M.J.CONDENSED MATTER VOLUME 4, NUMBER 1 1 DECEMBER 2001
*qachou@hotmail.com 4 17 2001 The Moroccan Statistical Physical and Condensed Matter Society The effect of the spin-orbit coupling in the relativistic contribution of
the atomic states overlapping.
M. Nejjar and A. Qachaou*
Laboratoire de Physique de la matière condensée (LPMC) B.P : 133-14000 Kenitra- Morocco
In this study, we made an explicit relativistic contribution to the overlapping of the atomic states. This contribution highlights the effect of the fine structure in this kind of overlapping. The weight of the relativistic term in the particular case of 3d transition metals is analyzed.
I. INTRODUCTION
The classification of the atomic states in core, valence and semicore states[1,2], enables us to treat differently these states in any single body calculation of electronic- structure in solids. Moreover, this classification depends primarily on the overlapping degree which binds these states in real space. The accuracy measurement of this overlapping is of capital importance in any calculation of prediction.
This classification was carried out on the basis of the expression [1]
R r R
E E
r R r
drr
= 1 2
1 2 2 1
∫0 2 1 2 2
− ∂ ϕ ϕ ∂
∂ − ϕ ϕ ∂ 2
=1 ϕ
ϕ (1)
connecting the atomic states overlapping to classical Wronskien corresponding to the second member of (1)[3], built in the particular case of the same angular momentum states. Expressed differently, the preceding expression lead to the Friedel sum rule used in the alloys study [4] and to the transferability criterion that control the scattering power of atomic pseudopotentials [5]. It was applied in the case of the linearized electronic-structure methods [6] to justify the valence and core states orthogonality in the muffin-tin sphere and of the waves functions in the interstitial region [7]. So, it turns out that it is very useful as well for the intra-atomic as for inter-atomic overlapping.
By construction, the equation (1) characterizes well the overlapping of the nonrelativistic atomic states, described by the principal and angular quantum numbers. Its extension to the relativistic states can be carried out in an implicit way by a simple substitution of the relativistic for the nonrelativistic states. However, the absence of an explicit relativistic contribution in the expression (1) remains not justified. This limits its application in the case of magnetic systems involving the spin-orbit coupling which refines the atomic states description.
In this work, we introduce an explicit relativistic contribution into the equation (1). We analyse the weight of this contribution for various relativistic atomic states.
Then, a test of the importance of this contribution is carried out on 3d transition metals.
II. ESTABLISHMENT OF THE RELATIVISTIC CONTRIBUTION
We start from the Dirac equation reduced to two coupled radial equations [ 8 ], expressed in the case of a spherical potential V(r).We use the atomicunits(c=α−1=137.037).
φ and F represent respectively the radial part of the major and minor components of the Dirac wave functions in the standard representation, k being the spin-orbit quantum number, equal to L ( angular momentum)
for 2
−1
=L
j and to – (L+1) for
2 +1
=L
j . The Dirac equations are given by:
) r r ( ) k r ( F ) r ( V dr E
) r ( d
) r ( )) r ( V E ( ) r ( r F k dr
) r ( dF
φ
−
+ −
α α 2 φ =
φ
− α
−
=
2
(2)
After substitution of the F expression, deducted from the second equation, in the first equation of the system (2), we obtain the second-order differential equation which generates major part φ, namely:
0
=
φ
+1 +2 −
+
φ + φ 2
− α
− φ
2 2 2 2
) r ( ) E ) r ( V ( M r
) L ( L
) r r ( k dr
) r ( d dr
)) r ( V ( d dr M
) r ( d
(3)
Where φ(r) is the r time radial wave function.
Equation ( 3 ) is more general; scalar relativistic case ensues directly from it if one replaces k by -1 (the spin- orbits term is being averaged). We note:
) E ) r ( V (
M −
2
−α 1
= 2 , where α is the fine structure
constant.
From the equation (3), we deduce after integration, a new expression of the overlapping between a wave function ϕ1 having the eigenvalue E1 and another wave function ϕ2 having the eigenvalue E2 in a sphere of R ray (R being the atomic core ray):
18 M. NEJJAR AND A. QACHAO 4
E RT E
r R r
) E r
E
) L ( L ) L ( ( L drr
R r R
− ∂ + ϕ ϕ ∂
∂ − ϕ ϕ ∂ 2 1
2 = 1
−
1 +
− 1
− + 1 ϕ ϕ
= 1 2
1 2 2 1
2
∫0 2 1 2
1 1 2
2 2 2 1
(4)
Where an additional terms RT (relativistic term) representing relativistic effect appears. For reasons of convenience this RT term is decomposed into four RTi parts such as RT=∑4 1i= RTi, with:
( )
∫0 1 2
2 1 1
2 2 2 1 4
1 2
1 2 2
∫0 1
2 1
2 1 1
2 4 3
∫0 2
1 2
1 2 1 2
1 2 2 2
∫0 2 1 2 2 1
2 1
ϕ ϕ
−
− 4
=α
−
+ ϕ −
ϕ
− 1 8
=α
∂ ϕ ϕ ∂
∂ − ϕ ϕ ∂
− 1 4
=α
2
− + ϕ 2 ϕ
−α
=
R R R R
M M dr
) r ( drrdV E E
k RT k
and )]
r ( V ) k k (
E k E k M [ M dr
) r ( drrdV E RT E
M ; r M
r dr
) r ( drr dV E RT E
; ) r ( V E E drr RT
(5)
Expression (4) then highlights, in addition to the nonrelativistic term, noted thereafter NRT, the relativistic correction RT. This relativistic term, unlike the NRT, expressed in an integral form depends, in addition to the waves functions and corresponding eigenvalues, on the concerned atomic potential and the relativistic quantum number. This correction can be regarded thus formally as a correction of the first rather than the second member of the equation (1). Let us note that the first member of the equation (4) differs from that of the equation (1) by the existence of an additional corrective term which holds in account the overlappings of different angular momenta.
III. APPLICATION
In the following, the evaluation of the relativistic contribution will be tested in some particular cases of 3d transition metals.
A. IMPORTANCE OF THE RELATIVISTIC CONTRIBUTION
For better determining the respective weight of NRT and RT terms, we carried out their comparison in terms of absolute values measured by the relative variation
NRT NRT RT −
=
∆1 . This informs us about the importance of the relativistic effect.
∆
1 depends obviously on widening of the atomic sphere. Thus, on figure1 we presented the evolution of∆
1 versus R in the case of the iron.0 ,5 1 ,0 1 ,5 2 ,0 2 ,5 3 ,0
0 ,0 0 ,5 1 ,0 1 ,5 2 ,0
3d5/2-3d
3/2, 3p
3/2-3p
1/2, 3s
1/2-3p
1/2, 3p
3/2-3d
3/2, 4s
1/2-3d 3/2
∆ 1
R(Bohr)
FIG1. The weight of the relativistic contribution calculated for iron.
According to the concerned overlapping it appears that the weight of this relativistic improvement evolves differently with the extent of the atomic core. The designation of an R ray delimits an atomic charge and consequently a degree of atomic orbital overlapping.
Beyond this ray one generally admits that the corresponding part of the atomic charge is negligible. Our analysis relates mainly to the internal area below R.
In the nonrelativistic case, this overlapping is related directly to the scattering power of the atomic core expressed by NRT at R. However, from this study it arises that relativistic contribution modifies the overlapping of orbital for suitable R rays. Indeed, the figure1 shows impossibility of choosing an adequate R ray for all the overlappings concerned in order to neglect the intervention of the relativistic contribution. Then, a good representation of the atomic overlapping would require an analysis of the weight of the relativistic contribution.
To try to better account for the effect of this contribution, we used the algebraic ratio
NRT
= RT
∆2
giving a direct measurement of this weight and constituting another means of comparison between RT and the NRT.
The results of a test of this ratio carried out on Ni, Fe and Ti atoms are presented on figures 2 and 3.
0 1 2 3 4
-1,0 -0,8 -0,6 -0,4 -0,2 0,0
Ni Ti
∆ 2 Fe
R(Bohr)
FIG2. The calculated relative ratio ∆2 versus R of the overlapping 3d5/2-3d3/2 for iron, titanium and nickel.
4 THE EFFECT OF THE SPIN-ORBIT COUPLING IN THE RELATIVISTIC CONTRIBUTION OF … 19
0,5 1,0 1,5 2,0 2,5
-0,9 -0,8 -0,7 -0,6 -0,5 -0,4 -0,3 -0,2 -0,1 0,0
Ni Ti
Fe
∆ 2
R(Bohr)
FIG 3. The calculated relative ratio ∆2 versus R of the overlapping 3p3/2-3p1/2 for iron, titanium and nickel.
The examination of these figures shows the existence of three rather distinct zones. A first appearing for the weak R rays is characterized by a strong RT contribution for two overlappings 3d5/2-3d3/2 and 3p3/2-3p1/2. This could be due to the effect of a kinematic relativistic contribution rather significant for the electrons in the vicinity of the atomic nucleus. The second zone, appearing in the shape of a landing, may be reflecting a predominance of NRT on RT.
This zone is delimited by a critical ray noted RC. This singular point marks consequently a separation, in real space, between the area of the core (R<RC) slightly influenced by the external effects and a valence area easily perturbed (R>RC). The position of RC depends obviously on the angular momentum (as shown in the pseudopotential theory [5]) and on the atomic number Z.
This singularity of ∆2 at RC could be explained by the strong electronic scattering at this critical ray. The area located beyond RC and constituting the third zone, casts light on the importance of the relativistic contribution. The weight of this contribution varies strongly according to Z.
Note that from the results shown on figures 2 and 3 illustrating the particular cases of the overlappings with zero parity, we obtain a clear relativistic contribution. In addition, the intensity of this contribution is more significant for 3d5/2-3d3/2 than for 3p3/2-3p1/2.
The results of the analysis of the nonnull parity overlappings are presented on the figure 4, for the iron case. The two cases of Ni and Ti correspond to a similar behavior.
0,5 1,0 1,5 2,0 2,5 3,0
0,00 0,02 0,04 0,06 0,08 0,10 0,12 0,14
3s1/2-3p
1/2, 3p
3/2-3d
3/2, 4s
1/2-3d
3/2
∆ 2
R(Bohr)
FIG 4. The calculated relative ratio ∆2 versus R of the nonnull parity overlapping for iron.
So, on this figure we note also the presence of the three zones above. This is more marked for overlapping 3s1/2- 3p1/2 than for 3p3/2-3d3/2, and particularly 4s1/2-3d3/2. This is may be expected because of the larger parity (∆L = 2) for the overlapping 4s1/2-3d3/2 and that the 3d states are more localized. Moreover, the last overlapping 4s1/2-3d3/2 does not show three zones differently to others (3s1/2-3p1/2 and 3p3/2-3d3/2). It corresponds also to ∆2 practically null. This suggests that the importance of the relativistic contribution could be the origin of the appearance of a ∆2 behavior into three zones.
Otherwise, by comparing the effect of the parity on this contribution, we note that the ∆2 singularity obtained at RC and observed for ∆L = 0 (figures 2 and 3) disappears completely in the case of ∆L > 0 (figure 4). The existence of a RC in the first case could be related to the states orthogonality condition.
It results from that the null parity overlappings, more interesting in the magnetic studies [9], require the introduction of RT for renormalizing the states overlappings.
B. ANALYSIS OF THE RELATIVISTIC TERM RT
The analysis of the relative weight of the various RTi terms enabled us to note the larger importance of RT4 term.
An application carried out on iron illustrated on figure 5 describing the evolution of the RT4/RT ratio according to the R ray for various overlappings shows a predominance of RT4 on the RT remaining parts for 3d5/2-3d3/2 and 3p3/2- 3p1/2 cases.
20 M. NEJJAR AND A. QACHAO 4
0 ,5 1 ,0 1 ,5 2 ,0 2 ,5 3 ,0 3 ,5 4 ,0
0 ,0 0 ,5 1 ,0 1 ,5 2 ,0
3d5/2-3d
3/2, 3p
3/2-3p
1/2, 3s
1/2-3p
1/2, 3p
3/2-3d
3/2, 4s
1/2-3d 3/2
RT4 / RT
R(bore)
FIG 5. The calculated RT4/RT ratio according to R for iron.
This predominance is maintained for all the considered R rays. However, for the other overlappings it decreases gradually and becomes minimal for 4s1/2-3d3/2. This suggests that, by taking in account the concerned overlappings parity, the RT4 contribution decreases when this parity increases. Otherwise, the strong relativistic contribution of the overlapping for vanishing parity must mainly du to RT4. This kind of overlappings is more important in the particular case of magneto-optical Kerr effect[10 ] and in the magnetic dichroism in x-rays absorption[ 9,11 ].
Bibliography:
[1] S. Goedecker, Phys. Rev. B47, (1993)9881.
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Let us note that the RT4 importance can be easily deduced from the presence of the factor (k1-k2)/(E2-E1) in its expression. Indeed, this factor increases
for the states characterized by a weak spin-orbit coupling.
This is sensitive enough for the energetically tighter valence states such as 3d5/2-3d3/2 and 3p3/2-3p1/2. The relativistic contribution is consequently very weak in any scalar relativistic calculation and fo r the states having k1 = k2. In addition the strong RT4/RT dependence versus R, for the weak atomic rays and for high parity
overlappings, elucidates the strong locality of the atomic charge when the factor (k1-k2)/(E2-E1) decreases.
IV.CONCLUSION
An expression taking account of the atoms fine structure was established for the states overlappings. It highlights a new explicit relativistic contribution which is more important when it is dominated by a depending term on the difference between the quantum spin-orbit numbers of the two states: ∆k=k1-k2. For transition metals treated in this work, we reported the importance of this contribution especially for the null parity overlappings. This relativistic correction allows the normalization of the atomic states overlappings inside the atomic spheres. The formalism established in this study remains adapte as well for the free atoms as for the atoms in a condensed system.
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