IACS Institutional Repository

Front Contents

Varma, H R

Banerjee, T

Deshmukh, P C

Indian J. Phys. 81(10) 1069-1077(2007) 
Selective interchannel coupling as computational strategy 
to interpret complex atomic spectrum 
H R Varma, T Banerjee* and P C Deshmukh 
Department of Physics, Indian Institute of Technology Madras, Chennai-600 036, India 
E-mail • pcd@physics.iitm.ac.in 
Abstract : Selective interchannel coupling has been used in the present work as an effective computational 
strategy to identify specific autoionization resonances and other correlation effects in the photoionization 
angular distribution asymmetry parameter for the case of valence ns subshell photoionization of atomic krypton 
and mercury. It is found that in the case of 6s photoionization of Hg, deviation of (3 from the non-relativistic value 
2 in the energy region below the 5d ionization thresholds is a cumulative effect of the Cooper minimum in 6s 
photoionization and resonant excitations from the 5d subshells. 
Keywords : autoionization resonances, relativistic random phase approximations, relativistic multichannel 
quantum defect theory 
PACSNo. : 32.80.Dz 
1. Introduction 
Autoionization is a many-body phenomenon in the photoabsorption process that occurs 
as a result of the interference between the degenerate ionization and excitation 
channels [1]. It causes significant fluctuations in the photoionization parameters. 
Understanding the process of autoionization is very crucial to extract information about 
the atomic spectrum and it can provide a great deal of information about relativistic and 
many-body effects. In addition, study of autoionization phenomena provides a good 
understanding of several related processes such as dielectronic recombination, stellar 
and tokamak plasma processes etc. Autoionization studies are also important to get 
information about metastable states that may have specific application in the production 
of lasers [2]. 
Atomic spectrum in the region of autoionization is extremely complex. Identification 
of resonances is a tedious work. Theoretical techniques come handy in this regard 
•Corresponding Author 
Present address : Department of Physics,
 a 9 m 7 , A r o 
Manipal Institute of Technology, Manipal-576 104, Karnataka. India ** dVUf I M O ° 
1070 H R Varma, TBanerjee and PC Deshmukh 
since they enable selective treatment of the final state correlations. Relativistic Random 
Phase approximation (RRPA) is one such powerful many-body technique which include 
final state correlations through interchannel coupling [3]. The loss of gauge invariance 
of RRPA resulting from the exclusion of channels that are left out is rather minor [4] 
Thus, by making use of selective truncation of the RRPA, the atomic spectra can be 
characterized since channels that are excluded can only be associated with missing 
features in the computed spectrum. The Relativistic Multichannel Quantum Defect 
Theory (RMQDT) when used in conjunction with the RRPA, provides additional strength 
to the computational strategy [5]. The RMQDT allows one to characterize the dynamics 
of photoionization processes in terms of a few parameters, namely the eigen quantum 
defects JJ„, the eigen-dipole amplitudes Dc>, and the transformation matrices Uky. The 
quantum defects parameters depend only on the short-range part of the potential and 
are nearly insensitive to energy. The RMQDT parameters are obtained from RRPA by 
doing calculations at a few energy points in the vicinity of the inner-shells which are 
closed for photoionization, and their insensitivity to energy enables interpolation in the 
autoionization energy region of interest. 
In the present work is studied the photoionization angular distribution asymmetry 
parameter 0 for valence ns subshells of atomic krypton and mercury in the energy 
region of autoionization resonances generated by the interference of photoionization 
channels from ns subshell and excitation channels from (n - 1)cf subshells. The 
relativistic expression of 0 is : 
_ |^1/2-3/21 + ^2(D1/2_i/201/2-,3/2 + CC) 
l ^ I2 [^ I 1 ' <1 ) 
I ^ 1/2-3/2 | + | ^ 1/2-3/2 I 
In the non-relativistic limit 0 is independent of photon energy and takes a constant 
value 0 = 2. The deviation of 0 from its non-relativistic limit 0 = 2 occurs in the vicinity 
of Cooper minima, as the two relativistic amplitudes go through respective 'zero* at 
different energies [6]. At high photon energies, where /-dependent relativistic effects can 
become large, strong interchannel coupling [7,8] can cause significant differences in the 
relativistic partial wave amplitudes, and may result in a departure of 0 from the non-
relativistic value '2\ Finally, in the autoionization resonance regions [9,10], selective 
excitations of one relativistic transition amplitude over the other also causes 0 to depart 
from '2\ 
We first illustrate the methodology by reviewing our recent work on the 
autoionization resonances in Kr angular distribution asymmetry parameter 0 [11,12]. The 
resonances have been identified successfully by controlling in our computational 
strategy the extent of final state correlations. 
In the 4s photoionization of atomic krypton, autoionization resonances occur 
below the 3d ionization thresholds as a result of interchannel coupling between the two 
Selective interchannel coupling as computational strategy to interpret complex atomic spectrum 1071 
,onization channels (4s -> 6 f t / 2 and 4s ep1 / 2 ) from the 4s subshell and the six 
excitation channels (3of5/2 -+ nf7/2l 3d 5 / 2 -* nf5/2, 3cf5/2 -> n ^ / 2 , 3d , / 2 -> nf5/2, 3d3 / 2 
> np3/2 and 3d 3 / 2 - • npy2) from the 3 d subshells. For preliminary investigation of the 
autoionization region, the RRPA technique was used. It is difficult to identify all the six 
series of resonances from 3d subshells in the same run. Therefore, to characterize the 
resonances RRPA was employed at three different levels of truncations : 
(i) 5 channel calculations : 
4s -+ 03 /2, p1/2, 
3d5 /2 —• f7/2, 4/2f ft/2» 
(ii) 5 channel calculations : 
(iii) 
4S - • 
3cf3/2 
fl3/2» P 
""* 4/2. 
4/21 
03/2» Pi/2. 
20 channel calculations : 
5P3/2 
5Pl/2 
4s -+ 
3Gf5/2 
3^3/2 
3P3/2 
"-* ^3/2. 
~+ ^3/2. 
^1/2. 
s, 
P3/2. Pi/2. 
~ * ^7/2. 
—> 4/2. 
"~* ^5/2. 
4/2. 
P3/2. 
^3/2. 
s, 
P3/2. 
P1/2. 
s, 
3p 1 / 2 -» d3 / 2 , s, and 
3s —» p ^ , Pi/2-
While the set (i) and set (ii) enabled isolating specific correlations, the set (iii) enabled 
extensive inclusion of most of the RPA correlations that have been included in the 
present study. 
In the present work, experimental thresholds have been used instead of Dirac-
Fock (D.F.) thresholds. Certain many-body correlations that are left out in the RRPA 
can be well accounted using experimental thresholds [14,15]. The experimental thresholds, 
along with the corresponding Dirac-Fock thresholds, are presented in Table 1 . 
Figure 1 shows the truncated RRPA results obtained using selective interchannel 
coupling given in (i) and (ii). At level (i) of truncation, excitation channels only from 
3d5 / 2 are included in the calculations whereas at level (ii), excitations only from 3d 3 / 2 
are included. From the information obtained from level (i) and (ii) resonance structures 
are characterized. Finally, using truncation level (iii), the resonances are labeled and 
compared with the experimental results as shown in Figure 2. The RRPA curve is 
broken in regions close to the resonances since in these energy regions, the RMQDT 
technique is better suited and the results are presented separately. 
1072 HR Varm, TBaner/eeandPC Deshmukh 
Table 1. Experimental and D F thresholds in eV 
Shell 
Expt 
Kr 
D.F 
4PI/2 
As 
3d6 /2 
3d3 /2 
3P3/2 
3Pi / 2 
3s 
140 
147 
27 5 
94 2 
95 4 
214 8 
222 4 
2931 
140 
14.74 
32.32 
101 41 
102 79 
226.20 
234 56 
305 43 
200-
1 t t -
197-
1 M -
3d, 
- ^ Ss?/ty 
Kr4s 
3d 
"1 ;|; 
j 
01 92 93 
Photon EnergyfeV) 
Figure 1. Kr 4s photoionization cross-section at 
two levels of truncations (i) 4s + 3<4/2 (solid), (u) 
4s + 3dj /2 (dotted) 
Photon Energy(eV) 
Figure 2. Comparison between theoretical Kr 4s /J 
values and experimental values 
As can be seen from this Figure, The positions of resonances show excellent 
agreement with experiment which is evident from the positions of 3d5/2 - • 5p and 
3d3/2 -+ 5 P resonances. This is further confirmed in the next section using RMQDT. 
The calculations clearly indicate deviations of f3 from 2.0, in qualitative agreement with 
experiment. 
Notice that the RRPA calculations overestimate width of these resonances and 
also overestimate the deviation of (3 from 2.0. This can be attributed to the omission 
of ionization-plus-excitation channels (photoionization satellites) in the calculations. In 
the next section a detailed study of the autoionization resonances is carried out using 
RMQDT in conjunction with RRPA. 
It may be added that when the channels included at different levels of truncation 
are all put together, as at level (iii) of truncations certainly the many-body correlation 
will get affected, but only marginally so. 
Selective interchannel coupling as computational strategy to interpret complex atomtc spectrum 1073 
For the RMQDT analysis, first a few energy points were selected near the 3c/5/2 
threshold to extract the ab initio RMQDT parameters. Again the methodology of 
'selective coupling1 of channels to identify the resonances has been used and RRPA 
+ RMQDT has been employed at the following three levels of truncation : 
(i) 5ch: channels from 4s and 3d5 / 2 are coupled, 
(ii) 5ch: channels from 4s and 3d3/2 are coupled, 
(iii) 8ch: channels from 4s, 3d 5 / 2 and 3cf3/2 are coupled. 
The quantum defect parameters have been obtained for each case (not shown) 
and used for interpolation in the autoionization region. In Figures 3-5 is shown the 
angular distribution asymmetry parameter /3 obtained using RMQDT In the same figures 
3d„ * . 
MO M S 010 019 920 S2J MO 03ft 040 045 
Photon Enorgy(eV) 
Figure 3. Kr 4s (3 in the truncated RRPA + RMQDT, 
coupling two ionization channels from 4s and three 
excitation channels from 3d5/2 along with the 
experimental results. 
0 1l 
1985 
1 9 M 
1975 
1970 
lAAtr 
] 
JKr 
j RMQOT 
J—•— Expt 
f\ 
\ 
^um 
* 4 
ll s 1 1 
J 5 n 
1 3 3 3 J | 1 )|1|J M 0 M S 010 019 020 02-5 030 03 0 040 04 0 0S0 009 
Photon Energy (eV) 
Figure 4. Kr 4s j3 in the truncated RRPA + RMQDT, 
coupling two ionization channels from 4s and three 
excitation channels from 3tf3/2 along with the 
experimental results 
2.00 
1.98-1 
1.96 
1.94 
1.92 
1.90 
1.88 
1.88 
a 
3 
Kr 
—•— Exp»riin«nt 
RRPA+RMQDT 
— - Convouted 
92.0 92.5 
I 
I 
I 
5 
1 
I f 
f t 
H 
93.0 93.5 
Photon Energy(eV) 
94.0 
Figure 5. Kr 4s j3 in the truncated RRPA+RMQDT, coupling two ionization channels from 4s and six excitation 
channels fro 3cfe/2 and 3d3/2 together along with the experimental data. Dashed curve is the convoluted 0 with 
a Gaussian of FWHM = 0.50 eV. 
1074 H R Varma, TBanerjee and PC Deshmukh 
are also shown the experimental results obtained by our collaborators [12]. 
In Figure 5 are presented the results for the angular distribution asymmetry 
parameter 0 obtained using level (iii). The resonances shown are the 3c/5/2 excitations 
to members of the Rydberg series with n > 6 since the RMQDT parameter vary 
somewhat for the lowest resonances making the RMQDT method inapplicable [13]. 
There is a good agreement with respect to position of the resonances obtained from 
RRPA + RMQDT with the experimental results. This methodology illustrates the 
success of the computational strategy adopted to characterize complex autoionization 
spectrum. 
There is a strong deviation of 0 from 2.0 in the resonance region. Akin to the 
RRPA calculations, the deviation of 0 from 2.0 is significantly greater than what is 
observed. This problem can be alleviated, at least partially, by convoluting the 
calculated 0 profiles with Gaussian of FWHM 50 meV which corresponds to the 
experimental resolution. The result is shown as dashed curve. This, however, does not 
bring theory into complete quantitative agreement with experiment. Overestimation of the 
deviation of 0 from 2 is attributed to correlations that are left out in the theoretical 
formalism. Strong correlation effects (interchannel coupling) causes pronounced variation 
in the relativistic channels in the region of autoionization resonances making 0 * 2. 
In the next section is discussed the autoionization resonance region below the 
5d ionization threshold of atomic mercury. 
2. Photoionization for 6s subshell of atomic mercury below 5d ionization 
thresholds 
Selective interchannel coupling was employed to identify the 6s autoionization spectrum 
of Hg. Preliminary studies using RRPA were carried out using four different levels of 
truncation : 
(i) 2 ch: channels only from 6s are coupled (intra-shell correlation), 
(ii) 5 ch: two channels from 6s and three channels from 5cf5/2 are coupled, 
(iii) 5 ch: two channels from 6s and three channels from 5d3/2 are coupled and 
(iv) 8 ch: two channels from 6s, three channels from 5cf3/2 and three channels 
from 5(^/2 are coupled. 
The analysis of the resonances was thus carried out separately to identify the 
resonances and then a detailed study was carried out using RMQDT. 
In Figure 6 and Figure 7 are shown the RMQDT results obtained from the 5 
channel calculations. The 6s angular distribution asymmetry parameter was obtained as 
a function of photon energy, and of the effective quantum number n\ The resonance 
structures are periodic, as a function of n*. In Figures 6 and 7 are also shown the 
results obtained from the RRPA alone, that is without using the RMQDT It can be 
Selective mterchannel coupling as computational strategy to interpret complex atomic spectrum 1075 
(a) (b) 
RRPA+RMQDT.5Ch-Hg (2o,3c,0. 6s, 5 c y 
^ 6 . 0 5 
-05 
2 0 
RRPA 6* Sdv 
RMQDT 6ft-op.5d -cl 
14 9 15 0 155 151 15 2 15 3 M 
Photon Enorgy (tV) 
Figure 6. (a) Hg 6s angular distribution asymmetry parameter ft as a function of effective principal quantum 
number nMrom RMQDT + RRPA coupling two ionization channels from 6s (6s -» €fo/2 and 6s - • 6Pi/2) and three 
excitation channels from 5c/5/2 (5cfe/2 - • nf7/2, 5cfe/2 —• nf5/2t and Sd5f2 - • nfhtz (b) Continuous line corresponds 
to Hg 6s ft as a function of photon energy from RMQDT + RRPA coupling two ionization channels from 6s and 
three excitation channels from 5tf5/2 as in (a), and dashed line ft from RRPA coupling five photoionization 
channels from 6s and 5d5/2 subshells 
(a) 
RRPA+RMQDT-5Ch-Hg (2o,3cf0, 6ft, 5dJ 
(b) 
RRPA 6s,SdM 
RMQDT 6»-op,5dw-cl 
17 4 17 5 17 6 
Photon Energy («V) 
Figure 7. (a) Hg 6s angular distribution asymmetry parameter ft as a function of effective principal quantum 
number n*from RMQDT + RRPA coupling two photoionization channels from 6s (6s -+ ep^/2 and 6s -* ep^/2) and 
three excitation channels from 5d3 / 2 (5cf3/2 - • nfhf2% 5d3 /2 - • np3/2, and 5d3/2 -+ npy2 (b) Continuous line 
corresponds to Hg 6s ft as a function of photon energy from RMQDT + RRPA coupling two ionization channels 
from 6s and three excitation channels from 5d3/2 as in (a) and dashed line corresponds to ft from RRPA coupling 
five photoionization channels from 6s and bd^^ subshells 
seen that the resonance positions are identified well in the RRPA calculations The 
quantum defect values are presented in Table 2. Also presented in the same Table are 
1076 H R Varma, TBaner/ee and PC Deshmukh 
Table 2. Quantam defects /*p and », for atomic mercury. 
RMQDT (present) HS[16] 
4.22 4.01 
1.03 1.02 
the quantum defects from an earlier study, using the Hartree-Slater (HS) approximation 
[16]. As seen from this table, the present results and the HS results [16] show a 
reasonably good agreement. 
The main focus in the present work is on the deviation of 6s 0 from its non-
relativistic value 0 = 2. At energies of the autoionization resonances, 0 deviates from 
'2'. In the case of autoionization resonances seen in the photoionization of the 4s 
subshell of atomic krypton, reported in Section 1, the background value of 0 was close 
to -2 . In the case of the 6s photoionization of atomic mercury however, 0 for 6s 
photoionization in the region of 5d excitations deviates significantly from '2' even in the 
background region, not merely at resonances. This dramatic deviation of 0 in the 
background (non-resonant) energy region is due to the RRPA Cooper minimum in the 
6s photoionization channels. The deviation of 0 = 2 in this energy region is therefore 
a cumulative effect of the Cooper minimum in 6s photoionization, and the resonant 
excitations from 5tf subshells. 
3. Conclusion 
The present work demonstrates the success of employing selective interchannel 
coupling as a computational strategy to extract detailed physical information in complex 
atomic spectra. The RRPA + RMQDT method is especially well adapted to use this 
technique. 
4. Acknowledgment 
This work was partially supported by Board of Research in Nuclear Sciences (BRNS), 
Department of Atomic Energy (DAE) and Department of Science and Technology (DST). 
References 
[1 ] U Fano Phys. Rev. 124 1866 (1961) 
[2] J Bokor. R R Freeman and W E Ccx>ke Phys. Rev. Lett. 48 1242 (1982) 
[3] W R Johnson and C D Lin Phys. Rev. A20 964 (1979) 
[4] D L Lin Phys. Rev. A16 600 (1977) 
[5] C M Lee and W R Johnson Phys. Rev. A22 979 (1980) 
[6] J Tulkki Phys. Rev. Lett. 62 2817 (1989) 
[7] O Hemmers, S T Man son, M M Sant'Anna, P Focke, H Wang, I A Sellin and D W Lindie Phys. Rev. A64 
022507 (2001) 
[8] Z Altun and S T Manson Phys. Rev. A61 030702 (2000) 
1077 
9) D Dill Phys. Rev. A71976 (1973) 
0] S B Whitfield, B D Krosschell and R Wehlitz J. Phys. B37 3435 (2004) 
S B Whitfield. R Wehlitz, H R Varma, T Banerjee, P C Deshmukh and S T Manson J. Phys. B39 L335 (2006) 
2] Hari R Varma PhD Thesis (Submitted to Indian Institute of Technology Madras, India) (2007) 
3] P C Deshmukh and S T Manson Phys. Rev. A28 209 (1983) 
4] W R Johnson, V Radojevid, Pranawa Deshmukh and K T Cheng Phys. Rev. A25 337 (1982) 
5] Steven L Carter and Hugh P Kelly J. Phys. B112467(1978) 
6] S T Manson Phys. Rev. 182 97 (1969) 


Selective Interchannel Coupling as Computational Strategy to Interpret Complex Atomic Spectrum

Advisory Board and Editors

Raman, C V

A New Radiation

Khan, F Liakath Ali

Sivagurunathan, P

Kamil, S Raja Mohamed

English

Infrared Studies on Hydrogen Bonding Interaction Between Alkyl Esters and p-Cresol in Carbon Tetrachloride

Bhaumik, Uma

IndianJ.Phys. 81(1) 75-81(2007) 
Study of DM interaction in triangular lattice 
Uma Bhaumik 
Vidyasagar College, 39 Sankar Ghosh Lane, Kolkata-700 006, India 
E-mail : umabhaumik@lycos.com 
• ••• • • - i — . . « i , , . — _ . i «,n , i , , , 
Abstract : We have studied the triangular lattice using linear spin-wave theory with Heisenberg antiferromagnets in addition 
to Dzyaloshinskii-Moriya interaction and weak magnetic field. The quantum corrections to the ground state energy and sublattice 
magnetization has been calculated analytically as a function of DM interaction strength and also of the magnetic field. The gap 
in the excitation spectrum has also been calculated. Finally, we come to the conclusion that the DM interaction stabilizes the LRO 
reducing the effect of the quantum fluctuation. 
Keywords : Triangular lattice, Dzyaloshinskii-Moriya interaction. 
PACSNos. : 75.10.Jm, 75.40.Qb 
1. Introduction 
In the last few decades, geometrically frustrated antiferromagnets (AFM) have come to 
be very important for both experimental and theoretical research. The most extensively 
studied systems are the triangular and kagom6 lattice AFM in two dimensions and 
pyrochlore lattice in three dimensions. A large number of studies have been devoted 
to the triangular lattice. P W Anderson [1] first proposed that the system has a spin-
disordered ground state similar to the frustrated square lattices with further neighbor 
exchange, along with the usual nearest neighbor interaction. The resonating valence 
bond (RVB) state is one of the possible candidates for the ground state in this regime. 
An estimate of ground state energy has been obtained from various RVB-type 
variational wavefunctions [2,3] and also from the variational values of Huse and Elser. 
But several other methods, such as spin wave theory [4-6], variational calculation [7], 
exact diagonalization of small clusters [8] and Monte Carlo numerical method indicated 
the possibility of long range order with the ground state energy lower than the spin 
disordered states. The sublattice magnetization is, however, reduced considerably 
(-0.239) from its classical values (-0.5) due to the zero point quantum fluctuations. It 
is generally believed that the frustrated triangular lattice with Heisenberg antiferromagnet 
(HAFM) is quite similar to the square lattice i.e. a ground state with long range N6el 
ordered. Experimental realizations of triangular lattice HAFM are materials like VCI2l 
VBr2, NaNi02 etc. Possible ground state orderings in the quantum antiferromagnets 
© 2007 IACS 
76 Uma Bhaumik 
(QAFM) include Neel, helical, spin liquid, spin nematic, dimer or chiral liquid. The 
triangular QAFM is thought to exhibit long range 120° Neel order at T = 0. Dimer 
ordering is found for various SU(n) models for large n and may even survive in the 
n -> 2 limit, te. for S = 1/2, for models with competing further neighbor interactions 
Indeed there is no experimental realization or proof of LRO or spin liquid ground states 
The system like Cs2CuCI4 which is quasi-two dimensional S = 1/2 system on triangular 
lattice [9] shows a LRO at below the temperature 0.62 K and supposed to have the 
anisotropic Dzyaloshmskii-Monya (DM) interaction. The DM interaction produces canting 
between the spins and a LRO in the form of an incommensurate spiral spin structure 
occurs below the Neel temperature at zero field. 
There is a lot of controversy about the ground state of such frustrated lattices 
A large number of studies have been done in these lattices with Heisenberg 
Hamiltonian. In this work we have seen that anisotropic Heisenberg Hamiltonian 
stabilizes the LRO. In view of that, we have undertaken this study of Heisenberg model 
along with anisotropic DM interaction on triangular lattice It has been observed that the 
effect of quantum fluctuation in ground state energy and sublattice magnetization 
decreases as the strength of DM interaction increases In particular, we would like to 
study the effect of this anisotropic DM interaction on the ground state energy 
sublattice magnetization and gap in the excitation spectrum. Effect of external magnetic 
field has also been taken into account. This paper is organized as follows In Section 
II, we write the general Hamiltonian and the ground state of the lattice. In Section HI 
we shall have the major steps of the calculation and the results. Then, in the last 
Section, we shall conclude with our results. 
2. The Hamiltonian 
The most general spin Hamiltonian for two neighboring spin-1/2 magnetic ions in case 
of the insulator is given by 
Hy=J„St SJ + DI S.xSj+S, An S, + hSt. (1) 
The first term is the Heisenberg term. It is the symmetric part of the Hamiltonian. The 
second term is the anisotropic DM interaction and the in the last term Af/ is the 
anisotropic symmetric exchange interaction. The last term is the zeeman term. Here, 
the consequences of the DM interaction on the low temperature magnetic structure are 
explored in the case where D is perpendicular to the lattice plane. Taking into account 
the super exchange mechanism, the isotropic exchange JH is proportional to f / / l / where 
tH is the inter site hopping and U is the onsite Coulomb repulsion. It was shown by 
Monya that \DJ is proportional to (Xtf lAU) where A is the spin orbit coupling and A 
is the crystal field splitting and AH is proportional to (A%2/42(J). The third term, for a 
small value of A, being one order of magnitude smaller than DM interaction is 
neglected. The magnetic field is acting on the z-direction. Then finally we take the 
Hamiltonian as, 
H - J% SrSj + D. £ SfxSj + h2st . (2) 
Study of DM Interaction in Triangular Lattice 11 
We have taken this Hamiltonian on this lattice. At first the classical ground state has 
been investigated in the following. 
In the classical limit (S -> <*), the spins are treated as the classical vector 
oriented in the x-z plane. The DM interaction is along the y-direction. In a single 
triangular plaquette three spin vectors make angles 0, #, an<# <% with respect to some 
reference direction. Then the classical Hamiltonian in the ansefice of the magnetic field, 
can be written as \ 
S2 
En= —7 [cos(0~^1)-fcos(0-i92) + cos(0^i?2 + t91)] i (3) 
COS0 
where tp = tan*"1 (D/J). To obtain the classical ground state one has to minimize the 
above expression with respect to #, and 6fe. Then two possible orientations are obtained 
as (i) # = 2TC/3, 02 = 4;r/3 for D is in negative y-direction and (ii) #, = 4^/3, 82 = 
2;r/3 for D is in positive y-direction. Thus, we can conclude that the classical ground 
state is 120° Neel ordered and the direction of the DM interaction changes the 
chiralities of the ordering. In Figure 1 two possible ground states are shown. Now we 
study the effect of quantum fluctuation using the linear spin-wave theory in the 
presence of small magnetic field. 
mm 
D > 0 D < 0 
Figure 1. Two possible Neel orderings for positive and negative values of DM interaction acting perpendicular 
to the lattice plane in a triangular lattice. 
3. Linear spin-wave theory 
We consider the canted spin ordering, where spin vectors are oriented in the x-z plane. 
To bring the neighboring spins in the direction of same magnetization axis (say z-axis), 
one should rotate all the spins about y-axis and in the new reference frame the spin 
are defined as 
S f = cosQS? -f sinQSf, 
Sf = -sinQS? +• cosQSf, (4) 
where Q is the angle of the spin vectors with respect to the z-direction. In this new 
description we take the Holstein Primakoff transformation, as 
S?=±j2S(af+a;); Sf =^>/2S(a^a;) ; S f . f S - * * * ) . (5) 
78 UmaBhaumik 
To study the excitation spectrum, these bosons are again transformed to the Fourier 
space by 
a * ^ S > * ' < a , (6) 
where N is the number of lattice points belonging to one sublattice. Here, we consider 
the nearest neighbor interaction only. Let J and D be the strength Heisenberg 
interaction and DM interaction strength. Triangular lattice is described as 120° Neel 
ordering and three types spins A(Q = 0), B(Q = 120°) and C(Q = 240°) are possible 
belonging to three different sublattices. So for D positive, Qa0 is taken as 2nlZ and for 
D negative, Qa0 is 4^/3 and in this way, another ground state for different chirality has 
been taken into account. Every spin belonging to one particular sublattice, interacts 
with the other through the bond directions rt = (1.0), r2 = (-1/2.V3/2) and 
r3 s (-1/2, -V3/2). We define the bosons of the sublattices At B and C as a, b and 
c then we have the Hamiltonian as 
"*4 s 2 ( - J i + ^° i ) + 3 > , s + ? 2 K«itA)wo(«[^] 
cx,p,k 
-3CJ . 
where 
H0(*) = M2 M, 
(7) 
(8) 
M,= 
M 2 = 
" C 
C 
c3 
:, c2z czz* 
z* C, Cj,* 
2z C2z' C, 
"0 z z* 
z' 0 z 
Z 2* 0 
J 
2 = rr («~*'' + e * r« + e~*'»), 
(9) 
(10) 
3S 
a; ; & - « . 
(11) 
(12) 
Study of DM Interaction in Triangular Lattice 79 
Here, the 6 x 6 matrix H0(k) can be diagonalized analytically by general Bogoliubov 
transformation. It is interesting to note that all the 3 x 3 blocks building the matrix 
H(/c) are the permutation matrices and thus can be diagonalised simultaneously in the 
basis 
ui«(i 1 1 ) ; "2 = 0 J y2); ^3=0 f ') a n d i-e^f. (13) 
The appearance of cubic roots of the unity is the manifestation ;^of the ternary symmetry 
of the problem. Now we introduce the generalized Bogoliubo/ transformation as 
= r
 (14) 
To preserve the Boson commutation relations and to map H0(§ onto a diagonal matrix 
the column of the transformation matrix is written as 
A,u, 
, where the coefficients X, 
and jut satisfy the hyperbolic orthonormalization conditions |A, |2 - | / i , |2=:1, etc. So once 
three column vectors are found, the remaining three vectors are obtained by the action 
of -<rx. In this way the transformation matrix 7 is obtained. So after the diagonalisation 
the Hamiltonian equation can be written as 
Hk^S^J^SD^3hS^^^H^k-3C,] (15) 
or 
H^-tfi-J^&DJ + ShS+SS £ M L A + — S ^ + ^ + ^ - S C , ) ' (16) 
2 k.L*\,2,3 3 k 
where 
o>i = Vfc + A(C2 + CMS + Pi(C2 -C 3 ) ] , (17) 
and 
p}=z+z\ pz=zj + z*j2, p3=z-j + zj2. (18) 
A three-dimensional excitation spectrum has been drawn in the Figure 2. It is 
seen that the excitation spectrum is real and positive over the Brillouin zone. There is 
the gap in the excitation spectra which has been calculated as 
(19) 
a>!°-0> r-&*°*&-*J**°:£ • 
80 UmaBhaumik 
Figure 2. The three dimensional excuanon specira Tor u • u.o ana n = i for S » 0.5. 
The ground state energy including the zero point motion is 
j l ( Bbr^ M ^ (20) 
In the above expression, the last two terms give the quantum correction to the 
classical ground state energy. In the Figure 3(a) we plot the ground state energy as 
a function of D and it is observed that total ground state energy decreases as D 
increases. 
0 3 & 
036 
034 
032 
N 03 
V 028 
020 
0 24 
022 
02 
' _ _ _ _ ^ — — ^ ^ ~ ~ ~ ~ ^ 
""~~"~"~~~ ' — - ^ 
^ ' 1 
^"^^^ 1 y^ 
• f 
/ 
< , it , 1 1 I . ! ! , ! 1 1 1 
0 006 01 015 02 025 0.3 039 04 046 0ft 
d 
(b) 
Figure 3- (a) Variation of ground state energy w.r.t. D at h = 0 (upper) and h » -1.0 (lower); (b) the variation 
of the subiattice magnetization with the DM interaction for h = 0 (lower curve) and for h = -0.3 (for upper 
curve). $ = 1/2 value is considered. 
There is also a reduction in the ground state subiattice magnetization due 
quantum mechanical fluctuations which is calculated per site as 
2 2Nfi utk) (21) 
In the Figure 3(b) we plot the subiattice magnetization as a function of D and 
it is observed that this also increases as D increases. 
Study of DM Interaction in Triangular Lattice 81 
4. Conclusion 
In the previous section, we have studied the triangular lattice with spin wave theory. 
These lattices have been widely studied with Heisenberg Hamiltonian. But our calculation 
includes the anisotropic DM interaction. So the models become interesting because 
these are frustrated as well as anisotropic. The results a^ e very much interesting 
because ground state energy and the reduction in sublattice magnetization reduces as 
the strength of DM interaction increases. So, the effect of Ihis anisotropic interaction 
is to reduce the effect of quantum fluctuation in the triangular lattice. There are some 
magnetic systems, which are frustrated and no conclusion pan be drawn about the 
ground state with the isotropic Heisenberg Hamiltonian. In these cases, due to large 
quantum fluctuation, the ground state is highly degenerate. But our calculation 
concludes that DM interaction reduces the quantum fluctuation and 120° ordering is 
favored. 
Acknowledgments 
The above work is financially supported by UGC minor research scheme (Letter No. 
F. PSW-05/04-05(ERO)). The author is grateful to Prof. Arghya Taraphder, IIT Kharagpur 
for useful suggestions and comments. 
References 
[1] P W Anderson Mater. Res. Bull. 8 153 (1973) 
[2] T Oguchi, H Nishimori and Y Taguchi J. Phys. Soc. Jpn. 55 2095 (1987) 
[3] V Kalmeyer and R B Laughlin Phys. Rev. Lett. 59 2095 (1987) 
[4] H Nishimori and S J Miyake Prog. Theor. Phys. 73 18 (1985) 
[5] S J Miyake Prog. Theor. Phys. 74 468 (1985) 
[6] Th Jolicoeur and J C Le Guillou Phys. Rev. B40 2727 (1989) 
[7] D Huseand and V Elser Phys. Rev. Lett. 60 2531 (1988) 
[8] H Nishimori and H Nakanishi J. Phys. Soc. Jpn. 57 626 (1988) 
[9] R Coldea, D A Tennant, R A Coeley, D F McMorrow, B Dorner and Z Tylczynski Phys. Rev. Lett. 79 151 
(1997) 


Study of DM Interaction in Triangular Lattice

Rout, S K

Badapanda, T

Sinha, E

Panigrahi, S

150 S K Rout, 7 Badapanda, E Stnha and S Panigraht 
Finally, the Zr-Ti complex sol was added drop wise to barium acetate solution at room 
temperature through continuous stirring to get the desired composition. The compositions 
were diluted to a concentration of 0 35 mol/lit with the addition of acetic acid. Thtn 
films were prepared on platinised silicon wafer by spin coating technique using a 
Milman spin coater The films were deposited at spinning speed of 3000 rpm for 5 mm 
Just after deposition the films were fired at 600°C for 5 mm for organic removal and 
crystallization in to perovskite phase The coating and fairing sequence was repeated 
20 times to attain films approximately 800 nm thick Finally, these were annealed at 
1100°C for 2 h for perovskite phase formation and improvement in crystallmity The 
phase formation behavior and surface morphology of these films were analyzed by X-ray 
diffraction The surface morphology and film thickness were studied by SEM For 
electrical measurement a gold electrode of 400 //m square were deposited on the top 
of the film by dc sputtering The substrate platinum was used as bottom electrode 
The room temperature frequency dependence of permittivity and loss tangent was 
studied by HP 9142A low frequency impedance analyzer 
3. Results and discussion 
Figure 1 shows the XRD patterns of Zr substituted BaTi03 thin films at room 
temperature The XRD patterns were indexed considering the cubic structure of BaTi03 
It is evident from the plot that all the films were crystallized in to single phase cubic 
perovskite structure Substitution of Ti4+ (0 61 A) by higher radii Zr44 (0 72 A) increases 
the d spacing, which shift of diffraction peaks towards lower angle This is a clear 
indication that Zr is systematically dissolved in BaTi03 lattice in the studied composition 
range 
20 30 40 50 GO 70 80 
20 (degree)-* 
Figure 1. X-ray diffraction pattern of Zr modified BaTi03 thin films at room temperature. 
Figure 2 shows SEM micrograph of sol gel derived BTZ thin film on platinised 
silicon wafer. All the films have subjected to a final anneal to 1100°C. The addition of 
Zr to the BaTi03 lattice decreased the grain size of the crystallized films. The 
decreasing gram size with increasing zirconium may be attributable to lower grain 
growth rates from the more slowly diffusion of the Zr4* ion, which has a bigger ionic 
Dielectric study on sol gel derived BTZ film deposited by spin coating technique 151 
Figure 2. SEM micrographs of sol-gel derived thin film (a) BaTi03, (b) Ba*r\,8Zr02O3t (c) BaTi06Zr04O3 and (d) 
BaTi03 (cross sectional view ) annealed at 1100°C. 
radius than Ti4+. Zirconium content may be attributed to lower grain growth for the 
same heat treatment [3]. 
Figure 3 shows the frequency dependency permittivity and dielectric loss of 
BaTi^xZry03 compositions. Permittivity of all the compositions was very stable in the 
whole frequency range. In general, permittivity (e) decreases with the increase in Zr-
substitution due to; (i), decrease in concentration of high permittivity material BaTi03> (ii) 
decrease in grain size resulting to the decrease in polarizability of the atoms in the 
structure. At very low frequency (Figure 3(b)), high dielectric loss is observed, which 
may be due to presence of all type of polarization. That also quickly decreases up to 
about 1 MHz due to the decrease in space charge polarization. The dielectric loss of 
all the compositions was very stable in the frequency range 1 MHz to about 8 MHz 
and this loss is very minimum indicating the good quality of the materials as well as 
the thin film. At high frequency a dielectric loss peak is observed. Several possible 
1000
 * ioT 
0.1 i 
8. 
I I I'l I H I | 
1000 10000 
0.01 
0.001 
100000 
1—i t r i i l l 
10000 100 
Frequecny (kHz) Frequecny (kHz) 
Figure 3. Frequency dependency of (a) permittivity and (b) dielectric loss of different Ba(Ti,_xZrx)03 compositions. 
152 S K Rout, TBadapanda, E Sinha and S Panigrahi 
causes exist for such dispersion including the hypothesis of the influence of the 
contact resistance between the probe and electrode, resonance due to high dielectric 
constant. Similar frequency dispersion behavior was also reported for other ferroelectric 
thin film [4,5]. 
AC conductivity (<r) of the samples was calculated using the formula 
a = 2a fC{d/A) land] here symbols have there usual meaning. The variation of ac 
conductivity with Zr-content is plotted in the Figure 4. The conductivity was found to 
Figure 4. Variation of ac conductivity Ba(T\{.£rx)Oz thin films with Zr-content (x) at different frequencies 
decrease with increase in Zr-concentration and increase with frequency. As a notion 
impurities and defects are considered to have major effect in titanate ceramics, i.e. 
higher the defect higher will be the conductivity. Here, as Zr-addition reduces the defect 
in the ceramic, that may be the reason of decrease in conductivity with substitution. 
As the grain size decreases, the number of grain boundary per unit thickness 
increases. It is well known [6,7] that in titanate ceramic, grain boundary are highly 
resistive than the grain. The decrease in conductivity is due to decrease in grain size. 
4. Conclusions 
From the above study, it may be concluded that 
(i) Zr is systematically dissolves in BaTi03 lattice in the studied composition range. 
(ii) Diffusion of Zr is relatively slower than Ti due to its higher ionic radii, resulting 
lower grain growth. 
(iii) Permittivity of all the compositions was very stable in the frequency range 1 kHz 
to about 8 MHz. In general permittivity (e) decreases with the increase in Zr 
substitution due the decrease in polarizability of the atoms in the structure. 
(iv) In general tan S decreases with Zr-substitution. 
(v) Dielectric conductivity decreases with Zr-substitution due to increase in number 
of grain boundary per unit thickness resulted from decrease in grain size. 
Dielectric study on sol gel derived BTZ film deposited by spin coating technique 153 
References 
[1] Y Zhi, A Chen, R Guo and A S Bhalla AppI. Phys. Lett. 81 1285 (2002) 
[2] X G Tang, H L W Chan and A L Ding Thin Solid Films 460 227 (2004) 
[3] M C Gust, L A Momoda, N D Evans and M L Mecartney J. Am. Cerafa. Soc. 84 1087 (2001) 
[4] R R Das, P Bhattacharya and R S Katiyar AppI. Phys. Lett 81 1672 {2002) 
[5] F M Pontes, DSL Pontes, E R Leite and E Longo J. AppI. Phys. 91 6^50 (2002) 
[6] S K Rout and J Bera Ferroelectrics 323 79 (2005) 
[7] S K Rout, S Panigrahi and J Bera Bull. Mater Sci. 28101 (2005) 


Dielectric Study on Sol Gel Derived BTZ Film Deposited by Spin Coating Technique

Nishti, S V

Kunabenchi, R S

Suresh Kumar, H M

Hanagodimath, S M

Biradar, J S

Electronic Excitation Energy Quenching of Indole Derivative By CCl<SUB>4</SUB> in Different Solvents

Indian J. Phys. 81(3) 405-408 (2007) 
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Notes for Contributors

Yadav, Mahendra Pratap

Kumar, Santosh

Cosmic Ray Decreases and Geomagnetic Storms During 1975-1994

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