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I318T011015 . 318 T011015d.318
Surveys in High Energy Physics, 2001, Vol. 00, pp. 1 ± 17 Reprints available directly from the publisher Photocopying permitted by license only # 2001 OPA (Overseas Publishers Association) N.V. Published by license under the Harwood Academic Publishers imprint, part of The Gordon and Breach Publishing Group. Printed in Singapore.

CONSTRAINING PARAMETERS OF MAGNETIC FIELD DECAY FOR ACCRETING ISOLATED NEUTRON STARS
S. B. POPOVa,* and M. E. PROKHOROV
a

b,y

Sternberg Astronomical Institute, Universitetskii pr. 13, 119899, Moscow, Russia; bSternberg Astronomical Institute, Universitetskii pr. 13, 119899, Moscow, Russia
(Received in ®nal form July 14 2000)

The in¯uence of exponential magnetic ®eld decay (MFD) on the spin evolution of isolated neutron stars is studied. The ROSAT observations of several X-ray sources, which can be accreting old isolated neutron stars, are used to constrain the exponential and power-law decay parameters. We show that for the exponential decay the ranges of minimum value of magnetic moment, "b, and the characteristic decay time, td, $ 1029.5 ! "b ! 1028 G cm3, $ 108 ! td ! 107 yrs are excluded assuming the standard initial magnetic moment, "0 1030 G cm3. For these parameters, neutron stars would never reach the stage of accretion from the interstellar medium even for a low space velocity of the stars and a high density of the ambient plasma. The range of excluded parameters increases for lower values of "0. We also show, that, contrary to exponential MFD, no signi®cant restrictions can be made for the parameters of power-law decay from the statistics of isolated neutron star candidates in ROSAT observations. Isolated neutron stars with constant magnetic ®elds and initial values of them less than "0 $ 1029 G cm3 never come to the stage of accretion. We brie¯y discuss the fate of old magnetars with and without MFD, and describe parameters of old accreting magnetars. Keywords: Stars; Neutron stars ± magnetic ®elds; Decay ± accretion

*Corresponding author. e-mail: polar@sai.msu.ru y e-mail: mystery@sai.msu.ru 1


I318T011015 . 318 T011015d.318 2 S. B. POPOV AND M. E. PROKHOROV

1. INTRODUCTION Astrophysical manifestations of neutron stars (NSs) are determined by their periods and magnetic ®elds. Four main evolutionary stages of isolated NSs can be singled out (see e.g., Lipunov, 1992 for more details): the ejector, the propeller, the accretor and the georotator. On average NSs should have high spatial velocities due to an additional kick obtained during the supernova explosion (Lyne and Lorimer, 1994; Lorimer et al., 1997) . The interstellar medium (ISM) accretion rate for high velocity objects should be rather low. However, recent population synthesis calculations (Popov et al., 2000) indicate that several old accreting NSs can be observed in the solar vicinity even for the space velocity distribution similar to one derived from radio pulsar observations. Magnetic ®eld decay (MFD) in NSs is a matter of controversy. Many models of the MFD have been proposed starting from the ®rst simple models (Gunn and Ostriker, 1970) up to the recent calculations (Sang and Chanmugam, 1990; Urpin and Muslimov, 1992) . Observations of radio pulsars (Lyne et al., 1998) give no evidence for MFD with characteristic time scales td shorter than $ 107 yrs. Here we suggest to use old accreting isolated neutron stars as probes of the models of ®eld decay and try to put some limits on the parameters of the exponential and power-law MFD on a longer time scale assuming that some X-ray sources observed by ROSAT are indeed old accreting isolated NSs (Haberl et al., 1998; Neuhauser and Trumper, 1999; Schwope et al., 1999) . This presentation is based mainly on two our recent papers (Popov and Prokhorov, 2000a, b). 2. CALCULATIONS AND RESULTS The main idea is to calculate the ejector time, i.e., a time interval spent by a NS on the ejector stage, for dierent parameters of the MFD and, using standard assumptions for the initial NS parameters, to compare this time with the Hubble time, tH. The ejector time, tE, monotonically increases with increasing velocity of NS, v, and decreasing density of the ISM, n. For a constant


I318T011015 . 318 T011015d.318 MAGNETIC FIELD DECAY 3

magnetic ®eld of a NS this relation takes the simple form: tE " const $ 109 "þ1 n 30
þ1 a 2

v10 yrsX

1

Using a high mean ISM density n $ 1 cm þ 3 and a low space velocity of NSs (about the sound speed in the ISM), v $ 10 kms þ 1, we arrive at the lower limit of tE. After the ejection stage has been over, the NS passes to the propeller stage and only after that can become an accreting X-ray source. The duration of the propeller stage tP is poorly known, but for a constant magnetic ®eld tP is always less than tE, (see Lipunov and Popov, 1995). Therefore if for some parameters of a NS tE exceeds the Hubble time tH 9 1010 yrs, it can not come to the accretion stage and hence can not underly the ROSAT INS candidate. We note, that if the initial magnetic moment of a NS is about "0 $ 1029 G cm3 or smaller, than (in the case of constant ®eld) this star never leave the ejector stage even for low velocity and high ISM density! So, signi®cant part of INS for any velocity distribution can't become accretors at all. In addition, we assumed that NSs are born with suciently small rotational periods, p0, and all have the same parameters of the MFD. We shall consider dierent initial surface magnetic ®eld values. 2.1. Exponential Decay The ®eld decay in this subsection is assumed to have an exponential shape: " "0 à e
þtat
d

Y

for " b "b

2

where "0 is the initial magnetic moment " 1a2Bp R3 , here Bp is the NS polar magnetic ®eld, RNS is the NS radius), td is the characteristic time scale of the decay, and "b is the bottom value of the magnetic moment which is reached at the time tcr: "0 Y 3 tcr td à ln "b and does not change after that. In Figure 1 we show as an illustration the evolutionary tracks of NSs on P-B diagram for v 10 km s þ 1 and n 1 cm þ 3. Tracks start at


I318T011015 . 318 T011015d.318 4 S. B. POPOV AND M. E. PROKHOROV

FIGURE 1 Tracks on P-B diagram. Tracks are plotted for bottom polar magnetic ®eld 8 à 1010 G, initial polar ®eld 2 à 1012 G, NS velocity 10 kms þ 1, ISM density 1 cm þ 3 and dierent td. The last point of tracks with dierent td corresponds to the following NS ages: 1010 yrs for td 107 and td 108 yrs; 1.5 á 109 yrs for td 109 yrs; $ 2 à 109 yrs for td 1010 yrs. The initial period is assumed to be p0 0.020 s. The line with diamonds shows the ejector period, pE.

t 0 when p 20 ms and " 1030 G cm3 and end at t tH 1010 yrs ( for td 107 yrs and td 108 yrs) or at the moment when p pE ( for td 109 yrs, td 1010 yrs and for a constant magnetic ®eld ). The line with diamonds shows p pE(B). The ejector stage ends when the critical ejector period, pE, is reached: sY 4 q where v10 v2 v2 a10 km sþ1 à vp is the NS's spatial velocity, vs and p s n are the sound velocity and density of the ISM, respectively. In the estimates below we shall assume v 10 km s þ 1 and n 1 cm þ 3. The initial NSs' spin periods should be taken much smaller than pE. Here to calculate duration of the ejection stage we assume p0 0 s. To pE 11X5 "30 n
1a2 þ1a4 1a2 v10


I318T011015 . 318 T011015d.318 MAGNETIC FIELD DECAY 5

compute this time we used the magnetodipole formula: dp 2 4%2 "2 Y dt 3 pIc3 5

where " can be a function of time. After a simple algebra we arrive at the following expression for tE: ! V b þt à ln T at p1 t þ1 Y bd tE ` tcr ` d T 6 tE b tcr T "0 a"b þ td 1a2"0 a"b 2 1 þ eþ2tcr atd Y b X tE b tcr
2 d 2

where the coecient T (which is just tE for constant magnetic ®eld ) is determined by the formula: T 3Ic 1 p 9 1017 I45 "þ30 v 0 2" 0 2vM
þ1a2 10

M

þ1a2 11

sX

7

Here M can be formally determined according to the Bondi equation for the mass accretion rate even if the NS is not at the accretion stage: M 9 1011 nvþ3 g sþ1 X 10 8

The results of calculations of tE for several values of "0 and td are shown in Figure 2. The right end points of all curves are limited by the values "b "0. These points correspond to the evolution of a NS with constant magnetic ®eld (see Eq. (2)) and for them tE T. One can see the increase of tE for evolution with a constant ®eld for smaller initial ®elds. If "b is small enough, the NS ®eld has no time to reach the bottom value. In this case tE is determined by the 1st branch of Eq. (6) and does not depend on "b. In Figure 2 this situation corresponds to the left horizontal parts of the curves. At s ! t2 T "b b "0 1 þ d2 þ 1 T td the situation changes so that tE starts to depend on "b. In this region two counter-acting factors operates. On the one hand, the NS braking becomes slower with decreasing " (see Eq. (5)). On the other hand, the


I318T011015 . 318 T011015d.318 6 S. B. POPOV AND M. E. PROKHOROV

FIGURE 2 Ejector time tE (in billion years) vs. the bottom value of the magnetic moment. The curves are shown for two values of the initial magnetic moment: 1030 G cm3 (upper curves) and 1031 G cm3.

end period of the ejection pE becomes shorter (4). Since tE ` T at the left hand side horizontal part and dTE ad "b j"0 ` 0, the right hand side of the curve must have a maximum. The ®rst factor plays the main role to the right of the maximum. The magnetic ®eld there rapidly falls down to "b at p(PE and most time NS evolves with the minimum ®eld " "b (this time period increases with decreasing "b). To the left of the maxium but before the horizontal part the NS's magnetic ®eld reaches " "b with the spin period close to pE (the smaller "b, the closer) and soon after t tcr the NS leaves the ejection stage. As it is seen from this Figure, for some combination of parameters tE is longer than the Hubble time. It means that such NSs never evolve further than the ejection stage. We argue that since accreting isolated NSs are really observed, the combinations of td and "b for which no accreting isolated NS appear


I318T011015 . 318 T011015d.318 MAGNETIC FIELD DECAY 7

can be excluded for the progenitors of ROSAT X-ray sources. The regions of excluded parameters are plotted in Figs. 3 and 4. The hatched regions correspond to parameters for which tE is longer than 1010 yrs, so a NS with such parameters never comes to the accretor stage and hence can not appear as an accreting X-ray source. In view of the fact of observations of accreting old isolated NSs by ROSAT satellite, this region can be called ``forbidden'' for a given "0. In the ``forbidden'' region in Figure 3, which is plotted for "0 1030 G cm3, all NSs reach the bottom ®eld in a Hubble time or faster, and the evolution on late stages proceeds with the minimal ®eld.

FIGURE 3 The characteristic time scale of the MFD, td, vs. bottom magnetic moment, "b. In the hatched region tE is greater than 1010 yrs. The dashed line corresponds to tH td à ln("0/"b), where tH 1010 years. The solid line corresponds to pE("b) p(t tcr), where tcr td à ln("0/"b). Both the lines and hatched region are plotted for "0 1030 G cm þ 3. The dash-dotted line is the same as the dashed one, but for "0 5 à 1029 G cm3. The dotted line shows the border of the ``forbidden'' region for "0 5 à 1029 G cm3.


I318T011015 . 318 T011015d.318 8 S. B. POPOV AND M. E. PROKHOROV

FIGURE 4 The characteristic time scale of the MFD, td, vs. bottom magnetic moment, "b. In the hatched region tE is greater than 1010 yrs. The dashed line corresponds to tH td à ln("0/"b), where tH 1010 yrs. The solid line corresponds to pE("b) p(t tcr), where tcr td à ln("0/"b). Both lines and region are plotted for "0 1029 G cm þ 3.

The left hand side of the forbidden region is determined approximately by the condition pE "b pt tcr X 9

The right hand side of the region is roughly determined by the value of "b, with which a NS can reach the ejection stage for any td, i.e., this "b corresponds to the minimum value of "0 with which a NS reaches the ejection stage without MFD. In Figure 3 we also show the ``forbidden'' region for "0 0.5 à 1030 G cm3 (dotted line). The dashed line in Figure 3 shows that for all interesting parameters a NS with "0 1030 G cm3 reaches "b in less than 1010 yrs. The dash-dotted line shows the same for "0 0.5 à 1030 G cm3. The solid line corresponds to pE("b) p(t tcr), where tcr td à ln ("0/"b). The physical sense of this line can be described in the following


I318T011015 . 318 T011015d.318 MAGNETIC FIELD DECAY 9

way. This line divides two regions: in the upper left region td are relatively long and "b relatively low, so NS can't reach bottom ®eld during ejector stage; in the lower right region td are short and "b relatively high, so NS reach "b at the stage of ejection. Figure 4 is plotted for "0 1029 G cm3. For long td( b 4 à 109 yrs) the NS cannot leave the ejection stage for any "b "0. That's why in the upper part of the ®gure a horizontal ``forbidden'' region appears. 2.2. Power-law Decay Power-law (as also exponential) MFD is a widely discussed variant of NSs' ®eld evolution. Power-law is a good ®t for several dierent calculations of the ®eld evolution (Goldreich and Reisenegger, 1992; Geppert et al., 2000) . The power-law MFD can be described with the following simple formula: dB þaB dt
1

X

10

So, we have two parameters of decay: a and . As far as this decay is relatively slow for the most interesting values of greater/about 1 (we use the same units as in (Colpi et al., 2000)), we don't specify any bottom magnetic ®eld, contrary to what we made for more rapid exponential decay (Popov and Prokhorov, 2000a). Even for the Model C from (Colpi et al., 2000) (see Tab. I) with relatively fast MFD the magnetic ®eld can decrease only down to $ 108 G in 1010 yrs (see Fig. 5). But for very small the magnetic ®eld can decay signi®cantly during the Hubble time for any reasonable value of a. And, probably, it is useful to introduce in the later case a bottom ®eld. At the stage of ejection an INS is spinning down according to the magnetodipole formula: PP % bB2 . Here b 3, values of magnetic ®eld, B, BI and B0, are taken in units 1013 G and time, t, in units 106 yrs (as in Colpi et al., 2000).
TABLE I Models A,B,C from (Colpi et al., 2000) Model a B A 0.01 5/4 % 1.9 à 1011 G B 0.15 5/4 % 2.4 à 1010 G C 10 1 % 108 G

I


I318T011015 . 318 T011015d.318 10 S. B. POPOV AND M. E. PROKHOROV

FIGURE 5 Power-law MFD. Model A: a 0.01, 1.25; solid line with circles. Model B: a 0.15, 1.25; dashed line with squares. Model C: a 10, 1; longdashed line with diamonds. Models were described in details in Colpi et al. (2000) (see also Tab. I).

In the table we show parameters of the Models A, B, C from (Colpi et al., 2000). BI is the magnetic ®eld calculated for t tHubble 1010 yrs and for the initial ®eld B0 1012 G. Models A and B correspond to ambipolar diusion in the irrotational and the solenoidal modes respectively. Model C describes MFD in the case of the Hall cascade (models are valid mostly for relatively high values of magnetic ®eld ). In Figure 6 we show dependence of the ejector period, pE, and the asymptotic period, pI, on the parameter a for 1 for dierent values of the initial magnetic ®eld, B0: pE 25X7B p
2 I 1a2 þ1a4 1a2 v10 In 2þ 0

sY

11 12



2b B 2 þ a

X


I318T011015 . 318 T011015d.318 MAGNETIC FIELD DECAY 11

FIGURE 6 Periods vs. parameter a for dierent values of the initial magnetic ®eld: 1011, 1012, 1013, 1014 G.

Here pE was calculated for t tHubble 1010 yrs, i.e., for the moment, when B BI. It is clear from Figure 6, that for the initial ®eld greater/about 1011 G low velocity INSs are able to come to the stage of accretion: for B0 1011 G lines for and pI and pE for the lowest possible velocity, 10 km/s, coincides. For power-law decay we can also plot ``forbidden'' regions on the plane a þ , where an INS for a given velocity for sure cannot come to the stage of accretion in the Hubble time (see Popov and Prokhorov, 2000b). If one also takes into account the stage of propeller (between ejector and accretor stages) it becomes clear, that ``forbidden'' regions for an INS which cannot reach the stage of accretion are even larger. For the most interesting cases (Models A, B, C from (Colpi et al., 2000)) and v ` 200 km/s INSs can reach the stage of accretion. It is an important point, that fraction of low velocity NSs is very small (Popov et al., 2000) and most of NSs have velocities about 200 km/s.


I318T011015 . 318 T011015d.318 12 S. B. POPOV AND M. E. PROKHOROV

3. EVOLVED MAGNETARS In the last several years a new class of objects - highly magnetized NSs, ``magnetars'' (Duncan and Thompson, 1992) ± became very popular in connection with soft -repeaters (SGR) and anomalous X-ray pulsars (AXP) (see Mereghetti and Stella, 1995; Kouveliotou et al., 1999; Mereghetti, 1999 and recent theoretical works Alpar, 1999; Marsden et al., 2000; Perna et al., 2000) . Magnetars come to the propeller stage with periods $ 10 ± 100 s in the Models A, B, C (see Fig. 2 in Colpi et al., 2000). Then their periods quickly increase, and NSs come to the stage of accretion with signi®cantly longer periods, and at that stage they evolve to a so-called equilibrium period (Lipunov and Popov, 1995; Konenkov and Popov, 1997) due to accretion of the turbulent ISM: peq $ 2800B
2a3 1a3 þ2a3 13a3 þ2a3 v10 vt10 13 I45 n

M

þ8a3 1X4

s

13

Here vt is a characteristic turbulent velocity, I - moment of inertia, M INS's mass. This formula underestimate the period for relatively high vt, and relatively low v, because it assumes, that all external angular moment can be accreted by a INS. Isolated accretor can be observed both with positive and negative sign of p (Lipunov and Popov, 1995). Spin periods of INSs can dier signi®cantly from peq contrary to NSs in disc-fed binaries, and similar to NSs in wide binaries, where accreted matter is captured from giant's stellar wind. It happens because spin-up/spin-down moments are relatively small. As the ®eld is decaying the equilibrium period is decreasing, coming to $ 28 sec when the ®eld is equal to 1010 G (we note here recently discovered objects RX J0420.0-5022 (Haberl et al., 2000) with spin period $ 22.7 s). It is important to discuss the possibility, that evolved magnetar can appear also as a georotator. It happens if: v b 300B
þ1a5 1a10 13 n

kmasX

14

For all values of a and that we used NSs at the end of their evolution (t 1010 yrs) have magnetic ®elds ` 1012 G for wide range of initial ®elds, so they never appear as georotators if v ` 480 km/s for


I318T011015 . 318 T011015d.318 MAGNETIC FIELD DECAY 13

n 1 cm þ 3. But without MFD magnetars with B b 1015 G and velocities v b 100 km/s can appear as georotators. Popov et al. (2000) showed, that georotator is a rare stage for INSs, because an INS can come to the georotator stage only from the propeller or accretor stage, but all these phases require relatively low velocities, and high velocity INSs spend most of their lives as ejectors. This situation is opposite to binary systems, where a lot of georotators are expected for fast stellar winds (wind velocity can be much faster than INS's velocity relative to ISM). Without MFD magnetars also can appear as accreting sources. In that case they can have very long periods and very narrow accretion columns (that means high temperature). Such sources are not observed now. Absence of some speci®c sources associated with evolved magnetars (binary or isolated) can put some limits on their number and properties (dr. V. Gvaramadze drew our attention to this point). At the accretion part of LNSs' evolution periods stay relatively close to peq (but can ¯uctuate around this value), and INSs' magnetic ®elds decay down to $ 1010 ± 1011 G in several billion years for the Models A and B. It corresponds to the polar cap radius about 0.15 km and temperature about 250 ± 260 eV (the same temperature, of course, can appear for INSs evolving with constant ®eld ), higher than for the observed INS candidates with temperature about 50 ± 80 eV. We p calculate the polar cap radius, Rcap RNS RNS aRA (RA ± Alfven radius), with the following formula: R
cap

6 à 103 B

þ2a7 1a7 þ3a7 3a2 v10 RNS6 13 n

cmX

15

The temperature can be even larger, than it follows from the formula above as far as for very high ®eld matter can be channeled in a narrow ring, so the area of the emitting region will be just a fraction of the total polar cap area. As the ®eld is decreasing the radius of the polar cap is increasing, and the temperature is falling. Sources with such properties (temperature about 250 ± 260 eV) are not observed yet (Schwope et al., 1999). But if the number of magnetars is signi®cant (about 10% of all NSs) accreting evolved magnetars can be found in the near future, as far as now we know about 5 accreting INS candidates (Treves et al., 2000; Neuhauser and Trumper, 1999) , and their number


I318T011015 . 318 T011015d.318 14 S. B. POPOV AND M. E. PROKHOROV

can be increased in future. p measurements are necessary to understand the nature of such sources, if they are observed. Recently discovered object RX J0420.0-5022 (Haberl et al., 2000) with the spin period $ 22.7 s, can be an example of an INS with decayed magnetic ®eld accreting from the ISM, as previously RX J0720.4-3125. Due to relatively low temperature, 57 eV, its progenitor cannot be a magnetar for power-law MFD (Models A,B,C ) or similar sets of parameters, because a very large polar cap is needed, which is dicult to obtain in these models. Of course RX J0420.0-5022 can be explained also as a cooling NS. The question ``are the observed candidates cooling or accreting objectsc'' is still open (see Treves et al., 2000). If one ®nds an object with p b 100 s and temperature about 50 ± 70 eV it can be a strong argument for its accretion nature, as far as such long periods for magnetars can be reached only for very high initial magnetic ®elds for reasonable models of MFD and other parameters.

4. DISCUSSION AND CONCLUSIONS We tried to evaluate the region of parameters which are excluded for models of the exponential and power-law MFD in NSs using the fact of observations of old accreting isolated NSs in X-rays. For the exponential decay the intermediate values of td ( $ 107 ± 108 yrs) in combination with the intermediate values of "b( $ 1028 ± 1029.5 G cm3) for "0 1030 G cm3 can be excluded for progenitors of isolated accreting NSs because NSs with such parameters would always remain on the ejector stage and never pass to the accretion stage. For high "0 NSs should reach tE even for td ` 108 yrs. For weaker ®elds the ``forbidden'' region becomes wider. The results are dependent on the initial magnetic ®eld "0, the ISM density n, and NS velocity v. In fact the limits obtained are very strong, because we did not take into account that NSs can spend some signi®cant time (in the case of MFD) at the propeller stage (the spin-down rate at this stage is very uncertain, see the list of formulae, for example, in (Lipunov and Popov, 1995) or (Lipunov, 1992)) .


I318T011015 . 318 T011015d.318 MAGNETIC FIELD DECAY 15

Note that there is another reason for which a very fast decay down to small values of "b can also be excluded, as far as this would lead to a huge amount of accreting isolated NSs in drastic contrast with observations. This situation is similar to the ``turn-o'' of the magnetic ®eld of a NS (i.e., quenching any magnetospheric eect on the accreting matter). So for any velocity and density distributions we should expect signi®cantly more accreting isolated NS than we know from ROSAT observations (of course, for high velocities X-ray sources will be very dim, but close NSs can be observed even for velocities $ 100 km s þ 1). For power-law MFD (contrary to exponential decay) we cannot put serious limits on the parameters of decay with the ROSAT observations of INS candidates as far as for all plausible models of power-law MFD INSs from low velocity tail are able to become accretors. For more detailed conclusions a NS census for power-law MFD is necessary, similar to non-decaying and exponential cases (Popov et al., 2000). An interesting possibility of observing evolved accreting magnetars appear both for the case of MFD and for constant ®eld evolution. These sources should be dierent from typical present day INS candidates observed by ROSAT. Existence or absence of old accreting magnetars is very important for the whole NS astrophysics. We conclude that the existence of several old isolated accreting NSs observed by ROSAT (if it is the correct interpretation of observations), can put important bounds on the models of the MFD for isolated NSs for exponential decay (without in¯uence of accretion, which can stimulate ®eld decay). These models should explain the fact of observations of $ 10 accreting isolated NSs in the solar vicinity. Here we can not fully discuss the relations between decay parameters and X-ray observations of isolated NSs without detailed calculations. What we showed is that this connection should be taken into account and made some illustrations of it, and future investigations in that ®eld would be desirable. Acknowledgements We thank Monica Colpi, Denis Konenkov, Konstantin Postnov, George Pavlov and Roberto Turolla for comments on the text and


I318T011015 . 318 T011015d.318 16 S. B. POPOV AND M. E. PROKHOROV

discussions. We want to thank Vladimir Lipunov and Aldo Treves for advices and attention to our work. We also thank University of Milan, University of Padova, University of Como and Brera Observatory (Merate) for hospitality. This work was supported by the RFBR (98-02-16801) and the INTAS (96-0315) grants. References
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