The Connection of Sum Rule and Branching Ratio Analyses of Magnetic X-Ray Circular Dichroism in 3d Systems
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"ALawrence Livermore National Laboratory, Chemistry and Materials Science Department, Livermore, CA 94550 - USA BUniversity of Missouri-Rolla, Department of Physics, Rolla, MO 65401-0249 • USA CUniversity of California-Davis, Department of Physics, Davis, CA 95616 *USA
DVirginia Commonwealth University, Department of Physics, Richmond, VA 23284-2000 - USA
In the recent past, Carra, et al. 1- 3 derived a sum rule for electric dipole transitions in a single ion model that could be used to extract an elementally-specific spin-magnetic-moment (l'SpN) from magnetic x-ray circular dichroism (MXCD) spectra. Earlier, we proposed the utilization of a branching ratio analysis 4 for the determination of gSPIN, based upon a simplified oneelectron, atomic picture which assumed complete orbital quenching. Here, it will be shown that these two approaches are essentially related in the case of 3d ferromagnetic materials. Both methods are based upon a comparison of the integrated intensity in the L 3 (J=3/2) white line peak versus the sum of the intensities in the L 3 (J=3/2) and L 2 (J=l/2) peaks, after background removal. An error estimate will also be presented. A more complete description of our work is under preparation 5 . Consider Equations 1 and 2, taken from references 1, 2, and 3.
Jj+
P=
d (,+j-da('+--) 1 e(e+1)+ 2 -c(c+1) j++j-do(4-+ +g-+0) 2 i(- +1)(4t + 2- n)
[
i+ j.jdog f .j+do(Rt+- _R_)_ c +__1_ . i(t + 1)- 2 -c+ =d1) (Sz-+-A(c,-,m)(Tz)_ c j
_I-)
[2]
f j++j-dco(i+ +g[+g0o)
3c(4e+2-n)
For the case of 3d magnetic materials and using the 2p -- 3d transition, i = 2 and c = 1. The number of 3d electrons (holes) is n (10-n). In our notation: dog+an1-/ -, 1+ f+ I3/2 = f j+dco 9+ 13-/ ,I/2 = J j.do) and I7 2 =f jdo 13/2=j+dot i1/3/2
-.
Switching to our notation and using R* = 1/2(lt+ + g-) as in Reference 2, a combination and rearrangement of Equations 1 and 2 gives us Equation 3, shown below.
457
Mat. Res. Soc. Symp. Proc. Vol. 384 01995 Materials Research Society
2(Sz) + 3(Lz) = 6(10 - n)
[+ ,2-
13/2
]
tI3/2 + 1+1/2 + I1/2 + 11/2]
[31
Here, we have also taken the liberty of dropping (T7 ) as done previously in References 1, 2, and 3. Again, notaion Equation 3 is merely a restatement of the sum rules • of,SRCarra, et al., using our notation and explicitly showing the spin moment =2(Sz)) -tSPiN and orbital moment RtR = (Lz)). The super script SR stands for sum rule. Now consider the branching ratio (BR) analysis previously proposed in Reference 3.
BR
4(10-n) BR-LBR +BR-1 rh, [BR+BRUN J 4(10-n) hv [BR+'--BRLN
=P
[4]
Here, we use BR+ + BR- = 2 BRLIN. Again LIN, +, and - denote polarization: linear, left circular, and right circular. (Br+-- = BR+ or BR-, with a sign change in Phv.) Phv is the circular polarization (+1 for left, 0 for linear, and -I for right circular). (Note: If Phv = 0, then BR+ = BR- = BRLIN and this equation is meaningless.) For the remainder of this work, IPhvl = 1, to be consistent with Carra, et al. As noted earlier 4 : BR = 13/2/(13/2 + 11/2)
[5]
The BR has the advantage of
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