\section{Introduction}

The first report of a detection of \igr\ in the hard X-ray band is found in
the {\it INTEGRAL}/IBIS all-sky survey catalogues, which are based on data
taken before the end of 2006 \citep{krivonos07,bird07}. At that time, its
nature was unknown, other than it was a transient source, detected during a
series of observations between December 2002 and February 2004 but which
was below the threshold of the {\it INTEGRAL} detectors between 2004-2007
\citep{bikmaev08}. A {\it Chandra} observation was performed on 18 December
2006, that is, during the off state of the {\it INTEGRAL} instruments.
However, a weak source consistent with the position of \igr\ was detected.
The {\it Chandra} observation allowed the refinement of its X-ray position
and the suggestion of an optical counterpart \citep{sazonov08}.
Low-resolution ($FWHM \sim 15$ \AA) optical spectroscopic observations of
the likely counterpart indicated a B3 star. Although the
H$\alpha$ line was found in absorption, \igr\ was proposed to be a
high-mass X-ray binary with a Be star companion. It was argued that the
star was going through a disc-loss episode at the time of the observations
\citep{bikmaev08}.

Be/X-ray binaries are a class of high-mass X-ray binaries that consist of a
Be star and a neutron star \citep{reig11}.  The mass donor in these systems
is a relatively massive ($\simmore 10 ~{\rm M}_\odot$) and fast-rotating
($\simmore$80\% of break-up velocity) star, whose equator is surrounded by
a disc formed from photospheric plasma ejected by the star.  \ha\ in
emission  is typically the dominant feature in the spectra of such stars.
In fact, the strength of the Balmer lines in general and of \ha\ in
particular (whether it has ever been in emission) together with a
luminosity class III-V constitute the defining properties of this class of
objects.  The equatorial discs are believed to be quasi-Keplerian and
supported by viscosity \citep{okazaki01}.  The shape and strength of the
spectral emission lines are useful indicators of the state of the disc.
Global disc variations include the transition from a Be phase, i.e., when
the disc is present, to a normal B star phase, i.e., when the disc is
absent and also cyclic V/R changes, i.e., variation in the ratio of the
blue to red peaks of a split profile that are attributed to the precession
of a density perturbation inside the disc \citep{okazaki91,papaloizou06}.

In this work we present the first long-term study of the optical
counterpart to the X-ray source \igr\ and report a disc-loss episode. The
absence of the disc allows us to derive some of the fundamental physical
parameters such as reddening, distance, and rotation velocity, without
contamination from the disc. We also confirm that \igr\ is a Be/X-ray
binary, although with an earlier spectral type than the one suggested by
\citet{bikmaev08}.

\begin{table}
\caption{Log of the spectroscopic observations in the blue region.}
\label{blue}
\centering
\begin{tabular}{@{~~}l@{~~}c@{~~}l@{~~}l@{~~}c}
\noalign{\smallskip}	\hline \noalign{\smallskip}
Date	&JD 			&Telescope  	&Wavelength	&num. of	       \\
	&(2,400,000+)		&		&coverage (\AA)	&spectra	       \\
\noalign{\smallskip}\hline\noalign{\smallskip}
13-09-2012  	&56184.34	&SKO    &3810--5165	&3 \\
14-09-2012	&56185.31	&SKO    &3800--5170	&5 \\
15-10-2012  	&56216.74	&FLWO	&3890--4900	&5 \\
26-12-2012  	&56288.36	&WHT	&3870--4670	&3 \\
15-07-2013	&56488.87	&FLWO	&3900--4900	&3 \\
23-08-2013	&56528.58	&WHT	&3840--4670	&3 \\
\noalign{\smallskip}	\hline
\end{tabular}
\end{table}

\begin{figure*}
\resizebox{\hsize}{!}{\includegraphics{./fig1.eps}}
\caption[]{Evolution of the \ha\ and He I 6678 \AA\ lines. Absorption by
the disc along the line of sight produces very narrow lines (shell
profiles). }
\label{haprof}
\end{figure*}

\section{Observations}

Optical spectroscopic and photometric observations of the optical
counterpart to the INTEGRAL source \igr\ were obtained from the 1.3m
telescope of the Skinakas observatory (SKO) in Crete (Greece) and from the
Fred Lawrence Whipple Observatory (FLWO) at Mt. Hopkins (Arizona).  In
addition, \igr\ was observed in service time with the 4.2-m William
Herschel Telescope telescope (WHT) of El Roque de los Muchachos observatory
in La Palma (Spain).

The 1.3\,m telescope of the Skinakas Observatory was equipped with a
2000$\times$800 ISA SITe CCD and a 1302 l~mm$^{-1}$ grating, giving a
nominal dispersion of $\sim$1.04 \AA/pixel. On the nights 29 September 2009
and 6 September 2011, a 2400 l~mm$^{-1}$ grating with a dispersion of
$\sim$0.46 \AA/pixel was used. We also observed \igr\ in queue mode with
the 1.5-m telescope at Mt. Hopkins (Arizona), and the FAST-II spectrograph
\citep{fabricant98} plus FAST3 CCD, a backside-illuminated 2688x512 UA
STA520A chip with 15$\mu$m pixels and a 1200 l~mm$^{-1}$ grating (0.38
\AA/pixel). The WHT spectra were obtained in service mode on the nights 26
December 2012 and 23 August 2013 with the ISIS spectrograph and the R1200B
grating plus the EEV12 4096$\times$2048 13.5-$\mu$m pixel CCD (0.22
\AA/pixel) for the blue arm and the R1200R grating and REDPLUS
4096$\times$2048 15-$\mu$m pixel CCD (0.25 \AA/pixel) for the red arm.  The
spectra were reduced with the dedicated packages for spectroscopy of the
{\tt STARLINK} and  {\tt IRAF} projects following the standard procedure.
In particular, the FAST spectra were reduced with the FAST pipeline
\citep{tokarz97}. The images were bias subtracted and flat-field corrected.
Spectra of comparison lamps were taken before each exposure in order to
account for small variations of the wavelength calibration during the
night. Finally, the spectra were extracted from an aperture encompassing
more than 90\% of the flux of the object. Sky subtraction was performed by
measuring the sky spectrum from an adjacent object-free region. To ensure
the homogeneous processing of the spectra, they were normalized with
respect to the local continuum, which  was rectified to unity by employing
a spline fit.

The photometric observations were made from the 1.3-m telescope of the
Skinakas Observatory. \igr\ was observed through the Johnson/Bessel $B$,
$V$, $R$, and $I$ filters \citep{bessel90}. For the photometric
observations the telescope was equipped with a  2048$\times$2048 ANDOR CCD
with a 13.5 $\mu$m pixel size (corresponding to 0.28 arcsec on the sky) and
thus provides a field of view of 9.5 arcmin $\times$ 9.5 arcmin. The gain
and read out noise of the CCD camera at a read-out velocity of 2
$\mu$s/pixel are 2.7 $e^{-}$/ADU and 8 $e^{-}$, respectively. The FWHM
(seeing estimate) of the point sources in the images varied from 4 to 6
pixels (1.1''--1.7'') during the different campaigns.    Reduction of the
data was carried out in the standard way using the IRAF tools for aperture
photometry. First, all images were bias-frame subtracted and flat-field
corrected using twilight sky flats to correct for pixel-to-pixel variations
on the chip. The resulting images are therefore free from the instrumental
effects. All the light inside an aperture with radius 4.5'' was summed up
to produce the instrumental magnitudes. The sky background was determined
as the statistical mode of the counts inside an annulus 5 pixels wide and
20 pixels from the center of the object. The absorption caused by the
Earth's atmosphere was taken into account by nightly extinction corrections
determined from measurements of selected stars that also served as
standards. Finally, the photometry was accurately corrected for colour
equations and transformed to the standard system using nightly observations
of standard stars from Landolt's catalogue \citep{landolt92,landolt09}. The
error of the photometry was calculated as the root-mean-square of the
difference between the derived final calibrated magnitudes of the standard
stars and the magnitudes of the catalogue.

The photometric magnitudes are given in Table~\ref{phot}, while information
about the spectroscopic observations can be found in Tables~\ref{red} and
\ref{blue}.

\begin{figure}
\resizebox{\hsize}{!}{\includegraphics{./fig2.eps}}
\caption[]{From top to bottom, evolution of the \ha\ equivalent width,
V/R ratio, $V$ magnitude, $(B-V)$ colour, and velocity shift with time.}
\label{specpar}
\end{figure}

\begin{figure*}
\begin{center}
\includegraphics[width=16cm,height=10cm]{./fig3.eps}
\caption[]{WHT spectrum of \igr\ and identified lines used for spectral
classification. The spectrum was smoothed with a Gaussian filter (FWHM=1).}
\label{wht}
\end{center}
\end{figure*}

\section{Results}

\subsection{The \ha\ line: evolution of spectral parameters}
\label{haevol}

The \ha\ line is the prime indicator of the circumstellar disc state. In
particular, its equivalent width (\ew)  provides a good measure of the size
of the circumstellar disc \citep{quirrenbach97,tycner05,grundstrom06}. \ha\
emission results from recombination of photoionised electrons by the
optical and UV radiation from the central star. Thus, in the absence of the
disc, no emission should be observed and the line should display an
absorption profile. The \ha\ line of the massive companion in \igr\ is
highly variable, both in strength and shape. When the line appears in
emission it always shows a double-peaked profile, but the relative
intensity of the blue (V) over the red (R) peaks varies. The central
absorption that separates the two peaks goes beyond the continuum, placing
\igr\ in the group of the so-called {\em shell} stars
\citep{hanuschik95,hanuschik96a,hummel00,rivinius06}. Figure \ref{haprof}
displays the evolution of the line profiles. V/R variability is clearly
seen, indicating a distorted disc \citep{hummel97}. Significant changes in
the structure of the equatorial disc on timescales of months are observed.
In addition, a long-term growth/dissipation of the disc is suggested by the
increase of the equivalent width and subsequent decrease.

Table~\ref{red} gives the log of the spectroscopic observations and some
important parameters that resulted from fitting  Gaussian functions to the
\ha\  line profile. Due to the deep central absorption, three Gaussian
components (two in emission and one in absorption) were generally needed to
obtain good fits. Column 5 gives the equivalent width of the entire (all
components) \ha\ line. The main source of uncertainty in
the equivalent width stems from the always difficult definition of the
continuum. The \ew\ given in Table~\ref{red} correspond to the average of
twelve measurements, each one from a different definition of the continuum
and the quoted error is the scatter (standard deviation) present in those
twelve measurements.

Column 6 shows the ratio between the core intensity of the blue and red
humps. The V/R ratio is computed as the logarithm of the ratio of the
relative fluxes  at the blue and red emission peak maxima. Thus negative
values indicate a red-dominated peak, that is, $V<R$, and positive values a
blue-dominated line, $V>R$ .

Columns 7 and 8 in Table~\ref{red} give the ratio of the peak flux
of each component over the minimum flux of the deep absorption core. This
ratio simply serves to confirm the shell nature of \igr\ in a more
quantitative way. \citet{hanuschik96a} established an empirical {\em shell
criterion} based on the H$\alpha$ line , whereby shell stars are those with
$F_{\rm p}/F_{\rm cd}\simmore 1.5$, where $F_{\rm p}$ and $F_{\rm cd}$ are
the mean peak and trough flux, respectively.

Column 9 is the velocity shift of the central narrow absorption shell
feature when the disc is present or of the absorption profile of the \ha\
line in the absence of the disc. Prior to the measurement of the velocity
shift, all the spectra were aligned taking the value of the insterstellar
line at 6612.8 \AA\ as reference.  Note that these shifts do not
necessarily represent the radial velocity of the binary, as the \ha\ line
is strongly affected by circumstellar matter \citep[see e.g.,][for a
discussion of the various effects when circumstellar matter is present in
the system]{harmanec03}. Typically, the He I lines in the blue end part of
the spectrum are used for radial velocity studies. Note, however, that in
\igr, even these lines are affected by disc emission (see
Sect.~\ref{diskc}). Nevertheless, we measured the radial velocity of the
binary by cross-correlating the higher resolution blue-end spectra obtained
from the FLWO and WHT (Table~\ref{blue}) with a template using the {\em
fxcor} task in the {\it IRAF} package. This template was generated from the
BSTAR2006 grid of synthetic spectra \citep{lanz07} and correspond to a
model atmosphere with $T_{\rm eff}=25000$ K, $log g=3.75$ convolved by a
rotational profile with $v \sin i=380$ km s$^{-1}$. The results for the
July and August 2013 observations, when the contribution of the disc is
expected to be minimum, are $v_r=-127\pm30$ km s$^{-1}$ (HJD 2,456,488.863)
and $v_r=-120\pm10$ km s$^{-1}$ (HJD 2,456,528.582), respectively.

Figure \ref{specpar} shows the evolution of \ew, the V/R ratio, the V
magnitude, the $(B-V)$ colour, and the velocity shift with time. In the
top panel of this figure, different symbols represent the equivalent width
of the different components of the line. Open circles give the sum of the
equivalent widths of the individual  V and R peaks, while the squares are
the equivalent width of the deep central absorption. These values were
obtained from the Gaussian fits. The overall equivalent width (filled
circles) was measured directly from the spectra.  In the bottom panel
of Fig.~\ref{specpar}, open symbols correspond to the velocity shifts
measured from the \ha\ line, while filled symbols are radial velocities
obtained by crossc-orrelating the 3950--4500 \AA\ spectra with the
template.  Black circles denote SKO spectra, blue triangles correspond to
data taken from the FLWO, and red diamonds spectra obtained with the
WHT.

\subsection{Spectral classification}
\label{specl}

Figure~\ref{wht} shows the average blue spectrum of \igr\ obtained with the
4.2-m WHT on the night 23 August 2013. The main spectral features have been
identified. The 3900--4600 \AA\ spectrum  is dominated by hydrogen and
neutral helium absorption lines, clearly indicating an early-type B star.
The earliest classes (B0 and B0.5) can be ruled out because no ionised
helium is present. However, \ion{Si}{III} 4552-68-75 is crearly detected,
favouring a spectral type B1-B1.5. The relatively weakness of of
\ion{Mg}{II} at 4481 \AA\ also indicates a spectral type earlier than B2.
The strength of the \ion{C}{III}+\ion{O}{II} blend at 4070 \AA\ and 4650
\AA\ agrees with this range (B1--B2) and points toward an evolved star. A
subgiant or giant star, i.e., luminosity class IV or III, is also favoured
by the presence of \ion{O}{II} at 4415-17 \AA. However, in this case, the
triplet {\ion{Si}{III} 4552-68-75 \AA\ and \ion{Si}{IV} 4089 \AA\ should be
stronger than observed. We conclude that the spectral type of the optical
counterpart to \igr\ is in the range B1--B1.5 V--III, with a prefered
classification of B1IV.

\begin{figure}
\resizebox{\hsize}{!}{\includegraphics{./fig4.eps}}
\caption[]{Comparison of two spectra of \igr\ at different epochs, when the
disc was present (solid black line) and when the disc presumably had
vanished (dashed red line). The spectra were taken from the FLWO on 15
October 2012 and 15 July 2013, respectively.}
\label{widthcomp}
\end{figure}

\subsection{Contribution of the disc to the spectral lines and colours}
\label{diskc}

Figure~\ref{widthcomp} shows a comparison of two blue-end spectra of \igr\
taken from the FLWO at two different epochs. The October 2012 spectrum
corresponds to a Be phase when the \ha\ line was strongly in emission,
while the July 2013 spectrum corresponds to a B phase when this line
displayed an absorption profile. One of the most striking results that can
be directly derived from the visual comparison of the spectra is the
significantly narrower width of the spectral lines, particularly those of
the Balmer series and the He I lines, when the disc is present. To estimate
the contribution of the disc to the width of the lines we measured the FWHM
of the  \ion{He}{I} at 6678 \AA\ (Fig.~\ref{haprof}) as a function of time.
The result can be seen in Fig.~\ref{hei}, where the evolution of \ew\ with
time is also plotted.  The width of the helium line was significantly
narrower and the core deeper during the strong shell phase (spectra taken
during 2012, MJD $\sim$56100--56200) than at instances where the disc was
weak, in 2009 and 2013).

The disc emission also affects the photometric magnitudes and colours. The
observed $(B-V)$ colour when the disc was present (observation taken in
2011) is 0.05 mag larger than during 2013 when the disc disappeared. That
is, the disc introduces and extra reddening component.

\begin{figure}
\resizebox{\hsize}{!}{\includegraphics{./fig5.eps}}
\caption[]{Evolution of the full width half maximum of the He I
$\lambda$6678 line. Note the progressive narrowing of the line as the shell
phase develops and the large values when \ew\ is positive (interpreted as
the absence of the disc).}
\label{hei}
\end{figure}

\subsection{The He I lines: rotational velocity}
\label{rotvel}

Shell stars are Be stars with strongly rotationally broadened photospheric
emission lines with a deep absorption core \citep{rivinius06}.  The
rotational velocity is believed to be a crucial parameter in the formation
of the circumstellar disc. A rotational velocity close to the break-up or
critical velocity (i.e. the velocity at which centrifugal forces balance
Newtonian gravity) reduces the effective equatorial gravity to the extent
that weak processes such as gas pressure and/or non-radial pulsations may
trigger the ejection of photospheric matter with sufficient energy and
angular momentum to make it spin up into a Keplerian disc. Because stellar
absorption lines in Be stars are rotationally broadened, their widths can
be used to estimate the projected rotational velocity, $v \sin i$, where
$v$ is the equatorial rotational velocity and $i$ the inclination angle
toward the observer.  However, to obtain a reliable measurement of the
rotational velocity of the Be star companion the He I lines have to be free
of disc emission. As shown in the previous section, the width of the
line becomes narrower as the disc grows, underestimating the value of the
rotational velocity.

We estimated the projected rotational velocity of \igr\ by measuring
the full width at half maximum (FWHM) of He I lines, following the
calibration by \citet{steele99}. These authors used four neutral helium
lines, namely 4026 \AA, 4143 \AA, 4387 \AA, and 4471 \AA, to derive
rotational velocities. We measured the width of these lines from the August
2013 WHT spectrum as it provides the highest resolution in our sample and
correspond to a disc-loss phase, as indicated by the absorption profile of
the \ha\ line. We made five different selections of the continuum and
fitted Gaussian profiles to these lines.  We also corrected the lines for
instrumental broadening by subtracting in quadrature the FWHM of a nearby
line from the calibrated spectra.  The projected rotational velocity
obtained as the average of the values from the four He I lines was
$v \sin i=365\pm15$ km s$^{-1}$. The quoted errors are the standard
deviation of all the measurements.

The rotational velocity can also be estimated by comparing the
high-resolution August 2013 WHT spectrum with a grid of synthetic spectra
broadened at various values of the rotational velocity.  We employed the
BSTAR2006 grid \citep{lanz07}, which uses the code TLUSTY
\citep{hubeny88,hubeny92,hubeny94} to create the model atmosphere and
SYNSPEC\footnote{http://nova.astro.umd.edu} to calculate the emergent
spectrum. We assumed a model atmosphere with solar composition, $T_{\rm
eff}=25000$ K and $\log g=3.50$ and a microturbulent velocity of 2 km
s$^{-1}$. This spectrum was convolved with rotational and intrumental
(gaussian) profiles using ROTIN3.  Thirteen rotational velocities from 300
km s$^{-1}$ to 420 km s$^{-1}$ with steps of 10 km s$^{-1}$ were
considered. The rotational velocity that minimises the sum of the squares
of the difference between data and model corresponded to $v \sin i=380$ km
s$^{-1}$, consistent with the previous value.

\subsection{Reddening and distance}

To estimate the distance, the amount of reddening to the source has to be
determined. In a Be star,  the total measured reddening is made up of two
components: one produced mainly by dust in the interstellar space through
the line of sight and another produced by the circumstellar gas around the
Be star \citep{dachs88,fabregat90}. Although the  physical origin and
wavelength dependence of these two reddenings is completely different,
their final effect upon the colours is very difficult to disentangle
\citep{torrejon07}. In fact, interstellar reddening is caused by {\em
absorption} and {\em scattering} processes, while circumstellar reddening
is due to extra {\em emission} at longer wavelenths. The disc-loss episode
observed in \igr\ allows us to derive the true magnitudes and colours of
the underlying Be star, without the contribution of the disc. Thus the
total reddening measured during a disc-loss episode corresponds entirely to
interstellar extinction.

The observed colour of \igr\ in the absence of the disc is
$(B-V)=0.50\pm0.02$ (Table~\ref{phot}), while the expected one for a
B1--1.5V--IV star is $(B-V)_0=-0.26$
\citep{johnson66,fitzgerald70,gutierrez-moreno79,wegner94}. Thus we derive
a colour excess of $E(B-V)=0.76\pm0.02$ or visual extinction $A_{\rm V}=R
\times E(B-V)= 2.4\pm0.1$, where the standard extinction law $R=3.1$ was
assumed. Taking an average absolute magnitude  of $M_V=-3.0$, typical of a
star of this spectral type \citep{humphreys84,wegner06}, the distance to
\igr\ is estimated to be 8.7$\pm$1.3 kpc. The final error was obtained by
propagating the errors of $B-V$  (0.02 mag), $A_V$ (0.1 mag) and $M_V$ (0.3
mag).

\section{Discussion}
\label{discussion}

We have investigated the long-term variability of the optical counterpart
to the X-ray source \igr. \citet{bikmaev08} suggested that \igr\ is a
high-mass X-ray binary with a B3e companion, even though their
spectroscopic observations showed \ha\ in absorption. They argued that the
system could be in a disc-loss phase. We confirm the Be nature of \igr, but
suggest an earlier type companion, in agreement with the spectral type
distribution of Be/X-ray binaries in the Milky Way.  All spectroscopically
identified optical companions of  Be/X-ray binaries in the Galaxy do not
have spectral type later than B2 \citep{negueruela98b}.

\begin{table*}
\caption{Comparison of the characteristic time scales of \igr\ with other
Be/X-ray binaries. $P_{\rm orb}$ is the orbital period, $T_{\rm V/R}$is
the time needed to complete a V/R cycle, and $T_{\rm disc}$ is the time
for the formation and dissipation of the disc. }
\label{comp}
\centering
\begin{tabular}{@{~~}l@{~~}l@{~~}c@{~~}c@{~~}c@{~~}c@{~~}c@{~~}l}
\noalign{\smallskip} \hline \noalign{\smallskip}
X-ray		&Spectral	&Disc-loss      &P$_{\rm orb}$   &$T_{\rm V/R}$	&$T_{\rm disc}$	&Reference \\
source		&type		&episode\tablefootmark{$^\dag$}&(days)	&(year)	&(year)	& \\
\noalign{\smallskip} \hline \noalign{\smallskip}
\igr		&B1V-IV	 	&yes		&--		&0.5-0.8	&5--6	&This work \\
4U 0115+634	&B0.2V  	&yes		&24.3		&0.5--1 	&3--5	&1,2      \\
RX J0146.9+6121	&B1III-V	&no		&--		&3.4    	&--	&3    \\
V 0332+53	&O8-9V  	&no		&34.2		&1      	&--	&4    \\
X-Per		&O9.5III	&yes		&250		&0.6--2 	&7	&5,6,7  \\
RX J0440.9+4431	&B1III-V	&yes		&--		&1.5--2     	&$>10$	&8      \\
1A 0535+262	&O9.7III	&yes		&111		&1--1.5 	&4--5	&9,10,11 \\
IGR J06074+2205	&B0.5IV 	&yes		&--		&--      	&4--5	&12      \\
RX J0812.4-3114	&B0.5III-V	&yes		&81.3		&--     	&3--4	&13 \\
4U 1145-619	&B0.2III	&no		&187		&3      	&--	&14  \\
4U 1258-61	&B2V		&yes		&132		&0.36     	&--	&15  \\
SAX J2103.5+4545&B0V		&yes		&12.7		&--		&1.5--2	&16 \\
\noalign{\smallskip} \hline
\end{tabular}
\tablefoot{
\tablefoottext{$^\dag$}{By disc-loss episodes we mean periods when the \ew\ was seen to be positive.}
}
\tablebib{
(1) \citet{negueruela01}; (2) \citet{reig07b}; (3) \citet{reig00} ;
(4) \citet{negueruela98a}; (5) \citet{lyubimkov97} ;
(6) \citet{delgado01}; (7) \citet{clark01}; (8) \citet{reig05b} ;
(9) \citet{clark98}; (10) \citet{haigh04}; (11) \citet{grundstrom07} ;
(12) \citet{reig10b}; (13) \citet{reig01};
(14) \citet{stevens97} ; (15) \citet{corbetgx86} ; (16) \citet{reig10a}
}
\end{table*}

\subsection{Spectral evolution and variability time scales}

Our monitoring of \igr\ reveals large amplitude changes in the shape and
strength of the spectral lines and two different time scales
associated with the variability of the disc: disc formation/dissipation is
estimated to occur on time scales of years, while V/R variability is seen
on time scales of months.

Our first observation was performed in July 2009 and shows the contribution
of a weak disc. Although the \ew\ is positive, indicating that absorption
dominates over emission, its value is smaller than that expected from a
pure photospheric line, which according to \citet{jaschek87} should be
$\sim$3.5--4 \AA. Also, the two peaks, V and R, separated by the central
depression can already be distinguished in our first spectrum.  The
strength of the \ha\ line increased and its shape changed from an
absorption dominated profile into an emission dominated one during the
period July 2009--September 2012. As the intensity increased, the line
became progressively more asymmetric. After September 2012, \ew\ began to
decrease with a faster rate than the rise. By Summer 2013, the \ha\ line
profile had turned into absorption, that is, the system entered a disc-loss
episode.

Due to the observational gaps, it is difficult to determine the overall
time scale for the formation and dissipation of the circumstellar disc.
The observations of \igr\ by \citet{bikmaev08} were made in Spring 2007
(low resolution) and Autumn 2007 (high-resolution) and show the \ha\ line
in absorption, although in the high-resolution spectrum the shape of the
line is reminiscent of a shell profile, i.e., with the contribution of a
small disc. If the Spring 2007 spectrum really showed an absorption profile
similar to our latest observations, then we can estimate the
formation/dissipation cycle to be about six years.

In addition to large amplitude changes in the strength of the \ha\ line,
\igr\ also display marked variations in the shape of the spectral lines.
The most prominent spectroscopic evidence of disc activity is the long-term
V/R variability, that is, the cyclic variation of the relative intensity of
the blue ($V$) and red ($R$) peaks in the split profile of the line. The V/R
variability is believed to be caused by the gradual change of the amount of
the emitting gas approaching the observer and that receding from the
observer due to the precession of a density perturbation in the disc
\citep{kato83,okazaki91,okazaki97,papaloizou06}. Double-peak symmetric
profiles are expected when the high-density part is behind or in front of
the star, while asymmetric profiles are seen when the high-density
perturbation is on one side of the disc \citep{telting94}.

In principle, it is possible from the data themselves to find out whether
the perturbation travels in the same direction as Keplerian orbits of the
material in the disc (prograde precession) or in opposite direction
(retrograde precession). If the motion of the density perturbation is
prograde and the disc is viewed at a high inclination angle (as in \igr),
then the $V>R$ phase should be followed by a $V=R$ phase with a strong
shell profile corresponding to the case where the perturbation lies between
the star and the observer \citep[see][for a sketch of prograde
motion]{telting94}.

In \igr, as the disc grew, the \ha\ line changed from a symmetric to an
asymmetric profile. When the disc was weak, that is, when the equivalent
width of the \ha\ line was a few Angstrom (before 2012), the intensity of
the blue and red peaks was roughly equal, $V=R$. The data reveal that the
perturbation developed between the end of 2011 and beginning of 2012. From
August 2012, a blue dominated profile is clearly present ($V>>R$). However,
the V/R ratio showed a fast decrease through the 2012 observations
(Fig.~\ref{specpar}), indicating that the V$>$R phase was coming to an end.
This was confirmed by the January 2013 observations, where an extreme
red-dominated profile is seen, $V/R\approx -0.9$.

It is worth noticing the extremely fast V/R time scales.  Although it is
not possible to pin down the exact moment of the onset of the V/R cycle due
to an observational gap of about a year (September 2011-August 2012), the
changes occurred very rapidly once the cycle started. During the $V>R$
phase, the V/R ratio changed from $\sim+0.4$ to $\sim+0.1$ in less than one
month, September-October 2012 (see Fig.~\ref{haprof} and Table~\ref{red}).
Likewise, the change from a blue-dominated $V>R$ to a red-dominated $V<R$
profile occurred in just two months (October-December 2012). If the motion
is prograde, a $V=R$ phase with a strong shell profile should be observed
in between the blue-dominated and red-dominate phases. We seem to have
missed most of this phase, which would have occurred between October and
December 2012. That is, in just two months the density perturbation must
have gone through the shell $V=R$ phase and most of the $V<R$ phase.
Extrapolating this behaviour, we estimate the duration of a whole
revolution to be of 6--9 months.  These changes are among the fastest in a
BeXB.  The very short time scale of the observed V/R variations
rises the question of whether these spectral changes are modulated by the
orbital period. Phase-locked V/R variations have been observed for various
Be binaries with hot companions \citep[][and references
therein]{harmanec01}, but possibly only on one BeXB (4U\,1258--61).
\citet{corbetgx86} found that the probability that the V/R variability
observed in 4U\,1258--61 was modulated by the X-ray flare period of $\sim
132$ d, which was proposed to be the orbital period of the system, was
$\sim$87\%.  Table~\ref{comp} gives a few well-studied characteristic time
scales of BeXBs: the orbital period, $P_{\rm orb}$, the V/R quasi-periods,
$T_{\rm V/R}$, and the approximate duration of the formation/dissipation of
the disc, $T_{\rm disc}$.

In common with other BeXBs \citep{reig05b,reig10b},
asymmetric profiles are not seen until the disc reaches certain size and
density. During the initial stages of disc growth, the shape of the \ha\
line is always symmetric. Only when \ew\ $\simmore -6$ \AA\ the \ha\ line
displays an asymmetric profile. \igr\ also agrees with this
result. As can be seen in Fig.~\ref{haprof}, all the spectra during the
period 2009--2011 show a symmetric profile and a small \ew. The star took
all this time to build the disc. Once a critical size and density was
reached (some time at the beginning of 2012),  the density perturbation
developed and started to travel around in the disc. This result is also
apparent from a comparison of the two panels in Fig.~\ref{specpar}. Before
MJD 55800, $V\approx R$ and \ew\ low. After MJD 56000, a strong asymmetric
emission profile is seen.

Although the maximum \ew\ measured is relatively small, \ew $\approx -8$
\AA, compared to other BeXB, it is consistent with a well developed disc.
In systems viewed edge-on, the maximum \ew\ is much smaller than in the
face-on case, because the projected area of the optically thick disc on the
sky is much smaller \citep{hummel94,sigut13}.  On the other hand, although
a fully developed disc must have been formed, it may not extend too far
away from the star. The fast V/R changes favoured a relatively compact
disc, where the density perturbation is capable to achieve a complete
revolution in a few months.

\subsection{Shell lines and disc contribution}

The Be star companion in \igr\ is a shell star as implied by the deep
central absorption between the two peaks of the \ha. This central
depression clearly goes beyond the continuum. The shell profiles are
thought to arise when the observer's line of sight toward the central star
intersects  parts of the disc, which is cooler than the stellar photosphere
\citep{hanuschik95,rivinius06}.   Statistical studies on the distribution
of rotational velocities of Be stars are consistent with the idea that Be
shell stars are simply normal Be stars seen near edge-on, that is, seen at
a large inclination angle \citep{porter96}. Our observations agree with
this idea. Figure~\ref{widthcomp} clearly shows that the spectral lines are
significantly narrower and deeper when the disc is present. The narrower
lines would result from the fact that the disc conceals the equator of the
star, where the contribution to the rotational velocity is largest. The
deeper lines would result from absorption of the photospheric emission by
the disc. Both circumstances require a high inclination angle. Further
evidence for a high inclination angle is provided by the photometric
observations. Both, positive and negative correlations between the
emission-line strength and light variations have been observed and
attributed to  geometrical effects \citep{harmanec83,harmanec00}.  Stars
viewed at very high inclination angle show the inverse correlation because
the inner parts of the Be envelope  block partly the stellar photosphere,
while the small projected area of the disc on the sky keeps the disc
emission to a minimum. In stars seen at certain inclination angle,
$i\simmore i_{\rm crit}$, the effect of the disc is to increase the
effective radius of the star, that is, as the disc grows an overall (star
plus disc) increase in brightness is expected. The value of the critical
inclination angle is not known but a rough estimate based on available data
suggest $i_{\rm crit}\sim 75^{\circ}$ \citep{sigut13}. \igr\ exhibits the
inverse correlation, that is, it becomes fainter at the beginning
of a new emission episode ($\sim$ JD 2,455,800, see Fig.~\ref{specpar}).

We have assesed the contribution of the disc on the width of helium lines,
which is the main parameter to estimate the star's rotational velocity, by
measuring the FWHM of the \ion{He}{I} 6678 \AA\ over time and by
determining the rotational velocity when the disc was present. A difference
of up to 7 \AA\ was measured between the witdh of the \ion{He}{I} 6678 \AA\
line with and without disc (see Fig.\ref{hei}). Repeating the calculation
performed in Sect.~\ref{rotvel} on the \ion{He}{I} 4026 \AA, 4387 \AA, and
4471 \AA\ when the disc was present and using the WHT December 2012
spectra, we obtain  $v\sin i=170\pm20$ km s$^{-1}$. Thus, the contribution
of the disc clearly underestimates the true rotational velocity.

In \igr, the shell profile seems to be a permanent feature.  We do not
observe any transition from a shell absorption profile to a Be "ordinary"
(i.e., pure emission) profile. In models that favour geometrically thin
discs with small opening angles, this result implies that the inclination
angle must be well above  $70^{\circ}$ \citep{hanuschik96a}. Thus, with
such a large inclination angle, the true rotational velocity is estimated
to be $v_{\rm rot}\sim$380-400 km s$^{-1}$, and the ratio of the equatorial
rotational velocity over the critical break-up velocity, $w=v_{\rm
rot}/v_{\rm crit}\sim0.8$\footnote{The break-up velocity of a B1Ve star
is $\sim 500$ km s$^{-1}$ \citep{porter96,townsend04,cranmer05}.}

If gravity darkening is taken into account, then the fractional rotational
velocity would be even larger. Gravity darkening results from fast
rotation. Rapidly rotating B stars have centrifugally distorted shapes with
the equatorial radius larger than the polar radius. As a result, the poles
have a higher surface gravity, and thus higher temperature. Gravity
darkening breaks the linear relationship between the line width and the
projected rotational velocity and makes fast rotators display narrower
profiles, hence underestimate the true rotational velocity. The reduction
of the measured rotational velocities with respect to the true critical
velocity amounts to 10--30\%, with the larger values corresponding to the
later spectral subtypes \citep{townsend04}. Correcting for gravity
darkening, the rotational velocity would be $v_{\rm rot} \sim \approx 450$
km s$^{-1}$ (assuming $i=80^{\circ}$) and the fractional rotational
velocity of the Be companion of \igr\  $w \approx0.9$, confirming the idea
that shell stars are Be stars rotating at near-critical rotation limit. We
caution the reader that the values of the break-up velocity assume that it
is possible to assign to each Be star a mass and radius equal to that of a
much less rapidly rotating B star, e.g., from well studied eclipsing
binaries. Given that there is not a single direct measurement of the mass
and radius for any known Be star, the break-up velocity should be taken as
an approximation \citep[see e.g][]{harmanec00}.

\section{Conclusion}

We have performed optical photometric and spectroscopic observations of the
optical counterpart to \igr. Our observations show that \igr\ is a
high-mass X-ray binary with a Be shell type companion.  Its long-term
optical spectroscopic variability is characterised by global changes in the
structure of the equatorial disc. These global changes manifest
observationally as asymmetric profiles and significant intensity
variability of the \ha\ line. The changes in the strength of the line are
associated with the formation and dissipation of the circumstellar disc. At
least since 2009, \igr\ has been in an active Be phase that ended in mid
2013, when a disc-loss episode was observed. The entire
formation/dissipation cycle is estimated to be six years, although given
the lack of data before 2009, this figure needs to be confirmed by future
observations. In contrast, the  V/R variability is among the fastest
observed in Be/X-ray binaries with characteristic timescales of the order
of few weeks for each V/R phase.

The absence of the disc left the underlying B star exposed, allowing us to
derive its astrophysical parameters. From the ratios of various metallic
lines we have derived a spectral type B1IVe. The width of He I lines imply
a rotational velocity of $\sim370$ km s$^{-1}$. Using the photometric
magnitudes and colours we have estimated the interstellar colour excess
$E(B-V)\sim0.76$ mag, and the distance $d\sim$ 8.5 kpc.

The presence of shell absorption lines indicate that the line of sight to
the star lies nearly perpendicular to its rotation axis. Although the
Balmer lines show the most clearly marked shell variability, the helium lines
are also strongly affected by disc emission, making them narrower than in
the absence of the disc.

\begin{acknowledgements}

We thank the referee P. Harmanec for his useful comments and suggestions
which has improved the clarity of this paper. We also thank observers P.
Berlind and M. Calkins for performing the FLWO observations and I.
Psaridaki for helping with the Skinakas observations.  Skinakas Observatory
is a collaborative project of the University of Crete, the Foundation for
Research and Technology-Hellas and the Max-Planck-Institut f\"ur
Extraterrestrische Physik. The WHT and its service programme (service
proposal references SW2012b14 and SW2013a19) are operated on the island of
La Palma by the Isaac Newton Group in the Spanish Observatorio del Roque de
los Muchachos of the Instituto de Astrof\'{\i}sica de Canarias. This paper
uses data products produced by the OIR Telescope Data  Center, supported by
the Smithsonian Astrophysical Observatory. This work has made use of NASA's
Astrophysics Data System Bibliographic Services and of the SIMBAD database,
operated at the CDS, Strasbourg, France.

\end{acknowledgements}

\bibliographystyle{aa}


\subsection{Band structure and Fermi surface}

The evolution of the band structure as a function of pressure is presented in
FIG.~\ref{bands}, showing no qualitative change in the
electronic states at the Fermi surface up to about 38.8 GPa.
From 56.7~GPa to 78.4~GPa, we observe some changes along the $\Gamma$-H
and $\Gamma$-N lines.

A 3D picture of the Fermi surface at ambient pressure and 56.7~GPa is drawn
in FIG.~\ref{Fermi}.  The lower energy band
forms the octahedron centered at the $\Gamma$-point. With increasing pressure,
this octahedron shrinks and it becomes surrounded
by six little ellipsoids when the previously described band-structure changes
on the $\Gamma$-H line appear.
Around N-point, the Fermi surface forms ellipsoids that are disconnected at
lower pressure and becomes connected by necks to the four neighboring
ellipsoids
above 56.7~GPa. In addition, a complicated open sheet structure, often
referred as "jungle gym", extends from $\Gamma$ to the H points in the BZ.

Our results are in good agreement with
the detailed studies of Anderson {\em et al.}~\cite{Anderson-1}
for the band structure and of Ref.~[\onlinecite{Anderson-2}] for the
Fermi surface.

\begin{figure} \epsfxsize=8cm \centerline{
\includegraphics[scale=0.33,angle=-90]{bands.ps}} \caption{\label{bands}
The band structures of niobium at several pressures (in GPa).}
\end{figure}

\begin{figure}
\caption{\label{Fermi} Fermi surfaces from two bands (the lower energy
band in the left panels and the higher energy band in the right panels)
at ambient pressure (top panels) and at 56.7
GPa (bottom panels).  Pictures obtained with
the XCrySDen package.\cite{xcrysden}} \end{figure}

\section{Results for niobium under pressure}

\subsection{Technical details}

The calculations of the ground-state electronic and vibrational
properties of Nb were performed using the Local-Density Approximation
(LDA) and an ultrasoft pseudopotential.
A kinetic energy cut-off of 45~Ry (270~Ry) was chosen for the expansion
into plane waves of the wavefunctions (density). Such high cut-offs were
necessary to obtain accurate values for some "strategic" low-frequency
phonons, located mostly near the $\Gamma$-point. In fact, even small
errors in this region of the spectrum lead to large relative errors in
the estimate of the  $\alpha^{2}F$ function  and of $\lambda$.

The integration over the Brillouin zone (BZ) requires special techniques
to account for the Fermi surface. We used the broadening technique proposed
in Ref.~[\onlinecite{Paxton}] with a smearing parameter of 0.03 Ry
(which was tested\cite{SdG-broad} to reproduce well
the experimental spectra).
The grids for the electronic BZ integration ({\bf k}-grid)
and for the phononic BZ integration ({\bf q}-grid)
have been chosen according to the Monkhorst-Pack scheme.\cite{mesh}

Details of the numerical quadrature used to evaluate the double-delta
term appearing in Eq.(2), together with convergence tests,
are given in the Appendix.

All calculations were performed using the
{\sc quantum-espresso}\cite{espresso}  suite of codes.

\subsection{Structural properties}

Niobium in the body-centered cubic structure was studied at eight values
of the lattice parameter from 6.34~a.u. to
5.64~a.u in steps of 0.1 a.u.  These lattice parameters correspond to
pressures ranging from -16.6~GPa to 78.4~GPa. The results are reported
in TABLE~\ref{units}. The calculated static equilibrium lattice constant
(zero-point motion and thermal effects not included) is about 6.14~a.u.,
slightly underestimating (as it usually happens within LDA)
the experimental value\cite{Ashcroft} of 6.24~a.u. The calculated bulk
modulus at the theoretical equilibrium lattice is 192~GPa, versus a
room-temperature
experimental value\cite{bulkmod} of 170~GPa and a calculated value of
162~GPa at the experimental lattice parameter of 6.24 a.u.
The calculated bulk modulus is very sensitive to the volume: it varies
by more than a factor three in the considered range of pressures.

\begin{table}
\begin{tabular}{cccc}
\hline \hline
\\[0.01mm]
  $\;\;\;$ $a$ $\;\;\;$ & $\;\;\;$  V/V$_{0}$ $\;\;\;$ &
$\;\;\;$ $P$ $\;\;\;$ & $\;\;\;$ $B$ $\;\;\;$ \\[0.2cm]
\hline \\[0.01mm]
  6.34 &  1.10 & -16.6 & 134  \\
  6.24 &  1.05 & -9.5 & 162  \\
  6.14 &  1.00 &  -0.6 & 192  \\
  6.04 &  0.95 &  10.0 & 220  \\
  5.94 &  0.91 & 22.9 & 249  \\
  5.84 &  0.86 &  38.8 & 308  \\
  5.74 &  0.82 & 56.7 & 354  \\
  5.64 &  0.78 &  78.4 & 424  \\[0.05cm]
\hline \hline
\end{tabular}
\caption{\label{units} The lattice constant $a$ (in a.u.), the corresponding
volume ratio V/V$_{0}$,
the pressure $P$ (in GPa), and bulk modulus $B$ (in GPa) calculated for BCC
niobium crystal at several pressures.
The experimental lattice constant\cite{Ashcroft} is 6.24 a.u.
and the bulk modulus\cite{bulkmod} is 170 GPa.}
\end{table}

\section{Electron-phonon coupling}

\subsection{Definitions}

The Hamiltonian for the electron-phonon interaction is given
in second quantization by
\begin{equation}
H_{el-ph} = \sum_{{\bf k}{\bf q }\nu}
g_{{\bf k}+{\bf q},{\bf k}}^{{\bf q}\nu ,mn} \;
c_{{\bf k}+{\bf q}}^{\dagger m}c_{{\bf k}}^{n} \; (b_{-{\bf q}\nu}^{\dagger} + b_{{\bf q}\nu})
\end{equation}

\noindent
where $c_{{\bf k}+{\bf q}}^{\dagger m}$ and $c_{{\bf k}}^{n}$ are the creation and
the annihilation operators for the quasiparticles with  energies
$\varepsilon_{{\bf k}+{\bf q},m}$ and  $\varepsilon_{{\bf k},n}$
in bands $m$ and $n$ with wavevectors ${\bf k}+{\bf q}$ and ${\bf k}$,
respectively;
$b_{{\bf q}\nu}^{\dagger}$ and  $b_{{\bf q}\nu}$ are the creation
and the annihilation operators for
phonons with energy $\omega_{{\bf q}\nu}$ and wavevector ${\bf q}$;
the matrix element $g_{{\bf k}+{\bf q},{\bf k}}^{{\bf q}\nu, mn}$
describes the electron-phonon coupling.
The coupling constants $g_{{\bf k}+{\bf q},{\bf k}}^{{\bf q}\nu, mn}$
define the spectral function, $\alpha^{2}F( \omega )$, its first reciprocal
momentum, $\lambda$, and the superconducting electron-phonon coupling constant,
$\lambda_{{\bf q}\nu}$, by the following set of equations:
\begin{eqnarray}
\alpha^{2}F( \omega) &=& \frac{1}{N(\varepsilon_{F})} \sum_{mn}\sum_{{\bf q}\nu}
\delta (\omega - \omega_{{\bf q}\nu})\sum_{\bf k}
|g_{{\bf k}+{\bf q},{\bf k}}^{{\bf q}\nu, mn}|^{2} \nonumber \\
&& \times \delta (\varepsilon_{{\bf k}+{\bf q},m}-\varepsilon_{F})
\delta (\varepsilon_{{\bf k},n}-\varepsilon_{F}), \label{a2F} \\
\lambda & = &
2 \int \frac{\alpha^{2}F( \omega)}{\omega} d\omega =  \sum_{{\bf q}\nu} \lambda_{{\bf q}\nu}, \label{lambda} \\
\lambda_{{\bf q}\nu} & = & \frac{2}{N(\varepsilon_{F})\omega_{{\bf q}\nu}} \sum_{mn} \sum_{\bf k}
|g_{{\bf k}+{\bf q},{\bf k}}^{{\bf q}\nu,mn} |^{2} \nonumber \\
 &&  \times \delta ( \varepsilon_{{\bf k}+{\bf q},m} - \varepsilon_{F})
\delta ( \varepsilon_{{\bf k},n} - \varepsilon_{F} ). \label{dirlam}
\end{eqnarray}

\noindent
The quantity $N(\varepsilon_{F})$ is the density of states at the Fermi energy,
$\varepsilon_{F}$, per both spins.

We introduce, after Allen,\cite{AllenGamma} the phonon linewidth
$\gamma_{{\bf q}\nu}$:
\begin{eqnarray}
\gamma_{{\bf q}\nu} & = & 2 \pi \omega_{{\bf q}\nu} \sum_{mn}\sum_{{\bf k}}
|g_{{\bf k}+{\bf q},{\bf k}}^{{\bf q}\nu, mn}|^{2} \; \nonumber \\
&& \times \delta (\varepsilon_{{\bf k}+{\bf q},m}-\varepsilon_{F})
\delta (\varepsilon_{{\bf k},n}-\varepsilon_{F}) \label{eq5}
\end{eqnarray}

\noindent
which enters the Eliashberg function, $\alpha^{2}F$, and the
electron-phonon coupling constant, $\lambda_{{\bf q}\nu}$, as follows:

\begin{eqnarray}
\alpha^{2}F( \omega) & = & \frac{1}{2 \pi \; N(\varepsilon_{F})} \sum_{{\bf q}\nu}
\frac{\gamma_{{\bf q}\nu}}{\omega_{{\bf q}\nu}} \delta (\omega - \omega_{{\bf q}\nu}),
\label{a2Feq}   \\
\lambda_{{\bf q}\nu} & = & \frac{\gamma_{{\bf q}\nu}}
{\pi \; N(\varepsilon_{F}) \; \omega_{{\bf q}\nu}^{2}}. \label{lambdaeq}
\end{eqnarray}

\subsection{Matrix elements of the electron-phonon interactions}

Within DFPT\cite{DFPT,Review} the electron-phonon matrix elements
can be obtained from the first-order derivative of the self-consistent
Kohn-Sham\cite{DFT} (KS) potential, $V_{KS}$,
with respect to atomic displacements, $\vec{u}_{s{\bf R}}$ for the
$s-$th atom in lattice position ${\bf R}$, as:
\begin{equation}
g_{{\bf k}+{\bf q},{\bf k}}^{{\bf q}\nu, mn} = \left( \frac{\hbar}{2\omega_{{\bf q}\nu}} \right)^{1/2}
\langle \psi_{{\bf k}+{\bf q},m} | \Delta V_{KS}^{{\bf q}\nu}  | \psi_{{\bf k},n} \rangle ,
\label{gg}
\end{equation}
where $\psi_{{\bf k},n}$ is the $n-$th valence KS orbital of wavevector
${\bf k}$ and
\begin{equation}
\Delta V_{KS}^{{\bf q}\nu} = \sum_{\bf R} \sum_{s} \frac{\partial V_{KS}}{\partial \vec{u}_{s{\bf R}}}
\cdot \vec{u}_{s}^{{\bf q}\nu} \frac{e^{i{\bf qR}}}{\sqrt{N}}
\label{dV}
\end{equation}
is the self-consistent first order variation of the KS potential,
$N$ is the number of cells in the crystal, and
$\vec{u}_{s}^{{\bf q}\nu}$ is the displacement pattern for phonon mode
$\vec{v}_{s}^{{\bf q}\nu}$:
\begin{equation}
\vec{u}_{s}^{{\bf q}\nu} = \frac{\vec{v}_s^{{\bf q}\nu}}{\sqrt{M_{s}}}
\end{equation}
The latter is obtained from the diagonalization
of the dynamical matrix, $\varPhi^{\alpha\beta}_{ss'}({\bf q})$:
\begin{equation}
\sum_{s'\beta} \frac{\varPhi^{\alpha\beta}_{ss'}({\bf q})}{\sqrt{M_{s}M_{s'}}}
v_{s'\beta}^{{\bf q}\nu}
 =  \omega_{{\bf q}\nu}^{2} v_{s\alpha}^{{\bf q}\nu}. \label{eq11}
\end{equation}
$M_{s}$ is the mass of atom $s$, $\alpha, \beta$ denote cartesian coordinates.

\section{Discussion}

After examination of the phonons and the electron-phonon coupling,
we notice that the decrease of $\lambda$ at high pressure
can be easily related to the  decrease of the $\alpha^{2}F(\omega)$ peak
and to its shift toward higher frequencies for all modes.

The origin of the increase of $\lambda$ at low pressure between
$\approx$0 GPa and $\approx$10 GPa is instead more difficult to trace.
We found out that it is  mainly determined by the anomalous dispersion
of the T1 and T2 modes close to $\Gamma$-point (see FIG.~\ref{ph-p})
in the frequency region below 1~THz,
that determines a low-frequency peak in the Eliashberg function of
the T1 mode above 10 GPa. This peak is instead absent at ambient pressure
and reappears for an expanded lattice only at about -16 GPa.

It has been shown \cite{Vonsovsky} for the Eliashberg model
that the contribution to $T_{c}$ from acoustic modes close to the $\Gamma$-point
vanishes. For the low-frequency modes which are associated to Kohn anomalies,
however, we can expect important contribution to $T_{c}$ because phonon
softening may occur with a finite phonon linewidth.
Therefore, the region near $\Gamma$ can give a very large contribution
to the electron-phonon coupling in niobium for all studied pressures
(see TABLE~\ref{DOS-lambda}).

Let us now consider the band structure.
In order to explain low-pressure anomalies in $T_c$,
Struzhkin {\em et al.}\cite{Struzhkin} proposed
the existence of necks between the
ellipsoids around N and the "jungle gym" open sheet extending from
$\Gamma$ to H along the $\Gamma$-$\Sigma$ line at a pressure below 5~GPa,
and the disappearance of these features at the higher pressures.
Our calculations do not support this suggestion.
The detailed analysis of the Fermi surface reported by Ostanin
{\em et al.} \cite{Ostanin} gives results close to ours.
Previous theoretical
\cite{Solanki,Anderson-2,Neve,lambda-1.52,fermi-1,fermi-2} and
experimental \cite{Anderson-2,fermi-3,fermi-4} investigations of the
Fermi surface also did not detect any changes at low pressure.

One can, therefore, connect the high-pressure decrease
in electron-phonon coupling to changes in the band structure, while
the origin of the low-pressure anomaly remains unclear.
Therefore, we need a different tool in order to detect tiny features
of the Fermi surface.

\begin{figure}
\epsfxsize=8cm
\includegraphics[scale=0.25,angle=-90.0]{nest-a.ps}
\includegraphics[scale=0.25,angle=-90.0]{nest-b.ps}
\vspace{-0.4cm}
\caption{\label{nest} The nesting factor, $X_{\bf q}$,
of the Fermi surface in niobium at eight studied pressures (in GPa),
for selected high-symmetry lines (top panel) and
high-symmetry points (bottom panel).}
\end{figure}
\begin{figure}
\caption{\label{nest-BZ} The isosurface of the nesting factor in the whole BZ,
$X_{\bf q}=1.0$ (arbitrary units), for niobium at pressures of -0.6 GPa (left panel)
and 10.0 GPa (right panel).
Pictures obtained with the XCrySDen package.\cite{xcrysden}} \end{figure}

We propose to look closer at Fermi surface nesting by plotting the
dispersion of the nesting factor:
\begin{equation}
 X_{\bf q}=\sum_{\bf k} \delta (\varepsilon_{{\bf k}}-\varepsilon_{F})
            \delta (\varepsilon_{{\bf k}+{\bf q}}-\varepsilon_{F}).
\end{equation}
Large nesting factors correspond to large regions of the Fermi surface
being connected by the nesting vector, $\bf q$, and are expected to correspond
to large electron-phonon couplings.

In FIG.~\ref{nest}, the factor $X_{\bf q}$ is reported as a function of pressure.
This quantity has been computed numerically using the smearing
technique with the broadening of 0.03 Ry.  The maximal nesting takes place at
the $\Gamma$-point with much smaller maxima at the high-symmetry points
H, P and N, and in the middle of the lines: $\Gamma$-H, H-P, P-$\Gamma$
and $\Gamma$-N. The nesting factor decreases monotonically with
increasing pressure in the whole BZ except around ambient pressure
(0\--10~GPa).
At the equilibrium lattice constant, a large damping
of the nesting factor moves the whole nesting curve  below its value
at 38.8~GPa and makes it even similar to the
curve drawn for the pressure of 56.7~GPa.

The damping of the nesting factor close to ambient pressure
occurs in the whole BZ, as we can see in FIG.~\ref{nest-BZ} in comparison
to nesting for a pressure of 10.0 GPa.
This damping explains the jump in the total electron-phonon coupling constant
which happens below and above the ambient pressure (see TABLE~\ref{DOS-lambda}).

\section{Introduction}

Niobium is a superconductor with a quite high critical temperature,
$T_{c}$=~9.25~K, for a simple metal. The experiments under pressure
by Struzhkin {\em et al.} \cite{Struzhkin} show discontinuities of $T_{c}$
at about 5~GPa and at 50-60~GPa.
The low-pressure discontinuity manifests itself as an increase of $T_{c}$
by about 1~K. The high-pressure anomaly is associated with a decrease
of the critical temperature. To date, the nature of these pressure-induced
discontinuities is not clear. Previous theoretical
studies\cite{Ostanin,Ma,Anderson-1,Anderson-2} agree in attributing the
high-pressure anomalies to some visible change in the band structure.
The low-pressure discontinuity of $T_{c}$, however, remains mysterious.
The goal of this work is to give more information about the nature and
the origin of the anomalous behavior of $T_{c}$ in niobium under pressure.
In particular, we want to understand the role of Kohn
anomalies\cite{anomalies} of the phonon spectra. Kohn anomalies are known
to drastically change the critical temperature in superconductors and are
believed to play a role in all body-centered cubic (bcc) metals
Nb, Mo, V, Ta.\cite{anomalies-exp}

Therefore, we look to the details of the electronic structure and dynamical
properties of Nb at eight pressures in the range from -16 GPa to 78 GPa.
We show that  Kohn anomalies are responsible for both discontinuities,
but the origin of the low-pressure Kohn anomaly is different
from that of the high-pressure one.
Both can be identified  by a closer study of the Fermi-surface
nesting and of the band structure.

The accurate calculation of the electron-phonon coupling $\lambda$ and of the
spectral function $\alpha^{2}F$ is crucial for our problem. To this end
we use density-functional perturbation theory\cite{DFPT,Review} (DFPT)
in a pseudopotential plane-wave approach. The Eliashberg function and the nesting
factor require an integration of the double delta over the Fermi surface, which
needs to be done with a high numerical accuracy.
We give a few advices for an efficient calculation of
the electron-phonon coupling. Some of the technical details, however,
can be used in general for calculations of other properties which
require an accurate numerical integration with the delta function.

This paper is organized as follows: In the next Section we remind the
physical definitions and give some details of the calculation of
electron-phonon interaction coefficients using Vanderbilt's ultrasoft
pseudopotentials.\cite{Vanderbilt}
In Sec. III, we give the technical details (Subsec. A) and present results
for several properties under pressure:
the lattice constant and bulk modulus (Subsec. B),
the band structure and Fermi surface (Subsec. C),
the phonon frequencies and linewidths (Subsec. D), and
the Eliashberg function and electron-phonon coupling constant (Subsec. E).
In Sec. IV, we discuss the origin of the anomalies, and we summarize in Sec. V.
In the Appendix, we give numerical details for the
calculation of the Eliashberg function.

\subsection{Matrix elements with ultrasoft pseudopotentials}

The use of  ultrasoft (US) pseudopotentials (PPs)\cite{Vanderbilt} allows
in many cases a significant reduction of the needed plane-wave kinetic
energy cutoff, as compared with standard norm-conserving pseudopotentials.
\cite{HBS1,KB,MT} This enables a more efficient calculation, at the price of
introducing additional terms originating from the augmentation charges
employed in this scheme.\cite{Vanderbilt}
A detailed description of DFPT with US PPs has been given
elsewhere by Dal Corso.\cite{AndreaUS}  Here, we only briefly describe
terms appearing in the electron-phonon coupling.

With US PPs the KS orbitals, $\psi_{{\bf k},n}$, satisfy
a generalized eigenvalue problem
\begin{equation}
\left( -\frac{\nabla^{2}}{2} + V_{KS} - \varepsilon_{{\bf k},n}
S \right) \psi_{{\bf k},n}=0
\end{equation}
where the overlap matrix, S, is given by
\begin{eqnarray}
S({\bf r}_{1},{\bf r}_{2}) &=& \delta ({\bf r}_{1}-{\bf r}_{2}) +  \sum_{snm} q_{nm} \nonumber \\
&& \times \beta_{n}({\bf r}_{1}-{\bf R}_{s})
\beta_{m}^{\ast}({\bf r}_{2}-{\bf R}_{s}).
\label{SS}
\end{eqnarray}
The charge correction $q_{nm}$, in the above formula, is defined with the
augmentation functions,
$Q_{nm}({\bf r}-{\bf R}_{s})$, as follows
\begin{equation}
q_{nm} = \int d^{3}r \; Q_{nm}({\bf r}-{\bf R}_{s}),
\label{charge}
\end{equation}
and the projector functions $\beta_{n}({\bf r}-{\bf R}_{s})$ are specific for the
type of atom at the position ${\bf R}_{s}$, and are obtained from the atomic calculations.

The valence charge density is then computed as
\begin{eqnarray}
\rho({\bf r}) & = & \sum_{{\bf k},i} (|\psi_{i}({\bf r})|^{2} + \sum_{smn} Q_{mn}({\bf r}-{\bf R}_{s})
\langle \psi_{{\bf k},i} | \beta_{m} \rangle \langle \beta_{n}| \psi_{{\bf k},i} \rangle ) \nonumber \\
& = & \sum_{{\bf k},i} \int \int d^{3}r_{1}d^{3}r_{2}
\psi_{{\bf k},i}^{\ast} ({\bf r}_{1}) K({\bf r}; {\bf r}_{1},{\bf r}_{2})
\psi_{{\bf k},i} ({\bf r}_{2}),
\end{eqnarray}
where the sum over ${\bf k}$ and $i$ runs on occupied KS orbitals,
the kernel
\begin{eqnarray}
K({\bf r}; {\bf r}_{1},{\bf r}_{2}) & = & \delta ({\bf r}-{\bf r}_{1})
\delta ({\bf r}-{\bf r}_{2}) + \sum_{snm} Q_{nm}({\bf r}-{\bf R}_{s}) \nonumber \\
&& \times  \beta_{n}({\bf r}_{1}-{\bf R}_{s})
\beta_{m}^{\ast}({\bf r}_{2}-{\bf R}_{s})
\end{eqnarray}
has been introduced for later convenience.

The KS selfconsistent potential in the US-PP scheme reads
\begin{eqnarray}
V_{KS} ({\bf r}_{1},{\bf r}_{2}) & = & V_{NL}({\bf r}_{1},{\bf r}_{2}) \nonumber \\
 && + \int d^{3}r  \; V_{eff}({\bf r})\; K({\bf r}; {\bf r}_{1},{\bf r}_{2}) .
\label{VNL}
\end{eqnarray}
The effective potential, $V_{eff}$, contains the local,
the Hartree and the exchange-correlation (xc) terms
\begin{equation}
V_{eff} ({\bf r}) = V_{loc} ({\bf r}) + \int d^{3}r_{1} \;
\frac{\rho({\bf r}_{1})}{|{\bf r}_{1}-{\bf r}|} + V_{xc} ({\bf r}),
\end{equation}
while the nonlocal term generalizes the usual Kleinman-Bylander\cite{KB}
form allowing several projectors for a given angular momentum component
\begin{equation}
V_{NL} ({\bf r}_{1},{\bf r}_{2}) = \sum_{snm} D_{nm}^{0} \;
\beta_{n}({\bf r}_{1}-{\bf R}_{s}) \beta_{m}^{\ast}({\bf r}_{2}-{\bf R}_{s}).
\end{equation}
When augmentation charges vanish ($Q_{nm}=0$) the above formulas
reduce to the standard norm-conserving formulation.

In order to generalize Eq.~(\ref{gg}) to the case of US PPs, one needs to
compute
first order perturbation theory in presence of overlap matrix, $S$, as below
\begin{equation}
g_{{\bf k}+{\bf q},{\bf k}}^{{\bf q}\nu, mn} =
\langle \psi_{{\bf k}+{\bf q},m} | \Delta V_{KS}^{{\bf q}\nu}
- \varepsilon_{{\bf k},n} \Delta S | \psi_{{\bf k},n} \rangle ,
\end{equation}
where
\begin{equation}
\Delta S = \sum_{\bf R} \sum_{s} \frac{\partial S}{\partial \vec{u}_{s{\bf R}}}
\cdot \vec{u}_{s}^{{\bf q}\nu} \frac{e^{i{\bf qR}}}{\sqrt{N}},
\end{equation}
and $\Delta V_{KS}^{{\bf q}\nu}$ is given by Eq.~(\ref{dV}).

The derivative of the Kohn-Sham potential is given in Ref.~[\onlinecite{AndreaUS}]
and for gradient-corrected functionals in Ref.~[\onlinecite{GGA}].

\subsection{Phonon frequencies and linewidths}

\begin{figure*}
\epsfxsize=8cm
\centerline{
\includegraphics[scale=0.45,angle=-90]{ph-6.ps}}
\caption{Phonon dispersions in niobium at several pressures (in GPa),
for the two transverse modes (left and medium panels, notation "T1" and "T2")
and the longitudinal mode (right panels, notation "L");
the scale on the vertical axis is the same for all three modes.}
\label{ph-p}
\end{figure*}
\begin{figure*}
\epsfxsize=8cm
\centerline{
\includegraphics[scale=0.45,angle=-90]{gam-6.ps}}
\caption{Phonon linewidths (in THz) in niobium
at several pressures (in GPa).
The three modes of $\gamma_{{\bf q}\nu}$
correspond to the three phonon modes shown in FIG.~\ref{ph-p};
the scale on the vertical axis is the same for all three modes.}
\label{gam-p}
\end{figure*}
\begin{figure*}
\epsfxsize=8cm
\leftline{
\includegraphics[scale=0.45,angle=-90]{a2F-6.ps}}
\caption{Eliashberg function, $\alpha^{2}F$, in niobium at several pressures (in GPa);
the scale on the horizontal axis is the same,
within a given mode, for all pressures.}
\label{a2F-p}
\end{figure*}

The phonon spectra are presented in FIG.~\ref{ph-p}. We observe
an overall increase in phonon frequencies with pressure,
especially at H, P and N high-symmetry points.
Close to the $\Gamma$-point along the $\Gamma$-H symmetry
line, very low frequency phonon modes with an  anomalous
pressure dependence can be observed. At both the experimental and
the calculated equilibrium lattice constants, the two
transverse modes (T1 and T2 in the following) display a
very flat dispersion close to  $\Gamma$-point.
At variance with all other modes in the BZ, the T1 and T2 modes along $\Gamma$-H
line soften with pressure (between $\approx$10 to $\approx$55 GPa in
our calculations).
This anomalous, non-monotonic, dispersion relation
eventually disappears and become a smooth curve
at the highest pressure we have considered.

The phonon linewidths $\gamma_{{\bf q}\nu}$ have a weak dependence upon
pressure (see FIG.~\ref{gam-p}), with two important exceptions:
i) at low pressure,
large variations in phonon linewidth occur for the T2 and L modes near the N
high-symmetry point, and ii) at high pressure (above $\approx$55 GPa),
a significant reduction in linewidth is observed in many parts of the BZ,
especially along the $\Gamma$-H direction.

\subsection{Eliashberg function and electron-phonon coupling constant}
\label{sec-lambda}

By integrating the calculated phonon linewidths and frequencies we can
obtain Eliashberg $\alpha^{2}F(\omega)$ function, Eq.\ (\ref{a2Feq}), that
we present in FIG.~\ref{a2F-p} for the three phonon branches separately.
As a general feature, all peaks in  $\alpha^{2}F(\omega)$ move to higher
frequency with pressure, as expected from the global positive frequency shift
with pressure visible in FIG.~\ref{ph-p}.
For the T1 and T2 modes the height of the main peak decreases with pressure,
while for the L mode  the maximal height of the peak appears at the equilibrium
lattice constant.

Our calculated $\lambda$ and the density of states at the studied pressures
are presented in TABLE~\ref{DOS-lambda}.
The pressure behavior of the electron-phonon coupling constant
is similar to what could be expected from the experimental pressure
dependence for $T_{c}$:  $\lambda$ shows a positive jump at low pressure
and decreases significantly at high pressure.

\begin{table}
\begin{tabular}{ccccccc}
\hline \hline
\\[0.01mm]
$\;\;\;$  $P$ $\;\;\;$ & & $\;\;\;$ $N(\varepsilon_F)$ $\;\;\;$
&& $\;\;\;$ $\lambda$ $\;\;\;$ &&
$\;\;\;$ $\sim T_{c}^{exp}$  $\;\;\;$ \\[0.2cm]
\hline\\[0.01mm]
 -16.6 & &  11.41 && 1.91  && \-- \\
 -9.5  & &  10.80 &&  1.60 && \-- \\
 -0.6  & &  10.12 && 1.41  && 9.2 \\
 10.0   & &  9.69  &&  1.65 && 10.0 \\
 22.9  & &  9.16  &&  1.47 && 9.8   \\
 38.8  & &  8.55  &&  1.29 && 9.7  \\
 56.7  & &  7.71  &&  1.10 && 9.5 \\
 78.4  & &  6.55  &&  0.86 && 8.8 \\[0.05cm]
\hline \hline
\end{tabular}
\caption{The parameters of niobium under pressure, $P$ (in GPa):
the electronic density of states at the Fermi surface
$N(\varepsilon_F)$ (states per spin and per Ry),
the electron-phonon coupling constant $\lambda$,
and the experimental critical temperature $T_{c}^{exp}$
from Ref.~[\onlinecite{Struzhkin}].}
\label{DOS-lambda}
\end{table}

A good review of theoretical works on the electron-phonon coupling in Nb
at ambient pressure is given by Solanki {\it et al.} in Ref.~[\onlinecite{Solanki}],
where the reported values of the electron-phonon coupling constant range
from 0.59, obtained\cite{Anderson-1} from augmented plane-wave method
(APW), to 1.52, obtained\cite{lambda-1.52} later from the same  method.
More recently, Savrasov\cite{Sav-lambda} obtained a value of 1.26,
Bauer {\it et al.} \cite{Bauer} obtained a value of 1.33,
while our results for $\lambda$ at ambient pressure is 1.41.

We notice that the main reason for the spread in the
reported values of $\lambda$ is the difference in calculated values
for $N(\varepsilon_F)$, entering the definition of $\lambda$ in the
denominator (Eqs.~(\ref{a2F})-(\ref{dirlam})).
In fact a large value of 14.1 (states per spin and per Ry) for the DOS and a
small value of 0.59 for $\lambda$  have been
reported in Ref.~[\onlinecite{Anderson-1}], while
a value of 8.89 for the DOS and $\lambda$=1.52 have been reported in
Ref.~[\onlinecite{lambda-1.52}].
Consistent with this trend, our calculated DOS of 10.12 at ambient pressure
is somewhat lower than the DOS of 10.21 obtained by Savrasov,\cite{Sav-lambda},
while other authors obtained
$\sim 11$ (Ref.~[\onlinecite{Ostanin}]), 11.77 (Ref.~[\onlinecite{Singalas}]),
and 9.84 (Ref.~[\onlinecite{Crabtree}]).

The experimental values of $\lambda$ obtained from the electronic
tunneling spectroscopy\cite{lambda-exp1,lambda-exp2} are 1.04 and 1.22,
while Haas-van Alphen data\cite{Crabtree} yield $\lambda=1.33$.

\section{Summary}

We investigated the origin of the two discontinuities of the superconducting
critical temperature, observed in Niobum  at low pressure, about 5~GPa,
and at high pressure, about 60~GPa. \cite{Struzhkin}
For this purpose, we developed computational tools for the accurate
calculation of the electron-phonon coupling.

We find that the anomalous behavior of $T_{c}$ in Nb
under pressure originates in Kohn anomalies close to the $\Gamma$-point
in the BZ, so the measured discontinuities are caused by low-frequency phonons.
In agreement with previous authors,\cite{Ostanin,Ma,Anderson-1,Anderson-2}
we find that the high-pressure discontinuity of $T_{c}$ is associated to
a visible change in the band structure.
As for the low-pressure discontinuity, we notice that such anomaly
shows up as a general decrease of the nesting  factor without any visible change
in the shape of the Fermi surface. Such a decrease is uniform in the whole BZ,
explaining why previous calculations, Refs. [\onlinecite{Ostanin,Ma}], did not detect any
anomaly in the electronic structure of Nb near ambient pressure.
The total electron-phonon coupling constant varies with pressure
as expected from the measured critical temperatures.

In conclusion, both discontinuities of $T_{c}$ in niobium, at low and high pressures,
can be reproduced when the electron-phonon spectral function is calculated  accurately,
and the anomalies can be explained by the closer look into the details of
the Fermi-surface nesting and the band structure.

\section{Implementation details and test calculations}

The electron-phonon coupling constant $\lambda$ and the spectral
function $\alpha^{2}F(\omega)$ are defined by a double-delta
integration on the Fermi surface (Eqs.~(\ref{a2F}) and (\ref{dirlam})).
The accurate calculation of these integrands requires a very dense
sampling in both the electronic ({\bf k}) and the phononic
({\bf q}) grids. One can use either the broadening technique
\cite{broadening} or the tetrahedron method.\cite{tetra} We choose
the former to perform the quadrature on the Fermi surface, and
the latter to evaluate the electron-phonon and phonon densities of
states as functions of the vibrational frequencies.  In the broadening
scheme, a finite energy width is attributed to each state. For any
function $f$ which has to be integrated with the double-delta,
one can use the formula
\begin{eqnarray}
I  &=  &\int d{\bf k} \int d{\bf q} \; f({\bf k},{\bf q}) \;
\delta(\epsilon_{\bf k}-\epsilon_{F}) \delta(\epsilon_{{\bf k}+{\bf q}}-\epsilon_{F})
 \nonumber \\
&  \simeq & {\Omega_{BZ}^2\over N_k N_q}
   \sum_{\bf k} \sum_{\bf q} f({\bf k},{\bf q}) \;
  \frac{1}{\sqrt{2\pi}\sigma}
  exp \left( - \frac{(\epsilon_{\bf k}-\epsilon_{F})^{2}}{\sigma^{2}} \right) \nonumber \\
&&  \times \frac{1}{\sqrt{2\pi}\sigma}
  exp \left( - \frac{(\epsilon_{{\bf k}+{\bf q}}-\epsilon_{F})^{2}}{\sigma^{2}} \right),
\label{integral}
\end{eqnarray}
where $\sigma$ is the broadening, $N_k$ and $N_q$ the number of {\bf k}-
and {\bf q}-points, $\Omega_{BZ}$ the volume of the BZ.
For infinitely dense grids
of {\bf k}- and {\bf q}-points,
convergence of the integrand is achieved when $\sigma$ approaches zero.
For finite grids, however, one has to find a range of $\sigma$ values
yielding close results for the integrands at different grids.
In metals like Nb the presence of Kohn anomalies in the phonon
spectra sets additional requirements for the accuracy. Thus, it is expected
that the aforementioned integrands have to be calculated at very dense
{\bf k}-point and {\bf q}-point grids.

\begin{figure}
\epsfxsize=8cm
\centerline{
\includegraphics[scale=0.34,angle=-90]{fig-1.ps}}
\caption{The density of states and the double-delta integrand on the Fermi surface of Nb.}
\label{DOS}
\end{figure}
\begin{figure}
\epsfxsize=8cm
\centerline{
\includegraphics[scale=0.34,angle=-90]{fig-2.ps}}
\caption{The phonon linewidth $\gamma_{{\bf q}\nu}$ for two selected {\bf q}-vectors.}
\label{q2q4}
\end{figure}
\begin{figure}
\epsfxsize=8cm
\centerline{
\includegraphics[scale=0.40,angle=0]{a2F.ps}}
\vspace{-3mm}
\caption{The Eliashberg function of Nb for fixed broadening $\sigma$=0.03~Ry
and different q-grid (top panel) and the Eliashberg function for
fixed q-grid and different broadening $\sigma$ (bottom panel).}
\label{spectra}

\end{figure}
\begin{figure}
\epsfxsize=8cm
\centerline{
\includegraphics[scale=0.30,angle=-90]{lambda.ps}}
\caption{The total electron-phonon coupling $\lambda$ of Nb for different
q-grids as a function of
broadening $\sigma$.}
\label{lambda-tot}
\end{figure}
\begin{figure}
\epsfxsize=8cm
\centerline{
\includegraphics[scale=0.25,angle=-90.0]{a2F-0.ps}}
\caption{The Eliashberg function for Nb at calculated equilibrium lattice constant (6.14~a.u.).}
\label{a2F-0}
\end{figure}

Since the electron-phonon coupling matrix elements are smooth functions
of {\bf k} and {\bf q}, we resort to an interpolation procedure.
For a chosen {\bf k}- and {\bf q}-vector grid, we calculate matrix
elements $g_{\bf k+q, k}^{{\bf q}s\alpha,mn}$, defined as in Eq. (\ref{gg})
but with respect to the displacement of a single atom $s$ along
cartesian component $\alpha$. For each {\bf q}-vector we make a
linear interpolation in {\bf k}-space of the matrix elements to a
denser {\bf k}-vector grid. We then perform the integration, using
the denser grid and gaussian broadening as in Eq. (\ref{integral}),
of auxiliary phonon linewidths
$\widetilde\gamma^{\alpha\beta}_{ss'}({\bf q})$, defined as:
\begin{eqnarray}
\widetilde\gamma^{\alpha\beta}_{ss'}({\bf q}) & = & \sum_{mn}\sum_{{\bf k}}
(g_{{\bf k+q, k}}^{{\bf q}s\alpha, mn})^*
g_{{\bf k+q, k}}^{{\bf q}s'\beta, mn} \; \nonumber \\
&& \times \delta (\varepsilon_{{\bf k}+{\bf q},m}-\varepsilon_{F})
\delta (\varepsilon_{{\bf k},n}-\varepsilon_{F}).
\label{auxgamma}
\end{eqnarray}
These are related to the $\gamma_{{\bf q}\nu}$ of Eq. (\ref{eq5})
through the relation
\begin{equation}
\gamma_{{\bf q}\nu} = 2\pi\omega_{{\bf q}\nu} \sum_{s\alpha}\sum_{s'\beta}
(u^{{\bf q}\nu}_{s\alpha})^*\widetilde \gamma^{\alpha\beta}_{ss'}({\bf q})
u^{{\bf q}\nu}_{s'\beta}.
\end{equation}
Symmetry is exploited to reduce the number of {\bf k}- and
{\bf q}-points used in the calculation. A convenient way to achieve
such a goal is to perform {\em symmetrization}. Let us denote
with $T^{\alpha \beta}$ the symmetry operators of the small group
of {\bf q} (i.e. the subgroup of crystal symmetry that leaves {\bf q}
unchanged). We restrict summation on {\bf k}-points to the irreducible
BZ calculated with respect to the small group of {\bf q}. Then
symmetrization is performed, separately for each {\bf q}-vector,
as follows:
\begin{eqnarray}
\widetilde \gamma^{\alpha \beta}_{ss'}({\bf q}) & = &
\sum_{\alpha' \beta'} \; T^{\alpha \alpha'} T^{\beta \beta'}
\widetilde\gamma^{\alpha' \beta'}_{T(s)T(s')}({\bf q})  \nonumber \\
 & & \times exp \left( i \vec{\bf q}\cdot
     ( \vec{\tau}_{s}-\vec{\tau}_{T(s)}-\vec{\tau}_{s'}+\vec{\tau}_{T(s')}) \right).
\label{symmetrization}
\end{eqnarray}
In the above formula, atom $s$ with atomic position $\vec{\tau}_s$
transform into atom $T(s)$  with atomic position $\vec{\tau}_{T(s)}$
after application of operation $T$.
The symmetrized matrix at each {\bf q} is subsequently rotated,
using the remaining crystal symmetries that are not in the small
group of {\bf q}, and the symmetrized matrices at all {\bf q}-vectors
in the star of {\bf q} are thus obtained with minimal computational
effort. The same procedure can be applied to dynamical matrices,
Eq. (\ref{eq11}).

Once the electron-phonon coupling matrix of Eq. (\ref{auxgamma})
are calculated on a {\bf q}-vector grid, it is possible to perform
Fourier interpolation and to interpolate to a finer grid. In this way,
the integration in the {\bf q} space needed to calculate
$\alpha^{2}F(\omega)$, Eq. (\ref{a2F}), and $\lambda$, Eq. (\ref{lambda}),
can be accurately performed with a reasonable computational effort.

Let us turn now to some numerical experiments.
FIG.~\ref{DOS} shows the density of states for Nb (left panel)
and the double-delta integrand of a constant function (right panel)
at the Fermi surface for a selected {\bf q}-vector, as a function
of the broadening $\sigma$ and of the Monkhorst-Pack\cite{mesh}
{\bf k}-point grid.  For large enough $\sigma$ the results
for different grids -- except the (8,8,8) grid which is too coarse --
converge to the same values, which however depend on $\sigma$.
For small enough $\sigma$ different grids yield different results.
Since we are interested in the $\sigma\rightarrow 0$ limit, we have
to choose a grid of an affordable size that yields converged results
for a $\sigma$ as small as possible. A reasonable choice is the
(64,64,64) grid with $\sigma$=0.02~Ry, yielding a DOS at the Fermi
energy about 10.1 states per spin and per Ry.
The convergence of the double-delta function is a little bit slower
than that of a single-delta.

The phonon linewidths, $\gamma_{{\bf q}\nu}$, for two selected phonons
{\bf q}$\nu$ are displayed in FIG.~\ref{q2q4}.  The self-consistent
calculations were performed i) at the {\bf k}-grids of (16,16,16) and
(32,32,32), interpolated to a denser (64,64,64) grid; ii)
at the (24,24,24) {\bf k}-grid, interpolated to (72,72,72); iii)
at the (48,48,48) {\bf k}-grid, interpolated to (96,96,96).
The integration weights for the {\bf k}-space quadrature,
i.e. the gaussians centered around the single-particle energies, were
obtained from the accurate self-consistent calculation at the
corresponding dense grids.  As one can see in FIG.~\ref{q2q4}, the
convergence in {\bf k}-points is obtained quite easily even for the
SCF calculation at the grid (16,16,16).

In order to obtain the spectral function $\alpha^{2}F(\omega)$ one
needs to perform the {\bf q}-space quadrature of the phonon
linewidths; Eq. (\ref{a2Feq}). For the phonon and the electron-phonon
densities of states at given frequency $\omega$, we employ the
tetrahedron method within the scheme proposed by
Bl\"ochl.\cite{Bloechl} FIG.~\ref{spectra} (upper panel) shows the
Eliashberg function, $\alpha^{2}F$, of Nb for {\bf q}-grid (8,8,8),
without interpolation and interpolated into denser grids.
This has been done for a fixed broadening, $\sigma$=0.03~Ry.  The
lower panel of the same figure shows the $\alpha^{2}F$ function at the
{\bf q}-grid of (8,8,8) interpolated to (20,20,20)-point grid for
several broadenings $\sigma$.

In FIG.~\ref{lambda-tot}, we report the variation of total
electron-phonon coupling constant $\lambda$ with the broadening
$\sigma$ for three {\bf q}-meshes.

In FIG.~\ref{a2F-0}, we present the electron-phonon density of states
for niobium under ambient pressure.  We fit the curve of the spectral
function with cubic splines \cite{NR} for a finer plot.