\section{Introduction}
\label{intro}

Evidence has been accumulating in recent years that at least a subclass of
Gamma--ray bursts (GRBs), the ones with a long ($\lower.5ex\hbox{\gtsima} 2$ s) burst event, are
associated with deaths of massive stars (e.g. Woosley 1993; Paczy\'nski 1998;
MacFadyen \& Woosley 1999). This evidence was initially based on the
relatively small offset of the GRB location with respect to the center of the
host galaxy (Bloom, Kulkarni \& Djorgovski 2002). Moreover, decisive
supernova features have been observed in the afterglow of a few nearby GRBs
(Galama et al. 1998, Della Valle et al. 2003; Stanek et al. 2003;
Malesani et al. 2004), directly linking long GRBs to massive stars. This also
provides strong observational evidence for the connection of GRBs to star
formation (Djorgovski et al. 1998; Fruchter et al. 1999; Prochaska et
al. 2004). A study on the GRB host galaxies
by Le Floc'h et al. (2003) found that these hosts have very blue colours,
comparable to those of the faint blue star-forming sources at high
redshift. The association of long GRBs with star forming regions supports
the idea that a large fraction of the optically-dark GRBs (i.e. GRBs without
an optical afterglow), as well, are due to high (dust) absorption
(Lazzati, Covino \& Ghisellini 2002; Rol et al. 2005; Filliatre et al. 2005).

Together with optical studies, which probe the dust content of the GRB
environment, X--ray studies of the GRB afterglows can give insight on the
ionization status and metal abundances of the matter in the GRB environment.
This can be done by using either emission or absorption features (e.g.,
B\"ottcher et al. 1999; Ghisellini et al. 2002). Although emission features are
more apparent, the cumulative effect of low-energy cutoff is easier to
detect in the relatively low signal to noise spectra of X--ray afterglows.
Moreover, if the absorbing material is located close to the GRB site ($\sim
0.1-10$ pc), it is expected that GRB photons may lead to a progressive
photoionization of the gas, gradually reducing the effect of low energy
absorption (Lazzati \& Perna 2002; Perna \& Lazzati 2002).

X--ray absorption in excess of the Galactic value has been
reported for a handful of GRB afterglows (Owens et al. 1998; Galama \& Wijers
2001; Stratta et al. 2004; De Luca et al. 2005; Gendre, Corsi \& Piro 2005).
Evidence for a decrease of the intrinsic column density with time in the
X--ray prompt emission of some GRBs has also been found (GRB980506, Connors \&
Hueter 1998; GRB980329, Frontera et al. 2000; GRB011211, Frontera et
al. 2004). Lazzati \& Perna (2002) interpreted them as evidence for GRBs
occurring within overdense regions in molecular clouds similar to star
formation globules in our Galaxy.

Stratta et al. (2004) presented a systematic analysis of a sample of 13 bright
afterglows observed with BeppoSAX narrow field instruments. They found a
significant detection of additional intervening material in only two cases
(namely, GRB990123 and GRB010222), but, owing to the limited photon
statistics, they could not exclude that intrinsic X--ray absorption is also
present in the other bursts.
Chandra observations of GRB afterglows have yielded a few detections and
constraints of the presence of intrinsic X--ray absorption (Gendre et
al. 2005). XMM-Newton observed 9 GRB
afterglows (for a review see De Luca et al. 2005 and Gendre et
al. 2005). These are mainly INTEGRAL GRBs, the large majority of which has
been discovered close to the Galactic plane (i.e. are characterized by a
relatively high Galactic column density). From XMM-Newton data one can
gather evidence that at least several GRBs occur in high density regions
within their host galaxies (e.g, De Luca et al. 2005).

Here we investigate the presence of intrinsic absorption in the complete set
of all the 17 GRBs promptly observed by Swift up to April 15, 2005.
The paper is organized as follows. In section 2 we present the data collected
by Swift and the analysis procedure. In section 3, we derive values and/or
upper limits on the column density in excess of the Galactic value (based on
the maps by Dickey \& Lockman 1990). For 6 GRBs a redshift has also been
determined through spectroscopic observations. For these we investigate the
intrinsic excess of absorption. In section 4 we discuss our results.
Section 5 is dedicated to our conclusions.

\section{Swift data}
\label{data}

The X--Ray Telescope (XRT, Burrows et al. 2005a) on board Swift (Gehrels et
al. 2004) is a focusing X--ray telescope with a 110 cm$^{2}$ effective
area at 1.5 keV, 23 arcmin field of view, 18 arcsec resolution (half-power diameter) and
0.2--10 keV energy range. The first GRB followed by XRT was GRB041223 (Burrows
et al. 2005b). Since then 22 other GRBs were observed by Swift up to April
15, 2005, together with one discovered by HETE II.
Of these 24 GRBs, 19 were observed by XRT and only two of them [GRB050410 and
GRB050117a (see also Hill et al. 2005)] were not detected or have
too few photons to perform a meaningful spectral analysis.
For eight of them Swift was able to repoint within a
few hundred seconds, whereas the remaining nine were observed at later
times ($>30$ min). In Table 1 we present a log of the XRT observations
used in the present work.

 \begin{table}
\label{datalog}
\caption{Observation log.}
\begin{tabular}{ccccc}
\hline
GRB    & Start time&Mode& Region$^{\$}$& Exp. time\\
       & (s)$^*$   &    &        & (s)\\
\hline
041223 & 16661	   & PC & 0, 20 & 4018 \\
050124 & 11113     & WT & 40x20 & 7351 \\
       &           & PC & 0, 20 & 11066\\
050126 &   131     & PC & 2, 20 & 278 \\
       &           & PC & 0, 13 & 8720 \\
050128 &   108     & PC & 5, 20 & 2878 \\
       &           & PC & 0, 15 & 14316 \\
050215b&  2100     & PC & 0, 10 & 35563 \\
050219a&    92     & WT & 40x20 & 5003 \\
       &           & PC & 0, 20 & 2570  \\
050219b&  3130     & WT & 40x20 & 15051 \\
       &           & PC & 0, 20 & 5724 \\
050223 &  2875     & PC & 0, 10 & 2337 \\
050306 &127390     & PC & 0, 10 & 12284 \\
050315 &    83     & WT & 40x20 & 748  \\
       &           & PC & 7, 40 & 690 \\
       &           & PC & 0, 20 & 9642 \\
050318 &  3277     & PC & 2, 20 & 395 \\
       &           & PC & 0, 20 & 3938 \\
050319 &    87     & PC & 3, 40 & 1495 \\
       &           & PC & 0, 40 & 2511 \\
050326 &  3258     & PC & 1, 16 & 60 \\
       &           & PC & 0, 15 & 40723\\
050401 &   131     & WT & 40x20 & 2292 \\
       &           & PC & 3, 20 & 276\\
       &           & PC & 0, 15 & 2079\\
050406 &    92     & WT & 40x20 & 109 \\
       &           & PC & 2, 30 & 70 \\
       &           & PC & 0, 10 & 49050 \\
050408 &  4653     & PC & 0, 11 & 1041 \\
       &           & PC & 0, 5  & 57532 \\
050412 &    89     & WT & 40x20 & 168 \\
       &           & PC & 2, 30 & 145 \\
\hline
\end{tabular}

$^*$ Time from the BAT trigger time.

$^{\$}$ Extraction region. In PC mode it is a circular/annular region with inner
and outer radii reported in the column. In the case of WT mode the region is
always a $40\times20$ region along the column centered on source.
\end{table}

\begin{table}
\label{values}
\caption{Results of the spectral analysis.}
\begin{tabular}{cccc}
\hline
GRB    &$N_H$ Gal.$^a$        & $N_H$ obs.$^b$   & $\chi^2_{\rm red}$ \\
       &$10^{20}\rm \ cm^{-2}$         & $10^{20}\rm \ cm^{-2}$    & (dof)	      \\
\hline
041223 & 10.9 (9.9) [5.6]    &$16.8^{+5.2}_{-4.2}$ & 0.8 (26)      \\
050124 & 5.2 (2.6) [1.7]     &$6.2^{+3.9}_{-2.5}$  & 1.3 (35)      \\
050126 & 5.3 (3.2) [2.6]     &$4.1^{+2.7}_{-2.6}$  & 0.9 (10)      \\
050128 & 4.8 (4.9) [3.8]     &$12.5^{+1.4}_{-1.3}$ & 1.3 (105)     \\
050215b& 2.1 (2.0) [0.9]     &$<3.4$            & 1.0 (4)       \\
050219a& 8.5 (10.1) [8.1]    &$30.1^{+6.5}_{-5.9}$ & 1.0 (57) \\
050219b& 3.8 (3.0) [1.7]     &$23.8^{+4.0}_{-3.7}$ & 1.0 (98)     \\
050223 & 7.1 (6.6) [4.4]     &$9.5^{+23.8}_{-7.3}$ & 1.2 (3)      \\
050306 & 3.1 (2.9) [3.5]     &$46.1^{+35.6}_{-28.8}$& 1.0 (3)      \\
050315 & 4.3 (3.3) [2.5]     &$14.9^{+3.9}_{-2.2}$ &  1.3 (42)     \\
050318 & 2.8 (1.8) [0.9]     &$4.2^{+1.9}_{-1.5}$  & 0.8 (39)      \\
050319 & 1.1 (1.2) [0.5]     &$3.0^{+0.9}_{-0.8}$  & 1.3 (29)      \\
050326 & 4.5 (3.8) [1.8]     &$18.9^{+7.1}_{-6.0}$ & 1.0 (25)      \\
050401 & 4.8 (4.4) [3.3]     &$21.1^{+2.2}_{-1.8}$ & 1.0 (297)     \\
050406 & 2.8 (1.7) [1.1]     &$<6.6$            & 1.2 (10)      \\
050408 & 1.7 (1.5) [1.3]     &$30.7^{+5.5}_{-4.9}$ & 0.9 (45)   \\
050412 & 2.2 (1.7) [1.0]     &$26.4^{+14.9}_{-12.4}$& 1.4 (11)    \\
\hline
\end{tabular}

$^a$ Column density values are from Dickey \& Lockman (1990). Values between
parentheses are from Kalberla et al. (2005) and in square parentheses from
Schlegel, Finkbeiner \& Davis (1998), using the usual conversion
$N_H=5.9\times 10^{21}\,E(B-V)\rm \ cm^{-2}$.

$^b$ The values of the column density have been computed at $z=0$ since we do
not have any knowledge of the GRB redshift for most of them (but see Table \ref{zz}).
\end{table}

GRBs are observed by XRT with different observing modes and source count
rates. These modes were designed to minimize photon pile-up when observing
the highly variable flux of GRB afterglows. The change between observing modes
should occur automatically when the XRT is in the so-called Auto State (for a
thorough description of XRT observing modes see Hill et al. 2004).
Many of these early bursts were observed in Manual State instead, with the
observing mode fixed, during the calibration phase (before April 5, 2005). For
these GRBs observations were often carried out in Photon Counting (PC, the usual
mode providing 2D images) and for some bright bursts the initial data are
piled-up.
This effect can be corrected by extracting light curves and spectra from an
annular region around the source center (rather than a simple circular region),
with a hole size that depends on the source brightness. As the afterglows
decays the pile-up effect becomes negligible and extraction from a circular
region is feasible. For bursts observed in Auto State observations started in
Window Timing (WT) mode (providing just 1D imaging).
Cross-calibration between modes assures that the two modes, PC and WT, provide
the same rate (within a few percent) on steady sources.

Here we analyzed the dataset shown in Table 1. All data were processed with
the standard XRT pipeline within
FTOOLS 6.0 ({\tt xrtpipeline} v. 0.8.8) in order to produce screened event
files. WT data were extracted in the 0.5--10 keV energy range, PC data in the
0.2--10 keV range. Standard grade filtering was adopted (0--2 for WT and
0--12 for PC, according to XRT nomenclature, see Burrows et al. 2005a). From these we
extracted spectra using regions selected to avoid pile up and to maximize the
signal to noise (see Table 1). In WT mode we adopted the standard
extraction region of $40\times 20$ pixels along the WT line. In PC we used an
annular region when the inner core of the Point Spread Function (PSF) was
piled-up and circular regions otherwise. The size of the extraction region
depends on the source strength and background level. Appropriate ancillary response
files were generated with the task {\tt xrtmkarf}, accounting for PSF
corrections. The latest response matrices (v.007) were used. The data were
rebinned to have 20 counts per energy bin (in some cases with few photons
energy bins with as low as 10 counts per bin were used).

The data were fitted with a simple absorbed power law model. We have adopted
the usual photoelectric absorption using Wisconsin (Morrison \& McCammon 1983)
cross-sections ({\tt wabs} model within XSPEC). Normalizations were
left free to vary. Power law photon index and absorbing column densities were
first tied across the different observations and observing modes, but if the
fit was not satisfactory the photon index was allowed to vary between
observations (indicating a spectral evolution). We searched for variations of
the column density with time by allowing the column density to vary across
different observations.
For GRBs with known redshift we also fitted an absorbed power law model with a
fixed Galactic column density component plus a free column density at the redshift
of the GRB. Results are shown in Tables 2 and 3 and in
Fig. \ref{nh}, where GRB total column densities are plotted against the Galactic
column densities.

\begin{figure}
\centering
\includegraphics[height=8cm,angle=-90]{nh.ps}
\caption{Galactic column density versus column density obtained from spectral
fit of the X--ray afterglow. Open circles are values obtained without any
redshift information. Filled circles indicate values for the six GRBs with
known redshift at the redshift of the host galaxy. Upper limits are also
indicated with filled and open circles, as above. The line represents the
prints of equal values between the Galactic and the total column density
(i.e. no intrinsic absorption).
}
\label{nh}
\end{figure}

\begin{table*}
\label{zz}
\caption{Results of the spectral analysis of GRB with known redshift.}
\begin{center}
\begin{tabular}{cccccc}
\hline
GRB    &Redshift & $N_H$ Gal.  & $N_H$ obs.$^*$   & $\chi^2_{\rm red}$ & $N_H$
DLA ($90\%$)\\
       & (ref.)  &$10^{20}\rm \ cm^{-2}$& $10^{20}\rm \ cm^{-2}$    & (dof)	          & $10^{20}\rm \ cm^{-2}$  \\
\hline
050126 &1.29 (1) & 5.3       &$<9.4          $    & 1.0 (10)         & 0.8 (6.4)\\
050315 &1.95 (2) & 4.3	     &$83.7^{+19.7}_{-17.4}$ &  1.4 (43)      & 1.4 (5.7)\\
050318 &1.44 (3) & 2.8       &$11.3^{+10.7}_{-8.9}$ & 0.6 (39)        & 0.9 (12.9)\\
050319 &3.24 (4) & 1.1       &$38.6^{+18.0}_{-16.0}$& 1.4 (29)        & 3.6 (447)\\
050401 &2.90 (5) & 4.8       &$366^{+47}_{-46}$    & 1.1 (294)      & 3.0 (272)\\
050408 &1.24 (6) & 1.7       &$134^{+34}_{-28}$    & 1.0 (45)       & 0.8 (4.7)\\
\hline
980703 &0.97     & 5.8       &$290^{+71}_{-27}$    &            & 0.6 (1.7) \\
990123 &1.60     & 2.1       &$30^{+70}_{-20}$   	&            & 1.1 (23.1) \\
990510 &1.63     & 9.4       &$160^{+19}_{-13}$    &            & 1.1(25.1)\\
000210 &0.85     & 2.5       &$50^{+10}_{-10}$    &             & 0.0 (0.0) \\
000214 &0.47     &5.8        &$<2.7$            &            & 0.0 (0.0) \\
000926 &2.07     & 2.7       &$40^{+35}_{-25 }$    &            & 1.6 (73.9)\\
010222 &1.48     & 1.6       &$120^{+70}_{-60 }$   &             & 1.0 (15.1)\\
001025a&1.48     & 6.1       &$66^{+30}_{-30}$    &             & 1.0 (15.1) \\
020322 &1.80     & 4.6       &$130^{+20}_{-20}$    &            & 1.3 (40.7)\\
020405 &0.70     & 4.3       &$47^{+37}_{-37}$     &            & 0.0 (0.0) \\
020813 &1.25     &7.5        &$<36.5$           &            & 0.8 (5.0)\\
021004 &2.33     &4.3        &$<34$             &            & 2.0 (118)\\
030226 &1.98     & 1.6       &$68^{+41}_{-33}$     &            & 1.5 (60.8)\\
030227 &3.90     & 22        &$680^{+18}_{-38}$    &            & 5.2 (649)\\
030328 &1.52     &4.3        &$<44.3$           &            & 1.0 (17.3)\\
\hline
\end{tabular}
\end{center}

$^*$ Column density values have been computed at the GRB redshift.

Refs.: 1) Berger, Cenko, Kulkarni 2005; 2) Kelson \&  Berger 2005; 3) Berger
\& Mulchaey 2005; 4) Fynbo et al. 2005; 5) Fynbo et al. 2005; 6) Berger,
Gladders \& Oemler 2005.
Values in the second part of the table are from Stratta et al. (2004), De Luca
et al. (2005) and Gendre et al. (2005).
\end{table*}

\section{Discussion}
\label{discu}

We have analyzed the X--ray spectra of 17 GRB afterglows observed with Swift
up to April 15, 2005. In at least 10 of them we find significant evidence that
the observed column density is higher than the Galactic value. In
contrast to previous investigations (De Luca et al. 2005; Stratta et
al. 2004; Gendre et al. 2005) based on BeppoSAX, INTEGRAL and HETE II, the
Swift sample has GRBs at low Galactic extinction (all below $\lower.5ex\hbox{\ltsima}
10^{21}\rm \ cm^{-2}$), therefore more effectively probing the presence of absorption
due to the GRB environment or host galaxy.

The evidence that a large fraction of GRBs is characterized by an absorbing
column density larger than the Galactic one clearly points to a high density
interstellar medium in the proximity of the GRB (in fact with X--rays we
directly probe the GRB line of sight, whereas in the optical the observations
might be contaminated by the host galaxy contribution).
Dense environments in the host galaxy, possibly associated with star forming
regions, provide a further clear signature in favour of the association
of long GRBs to the death of massive stars.
For GRBs characterized by a low number of counts no firm conclusions can be
drawn. Moreover, the effect of an intrinsic column density can be hidden
either by a large Galactic absorber (as often occurs for INTEGRAL GRBs) or by
a large redshift shifting the energy scale by $(1+z)$ and the column density
effective value by $\sim (1+z)^{2.6}$.

\begin{figure}
\centering
\includegraphics[height=8cm,angle=-90]{histo.ps}
\caption{Distribution (solid histogram) of intrinsic column density in a sample of
18 GRBs observed by Swift and other satellites (Stratta et al. 2004; De Luca
et al. 2005). This is compared with the expected distribution of column
density for GRB that occur in Galactic-like molecular clouds (dashed
histogram, from Reichart \& Price 2002).
}
\label{histo}
\end{figure}

We combine our sample of GRBs with known redshift with other intrinsic column densities
available in the literature (Stratta et al. 2004; De Luca et al. 2005; Gendre
et al. 2005), obtaining a sample of 21 GRBs (see Table \ref{zz}). In principle
the absorption excess found in most GRBs might not be local to the host galaxy
but may come from a line-of-sight interlooper. This often occurs in optical
studies of quasar with dumped Lyman absorber (DLA). Based on quasar studies
(Wolfe, Gawiser \& Prochaska 2005; P\'eroux et al. 2003) we simulated for each
of our bursts a distribution of line-of-sights (10000 trials),
evaluating a mean absorption (weighted as $(1+z)^{2.6}$) and $90\%$ confidence
value. These values are reported in column six of Table \ref{zz}. The
influence of DLA systems is marginal in our sample even if there are a few GRBs
in which the observed absorption excess may come from intervening DLAs.
Indeed, such systems have recently been found in few Swift GRBs
(e.g. GRB050730 with $\log(N_H)=22.3$, Starling et al. 2005, Chen et al. 2005,
and GRB 050401, Watson et al. 2005, $\log(N_H)=22.5$). However, they are due
to the interstellar medium in the GRB host. One of these GRBs is part of our
sample, GRB 050401, and indeed we obtained a high value of the instrinsic column
density.

In order to understand the origin of the absorption excess,
we compared the distribution of measured intrinsic column densities with the
distribution expected for bursts occurring in Galactic-like molecular clouds
(Reichart \& Price 2002) and with the one expected for bursts
occurring following a host galaxy mass distribution using the Milky Way as a
model (Vergani et al. 2004). For each of
these two column density distributions we simulated 10000 GRBs and compared,
by means of a Kolmogorov-Smirnov (KS) test, their instrinsic absorption
distribution to the observed distribution.
We found that the observed distribution is inconsistent with
the galaxy column density distribution (Vergani et al. 2004), with a KS null
hypothesis probability of $10^{-11}$, but it is consistent with GRBs
distributed randomly in molecular clouds (KS null hypothesis probability of 0.61).
The simulated distribution in the case of bursts occurring in Galactic-like
molecular clouds and the observed one are plotted in Fig. \ref{histo}. This results
would support an origin of long GRBs within high density regions such as
molecular clouds.
We stress that our sample also contains upper limits and that we are
sensitive to low values of column density, which however are found only in a
small fraction of the total sample.

\section{Summary and conclusions}

The main goal of the present paper is to investigate the presence of
intrinsic absorption in the X--ray spectra of GRB afterglows.
We analyzed a complete set of 17 afterglows observed by Swift XRT before
April 15, 2005. In 10 of them we found clear
signs of intrinsic absorption, i.e. with a column density higher than the
Galactic value estimated from the maps by Dickey \& Lockman (1990) at $>90\%$
confidence level (and with low probability of contamination from intervening
DLA systems).
For the remaining 7 cases, the statistics are not good enough to draw firm
conclusions. This clearly suggests that long GRBs are associated with high
density regions of the interstellar medium, supporting the idea
that they are related to the deaths of massive stars.

For the 6 GRBs with known redshift, together with 15 already known, we
can have an unbiased view of the intrinsic absorption in the host galaxy rest
frame. We found a range of $(1-35)\times 10^{21}\rm \ cm^{-2}$ for all
GRBs. This range of values is consistent with the hypothesis that GRBs occur
within giant molecular clouds, spanning a range of column density depending on
their exact location (Reichart \& Price 2002). In our rest frame this column
density is then reduced by a factor $\sim (1+z)^{2.6}$, making it more difficult to
determine the intrinsic column density, especially for distant GRBs or for GRBs
occurring at large Galactic column densities.

Finally, we compared the distribution of GRB column densities with known
redshift with theoretical predictions available in the literature finding good
agreement with the expectation (Reichart \& Price 2002) for bursts occurring
in molecular clouds.

\begin{acknowledgements}
This work is supported at OAB by funding
from ASI on grant number I/R/039/04, at Penn State by NASA contract NAS5-00136
and at the University of Leicester by the PPARC on grant numbers PPA/G/S/00524
and PPA/Z/S/2003/00507. We gratefully acknowledge the contributions of dozens
of members of the Swift team, who helped make this Observatory possible.
\end{acknowledgements}


\section{Introduction}

Measurements of the $g$ factor of low- and middle-$Z$ H- and Li-like ions
\cite{haf00,ver04,stu11,wag13,lin13,stu13,stu14,koel16}
have reached an accuaracy of a few $10^{-10}$. From the theoretical
side, to get this accuracy we need to evaluate various contributions to the $g$-factor value
\cite{blu97,per97,bei00a,bei00b,sha01,sha02b,nef02,yer02,gla02,sha02c,gla04,lee05,pac05,jen09,vol14,sha15,cza16,yer17a,yer17b,zat17,sha17,mal17,gla18,kar18,cza18}.
The comparison of theory and experiment on the $g$ factors of H- and Li-like
silicon has provided the most stringent tests
of bound-state quantum electrodynamics (QED) in presence of a magnetic field, while
the combination of the experimental and theoretical results on the $g$ factor
of  H-like ions with $Z=6,8,14$ lead to
the most precise determination of the electron mass \cite{stu14,zat17}.
The measurement of  the isotope shift of the $g$ factor of Li-like
$^{A}$Ca$^{17+}$ with $A=40$ and $A=48$ \cite{koel16} has triggered a special
interest to the calculations of the nuclear recoil contributions to the $g$ factor.

The fully relativistic theory of the nuclear recoil effect to the first order in
the electron-to-nucleus mass ratio, $m/M$,
on the $g$ factor
of atoms and ions was formulated in Ref. \cite{sha01}, where it was used to derive
closed formulas for the recoil effect on the $g$ factor of H-like ions to all orders in
$\alpha Z$. These formulas remain also valid for an ion with one electron over
closed shells (see, e.g., Ref. \cite{sha02c}), provided the electron propagators
are redefined for the vacuum with the closed shells included. In that case, in addition
to the one-electron contributions, one obtains two-electron recoil corrections
of the zeroth order in $1/Z$ which can be used to derive  effective four-component
recoil operators within the Breit approximation \cite{sha17}.
The one-electron recoil contribution was evaluated numerically to all
orders in $\alpha Z$ for the $1s$ and $2s$ states in Refs. \cite{sha02b} and \cite{sha17},
respectively. The calculations were performed for the point-nucleus case.
For the ground state of Li-like ions, the two-electron recoil contribution
vanishes to the zeroth order in $1/Z$.  However, the effective recoil
operator can be used to evaluate the recoil corrections of the first
and higher orders in $1/Z$ within the framework of the Breit approximation.
These calculations were carried out for $Z=3-20$ in Ref. \cite{sha17},
where it was found a large discrepancy of the obtained results
with the previous Breit-approximation calculations based on the two-component
approach \cite{yan01,yan02}. As was shown in Ref. \cite{sha17}, this discrepancy
was caused by omitting some important terms in the calculation scheme
formulated within the two-component approach for $s$ states in Ref. \cite{heg75}.
Later  \cite{gla18}, the four-component approach was also used to calculate the recoil effect
within the Breit approximation for middle-$Z$  B-like ions.

Special attention should be paid to probing
the QED nuclear recoil effect in experiments with heavy ions,
which are anticipated in the nearest future
at the Max-Planck-Institut f\"ur Kernphysik in Heidelberg
and at the HITRAP/FAIR facilities in Darmstadt.
This would provide
an opportunity for tests of QED at strong coupling regime
beyond the Furry picture. To this end, in Ref. \cite{mal17}
the nuclear recoil effect on the $g$ factor of
H- and Li-like Pb and U was calculated and it was shown that
the QED recoil contribution can be probed on a few-percent level
in a specific difference of the $g$ factors of heavy H- and Li-like lead.

In the present paper we extend the calculations of the recoil effect on the $g$ factor of Li-like ions
performed in Refs. \cite{sha17,mal17} to the range $Z=10-92$.  The one-electron
recoil contribution is calculated in the framework of the rigorous QED approach
with the wave functions which partly account for the screening of the Coulomb
potential by the closed shell electrons. As to the two-electron recoil contribution,
it is evaluated within the Breit approximation to all orders in $1/Z$.
All the calculations also partly account for the nuclear size corrections
to the recoil effect.

The relativistic units ($\hbar=c=1$) are used throughout the paper.

\section{ Basic formulas}

Let us consider a Li-like ion
which is put into a
homogeneous magnetic field, ${\bf A}_{\rm cl}({\bf r})=[
\mbox{\boldmath$\cal H$}
  \times {{\bf r}}]/2$ with
$ \mbox{\boldmath$\cal H$}$  directed along the $z$ axis.
To zeroth order in $1/Z$, the $m/M$ nuclear recoil contribution to the $g$ factor
is given by a sum of one- and two-electron contributions.  In case of the ground
$(1s)^2 2s$ state the two-electron contribution of zeroth order in $1/Z$
is equal to zero and one has to consider the one-electron term only.
The one-electron recoil contribution to the $g$ factor
 is given by \cite{sha01}
\begin{eqnarray} \label{rec_tot}
\Delta g&=&\frac{1}{\mu_0 m_a}\frac{i}{2\pi M}
\int_{-\infty}^{\infty} d\omega\;
\Biggl[\frac{\partial}{\partial {\cal H}}
\langle \tilde{a}|[p^k-D^k(\omega)+eA_{\rm cl}^k]
\nonumber\\
&&\times\tilde{G}(\omega+\tilde{\varepsilon}_a)
[p^k-D^k(\omega)+eA_{\rm cl}^k]
|\tilde{a}\rangle
\Biggr]_{{\cal H}=0}\,.
\label{06recoilt}
\end{eqnarray}
Here $a$ denotes the one-electron $2s$ state,
$\mu_0$ is the Bohr magneton, $m_a$ is the angular momentum
projection of the state under consideration, $M$ is the nuclear mass,
 $p^k=-i\nabla^k$ is the momentum operator,
$D^k(\omega)=-4\pi\alpha Z\alpha^l D^{lk}(\omega)$,
\begin{eqnarray} \label{06photon}
D^{lk}(\omega,{\bf r})&=&-\frac{1}{4\pi}\Bigl\{\frac
{\exp{(i|\omega|r)}}{r}\delta_{lk}\nonumber\\
&&+\nabla^{l}\nabla^{k}
\frac{(\exp{(i|\omega|r)}
-1)}{\omega^{2}r}\Bigr\}\,
\end{eqnarray}
is the transverse part of the photon propagator in the Coulomb
gauge,  ${\mbox{\boldmath$\alpha$}}$ is a vector
incorporating the Dirac matrices, and
the summation over the repeated indices is implied.
The tilde sign means that the corresponding quantity
(the wave function, the energy, and the Dirac-Coulomb Green's function
$\tilde{G}(\omega)=\sum_{\tilde{n}}|\tilde{n}\rangle \langle \tilde{n}|[\omega-\tilde{\varepsilon}_n(1-i0)]^{-1}$)
must be calculated in presence of the
magnetic field.

For the practical calculations, the one-electron contribution
is conveniently
represented by a sum of low-order
and  higher-order
terms,
$\Delta g=\Delta g_{\rm L}+\Delta g_{\rm H}$, where

\begin{eqnarray} \label{low}
\Delta g_{\rm L}&=&\frac{1}{\mu_0 {\cal H}  m_a}
\frac{1}{M} \langle \delta a|
\Bigr[ {\bf p}^2
 -\frac{\alpha Z}{r}\bigr({\mbox{\boldmath$\alpha$}}+\frac{({\mbox{\boldmath$\alpha$}}\cdot{\bf r}){\bf r}}{r^2}\bigr)
  \cdot{\bf p}
  \Bigr]|a\rangle\nonumber\\
&& -\frac{1}{ m_a}\frac{m}{M}\langle
a|\Bigl([{\bf r} \times {\bf p}]_z -\frac{\alpha Z}{2r}[{\bf r} \times {\mbox{\boldmath$\alpha$}} ]_z\Bigr)|a\rangle\,,
\end{eqnarray}
\begin{eqnarray}  \label{high}
\Delta g_{\rm H}&=&\frac{1}{\mu_0  {\cal H} m_a}
\frac{i}{2\pi M} \int_{-\infty}^{\infty} d\omega\;\Bigl\{ \langle
\delta a|\Bigl(D^k(\omega)-\frac{[p^k,V]}{\omega+i0}\Bigr)\nonumber\\
&&\times G(\omega+\varepsilon_a)\Bigl(D^k(\omega)+\frac{[p^k,V]}{\omega+i0}\Bigr)|a\rangle \nonumber\\
&&+\langle a|\Bigl(D^k(\omega)-\frac{[p^k,V]}{\omega+i0}\Bigr)
G(\omega+\varepsilon_a)\nonumber\\
&&\times\Bigl(D^k(\omega)+\frac{[p^k,V]}{\omega+i0}\Bigr) |\delta
a\rangle\nonumber\\ &&+ \langle a|\Bigl(D^k(\omega)-\frac{[p^k,V]}{\omega+i0}\Bigr)
G(\omega+\varepsilon_a)(\delta V-\delta \varepsilon_a)\nonumber\\ &&\times G(\omega+\varepsilon_a)
\Bigl(D^k(\omega)+\frac{[p^k,V]}{\omega+i0}\Bigr)|a\rangle \Bigr\}\,. \label{eq3}
\end{eqnarray}
Here
$V(r)$ is the potential of the nucleus or an effective local potential which
is the sum of the nuclear and screening potentials,
 $\delta V({\bf r})=-e{\mbox{\boldmath$\alpha$}} \cdot{\bf A}_{\rm cl}({\bf r})$,
$G(\omega)=\sum_n|n\rangle \langle n|[\omega-\varepsilon_n(1-i0)]^{-1}$ is the Dirac-Coulomb Green's
function, $\delta \varepsilon_a=\langle a|\delta V|a\rangle$, and
$|\delta a\rangle=\sum_n^{\varepsilon_n\ne  \varepsilon_a}|n\rangle\langle n|\delta V|a\rangle (\varepsilon_a-\varepsilon_n)^{-1}$.
The low-order term corresponds to the Breit approximation,
while the higher-order term  defines the QED one-electron recoil
contribution.

The recoil contributions of the first and higher orders
in $1/Z$ can be evaluated within the Breit approximation with the use of
the four-component recoil operators \cite{sha17}. The total Breit-approximation
recoil contribution can be represented as a sum of two terms. The first
term is obtained as a combined interaction due to $\delta V$ and
 the effective recoil Hamiltonian
(see Ref. \cite{sha98} and references therein):
\begin{eqnarray} \label{br1}
H_M=\frac{1}{2M}\sum_{i,k}\Bigl[{\bf p}_i\cdot {\bf p}_k
  -\frac{\alpha Z}{r_i}\bigr({\mbox{\boldmath$\alpha$}}_i+\frac{({\mbox{\boldmath$\alpha$}}_i\cdot{\bf r}_i){\bf r}_i}{r_i^2}\bigr)
  \cdot{\bf p}_k\Bigr]\,.
\end{eqnarray}
The second term is defined by the magnetic recoil
operator:
\begin{eqnarray} \label{br2}
H_M^{\rm magn}&=&-\mu_0
\frac{m}{M} \mbox{\boldmath$\cal H$} \cdot
\sum_{i, k}\Bigl\{[{\bf r}_i\times {\bf p}_k]\nonumber\\
&&-\frac{\alpha Z}{2r_k}\Bigl[{\bf r}_i\times\Bigl({\mbox{\boldmath$\alpha$}}_k
   +\frac{({\mbox{\boldmath$\alpha$}}_k\cdot{\bf r}_k){\bf r}_k}{r_k^2}\Bigr)\Bigr]
  \Bigr\}\,.
\end{eqnarray}
To zeroth order in $1/Z$, the one-electron parts of the operators (\ref{br1}) and (\ref{br2})
lead to the low-order contribution defined by Eq. (\ref{low}).

Thus, within the four-component Breit-approximation approach
the $m/M$ recoil effect on the $g$ factor can be evaluated
by adding the operators (\ref{br1}) and (\ref{br2})
to the Dirac-Coulomb-Breit Hamiltonian, which includes
the interaction with the external magnetic field.

\section{Numerical calculations}

Let us consider first the calculations within the Breit approxmation.
For these calculations we use  the
operators (\ref{br1}), (\ref{br2}), and the standard
Dirac-Coulomb-Breit Hamiltonian:
\begin{eqnarray} \label{dcb}
H^{\rm DCB}=
\Lambda^{(+)}\Bigl[\sum_{i}h_i^{\rm D}  +\sum_{i<k}V_{ik}\Bigr]\Lambda^{(+)}\,,
\end{eqnarray}
where the indices $i$ and $k$ enumerate the atomic electrons,
$\Lambda^{(+)}$ is the
projector on the positive-energy states, calculated including
 the interaction with external magnetic field  $\delta V$,
$h_i^{\rm D}$ is the one-electron Dirac Hamiltonian
including $\delta V$, and
\begin{eqnarray} \label{br}
  V_{ik} &=& V^{\rm C}_{ik}+ V^{\rm B}_{ik}\nonumber \\
&=&\frac{\displaystyle \alpha}{\displaystyle r_{ik}}
-\alpha\Bigl[\frac{\displaystyle{ {{\mbox{\boldmath$\alpha$}}}_i\cdot {{\mbox{\boldmath$\alpha$}}}_k}}
{\displaystyle{ r_{ik}}}+\frac{\displaystyle 1}{\displaystyle 2}
( {{\mbox{\boldmath$\alpha$}}}_i\cdot{{\mbox{\boldmath$\nabla$}}}_i)({ {{\mbox{\boldmath$\alpha$}}}_k\cdot{\mbox{\boldmath$\nabla$}}}_k)
r_{ik}\Bigr]
\,
\end{eqnarray}
is the electron-electron interaction operator within the Breit approximation.
The numerical calculations have been performed using the approach based on
the recursive representation of the perturbation theory \cite{gla17}.
The key advantages of the recursive perturbation approach over the standard one
are the universality and the computational efficiency. In Refs. \cite{roz14,var18},
this method was applied to find the higher-order contributions to the Zeeman
and Stark shifts in H-like and B-like atoms. The perfect agreement
of the obtained results with the calculations by other methods was demonstrated.
In Refs. \cite{gla17,mal17a}, the recursive method was used to calculate
the higher-order contributions of the interelectronic interaction in few-electron ions.

The total Breit-approximation recoil contribution for the state under consideration
can be  expressed as
\begin{eqnarray} \label{ABC}
\Delta g_{\rm Breit}&=&\frac{m}{M}(\alpha Z)^2 \Bigl[A(\alpha Z)+
  \frac{1}{Z} B(\alpha Z)\nonumber\\
&& + \frac{1}{Z^2} C(\alpha Z,Z)\Bigr]\,,
\end{eqnarray}
where the coefficients $A(\alpha Z)$ and $B(\alpha Z)$
define the recoil contributions
of the zeroth and first orders in $1/Z$, respectively, while
$C(\alpha Z,Z)$ incorporates the recoil corrections
of the second and higher orders in $1/Z$.
In this work, in the calculation of $C(\alpha Z,Z)$
we have taken into account the terms of  the orders $1/Z^2$, $1/Z^3$, and $1/Z^4$.
The contribution of the terms of  higher orders is much smaller than the present numerical uncertainty.

For the point-nucleus case,
the  coefficient  $A(\alpha Z)$, which is determined by the one-electron
low-order term (\ref{low}),
can be evaluated
analytically \cite{sha01}. In case of the $2s$ state it is given by
\begin{eqnarray} \label{06shabaeveq112}
  A(\alpha Z)= \frac{1}{4}\frac{8(2\gamma+1)}
  {3(1+\gamma)(2\gamma+\sqrt{2(1+\gamma)}}  \,,
\end{eqnarray}
where $\gamma=\sqrt{1-(\alpha Z)^2}$.
To the leading orders in $\alpha Z$, it leads to
\begin{eqnarray} \label{06shabaeveq113}
A(\alpha Z)=\frac{1}{4} + \frac{11}{192}(\alpha Z)^2+ \cdots \,.
\end{eqnarray}

The calculations to all orders in $1/Z$
have been performed with the point-nucleus recoil operators
defined by Eqs.  (\ref{br1}) and (\ref{br2}) but with the wave functions
evaluated for extended nuclei. This corresponds to a partial treatment
of the nuclear size corrections to the recoil effect. The Fermi
model of the nuclear charge distribution was used and
the nuclear charge radii were
taken from Ref. \cite{ang13}. The results of the calculations are presented in
Table \ref{ABC_tab}. The indicated uncertainties are due to the numerical
computation errors.

For the point-nucleus case, the higher-order one-electron contribution (\ref{high})
was calculated for the $1s$ and $2s$ states in a wide range of the  nuclear charge number
in Refs. \cite{sha02b,sha17}. In the present paper we perform  the calculations
for extended nuclei and effective potentials which partly account for the electron-electron
interaction effects.
Our first results for  $Z=82,92$
were presented
in Ref. \cite{mal17}, where they were used to search for a possibility
to test QED beyond the Furry picture. In the present paper we extend these calculations to
the range $Z=10-92$.
Since the inclusion of the screening potential into the zeroth-order
Hamiltonian allows one to account for the interelectronic-interaction
effects only partly, we perform the calculations with several
different effective potentials to keep better under control
the uncertainty of the corresponding contribution.
The calculations have been performed for the core-Hartree (CH),
local Dirac-Fock (LDF), and Perdew-Zunger (PZ) effective potentials.
The CH screening potential is derived from the radial charge density of two
$1s$ electrons,
\begin{eqnarray}
V_{\rm CH}(r)=\alpha\int_0^{\infty} dr'\frac{1}{r_>} \rho_{\rm CH}(r'),
\end{eqnarray}
where $r_>={\rm max}(r,r')$,
\begin{eqnarray}
 \rho_{\rm CH}(r)=2[G_{1s}^2(r) + F_{1s}^2(r)]\,,\;\;\int_0^{\infty} dr \rho_{\rm CH}(r) =2\,,
\end{eqnarray}
and $G/r$ and $F/r$ are the large and small components  of the radial Dirac wave function.
The LDF potential is constructed
by inversion of the radial Dirac equation with the radial wave functions obtained in the Dirac-Fock
approximation. The corresponding procedure
is described in detail in Ref. \cite{sha05}.
The last potential
applied in our work is the Perdew-Zunger potential \cite{per81} which
was widely employed in molecular and
cluster calculations.

In Eq. (\ref{high}), the summation over the intermediate electron states
was performed employing
the finite basis set method.  The basis functions were constructed from B-splines \cite{sap96}
within the framework of the dual-kinetic-balance approach \cite{sha04}.
The integration over $\omega$
was carried out analytically for the ``Coulomb'' contribution (the term without
 the ${\bf D}$ vector)
and numerically for the ``one-transverse'' and ``two-transverse'' photon contributions
(the terms with one and two ${\bf D}$ vectors, respectively)
using the Wick's rotation.
The total QED recoil contribution
$\Delta g_{\rm H}$ for the $2s$ state
is conveniently
expressed in terms of the function $P^{(2s)}(\alpha Z)$,
\begin{eqnarray}
\label{P}
\Delta g_{\rm H}^{(2s)}=\frac{m}{M}\frac{(\alpha Z)^5}{8} P^{(2s)}(\alpha Z)\,.
\end{eqnarray}
The numerical results are presented in  Table~\ref{H-like}.
For comparison, in the second column we list the point-nucleus results which
were partly presented in Ref. \cite{sha17}.

In Table~\ref{F_tab}, we present the total values of the recoil corrections
to the $g$ factor of the ground $(1s)^2 2s$ state of Li-like ions.
They are expressed in terms of the function $F(\alpha Z)$, defined by
\begin{eqnarray} \label{F}
\Delta g=\frac{m}{M}(\alpha Z)^2 F(\alpha Z)\,.
\end{eqnarray}
The Breit-approximation recoil contributions are obtained by Eq. (\ref{ABC})
with the coefficients given in Table \ref{ABC_tab}.
The uncertainties include both the error bars presented
in Table \ref{ABC_tab} and the uncertainties due to  the
approximate treatment  of the nuclear size correction to
the recoil effect. We have assumed that the relative value of the latter uncertainty is equal to the related
contribution to the binding energy which was evaluated within the Breit approximation in Ref. \cite{ale15}.
For the QED recoil contribution we use
the LDF values from Table~\ref{H-like}.
The  uncertainty of this term
is estimated as a sum of two contributions.
The first one is due to  the approximate treatment of the electron-electron
interaction effect on the QED recoil contribution.
This uncertainty was estimated by performing the calculations of the low-order
(non-QED) one-electron recoil contribution
with the LDF potential and comparing the obtained result with the total Breit recoil value
evaluated above.
The ratio of the obtained difference  to the non-QED LDF result was chosen
as the relative uncertainty of the corresponding correction to the
QED recoil contribution. It should be noted that this uncertainty exceeds
 the difference between the results obtained for the different
screening potentials presented in Table~\ref{H-like}.
The second contribution to the uncertainty is
caused by the approximate treatment of the nuclear size correction to the recoil effect.
It was estimated in the same way as for the Breit recoil contribution.
As one can  see from Table~\ref{F_tab}, for very heavy ions the QED recoil effect
becomes even bigger than the Breit recoil contribution.

The total recoil contribution to the $g$ factor
should also include small corrections of orders $\alpha (\alpha Z)^2(m/M)$
and $ (\alpha Z)^2(m/M)^2$ and the related corrections of higher
orders in $\alpha Z$ and in $1/Z$.
To the lowest order in $\alpha Z$ the corresponding one-electron corrections were evaluated
in Refs. \cite{gro71,clo71,pac08,eid10}.

\section{Conclusion}

In this paper we have evaluated the nuclear recoil effect of the first
order in $m/M$  on the ground-state $g$ factor of highly charged Li-like ions.
The Breit-approximation contributions have been calculated to all orders
in $1/Z$ employing the recursive perturbation theory. The one-electron
higher-order (QED) recoil contribution was evaluated to all orders in $\alpha Z$
with the wave functions which partly account for the electron-electron
interaction effects. As the result, the most precise theoretical
predictions for the recoil effect on the  $g$ factor of highly charged
Li-like ions are presented.

\section{Acknowledgments}
This work was supported by the Russian Science Foundation (Grant No. 17-12-01097).

\begin{table}
  \caption{The Breit-approximation recoil contributions to the $g$ factor of the
    $(1s)^2 2s$ state of  Li-like ions
    expressed in terms of
    the coefficients    $A(\alpha Z)$,  $B(\alpha Z)$, and  $C(\alpha Z, Z)$  defined
    by  Eq. (\ref{ABC}).}
\label{ABC_tab}
\begin{tabular}{cr@{.}lr@{.}lr@{.}lr@{.}lr@{.}l} \hline
$Z$& \multicolumn{2}{c}{$A(\alpha Z)$}
                    & \multicolumn{2}{c}{$B(\alpha Z)$}
  & \multicolumn{2}{c}{$C(\alpha Z, Z)$ }
\\
\hline
10  &   0&2503  &   $-$0&5172    &  $-$0&236(4)  \\
12  &   0&2504  &   $-$0&5179    &  $-$0&243(3)  \\
14  &   0&2506  &   $-$0&5187    &  $-$0&245(3)  \\
16  &   0&2508  &   $-$0&5197    &  $-$0&248(3)  \\
18  &   0&2510  &   $-$0&5207    &  $-$0&250(2)  \\
20  &   0&2512  &   $-$0&5219    &  $-$0&250(2)  \\
24  &   0&2517  &   $-$0&5247    &  $-$0&250(1)  \\
28  &   0&2524  &   $-$0&5279    &  $-$0&247(1)  \\
30  &   0&2527  &   $-$0&5297    &  $-$0&245(1)  \\
32  &   0&2531  &   $-$0&5315    &  $-$0&243  \\
40  &   0&2548  &   $-$0&5402    &  $-$0&228  \\
48  &   0&2567  &   $-$0&5504    &  $-$0&205  \\
50  &   0&2572  &   $-$0&5531    &  $-$0&198  \\
56  &   0&2588  &   $-$0&5618    &  $-$0&177  \\
60  &   0&2599  &   $-$0&5677    &  $-$0&160  \\
64  &   0&2607  &   $-$0&5734    &  $-$0&141  \\
70  &   0&2616  &   $-$0&5813    &  $-$0&105  \\
72  &   0&2617  &   $-$0&5836    &  $-$0&092  \\
80  &   0&2606  &   $-$0&5886    &  $-$0&037  \\
82  &   0&2597  &   $-$0&5881    &  $-$0&019  \\
90  &   0&2510  &   $-$0&5721    &  0&051  \\
92  &   0&2471  &   $-$0&5629    &  0&065  \\

\hline
\end{tabular}
\end{table}

\begin{table}
\caption{The higher-order (QED) recoil contribution to the $2s$ $g$ factor
expressed in terms of the function $P^{(2s)}(\alpha Z)$ defined by
Eq. (\ref{P}).
The indices Coul, CH, LDF and PZ refer to the Coulomb and various screening potentials, see the text.
The indices p.n. and f.n. correspond to the point-like and finite-size nuclear models.}
\label{H-like}
\begin{tabular}{cr@{.}lr@{.}lr@{.}lr@{.}lr@{.}l} \hline
  $Z$& \multicolumn{2}{c}{$P^{\rm (p.n.)}_{\rm Coul}(\alpha Z)$}
& \multicolumn{2}{c}{$P^{\rm (f.n.)}_{\rm Coul}(\alpha Z)$}
                    & \multicolumn{2}{c}{$P^{}_{\rm CH}(\alpha Z)$}
                                     & \multicolumn{2}{c}{$P^{}_{\rm LDF} (\alpha Z)$ }
                                                     & \multicolumn{2}{c}{$P^{}_{\rm PZ}(\alpha Z)$}
\\
\hline
10 & 8&8762(1)  &   8&9145   &   6&2670      &   6&1840   &    6&6098  \\

12 & 8&1943(1)  &   8&2333   &   6&1987      &   6&1326   &    6&4787  \\

14 & 7&6447(1)  &   7&6847   &   6&0614      &   6&0069   &    6&2955  \\

16 & 7&1911(1)  &   7&2325   &   5&8998      &   5&8539   &    6&0995  \\

18 & 6&8101(1)  &   6&8539   &   5&7349      &   5&6953   &    5&9081  \\

20 & 6&4860(1)   &   6&5309   &   5&5740   &    5&5395     &    5&7264   \\

24 & 5&9670(1)   &   6&0151   &   5&2844   &    5&2571     &    5&4065   \\

28 & 5&5753(1)   &   5&6267   &   5&0429   &    5&0205     &    5&1446   \\

30 & 5&4160(1)   &   5&4703   &   4&9412   &    4&9208     &    5&0351   \\

32 & 5&2771(1)   &   5&3341   &   4&8509   &    4&8322     &    4&9382   \\

40 & 4&8840(1)   &   4&9487   &   4&5900  &    4&5760     &    4&6588   \\

48 & 4&6937(1)   &   4&7686   &   4&4789  &    4&4680     &    4&5375   \\

50 & 4&6727(1)    &   4&7499   &   4&4723  &    4&4619     &    4&5290   \\

56 & 4&6697(1)    &   4&7530   &   4&5028  &    4&4940     &    4&5557   \\

60 & 4&7182(1)  &   4&8035   &   4&5658  &    4&5578     &    4&6171   \\

64 & 4&8098(2)  &   4&8958   &   4&6670  &    4&6596     &    4&7173   \\

70 & 5&039(1)  &   5&1144   &   4&8928  &    4&8865     &    4&9429   \\

72 & 5&144(1)  &   5&2114   &   4&9908  &    4&9847     &    5&0411  \\

80 & 5&753(3)  &   5&7437   &   5&5200  &    5&5152     &    5&5728   \\

82 & 5&965(3)  &   5&9188   &   5&6926  &    5&6881     &    5&7464   \\

90 & 7&19(2)  &   6&8284   &   6&5850  &    6&5818     &    6&6449   \\

92 & 7&64(2)  &   7&1187   &   6&8689  &    6&8662     &    6&9309   \\

\hline

\end{tabular}
\end{table}

\begin{table}
  \caption{The Breit, QED, and total recoil contributions to the $g$ factor of the
    $(1s)^2 2s$ state of  Li-like ions
    expressed in terms of
    the function $F(\alpha Z)$ defined by Eq. (\ref{F}).
  }
\label{F_tab}
\begin{tabular}{cr@{.}lr@{.}lr@{.}lr@{.}lr@{.}l} \hline
$Z$& \multicolumn{2}{c}{$F_{\rm Breit}$}
                    & \multicolumn{2}{c}{$F_{\rm QED}$}
  & \multicolumn{2}{c}{$F_{\rm total}$ }
\\
\hline
10  &   0&1962(1)  &   0&0003(1)    &  0&1965(1)  \\
12  &   0&2056  &   0&0005(1)    &  0&2061(1)  \\
14  &   0&2123  &   0&0008(1)    &  0&2131(1)  \\
16  &   0&2173  &   0&0012(1)    &  0&2185(1)  \\
18  &   0&2213  &   0&0016(2)    &  0&2229(2)  \\
20  &   0&2245  &   0&0022(2)    &  0&2266(2)  \\
24  &   0&2294  &   0&0035(3)    &  0&2330(3)  \\
28  &   0&2332  &   0&0054(3)    &  0&2385(3)  \\
30  &   0&2348  &   0&0065(4)    &  0&2412(4)  \\
32  &   0&2362  &   0&0077(4)    &  0&2439(4)  \\
40  &   0&2411  &   0&0142(6)    &  0&2553(6)  \\
48  &   0&2452  &   0&0240(8)    &  0&2692(8)  \\
50  &   0&2461  &   0&0271(9)    &  0&2732(9)  \\
56  &   0&2487(1)  &   0&0383(11)    &  0&2871(11)  \\
60  &   0&2503(1)  &   0&0478(13)    &  0&2982(13)  \\
64  &   0&2517(2)  &   0&0593(15)    &  0&3110(16)  \\
70  &   0&2533(3)  &   0&0814(20)    &  0&3347(20)  \\
72  &   0&2536(4)  &   0&0904(22)    &  0&3440(22)  \\
80  &   0&2533(10)  &   0&1372(30)    &  0&3904(32)  \\
82  &   0&2525(12)  &   0&1523(33)    &  0&4048(36)  \\
90  &   0&2446(28)  &   0&2331(54)    &  0&4777(61)  \\
92  &   0&2410(35)  &   0&2597(65)    &  0&5007(73)  \\

\hline
\end{tabular}
\end{table}


\section{Introduction}

The star formation history and the gas flows determine the distribution of chemical abundances in galaxies. The existence of radial gradients of chemical abundances in spiral galaxies is a well known fact, and is thought to be caused by 'inside-out' galaxy formation scenario, where the number of generations of stars that have existed is smaller as the galactocentric distance increases.\\

Almost no metals were created in the beginning of the Universe. Thus, the difference in metallic content between two stellar generations is noticeable and a radial gradient can be traced. On the other hand, the biggest fraction of helium was created during the Primordial Nucleosynthesis and the determination of its radial gradient is highly sensitive to errors and uncertainties in the calculation of He/H ratio.\\

To determine a radial gradient in the Galactic distribution of helium requires a sample of objects with good quality spectra that covers a wide range of Galactocentric distances. HII regions are suitable for this purpose since they trace the present-day chemical composition of the Galaxy.

\section{Procedure}
\label{procedure}

\begin{figure}[t]
\begin{center}
\hspace{0.25cm}
\includegraphics[height=5.0cm]{mendez-fig1.eps}
\caption{Distribution of HII regions of the sample. The Galactocentric distance is presented in kpc. The yellow dot indicates the position of the Sun (8 kpc).}
\label{hiimap}
\end{center}
\end{figure}

Our sample consists on 37 deep spectra of 23 Galactic HII regions, covering a range of radial distance between 5 and 17 kpc. In Figure \ref{hiimap} the distribution of the HII regions in the Galaxy disk is shown.\\

The data have been obtained with the high-resolution spectrograph UVES at VLT, OSIRIS at GTC and MagE at Magellan Telescope \citep{gr_04_NGC3576, ce_04_ORION, gr_05_S311, gr_06, gr_07, ce_13_NGC2579, ce_16, ce_17}. The spectral resolution and wavelength coverage allowed us to detect several He$\thinspace$I recombination lines in each spectrum in both spin configurations: triplet and singlet. \\

We calculated the abundance of He$^+$ using PyNeb \citep{pyneb_ref} with all the detected He$\thinspace$I lines in the sample. Some lines as $\lambda$3889, $\lambda$7065, $\lambda$7281 and $\lambda$9464 were discarded from the analysis for being deeply affected by line blending, self-absorption, collisions or blending with sky lines. Some other lines were also discarded in particular cases, as $\lambda$5016 in the Orion Nebula \citep{ce_04_ORION}. The effective recombination coefficients used were those calculated by \cite{porter_2013}, which include corrections for collisional effects. \\

To calculate the total helium abundance, the He$^0$/H$^+$ ratio was estimated using four Ionization Correction Factors (ICF) based on similarities between ionisation potentials of sulfur and/or oxygen and helium \citep{peimbert_77, kunth_83, peimbert_92, zhang_03}. Using Galactocentric distance estimates from the literature for each region of the sample, we derived the radial distribution of helium, analysing separately the results based on singlet and triplet lines.

\section{Results and conclusions }
\label{results}

Estimates of He$^+$/H$^+$ and their associated dispersions were comparatively higher using triplet lines.  As an example, Figure \ref{orion} shows calculations of He$^+$/H$^+$ obtained with every detected He$\thinspace$I line in the Orion Nebula. Figure \ref{lineas} shows the overestimate of He$^+$/H$^+$ in calculations based on the triplet He$\thinspace$I $\lambda$5876 line compared to the average value from singlets.\\

\begin{figure}[h]
\begin{center}
\hspace{0.25cm}
\includegraphics[height=4.5cm]{mendez-fig2.png}
\caption{Abundance of He$^+$ obtained with several recombination lines. Horizontal lines represent the average value for the singlets and triplets. The coloured band indicates the associated uncertainty. The red symbols represent triplets while blue ones represent singlets. Crosses indicate discarded lines.}
\label{orion}
\end{center}
\end{figure}

\begin{figure}[h!]
\begin{center}
\hspace{0.25cm}
\includegraphics[height=4.5cm]{mendez-fig3.eps}
\caption{Representation of the He$^{+}$/H$^+$ ratio determined for He$\thinspace$I 5876\AA$\thinspace$ with respect to the mean He$^+$/H$^+$ ratio determined from singlets (indicated by the color of the circles) as a function of T$_e$ and $n_e$.}
\label{lineas}
\end{center}
\end{figure}

The resulting radial distribution of helium has negative slope when singlet lines were used regardless of the ICF used. In the case of triplet lines, the slope depends on the ICF used. Calculations based on triplet lines are affected by self-absorption mechanisms that are not easy to correct. Figure \ref{radial} shows the radial distribution of helium based on singlet lines and the average value if all the ICF are considered.

\begin{figure}[h]
\begin{center}
\hspace{0.25cm}
\includegraphics[height=4.5cm]{mendez-fig4.eps}
\caption{Radial distribution of He/H calculated with singlet lines and the average value of all ICF used. Red symbols have ICF$\geq$1.2 and were not
used for the linear fit.}
\label{radial}
\end{center}
\end{figure}

Our preliminar results are summarised as follows:
\begin{enumerate}
\item{Almost all triplet lines comparatively provide overabundances in the He$^{+}$/H$^{+}$ ratio. The dispersion is also larger. Corrections for collisional processes are considered in the He abundance determinations. This suggests that the effects of self-absorption are not negligible in most triplet lines.}
\item{Discarding triplet lines, the radial distribution of helium has a negative slope regardless of the ICF used. Nevertheless, due to the uncertainty introduced by the ICFs, errors in the slope are still consistent with a flat distribution.}
\item{HII regions associated with Wolf-Rayet or evolved O stars present helium overabundances.}
\end{enumerate}

\acknowledgments CEL and JRG acknowledge support from the project AYA2015-65205-P. JGR acknowledges support from an Advanced Fellowship from the Severo Ochoa excellence program (SEV-2015-0548).

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