- Email: [email protected]
Paramagnetic form factors of hcp transition metals
Journal of Magnetism and Magnetic Materials 54-57 (1986) 953-954 PARAMAGNETIC
FORM
FACTORS
953
OF hcp TRANSITION
METALS
S.H. L I U , A.J. L I U * a n d J . F . C O O K E Solid State Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA
Recently we have developed a numerical technique to compute the total form factor of the itinerant electrons in hcp transition metals. The calculated orbital contributions deviate from atomic orbital form factors. There is reasonable overall agreement between theory and elastic neutron scattering experiments.
In the past two decades the induced m o m e n t form factor has been m e a s u r e d for a large n u m b e r of transition metals, a n d these data provide detailed i n f o r m a t i o n on the response of the metals to a n external magnetic field. The m o m e n t distribution is d e t e r m i n e d by the energy levels a n d wave functions of the metallic electrons. The induced m o m e n t consists of an orbital contribution a n d a spin contribution. T h e spin c o n t r i b u t i o n has been calculated for most transition metals. F o r a long time it was n o t possible to c o m p u t e the orbital c o n t r i b u t i o n for itinerant electrons, and one traditionally analyzes this p a r t by using the atomic model. As pointed out in a n u m b e r of recent review articles [1], realistic calculations of the orbital c o n t r i b u t i o n are of f u n d a m e n t a l i m p o r t a n c e in assessing the validity of theoretical models. Recently, we have been able to formulate the orbital form factor p r o b l e m in such a way that it can be calculated accurately for cubic transition metals [2]. In this p a p e r we extend the same consideration to hcp transition metals. In a magnetic form factor m e a s u r e m e n t one detects the spin-flip ratio of polarized neutrons. If the lattice has one a t o m in a unit cell, the nuclear scattering a m p l i t u d e is a constant, a n d the result of the measurem e n t can be directly c o m p a r e d with the Fourier transform of the magnetization density. The hcp structure has two atoms in a unit cell. We choose the unit cell such that the atoms are situated one at the origin a n d the other at 8. The nuclear c o n t r i b u t i o n has the expression F n ( G ) = b[1 + e x p ( i G - 6 ) ] , where Fv4 is the nuclear structure factor, b is the nuclear scattering a m p l i t u d e a n d G is a reciprocal lattice vector. The Fourier t r a n s f o r m of the magnetization density for the unit cell is p r o p o r t i o n a l to the generalized susceptibility function x(G). T h e spin-flip ratio F(G) is given by F(G) ec 4 Re(b*(G)x(G)/I b(G) 12). If we normalize F(G) in the forward direction to the static susceptibility, we can write
X(G), respectively. In general, x ( G ) can be written in terms of spin a n d orbital c o n t r i b u t i o n s x ( G ) = a s x s ( G ) + Xo(G), where a s is the a p p r o p r i a t e enhancem e n t factor. The expression for x 0 ( G ) is given in ref. [2]: 8N# 2
fnk - f,~'k
×
ia"ln'k )
k. we
× (n'k I(~; × k. v')(d, r) I nk>,
(2)
where B is the unit vector along the external field, G ± B, /~B is the Bohr magneton, N is the n u m b e r of unit cells in the sample, Enk is the energy of the eigenstate Ink) of the electron in b a n d n with crystal m o m e n t u m k, a n d fnk is the Fermi factor. The matrix elements are calculated within the unit cell. The energy b a n d s a n d Bloch wave functions for the hcp metals were calculated by the selfconsistent non-relativistic K K R m e t h o d in the muffin-tin approximation. T h e K o h n - S h a m a p p r o x i m a t i o n for the exchange-correlation potential was used. The energy levels a n d wave functions were determined in the 1 / 2 4 irreducible sector of the Brillouin zone over a mesh of 990 points. The 1.0-
x
x
•
o.o- ~
ID
ID
&
ITINERANT MIC
0.6-
0.4--
t~ O
02 -
•
e'~,..~_
sin G . 8
F(G) = x'(G) + 1 + cos G. 6 X"(G)"
(1)
0.0
where X' a n d X" are the real a n d imaginary parts of * Present address: Cornell University, Ithaca, New York, USA. 0304-8853/86/$03.50
o.o
i
o.,
I
o.~
~.~
d.,
~in(o)/x Fig. 1. Comparison of orNtal part of the form factor calculated from itinerant and atomic theory for zirconium.
© E l s e v i e r S c i e n c e P u b l i s h e r s B.V.
954
S.H. Liu et al. / Form factor of hop transition metals
Table 1 Orbit (Fo) and unenhanced spin ( ~ ) contributions to the itinerant electron form factor in units of 1 0 - 6 emu/mol
1.0 ")
a
F~(G)
F,)(6)
.~,
000 100 002 10 1 10 2
47.49 9.86 6.03 3.38 -0.22 -0.01 1.29 1.51 1.74
65.81 33.05 23.90 22.44 11.42 10.18 5.44 5.56 4.05
~.. 13
1 10
10 3 200 20 1
.~
x THEORY T EXP.
0,8-
0.6-
0.4--
I
0.2-
X
0,0-
wave functions consist of two sets of spherical h a r m o n ics with l up to 2, one centered a r o u n d each atom. F o r the matrix elements we extend the muffin-tin orbitals into two overlapping atomic spheres a n d app r o x i m a t e the unit cell integral by the sum of the atomic sphere integrals. F o r states outside the irreducible sector, we use the group of symmetry operations of the hcp structure to rewrite the matrix element as one whose states are within the irreducible sector b u t the corres p o n d i n g o p e r a t o r is transformed. The final sum over the Brillouin zone is calculated by using the t e t r a h e d r o n technique. A c o m p a r i s o n of results for the orbital part of the form factor calculated from the atomic a n d itinerant models for zirconium is given in fig. 1. These results are typical for the hcp materials we have studied thus far. The itinerant theory predicts a more rapid decrease with wave vector t h a n atomic theory a n d a definite asymmetry associated with wave-vector direction. This behaviour is similar to that f o u n d for the spin contribution. In contrast to the atomic model, numerical results can be o b t a i n e d o n an absolute scale from itinerant theory for p a r a m a g n e t i c metals. A n example of results for the orbit and u n e n h a n c e d spin c o n t r i b u t i o n s for zirconium are given in table 1. Notice that the orbital a n d u n e n h a n c e d spin c o n t r i b u t i o n s are comparable. Z i r c o n i u m is u n i q u e in this respect. Our results for Sc, Y, a n d Lu indicate that the orbital c o n t r i b u t i o n is approximately 10% or less of the u n e n h a n c e d spin contribution at I G] = 0. In all cases, the spin a n d orbital terms b e c o m e c o m p a r a b l e as [G[ is increased. If for zirconium we choose c% = 1.1, which yields the measured static susceptibility at [ G i = 0, then we o b t a i n the results for the total form factor shown in fig. 2. As can b e seen, the agreement with experiment is quite good.
-0.2
o.o
~.2
&
X
&
&
sin(O)/X Fig. 2. Comparison of itinerant (total) form factor and experimental results of Stassis et al. [3] for zirconium.
C o m p a r i s o n s of the type shown in fig. 2 can be misleading. The agreement one gets is dependent, to a great extent, on the way the experimental data are normalized. Realistic tests of theoretical predictions can only be m a d e on the basis of a c o m p a r i s o n to experim e n t a l data o b t a i n e d o n an absolute scale. A more sensitive test of the theory could b e m a d e if the spin a n d orbital c o n t r i b u t i o n s could be separated, because of the theoretical p r o b l e m s associated with calculating a s a n d the relatively small orbital c o n t r i b u t i o n (particularly a t small [ G I ) f o u n d for a n u m b e r of materials. While we conclude from our work thus far that calculations based o n itinerant electron theory are in reasonable agreement with experimental, we anticipate that more sensitive tests will be possible in the near future. This research was sponsored by the Division of Materials Sciences, US D e p a r t m e n t of Energy u n d e r c o n t r a s t DE-AC05-840R21400 with M a r t i n Marietta Energy Systems, Inc. [1] R.M. Moon, W.C. Koehler and J.W. Cable, Proc. Conf. on Neutron Scattering, Gatlinburg, (CONF-760601-P2) (1976) p. 577. C. Stassis, Nukleonika 24 (1979) 765, R.M. Moon, J. de Phys. C7 (1982) 187. [2] K.H. Oh, B.N. Harmon, S.H. Liu and S.K. Sinha, Phys. Rev. B 14 (1976) 1283. [3] C. Stassis, G. Kline, B.N. Harmon, R.M. Moon and W.C. Koehler, J. Magn. Magn. Mat. 14 (1979) 303.
FORM
FACTORS
953
OF hcp TRANSITION
METALS
S.H. L I U , A.J. L I U * a n d J . F . C O O K E Solid State Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA
Recently we have developed a numerical technique to compute the total form factor of the itinerant electrons in hcp transition metals. The calculated orbital contributions deviate from atomic orbital form factors. There is reasonable overall agreement between theory and elastic neutron scattering experiments.
In the past two decades the induced m o m e n t form factor has been m e a s u r e d for a large n u m b e r of transition metals, a n d these data provide detailed i n f o r m a t i o n on the response of the metals to a n external magnetic field. The m o m e n t distribution is d e t e r m i n e d by the energy levels a n d wave functions of the metallic electrons. The induced m o m e n t consists of an orbital contribution a n d a spin contribution. T h e spin c o n t r i b u t i o n has been calculated for most transition metals. F o r a long time it was n o t possible to c o m p u t e the orbital c o n t r i b u t i o n for itinerant electrons, and one traditionally analyzes this p a r t by using the atomic model. As pointed out in a n u m b e r of recent review articles [1], realistic calculations of the orbital c o n t r i b u t i o n are of f u n d a m e n t a l i m p o r t a n c e in assessing the validity of theoretical models. Recently, we have been able to formulate the orbital form factor p r o b l e m in such a way that it can be calculated accurately for cubic transition metals [2]. In this p a p e r we extend the same consideration to hcp transition metals. In a magnetic form factor m e a s u r e m e n t one detects the spin-flip ratio of polarized neutrons. If the lattice has one a t o m in a unit cell, the nuclear scattering a m p l i t u d e is a constant, a n d the result of the measurem e n t can be directly c o m p a r e d with the Fourier transform of the magnetization density. The hcp structure has two atoms in a unit cell. We choose the unit cell such that the atoms are situated one at the origin a n d the other at 8. The nuclear c o n t r i b u t i o n has the expression F n ( G ) = b[1 + e x p ( i G - 6 ) ] , where Fv4 is the nuclear structure factor, b is the nuclear scattering a m p l i t u d e a n d G is a reciprocal lattice vector. The Fourier t r a n s f o r m of the magnetization density for the unit cell is p r o p o r t i o n a l to the generalized susceptibility function x(G). T h e spin-flip ratio F(G) is given by F(G) ec 4 Re(b*(G)x(G)/I b(G) 12). If we normalize F(G) in the forward direction to the static susceptibility, we can write
X(G), respectively. In general, x ( G ) can be written in terms of spin a n d orbital c o n t r i b u t i o n s x ( G ) = a s x s ( G ) + Xo(G), where a s is the a p p r o p r i a t e enhancem e n t factor. The expression for x 0 ( G ) is given in ref. [2]: 8N# 2
fnk - f,~'k
×
ia"ln'k )
k. we
× (n'k I(~; × k. v')(d, r) I nk>,
(2)
where B is the unit vector along the external field, G ± B, /~B is the Bohr magneton, N is the n u m b e r of unit cells in the sample, Enk is the energy of the eigenstate Ink) of the electron in b a n d n with crystal m o m e n t u m k, a n d fnk is the Fermi factor. The matrix elements are calculated within the unit cell. The energy b a n d s a n d Bloch wave functions for the hcp metals were calculated by the selfconsistent non-relativistic K K R m e t h o d in the muffin-tin approximation. T h e K o h n - S h a m a p p r o x i m a t i o n for the exchange-correlation potential was used. The energy levels a n d wave functions were determined in the 1 / 2 4 irreducible sector of the Brillouin zone over a mesh of 990 points. The 1.0-
x
x
•
o.o- ~
ID
ID
&
ITINERANT MIC
0.6-
0.4--
t~ O
02 -
•
e'~,..~_
sin G . 8
F(G) = x'(G) + 1 + cos G. 6 X"(G)"
(1)
0.0
where X' a n d X" are the real a n d imaginary parts of * Present address: Cornell University, Ithaca, New York, USA. 0304-8853/86/$03.50
o.o
i
o.,
I
o.~
~.~
d.,
~in(o)/x Fig. 1. Comparison of orNtal part of the form factor calculated from itinerant and atomic theory for zirconium.
© E l s e v i e r S c i e n c e P u b l i s h e r s B.V.
954
S.H. Liu et al. / Form factor of hop transition metals
Table 1 Orbit (Fo) and unenhanced spin ( ~ ) contributions to the itinerant electron form factor in units of 1 0 - 6 emu/mol
1.0 ")
a
F~(G)
F,)(6)
.~,
000 100 002 10 1 10 2
47.49 9.86 6.03 3.38 -0.22 -0.01 1.29 1.51 1.74
65.81 33.05 23.90 22.44 11.42 10.18 5.44 5.56 4.05
~.. 13
1 10
10 3 200 20 1
.~
x THEORY T EXP.
0,8-
0.6-
0.4--
I
0.2-
X
0,0-
wave functions consist of two sets of spherical h a r m o n ics with l up to 2, one centered a r o u n d each atom. F o r the matrix elements we extend the muffin-tin orbitals into two overlapping atomic spheres a n d app r o x i m a t e the unit cell integral by the sum of the atomic sphere integrals. F o r states outside the irreducible sector, we use the group of symmetry operations of the hcp structure to rewrite the matrix element as one whose states are within the irreducible sector b u t the corres p o n d i n g o p e r a t o r is transformed. The final sum over the Brillouin zone is calculated by using the t e t r a h e d r o n technique. A c o m p a r i s o n of results for the orbital part of the form factor calculated from the atomic a n d itinerant models for zirconium is given in fig. 1. These results are typical for the hcp materials we have studied thus far. The itinerant theory predicts a more rapid decrease with wave vector t h a n atomic theory a n d a definite asymmetry associated with wave-vector direction. This behaviour is similar to that f o u n d for the spin contribution. In contrast to the atomic model, numerical results can be o b t a i n e d o n an absolute scale from itinerant theory for p a r a m a g n e t i c metals. A n example of results for the orbit and u n e n h a n c e d spin c o n t r i b u t i o n s for zirconium are given in table 1. Notice that the orbital a n d u n e n h a n c e d spin c o n t r i b u t i o n s are comparable. Z i r c o n i u m is u n i q u e in this respect. Our results for Sc, Y, a n d Lu indicate that the orbital c o n t r i b u t i o n is approximately 10% or less of the u n e n h a n c e d spin contribution at I G] = 0. In all cases, the spin a n d orbital terms b e c o m e c o m p a r a b l e as [G[ is increased. If for zirconium we choose c% = 1.1, which yields the measured static susceptibility at [ G i = 0, then we o b t a i n the results for the total form factor shown in fig. 2. As can b e seen, the agreement with experiment is quite good.
-0.2
o.o
~.2
&
X
&
&
sin(O)/X Fig. 2. Comparison of itinerant (total) form factor and experimental results of Stassis et al. [3] for zirconium.
C o m p a r i s o n s of the type shown in fig. 2 can be misleading. The agreement one gets is dependent, to a great extent, on the way the experimental data are normalized. Realistic tests of theoretical predictions can only be m a d e on the basis of a c o m p a r i s o n to experim e n t a l data o b t a i n e d o n an absolute scale. A more sensitive test of the theory could b e m a d e if the spin a n d orbital c o n t r i b u t i o n s could be separated, because of the theoretical p r o b l e m s associated with calculating a s a n d the relatively small orbital c o n t r i b u t i o n (particularly a t small [ G I ) f o u n d for a n u m b e r of materials. While we conclude from our work thus far that calculations based o n itinerant electron theory are in reasonable agreement with experimental, we anticipate that more sensitive tests will be possible in the near future. This research was sponsored by the Division of Materials Sciences, US D e p a r t m e n t of Energy u n d e r c o n t r a s t DE-AC05-840R21400 with M a r t i n Marietta Energy Systems, Inc. [1] R.M. Moon, W.C. Koehler and J.W. Cable, Proc. Conf. on Neutron Scattering, Gatlinburg, (CONF-760601-P2) (1976) p. 577. C. Stassis, Nukleonika 24 (1979) 765, R.M. Moon, J. de Phys. C7 (1982) 187. [2] K.H. Oh, B.N. Harmon, S.H. Liu and S.K. Sinha, Phys. Rev. B 14 (1976) 1283. [3] C. Stassis, G. Kline, B.N. Harmon, R.M. Moon and W.C. Koehler, J. Magn. Magn. Mat. 14 (1979) 303.