Conference-ISSS-7-Positron Energy Loss and the Interaction between Positrons and Soliton-Like Electron Density in Graphite-Alkali Metal Intercalation Compounds

It is known that alkali-metal graphite intercalation compounds (AGIC) have catalytic activities for hydrogen layer of the compounds. The behaviors of these compounds have been known to depend on the structure, sort of metals and temperature. Hydrogens are physisorptively accommodated in the molecules in the interstices among the alkali-metal ions in intercalant layers of stage-2 compounds C24M (M=K, Rb and Cs) at temperatures below 200 K [1–3]. Chemisorption of hydrogen takes place in both first and second stage compounds at higher temperatures [4, 5]. In order to clarify the mechanism of hydrogen chemisorption in the graphite compounds and properties of hydrogen-absorbed graphite compounds, several studies have been carried out. The change in magnetic susceptibilities as a function of hydrogen content has been investigated for hydrogen-chemisorbed C8K [6]. It has been suggested that dissolved hydrogen in C8K is paramagnetic by means of magnetic resonance studies [7]. The studies of means of ESR and electrical conductivity [8] have shown that in C8K, C24K, and C24Rb, the hydride ions are stabilized after the dissociation of the hydrogen molecules into atoms and the subsequent charge transfer, and in C8Rb hydrogens are absorbed in the atomic form. Recently the present authors have discussed the mechanism of anomalous magnetic effect due to hydrogen uptake in C8RbHx with the theoretical formula, which is extended from “topological quasi-hydrogen” model [24]. It is suggested strongly that the hydrogen state in C8RbHx might have the Kondo-like property [24]. Positron annihilation spectroscopy is a useful method in the study of electronic structures of materials. This spectroscopy has been used in the investigation of electronic structures in graphite and AGICs. The momentum distribution of σand π-electrons in a graphite crystal have been studied by angular correlation of positron annihilation radiation (ACPAR) [9–12], and by Dopplerbroadening positron annihilation radiation (DBPAR) [13]. ACPAR spectra of C8K and C24K, after subtraction of a broad contributions which accounts for annihilation with the graphite σ and π-electrons. Features of the nar-


INTRODUCTION
It is known that alkali-metal graphite intercalation compounds (AGIC) have catalytic activities for hydrogen layer of the compounds.The behaviors of these compounds have been known to depend on the structure, sort of metals and temperature.Hydrogens are physisorptively accommodated in the molecules in the interstices among the alkali-metal ions in intercalant layers of stage-2 compounds C 24 M (M=K, Rb and Cs) at temperatures below 200 K [1][2][3].Chemisorption of hydrogen takes place in both first and second stage compounds at higher temperatures [4,5].In order to clarify the mechanism of hydrogen chemisorption in the graphite compounds and properties of hydrogen-absorbed graphite compounds, several studies have been carried out.The change in magnetic susceptibilities as a function of hydrogen content has been investigated for hydrogen-chemisorbed C 8 K [6].It has been suggested that dissolved hydrogen in C 8 K is paramagnetic by means of magnetic resonance studies [7].The studies of means of ESR and electrical conductivity [8] have shown that in C 8 K, C 24 K, and C 24 Rb, the hydride ions are stabilized after the dissociation of the hydrogen molecules into atoms and the subsequent charge transfer, and in C 8 Rb hydrogens are absorbed in the atomic form.Recently the present authors have discussed the mechanism of anomalous magnetic effect due to hydrogen uptake in C 8 RbH x with the theoretical formula, which is extended from "topological quasi-hydrogen" model [24].It is suggested strongly that the hydrogen state in C 8 RbH x might have the Kondo-like property [24].
Positron annihilation spectroscopy is a useful method in the study of electronic structures of materials.This spectroscopy has been used in the investigation of electronic structures in graphite and AGICs.The momentum distribution of σand π-electrons in a graphite crystal have been studied by angular correlation of positron annihilation radiation (ACPAR) [9][10][11][12], and by Dopplerbroadening positron annihilation radiation (DBPAR) [13].ACPAR spectra of C 8 K and C 24 K, after subtraction of a broad contributions which accounts for annihilation with the graphite σ and π-electrons.Features of the nar-row components in ACPAR spectra are in good agreement with the quasi-two dimensional electronic structures, which might correspond to the interlayer state with quasi-two dimensional free electron character parallel to the carbon planes [16].In addition, hydrogen physisorption and chemisorption effects in AGICs have been studied by DBPAR [17][18][19][20].The DBPAR spectra line shape of C 8 K became sharp through hydrogen chemisorption at 300 K [17,18], while the spectral lineshape of C 24 Cs became broad through the physisorption of hydrogen molecules at 77 K [18,19].The intensity of the narrow component of DBPAR spectrum of C 8 Rb was suppressed through hydrogen absorption at 300 K, while it decreased at first and then increased through the absorption at the 373 K [20].From the change in DBPAR spectral line-shape of C 8 Rb, the hydrogen accommodated in C 8 Rb is consider to be atomic at 300 K and to be in the form of the hydride ion at 373 K.There exist some problems in the assumption that the narrow component of positron annihilation spectra corresponds to the interlayer state in AGICs.That is, the energy level of the interlayer state in the second-stage AGICs is above Fermi energy EF.Furthermore, it looks like that the narrow component is not attributed to the simple positronium Ps in AGICs, because simple Ps cannot exist in the metallic state of AGICs.
In this study, we analyze the ACPAR spectra in C 24 K [21] with the theoretical formula, which is extended from "topological quasi-positronium" model [24] and discuss the origin of the anisotropic narrow components in ACPAR spectra.In addition, we will discuss the positron energy loss due to the interaction between positrons and skyrmion-like electron density in graphite-alkali metal intercalation compounds.

II. TOPOLOGICAL QUASI-POSITRONIUM AND POSITRON ENERGY LOSS
The structure of the stage-1 and stage-2 alkali-metal-GIC's C 8 K and C 24 K are shown in Figure 1.In the case of the Stage-1 compound C 8 K, the alkali-metal atoms form a closed-packed two-dimensional triangular lattice structure between the graphite layers.
In the stage-2 GIC C 24 K, the atomic density of potassium metal atoms intercalated between graphite layers is reduced to 2/3 of the density of the close-packed structure in the stage-1 compound, taking into account the difference in composition between the stage-1 compound   C 8 K and the stage-2 compound C 24 K.According to the electronic structure model [16], there coexist quasi-two dimensional graphite π-bands, bands originating from potassium-metal s electrons, and the electronic interlayer state in the stage-2 GIC C 24 K.The Fermi energy EF and the location of the bands are determined by a balance between electronic and lattice energies.The electronic interlayer states were introduced by Posternak et al. [16].These interlayer states, which exhibit free-electron character parallel to the layers, form a quasi-two dimensional band close to the Fermi energy and provide for alkali graphite intercalation compounds a new understanding of the origin of bands near the Fermi energy.Since the energy level of the interlayer state is close to the Fermi energy level, SU(2) pseudospin structure can be introduced by assigning "up" and "down" pseudospins to the electron on the interlayer state level and the Fermi energy level, respectively.A skyrmion is a topological soliton spread over two energy levels.Now, we shall consider the quantum fluctuation in quasi (2+1) electron state's system.A charged skyrmion-like soliton in quasi (2+1) intercalate state in AGICs might be described as a topological soliton [22,23]."Topological quasi-positronium" is regarded as the skyrmion-like soliton, which traps a positron, of the electron-density.The electron number associated with the skyrmion-like soliton can be derived from the induced current J µ sky on the SU(2) pseudospin-like skyrmion configuration [25][26][27][28].The electron number N sky is where The electron-density distribution parallel to the quasitwo dimensional electron layer in the interlayer states in C 24 K can be represented by the SU(2) pseudospin-like skyrmion configuration.
Figure 2(a) shows the annihilation spectra for pairmomentum p perpendicular to the crystallographic c-axis in C 24 K.The triangle in Figure 2(a) show p-distribution of the narrow component (p⊥c), taking into account that pdistribution of the narrow component (p⊥c) of the annihilation spectra is isotropic approximately on the direction parallel to the graphite plane.Figure 2(b) shows the annihilation spectra for pair-momentum p parallel to the crystallographic c-axis in C 24 K.The triangle in Fig. 2(b) show p-distribution of the narrow component (p∥c), assuming that p-distribution of the narrow component (p∥c) of the annihilation spectra is in elliptic circle approximately.
Figure 3(a) and (b) show the electron-density distribution perpendicular to the c-axis and parallel to the caxis, which is Fourier-transformed from the narrow component for pair-momentum p perpendicular to the c-axis and parallel to the c-axis, respectively.The solid line in Figure 3(a) is fitted one by the function J 0 sky in eq.( 2).The value of ξ is estimated to be ∼2.1 Å.The solid line in Figure 3(b) is fitted one by the Gaussian function ∝ exp(−r 2 /2δ 2 ).The value of δ is estimated to be ∼1.1 Å.As the origin of the anisotropic narrow component of ACPAR for AGICs, we can consider the contribution of "topological quasi-positronium", from this http://www.sssj.org/ejssnt(J-Stage: http://www.jstage.jst.go.jp/browse/ejssnt/) e-Journal of Surface Science and Nanotechnology Volume 13 ( 2015) analysis."Topological quasi-positronium" is regarded as the skyrmion-like soliton, which traps a positron, of the electron-density.Now, we shall consider the positron energy loss due to the interaction between positrons and skyrmion-like electron density.The rate for the positron energy loss is defined by is the positron energy with momentum p, in the case of low energies ( E posi p < ∼2000 eV).W q is the transition probability.The transition probability W q is related to the imaginary part of the positron self-energy |Σ(p, E posi p )| as follows.
This estimation has been made within the so-called random-phase approximation (RPA) as shown in Eq.( 6).The dashed line represents the electron-propagator in the skyrmion-like state, which is introduced in the present theoretical formula.In RPA approximation here v q = − 4πe 2 ε(q, 0) .(7) ε(q, 0) is the dielectric function due to π * band.Since electron density of the Fermi energy in π * band is low, it is suspected that the value of ε(q, 0) is not large.But in the case of C 24 K, the location of the interlayer state above E F has not been established [16].The energy level, E sky , of the skyrmion-like state seems to be located in the energy region ≥ E F .Equation ( 6) shows a logarithmic diverging frequency W → 0. That is, if E sky −E F ∼W is very small, we might be able to observe the anomalous property of the positron energy loss in C 24 K.When the effect of quasiparticle damping on the singularities is large, the singularities will not be observable.
To include the positron e + -e − correlation in this regime we replace a single screened interaction by an energyindependent t matrix which sums the diagrams shown in Figure 4(b).In this case,

III. CONCLUSION
The static t matrix t(p, k; p − q, k + q) is the scattering amplitude for a single e + of momentum p scattering from a single e − with momentum k scattering to the state p − q and k + q.In the case of C 24 K, it is deduced that the interlayer state is located 0.2 eV above E F [16].
The narrow component in the positron annihilation angular correlation spectra in the second stage graphitepotassium intercalation C 24 K is analyzed from the viewpoint of the topological quasi-positronium model.We have considered the positron energy loss due to the interaction between positrons and skyrmion-like electron density within the random-phase approximation(RPA).The anomalous property of the positron energy loss in C 24 K has been suggested.
FIG. 4. A Feynman diagram of the positron proper selfenergy within the random-phase approximation (RPA).A double solid line denotes a free positron Green's function.A single solid line denotes a hole Green's function.A single dotted-line denotes the electron Green's function.A dashed-dotted line denotes a bare positron-electron Coulomb interaction.(b) A Feynman diagram of the positron proper self-energy with the effective Coulomb interaction.(c) A Feynman diagram of the effective Coulomb interaction.

Figure 4 (
a), (b), (c) shows the approximate self-energy for low-energy positrons, where the dashed-dotted line represents the statically screened Coulomb potential.The solid line shows the hole-propagator in π * band in C 24 K.