B Baltz

Rodmacq

M Cormier
email: m.cormier@science.ru.nl

A Mougin

J Ferré

J-P Jamet

R Weil

J Fassbender

V Baltz

B Rodmacq

Fast propagation of weakly-pinned domain walls and current-assisted magnetization reversal in He + -irradiated Pt/Co/Pt nanotracks

Keywords: numbers: 75.60.Ch, 75.75.-c, 61.80.Jh, 85.70.Kh

Fast propagation of weakly pinned domain walls and current-assisted magnetization reversal in

He+-irradiated Pt/Co/Pt nanotracks

Introduction

Current developments of magnetic data storage and processing technologies make highly desirable to control fast and reproducible magnetic-domain-wall motion in narrow magnetic tracks [1,2], using either magnetic field [1,3,4,5], or electrical current [2,[START_REF] Thomas | Current induced domain-wall motion in magnetic nanowires Handbook of Magnetism and Advanced Magnetic Materials[END_REF][START_REF] Lindner | [END_REF]. For this purpose, nanotracks defined in ultrathin magnetic films with out-of-plane anisotropy seem to be particularly interesting [8,9]. Especially, confinement effects due to the low film thickness lead to a lowered Walker breakdown [8,9,10], which allows to access the so-called precessional-propagation regime under a reduced excitation. In this regime, the continuous evolution of the domain-wall micromagnetic structure during propagation is expected to lead to a reduced wall pinning by the sample's intrinsic defects. However, in most of the out-of-plane metallic nanosystems which were studied up to now, domain-wall pinning was shown to still play a predominant role, which resulted in relatively low domain-wall velocities, as compared to the ones observed in the corresponding plain magnetic films [4]. In this paper, we show that in an etched Pt/Co/Pt nanotrack with out-of-plane magnetic anisotropy, where pinning has been artificially reduced by He + -irradiation [11,12], weakly-pinned domain-walls can propagate as fast and under magnetic field as low as in the corresponding plain irradiated film. Moreover, when magnetic-field and electrical-current pulses are simultaneously applied to the track, a considerably faster magnetization reversal is observed, which is due to a Joule-heating-induced thermomagnetic effect.

Sample preparation

Magnetic nanotracks were patterned, by a conventional process of e-beam lithography and argon-ion-beam etching, in Pt(4.5 nm) / Co(0.5 nm) / Pt(3.5 nm) films, sputterdeposited onto thermally-oxidized Si(100) / SiO 2 (500 nm) substrates. The as-grown system presents a strong out-of-plane magnetic anisotropy, and reverses its magnetization by few domain nucleations, and predominant easy domain-wall propagation [13]. Before patterning, the films were uniformly irradiated at very low doses of 4 to 5 × 10 15 He + cm -2 , with an energy of 30 keV. This irradiation resulted in a strong reduction of the out-of-plane magnetic anisotropy [11,14], and, consequently, of the coercive and domain-wall-propagation fields [5,14,15]. The widths of the tracks described in the following range from 500 to 750 nm, for a typical length of 100 µm. At one end of each track (figure 1(a)), a trumpet-like enlargement was intended to enhance the domain-nucleation probability, whereas the other end has a pointed shape.

Field-induced domain-wall propagation

Domain-wall propagation in the nanotracks was studied, at room temperature, by magneto-optical polar Kerr microscopy. The sample was excited by magnetic-field pulses (figure 1(l)) from a small inductive coil. After each pulse, a magneto-optical image was taken, and the magnetic state of the track was deduced from an image difference with the initial, magnetically-saturated state. Figures 1(b)-(k) show an example of typical experimental results. A reversed domain is nucleated near the end of the nucleation reservoir (figure 1(b)), by the first magnetic-field pulse (about 170 Oe, 1 µs). Further pulses (here about 150 Oe, 1 µs) provoke a propagation of the corresponding domain walls, until the track is entirely reversed (figure 1(k)). Propagation is very regular, without any apparent pinning, as can be seen in figure 1(m), where the propagation distance is shown to be directly proportional to the number of applied field pulses. The same kind of experiment was performed for different combinations of the magnetic-fieldpulse amplitude and duration. Doing this, we could estimate, for each pulse duration, an apparent domain-wall-propagation field, H prop , above which a domain wall reproducibly propagates (figure 1(n)): for a 1 µs pulse, H prop ≈ 120 Oe only, while a 1 Oe dc magnetic field is sufficient to make a domain wall move, which is comparable to recent results obtained in permalloy nanotracks with in-plane magnetic anisotropy [16].

It was carefully checked that when no magnetic field is applied, domain walls do not move whithin the characteristic duration of the measurements. Besides, no further domain nucleation was observed due to propagation-field pulses, which means that the critical propagation field is much lower than the critical nucleation field. This comes along with the extremely soft magnetic properties of the films after He + -irradiation, which are obviously conserved after patterning. Moreover, the very good homogeneity of domain-wall propagation (figure 1(m)) indicates the absence of strong extrinsic pinning defects which could have been introduced during the patterning process, as well as a good homogeneity of the track edges.

Figure 2 shows the measured domain-wall velocity, v, as a function of the applied magnetic field, H, in both an irradiated film (before patterning) and a nanotrack. So as to avoid any artifact due to transient domain-wall propagation during the rising and falling edges of the magnetic-field pulses (figure 1(l)), several measurements were performed for each field value, with different field-pulse durations. The actual "plateau" domain-wall velocity was deduced from the slope of the domain-wall-displacement vs field-pulse-duration linear dependence. Similar data measured on the as-grown film (before irradiation) can be found elsewhere [13]. The velocity curves corresponding to the irradiated film and to the nanotrack, respectively, are very similar, which is a further indication of the preservation of the irradiated-film properties after patterning. The lowfield v(H) dependence is characteristic of domain-wall creep [13]. This is confirmed in the inset of figure 2, where ln v is shown to vary linearly with H -1/4 , in accordance with the creep-velocity-law of a 1D elastic interface moving in a 2D weakly-disordered medium [17,18]:

v = v 0 exp   - T dep T H dep H 1/4   , ( 1
)
where T dep and H dep are the critical depinning temperature and field, respectively, T is the measurement temperature, and v 0 is a numerical prefactor. Fitting the velocity data to (1) (inset of figure 2) leads to T dep /T ≈ 11 in the track. This is slightly higher than in the film, where T dep /T ≈ 9, which indicates that pinning is slightly stronger in the track than in the film. This slight increase of the pinning strength in the track is coherent with previous studies of tracks similar to ours, in which artificial edge roughness was deliberately added during patterning [4]. In our tracks, pinning remains weak enough not to significantly perturb domain-wall propagation (figure 1). The limit field, H * , up to which the creep-velocity-law ( 1) is valid, is slightly reduced in the track (60 Oe, see inset of figure 2) as compared to the film (100 Oe). Under a stronger magnetic field, domain-wall propagation rapidly becomes insensitive to pinning, and viscous propagation is obtained after a so-called depinning transition [13]. Unlike in the film, the viscous regime could not be clearly evidenced in the track, due to excessive nucleation under high field [19], which makes impossible to reliably measure domain-wall propagation above H max ≈ 200 Oe. Nucleation could not be avoided by simply reducing the duration of magnetic-field pulses, which could not be made shorter than 50 ns on our experimental setup.

After all, combining the initial properties of the magnetic films, their optimization using ion-irradiation, and a low-damage patterning process, we could reach domain-wall velocities in the tracks at least two orders of magnitude higher than those previously reported in comparable non-irradiated systems [4], while applying magnetic fields more than ten times smaller.

Current-assisted magnetization reversal

A further improvement in the track-magnetization reversal speed was obtained by combining the effects of magnetic-field and electrical-current pulses to induce reversal. Namely, we simultaneously applied a field pulse, and injected a (shorter) current pulse into a track containing domain walls (figure 3). Further details on the experimental procedure for current injection can be found elsewhere [20]. In particular, we checked that the current pulse alone did not have any effect on the domain structure in the track. Following this procedure, a much greater track length was reversed (up to 45 µm, see left domain in figure 3) than with the same field pulse alone (10 µm, see figure 2), leading to an average track-reversal speed as high as 45 m s -1 under less than 200 Oe. For more intense current pulses (for example about 8×10 11 A m -2 during 100 ns), the entire track (about 100 µm) could even be reversed by one single field pulse. In these nanotracks, the only measurable effect of electrical current was demonstrated to be Joule heating, which totally hinders any possible spin-transfer effect [20]. Therefore, the higher reversal speed reported here is purely of thermomagnetic origin: Joule-heating during the current pulse (estimated to be about 20 K for figure 3 [21]) thermally activates magnetic-field-induced domain nucleation and domain-wall propagation, which probably both contribute to the observed reversal [19]. This situation is very similar to the process involved in laserassisted thermomagnetic recording [START_REF] Mansuripur | The Physical Principles of Magneto-Optical Recording[END_REF][START_REF] Shiroishi | [END_REF].

Conclusion

We demonstrated that in He + -irradiated Pt/Co/Pt etched nanotracks with out-of-plane magnetic anisotropy, considerable magnetization-reversal speeds can be obtained under unprecedently-weak magnetic fields. Field-induced domain-wall propagation in the tracks was shown to be as fast and under fields as low as in the corresponding nonpatterned films. Taking advantage of the Joule-heating-induced thermomagnetic effect occuring when a current flows into the track, a strong further enhancement of the trackreversal speed was demonstrated. These results reflect a major improvement compared with previous results obtained in similar systems [4,5]. They demonstrate once more the importance of ion-irradiation [5], and of a controlled and optimized patterning process [16], to achieve fast and reproducible domain wall propagation in magnetic nanotracks, known as crucial for technological applications [1,2].

Figure 1 .

 1 Figure 1. Magnetic-field-induced domain-wall propagation in an etched nanotrack. (a) Optical picture of the track (510 nm-wide, patterned in a film which was previously irradiated at a dose of 4 × 10 15 He + cm -2 ). (b)-(k) Magneto-optical images of: (b) the magnetic state of the track after nucleation of a domain out of the initially-saturated state; (c)-(k) successive magnetic states obtained starting from state (b), by applying a series of identical magnetic-field pulses (≈ 150 Oe, 1 µs), whose typical shape is depicted in (l). Dark and clear areas respectively correspond to areas of the sample which were magnetically reversed, or which are still in their initial magnetic state. (m) Distance, d (+/-0.5 µm), travelled by the wall, as a function of the number, n, of applied field pulses. (n) Apparent domain-wall-propagation field, H prop , as a function of the field-pulse duration, ∆t, in a similar, 750 nm-wide nanotrack, patterned in a film which was irradiated at a dose of 5 × 10 15 He + cm -2 .

Figure 2 .

 2 Figure2. Domain-wall-propagation velocity, v, as a function of the applied magnetic field, H, in a plain film irradiated at a dose of 5×10 15 He + cm -2 , and in a 750 nm-wide nanotrack patterned in this film. The dotted line is a linear fit to the high-field velocity values in the film. Inset: Same data plotted in a coordinate system which allows to evidence the low-field creep-velocity-law (1). The dotted lines are fits to(1).

Figure 3 .

 3 Figure3. Magneto-optical images of a 750 nm-wide nanotrack, patterned in a film irradiated at a dose of 5 × 10 15 He + cm -2 , before (top), and after (bottom), the simultaneous application of a magnetic-field pulse (≈ 190 Oe, 1 µs), and an electricalcurrent pulse (≈ 3×10 11 A m -2 , 500 ns). Dark and clear areas respectively correspond to areas of the sample which were magnetically reversed, or which are still in their initial magnetic state. The reversed domain appearing on the right in the lower picture either already existed (outside of the field-of-view) when the upper picture was taken, or was nucleated under the joint effects of field and current.

Acknowledgments

J.-P. Adam is gratefully acknowledged for assistance with the measurements. This work was partially supported by the EU-"Research Infrastructures Transnational Access" program "Center for Application of Ion Beams in Materials Research" under contract no. 025646, and by the French ANR-07-NANO-034 "Dynawall" project. M. C. thanks ALTIS Semiconductor and the French A.N.R.T. for CIFRE financial support.


Markus Meinert
email: meinert@physik.uni-bielefeld.de

Jan-Michael Schmalhorst

Günter Reiss

Elke Arenholz

Ferrimagnetism and disorder of epitaxial Mn 2-x Co x VAl Heusler compound thin films

The quaternary full Heusler compound Mn 2-x Co x VAl with x = 1 is predicted to be a half-metallic antiferromagnet. Thin films of the quaternary compounds with x = 0 . . . 2 were prepared by DC and RF magnetron co-sputtering on heated MgO (001) substrates. The magnetic structure was examined by x-ray magnetic circular dichroism and the chemical disorder was characterized by x-ray diffraction. Ferrimagnetic coupling of V to Mn was observed for Mn 2 VAl (x = 0). For x = 0.5, we also found ferrimagnetic order with V and Co antiparallel to Mn. The observed reduced magnetic moments are interpreted with the help of band structure calculations in the coherent potential approximation. Mn 2 VAl is very sensitive to disorder involving Mn, because nearest-neighbor Mn atoms couple antiferromagnetically. Co 2 VAl has B2 order and has reduced magnetization. In the cases with x ≥ 0.9 conventional ferromagnetism was observed, closely related to the atomic disorder in these compounds.

Introduction

Half-metallic fully compensated ferrimagnets (HMFi), which are also known as halfmetallic antiferromagnets, attracted large interest during the past years. A material with this property would exhibit full spin polarization at the Fermi level, but the magnetization would be effectively zero. It was first predicted for Mn and In doped FeVSb [1]. Among others, La 2 VMnO 6 and related double perovskites [2], and certain diluted magnetic semiconductors have been later predicted to be half-metallic antiferromagnets as well [3]. This is interesting for technological applications, e.g. for spin transfer torque switching, which depends on the magnetization and the spin polarization of the material to be switched.

Galanakis et al. pointed out that it may be possible to synthesize a HMFi by substituting Co for Mn in the Heusler compound Mn 2 VAl [4]. Mn 2 VAl is a (potentially half-metallic) ferrimagnet with antiparallel coupling of Mn and V moments and a total moment of -2 µ B per formula unit. The high Curie temperature of 760 K makes it interesting for practical applications. Numerous experimental [5,6,7,8,9] and theoretical [10,11,12,13,14] studies are found in the literature. Following the Slater-Pauling rule for Heusler compounds, m = N V -24 [15], the magnetic moment m is to be taken as negative, because the number of valence electrons N V is 22. Thus, by adding effectively two electrons per unit cell, the magnetization should vanish. This can be achieved by substituting one Mn with one Co atom, which has two additional electrons. Ab initio simulations were carried out on this system in the L2 1 structure with Mn and Co randomly spread across the Wyckoff 8c sites and V and Al on the 4a and 4b sites. Indeed, a HMFi is found with magnetic moments of: -1.388 (Mn), 0.586 (Co), 0.782 (V), 0.019 (Al) [4]. It was shown by Luo et al. that the site occupation preference in Mn 2 Y Al depends on the number of valence electrons of Y : if it is lower than the one of Mn, Y would preferentially occupy the 4a/b sites, but if it is higher, Y would rather occupy the 8c sites together with Mn, changing the structure to the Hg 2 CuTi type [16]. Accordingly, one can expect an occupation as proposed by Galanakis et al. in Mn 2-x Co x VAl (MCVA).

Disorder is a major concern when dealing with half-metallic Heusler compounds. The presence of disorder has been repeatedly demonstrated for Co 2 -based Heusler bulk and thin films (see, e.g., [17,18,19,20]). Theoretical studies have investigated the impact of disorder on the magnetic properties and the half-metallicity of the compounds [21,22,23,24,25,26]. For some compounds major impact of disorder on the halfmetallicity is observed, which also depends on the type of disorder. Particularly, Picozzi et al. [21] found that a Mn atom substituting a Co atom in Co 2 MnSi, which has Mn as nearest neighbors, would couple antiparallel to the surrounding Mn atoms, and thus reduce the total magnetization drastically. The strong dependence of the magnetic moment of Mn on its chemical and magnetic environment has been demonstrated by, e.g., Rader et al. [27]. Hence, disorder bringing Mn into nearest-neighbor positions has to be controlled.

For many practical applications it is necessary to prepare high quality thin films of the magnetic materials. Therefore one has to find suitable deposition techniques and optimize the parameters. The parent compounds Mn 2 VAl and Co 2 VAl [28,29] have been successfully synthesized in the bulk and epitaxial growth of Mn 2 VAl films with L2 1 ordering on MgO (001) single crystals was also demonstrated [30,31]. In this paper we present experimental results on the structural and magnetic properties of epitaxial Mn 2-x Co x VAl thin films.

Methods

Experimental details

The samples were deposited using a UHV co-sputtering system equipped with five DC and two RF 3" magnetron sputtering sources, arranged in a confocal sputter-up geometry. Up to four sources can be used simultaneously. The target-to-substrate distance is 21 cm and the inclination of the sources is 30 • . The substrate carrier can be heated to 1000 • C by an infrared heater from the backside. An electron beam evaporator with one crucible is placed in the center of the chamber at a distance of 50 cm to the sample. The base pressure of the system was typically 5 • 10 -10 mbar.

Elemental targets of Mn, Co, V, and Al of 99.95 % purity were used. The sputtering pressure was set to 2 • 10 -3 mbar. The correct sputter power ratios were set up using a combined x-ray reflectivity and x-ray fluorescence technique.

All samples used in this study had the following stack sequence: MgO (001) single crystal / Mn 2-x Co x VAl 18 nm / Mg 0.5 nm / MgO 1.5 nm with x = 0 / 0.5 / 0.9 / 1.0 / 1.1 / 1.5 / 2. The upper MgO was deposited by e-beam evaporation. Diffraction measurements on Mn 2 VAl films deposited at various temperatures revealed that a substrate carrier temperature of at least 600 • C was necessary to obtain good order, but temperatures above 700 • C lead to strong Mn sublimation, which can not be reliably compensated by higher sputtering power (compare with [30]). Therefore all samples discussed in this paper were deposited at a carrier temperature of 700 • C. The protective Mg / MgO bilayer was deposited after cooling the samples to prevent oxidation and interdiffusion.

X-ray diffraction (XRD), reflectometry (XRR), and fluorescence (XRF) were performed in a Philips X'Pert Pro MPD diffractometer with Cu anode, Bragg-Brentano and collimator point focus optics, an open Euler cradle and an Amptek fluorescence detector in a He enclosure.

X-ray magnetic circular dichroism (XMCD) was measured at beamline 6.3.1 of the Advanced Light Source (Berkeley, CA, USA). A magnetic field of ± 1.6 T parallel to the incoming x-ray beam was applied, the sample surfaces were inclined by 30 • with respect to the incoming beam. Element specific magnetic hysteresis loops were taken with a magnetic field of up to ± 2 T. The magnetic field was switched for every energy point to obtain the dichroic signal. Data were taken at 20 K, 150 K, 200 K, and 300 K. All XMCD spectra were taken at least twice, with circular polarization degrees of +60 % and -60 %, respectively. Systematic measurements were performed in the surface sensitive total electron yield mode. Additionally, following an idea by Kallmayer et al. [32], the visible light fluorescence of the MgO substrate was detected by a photo diode behind the sample. Thus, bulk information of the films could be obtained in x-ray transmission.

Electronic structure calculations

Electronic structure calculations of disordered compounds were performed with the Munich SPRKKR package, a spin-polarized relativistic Korringa-Kohn-Rostoker code [START_REF]The Munich SPR-KKR package[END_REF]. The ground state self-consistent potential calculations were performed on 834 k points in the irreducible wedge of the Brillouin zone. The exchange-correlation potential was approximated by the Perdew-Burke-Ernzerhof implementation of the generalized gradient approximation [34], the Fermi energy was determined using Lloyd's formula [35,36]. The angular momentum expansion was taken up to l max = 3. A scalar relativistic representation of the valence states was used in all cases, thus neglecting the spin-orbit coupling. For Mn 2 VAl the atomic spheres approximation was applied and Co 2 VAl was treated with full potential calculations. Half-metallic ground states were obtained for Mn 2 VAl and Co 2 VAl with their respective bulk lattice parameters. To account for disorder, the coherent potential approximation (CPA) was used. In our calculations with the ideally ordered L2 1 structure, Mn 2 VAl has a total moment of 2.01 µ B /f.u., with 1.54 µ B on Mn and -1.03 µ B on V. Co 2 VAl has a total moment of 1.99 µ B /f.u., with 0.87 µ B on Co and 0.28 µ B on V. These values are in good agreement with calculations presented by other authors [37].

Experimental results and discussion

Lattice structure

All MCVA films were found to be highly epitaxial with MCVA [001] MgO [001], rocking curve widths of 0.6 • to 1.5 • , and an MCVA [100] MgO [110] in-plane relation. Laue oscillations observed at the (002) reflections demonstrate the lattice and interface coherence of the films in the two limiting cases of Mn 2 VAl and Co 2 VAl (Fig. 1(a)). For x = 1, however, the oscillations are less pronounced.

Figure 1(b) displays the out-of-plane lattice parameter c as a function of x. According to Vegard's law [38], a linear decrease of the lattice parameter with increasing x can be expected for a simple substitutional model. However, a significant deviation from this law is observed at x = 1. This indicates, as we will see in detail later, a structural and magnetic order-disorder transition. For Mn 2 VAl, c is slightly lower than the bulk value of 5.875 Å [8]; Co 2 VAl has also a slightly reduced c compared to the bulk value of 5.77 Å [28]. This is compatible with a tetragonal distortion caused by the epitaxial matching with the substrate: the lattice is expanded in the plane and shrinks in the out-of-plane direction. For the case of Co 2 TiSn we have recently performed first principles calculations of the change in total energy for this type of lattice distortion.

In this case it is of the order of 25 -50 meV, and is thus easily activated during the film growth [39]. For the compounds presented here, we expect a similar energy range. Takamura's extended order model for Heusler compounds [40] was applied to obtain the order parameters S B2 and S L2 1 from the measured XRD peak intensities. Unlike Webster's model [44], this model takes the dependence of S L2 1 on S B2 into account. The structure factors were obtained from the measured intensities by correcting for the Lorentz-Polarization term and the temperature factor with an effective Debye-Waller factor of B eff = 0.4. S B2 is calculated from the four structure factor ratios of (002) and ( 222) versus (022) and (004), respectively. S L2 1 is calculated as the average of the (111) structure factor versus (022) and (004). The full atomic scattering factors including angular dependence and anomalous corrections were used in the numerical model calculations. As shown in Fig. 1(c), the Mn 2 VAl films are ordered in the L2 1 structure with significant V-Al disorder (S L2 1 ≈ 0.4). With increasing Co content, the L2 1 order disappears in the alloy system; Co 2 VAl does not show any sign of L2 1 ordering.

On the other hand, the degree of B2 order increases slightly with increasing Co content, from S B2 = 0.7 to S B2 = 0.8, i.e., 85 % to 90 % of the Co atoms are on the 8c sites. However, we note here that disorder between Co, Mn, and V can not be identified with this method, because the atomic form factors are too similar.

A Williamson-Hall analysis [45] of the integral peak widths of the (002), (004), and (006) reflections was performed. A Gaussian instrumental peak broadening and a Lorentzian convolution of grain size and strain effects were assumed, i.e., the contributions were separated with

B 2 obs = B 2 inst + B 2 ss (1)
and

B ss • cos θ = kλ D + 4 ε [001] sin θ (2)
with B obs being the observed width, B inst the instrumental width, B ss the size-strain width, the shape factor k = 0.9, the coherence length (grain size) D and the averaged [001] component of the strain tensor ε [001] . The analysis results are displayed in Fig. 1(d) and (e). The measured coherence length matches the film thicknesses quite well within the accuracy of the measuring and fitting procedure. A clear trend of increasing strain can be observed, from 0.18 % to 0.47 %. The lattice mismatch of Co 2 VAl (3.1 %) is about 2.4 times as large as the mismatch of Mn 2 VAl (1.3 %) with MgO. The same factor applies to the strain values, which verifies the high quality of the epitaxy. The lower degree of film coherence, the deviation from Vegard's law and the rather low strain in spite of the large lattice mismatch indicate an increased density of lattice defects in Mn 1 Co 1 VAl. The defects allow for relaxation of the film, which can reduce the microstrain at a loss of coherence. Ziebeck and Webster found that Co 2 VAl crystallizes in the L2 1 phase, but exhibits some preferential V-Al disorder [28]. The samples measured by them were annealed at 800 • C for 24h. The samples by Kanomata et al. were annealed at up to 1200 • C, and still exhibited a complex grain structure consisting of L2 1 and B2 ordered fractions. Deposition at 700 • C may thus be insufficient to promote L2 1 order in Co 2 VAl. However, as stated initially, a higher deposition temperature was not usable because of Mn sublimation.

Magnetic and electronic structure

We begin with a discussion of the XMCD spectra in dependence on x, which are shown in Fig. 2. For x = 0, i.e., for pure Mn 2 VAl, we find an antiparallel alignment of the Mn and V moments, which was verified with element specific hysteresis loops (not shown). This is preserved up to x = 0.5, going along with an antiparallel coupling of Co to Mn. Here, we find the predicted ferrimagnetic order with the Co and V moments pointing opposite to the Mn moments. With further increasing x, all magnetic moments point in the same direction; the alloys become ferromagnets. This transition is closely related to chemical disorder which is indicated by the deviation of the lattice parameter from Vegard's law. Across the stoichiometry series the shape of the spectra changes significantly. Most prominently, the splitting of the V and Mn lines vanishes at x = 0.9 and above. The appearance of this splitting is directly correlated with the appearance of ferrimagnetism. The line shape of the Mn XMCD for x = 1.5 is very similar to the Mn line shape in Co 2 MnAl or Co 2 MnSi [46]. For the ferrimagnetic coupling of Co and Mn, they have to be second nearest neighbors on octahedral positions. Co and Mn on tetrahedral nearest-neighbor positions couple ferromagnetically, as in Co 2 MnGe [44] and the other Co 2 Mn-based Heusler compounds.

To assert that the complex shape of the Mn and V spectra is not a surface effect, we have measured the transmitted x-ray intensity in luminescence detection at room temperature for Mn 2 VAl. The XMCD spectra are almost equal in total electron yield and in transmission (see Figure 3), although in both cases the L3 pre-peak is more pronounced in transmission. However, compared to the total area of the peaks, this deviation is small. The fine structure of the spectra is consequently related to the electronic structure of the films rather than to a surface effect.

Using the sum rule analysis we extracted the spin and orbital magnetic moments order is observed for x ≤ 0.5, ferromagnetic (FM) order is observed for x ≥ 0.9.

from the XMCD spectra [47]. Table 1 summarizes the total magnetic moments obtained from sum rule analysis and provides estimates of the Curie temperatures obtained from temperature dependent XMCD for x = 0, 1, 2 (the spectra are not shown here). Figure 4 displays the element specific total moments in dependence on x. Because of dynamical screening effects of the x-ray field, the sum rules fail for the early 3d transition metals [41]. To compensate the resulting spectral mixing effects, the apparent spin magnetic moments can be multiplied with correction factors as suggested by Dürr et al. and Scherz et al., i.e. 1.5 for Mn [42] and 5 for V [43]. Actually, the applied correction factors depend on the actual electronic structure and can not be simply transferred to different systems. However, we assume that this influence is rather small, so that quantitative results can be obtained.

In Mn 2 VAl we find a lowered Mn moment (1 µ B ) and an enhanced V moment (-1.1 µ B ), resulting in a total magnetization of 0.88 µ B /f.u. No change of the magnetic moments was observed at RT as compared to 20 K, hence the Curie temperature is much higher than RT. The film is not well described by a pure L2 1 order model. As discussed earlier, the film has some disorder between Mn and (V,Al). In this case, Mn atoms reside on sites surrounded by other Mn atoms, which couple antiferromagnetically at short distance. Indeed, by calculating the self-consistent potential in SPRKKR with 20% Mn-Al or Mn-V exchange, we find antiparallel coupling of the antisites, similar to the findings by Picozzi et al. for Co 2 MnSi [21]. For Mn-Al exchange, the Mn(8a) moment is reduced to 1.22 µ B and the Mn on the Al site has -2.48 µ B . The V moment is reduced to -0.83 µ B . This results in a total magnetization of 0.85 µ B /f.u., and the average Mn moment is consequently 0.85 µ B . In the case of Mn-V exchange, the Mn(8a) moment remains at 1.58 µ B and the Mn on the V site has -2.63 µ B . The V moment on the 4b site is -0.87 µ B and +0.84 µ B on the 8a site. In this case the total moment is 1.78 µ B /f.u., with an average Mn moment of 1.16 µ B . Further, the case of Mn-Al exchange is energetically preferred with respect to the Mn-V exchange. Seeing the low total and Mn moments and the high V moment, a preferential Mn-Al exchange in Mn 2 VAl is thus in good agreement with the structural and the magnetic data. Our calculations show that the 20 % Mn-Al disorder and B2 disorder barely influence the half-metallic gap of Mn 2 VAl. For B2 disorder, the total magnetic moment also remains unaffected. In contrast, 20 % Mn-V disorder destroy the gap. This is in contrast to the findings by Luo et al., obtained with a supercell approach in a pseudopotential code. They state that the gap is preserved under 25 % Mn-V disorder [16].

Co 2 VAl has a reduced Co moment (0.69 µ B ) and a V moment of 0.28 µ B , giving a total magnetization of 1.66 µ B /f.u. The film has B2 order, which is expected to reduce the magnetization from the highly ordered L2 1 case. We find magnetic moments of 0.75 µ B for Co and 0.4 µ B for V in a B2 ordered SPRKKR calculation, with a total moment of 1.86 µ B /f.u., in good agreement with our measurements. Some additional disorder involving Co and V could explain the further reduced moments. The Curie temperature is about 210 K (see Table 1), which is significantly lower than the value for bulk samples (310 K [28]). A calculation of the Curie temperature with SPRKKR within the mean field approximation (see [48] for details of the procedure) yields 352 K in the L2 1 case and 165 K in the B2 ordered case. The observed significant reduction of the Curie temperature in the disordered alloy is thus in agreement with theory. The half-metallic gap of Co 2 VAl vanishes in the B2 structure.

At x = 0.5, a nearly complete magnetic compensation with a total moment of only 0.1 µ B /f.u. is observed. Remarkably, at x = 1.5 the total magnetic moment becomes larger than 2 µ B /f.u., caused by the high Mn moment of 1.67 µ B . This is in agreement with the different Mn line shape: in, e.g., Co 2 MnAl, in which Mn has a similar line shape, Mn has a moment of about 3 µ B [44]. Thus, the mechanism mainly responsible for the ferromagnetic coupling of all moments is the preferentially tetrahedral (instead of octahedral) coordination of Mn atoms with Co.

Conclusions

Epitaxial thin films of Mn 2-x Co x VAl have been synthesized on MgO (001) substrates by DC and RF magnetron co-sputtering. It was intended to observe a ferrimagnetic compensation of the magnetization at x = 1. The films have significant chemical disorder, depending on the degree of Mn-Co substitution. Mn 2 VAl was found to be L2 1 ordered, with a preferential Mn-Al disorder and additional V-Al disorder. The Mn-Al disorder reduces the total moment considerably, because the nearest-neighbor Mn atoms couple antiferromagnetically in this configuration. Accordingly, the magnetization of Mn 2 VAl is very sensitive to disorder involving Mn. However, the band structure calculations suggest that only Mn-V disorder has an influence on the half-metallic gap. Because of the disorder, a nearly complete magnetic compensation was observed for Mn 1.5 Co 0.5 VAl. With further Co substitution, the electronic structure changes considerably, and a parallel coupling of Co, Mn, and V was observed. We suppose that Co and Mn become preferentially nearest neighbors, which leads to a parallel coupling of their magnetic moments.

The Co 2 VAl films, being the second extremum of the substitutional series, had B2 order. The band structure calculations with B2 order suggest reduced moments, but the experimentally determined moments are even lower, which indicates additional disorder involving Co. The Curie temperature was significantly reduced, which is in agreement with the trend observed in the mean field calculation. It is in principle possible to obtain a high degree of L2 1 order in bulk Co 2 VAl by appropriate thermal treatment, but our maximum substrate temperature was limited by Mn evaporation. While it may be possible to obtain the correct occupation for the ferrimagnetic compensation in the bulk, it seems not possible to obtain films with a high degree of order.

Figure 1 .

 1 Figure 1. (a): θ-2θ scans of the (002) reflections of Mn 2 VAl (x = 0), Mn 1 Co 1 VAl (x = 1), and Co 2 VAl (x = 2). Clear Laue oscillations are visible in both cases. (b): out-of-plane lattice parameter c as function of x. (c): Order parameters S B2 and S L2 1 as functions of x. (d): Microstrain ε [001] and (e): coherence length D and as functions of x. The dashed line in (e) denotes the film thickness.

Figure 2 .

 2 Figure 2.Experimental XMCD spectra for V, Mn, and Co at 20 K. The corresponding XAS spectra were normalized to a post-edge jump height of 1. The spectra for x = 0.9, 1.1 are similar to x = 1 and are omitted for clarity.

Figure 3 .

 3 Figure 3. Normalized XMCD spectra of Mn and V in electron yield and luminescence detection.

Figure 4 .

 4 Figure 4. Element specific magnetic moments as functions of x. Ferrimagnetic (FiM) order is observed for x ≤ 0.5, ferromagnetic (FM) order is observed for x ≥ 0.9.

Table 1 .

 1 Experimental total magnetic moments at 20 K (given in µ B / f.u.) and estimated Curie temperatures derived from temperature-dependent XMCD.

		m tot T C
	Mn 2 VAl	0.88	RT
	Mn 1.5 Co 0.5 VAl 0.1	-
	Mn 1.0 Co 1.0 VAl 1.09 ≈ 350 K
	Mn 0.5 Co 1.5 VAl 2.29 -
	Co 2 VAl	1.66 ≈ 210 K

Acknowledgements

The authors gratefully acknowledge financial support by the Deutsche Forschungsgemeinschaft (DFG) and the Bundesministerium für Bildung und Forschung (BMBF). They thank for the opportunity to work at BL 6.3.1 of the Advanced Light Source, Berkeley, USA, which is supported by the Director, Office of Science, Office of Basic Energy Sciences, of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231.


A Krugmann
email: krugmann@ikp.tu-darmstadt.de

Z P Li

J Meng

N Pietralla

D Vretenar

Comparison of the confined β-soft Rotor Model and a microscopic collective Hamiltonian based on the relativistic mean field model in 150,152 Nd

Keywords: numbers: 21.60.Ev, 21.10.Re, 21.60.Jz, 27.70.+q

A comparison between the analytical wavefunction of the Confined βsoft Rotor Model and a collective Hamiltonian based on the relativistic mean field model has been performed for the low lying states in the ground state band and βband of the nuclei 150,152 Nd. A remarkable similarity of the two models in energies, intra-and interband B(E2)-values and centrifugal stretching has been observed. In the transitional nucleus 150 Nd the relative stretching is about 0.025/ whereas the rotational nucleus 152 Nd with a much stiffer potential in the quadrupole deformation parameter β stretches with approximately 0.01/ . The centroids of the wave functions almost exactly coincide in the two models for the ground state band and also for the β-band. The Ansatz in the CBS model of an outer potential wall that stays almost constant with increasing number of valence neutrons and an inner wall that shifts to higher deformations has been successfully justified microscopically.

Introduction

The understanding of transitional nuclei has considerably deepened during the last decade. Inspired by the formulation of the E(5) and X(5) solution of the geometrical Bohr Hamiltonian [1,2,3] a great deal of experimental and theoretical work has been spent on the investigation of nuclei at the shape phase transitional points [4,5,6,7]. The remarkable quantitative success of a geometrical nuclear structure model using a square well potential has been demonstrated to microscopically originate from the fact that near the shape phase transitional point quite flat and soft potentials can develop [8,9]. Beyond the transitional point X (5) with an R 4/2 ratio of 2.9 deformed nuclei rapidly evolve into almost rigidly rotating bodies with R 4/2 values close to 3.33. The rigid rotor limit is, however, hardly reached exactly. Deformed nuclei typically exhibit R 4/2 values between 3.2 and 3.32 while very few deformed nuclei have R 4/2 values between 2.9 and 3.2. Inspired by the success of the X(5) model that was based on the phenomenon of intrinsic excitations and centrifugal stretching in a soft potential, the X(5) model has been generalized in terms of the Confined β-Soft (CBS) rotor model [10,11]. Similar to the X(5) model, the CBS rotor model considers a square-well potential, however, with the inner potential boundary shifted away from β=0 (see section 2.2). Alike X (5) the CBS rotor model is analytically solvable in terms of Bessel functions. It has been demonstrated to have a remarkable capability for quantitatively describing the evolution of excitation energies of rotational bands in deformed nuclei [12,13]. It is interesting to study whether and to what extent microscopic calculations justify or contradict the choice of simplistic potentials and resulting nuclear wave functions of the CBS rotor model. Within this article we compare the results for the transitional and deformed neighbouring even-even isotopes 150 Nd and 152 Nd for which microscopic calculations based on the Relativistic Mean Field approach are available [8,9]. Fig. 1 displays the effective potentials as a function of the quadrupole deformation parameter β for the description of 150,152 Nd. The square-well potentials of the CBS rotor model have been obtained from a fit to the R 4/2 ratios and the B(E2; 2 + 1 → 0 + 1 ) values, while the potential curves have been calculated microscopically from Relativistic Mean Field potentials. For a better comparison, the RMF curves are actually projections on the β-axis of a full triaxial calculation in the β -γ-plane. One obvious observation is that the inner potential boundary shifts much more as a function of neutron number than the outer boundary according to both models. The outer boundary seems to be almost insensitive to the transition from spherical to axially prolate nuclei. The purpose of this paper is to study the impact on the evolution of nuclear structure originating in the shift of the inner potential wall and to what extent the use of square well potentials for the description of transitional nuclei is microscopically justified. In the next section we will shortly summarize the basic concepts of the models before we discuss their application to the nuclei 150,152 Nd and the comparison of their results in section III.

Model definition

The Collective Hamiltonian

The collective Hamiltonian

Ĥ = Tvib + Trot + V coll (1)
in a very general form [14] describes the collective motion of a nucleus with the vibrational kinetic energy Tvib = -

2 2 √ wr 1 β 4 ∂ ∂β r w β 4 B γγ ∂ ∂β - ∂ ∂β r w β 3 B βγ ∂ ∂γ + 1 β sin 3γ - ∂ ∂γ r w sin 3γB βγ ∂ ∂β + 1 β ∂ ∂γ r w sin 3γB ββ ∂ ∂γ (2)
and the rotational kinetic energy Trot = 1 2

3 k=1 Ĵ2 k I k . ( 3
)
V coll is the collective potential. For this potential there are different ansatzes, and we'll discuss two of them in this paper.

Ĵk denotes the components of the angular momentum in the body-fixed frame of a nucleus, and the mass parameters B ββ , B βγ , B γγ [15], as well as the moments of inertia I k , depend on the quadrupole deformation variables β and γ:

I k = 4B k β 2 sin 2 (γ -2kπ/3) . (4)
Two additional quantities that appear in the expression for the vibrational energy: r = B 1 B 2 B 3 , and w = B ββ B γγ -B 2 βγ , determine the volume element in the collective space. In the next two chapters some assumptions for the CBS model and the Relativistic Mean Field approach will be made and the collective potentials will be specified in order to solve the Schrödinger Equation.

The Confined β-soft Rotor Model

The CBS rotor model represents an approximate analytical solution to the collective Hamiltonian in equation (1) as it was proposed by Bohr and Mottelson [14] in the quadrupole shape parameters β and γ. Taking the Hamiltonian in equation ( 1) and ( 2) and assuming that B k = B γγ = B ββ = B and B βγ = 0, the Hamiltonian simplifies to

H = - 2 2B 1 β 4 ∂ ∂β β 4 ∂ ∂β + 1 β 2 sin 3γ ∂ ∂γ sin 3γ ∂ ∂γ - 1 4β 2 k Ĵ2 k sin 2 (γ -2 3 πk) + V (β, γ). (5)
Assuming a separable potential V (β, γ) = u(β)+v(γ) the wave functions approximately separate into

Ψ(φ, θ, ψ, β, γ) = ξ L (β) η K (γ) Φ I MK (Ω). (6)
The angular part corresponds to linear combinations of the Wigner functions

Φ I MK (Ω) = 2I + 1 16π 2 (1 + δ K0 ) D I * MK (Ω) + (-1) I D I * M -K (Ω) (7)
and η K denotes the appropriate wave function in γ. For sufficiently axially symmetric prolate nuclei one might consider a steep harmonic oscillator in γ [2]. ξ L (β) describes the part of the wave function depending on the deformation variable β. The approximate separation of variables [2] leads to the differential equation

- 2 2B 1 β 4 ∂ ∂β β 4 ∂ ∂β - 1 3β 2 L(L + 1) + u(β) ξ L (β) = E ξ L (β) ( 8 )
for ξ L (β). It can be considered as the 'radial' equation in the space of quadrupole deformation parameters. It contains the angular momentum dependence through the centrifugal term. The CBS rotor model assumes for prolate axially symmetric nuclei an infinite square well potential u(β), with boundaries at β M > β m ≥ 0 . For this potential the differential equation ( 8) is analytically solvable. The ratio r β = β m /β M parameterizes the width of this potential, that is the stiffness of the nucleus in the β degree of freedom. For r β = 0 the X(5) limit is obtained with large fluctuations in β. The Rigid Rotor Limit without fluctuations in β corresponds to r β → 1. The quantization condition of the CBS rotor model is

Q r β ν(L) (z) = J ν(L) (z) Y ν(L) (r β z) -J ν(L) (r β z) Y ν(L) (z) = 0 (9)
with J ν and Y ν being Bessel functions of first and second kind of irrational order ν = [L(L + 1) -K 2 ] /3 + 9/4. For a given structural parameter r β and any spin value L the sth zero of equation ( 9) is denoted by z r β L,s . The full solution of equation( 8) with the aforementioned choice of the CBS square-well potential is then given as

ξ L,s (β) = c L,s β -3 2 J ν z r β L,s β β M + J ν r β z r β L,s Y ν r β z r β L,s Y ν z r β L,s β β M . ( 10
)
The normalization constant c L,s of the wave function ( 10) is given by the normalization condition

1 = β M βm β 4 [ξ L,s (β)] 2 dβ . ( 11
)
The eigenvalues of equation ( 8) are obtained as

E L,s = 2 2Bβ 2 M z r β L,s 2 . ( 12
)
The CBS rotor model well describes the evolution of low-energy 0 + bands [10], ground bands of strongly deformed nuclei [12], and the dependence of relative moments of inertia as a function of spin in deformed transitional nuclei [13]. The CBS rotor model has also been successfully used for studying relative as well as absolute E0 transitions in the region of the X(5) nuclei up to the rigid rotor limit [16].

Microscopic Relativistic Mean Field Approach

In this work a microscopic collective Hamiltonian based on the relativistic mean field model is used [8,9]. The kinetic energy part of the Hamiltonian is in the general form as already seen in equations ( 1) -( 3). The seven functions, i.e. the three moments of inertia I k , the three mass parameters B ββ , B βγ , B γγ , and the collective potential V coll , are determined by the choice of a particular microscopic nuclear energy-density functional or effective interaction. In the particle-hole channel we use the relativistic functional PC-F1 (point-coupling Lagrangian) [17], as we did in our previous studies of shape transitions in the Nd-chain [9]. Also a density-independent δ force in the particleparticle channel treated by the BCS approximation was used. The moments of inertia are calculated microscopically from the Inglis-Belyaev formula:

I k = i,j (u i v j -v i u j ) 2 E i + E j i| Ĵk |j | 2 k = 1, 2, 3, (13)
where the summation runs over the proton and neutron quasiparticle states, and k denotes the axis of rotation. The quasiparticle energies E i , occupation probabilities v i , and single-nucleon wave functions ψ i are determined by solutions of the constrained RMF+BCS equations. The mass parameters associated with the two quadrupole collective coordinates q 0 = Q20 and q 2 = Q22 are also calculated in the cranking approximation

B µν (q 0 , q 2 ) = 2 2 M -1 (1) M (3) M -1 (1) µν , (14)
with

M (n),µν (q 0 , q 2 ) = i,j i| Q2µ |j j| Q2ν |i (E i + E j ) n (u i v j + v i u j ) 2 . ( 15
)
In contrast to the CBS Model, the potential V coll in the collective Hamiltonian Equation ( 1) is obtained by subtracting the zero-point energy corrections from the total energy that corresponds to the solution of constrained RMF+BCS equations, at each point on the triaxial deformation plane. The Hamiltonian (1) describes quadrupole vibrations, rotations, and the coupling of these collective modes. The corresponding eigenvalue problem is solved using an expansion of eigenfunctions in terms of a complete set of basis functions that depend on the deformation variables β and γ, and the Euler angles φ, θ and ψ. The diagonalization of the Hamiltonian yields the excitation energies and collective wave functions

Ψ IM α (β, γ, Ω) = K∈∆I ψ I αK (β, γ)Φ I MK (Ω), (16)
that are used to calculate observables. The angular part corresponds to linear combinations of the Wigner functions [see equation (7)] and the summation in equation ( 16) is over the allowed set of K values: ∆I = 0, 2, . . . , I for I mod 2 = 0 2, 4, . . . , I -1 for

I mod 2 = 1 . ( 17
)
For a given collective state in equation ( 16), the probability density distribution in the β -γ-plane is defined by

ρ Iα (β, γ) = K∈∆I |ψ I αK (β, γ)| 2 β 3 | sin 3γ| . (18)
The normalization reads

∞ 0 βdβ 2π 0 dγ ρ Iα (β, γ) = 1 . ( 19
)
If we integrate the wave function ( 16) along γ, we obtain the projection [see equation (12)] has been fitted to the excitation energies of the 2 + 1 states and takes the values of 37.9 keV for 150 Nd and 30.7 keV for 152 Nd. The effective E2 charges of the first order (∆k = 0)-part of the transition operator T (E2) = 3 4π ZeR 2 β M (β/β M ) D 2 µ0 [10] have been scaled to the experimental B(E2; 2 + 1 → 0 + 1 ) values. For the calculations we used β M = 0.451 for 150 Nd and β M = 0.471 for 152 Nd. For the RMF model we have multiplied the Inglis-Belyaev moments of inertia in equation ( 13) with a common factor (1 + α) determined in such a way that the calculated energy of the 2 + 1 state coincides with the experimental value. This is because the well-known fact that the Inglis-Belyaev formula predicts effective moments of inertia that are considerably smaller than empirical values. For 150 Nd we used α = 0.40 and for 152 Nd α = 0.57. The transition rates are calculated in the full configuration space using bare charges [8,9]. The s=1 ground state band is very well reproduced by the two models. In the microscopic RMF approach the energies differ only by a few percent compared to the experimental energies. In the CBS rotor model the deviation is even less than 0.3 % [12]. In the s=2 band, which is the collective β-band for this nucleus, the rotational structure is very well reproduced in the two models and the deviations in energy primarily result from a mismatch of the rotational moment of inertia by up to 25 %. The B(E2) intraand interband transition strengths show a satisfactory agreement in both models. Note, that both, energies and B(E2) values for the excited k=0 band in the CBS model, represent model predictions without any further parameter adjustment.

ρ Iα (β) = K∈∆I 2π 0 |ψ I αK (β, γ)| 2 β 4 | sin 3γ|dγ . ( 20
Figure 3 shows the same spectra for the nucleus 152 Nd [19]. In this case, only few data is available. Here, the two models show again a nice agreement in the excitation energies and B(E2) values for the ground state band, as well as for the β-band.

One underlying mechanism at work in the CBS model is the centrifugal stretching of the nucleus which can be classically understood as an increase in the moment of inertia (MoI) as a function of the rotational angular momentum. In the upper graph in figure 4 the evolution of the MoI with spin, here plotted as the relative dynamical moment of inertia (see also [12])

θ(J) θ(2) = J(J + 1) 6
E(J = 2) E(J) (22)
is compared to the CBS and microscopic RMF predictions. Both models well agree with the data on the ground state bands of 150,152 Nd. The stretching for the two models has been calculated for the ground state rotational bands in the nuclei 150,152 Nd. Defining a dimensionless stretching parameter

S(J) = β J i -β 0 + 1 β 0 + 1 , (23)
one can quantitatively see in the lower graph of figure 4, that the two models show a very good agreement on the predicted centrifugal stretching. In this figure the centrifugal stretching goes more or less linearly with the spin, and an increasing potential stiffness means a decrease in centrifugal stretching. It is remarkable how well the experimental MoI graph in the upper part of the figure correlates with the theoretically calculated stretching parameters in the lower part. The slope for the centrifugal stretching in 150 Nd is for both models approximately 0.025/ . The much stiffer potential in β for 152 Nd creates only a slope of approximately 0.01/ . Looking at the evolution as a function of valence nucleons, the centrifugal stretching decreases approximately by a factor of 2.5, going from the X(5) nucleus 150 Nd to the well deformed rotor 152 Nd. The agreement of both models in figure 4 appears to be worse for 150 Nd than for 152 Nd. However, the disagreement between the models simply scales with the size of the effect.

In 150 Nd the dynamical MoI for the 10 + 1 state is 49 % larger than the one for the 2 + 1 state. The RMF model predicts an increase of 40 %, only, i.e., underestimates this increase by about one fifth of its size. In 152 Nd the dynamical MoI for the 10 + 1 state is 11 % larger than the one for the 2 + 1 state. Here, the RMF model predicts an increase of 14 %, i.e., again, at slight variance to the data.

Comparison of the collective wave functions

Figure 5 shows wave functions of the ground state bands in 150 Nd and 152 Nd as a function of β. The vertical bold lines in the first and third subfigure represent the potential boundaries of the square-well CBS potential. The vertical lines inside the wave functions represent the centroid of each wave function. Comparing these nuclides, one can easily see the similar amount of centrifugal stretching in the two models. The fact, that the potential stiffens with increasing neutron number can also be seen by comparing the upper and the lower two subfigures. The wavefunctions of the rotational nucleus 152 Nd are more compressed and shifted towards a larger average deformation. Figure 5 and 6 indicate that just a very small fraction of the wave functions computed microscopically in the relativistic mean field model turns out to lie outside of the CBS square well potential walls. In fact, the wave functions of the two models are very similar. They almost exactly show the same amplitude as well as the same centroid for the ground state band in 150 Nd and 152 Nd, respectively. Figure 6 shows the wave functions of the β-band heads. They have one node in the deformation coordinate β. One can clearly see the large overlap of the wave functions between the two models. The centroids of the two peaks in the wave functions of the βband head almost exactly coincide in the case of 152 Nd. However, the shapes of the wave functions in the β-band differ slightly for both models. Here, the CBS model shows two peaks in the β-band that have almost the same amplitude whereas the microscopic RMF model produces a smaller peak for the waist of the wave function at larger deformation than for the waist at smaller deformation. For the case of 150 Nd the difference between the prediction of the models for the 0 + 2 wave function is slightly larger than for the more rotational nucleus 152 Nd. The top of figure 6 shows, that the amplitudes of the two waists of the wave functions are very similar, and the centroids of the second waist are close, but here, the first maxima of the wave functions occur at different deformations. For this example the RMF potential is more repulsive for small deformations than the CBS potential, see figure 1, and hence, the RMF wave function is reduced for lower β in comparison to the CBS model.

Summary

In conclusion, a comparison between the analytical wave functions of the CBS rotor model and a microscopic Hamiltonian based on the relativistic mean field model has been performed for the low lying states in the ground state band and β-band of the nuclei 150,152 Nd. The energy levels of these states and the relative evolution of their MoIs agree satisfactorily with data. A remarkable similarity of the two models in energies, intra-and interband B(E2)- values and centrifugal stretching has been observed. In the transitional nucleus 150 Nd the stretching is about 0.025/ whereas the rotational nucleus 152 Nd with a much stiffer potential in β stretches with approximately 0.01/ . The centroids of the wave functions almost exactly coincide in the two models for the ground state band and also for the β-band. The wave functions, especially for the ground state band in 150,152 Nd, are very similar in the two models, although the two models base on completely different ansatzes.

In particular, the Ansatz in the CBS model of an outer potential wall that stays almost constant with a varying number of valence neutrons and an inner wall that shifts to higher deformations has been microscopically justified for the given example of the isotopes 150,152 Nd near the shape phase transitional point. In both compared models, the position of the outer potential wall seems to be almost independent of a transition from a spherical to an axially symmetric shape of the nucleus while the change in structure as a function of nucleon number is dominated by the change of the potential at small deformation.

Figure 1 .

 1 Figure 1. Relativistic Mean field Potentials compared to the CBS square well potentials that make an analytical solution of the Bohr Hamiltonian possible.

0 ρ 3 .Figure 2 2 M

 0322 Figure2shows the low lying excitation spectrum of the nucleus 150 Nd[18] in comparison to the two models. For the CBS rotor model we have fitted the structural parameter r β to the 4 + 1 /2 + 1 -energy ratio R 4/2 . For 150 Nd we used r β = 0.078 and for 152 Nd r β = 0.35. The energy-scale parameter 2 2Bβ 2 M

Figure 2 .

 2 Figure 2. Low lying energy levels in 150 Nd for the relativistic mean field Hamiltonian (left), the CBS model (middle) and experimental values (right). Energies and B(E2) values for Exp. and RMF were taken from reference [9].

Figure 3 .

 3 Figure 3. Low lying energy levels in 152 Nd for the relativistic mean field Hamiltonian (left), the CBS model (middle) and experimental values (right). Energies and B(E2)values for Exp. were taken from reference [19].

Figure 4 .

 4 Figure 4. Upper graph: Experimental Moments of Inertia and theoretical predictions for the ground state band in the nuclei 150,152 Nd. lower graph: Comparison of the centrifugal stretching parameter S(J) in 150,152 Nd for the two models as a function of spin.

1 Figure 5 .

 15 Figure 5. Wave functions of the groundstate band in 150,152 Nd for the CBS rotor model and the relativistic mean field model. The black borders show the potential walls in case of the CBS potential.

2 Figure 6 .

 26 Figure 6. Wave functions of the band heads of the β-band in 150,152 Nd. The black borders show the potential walls in case of the CBS potential.

Acknowledgments

This work is supported partly by the NSFC under Grant Nos. 10775004 and 10975008, the Major State 973 Program 2007CB815000, as well as by the Southwest University Initial Research Foundation Grant to Doctor (No. SWU110039). Support by the DFG under Grant No. SFB 634 and by the Helmholtz International Center for FAIR is gratefully acknowledged. This work was also supported in part by MZOS -project 1191005-1010.


Wael Sabbah
email: w.sabbah@ucl.ac.uk

Georgios ; Tsakos

Tarani Chandola

Tim Newton

Ichiro Kawachi

Aubrey Sheiham

Michael G Marmot

Richard G Watt

The relationship between

Keywords: Periodontal disease, social network, social support, older adults Periodontal disease, social network, social support, older adults

come

Introduction

Lower levels of social network, support and isolation are related to various aspects of physical and mental disease, and to rates of mortality [START_REF] Berkman | Social epidemiology[END_REF][START_REF] Stansfeld | Social support and social cohesion[END_REF]). There is an inverse relationship between higher levels of social network and support on the one hand and morbidity, coronary heart disease and cause specific mortality rates, on the other hand [START_REF] Ruberman | Psychosocial influences on mortality after myocardial infarction[END_REF][START_REF] Orth-Gomer | Social network interaction and mortality[END_REF][START_REF] Berkman | Emotional support and survival after myocardial infarction[END_REF][START_REF] Kawachi | A prospective study of social networks in relation to total mortality and cardiovascular disease in men in the USA[END_REF][START_REF] Eng | Social ties and change in social ties in relation to subsequent total and cause-specific mortality and coronary heart disease incidence in men[END_REF][START_REF] Stansfeld | Social support and social cohesion[END_REF][START_REF] Hoppmann | Spousal interrelations in old age--a mini-review[END_REF][START_REF] Mead | Effects of befriending on depressive symptoms and distress: systematic review and meta-analysis[END_REF]. Several theories have been postulated about the mechanism linking social network and support to health and mortality [START_REF] Berkman | The relationship of social networks and social support to morbidity and mortality[END_REF][START_REF] Stansfeld | Social support and social cohesion[END_REF]. They include direct financial and physical support to engage in health enhancing behaviours and to access care, gaining information from the social network pertaining to innovative health promoting behaviours and health hazards. Furthermore, social and emotional support could help mitigate the consequences of stressful events and coping with diseases, risk factors, and the negative effect of social isolation [START_REF] Berkman | The relationship of social networks and social support to morbidity and mortality[END_REF][START_REF] Kawachi | A prospective study of social networks in relation to total mortality and cardiovascular disease in men in the USA[END_REF] through alterations in neuroendocrine and immunological control systems [START_REF] Brunner | Social organization, stress and health[END_REF]. As there are common pathways affecting general and oral health, [START_REF] Sheiham | Evaluation of social and psychological factors in periodontal disease[END_REF][START_REF] Sabbah | Effects of allostatic load on the social gradient in ischaemic heart disease and periodontal disease: evidence from the third national health and nutrition examination survey[END_REF][START_REF] Borrell | Social disparities in periodontitis among US adults: the effect of allostatic load[END_REF], some dental studies have examined the relationship between social network and support and other societal characteristics with oral health in children and adolescents [START_REF] Hanson | Social network, social support and dental status in elderly Swedish men[END_REF][START_REF] Pattussi | Clinical, social and psychosocial factors associated with self-rated oral health in Brazilian adolescents[END_REF]2008;[START_REF] Aida | Contributions of social context to inequality in dental caries: a multilevel analysis of Japanese 3-year-old children[END_REF]2009) and in adults [START_REF] Mcgrath | Influences of social support on the oral health of older people in Britain[END_REF][START_REF] Merchant | A prospective study of social support, anger expression and risk of periodontitis in men[END_REF]Monteira da Silva 1996). One study examined the relationship between social networks and periodontitis among male health professionals in the USA and found that individuals having friends and who participated in social activities were less likely to have peridontitis [START_REF] Merchant | A prospective study of social support, anger expression and risk of periodontitis in men[END_REF].

Another study reported an association between social isolation and periodontitis (Monteiro da Silva 1996). However, these two studies on the relationship between periodontitis and social network and support were either conducted in a specific occupational group or on a relatively small sample.

While social support and networks were suggested to mediate the socioeconomic inequalities in general health [START_REF] Stansfeld | Social support and social cohesion[END_REF], no study has examined this possible role of social support and networks in relation to oral health. Therefore, we set out to examine the relationship between social network and support and periodontal disease in a nationally representative sample of American older adults, and to assess whether social network and support mediated the known socioeconomic inequality in periodontal diseases [START_REF] Stansfeld | Social support and social cohesion[END_REF].

We postulated that higher levels of social network and support were inversely related to periodontal disease among older American adults. Considering the different mechanisms through which social network and support influences health [START_REF] Berkman | The relationship of social networks and social support to morbidity and mortality[END_REF][START_REF] Kawachi | A prospective study of social networks in relation to total mortality and cardiovascular disease in men in the USA[END_REF], our second hypothesis was that social network and support mediates the relationship between socioeconomic position and periodontal disease in older adults. The objectives of the study were to test: (1) the relationship between three markers of social network and support [START_REF] Cohen | Social support measurement and intervention: a guide for health and social scientists[END_REF], namely number of close friends, need for emotional support and marital status, with periodontal disease; (2) whether social network and support mediated the socioeconomic disparities in periodontal disease, in a national representative sample of Americans aged 60 years and older. , 2005).. The survey collected data on individual characteristics such as gender, age, race/ ethnicity, education, income and marital status. The interviewer also asked questions about selected behaviours, including smoking and dental visits. Furthermore, the survey included questions on social network and support providing data on emotional, material, and network (the number of members in a network) support (CDC, 2005). The questions were selected from the Yale Health and Aging Study and the Social Network Index -Alameda County Study [START_REF] Ruberman | Psychosocial influences on mortality after myocardial infarction[END_REF].

The MEC examination included a detailed oral examination [START_REF] Dye | Trends in oral health status: United States, 1988-1994 and 1999-2004[END_REF]. Participation in the oral health component was limited to age groups considered most critical for monitoring oral health (13 years and older for periodontal examination). Information was also obtained from the home interview covering assessments on perceived oral health status (CDC, 2005). The NHANES periodontal examination protocol was based upon the (NIDCR) criteria with minor modifications [START_REF] Dye | Trends in oral health status: United States, 1988-1994 and 1999-2004[END_REF].

Periodontal disease variables:

Periodontal assessments in NHANES 2001-2004 were made at three facial sites (mesio-, disto-, and mid-facial) on each fully erupted permanent tooth, except third molars, in two randomly selected quadrants (one maxillary and one mandibular). Detailed information on the NHANES dental examinations for the survey periods are published elsewhere [START_REF] Drury | An overview of the oral health component of the 1988-91 National Health and Nutrition Examination Survey[END_REF]Dye et al 2007a;b). Periodontal data were used from the surveys 2001-2002 and 2003-2004 for consistency in the number of sites diagnosed for periodontal disease. Two variables were created: (1) extent of loss of periodontal attachment of 3mm+ (expressed as the percentage of affected sites), and (2) moderate periodontitis was defined by the presence of 2 or more inter-proximal sites with loss of attachment of 4mm or more (not on same tooth), or 2 or more inter-proximal sites with pocket depth 5mm or more (not on same tooth) [START_REF] Page | Case definitions for use in population-based surveillance of periodontitis[END_REF].

Social network and support variables:

Three variables indicating social network and support were used. Whether the participants needed more emotional help during the past year was used as a measure of social support. Number of close friends, and marital status were used as indicators of social networks. These three markers of social network and support have been repeatedly used in medical [START_REF] Manzoli | Marital status and mortality in the elderly: a systematic review and meta-analysis[END_REF][START_REF] Hoppmann | Spousal interrelations in old age--a mini-review[END_REF][START_REF] Mead | Effects of befriending on depressive symptoms and distress: systematic review and meta-analysis[END_REF], and dental studies [START_REF] Marcenes | The relationship between marital quality and oral health status[END_REF][START_REF] Merchant | A prospective study of social support, anger expression and risk of periodontitis in men[END_REF][START_REF] Aida | The different effects of vertical social capital and horizontal social capital on dental status: A multilevel analysis[END_REF]. The need for emotional support was a dichotomous variable (yes, no), the number of close friends was categorised into tertiles.

Marital status was grouped to indicate whether the participants were (1) married or live with a partner, (2) widowed, (3) divorced, separated or single.

Behavioural variables

These included smoking and dental visits. Smoking indicates whether the person is a current smoker, a former smoker who smoked at least 100 cigarettes, or never smoked. Dental visit indicates whether the participant visited a dentist at least once within the past 2 years.

Socio-demographic variables:

These included age (continuous variable), gender, and race/ ethnicity (White, African Americans, Mexican Americans, other Hispanic and other ethnicities). Income was indicated by family poverty-income ratio, which is the ratio between house hold income and poverty threshold, and is used to account for inflation throughout the years of the survey. Poverty income ratio was categorised into quartiles. Education reflects whether the participants had less than high school diploma, high school, or more than high school.

Statistical analysis

First we assessed the distribution of all explanatory variables by periodontal status (moderate periodontitis). A series of regression models were conducted for both periodontal disease variables alternatively adjusting for markers of social network and support. The first model was adjusted for gender, age and race/ ethnicity, the second model additionally adjusted for education, income, number of teeth and diabetes, the third model additionally adjusted for dental visits and smoking. The sequence of adjustment is based on the hypothesis that social network and support influence health through an impact on health-related behaviours [START_REF] Berkman | The relationship of social networks and social support to morbidity and mortality[END_REF]. This method was used in medical research to account for direct and indirect effect of explanatory factors [START_REF] Van Oort | Material, psychosocial, and behavioural factors in the explanation of educational inequalities in mortality in the Netherlands[END_REF]. Logistic regression was employed in the analysis pertaining to moderate periodontitis, and negative binomial regression in the analysis pertaining to extent of loss of periodontal attachment.

Other studies have indicated that men benefit more than women from social network and support [START_REF] Berkman | The relationship of social networks and social support to morbidity and mortality[END_REF][START_REF] Berkman | Social epidemiology[END_REF], hence we tested for interaction between F o r P e e r R e v i e w 10 social support and networks variables and gender. We also tested interaction between social networks and each of age, smoking and dental visits. To demonstrate that the relationship between social support and networks with periodontal disease operates via a behavioural pathway, we also examined the association between the three markers of social support and networks with each of dental visits and smoking, adjusting for sex and age.

Regression models were conducted to measure the variation in income and education disparities in periodontal disease and the impact of adjusting for social support and network on the socioeconomic disparities in periodontal disease. The first model adjusted for age, gender, race/ ethnicity, number of teeth, diabetes, smoking status and dental visits, in addition to income and education. The second model additionally adjusted for marital status, number of friends and needing more emotional support, in order to assess whether the relationship between income and education and periodontal disease is attenuated after adjustment for social networks and support. A four year weight variable based on the 2-years mobile examination weight was created and used to adjust for the complexity of the survey. Stata survey command was used throughout the analysis.

Results:

The analysis was conducted for 1632 older adults aged 60 years and over who had complete social network and support and periodontal disease data in NHANES 2001NHANES -2004 1).

The mean age of widowed individuals (74.8 years) was higher than that of married and cohabiting (68.9 years) and divorced and single (67.5). On the other hand, more women were widowed (37.5%) than men (10.8%). The mean age across tertiles of number of close friends was 69.7, 69.8 and 70.5 years in the highest, middle and lowest friend group.

There was a relationship between the three markers of social support and networks, and smoking and dental visits. Smoking was highest among divorced (22%) than widowed (8%) and married (7%).

Similarly, 13% of participants with least number of friends were smokers, compared to 8% among participants with more friends. Individuals who needed emotional support had higher prevalence of smoking (10%) than those who did not need emotional support (9%). Married participants reported more visits to a dentist (82%) than widowed (73%) and divorced (72%). Individuals with highest number of friends reported more dental visits (83%) than those in the middle (81%) and lowest (70%) friend groups.

Participants who did not need emotional support reported more dental visits (80%) than those who needed emotional support (71%).

Widowed and divorced participants were more likely to be a current or former smoker compared to married participants, with odds ratios of 1.58 (95%CI 1.16, 2.14) and 1.79 Table 2 shows the association between the three markers of social network and support (marital status, needing more emotional support and number of friends) with extent of loss of periodontal attachment of 3mm or more and case definition of moderate periodontitis [START_REF] Page | Case definitions for use in population-based surveillance of periodontitis[END_REF]. Widowed and divorced had higher levels of loss of periodontal attachment than married persons, with rate ratios 1.60 (CI 1.27, 2.01) and 1.33 (CI 1.09, 1.63), respectively. After adjusting for income, education, smoking and dental visits, widowed still had significantly higher levels of loss of periodontal attachment (Table 2). Widowed individuals were significantly more likely to have moderate periodontitis than married individuals (OR 1.65, CI 1.09, 2.05). However, this relationship lost significance after adjusting for income, education, smoking and dental visits. Participants with the least number of friends had significantly higher levels of loss of periodontal attachment than those with the highest number of friends. This relationship remained significant even after adjusting for socioeconomic and behavioural factors (rate ratio 1.22, CI 1.03, 1.45) ( adjusting for behaviours. Needing more emotional support did not appear to have an association with periodontitis or the extent of loss of periodontal attachment (Table 2).

(
While there was clear income gradients in moderate periodontitis, education did not seem to have a great impact on moderate periodontitis for this age group (Table 3). Overall, social network and support had a very modest impact on the association between income and education, and moderate periodontitis. Individuals in the lowest income quartile had odds ratios of 2.17 (CI 1.26, 3.73) for having moderate periodontitis. After adjusting for social network and support, this odds ratio hardly changed (Table 3).

On the other hand, participants in the lowest education group had higher levels of loss of periodontal attachment (rate ratio 1.42, CI 1.15, 1.76). After adjusting for social network and support the relationship was slightly attenuated and remained significant (Table 3). There was no interaction between social network variables with neither of age, gender, smoking nor dental visits.

Discussion

Marital status and numbers of friends were significantly associated with two indicators of periodontal disease measuring case definition and severity of the disease among older Americans aged 60 years and over. Considering the postulated mechanism for the relationship between social network, support and oral health through health behaviours [START_REF] Berkman | The relationship of social networks and social support to morbidity and mortality[END_REF], the aforementioned relationships lost significance after adjustment for behaviours, with the exception of the relationships between being widowed and having lowest number of friends with extent of loss of periodontal attachments. Furthermore, social network and support variables did not seem to have an impact on socioeconomic disparities in periodontal disease. In the current study we used number of friends, needing more emotional support and marital status to indicate social network and support [START_REF] Cohen | Social support measurement and intervention: a guide for health and social scientists[END_REF]. Numbers of friends and the need for emotional support have both been frequently used as markers of social network and support [START_REF] Ruberman | Psychosocial influences on mortality after myocardial infarction[END_REF][START_REF] Orth-Gomer | Social network interaction and mortality[END_REF][START_REF] Berkman | Emotional support and survival after myocardial infarction[END_REF][START_REF] Kawachi | A prospective study of social networks in relation to total mortality and cardiovascular disease in men in the USA[END_REF][START_REF] Eng | Social ties and change in social ties in relation to subsequent total and cause-specific mortality and coronary heart disease incidence in men[END_REF][START_REF] Stansfeld | Social support and social cohesion[END_REF][START_REF] Hoppmann | Spousal interrelations in old age--a mini-review[END_REF][START_REF] Mead | Effects of befriending on depressive symptoms and distress: systematic review and meta-analysis[END_REF][START_REF] Mead | Effects of befriending on depressive symptoms and distress: systematic review and meta-analysis[END_REF]. Furthermore, two dental studies found that having a greater number of friends was inversely related to periodontal disease (Monteiro da Silva 1996; [START_REF] Merchant | A prospective study of social support, anger expression and risk of periodontitis in men[END_REF]. The current study supports these findings by demonstrating them in a nationally representative sample of American older adults. On the other hand, the need for emotional support was not associated with periodontal disease in this study.

Marital status was an important determinant of health and mortality, especially among older adults [START_REF] Manzoli | Marital status and mortality in the elderly: a systematic review and meta-analysis[END_REF][START_REF] Hoppmann | Spousal interrelations in old age--a mini-review[END_REF]. Additionally, [START_REF] Marcenes | The relationship between marital quality and oral health status[END_REF] demonstrated the importance of the quality of marriage for oral health. In the current study, marital status appeared to be the most important social networks factor in relation to periodontal diseases among older adults with being widowed maintaining its significant relationship with the extent of loss of periodontal attachment even after adjusting for smoking and dental visits. It is worth noting that while widowed persons were generally older than the rest of the sample, which would imply that the observed relationship with periodontal disease is due to the accumulative effect of age. However, there was no interaction between marital status and age. Furthermore, the majority of widowed were women, who generally have lower levels of periodontal disease than men. This is a cross-sectional study which cannot provide evidence on causal inferences. Being widowed had a stronger relationship with periodontal disease than being divorced or single.

As the death of a spouse is very likely to be a stressful life event, this finding is consistent with psychological pathways affecting periodontal disease by reducing host resistance and the ability to cope with stressors [START_REF] Sheiham | Evaluation of social and psychological factors in periodontal disease[END_REF]. Being divorced may also be a stressful life event, though in this age group it is reasonable to speculate that becoming widowed was probably a relatively more recent life event than being divorced, and this may partly contribute to the stronger association with current periodontal status. On the other hand, periodontitis is a progressive, cumulative disease; hence the importance of more recent stressful events is questionable. The relatively stronger relationship between being widowed and periodontal disease could also be attributed to the older age of widowed participants, potentially indicating the possibility of residual confounding by age; however the analysis adjusted for the effect of age group.

In this study we also examined income and education gradients in periodontal disease and the impact of controlling for social network and support on the aforementioned relationship.

There were income and education gradients in periodontal disease. However social network and support appeared to have no impact on the social gradients in periodontal disease. [START_REF] Stansfeld | Social support and social cohesion[END_REF] found that social network and support had a limited effect on the social gradients in general health. Other studies have indicated that men benefit more than women from social network and support [START_REF] Berkman | The relationship of social networks and social support to morbidity and mortality[END_REF][START_REF] Berkman | Social epidemiology[END_REF], in this study there was no interaction between gender and social network and support.

The effect of social network and support on health was postulated to occur through four pathways [START_REF] Berkman | The relationship of social networks and social support to morbidity and mortality[END_REF][START_REF] Kawachi | A prospective study of social networks in relation to total mortality and cardiovascular disease in men in the USA[END_REF]: first, an instrumental support, for example health promoting habits and hazards to health from their social network; third, emotional support which help individuals to cope with the disease and with health hazards; fourth, through an influence on host resistance. It is highly likely that similar pathways affect periodontal disease in older adults [START_REF] Sheiham | Evaluation of social and psychological factors in periodontal disease[END_REF]. For example, social network and support could influence oral health through a change in behaviours [START_REF] Mcgrath | Influences of social support on the oral health of older people in Britain[END_REF], or by buffering the negative effect of stressful events on periodontal disease [START_REF] Sheiham | Evaluation of social and psychological factors in periodontal disease[END_REF]. These pathways linking social network and support and oral health could not be tested in this study due to data limitations.

This study demonstrated that social networks, indicated by number of friends and marital status, were significantly related to severity of loss of attachment in a nationally representative sample of older Americans aged 60 years and over. The importance of certain aspects of social networks, namely being widowed, and having few friends were further demonstrated as such individuals remained more likely to have greater levels of loss of periodontal attachment even after adjusting for behavioural determinants of periodontitis such as smoking and dental visits. Social network and support appeared to have no impact on the social gradients in periodontal disease in this age group.

The current study has the advantages of using a nationally representative sample of American older adults and assessing the impact of social network and support on the social gradients in periodontitis. However, considering the cross-sectional nature of the study, no causal association could be concluded from this study. Additionally, the dataset lacked a number of important behavioural factors, such as tooth-brushing, which could have influenced the results. Furthermore, the variable on the need for emotional support is a crude measure of [START_REF] Berkman | Social epidemiology[END_REF]. In this study, it is possible that even individuals who reported the need for emotional support still received some support.

  Data were on participants aged 60 years and over who participated in the National Health and Nutrition Examination Survey (NHANES) 2001 to 2004. NHANES 2001-2004 collected information on a nationally representative sample in each year using a stratified, multistage design to obtain a representative probability sample of the civilian noninstutionalized population of the United States (CDC

  second, informational support where individuals acquire knowledge about

  it indicates individuals' perception of the need for emotional support. Others have used a question asking whether an individual received any emotional support(Berkman

  . There was no significant difference in periodontal disease between the 1632 individuals included in the analysis and the 134 individuals excluded for having incomplete data in this age group. The mean age was 69.7 years. The prevalence of moderate periodontitis was 23.1% and 17.1% 11 among participants aged 60 to 70 years and 71 and older, respectively. Married and cohabiting individuals had lower prevalence of moderate periodontitis (18.6%) than widowed (23.7%), and divorced (19.1%). Similarly, individuals with the highest number of friends had the lowest prevalence of periodontitis (Table

	F o r
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  .38, 0.65), and 0.62 (95%CI 0.49, 0.78) respectively. On the other hand, participants in the middle and lowest number of friends groups had insignificant odds ratios of 1.07 (95%CI 0.86, 1.34) and 1.12 (95%CI 0.92, 1.39) respectively for being current or former smokers in a regression model adjusting for sex and age. Participants with the lowest number of friends were significantly less likely to visit a dentist than those with highest number of friends with odds ratios of 0.52 (95%CI 0.42, 0.64). The odds ratios for the associations between need for support and being a current or former smoker, and dental visits

	(95%CI 0were not significant with odds ratios of 0.93 (95%CI 0.74, 1.16) and 0.82 (95%CI 0.63, 1.05), F r o respectively.
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	12

95%CI 1.41, 2.27)

, respectively in a regression model adjusting for sex and age. Widowed and divorced individuals were also less likely to visit a dentist with odds ratios of 0.50

Table 2

 2

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	13

).

Similarly, persons who had the lowest number of friends were significantly more likely to have moderate periodontitis

(OR 1.53, CI 1.15, 2.04)

. The relationship was eliminated after

Table 1 :

 1 Social networks indicated by number of friends and marital status was significantly related to the extent of loss of periodontal attachment, but not the definition of moderate periodontitis, after adjusting for socioeconomic and behavioural factors in a nationally representative sample of older Americans aged 60 years and over. Social network and support appeared to have no impact on the social gradients in periodontal disease in this age group. Mean extent of loss of periodontal attachment and percentage of moderate periodontitis within groups of explanatory variables among older Americans (n=1632)

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Table 2 :

 2 Association between social network and support indicators and two measures of periodontal disease among Americans aged 60 years and over (n=1632)(NHANES 2001(NHANES -2004) ) Model 1 adjusted for age, gender and race/ ethnicity, in addition to the variables of social network/ support. Model 2 additionally adjusted for income, education, diabetes and number of teeth. Model 3 additionally adjusted for smoking status (current, former smoker, never-smoker) and dental visits.

			Marital status (Reference group: married)	Need more	Number of friends (Reference group:
					emotional support	highest Tertile)
			widowed	Divorced/ separated		Middle Tertile	Lowest Tertile
	Rate ratio for extent	Model 1	1.60 *** (1.27, 2.01)	1.33 ** (1.09, 1.63)	0.98 NS (0.74, 1.30)	1.05 NS (0.88, 1.26)	1.43 *** (1.21, 1.68)
	loss of attachment	Model 2	1.33 ** (1.09, 1.63)	1.14 NS (0.90, 1.44)	0.86 NS (0.69, 1.07)	1.02 NS (0.85, 1.22)	1.23 * (1.05, 1.46)
		Model 3	1.27 * (1.03, 1.58)	1.07 NS (0.86, 1.34)	0.87 NS (0.72, 1.-6)	0.99 NS (0.83, 1.19)	1.22 * (1.03, 1.45)
	Odds ratio for	Model 1	1.65 * (1.09, 2.50)	1.07 NS (0.70, 1.61)	0.86 NS (0.51, 1.44)	1.19 NS (0.93, 1.53)	1.53 ** (1.15, 2.04)
	moderate	Model 2	1.36 NS (0.88, 2.10)	0.88 NS (0.56, 1.37)	0.73 NS (0.45, 1.20)	1.14 NS (0.87, 1.78)	1.32 NS (0.97, 1.78)
	periodontitis	Model 3	1.28 NS (0.82, 2.00)	0.80 NS (0.52, 1.24)	0.76 NS (0.46, 1.25)	1.09 NS (0.84, 1.40)	1.29 NS (0.95, 1.77)

Journal of Clinical Periodontology -PROOFJournal of Clinical Periodontology -PROOF

Journal of Clinical Periodontology -PROOF

This study was supported by the National Institute of Dental and Craniofacial Research/ NIH grant number: 1 R21 DE019535-01.

The authors declare no conflict of interest.

Table 3: Income and education gradients in periodontal disease for Americans aged 60 years and over (n=1632) (NHANES 2001(NHANES -2004) )