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Effect of magnetic field on positron lifetimes of Fe, Co and Ni

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2015 J. Phys.: Condens. Matter 27 246001(http://iopscience.iop.org/0953-8984/27/24/246001)
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Journal of Physics: Condensed Matter
J. Phys.: Condens. Matter 27 (2015) 246001 (5pp)

doi:10.1088/0953-8984/27/24/246001

Effect of magnetic field on positron
lifetimes of Fe, Co and Ni
H Li1 , M Maekawa1 , A Kawasuso1 and N Tanimura2
1

Advanced Science Research Center, Japan Atomic Energy Agency, 1233 Watanuki, Takasaki, Gunma
JP-370-1292, Japan
2
Science Solution Division, Mizuho Information and Research Institute, 2-3, Kanda-Nishikicho,
Chiyoda-ku, Tokyo, 101-8443, Japan
E-mail: li.hui@jaea.go.jp
Received 26 December 2014, revised 9 April 2015
Accepted for publication 20 April 2015
Published 3 June 2015
Abstract

Positron lifetime spectra of Fe, Co and Ni were measured under magnetic field using a 22 Na
source. Very small but distinguishable difference of positron lifetime upon magnetic fieldreversal was observed suggesting the existence of two bulk lifetimes associated with majority
and minority spin electrons. Using two spin-dependent Fe bulk lifetimes, the difference
Doppler broadening of annihilation radiation spectra between majority and minority spinelectrons were also examined. Agreement between experiment and theory indicates thatspin-polarized positron annihilation spectroscopy may have potential in investigation of
spin-aligned electron momentum distribution.
Keywords: spin-polarized positron lifetime, Fe, Co and Ni, Doppler broadening ofannihilation radiation(Some figures may appear in colour only in the online journal)

easy because of the extremely low three-fold coincidence rate(∼1 cps). To determine the total annihilation rates of spinup and spin-down positrons, spin-polarized positron lifetime
measurement, which has not yet been carried out except forpreviously considering asymmetry of counts in the selected
region from the lifetime spectrum [30, 31], is important.
Recently, a theoretical work implied the observable differencesof positron lifetimes associated with majority and minority spinelectrons in Fe, Co and Ni [32].
In this study, we measured the spin-dependent positron
lifetimes in Fe, Co and Ni. Using the spin-dependent lifetimes
in Fe, we also examined the direct difference DBAR spectra
associated with majority and minority spin electrons.

1. Introduction

After the discovery of parity non-conservation in the weakinteraction [1, 2], spin-polarized positrons from radioisotopes
were used in the angular correlation of annihilation radiation(ACAR) experiments for studying ferromagnetic band
structures [3–20]. The half-metallicity of NiMnSb was shownfor the first time by using spin-polarized two-dimentional
ACAR [20, 21]. Some reports also show that even nonpolarized positrons are useful to probe Fermi surfaces offerromagnets [22, 23].
Recently, we demonstrated thatthe Doppler broadening of annihilation radiation (DBAR)
technique using 22 Na and 68 Ge–68 Ga sources are also usefulfor similar purposes [24–27]. These methods are analogous
to the magnetic Compton profiling (MCP) [28, 29]. However,as compared to MCP, in ACAR and DBAR cases, to obtain
direct difference spectra between majority and minority spinbands, the spectrum measured in magnetic field needs tobe renormalized using the total annihilation rate of spin-up
and spin-down positrons or by field-reversal of three-gammaannihilation probability. Berko and Mills attempted to measure
the latter [6]. However, its precise determination was not
0953-8984/15/246001+05$33.00

2. Theory

The total annihilation rate of spin-up (spin-down) positrons isgiven by [6]
1
↓(↑)
↓(↑)
↑(↓)
=
[λS wi + λT (wi + 2wi )],
2 i=1
occ

λ

1

⇑(⇓)

(1)

© 2015 IOP Publishing Ltd Printed in the UK

J. Phys.: Condens. Matter 27 (2015) 246001

H Li et al

where the majority spin band is described as ↓ (magnetic
momentum is up) and minority spin band is expressed as ↑(magnetic momentum is down); λS = 4πre2 c, where re is theclassical electron radius and c is light speed; λT = λS /1115,
where 1115 is the ratio of the decay constants between singlet↓(↑)
and triplet positron-electron pairs [33], and wi is the overlap
integral between the positron and the i-th majority(minority)spin-band wave functions, which is given by [6, 25]

↓(↑)
↓(↑)
= ni (r)n+ (r)γ [n− (r)]dr,(2)
wi

where N (pz ) = N+ (pz ) − N− (pz ), N (pz ) = N+ (pz ) +N− (pz ), Pλ = P (λ⇓ − λ⇑ )/(λ⇑ + λ⇓ ) and N± (pz )dpz isnormalized to unity. Berko and Mills [6] suggested that thefield-reversal of three-gamma annihilation probability, P 3γ =
3γ
3γ
3γ
3γ
(N+ − N− )/(N+ + N− ), is approximately equal to Pλ .
In this work, the difference DBAR spectra for individual
bands of Fe were theoretically calculated. The electron wave
functions were obtained from the ABINIT computation [35]with the projector-augmented-wave method [36]. The initialvalence electron configurations were assumed to be 3d 6 4s 2 .
The positron wave function was calculated based on the twocomponent density functional theory. The DBAR spectrum
was obtained by double-integrating the momentum densitywith a Gaussian convolution having the full width at half
maximum of 1.4 keV. The details of the calculation were
described elsewhere [25].

↓(↑)

where ni (r) is the electron density of the i-th
majority(minority) spin-band, n+ (r) is the positron densityand γ [n− (r)] is the enhancement factor [34]. In positive andnegative magnetic fields, positron lifetime spectrum is given by
λS  ↓⇑⇓↑[wi (1 ± P )e−λ t + wi (1 ∓ P )e−λ t ],4 i
occ

L± (t) =(3)
3. Experiment

where P is the positron spin polarization. The field direction isdefined as positive (negative) when its orientation is the sameas (opposite to) that of positron spin polarization. Due to thedifference of electron densities between majority and minority
spin bands in ferromagnets, corresponding two bulk lifetimes
exist.
The first and second terms in the brace of equation (1) correspond to two-gamma and three-gamma events, respectively.
Therefore, λ⇑(⇓) = λmaj(min) + (λmaj(min) + 2λmin(maj) )/1115,−1
where λmaj(min) (= τmaj(min)) denotes the two-gamma annihilation rate (inverse of lifetime) of positrons with majority (minority) spin electrons. Two-gamma positron annihilation rate
in positive(negative) magnetic field is given by
λ± (= τ±−1 ) =1±P
1∓Pλmaj +
λmin .
2
2

The samples used in this work were polycrystalline Fe(4N),
Co(5N), Ni(5N) and Cu(4N), and single crystal Fe(1 0 0) andFe(1 1 1) discs with 10 mm diameter and 1 or 2 mm thickness.
All the samples were electrochemically polished on one side.
The Fe, Co and Ni samples were annealed at 1200 ◦ C for 2 h
in vacuum. The Cu samples were annealed at 1000 ◦ C for 3 h
in vacuum.
A 22 Na positron source of 1 MBq was deposited onto a
Ti film (5 µm thick). The source was sandwiched by an Fe,
Co , Ni or Cu sample with a Cu sample and the sample-source
pairs were horizontally placed in the gap of two permanent
magnets arranged vertically where the magnetic field was±1 T. If both sides of the source are ferromagnetic samples,no field reversal asymmetry of positron lifetime should beobserved due to opposite positron spin polarizations for
opposite directions. Positron lifetime spectra were measured
using a fast–fast coincidence system and two Hamamatsu
H3378-50 photomultipliers coupled with plastic scintillators ofsizes of 30 mm in height and 20 mm in diameter. One spectrum
contained more than 1 × 106 counts. After finishing each
lifetime measurement, the sample-source face was reversedwith respect to the field direction. No serious influence ofthe fringing field to the lifetime measurement system wasconfirmed from the measurements without magnetic field.
Longitudinal spin polarization of positrons was P = 35% [37].
The obtained lifetime spectra were decomposed into three
lifetime components by the PATFIT program [38]. The firstcomponent (τ1 , I1 , where τ and I denote lifetime and intensity,
respectively) was attributed to positron annihilation in the bulk.
The second (τ2 ∼ 269 ps, I2 ∼ 26%) and third (τ3 ∼ 1.4 ns,
I3 ∼ 2.2%) components were due to annihilation of positrons
in the source and the space between the samples and thesource, respectively. The intensity of source annihilation in
thick Ti foil covering 22 Na was significant, however, these two
lifetimes were longer than the first lifetime and easily separated
from lifetime spectra. The half of positrons were assumed toannihilate in one of the sandwiching samples and hence thebulk lifetimes of ferromagnet samples in positive (negative)(4)

The DBAR spectrum in magnetic field is given by [6, 25]


occ
↓
↑
λS  (1±P )Ni (pz ) (1∓P )Ni (pz )N± (pz ) =
,
+
4 i=1
λ⇑
λ⇓
(5)
↓(↑)
where Ni (pz ) is the DBAR spectrum :
↓(↑)

Ni

 

(pz ) =↓(↑)

ρi(p)dpx dpy .

(6)

↓(↑)

Here, ρi (p) are the positron-electron-momentum densities
for the ith majority (and minority) spin bands given by [25]

2



↓(↑)
↓(↑)
−ipr

ρi (p) =  e
+ (r)i (r) γ [n− (r)]dr , (7)↓(↑)

where + (r) is the positron wave function, i (r) is theelectron wave function.
The difference DBAR spectrum between majority andminority spin bands is given by [25]
occ


↓

↑

[Ni (pz ) − Ni (pz )] ∝ N (pz ) − Pλ N(pz ),

(8)

i=1

2

J. Phys.: Condens. Matter 27 (2015) 246001

H Li et al

Table 1. Experimental and theoretical (density functional theory(DFT))lifetimes (τmaj and τmin ) associated with majority andminority spin electrons, field reversal of total annihilation rate (Pλ )determined from equation (8) and P = 0.35, of Fe, Co and Ni andthe field reversal of three-gamma annihilation probabilities (P 3γ ) in
the previous work [6].

Present

DFT [32] Previous [6]Feτmaj (ps) 109 ± 2
95.2
τmin (ps) 120 ± 3
107.0
Pλ−0.018 ± 0.005 −0.021
Coτmaj (ps) 100 ± 5
94.5
τmin (ps) 114 ± 598.2
Pλ−0.024 ± 0.012 −0.007
Ni
101.1
τmaj (ps) 106 ± 5τmin (ps) 100 ± 596.8
Pλ0.010 ± 0.011
0.008

P 3γ : 0.0012 ± 0.0009

Cu + Co, the lifetime shows no change upon field reversal. The
lifetime also agrees with the bulk lifetime determined withoutmagnetic field. In the other cases, the lifetimes change uponfield reversal. In the cases of Cu+Fe and Cu+Co, the lifetimes
in positive field are shorter than those in negative field, whereas,
the relationship is opposite for the Cu + Ni. Table 1 lists thebulk lifetimes associated with the majority and minority spinelectrons deduced from equations (4) and (7) and theoreticalvalues by Lin et al [32].
Apart from the absolute differences between experimentand theory, the signs agree with each other. The results for
the Fe and Co samples may be simply explained considering
the fact that the number of majority spin electrons is more thanthat of minority spin electrons. According to Lin et al [32], theopposite tendency for the Ni sample can be explained as due todelocalized distribution of minority spin electrons in Ni. Thatis, the minority spins are broadened into interstitial regions
and the minority spin densities there are larger than those ofmajority spins. According to the calculated MCP profiles forferromagnetic nickel by Dixon et al [43], the first four bands
are quite negative in the low momentum region, indicating thatthe delocalized states in real space are negatively polarized.
Positron and electron wavefunctions have more overlapping
in interstitial region. Hence, in the case of Ni, the lifetime
associated with minority spin electrons may be shorter thanthat of majority spin electrons.
The field reversal of Pλ determined from the above
positron lifetimes and P = 0.35 are also listed together withthose by Lin et al [32] and P 3γ by Berko and Mills (BM) [6] intable 1. Both the sign and magnitude of Pλ are in good agreement with those of the DFT calculation. However, two newworks give significantly larger Pλ as compared to BM’s P 3γ .
Figures 2(a)–(c) and 3(a)–(c) show the difference DBAR
spectra of Fe single crystal obtained from measured DBAR
spectra using equation (8) with the BM’s P 3γ [6] and Pλby Lin et al [32], respectively, of which P is replacedwith 0.6, along different momentum axes. The curves weresmoothed by the five-point interpolation. The curve shapes
in figures 2(a)–(c) are similar to what BM ever reported [6].

Figure 1. The first lifetimes (τ1,± ) for the Cu + Cu, Fe + Cu,
Co + Cu and Ni + Cu samples under magnetic field of ±1 T.

magnetic field were deduced by
τ± = 2τ1,± − τCu .

P 3γ : −0.0053 ± 0.0009(9)

The uncertainty due to subtraction of 50% Cu was low. We
calculated the transportation of error and obtained typical final
error bar to be less than 1 ps. DBAR spectra were measured
by using a high-purity Ge detector with an energy resolutionof 1.4 keV at 511 keV as described elsewhere [25]. The source
was a 20 MBq 68 Ge–68 Ga and the average spin polarization
was 60%. Here, gamma-ray energy of 1 keV corresponds toan electron momentum of p = 3.92 × 10−3 m0 c (0.54 a.u.).
In each spectrum, more than 106 events were accumulated.
4. Results and discussion

The bulk lifetimes obtained without magnetic field were
114 ± 1 ps (Cu), 109 ± 1 ps (Fe), 109 ± 1 ps (Co), and
101 ± 1 ps (Ni). These are in good agreement with previous
experiments [39–41] and theories [32, 39–42]. Therefore, thevacancy concentrations in these samples must be lower thanthe detection limit and the vacancy contribution was less than
1% as compared to the bulk.
Figure 1 shows the first lifetimes (τ1,± ) obtained fromthe Cu + Cu, Cu + Fe, Cu + Co and Cu + Ni systems inmagnetic field. The measurements were repeated by reversing
magnetic field for 10 ∼ 20 times, The average lifetimes wereused for the further analysis. In the cases of the Cu + Fe and3

J. Phys.: Condens. Matter 27 (2015) 246001

H Li et al

Figure 2. Experimental difference spectra between majority andminority spin bands of Fe using equation (8) and BM’s P 3γ [6]instead of Pλ [32] in different momentum axes of (a) [0 0 1], (b)[1 1 0] and [1 1 1].

Figure 3. Experimental and calculated difference spectra between
majority and minority spin bands of Fe using equation (8) andPλ [32] and theoretical curves in different momentum axes of(a) [0 0 1], (b) [1 1 0] and [1 1 1].

Contrarily, the curve shapes in figures 3(a)–(c) obtained using
the present Pλ are rather different from the BM’s data [6].
Therefore, the problem is that, which of BM’s P 3γ [6] or thepresent Pλ is more realistic. In figures 3(a)–(c), the calculateddifference spectra between majority and minority spin bands
are also shown. The experiment and the calculation are ingood agreement when using the present Pλ . Probably the
BM’s P 3γ [6] was underestimated due to the extremely low
counting rate. Nevertheless, in both figures 2 and 3, the curve
shape depends on the momentum axis suggesting different
momentum distributions along different directions.
In figures 3(a)–(c), along the [1 0 0] direction, a sharp peak
and a shoulder appear in the lower momentum region and ataround pz = 10×10−3 m0 c, respectively. Along the [1 1 0] and[1 1 1] directions, the shapes are more gentle. In figure 3(a), noclear evidence about negative spin polarization of s electrons
is seen since the spectrum shows no dip at around pz = 0. In
figures 3(b) and (c), although no dips appear at around pz = 0,
the intensities are reduced as compared to figure 3(a). Thismay be explained by negative spin polarization of s electrons.
From the comparison with the calculated curves, the bands
3 and 4 give rise to such features. The bands 1 and 2 havemore s-like character, but their contributions to the spectra arevery small probably due to the small density of state and morepositive spin polarization. The bands 5 and 6 have more d-like
dispersion along any directions making over all broad spectrum
shapes.

5. Conclusion

In conclusion, we measured spin-dependent positron lifetimes
for Fe, Co and Ni. The differences of bulk positron lifetimes
associated with majority and minority spin electrons wereconfirmed. The yielded difference DBAR spectra between
majority and minority spin electrons using the two spindependent Fe bulk lifetimes were well consistent with thetheoretical calculations. This study shows that spin-polarized
positron lifetime technique can be used in analysis of spinaligned electron momentum distribution.
Acknowledgments

We thank M Saito and J B Lin of Kanazawa University for
their valuable comments about theoretical calculations. This
work was supported by JSPS KAKENHI Grant No.24310072.
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