# Interaction effects on galaxy pairs with Gemini/GMOS- I: Electron density

A. C. Krabbe1, D. A. Rosa1, O. L. Dors Jr.1, M. G. Pastoriza2, C. Winge3, G.
F. Hägele4,5, M. V. Cardaci4,5 and I. Rodrigues1
1 Universidade do Vale do Paraíba, Av. Shishima Hifumi, 2911, Cep 12244-000,
São José dos Campos, SP, Brazil
2 Instituto de Física, Universidade Federal do Rio Grande do Sul, Av. Bento
Gonçalves, 9500, Cep 91359-050, Porto Alegre, RS, Brazil
3 Gemini Observatory, c/o AURA Inc., Casilla 603, La Serena, Chile
4 Instituto de Astrofísica de La Plata (CONICET La Plata–UNLP), Argentina.
5 Facultad de Ciencias Astronómicas y Geofísicas, Universidad Nacional de La
Plata, Paseo del Bosque s/n, 1900 La Plata, Argentina
<EMAIL_ADDRESS>
(Accepted -. Received -.)

###### Abstract

We present an observational study about the impacts of the interactions in the
electron density of H ii regions located in 7 systems of interacting galaxies.
The data consist of long-slit spectra in the range 4400-7300 Å, obtained with
the Gemini Multi-Object Spectrograph at Gemini South (GMOS). The electron
density was determined using the ratio of emission lines [S
II]$\lambda$6716/$\lambda$6731\. Our results indicate that the electron
density estimates obtained of H ii regions from our sample of interacting
galaxies are systematically higher than those derived for isolated galaxies.
The mean electron density values of interacting galaxies are in the range of
$N_{\rm e}=24-532$ $\rm cm^{-3}$, while those obtained for isolated galaxies
are in the range of $N_{\rm e}=40-137\>\rm cm^{-3}$. Comparing the observed
emission lines with predictions of photoionization models, we verified that
almost all the H ii regions of the galaxies AM 1054A, AM 2058B, and AM 2306B,
have emission lines excited by shock gas. For the remaining galaxies, only few
H ii regions has emission lines excited by shocks, such as in AM 2322B (1
point), and AM 2322A (4 points). No correlation is obtained between the
presence of shocks and electron densities. Indeed, the highest electron
density values found in our sample do not belong to the objects with gas shock
excitation. We emphasize the importance of considering theses quantities
especially when the metallicity is derived for these types of systems.

###### keywords:

galaxies: ISM

††pagerange: Interaction effects on galaxy pairs with Gemini/GMOS- I: Electron
density–LABEL:lastpage††pubyear: 2012

##  1 Introduction

The study of physical processes involved in galaxy collisions and mergers in
the local universe is fundamental to understand the formation and evolution of
these objects, providing important constraints in simulations of the universe
at large scale.

In particular, the chemical abundance is highly modified in
interacting/merging galaxies. Kewley et al. (2010) presented a systematic
investigation about metallicity gradients in close pairs of galaxies. These
authors determined the oxygen abundance (generally used as a tracer of the
metallicity $Z$) along the disk of eight galaxies in close pairs and found
metallicity gradients shallower than the ones in isolated galaxies. Similar
results have been reached by Krabbe et al. (2011, 2008), who built spatial
profiles of oxygen abundance of the gaseous phase of the galaxy pairs AM
2306-721 and AM 2322-2821. This flattening in the oxygen abundance gradient
reflects the effects of gas redistribution along the galaxy disk due to metal-
poor inflow of gas from outskirts of the centre of interacting galaxies (Rupke
et al., 2010).

The large-scale gas motion created by the interaction induces high star
formation rate and galactic-scale outflows (Veilleux et al., 2005), producing
shock excitation in star-forming regions, such as reported in recent studies
of Luminous Infrared Galaxies by Soto & Martin (2012) and Rich et al. (2012,
2011). In particular, Rich et al. (2011), through integral field spectroscopic
data of the Luminous Infrared Galaxies IC 1623 and NGC 3256, showed that broad
line profiles are often associated with gas shock excitation in H ii regions
located in mergers. Similar results were also found by Newman et al. (2012)
for the clumpy star-forming galaxy ZC 406690 (see also Soto et al. 2012).
These authors pointed out that the broad emission likely originates from
large-scale outflows with high mass rates from individual star-forming
regions. The changes in galaxies that experience an encounter seem to have a
relation with the separation among the objects interacting, such as showed by
Scudder et al. (2012). These authors, using spectroscopy data of a large
sample of objects with a close companion taken from Sloan Digital Sky Survey
Data Release, found that the metallicity gradient and the star formation rate
(SFR) are correlated with the separation of the galaxy pairs analysed, in the
sense that the gradients are flatter and the SFR are higher at smaller
separations.

Despite recent efforts to probe the properties of interacting galaxies, the
electron density of star-forming regions have been poorly determined in these
systems, as well as its correlation with other quantities (e.g. $Z$, SFR). In
galaxy disks of interacting galaxies, where gas motions and gas excited by
shock are present, high electron density is expected and, can be used as a
signature of the presence of these motions and shocks. In fact, Puech et al.
(2006), in a study about galaxy interaction, mapped electron densities in six
distant galaxies (z $\sim$ 0.55) and found that the highest electron density
values observed could be associated to the collision between molecular clouds
of the interstellar medium and gas inflow/outflow events. These authors
derived electron density values lower than 400 $\rm cm^{-3}$, typical of
classical H ii regions (Copetti et al., 2000; Castañeda et al., 1992).
However, Puech et al. (2006) used as a sensor the [O
ii]$\lambda$3729/$\lambda$3727 ratio, which underestimates the electron
density in relation to determinations via other line ratios (Copetti & Writzl,
2002). Most of oxygen determinations of the gas phase in interacting galaxies
(e.g. Scudder et al. 2012; Rich et al. 2012, 2011; Krabbe et al. 2011; Kewley
et al. 2010; Krabbe et al. 2008) are based on theoretical models that consider
low electron density values of 10-200 $\rm cm^{-3}$ (Krabbe et al., 2011; Dors
et al., 2011; Kewley & Dopita, 2002; Dopita et al., 2000). If the electron
density values considerably differ of those considered in the models, the
oxygen abundance estimations will be doubtful. In fact, Oey & Kennicutt (1993)
showed that systematic variations in the nebular density introduce significant
uncertainties into the abundances obtained using methods based on strong
emission lines. They found that differences between 10 and 200 $\rm cm^{-3}$,
a typical range for the electron density derived in giant H ii regions (e.g.
Copetti et al. 2000; Castañeda et al. 1992; Kennicutt 1984; O’dell & Castañeda
1984), reflect variations up to 0.5 dex in oxygen abundances, mainly for the
high metallicity regime. These variations can increase even more when higher
electron density values, such as the ones found in star-forming clumps (e.g.
300-1800 $\rm cm^{-3}$, Newman et al. 2012), are considered in abundance
determinations.

In this paper, we used long-slit spectroscopic data of a sample of seven pair
galaxies to verify the effects of the interaction on the electron density in
these systems. This work is organized as follows. In Section 2, we summarize
the observations and data reduction. In Section 3, the method to compute the
electron density is described. Results and discussion are presented in
Sections 4 and 5, respectively. The conclusions of the outcomes are given in
Section 6.

Table 1: Galaxy sample.

ID | Morphology |  $\alpha$(2000) |  $\delta$(2000) | $cz\,(\rm km/s)$ |  $m_{\rm B}$ (mag) | Others names
---|---|---|---|---|---|---
AM 1054-325 | Sm [2] | $10^{\rm{h}}56^{\rm{m}}58\aas@@fstack{s}2$ | $-33^{\rm{h}}09^{\rm{m}}52\aas@@fstack{s}0$ | 3 788 [10] | 14.55 [2] | ESO 376-IG 027
| Sa [5] | 10 57 04.2s |  $-$33 09 21.0 | 3 850 [5] | 15.41 [8] | ESO 376- G 028
AM 1219-430 | Sm [6] | 12 21 57.3 |  $-$43 20 05.0 | 6 957 [3] | 14.30 [7] | ESO 267-IG 041
| S? [6] | 12 22 04.0 |  $-$43 20 21.0 | 6 879 [3] | - | FAIRALL 0157
AM 1256-433 | E [3] | 12 58 50.9 |  $-$43 52 30.0 | 9 215 [3] | 14.75 [8] | ESO 269-IG 022 NED01
| E [3] | 12 58 50.6 |  $-$43 52 53.0 | 9 183 [3] | 16.17 [8] | ESO 269-IG 022 NED02
| SBC [3] | 12 58 57.6 |  $-$43 50 11.0 | 9 014 [3] | 16.41 [1] | ESO 269-IG 023 NED01
AM 2058-381 | Sbc [6] | 21 01 39.1 |  $-$38 04 59.0 | 12 383 [3] | 14.91 [1] | ESO 341- G 030
| ? | 21 01 39.9 |  $-$38 05 53.0 | 12 460 [3] | 16.24 [1] | ESO 341- G 030 NOTES01
AM 2229-735 | SO? [3] | 22 33 43.7 |  $-$73 40 47.0 | 17 535 [3] | 15.98 [1] | AM 2229-735 NED01
| ? | 22 33 48.3 |  $-$73 40 56.0 | 17 342 [3] | 17.36 [1] | AM 2229-735 NED02
AM 2306-721 | SAB(r)c | 23 09 39.3 |  $-$71 01 34.0 | 8 919 [4] | 14.07 [1] | ESO 077- G 003
| ? | 23 09 44.5 |  $-$72 00 04.0 | 8 669 [4] | 14.47 [1] | ESO 077-IG 004
AM 2322-821 | SA(r)c | 23 26 27.6 |  $-$81 54 42.0 | 3 680 [3] | 13.35 [1] | ESO 012- G 001, NGC 7637
| ? | 23 25 55.4 |  $-$81 52 41.0 | 3 376 [4] | 15.41 [1] | ESO 012- G 001 NOTES01

References: [1] Ferreiro & Pastoriza (2004); [2] Weilbacher et al. (2000); [3]
Donzelli & Pastoriza (1997); [4] Krabbe et al. (2011); [5] Lauberts (1982));
[6] Paturel et al. (2003); [7] de Vaucouleurs et al. (1991); [8] Lauberts &
Valentijn (1989); [9] Huchra et al. (2012); [10] Jones et al. (2009)

Conventions: $\alpha$, $\delta$: equatorial coordinates

##  2 Observations and Data Reduction

We have selected several systems from Ferreiro & Pastoriza (2004) to study the
effects of the kinematics, stellar population, gradient abundances, and
electron densities of interacting galaxies. The first results of this
programme were presented for AM2306-721 (Krabbe et al., 2008) and AM2322-821
(Krabbe et al., 2011). Table 1 summarizes the main characteristics of the
systems: identification, morphology, position, radial velocity, apparent B
magnitude, and other designations.

Long-slit spectroscopic data were obtained on May, June, and July 2006 and
2007; and July 2008 with the Gemini Multi-Object Spectrograph (GMOS-S)
attached to the 8 m Gemini South telescope, Chile, as part of the poor weather
programs GS-2006A-DD-6, GS-2007A-Q-76, and GS-2008A-Q-206. Spectra in the
range 4400-7300 Å were acquired with the B600 grating, and 1$\arcsec$ slit
width, assuming a compromise between spectral resolution (5.5 Å), spectral
coverage, and slit losses (due to the Image Quality = ANY constraint). The
frames were binned on-chip by 4 and 2 pixels in the spatial and spectra
directions, respectively, resulting in a spatial scale of 0.288 $\arcsec$px-1,
and dispersion of 0.9 Å px-1.

Spectra were taken at different position angles on the sky, with the goal of
observing the nucleus and the brightest regions of the galaxies. The exposure
time on each single frame was limited to 700 seconds to minimize the effects
of cosmic rays, with multiple frames being obtained for each slit position to
achieve a suitable signal. The slit positions for each system are shown in
Fig. 1, superimposed on the GMOS-S r$\arcmin$ acquisition images. Table 2
gives the journal of observations. Conditions during the observing runs were
not photometric, with thin cirrus and image quality in the range 0.6$\arcsec$
to 1.7$\arcsec$ (as measured from stars in the acquisition images taken just
prior to the spectroscopic observations).

The spectroscopic data reduction was carried out using the gemini.gmos package
and generic IRAF111Image Reduction and Analysis Facility, distributed by NOAO,
operated by AURA, Inc., under agreement with NSF. tasks. We followed the
standard procedure: (1) the data were bias subtracted and flat-fielded; (2)
the wavelength calibration was established from the Cu-Ar arc frames with
typical residuals of 0.2 Å and applied to the object frames; (3) the
individual spectra of same slit positions and wavelength range were averaged
with cosmic ray rejection; (4) the object frames were sky subtracted
interactively using the gsskysub task, which uses a background sample of off-
object areas to fit a function to the specified rows, and this fit was then
subtracted from the column of each spectra; (5) the spectra were relative flux
calibrated using observations of a flux standard star taken with the same set
up as the science observations; (6) finally, one-dimensional spectra were
extracted from the two-dimensional spectra by summing over four rows along the
spatial direction. Each spectrum, therefore, comprises the flux contained in
an aperture of 1$\arcsec\times 1.152\arcsec$.

The intensities of the H$\beta$, [O iii]$\lambda$5007, [O i]$\lambda$6300,
H$\alpha$, [N ii]$\lambda$6584, and [S ii]$\lambda\,$6716,$\lambda$ 6731
emission lines were measured using a single Gaussian line profile fitting on
the spectra. We used the IRAF splot routine to fit the lines, with the
associated error being given as
$\sigma^{2}=\sigma_{cont}^{2}+\sigma_{line}^{2}$, where $\sigma_{cont}$ and
$\sigma_{line}$ are the continuum rms and the Poisson error of the line flux,
respectively. Furthermore, we considered only measurements whose continuum
around $\lambda$ 6700 Å reach a signal-to-noise S/N $\geq$ 8\. The emission
line intensities were not corrected for the interstellar extinction, because
it is negligible due to the small separation between the [S ii]$\lambda 6716$
and $\lambda 6731$ emission lines.

Table 2: Journal of observations. Object | Date | Exposure | PA (°) |  $\Delta\,\lambda$ (Å)
---|---|---|---|---
|  | Time (s) |  |
AM 1054-325 | 2007-06-21 | 4 $\times$ 600 | 77 | 4280-7130
AM 1219-430 | 2007-06-06 | 4 $\times$ 600 | 25 | 4280-7130
| 2007-05-26 | 4 $\times$ 600 | 162 | 4280-7130
| 2007-06-22 | 4 $\times$ 600 | 341 | 4280-7130
AM 1256-433 | 2007-07-06 | 4 $\times$ 600 | 292 | 4280-7130
| 2007-06-21 | 4 $\times$ 600 | 325 | 4280-7130
AM 2058-381 | 2006-05-20 | 4 $\times$ 600 | 42 | 4351-7213
| 2007-05-26 | 4 $\times$ 600 | 94 | 4351-7213
| 2007-05-24 | 4 $\times$ 600 | 125 | 4351-7213
| 2007-05-30 | 4 $\times$ 600 | 350 | 4351-7213
AM 2229-735 | 2006-07-20 | 6 $\times$ 600 | 134 | 4390-7250
| 2006-07-16 | 6 $\times$ 600 | 161 | 4390-7250
AM 2306-721 | 2006-06-20 | 4 $\times$ 600 | 118 | 4280-7130
| 2006-06-20 | 4 $\times$ 600 | 190 | 4280-7130
| 2006-06-20 | 4 $\times$ 600 | 238 | 4280-7130
AM 2322-821 | 2006-07-01 | 3 $\times$ 700 | 59 | 4280-7130
| 2008-07-27 | 6 $\times$ 600 | 60 | 4280-7130
| 2006-06-30 | 6 $\times$ 600 | 318 | 4280-7130

Figure 1: The slit positions for each system are shown superimposed on the
GMOS-S r$\arcmin$ acquisition image. Figure 2: A sample of spectra in the
range of 6600 to 6800 Å from areas of different galaxies. The flux scale was
normalized to the peak of [S ii] $\lambda$ 6716.

##  3 Determination of the electron density

The electron density $N_{\rm e}$ was derived from the [S
ii]$\lambda\,$6716/$\lambda$ 6731 emission line intensity ratio by solving
numerically the equilibrium equation for a $n$-level atom approximation using
the temden routine of the nebular package of the STSDAS/IRAF, assuming an
electron temperature of 10 000 K, because temperature sensitive emission lines
were unobservable in our sample.

The references for the collision strengths, transition probabilities, and
energy levels are Ramsbottom et al. (1996), Verner et al. (1987), Keenan et
al. (1993), and Bowen et al. (1960). There are two main sources of errors in
the determination of electron densities. One is the dependence of the $N_{\rm
e}$ on the electron temperature $T_{\rm e}$ assumed. However, this dependence
is weak in the range of temperatures usually found in galactic H ii regions
(e.g. Copetti et al., 2000). We adopted a mean electron temperature of 10 000
K as a representative value, because it is a typical electron temperature
value for these kinds of objects and there are no estimations for our sample.
The other main source of error is the saturation of the line ratio for both
low and high values of the electron density, which makes the [S ii]$\lambda$
6716/$\lambda$ 6731 ratio a reliable sensor of the electron density in the
range of 2.45 $<$ log $N_{\rm e}$ (cm${}^{-3})$ $<$ 3.85 (Stanghellini et al.,
1989).

##  4 Results

Fig. 2 shows a sample of the spectra of some H ii regions of the galaxies
around the [S ii]$\lambda$6716 and [S ii]$\lambda$6731 emission lines. The
profiles of $\log$([O i]$\lambda$6300/H$\alpha$), [S
ii]$\lambda$6716/$\lambda$6731 ratio, and $N_{\rm e}$ as a function of
galactocentric radius for the galaxies are shown in Figs. 4-14. The intensity
of $\log$([O i]$\lambda$6300/H$\alpha$) was plotted only for the apertures for
which the electron density determination was possible. The galactocentric
radius are not corrected by galaxy inclination. In Fig. 1 the adopted centre
of each galaxy is marked with a red cross. Table 3 presents some statistics of
the [S ii] $\lambda$ 6716/$\lambda$ 6731 ratio and electron density
measurements, including the number N of distinct nebular areas, the mean, the
median, the maximum and minimum, and the standard deviation $\sigma$. The
results for each system are presented separately.

Table 3: [S ii] ratio and electron density statistics. |  |  [S ii] $\lambda\,6716/\lambda\,6731$ |  | $N_{{\rm e}}~{}(\mathrm{cm^{-3}})$
---|---|---|---|---
Objects |  | N | mean | median | max | min | $\sigma$ |  | N | mean | median | max | min | $\sigma$
AM 1054A |  | 16 | 1.19 | 1.08 | 1.70 | 0.97 | 0.93 |  | 13 | 434 | 462 | 681 | 65 | 191
AM 1054B |  | 3 | 0.86 | 0.85 | 0.92 | 0.79 | 0.07 |  | 3 | 1130 | 1082 | 1476 | 833 | 324
AM 1219A |  | 29 | 1.12 | 1.06 | 1.77 | 0.86 | 0.23 |  | 26 | 532 | 518 | 1073 | 85 | 286
AM 1219B |  | 5 | 0.70 | 0.82 | 0.92 | 0.23 | 0.27 |  | 4 | 1408 | 1294 | 2189 | 855 | 564
AM 1256B |  | 43 | 1.48 | 1.42 | 2.08 | 0.99 | 0.27 |  | 22 | 181 | 317 | 626 | 7 | 168
AM 2058A |  | 20 | 1.38 | 1.26 | 2.22 | 0.90 | 0.37 |  | 13 | 376 | 318 | 911 | 33 | 263
AM 2058B |  | 8 | 1.47 | 1.42 | 1.80 | 1.24 | 0.19 |  | 4 | 86 | 60 | 184 | 42 | 66
AM 2229A |  | 33 | 1.60 | 1.59 | 2.61 | 0.19 | 0.59 |  | 7 | 346 | 226 | 686 | 28 | 280
AM 2306A |  | 8 | 1.41 | 1.44 | 1.60 | 1.16 | 0.16 |  | 5 | 131 | 107 | 298 | 32 | 99
AM 2306B |  | 15 | 1.38 | 1.30 | 2.88 | 0.92 | 0.47 |  | 11 | 300 | 212 | 826 | 19 | 273
AM 2322A |  | 81 | 1.41 | 1.42 | 1.89 | 0.85 | 0.20 |  | 41 | 200 | 103 | 1121 | 11 | 259
AM 2322B |  | 23 | 1.47 | 1.43 | 1.74 | 1.35 | 0.10 |  | 12 | 24 | 15 | 75 | 3 | 23

###  4.1 AM 1054-325

This system is composed by a peculiar spiral with disturbed arms (hereafter AM
1054A) and a spiral-like object (hereafter AM 1054B). AM 1054A contains very
luminous H II regions along their galactic disk. As can be seen in the Fig. 1,
AM 1054A seems to have two nuclei. According to measurements obtained from
Weilbacher et al. (2000), the “main” nucleus of this galaxy (ESO 376-IG 027)
is the reddest [(B-V)=0.52], while the other (ESO-LV 3760271) has the blue
colours of a strong starburst [(B-V)=0.21]. Both nuclei names are marked in
the Fig. 1. The measured radial velocity is 3788 km/s (Jones et al., 2009) and
3853 km/s (Sekiguchi & Wolstencroft, 1993) for ESO 376-IG 027 and ESO-LV
3760271, respectively. Therefore, the small difference found between their
radial velocities together with the perturbed morphology of the galaxy seem to
indicate that these objects are gravitationally bound.

For AM 1054A, the electron density values estimated from [S
ii]$\lambda\,6716/\lambda\,6731$ ratio (see Fig. 4) present variations of
relatively high amplitude along the radius of the galaxy, with the minimum
value of $N_{\rm e}$= 65 $\mathrm{cm^{-3}}$ and the maximum of $N_{\rm e}$=
681 $\mathrm{cm^{-3}}$. We found a mean density of $N_{\rm e}=434\pm 53$
$\mathrm{cm^{-3}}$. In this galaxy, the slit position is cutting a bright
star-forming region, but does not cross the nucleus of the galaxy. For
AM1054B, only few apertures had the [S ii]$\lambda\lambda$ 6716, 6731 emission
lines with enough signal to be measured. A mean density of $N_{\rm
e}=1\,130\pm 187$ $\mathrm{cm^{-3}}$ was derived for this galaxy.

Figure 3: AM 1054-325. $\log$([O i $\lambda$ 6300/H$\alpha$) ratio, [S ii]
$\lambda$ 6716/$\lambda$ 6731 ratio, and $N_{\rm e}$ as a function of
galactocentric radius for AM 1054A.

Figure 4: Same as Fig. 4, but for AM 1054B.

###  4.2 AM 1219-430

This pair is composed by a disturbed spiral (hereafter AM 1219A) and a smaller
disk galaxy (AM 1219B). AM 1219A shows a tidal tail produced by the
interaction of the galaxies, with very bright H ii regions. Systemic
velocities of 6 957 km s-1 and 6 879 km s-1 were estimated by Donzelli &
Pastoriza (1997) for AM 1219A and AM1219B, respectively.

The distribution of electron densities exhibits variations of high amplitude
across the radius of the main galaxy in the range of $N_{\rm e}=85-1073$
$\mathrm{cm^{-3}}$. We found a mean density of $N_{\rm e}=532\pm 56$
$\mathrm{cm^{-3}}$. As in the case of AM 1054B, only for few apertures of AM
1219B the [S ii]$\lambda\lambda$ 6716,6731 emission lines have enough signal
to be measured. A mean density of $N_{\rm e}=1\,408\pm 282$ $\mathrm{cm^{-3}}$
was derived for this galaxy. Interestingly, the $N_{\rm e}$ increases toward
the outskirt of this galaxy. This region is at the end of the spiral arm to
the Northwest.

Figure 5: Same as Fig. 4, but for AM 1219A.

Figure 6: Same as Fig. 4, but for AM 1219B.

###  4.3 AM 1256-433

AM 1256-433 is a system constituted by three galaxies. Two are elliptical with
very bright nuclei, ESO 269-IG 022 NED01 and ESO 269-IG 022 NED02, and one
very disturbed spiral galaxy, ESO 269-IG 023 NED01, hereafter AM 1256B. In
addition, an isolated disk galaxy, ESO 269-IG 023 NED02/PGC 543979
($\alpha=12^{\rm{h}}59^{\rm{m}}00\aas@@fstack{s}6$ and
$\delta=-43^{\rm{h}}50^{\rm{m}}23^{\rm{s}}$ J2000), appears in the field of
view of this system, about 30$\arcsec$ to the Southeast of the centre of AM
1256B. From our data, we obtained for this isolated galaxy a heliocentric
velocity of 18 896 km s-1 indicating that it does not belong to this system,
and it was incorrectly associated with AM 1256-433 by Donzelli & Pastoriza
(1997); Ferreiro & Pastoriza (2004), and Ferreiro et al. (2008). In Fig. 1,
only AM 1256B and the isolated galaxy ESO 269-IG 023 NED02 is shown.

As can be seen in Fig. 7, some regions (for example at about 6 and 12 kpc from
the centre of the galaxy) present un-physically large values of the [S
ii]$\lambda$6716/$\lambda$6731 ratio, above the theoretical value of 1.4, the
value for the low density limit according to the Osterbrock & Ferland (2006)
curve for this relation. There could be some uncertainties associated with the
measurements of these sulphur emission lines, due to the placement of the
continuum and deblending of the lines, that might produce larger values of the
[S ii] ratio than the expected ones. Values of the [S ii] ratio larger than
the 1.4 upper limit were already observed in other studies using different
kinds of instruments (e.g. Kennicutt, Keel, & Blaha, 1989; Zaritsky,
Kennicutt, & Huchra, 1994; Lagos et al., 2009; Relaño et al., 2010; López-
Hernández et al., 2013). As pointed out by López-Hernández et al. (2013), the
theoretical density determination also needs to be adjusted to the sulphur
atomic data and deserves to be revisited. From a spatial distribution study of
the electron density in a sample of H ii regions in M33, these authors
highlighted that when values of the $\lambda$ 6716/$\lambda$ 6731 ratio above
the 1.4 limit are obtained, it is reasonable to assume that the electron
densities are lower than 10 $\mathrm{cm^{-3}}$. They also noted that a safe
way to proceed is to take $N_{\rm e}=100$ $\mathrm{cm^{-3}}$, because even
before reaching the 1.4 limit, the estimation of the electron density is very
uncertain. A mean density of $N_{\rm e}=181\pm 36$ $\mathrm{cm^{-3}}$ was
derived for this galaxy. Again, the $N_{\rm e}$ increases toward the outskirt
of this galaxy, corresponding to the end of the spiral arm at Southeast.

Figure 7: Same as Fig. 4, but for AM 1256B.

###  4.4 AM 2058-381

This system of galaxies is a typical M 51 type pair. It has a systemic
velocity of ${cz}$ = 12 286 km s-1 (Donzelli & Pastoriza, 1997) and consists
of a main galaxy with two spiral arms (hereafter, AM 2058A) and a companion
irregular galaxy (hereafter, AM 2058B).

The electron densities obtained for AM 2058A have variations across the galaxy
in the range of $N_{\rm e}=33-911$ $\mathrm{cm^{-3}}$, and these values are
not dependent upon the position. Due to the small radius of AM 2058B, only a
few apertures could be extracted for this galaxy. The electron densities (see
Fig. 9) are relatively low, with a mean value of $N_{\rm e}=86\pm 33$
$\mathrm{cm^{-3}}$, which is compatible with estimations for giant
extragalactic H ii regions (e.g. Castañeda et al. 1992).

Figure 8: Same as Fig. 4, but for AM 2058A.

Figure 9: Same as Fig. 4, but for AM 2058B.

###  4.5 AM 2229-735

This pair of galaxies consists of a main spiral galaxy strongly disturbed
(hereafter AM 2229A) and a smaller disk galaxy that could be connected to the
main one by a bridge. AM 2229A has a very massive nucleus of $M=5\times
10^{8}M_{\sun}$ (Ferreiro et al., 2008) and very bright H ii regions. Only the
primary galaxy was observed.

Most of observed regions in AM 2229A present un-physically large values of the
S ii ratio according to the Osterbrock & Ferland (2006) curve for the relation
between this ratio and the electron density. We derived a mean electron
density of $N_{\rm e}=346\pm 95$ $\mathrm{cm^{-3}}$.

Figure 10: Same as Fig. 4, but for AM 2229A.

###  4.6 AM 2306-721

AM 2306-721 is a pair composed by a spiral galaxy with disturbed arms
(hereafter AM 2306A) interacting with an irregular galaxy (hereafter AM
2306B). Both galaxies contain very luminous H ii regions with H$\alpha$
luminosity in the range of 8.30 $\times$1039 $\leq$ $L$ (H$\alpha$) $\leq$
1.32 $\times$1042 erg s-1 and high star-formation rate in the range of 0.07 -
10 $M_{\odot}$ yr-1, as estimated from H$\alpha$ images by Ferreiro et al.
(2008).

The few measurements of electron densities provide values in the range of
$N_{\rm e}=32-298$ $\mathrm{cm^{-3}}$ and $N_{\rm e}=19-826$
$\mathrm{cm^{-3}}$ for AM 2302A and AM 2306B, respectively. Although, we do
not have estimates of the electron density at the centre of the main galaxy,
the spatial profile seems to indicate an increasing of the $N_{\rm e}$ toward
the centre of the galaxy, which could be a consequence of gas inflow. Again,
in the secondary galaxy, the electron density smoothly increases from about 4
kpc toward the outer regions of the galaxy to the end of the spiral arm at the
Southeast.

Figure 11: Same as Fig. 4, but for AM 2306A.

Figure 12: Same as Fig. 4, but for AM 2306B.

###  4.7 AM 2322-821

AM 2322-821 is composed of a SA(r)c galaxy with disturbed arms (hereafter AM
2322A) in interaction with an irregular galaxy (hereafter AM 2322B). Both
galaxies contain very luminous H ii regions with 2.53$\times$1039 $\leq$ $L$
(H$\alpha$) $\leq$ 1.45$\times$1041 erg s-1 and star formation rates from 0.02
to 1.15 $M_{\odot}$ yr-1 (Ferreiro et al., 2008).

The distribution of electron temperatures exhibits variations of very low
amplitude across the radius of AM 2322A. One region (at about 2 kpc from the
centre of the galaxy) has four values of densities systematically higher than
the other apertures along the radius of galaxy. This region is marked in Fig.
15. In this region, the values of densities are in the range of $N_{\rm
e}=803-1121$ $\mathrm{cm^{-3}}$. We found a mean electron density of $N_{\rm
e}=200\pm 12$ $\mathrm{cm^{-3}}$. AM 2322B presents a relatively homogeneous
electron density distribution, with a mean density of $N_{\rm e}=24\pm 4.8$
$\mathrm{cm^{-3}}$. This is the galaxy with the lowest density in our sample.

Figure 13: Same as Fig. 4, but for AM 2322A.

Figure 14: Same as Fig. 4, but for AM 2322B. Figure 15: Image of AM 2322A with
the region of high density (see the text) marked with a circle.

##  5 Discussion

To verify if there are differences between the $N_{\rm e}$ values observed in
the H ii regions of our sample and those obtained in isolated galaxies, we
have calculated the electron densities from published measurements of the [S
ii] line-ratio for disk H ii regions in the isolated galaxies M 101, NGC 1232,
NGC 1365, NGC 2903, NGC 2997, and NGC 5236 and compared these values with our
results. The data of these objects were taken from Kennicutt et al. (2003) for
M 101 and from Bresolin et al. (2005) for the other galaxies. The same atomic
parameters and electron temperature adopted for our determinations were used.
The spatial profiles of the [S II]$\lambda$6716/$\lambda$6731 ratio and the
electron densities derived for some H ii regions in the isolated galaxies are
shown in Fig. 16.

As can be seen in this figure, the estimated electron densities are relatively
homogeneous along the radius of each isolated galaxy. The derived mean
electron densities are in the range of $N_{\rm e}=40-137\,\rm cm^{-3}$. Only
one high value of $N_{\rm e}\approx 900\>\rm cm^{-3}$ is derived in the
central region of NGC 5236. It is a metal-rich H ii region, with a low
electron temperature of $T_{\rm e}$(O iii)$=4\,000\pm 2\,000$ K and an oxygen
abundance of 12+log(O/H)$\approx$ 8.9 dex as derived by Bresolin et al.
(2005). This high value can be caused by mass loss and strong stellar winds
from embedded Wolf Rayet stars, which are common in metal-rich environments
(e.g. Pindao et al. 2002; Bresolin & Kennicutt 2002; Schaerer et al. 2000). If
the adopted electron temperature is $T_{\rm e}$(O iii)$=4\,000$ K, an
estimation of $N_{\rm e}\approx 623\>\rm cm^{-3}$ is obtained. This value is
about 30% lower than the one obtained assuming an electron temperature of
$T_{\rm e}$(O iii)$=10\>000$ K. Then, even though the dependence of the
$N_{\rm e}$ with the electron temperature is weak, it could have an important
effect when temperature fluctuations of high amplitude were observed in H ii
regions.

The values of the electron density obtained from our sample of interacting
galaxies are systematically higher than those derived for the isolated ones.
The mean electron density values derived by us for the interacting galaxies in
our sample are in the range of $N_{\rm e}=24-532\,\rm cm^{-3}$, which also
show higher values than for isolated galaxies. Newman et al. (2012), for the
clumpy star-forming galaxy ZC406690, also obtained high electron density
values ($N_{\rm e}=300-1800\,\rm cm^{-3}$). Moreover, several of our
interacting galaxies (AM 2306B, AM 1219A, and AM 1256B) show a slight
increment of the $N_{\rm e}$ in the outer parts of the galaxy, opposite of
what is observed in the isolated galaxies, where the electron density is
homogeneous along the radius. The high electron density values found in the
outlying parts for the majority of the objects of our sample would be due to
zones of induced star formation by direct cloud-cloud interaction (for a
review see Bournaud 2011). In these regions, turbulent flows can locally
compress the gas, forming over-densities that subsequently cool and collapse
into star-forming clouds (Duc et al., 2013; Elmegreen, 2002). Although we do
not have estimates of the electron density at the centre of AM 2306A, the
spatial profile seems to indicate an increasing of $N_{\rm e}$ toward the
centre of the galaxy, which could be due to inflowing gas. However, in only a
few regions in this galaxy were possible to estimate $N_{\rm e}$, therefore,
this is a marginal conclusion.

It is worth mentioning that H ii regions seemed to be inhomogeneous, and the
zones where most of the emission from the ionized gas is originated only
occupy a small fraction of the total volume (i.e., small filling factor).
Hence, our electron density values derived from the [S ii] emission lines are
representative of a fraction of the total volume of the H ii region (referred
as in situ electron densities). According to Giammanco et al. (2004), these
inhomogeneities, if optically thick, can modify the determinations of electron
temperatures and densities, ionization parameters, and abundances. Copetti et
al. (2000) presented a study on internal variation of the electron density in
a sample of spatially resolved galactic H ii regions of different sizes and
evolutionary stages. These authors found that the electron density within H ii
regions (e.g. S 307) can range from about 30 to 600 $\rm cm^{-3}$, and a
filling factor of the order of 0.1 is compatible with their data. Therefore,
the estimated electron densities could be about 10% of the in situ values
sampled by the sulphur line ratio.

Figure 16: Profiles of the electron density as a function of R/R0, where R0 is
the galactocentric distance deprojeted for the isolated galaxies M 101, NGC
1232, NGC 1365, NGC 2903, NGC 2997 and NGC 5236. Figure 17: Diagnostic diagram
of [O III]$\lambda$5007/H$\beta$ vs. [O I]$\lambda$6300/H$\alpha$. The black
solid line from Kewley et al. (2006) separates the objects ionized by massive
stars from the ones containing active nuclei and/or shock excited gas. The
data for distinct galaxies are marked by different symbols as indicated. The
typical error bar (not shown) of the emission line ratios is about 10 per
cent.

An important issue is to study the origin of the high electron density values
found in the H ii regions of our sample. The presence of gas shock excitation
in interacting galaxies is very important not only because they affect
quantities derived from spectroscopy, but also due to they act as a mechanism
for dissipating the kinetic energy and the angular momentum of the infalling
gas in merging systems, as discussed by Rich et al. (2011). The gas shock also
increases the density due to the compression of the interstellar material. To
analyse if the presence of shock-excited gas produces the high electron
density values, the diagnostic diagram [O III]$\lambda$5007/H$\beta$ vs. [O
I]$\lambda$6300/H$\alpha$ proposed by Baldwin et al. (1981) and Veilleux &
Osterbrock (1987), and used to separate objects ionized by stars, by shocks
and/or active nuclei (AGN) was considered.

In Fig. 17, the diagnostic diagram containing the data of the H ii regions
studied by us is shown. The galaxy nuclei data are not shown in this diagram.
We also show in this plot, the line proposed by Kewley et al. (2006) to
separate objects with distinct ionizing sources: shock gas and massive star
excitations. We can see that the all H ii regions in AM 1054A, AM 2058B, AM
2306B, and some regions in AM 2322A (3 apertures) and AM 2322B (1 aperture)
occupy the area where objects with shock as the main ionizing source are
located. The number of objects represented in Fig. 17 differ from those in the
profile figures (Figs. 4-14) because the [O iii]$\lambda 5007$/H$\beta$ ratio
could not be measured for all apertures. From the comparison of the spatial
profiles of the electron density and the logarithm of [O i]$\lambda
6300$/H$\alpha$ in the AM 1054A, AM 2058B, and AM 2306B galaxies (Figs. 4, 9
and 12, respectively) we can note the following:

  * •

AM 1054A: All regions of this galaxy have gas shock excitation and the values
of electron density are relatively high.

  * •

AM 2058B: It is a small galaxy and only a few apertures could be extracted. As
can be seen in Fig. 17, all four disk H ii regions of this galaxy have gas
shock excitation, and from Fig. 9, we can note that these regions present low
electron density values ($<200\>\rm cm^{-3}$).

  * •

AM 2306B: the regions with highest [O I]$\lambda$6300/H$\alpha$ and $N_{\rm
e}$ values ($\approx 700$ $\rm cm^{-3}$ ) lie in the outskirts of galaxy. As
can be seen in Fig. 12, it seems to be a trend in this object: from about 4
kpc both the $N_{\rm e}$ and the [O I]/H$\alpha$ ratio increase to the outer
parts of the galaxy; in the inner part (up to 2 kpc), the profiles of these
two quantities are almost flat showing low values.

The cause of the high electron density values associated with the shock
excitation region in interacting galaxies is essential to understand how the
flux gas works in them. High-velocity gas motions can destroy molecular clouds
and quench star formation (Tubbs, 1982). To investigate if the high electron
density values found in our sample are associated with the presence of
excitation by gas shock, we plotted in Fig. 18 the $N_{\rm e}$ versus the
logarithm of the observed [O i]$\lambda$6300/H$\alpha$ emission line ratio.
Objects with distinct gas excitation source, in according to Fig. 17, are
indicated by different symbols. No correlation is obtained between the
presence of shocks and electron densities. The highest electron density values
found in our sample do not belong to objects with gas shock excitation.
Therefore, the high electron density values found in the H ii regions of our
sample do not seem to be caused by the presence of gas shock excitation.
However, a deeper analysis such as investigating the presence of correlation
between the velocity dispersion of some emission line and its intensity (e.g.
Storchi-Bergmann et al. 2007) or the implications of multiple kinematical
components in the emission line profiles on the derived properties (Hägele et
al., 2013; Hägele et al., 2012; Amorín et al., 2012) is necessary to confirm
our result. Interestingly, the objects with the highest electron density
values present the smallest [O i]$\lambda$6300/H$\alpha$ line intensity
ratios.

Figure 18: Electron density values $N_{\rm e}$ derived for our sample versus
the observed [O i]$\lambda$6300/H$\alpha$ ratio. Squares represent regions
ionized by massive stars while triangles represent those with gas shock
excitation, according to the diagnostic diagram presented in Fig. 17.

##  6 Conclusions

An observational study of the effects of the interaction on the electron
densities from the H ii regions along the radius of a sample of interacting
galaxies is performed. The data consist of long-slit spectra of high signal-
to-noise ratio in the 4390-7250 Å obtained with the Gemini Multi-Object
Spectrograph at Gemini South (GMOS). The electron density was determined using
the ratio of lines [S II]$\lambda$6716/$\lambda$6731\. The main findings are
the following:

  * •

The electron density estimates obtained for some H ii regions of our sample of
interacting galaxies are systematically higher than those derived for isolated
galaxies in the literature. The mean electron density values of interacting
galaxies are in the range of $N_{\rm e}=24-532\,\rm cm^{-3}$, while those
obtained for isolated galaxies are in the range of $N_{\rm e}=40-137\,\rm
cm^{-3}$.

  * •

Some interacting galaxies: AM 2306B, AM 1219A, and AM 1256B show an increment
of $N_{\rm e}$ toward the outskirts of each system. This kind of relation is
not observed in isolated galaxies, where the electron density profile is
rather flat along the radius of each galaxy.

  * •

The galaxies where the mechanism of gas shock excitation is present in almost
all the H ii regions are AM 1054A, AM 2058B, and AM 2306B. For the remaining
galaxies, only few H ii regions has emission lines excited by shocks, such as
in AM 2322B (1 point) and AM 2322A (4 points). It is noteworthy that only in
three of all objects analysed here the main excitation mechanism for all of
their H ii regions is shocks.

  * •

No correlation is obtained between the presence of shocks and electron
densities. Indeed, the highest electron density values found in our sample do
not belong to the objects with gas shock excitation. Therefore, the high
electron density values found in the H ii regions of our sample do not seem to
be caused by the presence of gas shock excitation.

## Acknowledgements

Based on observations obtained at the Gemini Observatory, which is operated by
the Association of Universities for Research in Astronomy, Inc., under a
cooperative agreement with the NSF on behalf of the Gemini partnership: the
National Science Foundation (United States), the Science and Technology
Facilities Council (United Kingdom), the National Research Council (Canada),
CONICYT (Chile), the Australian Research Council (Australia), Ministério da
Ciencia e Tecnologia (Brazil), and SECYT (Argentina).

A. C. Krabbe, O. L. Dors Jr., and D. A. Rosa thank the support of FAPESP,
process 2010/01490-3, 2009/14787-7, and 2011/08202-6 respectively.

We also thank Ms. Alene Alder-Rangel for editing the English in this
manuscript.

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# Impurities in graphene and their influence on the Casimir interaction

N. Khusnutdinov<EMAIL_ADDRESS>D. Vassilevich<EMAIL_ADDRESS>Centro de Matemática, Computação e Cognição - Universidade Federal do ABC,
Santo André, SP, Brazil

###### Abstract

We study the influence of impurities in graphene described by a scattering
rate $\Gamma$ on the Casimir interaction between graphene and an ideal
conductor or between two identical sheets of graphene at zero temperature and
chemical potential. To this end, we compute the polarization tensor of
quasiparticles in graphene and corresponding conductivities for TE and TM
channels. The Casimir energy density is evaluated with the help of the
Lifshitz formula. We find that depending on the value of mass gap parameter
the presence of $\Gamma$ may lead to a slight damping or to a considerable
enhancement of the Casimir interaction.

##  I Introduction

The Casimir effect [1, 2] is an interaction of neutral bodies due to quantum
vacuum fluctuations. The study of Casimir interaction between advanced
materials is a new and promising area of research [3]. On one hand, the
unusual electronic properties of these materials lead to interesting effects
on the Casimir force. On the other hand, the improved quality of Casimir
experiments makes them a useful tool for exploration of the materials
themselves.

The Dirac materials (where the quasiparticles obey a quasirelativistic Dirac-
type equation at sufficiently low energies) provide us with an example of
interplay between Quantum Field Theory and Condensed Matter. Graphene is a
prominent representative of this family [4, 5]. Dealing with the Dirac
materials it is natural to describe the interaction with electromagnetic field
by the polarization tensor of quasiparticles and use this tensor to calculate
the Casimir interaction. In the case of graphene, such an approach was used in
[6] and in [7] at zero and non-zero temperature, respectively. Remarkably, the
polarization tensor approach to Casimir interaction of graphene was the only
one which was confirmed at experiments [8, 9, 10, 11].

All real materials contain impurities. Particular form of impurities can vary.
Impurities refer to a general form for breaking the cleanliness of pristine
materials. A classification of impurities and defects in graphene-like
materials may be found in the reviews [12, 13, 14, 15]. The two-dimensional
nature of graphene decreases the number of possible types of defects and
impurities. The point is that it is energetically favourable for adatoms or
substitutional impurities to reside outside of the graphene surface. They may
be charged [16, 17, 18], magnetic [15], isotopic [19, 20], topological such as
pentagons and heptagons [21, 13], or be a consequence of imperfections and
growth-induced defects like point [22] and cluster defects [12]. Intentional
impurity is usually called a dopant while the impurity itself can be both
intentional and unintentional (accidental). Doping is used to change the
physical or chemical properties of a material. Impurities in graphene [23, 24]
may transform a linear dispersion near Dirac points to a quadratic one which
signifies the appearance of a mass gap induced by impurities. There are
different approaches to describing impurities and their impact on the physical
properties of materials. The common is a tight-binding model with a shot- or
long-range potential [13], and scattering approach [25, 26].

With this large variety of the types of impurities in graphene we need a good
model which captures the universal properties of impurities while being
sufficiently simple for being used in calculation of the polarization tensor.
A successful way of describing impurities consists in adding to the propagator
of quasiparticles a parameter $\Gamma$ which describes the impurity scattering
rate. In other words, $\Gamma$ is an imaginary part of the fermion self-
energy. Such a description has been applied to graphene mostly in the presence
of an external magnetic field in [27, 28, 29, 30, 31]. The computations of
[31] are in a very good agreement with the measurements [32] of giant Faraday
rotation in graphene. In principle $\Gamma$ can depend on the frequency though
keeping it constant appeared to be a good approximation. In this work, we
neglect the other role of impurities which is their ability to create a non-
zero chemical potential $\mu$.

A particular form of impurities which were atoms (mostly sodium) on the
surface of graphene and their influence on the Casimir force were considered
in [11, 10]. According to these papers such impurities lead to a mass gap and
a non-zero chemical potential of graphene but not to the appearance of
impurity scattering described by scattering rate $\Gamma$.

The primary goal of this paper is to study the influence of impurity
scattering rate $\Gamma$ on the Casimir interaction between graphene and an
ideal metal and between two graphene sheets. We restrict ourselves to the case
of a zero temperature and vanishing chemical potential. This is a somewhat
simplified setting. Therefore, we will not try to compare our results to any
existing experiment (they were done at room temperature). Rather, we will
clarify some generic features of the Casimir interaction of graphene. Our main
conclusions is that turning on $\Gamma$ leads to a slight decrease of the
Casimir energy density for small masses and to a considerable enhancement
thereof for large masses. Thus, taking impurities into account may be
essential for analysing precision Casimir measurements.

This paper is organized as follows. In the next Section, we calculate the
polarization tensor of quasiparticles in graphene in the Pauli–Villars
subtraction scheme. In Section III, we analyse the conductivities following
from this polarization tensor and discuss their properties which are relevant
for the Casimir interaction. The Casimir interaction itself is studied in
Section IV. Section V contains some concluding remarks. We put long formulas
for the conductivities in the Appendix. We use the units $\hbar=c=1$.

##  II Polarization tensor

To fix our conventions, we start with the Dirac operator

$\not{D}=\mathrm{i}\tilde{\gamma}^{i}(\partial_{i}+\mathrm{i}eA_{i})-m,$ (1)

which describes the propagation of quasiparticles in graphene. The Latin
letters form the middle of Alphabet will are used to denote coordinates of
$2+1$ dimensional vectors, $i,j,k=0,1,2$. The letters from the beginning of
Alphabet will denote spatial components, $a,b,c=1,2$. A twiddle above a vector
means that the spatial components are rescaled with the Fermi velocity,

$\tilde{\gamma}^{0}=\gamma^{0},\qquad\tilde{\gamma}^{a}=v_{F}\gamma^{a}.$ (2)

The Dirac $\gamma$ matrices are $8$-dimensional (which corresponds to four
generations of fermions in graphene). They satisfy the condition
$\gamma^{i}\gamma^{j}+\gamma^{j}\gamma^{i}=2g^{ij}$ with the flat metric
$g^{ij}=(+,-,-)$. Besides, $\mathrm{tr}\,(\gamma^{i}\gamma^{j})=8g^{ij}$ and
$\mathrm{tr}\,(\gamma^{i}\gamma^{j}\gamma^{l}\gamma^{k})=8(g^{ij}g^{lk}-g^{il}g^{jk}+g^{ik}g^{jl})$.
We will use boldface to denote spatial components of the momenta 3-vectors,
$k=(k_{0},\mathbf{k})$, so that $p\cdot
k=p_{0}k_{0}-\mathbf{p}\cdot\mathbf{k}$ with
$\mathbf{p}\cdot\mathbf{k}=k_{a}p_{a}\delta^{ab}$ with $\delta^{ab}$ being the
Kronecker symbol.

The propagator $S(x)$ is the kernel of an inverse of a free Dirac operator
$\not{D}_{0}=\not{D}_{A=0}$, i.e. $\not{D}^{x}_{0}S(x-y)=\delta^{(3)}(x-y)$.
After a Fourier transform

$S(x)=\int\frac{\mathrm{d}^{3}k}{(2\pi)^{3}}e^{\mathrm{i}kx}S(k),$ (3)

one arrives at the following expression

$S(k)=-\frac{k_{0}\gamma^{0}+v_{F}k_{a}\gamma^{a}+m}{k_{0}^{2}-v_{F}^{2}k^{2}-m^{2}}.$
(4)

To introduce impurities, characterized by the scattering rate $\Gamma$, and
chemical potential $\mu$ one shifts the temporal component of the momentum $k$
in the propagator (4) as
$k_{0}\to\widehat{k}_{0}=k_{0}+\mathrm{i}\Gamma\mathrm{sgn}\,(k_{0})+\mu$,
(see [27, 28, 30, 29, 33]) For the rest of this work we set $\mu=0$. We denote
$\widehat{k}=(\widehat{k}_{0},\mathbf{k})$.

The introduction of $\Gamma$ into the Quantum Field Theory approach leads to
certain difficulties with the gauge invariance. The modification of fermionic
propagator can be understood as a consequence of an additional term in the
Dirac action containing
$\psi^{\dagger}\Gamma\mathrm{sgn}\,(-\mathrm{i}\partial_{0})\psi$. To maintain
gauge invariance, each partial derivative has to be accompanied by a gauge
field. That is, one should consider the expression
$\psi^{\dagger}\Gamma\mathrm{sgn}\,(-\mathrm{i}(\partial_{0}+\mathrm{i}eA_{0}))\psi$.
Although it is not clear how one should deal with the sign function of a
differential operator, new (and rather complicated) vertexes involving
$\psi^{\dagger}$, $\psi$, and $A_{0}$ seem to be inevitable. To overcome this
difficulty we proceed as in [34]. Namely, we consider only the diagrams
without $A_{0}$ lines which are thus independent of whichever vertices
involving $A_{0}$. I.e., we consider only the spatial components of
polarization tensor,

$\Pi^{ab}(p)=\mathrm{i}e^{2}\int\frac{\mathrm{d}^{3}k}{(2\pi)^{3}}\mathrm{tr}\,\left(S(\widehat{k})\tilde{\gamma}^{a}S(\widehat{k-p})\tilde{\gamma}^{b}\right).$
(5)

The full tensor may be recovered, if needed, with the help of the
transversality condition.

In the expression (5), one can change the integration variable
$k\to\tilde{k}$. The Jacobian factor $v_{F}^{-2}$ cancels $v_{F}^{2}$ coming
from $\tilde{\gamma}^{a}$ and $\tilde{\gamma}^{b}$. As a result, the whole
dependence on (5) $v_{F}$ remains in the external momentum only, so that

$\Pi^{ab}(p)=\Pi^{ab}_{v_{F}=1}(\tilde{p}).$ (6)

In other words, to compute the spatial components of polarization tensor it is
sufficient to do the computations for $v_{F}=1$ and then replace $p$ by
$\tilde{p}$. Till the end of this section, all computations will be done for
$v_{F}=1$. Thus,

$\Pi^{ab}(p)=\frac{\mathrm{i}e^{2}}{\pi^{3}}\int\mathrm{d}^{3}k\frac{-\delta^{ab}(m^{2}-\widehat{k}\cdot(\widehat{k-p}))+2k^{a}k^{b}-k^{a}p^{b}-k^{b}p^{a}}{(\widehat{k}^{2}-m^{2})((\widehat{k-p})^{2}-m^{2})}.$
(7)

By using the Feynman parametrization this integral can be rewritten as

$\Pi^{ab}(p)=\frac{\mathrm{i}e^{2}}{\pi^{3}}\int\mathrm{d}^{3}k\int_{0}^{1}\mathrm{d}x\,\frac{-\delta^{ab}(m^{2}-\widehat{k}\cdot(\widehat{k-p}))+2k^{a}k^{b}-k^{a}p^{b}-k^{b}p^{a}}{(x[\widehat{k}^{2}-m^{2}]+(1-x)[(\widehat{k-p})^{2}-m^{2}])^{2}}.$
(8)

The denominator of the integrand reads

$\displaystyle(x[\widehat{k}^{2}-m^{2}]+(1-x)[(\widehat{k-p})^{2}-m^{2}])^{2}=(\mathbf{q}^{2}-R)^{2},$
$\displaystyle\mathbf{q}=\mathbf{k}-(1-x)\mathbf{p},$ (9) $\displaystyle
R=x\widehat{k}_{0}^{2}+(1-x)(\widehat{k-p})_{0}^{2}-x(1-x)\mathbf{p}^{2}-m^{2}.$

We change the integration variable
$\mathrm{d}^{2}\mathbf{k}\to\mathrm{d}^{2}\mathbf{q}$. Note, that the terms in
numerator of (8) which are linear in $\mathbf{q}$ are integrated to zero.
Also, under the integral one may replace $\mathbf{q}^{a}\mathbf{q}^{b}$ by
$\tfrac{1}{2}\delta^{ab}\mathbf{q}^{2}$. After the integration over
$\mathbf{q}$ one arrives at the expression

$\Pi^{ab}(p)=\frac{\mathrm{i}e^{2}}{\pi^{2}}\int_{0}^{1}\mathrm{d}x\int\mathrm{d}k_{0}\frac{\delta^{ab}(m^{2}-\widehat{k}_{0}(\widehat{k-p})_{0}-x(1-x)\mathbf{p}^{2})+2x(1-x)p^{a}p^{b}}{R}.$
(10)

To integrate over $k_{0}$ one has to split the integration region into a union
of intervals where the signs of $k_{0}$ and $k_{0}-p_{0}$ are constant. If,
for simplicity, we restrict ourselves to the case $p_{0}>0$ these are the
intervals $(-\infty,0)$, $(0,p_{0})$, and $(p_{0},\infty)$. One can easily
show that the integrals over the first and the last intervals coincide. We
have,

$\displaystyle\Pi^{ab}(p)=\frac{\mathrm{i}e^{2}}{\pi^{2}}\int_{0}^{1}\mathrm{d}x\left[2\int_{-\infty}^{0}\mathrm{d}k_{0}\,\frac{\delta^{ab}(m^{2}-({k}_{0}-\mathrm{i}\Gamma)({k_{0}-p_{0}}-\mathrm{i}\Gamma)-x(1-x)\mathbf{p}^{2})+2x(1-x)p^{a}p^{b}}{x({k}_{0}-\mathrm{i}\Gamma)^{2}+(1-x)(k_{0}-p_{0}-\mathrm{i}\Gamma)^{2}-x(1-x)\mathbf{p}^{2}-m^{2}}\right.$
$\displaystyle\qquad\qquad\qquad+\left.\int_{0}^{p_{0}}\mathrm{d}k_{0}\,\frac{\delta^{ab}(m^{2}-({k}_{0}+\mathrm{i}\Gamma)({k_{0}-p_{0}}-\mathrm{i}\Gamma)-x(1-x)\mathbf{p}^{2})+2x(1-x)p^{a}p^{b}}{x({k}_{0}+\mathrm{i}\Gamma)^{2}+(1-x)(k_{0}-p_{0}-\mathrm{i}\Gamma)^{2}-x(1-x)\mathbf{p}^{2}-m^{2}}\right].$
(11)

The integral over $k_{0}$ is divergent, so that we need a regularization and
renormalization procedure for which we choose the Pauli–Villars subtraction.
We subtract from the integrand in (11) the same expression due to a regulator
field with a mass $M$, compute the integrals and then send $M\to\infty$.
Symbolically,

$\Pi^{ab}_{\mathrm{PV}}(p)=\lim_{M\to\infty}(\Pi^{ab}(p,m)-\Pi^{ab}(p,M)).$
(12)

We represent the polarization tensor through two form factors, $\alpha$ and
$\beta$, as

$\displaystyle\Pi^{ab}_{\mathrm{PV}}(p)=\alpha(p)\delta^{ab}+\beta(p)\,\frac{p^{a}p^{b}}{\mathbf{p}^{2}}.$
(13)

These form factors read

$\displaystyle\alpha(p)=\frac{2\mathrm{i}e^{2}}{\pi^{2}}\int_{0}^{1}\mathrm{d}x\left\\{\frac{-\mathrm{i}\pi
p^{2}(1-x)x}{\sqrt{-q}}-\frac{2p^{2}(1-x)x}{\sqrt{q}}\arctan\left(\frac{p_{0}x+\mathrm{i}\Gamma}{\sqrt{q}}\right)+\frac{2P^{2}(1-x)x}{\sqrt{Q}}\arctan\left(\frac{P_{0}x-\mathrm{i}\Gamma}{\sqrt{Q}}\right)\right.$
$\displaystyle\qquad\qquad+\left.\frac{1}{2}p_{0}(1-2x)\ln\left((p_{0}x+\mathrm{i}\Gamma)^{2}+q\right)-\frac{1}{2}P_{0}(1-2x)\ln\left((P_{0}x-\mathrm{i}\Gamma)^{2}+Q\right)\right\\},$
(14)
$\displaystyle\beta(p)=-\frac{4\mathrm{i}e^{2}\mathbf{p}^{2}}{\pi^{2}}\int_{0}^{1}\mathrm{d}x\,x(1-x)\left\\{\frac{\mathrm{i}\pi}{2\sqrt{-q}}+\frac{1}{\sqrt{q}}\arctan\left(\frac{p_{0}x+\mathrm{i}\Gamma}{\sqrt{q}}\right)-\frac{1}{\sqrt{Q}}\arctan\left(\frac{P_{0}x-\mathrm{i}\Gamma}{\sqrt{Q}}\right)\right\\},$

where $q=p^{2}(1-x)x-m^{2}$, $Q=P^{2}(1-x)x-m^{2}$,
$P^{2}=P_{0}^{2}-\mathbf{p}^{2}$, and $P_{0}=p_{0}+2\mathrm{i}\Gamma$. Note
that the integrals over $x$ in (14) are convergent.

In the limit $\Gamma\to 0$ our results agree with previous calculations,

$\Pi^{ab}_{\mathrm{PV}}(\Gamma=0)=-\frac{e^{2}}{\pi}\left(\delta^{ab}+\frac{p^{a}p^{b}}{p^{2}}\right)\frac{2m|p|-(p^{2}+4m^{2})\mathrm{arctanh}(|p|/2m)}{2|p|},$
(15)

see [35] and also [6].

##  III Conductivities

From now on we restore the dependence of polarization tensor on the Fermi
velocity according to Eq. (6).

In this Section, we consider the conductivity tensor defined as

$\sigma^{ab}=\frac{\Pi^{ab}_{\mathrm{PV}}}{\mathrm{i}p_{0}}.$ (16)

For vanishing spatial momenta, $p_{a}=0$, only a scalar conductivity
$\sigma(p_{0})$ entering the tensor conductivity as

$\delta^{ab}\sigma(p_{0})=\sigma^{ab}|_{p_{a}=0}$ (17)

is important. It is convenient to measure $\sigma(p_{0})$ in the units of
universal conductivity of graphene $\sigma_{\mathrm{gr}}=e^{2}/4$ which is
nothing else than the conductivity of a pristine graphene with $\Gamma\to+0$
at $m\to 0$. In terms of the form factors from the previous Section,
$\sigma(p_{0})=\alpha(p_{0},0)/(\mathrm{i}p_{0})$. For arbitrary values of the
parameters $\sigma(p_{0})$ can only be evaluated numerically. However, an
expansion in $p_{0}$ for $p_{0}\ll\Gamma$ and $p_{0}\ll m$ can be done
analytically,

$\frac{\sigma}{\sigma_{\mathrm{gr}}}=\frac{4}{\pi^{2}}\left\\{\frac{2\Gamma^{2}}{\Gamma^{2}+m^{2}}+\frac{\mathrm{i}p_{0}}{3m}\left(\frac{\Gamma
m\left(\Gamma^{2}-5m^{2}\right)}{\left(\Gamma^{2}+m^{2}\right)^{2}}+2\arctan\left(\frac{\Gamma}{m}\right)-\pi\right)+\mathcal{O}(p_{0}^{2})\right\\}.$
(18)

At $p_{0}=0$ this equation coincides with a relation obtained in [27].

Let us check whether at small frequencies $p_{0}$ the conductivity can be
approximated by the Drude formula

$\frac{\sigma_{\mathrm{Dr}}}{\sigma_{\mathrm{gr}}}=\frac{s\mathfrak{g}}{\mathrm{i}p_{0}+\mathfrak{g}}=\frac{s\mathfrak{g}^{2}}{p_{0}^{2}+\mathfrak{g}^{2}}-\mathrm{i}\frac{sp_{0}\mathfrak{g}}{p_{0}^{2}+\mathfrak{g}^{2}},$
(19)

where $\mathfrak{g}$ has the meaning of inverse scattering rate (similarly to
$\Gamma$) while $s$ defines the Drude weight. This is a phenomenological
formula valid mostly for metals. There is no profound reason why it should
describe the conductivity of graphene. However, this is an interesting check.

By identifying the leading terms in the small $p_{0}$ expansion of (19) with
(18) we obtain

$s=\frac{8\Gamma^{2}}{\pi^{2}(\Gamma^{2}+m^{2})},\qquad\mathfrak{g}=-\frac{6\Gamma^{2}m}{(\Gamma^{2}+m^{2})\left(\frac{\Gamma
m\left(\Gamma^{2}-5m^{2}\right)}{\left(\Gamma^{2}+m^{2}\right)^{2}}+2\arctan\left(\frac{\Gamma}{m}\right)-\pi\right)}.$
(20)

This quantities are plotted at Fig. 1 while the real and imaginary parts of
$\sigma$ are depicted at Fig. 2 in comparison with the Drude formula.

Figure 1: The plots of the parameters of Drude model given by Eq. (20)

Somewhat surprisingly, the Drude scattering rate $\mathfrak{g}$ may be
considerably larger or considerably smaller than the scattering rate $\Gamma$
of the quasiparticles. By looking at Fig. 2, we see that the agreement between
imaginary parts of $\sigma$ and $\sigma_{\mathrm{Dr}}$ is fairly good at low
frequencies though the deviations become larger at higher frequencies. The
real parts show qualitatively different behaviour. $\Re\sigma$ increases at
low frequencies while $\Re\sigma_{\mathrm{Dr}}$ decreases. Besides,
$\Re\sigma_{\mathrm{Dr}}$ does not have a jump at the threshold of pair
creation $p_{0}=2m$ which is a characteristic feature of the conductivity of
graphene. We conclude that the Drude formula fails to give a reasonable
approximation to the conductivity of graphene with impurities in the range of
frequencies which are relevant for the Casimir effect. The quality of
approximation may be improved by using the Drude–Lorentz formulas for
conductivities [36].

Figure 2: Real (a) and imaginary (b) parts of the conductivity as functions of
$p_{0}/\Gamma$. The solid lines are for Drude model (19) with parameters (20).
The dashed lines are the conductivity derived from the polarization tensor
with $\mathbf{p}=0$

For $\mathbf{p}\neq\mathbf{0}$, it is convenient to use two different
conductivities,

$\sigma_{\mathrm{te}}=\frac{\alpha}{\mathrm{i}p_{0}}\quad\mbox{and}\quad\sigma_{\mathrm{tm}}=\frac{\alpha+\beta}{\mathrm{i}p_{0}}.$
(21)

As we will see in the next Section, these conductivities describe the
reflection of TE and TM waves, respectively, on the surface of graphene. We
will also see that TM modes give a dominant contribution to the Casimir
interaction.

We would like to discuss a property of the conductivities which will be
important for the analysis of Casimir effect. When $\Gamma=0$ and the
$\tilde{p}$ is much smaller than $m$, the imaginary parts conductivities are
small while the real part vanishes, see (15). The smallness is a direct
consequence of the Pauli–Villars subtraction which ensures that the
polarization tensor vanished at $m\to\infty$. Since $\Gamma$ influences
conductivities, switching on $\Gamma$ at $\tilde{p}<2m$ should lead to an
increase of $\Re\sigma_{\mathrm{tm}}$. One should only find whether this
increase is large or small. Our numerical results are presented on Fig. 3
where $\Re\sigma_{\mathrm{tm}}(\Gamma)$ divided by the absolute values of
$\sigma_{\mathrm{tm}}$ at $\Gamma=0$ is depicted as a function of $\Gamma$ for
three different frequencies $p_{0}$ and a fixed $m=0.01$ eV. For $p_{0}$ much
smaller than $2m$ there is a strong enhancement of the conductivity. This
enhancement becomes weaker and disappears as $p_{0}$ approaches $2m$.

Figure 3: The normalized conductivity
$\Re\sigma_{\mathrm{tm}}(\Gamma)/|\sigma_{\mathrm{tm}}(\Gamma=0)|$ as a
function of $\Gamma$ for $m=0.01$ eV, $\mathbf{p}=0$, and three different
values of frequency $p_{0}$.

##  IV The Casimir energy

Let us consider two parallel infinite planes in the vacuum separated by a
distance $a$. One plane will be always occupied by graphene, while the other
plane will be either graphene or an ideal conductor. The Casimir energy per
unit surface for two interacting plane surfaces $I$ and $II$ is given by the
Lifshitz formula (see, for example, [1])

$\mathcal{E}^{I,II}=\int\frac{\mathrm{d}^{2}\mathbf{p}}{(2\pi)^{3}}\int_{0}^{\infty}\mathrm{d}\xi\left[\ln\left(1-e^{-2ap_{E}}r^{(I)}_{\mathrm{te}}r^{(II)}_{\mathrm{te}}\right)+\ln\left(1-e^{-2ap_{E}}r^{(I)}_{\mathrm{tm}}r^{(II)}_{\mathrm{tm}}\right)\right],$
(22)

where the integrations are done over the spatial momentum $\mathbf{p}$
parallel to the plates and over the imaginary frequency $\xi$,
$p_{0}=\mathrm{i}\xi$, and $p_{E}=\sqrt{\xi^{2}+\mathbf{p}^{2}}$. In (22),
$r_{\mathrm{te},\mathrm{tm}}^{I,II}$ denote the reflection coefficient for TE
and TM waves on the first and second surfaces, respectively. These
coefficients can be expressed through the polarization tensor as [7]

$r_{\mathrm{te}}=-\left(1+\frac{2p_{E}}{\xi\sigma_{\mathrm{te}}}\right)^{-1},\
r_{\mathrm{tm}}=\left(1+\frac{2\xi}{p_{E}\sigma_{\mathrm{tm}}}\right)^{-1},$
(23)

where $\sigma_{\mathrm{te},\mathrm{tm}}$ have been defined above in Eq. (21).
After continuation to the imaginary frequencies they read
$\sigma_{\mathrm{te}}=-{\alpha}/{\xi}$,
$\sigma_{\mathrm{tm}}=-{(\alpha+\beta)}/{\xi}$.

For an ideal conductor $r_{\mathrm{tm}}=-r_{\mathrm{te}}=1$ which can be
obtained from (23) by taking the limits
$\sigma_{\mathrm{te},\mathrm{tm}}\to\infty$. The Casimir energy density for
two ideal conductors reads

$\mathcal{E}_{\mathrm{id}}=-\frac{\pi^{2}}{720a^{3}}.$ (24)

We like to mention that Eq. (22) is valid only if the reflection matrix is
diagonal in the TE-TM basis. Otherwise, the Lifshitz formula is a bit more
complicated, see e.g., [37].

Since the reflection coefficients depend on $|\mathbf{p}|$ rather than on
individual components of $\mathbf{p}$ we can integrate over the angular
coordinates in the Lifshitz formula (22) which boils down to the replacement
$\mathrm{d}^{2}\mathbf{p}\to(2\pi)|\mathbf{p}|\mathrm{d}|\mathbf{p}|$. Next,
we introduce new variables, $y$ and $z$, by the formulas
$\mathbf{p}^{2}=p_{E}^{2}(1-y^{2})$, $\xi=p_{E}y$, and $p_{E}\to z/a$. In
these new variables,

$\mathcal{E}^{I,II}=\frac{1}{a^{3}}\int_{0}^{\infty}\frac{z^{2}\mathrm{d}z}{(2\pi)^{2}}\int_{0}^{1}\mathrm{d}y\left[\ln\left(1-\frac{e^{-2z}}{\left(1+\frac{2}{y\sigma_{\mathrm{te}}^{I}}\right)\left(1+\frac{2}{y\sigma_{\mathrm{te}}^{II}}\right)}\right)+\ln\left(1-\frac{e^{-2z}}{\left(1+\frac{2y}{\sigma_{\mathrm{tm}}^{I}}\right)\left(1+\frac{2y}{\sigma_{\mathrm{tm}}^{II}}\right)}\right)\right],$
(25)

One can notice that apart from an overall factor of $1/a^{3}$ the whole
dependence of Casimir energy (25) on the distance resides in the
conductivities where it appears in the combinations

$\tilde{m}=\frac{am}{z},\qquad\tilde{\Gamma}=\frac{a\Gamma}{z}.$ (26)

Explicit formulas for $\sigma_{\mathrm{te}}$ and $\sigma_{\mathrm{tm}}$ are
quite long and deferred to the Appendix A, see (33). The limit $a\to 0$ is
equivalent to taking $m\to 0$ and $\Gamma\to 0$ so that we have the case
gapless pristine graphene studied in [6] with $\mathcal{E}\sim 1/a^{3}$.

To analyze the opposite limit of large distance we have to take
$\tilde{m}\to\infty$ and $\tilde{\Gamma}\to\infty$ while keeping the fraction
$\tilde{\Gamma}/\tilde{m}=\Gamma/m$ fixed. In this limit,
$\sigma_{\mathrm{te}}$ and $\sigma_{\mathrm{tm}}$ behave identically,

$\displaystyle\frac{\sigma_{\mathrm{te}}}{\sigma_{\mathrm{gr}}},\
\frac{\sigma_{\mathrm{tm}}}{\sigma_{\mathrm{gr}}}\to$
$\displaystyle\frac{8\tilde{\Gamma}^{2}}{\pi^{2}\left(\tilde{\Gamma}^{2}+\tilde{m}^{2}\right)}=\frac{8}{\pi^{2}}G,$
(27)

where

$G\equiv\frac{\Gamma^{2}}{m^{2}+\Gamma^{2}}\leq 1.$ (28)

Thus, we have the case of constant conductivities analysed in [38]. The
Casimir energy decays as $1/a^{3}$ and

$\displaystyle\frac{\mathcal{E}^{\mathrm{g,g}}_{\mathrm{tm}}}{\mathcal{E}_{\mathrm{id}}}\approx
4.3\times
10^{-3}G,\qquad\frac{\mathcal{E}^{\mathrm{g,g}}_{\mathrm{te}}}{\mathcal{E}_{\mathrm{id}}}\approx
1.3\times 10^{-5}G^{2},$ (29)
$\displaystyle\frac{\mathcal{E}^{\mathrm{g,id}}_{\mathrm{tm}}}{\mathcal{E}_{\mathrm{id}}}\approx
2\times
10^{-2}G,\qquad\frac{\mathcal{E}^{\mathrm{g,id}}_{\mathrm{te}}}{\mathcal{E}_{\mathrm{id}}}\approx
2.1\times 10^{-3}G,$ (30)

where the superscripts ${\mathrm{g,g}}$ and ${\mathrm{g,id}}$ mean the
interaction between two identical sheets of graphene and between graphene and
an ideal conductor, respectively. We normalized the results to the Casimir
energy of two ideal conductors (24) and separated the contributions of TE and
TM modes. As usual [7], the contributions of TE modes are much smaller than
the TM contributions.

For $\Gamma=0$ the terms on the right hand sides of (27) vanish and the
leading terms in the $a\to\infty$ expansion read

$\frac{\sigma_{\mathrm{te}}}{\sigma_{\mathrm{gr}}}=\frac{4z{((1-y^{2}){v_{F}}^{2}+y^{2})}}{3\pi
y(am)},\qquad\frac{\sigma_{\mathrm{tm}}}{\sigma_{\mathrm{gr}}}=\frac{4z{y}}{3\pi(am)}.$
(31)

As a result, the Casimir energy of two graphene sheets behave at large
distances as $\mathcal{E}\sim 1/(a^{3}(am)^{2})$ while for the system of a
perfect metal and graphene the asymptotic behaviour is $\mathcal{E}\sim
1/(a^{3}(am))$, in agreement with [6].

The rest of the analysis will be done numerically. For orientation, we present
the value of $\Gamma=4.4\times 10^{-3}$ eV which gives the best fit [31]
between the Quantum Field Theory polarization tensor and the Faraday rotation
experiment [32] at $7$ T.111For a magnetic field of $3$T different values of
the parameters were obtained as well as a much worse fit. This can be
attributed to some specific physics in particular samples of graphene used in
the experiment, see [39]. The separation $a$ used in the Casimir experiments
with graphene [8, 10] is between $200$ nm and $2000$ nm. The value for $m$ for
the graphene sample used in the Casimir experiment [11] was $\sim 0.15$ eV.
The presence of a damping factor $e^{-2ap_{E}}$ in the Lifshitz formula (22)
indicates that the region of relevant Euclidean momenta is restricted by the
inequality $p_{E}\lesssim(2a)^{-1}$. E.g., for $a=500$ nm one has
$(2a)^{-1}\simeq 0.2$ eV.

Figure 4: The Casimir energy density as a function of the distance normalised
$\mathcal{E}_{\mathrm{id}}$ (Eq. (24)) for the system of graphene and an ideal
conductor (on (a) and (b)) and for two identical graphene sheets (c). The
impurity scattering rate is $\Gamma=5\times 10^{-3}$ eV on (a) and (c), and
$\Gamma=5\times 10^{-2}$ eV on (b).

The Casimir energy density normalized to the Casimir energy density between
two ideal conductors is plotted on Fig. 4. By comparing Fig. 4(a,b) with Fig.
4(c) we see that the Casimir interaction of graphene with graphene behaves
qualitatively similarly to the interaction of graphene with an ideal conductor
though the former is about 5 times weaker than the latter. Thus, in what
follows we will only consider the interaction of graphene with an ideal
conductor. On Figs. 4 (a) and (b) we see that the Casimir interaction is
damped by the mass exactly as it happens for $\Gamma=0$, see [6]. The line for
$m=0$ appears slightly upper for $\Gamma=0.005$ eV than for $\Gamma=0.05$ eV,
while the line for $m=0.05$ eV passes significantly lower on Fig. 4(a) than on
Fig. 4(b). Thus, the effect of $\Gamma$ strongly depends on the mass. To see
this effect more clearly, we depicted on Fig. 5 the relative variation of
Casimir energy density

$\delta_{\Gamma}\mathcal{E}^{\mathrm{g},\mathrm{id}}=\frac{\mathcal{E}^{\mathrm{g},\mathrm{id}}(\Gamma)-\mathcal{E}^{\mathrm{g},\mathrm{id}}(\Gamma=0)}{\mathcal{E}^{\mathrm{g},\mathrm{id}}(\Gamma=0)}$
(32)

at a fixed separation $a=500$ nm and three different values of the mass.

The large relative enhancement of the Casimir effect for $m=0.1$ eV can be
explained by the observation made at the end of the previous section that
turning on $\Gamma$ leads to a significant enhancement of
$\Re\sigma_{\mathrm{tm}}$ for the frequencies $p_{0}\ll m$. For a larger mass
the interval of frequencies where the enhancement of conductivity takes place
becomes larger which is translated to the enhancement of Casimir interaction.
Since the Casimir effect is an integral effect containing competing
contributions from all frequencies and tangential momenta it is impossible to
predict the amplitude of the enhancement basing on this type of arguments. For
the same reason, it is hard to give a precise explanation to the very moderate
damping of Casimir force by $\Gamma$ at very small masses like $m=10^{-3}$ eV
at Fig. 5. We can only suppose that at the absence of the mechanism described
above the impurity scattering should reduce the conductivities and thus the
Casimir interaction according to the high-school physics intuition. For an
intermediate mass, $m=10^{-2}$ eV, we see an intermediate behaviour.

Figure 5: The plots of relative variation of the Casimir energy density (32)
at $a=500$nm as a function of $\Gamma$ for three values of the mass.

Another characteristic feature of the Casimir interaction in the presence of
$\Gamma$ is a $1/a^{3}$ behaviour of the Casimir energy at large distances.
However, the asymptotic values (30) are reached at very large $a$. For
example, for $m=\Gamma=0.01$ eV at $a=10^{6}$ nm the difference between exact
and asymptotic values is still at the 10% level. For some other choices of $m$
and $\Gamma$ the agreement may be reached at somewhat smaller distances which
are anyway beyond the distances used at experiments.

##  V Conclusions

In this paper, we studied the Casimir interaction of graphene containing
impurities described by a scattering rate $\Gamma$. We evaluated the
polarization tensor of quasiparticles in the Pauli–Villars subtraction scheme
and computed the corresponding conductivities. Next, we used the
conductivities in the Lifshitz formula to calculate the Casimir energy density
for various values of parameters. Our main message is that the presence of
impurities can considerably enhance the Casimir interaction for large values
of the mass (about $m=0.1$ eV) and somewhat damp the Casimir interaction for
near zero mass. Thus, the non-zero impurity scattering rate $\Gamma$ should be
taken into account in the analysis of precision Casimir experiments with
graphene. To do this however one should also take into consideration the
chemical potential and temperature since both influence in an essential way
the Casimir force [7, 40, 11]. This will be our next task which we are going
to take up in the near future.

An additional motivation for considering the influence of impurities jointly
with the temperature comes from the old discussion on the Nernst theorem in
the Casimir physics [41, 42, 43], see [44] for a recent review. The violation
of Nernst theorem was blamed on the relaxation rate parameter present in the
Drude formula. It is interesting to check what happens if this rate appears in
a different model.

In [34] an unconventional version of the Pauli–Villars subtraction scheme in
the presence of impurities was suggested. It would be interesting to study the
influence of this scheme on Casimir interaction.

Finally, we would like to mention an approach to the Casimir interaction of
graphene based on the QFT effective action [6, 45]. This approach is
equivalent to a fine structure constant expansion of the Lifshitz formula.

###### Acknowledgements.

We are grateful to Ignat Fialkovsky for previous collaboration and discussions
on the impurities in graphene. This work was supported in parts by the São
Paulo Research Foundation (FAPESP) through the grants 2021/10128-0 (D.V.) and
2022/08771-5 (N.K.), and by the National Council for Scientific and
Technological Development (CNPq), grant 304758/2022-1 (D.V).

##  Appendix A Long but useful formulas

Here we present explicit form of the conductivities after the changes of
variables made above Eq. (25) and in (26).

$\displaystyle\frac{\sigma_{\mathrm{te}}}{\sigma_{\mathrm{gr}}}=\frac{8}{\pi^{2}y}\int_{0}^{1}$
$\displaystyle\mathrm{d}x\left\\{\frac{\pi{((1-y^{2}){v_{F}}^{2}+y^{2})}(1-x)x}{\sqrt{\tilde{m}^{2}+(1-x)x((1-y^{2}){v_{F}}^{2}+y^{2})}}\right.\allowdisplaybreaks$
$\displaystyle-\frac{2{((1-y^{2}){v_{F}}^{2}+y^{2})}(1-x)x}{\sqrt{\tilde{m}^{2}+(1-x)x((1-y^{2}){v_{F}}^{2}+y^{2})}}\arctan\left(\frac{\tilde{\Gamma}+yx}{\sqrt{\tilde{m}^{2}+(1-x)x((1-y^{2}){v_{F}}^{2}+y^{2})}}\right)\allowdisplaybreaks$
$\displaystyle-\left.\frac{2{((1-y^{2}){v_{F}}^{2}+(y+2\tilde{\Gamma})^{2})}(1-x)x}{\sqrt{\tilde{m}^{2}+(1-x)x((1-y^{2}){v_{F}}^{2}+(y+2\tilde{\Gamma})^{2})}}\arctan\left(\frac{\tilde{\Gamma}-x(y+2\tilde{\Gamma})}{\sqrt{\tilde{m}^{2}+(1-x)x((1-y^{2}){v_{F}}^{2}+(y+2\tilde{\Gamma})^{2})}}\right)\right.\allowdisplaybreaks$
$\displaystyle+\left.\frac{1}{2}y(1-2x)\ln\left((yx+\tilde{\Gamma})^{2}+\tilde{m}^{2}+(1-x)x((1-y^{2}){v_{F}}^{2}+y^{2})\right)\right.\allowdisplaybreaks$
$\displaystyle-\left.\frac{1}{2}(y+2\tilde{\Gamma})(1-2x)\ln\left((\tilde{\Gamma}-(y+2\tilde{\Gamma})x)^{2}+\tilde{m}^{2}+(1-x)x((1-y^{2}){v_{F}}^{2}+(y+2\tilde{\Gamma})^{2})\right)\right\\},\allowdisplaybreaks$
$\displaystyle\frac{\sigma_{\mathrm{tm}}}{\sigma_{\mathrm{gr}}}=\frac{8}{\pi^{2}y}\int_{0}^{1}$
$\displaystyle\mathrm{d}x\left\\{\frac{\pi{y^{2}}(1-x)x}{\sqrt{\tilde{m}^{2}+(1-x)x((1-y^{2}){v_{F}}^{2}+y^{2})}}\right.\allowdisplaybreaks$
$\displaystyle-\frac{2{y^{2}}(1-x)x}{\sqrt{\tilde{m}^{2}+(1-x)x((1-y^{2}){v_{F}}^{2}+y^{2})}}\arctan\left(\frac{\tilde{\Gamma}+yx}{\sqrt{\tilde{m}^{2}+(1-x)x((1-y^{2}){v_{F}}^{2}+y^{2})}}\right)\allowdisplaybreaks$
$\displaystyle-\left.\frac{2{(y+2\tilde{\Gamma})^{2}}(1-x)x}{\sqrt{\tilde{m}^{2}+(1-x)x((1-y^{2}){v_{F}}^{2}+(y+2\tilde{\Gamma})^{2})}}\arctan\left(\frac{\tilde{\Gamma}-x(y+2\tilde{\Gamma})}{\sqrt{\tilde{m}^{2}+(1-x)x((1-y^{2}){v_{F}}^{2}+(y+2\tilde{\Gamma})^{2})}}\right)\right.\allowdisplaybreaks$
$\displaystyle+\left.\frac{1}{2}y(1-2x)\ln\left((yx+\tilde{\Gamma})^{2}+\tilde{m}^{2}+(1-x)x((1-y^{2}){v_{F}}^{2}+y^{2})\right)\right.\allowdisplaybreaks$
$\displaystyle-\left.\frac{1}{2}(y+2\tilde{\Gamma})(1-2x)\ln\left((\tilde{\Gamma}-(y+2\tilde{\Gamma})x)^{2}+\tilde{m}^{2}+(1-x)x((1-y^{2}){v_{F}}^{2}+(y+2\tilde{\Gamma})^{2})\right)\right\\}.$
(33)

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