PHYSICAL REVIEW B 85, 024417 (2012)

Doppler broadening of annihilation radiation measurements on 3d and 4 f ferromagnets
using polarized positrons
A. Kawasuso,* M. Maekawa, Y. Fukaya, A. Yabuuchi, and I. Mochizuki
Advanced Science Research Center, Japan Atomic Energy Agency, 1233 Watanuki, Takasaki, Gunma JP-370-1292, Japan(Received 20 October 2011; published 23 January 2012)
We measured the Doppler broadening of annihilation radiation (DBAR) spectra of 3d (Fe, Co, and Ni) and4f (Gd, Tb, and Dy) ferromagnets under a magnetic field by using spin-polarized positrons from a 68 Ge68
Ga source. The results showed that the DBAR spectra of these metals have notably different magnetic-fielddependences. The differences among Fe, Co, and Ni reflect that the upper minority spin bands of Fe and Co
are nearly empty while those of Ni are still mostly occupied. For the rare-earth metals instead of the inner 4f
electrons, 5d electrons that mediate the exchange interaction of the 4f electrons are primarily responsible forthe magnetic-field effects on the DBAR spectra. Furthermore, the magnetic-field effects on the DBAR spectra ofGd, Tb, and Dy vanished above the Curie temperatures of the magnetic-phase transition for these metals.
DOI: 10.1103/PhysRevB.85.024417

PACS number(s): 78.70.Bj, 71.60.+z, 75.50.−y

I. INTRODUCTION

II. PRINCIPLE OF SP-DBAR

The angular correlation of annihilation radiation (ACAR)
is one method of positron annihilation spectroscopy (PAS)that provides an electron-momentum distribution. The ACAR
method has been used extensively for the study of Fermisurface topology. By using longitudinally spin-polarized
positrons emitted from radioisotopes, polarized band structures of ferromagnets have also been studied.1–18 Recently, wedemonstrated, by using highly spin-polarized positrons froma 68 Ge-68 Ga source, that the technique of Doppler broadeningof annihilation radiation (DBAR) can also provide knowledge
on polarized electrons.19
Spin-polarized positron annihilation spectroscopy (SPPAS) has several attractive points. Low-energy spin-polarized
positron beams20–23 are useful to study the magnetism andspin-related phenomena occurring at surfaces and interfaces
and in thin films. Considering that positrons are trapped byvacancy defects, SP-PAS can also be used in studying vacancyinduced magnetism.24 In this respect, to detect excess electron
spins at vacancy defects, positron-annihilation-lifetime measurements using spin-polarized positrons are also an intriguing
experiment.
To establish the SP-PAS method as a probe for ferromagnets
and spin-related phenomena, further fundamental studies areneeded. In our previous work, we found that the amplitudesof the differential DBAR spectra of Fe, Co, Ni, and Gd are
related to their electron polarizations. However, we observed
no major differences in their spectral shapes that should reflect
the different band structures. Since at that time we focused
only on the change in the DBAR spectra upon field reversal,
detailed changes of the spectral shapes depending on themagnetic field would have been overlooked. Here, to findstructures of DBAR spectra reflecting polarized electron states,
we investigated magnetic-field dependences of the DBAR
spectra for the above metals. To confirm the effect of theferromagnetic-phase transition on the DBAR spectra, we alsomeasured the temperature dependences of the DBAR spectra
for 4f (Gd, Tb, and Dy) ferromagnets having lower Curie
temperatures.

Before describing the experimental details, we briefly
summarize the principle of SP-DBAR. Denoting the majority
spin band by ↓ (magnetic moment is up) and minority spinband by ↑ (magnetic moment is down), the positron-electronmomentum densities for the ith majority (and minority) spin↓(↑)
bands ρi (p) are given by

2



↓(↑)
↓(↑)
ρi (p) =  e−ipr + (r)i (r) γ [n− (r)]dr , (1)1098-0121/2012/85(2)/024417(6)↓(↑)

where + (r) is the positron wave function, i (r) is the electron wave function, and γ [n− (r)] is the enhancement factor.25↓(↑)
The corresponding DBAR spectrum Ni (pz ) is given by
the convolution of the apparatus resolution function and the
double integral (dpx dpy ) of the above ith electron-positronmomentum density. This expression is valid for a single crystal.
For a polycrystal, the DBAR spectrum should be given by aweighted average between those in different momentum axes.
↓(↑)
↓(↑)
In this case, Ni (pz ) is simply replaced with Ni (p) =
↓(↑)
↓(↑)
↓(↑)
αNi (px = p) + βNi (py = p) + γ Ni (pz = p) with13
appropriate coefficients.
Two DBAR spectra, N+ (pz ) and N− (pz ), measured using
longitudinally polarized positrons under a magnetic field thatis parallel (+) or antiparallel (−) to the positron-polarization
vector (see Fig. 1), are given by

occ 
↓
↑
λS  (1±P )Ni (pz ) (1∓P )Ni (pz ), (2)N± (pz ) =
+
4 i=1
λ⇑
λ⇓
where the summation is over all occupied states; λS = 4π re2 c,
where re is the classical electron radius; P is the longitudinalpolarization of positrons; and λ⇑(⇓) is the total annihilation
rate of spin-up (spin-down) positrons
1
↓(↑)
↓(↑)
↑(↓)
[λS wi + λT (wi + 2wi )].
2 i=1
occ

λ⇑(⇓) =

(3)↓(↑)

Here, λT = λS /1115, and wi is the overlap integral between
the positron and ith majority (minority) spin-band wave

024417-1©2012 American Physical SocietyKAWASUSO, MAEKAWA, FUKAYA, YABUUCHI, AND MOCHIZUKIPHYSICAL REVIEW B 85, 024417 (2012)

examine the difference between two DBAR spectra in different
field conditions, both experimental and calculated spectra aresimply normalized to unity, and differential DBAR spectra are
obtained.
III. EXPERIMENTAL METHODS

FIG. 1. Experimental setup for spin-polarized DBAR measurements. Positrons emitted from the source are implanted in thesample. Positrons are longitudinally polarized in the direction of
the polarization vector. An external magnetic field from −1 T to
+1 T is applied with the electromagnet. Annihilation radiation fromthe source is shielded by lead blocks.

 +∞ ↓(↑)
functions, equaling −∞ Ni (pz )dpz . Equation (2) suggeststhat the DBAR spectrum exhibits the characters of more
majority and more minority spin bands under positive andnegative fields, respectively. Their difference is given by
occ  ↓↑λS P  Ni (pz ) Ni (pz ). (4)N+ (pz ) − N− (pz ) =
−
2 i=1
λ⇑
λ⇓
Thus, the differential DBAR spectrum is not a simple summa↓↑tion of partial differential spectra [Ni (pz ) − Ni (pz )] unlike
the case of the magnetic-Compton-profile measurement. Berko
and Mills7 proposed that by using field-reversal asymmetryof the three-gamma-annihilation probability, the following
relation is obtained:
occ

↓
↑
[Ni (pz ) − Ni (pz )] ∝i=1≈

N(pz ) + P

λ⇑ − λ⇓N (pz )λ⇑ + λ⇓N (pz ) + P 3γ N(pz ),

(5)

where
N(pz ) = N+ (pz ) − N− (pz ),N(pz ) = N+ (pz ) +
3γ
3γ
3γ
3γ
N− (pz ), and P 3γ = (N+ − N− )/(N+ + N− ). Therefore,
either through Eq. (4) or (5), one can extract the components
related to magnetic electrons. In the present experiment, to

The samples used in this study were polycrystalline, Fe(4N), Co (5N), Ni (5N), Gd (3N), Tb (3N), and Dy (3N),with dimensions of 10.0 mm × 12.5 mm × 2.0 mm. The
samples were mechanically and electrochemically polishedand subjected to heat treatment at 1100 ◦ C for 2 h in a
vacuum. Through the nuclear reaction of 69 Ga(p,2n)68 Ge
induced by 20-MeV proton irradiation of a GaN substrate(φ8 mm), a positron source (68 Ge-68 Ga, 20 MBq) was
produced (total fluence: 9 × 1017 protons). Through themagnetic-field dependence of the S parameter related to theself-annihilation of spin-singlet (para-)positronium in α-SiO2 ,the longitudinal spin polarization of positrons emitted fromthe source was determined to be 0.7.26 Basically, in thelongitudinal condition, the positron polarization is not lost
so much since the polarization vector and magnetic field
are parallel (or antiparallel). Nevertheless, depolarization
will occur during the thermalization and diffusion processes
because of inner magnetic fields.27 To preserve the positronpolarization as much as possible, it is important to use a source
emitting highly polarized positrons. The samples and thesource were placed at the center of a gap in an electromagnet(with a magnetic-field variable up to 1 T), maintaining a
distance of 7 mm. The sample temperature was changed from
5 K to 315 K. To detect annihilation radiation from only
the samples, the source was shielded by lead blocks. The
experimental arrangement is schematically shown in Fig. 1.
DBAR spectra were measured by using a high-purity-Ge
detector with an energy resolution of 1.4 keV at 511 keV. Here,the photon energy of Eγ = 1 keV corresponds to an electron
momentum of p = 3.92 × 10−3 m0 c (0.54 a.u.). In each spectrum, more than 106 events were accumulated. To compare thedifferences in spectral shapes under different field conditions
in detail, all the spectral area intensities were normalized tounity.
IV. RESULTS AND DISCUSSION
A. Magnetic-field dependence

Figure 2 shows the DBAR spectra of the Fe sample
obtained under positive and negative fields (B = ±0.8 T)
and the differential spectrum N+ (p) − N− (p). We call thefinite differential spectrum the field-reversal asymmetry. The
momentum resolution of DBAR is poorer than that of ACAR.
Nevertheless, such field effects are observable. The positive
intensity of the differential spectrum at |p|  5 × 10−3 m0 c
was attributed to the enhanced (reduced) annihilation between
spin-up positrons and spin-down (spin-up) 3d electrons having
broader momentum distributions under a positive (negative)
field. The negative intensity at |p|  5 × 10−3 m0 c was
explained as the relative decrease of low-momentum components. In our previous work, the shapes of the differential
spectra [N+ (p) − N− (p)] for Fe, Co, Ni, and Gd weresimilar to each other. No further fine structures reflecting

024417-2

DOPPLER BROADENING OF ANNIHILATION RADIATION . . .

FIG. 2. (a) DBAR spectra obtained for the Fe sample at B =±0.8 T and (b) the differential spectrum. These spectra are folded at
p = 0 to enhance the statistics. Shaded areas denote the windows of
the W parameter.

PHYSICAL REVIEW B 85, 024417 (2012)

positrons with polarized 3d electrons having higher momenta
are enhanced under positive fields and reduced under negative
fields. For the Co sample, a similar tendency is seen, but thelinearity seems to be a little lost under negative fields. In
the case of the Ni sample, the deviation from the linear
dependence is more pronounced to be somewhat concavedown-parabolic in shape, that is, in both positive and negative
fields, the annihilation probabilities of positrons and polarized
electrons having higher momenta seem to be enhanced though
the amounts are different. As for the the Gd sample, the fielddependence is again parabolic, but it is concave up, implyingthat the annihilation probabilities of positrons with polarized
electrons having lower momenta are enhanced under both
positive and negative fields. The above results imply that thechanges of spectral shapes from those at a zero field reflect thedifferent band structures of these metals.
Figure 4 shows the DBAR differential spectra from a zero
field [N+/− (p) − N0 (p)] for the Fe, Co, Ni, and Gd samples
at B = ±1 T. Under a positive field [N+ (p) − N0 (p)], theFe, Co, and Ni samples exhibit similar features. Under anegative field [N− (p) − N0 (p)], the spectra of the Fe and Co
samples have negative intensities at |p|  3 × 10−3 m0 c and
10 × 10−3  |p|  20 × 10−3 m0 c, respectively, and positive

N+/- (p) - N0(p) (arb. units)

the differences in the band structures of these metals wereobserved. As shown below, the line-shape parameters exhibit
somewhat different field dependences.
Figure 3 shows the so-called W parameter, which is definedas the area intensity in the higher-momentum region as shownin Fig. 2, as a function of the magnetic field. In the case of the
Fe sample, the W parameter behaves nearly linearly with themagnetic field, indicating that the annihilation probabilities of0

FIG. 3. W parameters as a function of the magnetic field at room
temperature obtained for the Fe, Co, Ni, and Gd samples. Solid lines
are guides for the eye.

10

Fe

Fe

N + - N0

N - - N0Co

Co

N + - N0

N - - N0

Ni

Ni

N + - N0

N- - N0

Gd

Gd

N + - N0

N- - N0

20

0

10

20
-3
Electron momentum p (10 m0c)

FIG. 4. Differential DBAR spectra between positive or negativeand zero fields [N+/− (p) − N0 (p)] for the Fe, Co, Ni, and Gd samples
at room temperature (B = ±1 T). These spectra are folded at p = 0 toenhance the statistics. Solid lines denote curves calculated by Berko’s
method. The amplitudes are adjusted to levels comparable with thosein experiments.

024417-3

KAWASUSO, MAEKAWA, FUKAYA, YABUUCHI, AND MOCHIZUKI

intensities at 3 × 10−3  |p|  10 × 10−3 m0 c. The spectrumof the Ni sample shows no significant change upon fieldreversal except for a small change of the intensity. The
spectrum of the Gd sample under a positive field exhibits
somewhat opposite features to those of the other samples, andits shape is conserved upon field reversal except for a smallchange in the intensity.
According to the theoretical band structure of Fe,16,28–30
the first to fifth majority spin bands are nearly fully occupied,
the sixth majority spin band is partially occupied, and thefourth to sixth minority spin bands are empty or only
partially occupied. The upper-band electrons have almost
d-like character while the lower-band electrons have both
sp-like and d-like characters. Hence, the spectrum under apositive field probably reflects the d-like characters of theupper majority spin bands while that under a negative fieldreflects the more sp-like characters of the lower minority spinbands.
In the case of Co, the first to tenth majority spin bands
are nearly fully occupied, the eleventh and twelfth majority
spin bands are partially occupied, and the seventh to twelfth
minority spin bands are empty or only partially occupied.31
Hence, in Fig. 4, the spectral shape under a positive field
probably reflects the d-like characters of the upper majority
spin bands while that under a negative field reflects the more
sp-like characters of the lower minority spin bands.
Both the six majority and six minority spin bands of Ni
are mostly occupied.32,33 Only small amounts of electrons
are transferred from the minority to the majority spin bands.
Hence, the spectral shapes under both positive and negative
fields in Fig. 4 give rise to analogous features, mainly reflecting
the fourth to sixth bands having d-like character. The slightdifference in the intensities upon field reversal is caused by the
relatively low occupations of the fifth and sixth minority spinbands.
As for Gd, the 4f majority spin bands are completely
occupied while its minority spin bands are nearly empty.34,35
The 5d majority spin bands are partially occupied while its
minority spin bands are empty. Both majority and minority
spin bands of the 6s electrons are nearly occupied. Hence,
the field effects are attributed to the 4f and/or 5d bands. The
annihilation probabilities of 5d electrons with positrons aremuch higher than those of inner 4f shell electrons. The 5delectrons result in narrower DBAR spectra as compared to
the 4f electrons. Therefore, the field effect on the spectral
shape in Fig. 4 likely arises from the 5d bands. Knownas the Ruderman-Kittel-Kasuya-Yoshida (RKKY) interaction,
the ferromagnetism of Gd appears due to indirect exchangeinteractions between 4f electrons mediated by 5d electrons.
Positrons probably see the polarization of such 5d electrons.
To elucidate details above, we calculated the differential
DBAR spectra for individual bands and the spin-averaged total
DBAR spectra. The electron wave functions were obtained
from the ABINIT computation36 with the projector-augmentedwave method.37 The initial valence-electron configurations
were assumed to be 3s 2 3p6 3d 6 4s 2 (Fe), 3s 2 3p6 3d 7 4s 2 (Co),
3s 2 3p6 3d 8 4s 2 (Ni), and 4f 7 5s 2 5p6 5d 1 6s 2 (Gd). The positronwave function was calculated based on two-component density
functional theory. The DBAR spectra were obtained from
Eq. (1) with a Gaussian resolution function having a full

PHYSICAL REVIEW B 85, 024417 (2012)

FIG. 5. Differential curves between the ith band [Ni (p)] and thetotal DBAR spectra [NT (p)] in the spin-averaged state calculated
for Fe and Gd. For Gd, the fourteen 4f bands are averaged. The
amplitudes here are only for comparison.

width at half maximum of 1.4 keV. Since the present samples
were polycrystals, the calculated DBAR spectra along thethree-momentum axes ([100], [100], and [111] for Fe andNi and [1100], [1120], and [0001] for Co and Gd) were
spherically averaged. The details of the calculations aredescribed elsewhere.38
Figure 5 shows the calculated differential spectra between
the spectra of individual bands and the spin-averaged total
spectra for Fe and Gd. In Fig. 4, the experimental spectrum of
the Fe sample under a positive field exhibits the features of the
fourth to sixth bands while under a negative field it exhibits
the features of lower bands. The results for the Co and Ni
samples may be explained similarly. The experimental spectraof the Gd sample under positive and negative fields in Fig. 4
are similar to those calculated for the first to fourth bands in
Fig. 5. (The first and second bands are 6s-like, and the thirdand fourth bands are 5d-like.) The solid lines in Fig. 4 are thedifferential spectra calculated using Berko’s method having
the occupations of individual bands.6,19 (The amplitudes of thecalculated spectra are adjusted to levels comparable with thosein experiments.) Although the calculated differential spectra
are not fully compatible with the experimental spectra, theiragreement is generally good.
In a previous work, we found that the strength of the
field-reversal asymmetry of the DBAR spectra for Fe, Co, andNi is approximately proportional to their magnetization. In
these metals, the annihilation probabilities between positrons
and valence (3d4s) electrons are comparable because of theirclose electron densities. Hence, the field-reversal asymmetry
observed for these metals is approximately proportional tothe averaged spin polarizations of the valence bands, i.e.,magnetizations.
Similar discussion may be allowed for Gd, Tb, and Dy. The
field-reversal asymmetry observed for the Tb and Dy samples
was much smaller (<1/7) as compared to the Gd sample. As
discussed above, in these metals, the field effect on the DBAR
spectra is caused by polarized 5d electrons. Therefore, the024417-4

DOPPLER BROADENING OF ANNIHILATION RADIATION . . .

PHYSICAL REVIEW B 85, 024417 (2012)shows the temperature dependences of the intensities of the
DBARspectra relative to those at the maximum temperature+∞
TM = −∞ |NT =T (p) − NT =TM (p)|dp, obtained for the Gd,
Tb, and Dy samples at B = ±0.8 T. The intensities under anegative field are greater than those under a positive field. Thisis consistent with the results shown in Fig. 4. The criticaltemperatures at which the field effects vanish seem to bein agreement with the Curie temperatures (TC ) associatedwith the ferromagnetic-phase transitions of these metals (Gd:293 K, Tb: 222 K, and Dy: 90 K). The absence of temperaturedependences for the Fe, Co, and Ni samples is explained bytheir high Curie temperatures.
The temperature dependence of magnetization (M) is givenby the Weiss equation
M
gJ μB J (B + λM)
= BJ
,
(6)
M0
kB T

FIG. 6. Area intensities of the differential DBAR spectra relative
to those at the highest temperatures obtained for the Gd, Tb, andDy samples at B = 0.8 T as a function of temperature. The Curie
temperatures (TC ) of the ferromagnetic-phase transition are indicated.
Solid lines are the calculated temperature dependences based on the
Weiss theory.

suppressed field-reversal asymmetry of the Tb and Dy samples
implies much lower polarizations of their 5d bands comparedto that of Gd.
B. Temperature dependence

Under a magnetic field, the DBAR spectra of the Fe, Co,
and Ni samples showed no apparent temperature dependences
below 315 K while the field effects on the DBAR spectra
for the rare-earth samples increased upon cooling. Figure 6*

kawasuso.atsuo@jaea.go.jp
S. S. Hanna and R. S. Preston, Phys. Rev. 106, 1363 (1957).
2
S. S. Hanna and R. S. Preston, Phys. Rev. 109, 716 (1958).
3
P. E. Mijnarends and L. Hambro, Phys. Lett. 10, 272 (1964).
4
P. E. Mijnarends, Physica (Amsterdam) 63, 248 (1973).
5
S. Berko and J. Zuckerman, Phys. Rev. Lett. 13, 339 (1964).
6
S. Berko, in Positron Annihilation, edited by A. T. Stewart andL. O. Roellig (Academic Press, New York, 1967), p. 61.
7
S. Berko and A. P. Mills, J. Phys., Colloq. 32, C1 (1971).
8
C. Hohenemser, J. M. Weingart, and S. Berko, Phys. Lett. A 28, 41(1968).
1

where M0 is the magnetization at T = 0 K, J is the total
angular momentum (Gd: 7/2, Tb: 6, and Dy: 15/2), gJ
is the Lande’s g factor (Gd: 2, Tb: 1.5, and Dy: 1.33),
μB is the Bohr magneton, BJ is the Brillouin function,
kB is the Boltzmann constant, and λ is the molecular-field
constant (3kB TC )/[NgJ2 μ2B J (J + 1)]. Solid lines M/M0 are
calculated assuming that the DBAR intensity is proportional
to the magnetization. (The amplitudes are adjusted to levels
comparable with those in experiments.) The experimental
temperature dependences are qualitatively reproduced. Thus,
the above results demonstrate that the polarizations of the 5dbands and hence the magnetizations are lost upon heating.
V. CONCLUSION

In conclusion, we examined the magnetic-field and temperature dependences of the DBAR spectra for 3d and 4f
ferromagnets. The different field dependences observed for Fe,
Co, Ni, and Gd were explained by considering their different
band structures. We found that the field effects on DBAR
spectra for Gd, Tb, and Dy disappear above ferromagneticphase-transition temperatures. These results suggest that spinpolarized positron annihilation spectroscopy can be used incharacterizing the polarized electrons of ferromagnets andalso in spin-related phenomena. A highly spin-polarizedpositron beam is required for performing such advanced
research.

9T. W. Mihalisin and R. D. Parks, Phys. Lett. 21, 610 (1966).
T. W. Mihalisin and R. D. Parks, Phys. Rev. Lett. 18, 210 (1967).
11
T. W. Mihalisin and R. D. Parks, Solid State Commun. 7, 33 (1969).
12
N. Shiotani, T. Okada, H. Sekizawa, T. Mizoguchi, and T. Karasawa,
J. Phys. Soc. Jpn. 35, 456 (1973).
13
M. Šob, S. Szuszkiewicz, and M. Szuszkiewicz, Phys. Status Solidi
B 123, 649 (1984).
14
T. Jarborg, A. A. Manuel, Y. Mathys, M. Peter, A. K. Singh, andE. Walker, J. Magn. Magn. Mater. 54–57, 1023 (1986).
15
S. Szuszkiewicz, M. Šob, and M. Szuszkiewicz, J. Magn. Magn.
Mater. 62, 202 (1986).
10024417-5

KAWASUSO, MAEKAWA, FUKAYA, YABUUCHI, AND MOCHIZUKI
16

P. Genoud, A. K. Singh, A. A. Manuel, T. Jarlborg, E. Walker,
M. Peter, and M. Welle, J. Phys. F 18, 1933 (1988).
17
P. Genoud, A. A. Manuel, E. Walker, and M. Peter, J. Phys.:
Condens. Matter 3, 4201 (1991).
18
H. Kondo, T. Kubota, H. Nakashima, T. Kawano, and S. Tanigawa,J. Phys.: Condens. Matter 4, 4595 (1992).
19
A. Kawasuso, M. Maekawa, Y. Fukaya, A. Yabuuchi, and
I. Mochizuki, Phys. Rev. B 83, 100406(R) (2011).
20
P. W. Zitzewitz, J. C. van House, A. Rich, and D. W. Gidley, Phys.
Rev. Lett. 43, 1281 (1979).
21
D. W. Gidley, A. R. Köymen, and T. W. Capehart, Phys. Rev. Lett.
49, 1779 (1982).
22
A. Kawasuso and M. Maekawa, Appl. Surf. Sci. 255, 108 (2008).
23
P. G. Coleman and A. Kallis, J. Phys.: Conf. Ser. 262, 012016(2011).
24
M. Alatalo, M. Puska, and R. M. Nieminen, J. Phys.: Condens.
Matter 5, L307 (1993).
25
E. Boroński and R. M. Nieminen, Phys. Rev. B 34, 3820 (1986).
26
Y. Nagai, Y. Nagashima, J. Kim, Y. Itoh, and T. Hyodo, Nucl.
Instrum. Methods Phys. Res., Sect. B 171, 199 (2000).

27

PHYSICAL REVIEW B 85, 024417 (2012)

J. Major, in Positron Beams and Their Applications, edited by
P. Coleman ( World Scientific, Singapore, 1999), p. 259.
28
S. Wakoh and J. Yamashita, J. Phys. Soc. Jpn. 21, 1712
(1966).
29
J. Callaway and C. S. Wang, Phys. Rev. B 16, 2095 (1977).
30
Y. Kubo and S. Asano, Phys. Rev. B 42, 4431 (1990).
31
M. Matsumoto, K. Tomimoto, and S. Wakoh, J. Phys. Soc. Jpn. 62,2734 (1993).
32
S. Wakoh, J. Phys. Soc. Jpn. 20, 1894 (1965).
33
Y. Kakutani, Y. Kubo, A. Koizumi, N. Sakai, B. L. Ahuja, andB. K. Sharma, J. Phys. Soc. Jpn. 72, 599 (2003).
34
J. Sticht and J. Kübler, Solid State Commun. 53, 529 (1985).
35
H. Yamagami, J. Phys. Soc. Jpn. 67, 3176 (1998).
36
X. Gonze, J.-M. Beuken, R. Caracas, F. Detraux, M. Fuchs, G.M. Rignanese, L. Sindic, M. Verstraete, G. Zerah, F. Jollet, M.
Torrent, A. Roy, M. Mikami, Ph. Ghosez, J.-Y. Raty, and D.C.
Allan, Comput. Mater. Sci. 25, 478 (2002).
37
P. E. Blöchl, Phys. Rev. B 50, 17953 (1994).
38
A. Kawasuso, M. Maekawa, and K. Betsuyaku, J. Phys.: Conf. Ser.
225, 012027 (2010).

024417-6