 2004, 
(eds) L M Zelling, M A Geller and J H Allen (2004) 
[21] S Basu, Su Basu, M Groves, H C Yeh, S Y Su, F J Rich, P J Sultan and M J Keskinen Geophys. Res. Lett 
18 3577 (2001) 
[22] M Blanc and A D Richmond J. Geophys. Res. 85 1669 (1980) 


IMF and Geomagnetic Field : A New Feature

Mirafzali, S Y

Sarbishaei, M

Indian J. Phys. 82 (1) 59-67 (2008) 
wft 
The effect of anisotropy and external magnetic field on 
the thermal entanglement in two spin-one system 
S Y Mirafzali* and M Sarbishaei 
Department of Physics, Faculty of Sciences. Fertlowsi University of Mashhad, Iran 
E-mail al_ as82'fl'stu-mail urn ac lr 
Received 3 April 2007. dccpeted 2b November 2007 
Abstract : We study the thermal entanglement in a two-spin-one system with anisotropy in the exchange 
coupling between two spins. We use the realignment criterion to distinguish the entangled states, and the negativity 
for measuring the entanglement in this system We see that the anisotropy can provide an additional parameter 
tor enhancing the entanglement. 
Keywords : Entanglement, negativity, realignment criterion 
PACS Nos. : 03.67.Lx , 03.65.Ud, 75 10.Jm 
1. Introduction 
Entanglement is a non-local correlation between quantum systems that does not exist 
classically. It plays a central role in quantum information processing such as quantum 
teleportation, dense coding and so on. 
The entanglement properties of various systems have been intensively studied. 
In most of these studies, spin 1/2 systems have been considered, but higher spin 
systems have been less investigated. The reason for this restriction is that a good 
operational entanglement measure, when the system is in a mixed state, has been 
known only for two spin-halves systems. However there is currently an effort to study 
entanglement in systems with higher spins. 
There are several articles on entanglement in spin-one chains [1-7]. Zhang et al 
[8] have studied the thermal entanglement of a system consisting two wells in the 
optical lattice with one spin-1 atom in each well. One can summarize the Hamiltonian 
of this system as follow [9]. 
•Corresponding Author © 2008 IACS 
60 S Y Mirafzali and M Sarbishaei 
H = t i J(St S 2 H *(Si S2)2 (1) 
The above Hamiltonian, in the absence and in the presence of an external magnetic 
field S, has been studied in detail in [10-13] However, the effect of anisotropy in 
exchange coupling between two atoms has not been considered It can provide an 
additional parameter (besides the other parameters) for controlling entanglement In 
this paper we add an isotropy to the above Hamiltonian and study the effects of 
anisotropy on thermal entanglement 
2. Negativity and realignment criterion 
As it was noticed by Peres [14], a necessary criterion for mixed state of a bipartite 
system to be separable is that its partial transpose with respect to one of the 
subsystems is positive Horodecki family [15] have shown that this condition is also 
sufficient if the Hilbert space of the bipartite system has dimension 2 x 2 or 2 x 3 
But for higher dimensions, the PPT (or Peres-Horodecki) criterion is not a sufficient 
one since there are entangled states that have positive partial transpose These states 
are called bound entangled states (BES) Nevertheless, the non-positive partial transpose 
(NPPT) gives a sufficient condition for entanglement 
Vidal and Werner [16] introduced a computable measure of entanglement called 
negativity The negativity of a state p is defined as 
w(/')=-]Ck (2) 
where //, is the negative eigenvalue of / / 1 , and 7i denotes the partial transpose with 
respect to the first system Negativity essentially measures the degree to which//1 
fails to be positive Therefore, it can be considered as a quantification of the PPT 
criterion for separability and because of that we cannot confirm whether the state is 
entangled or not in the region of zero negativity We must seek for another criterion 
that reveals entanglement where the negativity fails to distinguish between separable 
states and entangled states 
Another computable separability criterion was introduced by Rudolph [17] and 
Chen et al [18] called realignment criterion It is independent of Peres criterion, and 
turns out to be strong enough to detect entanglement of almost all known states, for 
which the former criterion fails [19] However, for some states it is weaker than PPT 
one [18] 
In order to make use of this criterion we only need to rearrange the entries of 
p. If p is separable, then the sum of the square roots of the eigenvalues for pp* 
should be < 1 , or equivalent^ 
* = log|2> <o 
The effect ofanisotropy and external magnetic field etc. 61 
where ]T) ^ , is the sum of the square roots of the eigenvalues for pp*. The 
realigned form of a m x m block matrix p with block size n x n is defined as 
vec(pv)T 
vec(pmA)T 
(4) 
where for each m x n matrix A = [ay] (a,; is the matrix entry of A), vec(A) is defined 
as vec{A) = [aUl...1amV...1 a12,...,a1n,...,amn]T. Then if R > 0, p is entangled otherwise 
this criterion fails to confirm whether the system is entangled or not. So we set R = 0 
in our calculation whenever R < 0. 
3. The model's Hamiltonian and thermal entanglement 
We consider the Hamiltonian of the form 
S]yS2y 
fS(S1z + S2Z) (5) 
where S,v(a = x, y,z) are the spin operators, J is the strength of Heisenberg 
nteraction, K is the nonlinear couplings constant and the magnetic field is assumed 
o be along the z-axes, and where we have also added an anisotropy in the z 
direction which is represented by A. When the spin for each site is one, its 
:omponent takes the form (/' = 1, 2) 
1 
f0 
^ 0 
1 
0 
1 
0 
1 
0 '
 S
'
y
 - V2 
0 
/ 
0 
-;' 
0 
i 
0 ' 
- / 
0 
• s* = 
'1 
0 
0 
0 
0 
0 
0 
0 (6) 
The state of the system described by the Hamiltonian H at thermal equilibrium is : 
p(T) = exp(-/3H)/Z (7) 
where Z = 7r[expH?H)] is the partition function and 0 = 1/KBT (KB is the Boltzmann's 
constant that for simplicity being set to be unit, KB = 1, and T is the temperature). 
A s
 p(T) represent a thermal state, the entanglement in the state is called the thermal 
entanglement. We analyze the dependence of thermal entanglement in this system on 
S Y Mirafzah and M Sarbishaei 62 
temperature and external field and study the effects of anisotropy on the thermal 
entanglement. 
4. The solutions 
We considered the two cases : 
Case 1 : K = 0 
The eigenvalues and eigenvectors of H are obtained as 
H\\1) = (JA f 2S)11,1), 
H\ - 1 , 1 ) - (JA- 2 e ) | - 1 , 1), 
H\*/) = (B-tJ)\<l',i), (8) 
| * i i ) = i ( | i . o > i | a i ) ) . 
H^^l-BiJ)^), 
|^>- i ( |o . i> + H.o», 
|# 4> = [ 8 ^ T / 4 ) 2 ] ^ ( 2 ^ - 1 ) + T, i |0,0) + 2 | - ^ l ) ) . 
tf - A±(A2 +8)^. 
H\<1>) = -JA\4>), 
where | / , / ) , ( / , / = 1,0, - 1 ) are the eigenstates of S1ZS2Z. 
We calculated numerically the negativity and R = log Ylai o f ^^")- F o r t n e 
case J = - 1 , we plotted negativity and R as a function of B and A for four typical 
temperatures and the results are shown in Figure 1. 
We see that almost N (negativity) and R behave alike. They are even functions 
of B and their maximum is at B = 0 (for fixed A and T). At T = 0.05 (Figure 1 (a)), 
they have three sharp peaks. The center of the middle peak locates at S = 0, but with 
The effect of anisotropy and external magnetic field etc. 63 
(a) (b) 
Figure 1. The negativity (A/) and Rasa function of Sand A at (a) T= 0 05, (b) T= 0 2, (c) T= 0 6 and (d) 7= 1 2 
increasing the temperature the left and the right peaks disappear (Figure 1 (b)-(d)). In 
general as the magnetic field and temperature increase, N and R decrease. At low 
temperatures, N and R are as strong as each other, but at higher temperatures (T > 
0.02), N is stronger than R (Figure 1 (c)-(d)). The threshold temperature, at which the 
negativity and R vanish, is T « 1.36667. 
The maximum values of N and R (for fixed B and T) occur when A is equal 
to J (see Figure 2), and they vanish as the anisotropy (A) tends to -J (in Figure 2, 
-J is equal to one). For A = J, in the interval - 3 < B < 3 N and R are nonzero, but 
as A tends to - J , this interval becomes smaller. 
For J = 1, we obtain exactly the same results as J = - 1 , and for this reason, 
we do not fix the sign of J in our descriptions. 
Case 2 : K * 0 
The eigenvalues and eigenvectors of H are obtained as 
64 S YMirafzali and M Sarbishaei 
1 
09 
08 
07 
06 
U5 
04 
03 
02 
01 
°-1 
[ B=1 95 
""' 05 
M B=1 
0 
A 
(a) 
\B=O 
05 1 
1 
09 
08 
07 
06 
05 
04 
03 
02 
01 
0 
x8*2 ^ ^ \ 1 
1 05 0 05 1 
A 
(b) 
Figure 2. The negativity (solid lines) and R (dashed lines) as a function of A for three typical S at (a) T= 0.05 and 
(b) 7=0.2 
H\ 1,-l) = ( J J - 2 8 + K 4 2 ) | - 1 , - l ) , 
w | * 1 i ) = ( e ± j + /c)|tf'1i), 
K)~i(Ko)±|o.i» 
H\^2l)-(~BU+K)\<P2i), 
\*2,)~js[\-\o)±\o.-i» 
H|«*> = ( 7 * / 2 ) | « i ) , 
|0 i> = (8 + ( 7 r ) 2 ) / 2 (2 | l , - l ) - / i |O > O) + 2|-1,l)) 
H~^hJ- KA) 
« - JA + KA2 + K, 
(9) 
V = o ^ K±yj(cv+K)2 \ 8{J-KA)2 
H\*) = (-J& + KA2)\4>), 
l*> = i(h.-1>-|-1.1» 
where | / , ; ) , ( / , ; = 1 . 0 . - 1 ) are the eigenstates of S^ZS2Z. 
We calculated numerically the negativity and R of p(T). In the former case, we 
7776 effect ofanisotropy and external magnetic field etc. 65 
saw the effect of temperature on entanglement. As we increase the temperature, N 
and R decrease. In this case the effect of temperature is the same as former one, 
therefore the temperature is taken constant in the following discussion (T = 0.2). 
For the case J = - 1 , we plotted the negativity and R as a function of K and 
A for four typical B and the results are shown in Figure 3 (N and R are even 
functions of B therefore we only considered B > 0). 
1 "-1 1 - 1 
B=0 B=0 5 
(a) (b) 
B=1 5 
(c) (d) 
Figure 3. The negativity (A/) and R as a function of K and J at (a) 8 = 0, (b) B = 0.5, (c) 8 = 1.5 and (d) 8 = 2.5 
(7-= 0.2). 
To discuss the behavior of anisotropy in this case we can distinguish four 
regions in Figure 3 : 
(i) The region where the sign of J, K and A axe the same : In this region, for fixed 
S, the values of N and R are larger than their values in the other regions (their 
maximum is at J = K = A). They monotonously decrease as B increases. 
(ii) The region where the sign of K and J are the same but not the same as A : 
66 S Y Mirafzali and M Sarbishaei 
as B increases from zero to 8 = 0.5, N and R increase, then they decrease 
as B increases further. 
(iii) The region where the sign of K and A are the same but not the same as J : 
in this case, N and R monotonously decrease as B increases. They vanish for 
B > 0.8. The maximum value of N and R (which occurs at B = 0) is about 0.7 
and 0.4 respectively. 
(iv) The region where the sign of A and J are the same but not the same as K : 
This region is similar to region 2. But, in this region, at some points, N is 
stronger than R. 
For the case of J = 1, we obtained the same results as J = - 1 . 
5. Conclusions 
We used the negativity, to measure entanglement in this system and therefore our 
conclusions about the measure of entanglement are restricted to non positive partial 
transpose states. We also used the realignment criterion to distinguish between 
separable states and entangled states. However, we see that in this system, the 
realignment criterion is weaker than negativity. 
We see that when J, K and A are equal, we have maximum entanglement. 
Therefore, the presence of anisotropy in the exchange coupling between two spins 
can enhance the entanglement. Moreover as the external magnetic field increases, 
negativity and R decrease. As for temperature, we see that increasing temperature will 
decrease entanglement as expected, since by increasing temperature we approach to 
classical situation. 
Acknowledgment 
One of the authors (SYM) would like to thank R A Zeinali for his helpful discussion. 
References 
[1] J Schliemann Phys. Rev. A68 012309 (2003) 
[2] J Schliemann Phys. Rev. A72 012307 (2005) 
[3] H Fan, V Korpin and V Roychowdhury Phys. Rev. Lett. 93 227203 (2004) 
[4] F Verstraete, M A Martin-Oelgado and J I Cirac Phys. Rev. Lett 92 087201 (2004) 
[5] L Zhou, X X Yi, H S Song and Y Q Quo Chinese. Phys. 14 1168 (2005) 
[6] X Wang, H B Li, Z Sun and Y Q Li J. Phys. A : Math. Gen. 38 8703 (2005) 
[7] Z Sun, X G Wang and Y Q Li New Journal of Physics. 7 838713 (2005) 
[8] G F Zhang, J Q Liang, G E Zhang and Q W Yan Eur. Phys. J. D32 409 (2005) 
[9] S K Yip Phys. Rev. Lett. 90 250402 (1998) 
[10] G F Zhang and S S Li Solid. State Commun. 138 17 (2006) 
[11] G F Zhang and S S Li Opt. Commun. 260 347 (2006) 
The effect of anisotropy and external magnetic field etc. 
[12] G F Zhang, S S Li and J Q Liang Opt Commun. 245 457 (2005) 
[13] G F Zhang and S S Li Eur. Phys. J. D37 123 (2006) 
[14] A Peres Phys. Rev. Lett. 77 1413 (1996) 
[15] M Horodecki, P Horodecki and R Horodecki Phys. Lett A223 1 (1996) 
[16] G Vidal and R F Werner Phys. Rev. A65 032314 (2002) 
[17] O Rudolph J. Phys. A : Math. Gen. 36 5825 (2003) 
[18] K Chen and L A Wu Quantum Information and Computation Vol. 3, No. 3 193 (2003) 
[19] M Horodecki, P Horodecki and R Horodecki Open. Syst. Inf. Dyn. 13 103 (2006) 


The Effect of Anisotropy and External Magnetic Field on the Thermal Entanglement in Two Spin-One System

Nath, P P

Joarder, R N

5V0t€ 
Indian J. Phys. 82(7) 913-917(2008) 
A note on the dynamical structure of liquid LiNa alloy 
PPNath^andRNJoarder 
department of Engineering Physics, Bengal Institute of Technology, Kolkata-700 150, India 
Department of Physics, Jadavpur University, Kolkata-700 332, India 
E-mail rnjoarder@yahoo co in 
Received 28 June 2007, accepted 16 April 2008 
Abstract : The dynamical structure factors of liquid LiNa alloy are calculated in the visco-elastic approximation 
using static properties like effective pair potential and static structure factors The results are compared with 
inelastic neutron scattering (INS) and molecular dynamics (MD) simulation results The specific features are 
discussed It is clearly shown that the effect of the concentration variation on the dynamical structure is small 
though the corresponding effect on the static structure is quite large 
Keywords : Dynamical structure, ordering potential, visco-elastic approximation. 
PACS Nos. : 61 20, 61 25 Mv, 63 50 
Based on the modified Young-Hoshino (MYH) potential for Li and simple Ashcroft 
model for Na, the successful calculation of Bhatia-Thornton (B-T) static structure 
factors and the thermodynamic properties of liquid LiNa system at different 
concentrations [1] and thereafter the dynamical properttos like diffusion constants [2], 
we are tempted to see the applicability of this model for the calculation of dynamical 
structure factors of this system at different concentrations. This is important because, 
as pointed out by Sinn et al [3], the precise measurements of the dynamical structure 
factors provide a stringent test of the inter-atomic interactions. For instance, the results 
of dynamical structure factors for liquid Li obtained with neutral pseudo-atom method 
[4] compare well with their experimental data, whereas Ashcroft's empty core model 
appears to be inadequate. Evidently the choice of the model potential is important. 
The liquid LiNa alloy is a typical phase separating (ps) system whose 
stoichiometric composition (Cy = 0.61) is also the null alloy composition in neutron 
diffraction. The total static structure factor, S(g) reduces to concentration-concentration 
w 
'Corresponding Author © 2008 IACS 
914 P P Nath and R N Joarder 
structure factor, SJqWq.^ and is dominated by a strong small angle scattering with 
subsequent weak oscillations [1]. The system possesses a typical ordering potential. 
It is therefore interesting to see if these features in the ordering potential and static 
structure factor have appreciable effects on the dynamical structure. 
For binary liquid alloy in the effective medium approximation (EMA) [5] we use 
the concept of Wigner-Seitz sphere treating them as a pseudo one component system 
with mean potential (<f>Jr) s </>NN(r)), obtainable from component pair potentials and 
the pure component parameters are replaced by effective values. For dynamical 
structure factor, S(g,u;) we consider visco-elastic approximation [6]. The visco-elastic 
model is simplest one to use and has been applied to a number of systems with 
reasonable success. It gives a method for connecting dynamical properties to static 
properties like liquid structure factor and effective pair potential. The method is based 
on phenomenological analysis of time correlation function and principal advantage is 
that it provides unified treatment of a wide variety of transport coefficients. In the 
model stress correlation functions can be related to intermediate dynamic scattering 
function and expression for S(qtJ) can be conveniently obtained. Thus, S(q,u>) is given 
by 
0 / s 1 T(q)v?hq2lu,2L(q)-u,20(q)) 
S(qtu) = - T L — — * - y (1) 
n§vrlq)iLjz-u>l(q)j\ +\u;2-uj*(q)} J 
with 
7 ^ = ^> /K(9) -^o(q) ) , Vo(q) = v*q/y[S(q) (2) 
where vm is the thermal velocity and uL{q) is the longitudinal frequency of the 
associated wave. S(q), the total structure factor for LiNa alloy can be obtained in the 
low order perturbation theory using appropriate pseudopotentials [1J. For uL(q) we 
follow Bhatia-Singh calculation procedure [7] primarily intended for amorphous systems. 
The relations can be extended to liquids and mixtures with some modifications. For 
liquid, the parameters 0 and 6 of the first and the second derivatives of the effective 
pair potential are related and the modified dispersion relations for liquid can be 
obtained [7]. 
We consider LiNa liquid alloy at T= 725 K and ionic density, p * 0.031083 A"*. 
In Figure 1 we show the calculated S(q) and S^tf/cuC^ at three different 
concentrations. The general features of the experimental (neutron) S(q) curves [8J are 
represented reasonably well by our model potentials as shown in our previous paper 
(Ref. [1] of this paper) for c^ = 0.61. The agreement with other curves is similar. For 
Cu » 0.61, S(q) based on MD simulation [9] has also similar features. In the model, 
A note on the dynamical structure of liquid LiNa alloy 915 
Figure 1. LiNa liquid alloy, structure factors vs. q: S(q) , Scdq)/^^ at (a) q_, = 0.3 (origin shifted by 
1.0) (b) CL, * 0.61 (null alloy) (origin shifted by 0.5) (c) cu = 0.8. 
at CL, = 0.61, S(q) and Scc(q)/cLicr4a are almost indistinguishable while for Cy = 0.3 and 
cu = 0.8 they are very different in their features. The calculated uL(q) (not shown) 
appears in some respect similar to that of pure liquid Li. 
The dynamical structure factors have been obtained for three g-values, q = 0.8, 
1.2 and 1.5 A*"1 and the results are shown in Figure 2. The results when compared 
with MD simulation results [10], we find reasonably good agreement for q = 0.8 and 
1.2 A""1. The experimental, INS data [11] are also shown and there is some 
agreement. The agreement is reasonable in the sense that the viscoelastic model is 
an appropriate model and no calculation even on liquid metals e.g. Li, Rb etc. could 
yield better agreement [12] with MD/experimental results. Further LiNa is a phase 
separating non-simple alloy where the application of the model is surely approximate. 
In this connection it is also to be noted that when the experimental, INS data is 
integrated over u to give the static S(q), the obtained results show some discrepancies 
with experimental S(q) result [10] while MD data give a better agreement. Evidently, 
the experimental (INS), S(q,u;) data are somewhat lower. We have also done 
calculations for qj = 0.3 and 0.8, to see the concentration dependence of S(q,u;). At 
q = 0.8 and 1.2 A"1, the effect of concentration on S(<J^ ;) is small while at q = 1.5 
A"1 the effect is significant. 
Though the total static structure factors are different considerably with 
concentration at these ^values S(q,J) are not affected much. Again at lower 
concentration, S(q,v) are mostly lower at q = 0.8 and 1.2 A"*1 while this trend is almost 
reversed at q = 1.5 A"1. At q = 1,5 A"1, however, our calculated S(qta;) are relatively 
not in good agreement with MD as well as INS results. Anyway, it is clear that though 
the effect of concentration is enormous on static structure factor and ordering 
potential, it is less prominent on dynamical structure factor. The possible explanation 
916 
P P Nath and R N Joarder 
Figure 2. S(q,u) vs. u{u; in p s_f) for liquid LiNa alloy for (a) q = 0.8 A"1, (b) 1 2 A-' and (c) 1.5 A-1 
Calculated — (at 0 61 Li concentration), (at 0.3 Li concentration) and (at 0.8 Li concentration), MD 
simulation result ooo and experimental result ••• both at 0.61 Li concentration 
may be as follows. S(q,u) in the explored (q,u) region decreases monotonically as 
function of u with no side peak. At null alloy (q., = 0.61), S[q,u) is entirely given by 
ScdQ'U) which therefore follow the same pattern of S(q,u>). Though SNfAq,u>) shows 
side peak [10] its contribution to S(q,u>) is very small at CJJ = 0.3 and 0.8. Again, like 
Scdq) (Figure 1 except at q -» 0), Scc{qtuj) is also expected to be not strongly 
concentration dependent. So it is expected that S(q,o;) is only weakly dependent on 
concentration. The concentration dependence is however prominent at larger 
Li-concentration and larger q-values. In case of static structure, however, the SNN(q) 
and ScdQ) have basic differences in shape. While SNtj(q) has a pattern more like that 
of pure Li or Na with a large first peak, Scdq) has a small double peak with low-q 
region changing enormously with concentration and this shoots up at CL, = 0.61. This 
difference in SN^q) and SCM structure {SM is very small at all q) is responsible 
for large change in S{q) with concentration. Further, the overall features of S(q,u;) 
curves are that unlike pure liquid Li, no side peaks revealing the existence of 
propagating sound modes are present. 
A note on the dynamical structure of liquid LiNa alloy 917 
Acknowledgment 
One of the authors (RNJ) acknowledges with thanks, the financial support from CSIR, 
New Delhi (India). 
References 
[1] P P Nath, S Sarkar and R N Joarder Indian J. Phys. 77A 609 (2003) 
[2] P P Nath and R N Joarder Indian J. Pure and Appl. Phys. 43 180 (2005) 
[3] S Sinn, F Sette, U Bergmann, Ch Halcoussis, M Krisch, R Verbeni and E Burkel Phys. Rev. Lett. 781715 
(1997) 
[4] M Canales, L E Gonzalez and J A Padro Phys. Rev. E50 3656 (1994) 
[5] A Pratap, D Bhandari, N S Saxena and M P Saxena Indian J. Pure Appl. Phys. 33 601 (1995) 
[6] K Hoshino, H Ugawa and M Watabe J. Phys. Soc. Jpn. 61 2182 (1992) 
[7] A B Bhatia and R N Singh Phys. Rev. B31 4751 (1985) 
[8] H Ruppersberg and W Knoll Z. Naturforsch a32 1374 (1977) 
[9] M Canales, D J Gonzalez, L E Gonzalez and J A Padro Phys. Rev. E58 4747 (1998) 
[10] N Anento, J Casas, M Canales, D J Gonzalez, L E Gonzalez and J A Padro J. Non-Cryst. Sol. 250 348 
(1999) 
[11] PRGartrell-Mills, RLMcGreevyand WVanderLugtPhysicaB154 1 (1988) 
[12] J Cases, D J Gonzalez and L E Gonzalez Phys. Rev. B60 10094 (1999); J P Hansen and IR McDonald 
Theory of Simple Liquids (London : Academic Press) (1986) 


A Note on the Dynamical Structure of Liquid Lina Alloy

Indian Association for the Cultivation of Science

Not a member yet