11institutetext:  European Synchrotron Radiation Facility, 71 Avenue des
Martyrs, 38000 Grenoble, France

# Beam focus modifications by cropping partially coherent X-ray beams

Manuel Sanchez del Rio 11 Rafael Celestre 11 Juan Reyes-Herrera 11 Philipp
Brumund 11
Marco Cammarata 1111

###### Abstract

We simulate the focusing of a partially-coherent X-ray beam emitted by an
undulator in a fourth-generation storage ring by performing a coherent mode
decomposition and wave optics propagation. The focus position is shifted, and
its size is enlarged when an entrance slit crops the beam. This is the usual
case when a slit is used to select the coherent fraction. The pairing of two
focusing elements (mirrors, lenses or transfocators) to ensure a fixed focal
position is also analyzed. Our results show that the image of a partially
coherent source, such an undulator in a low-emittance storage ring, is a non-
trivial function of the aperture used to control the coherence fraction.

##  1 Introduction

Fourth-generation storage-ring-based X-ray synchrotron sources deliver photon
beams with high brilliance and coherence. Although the transverse coherence of
these beams is highly improved in the horizontal direction, the overall
coherent fraction is of the order of a few per cent for hard X-rays ($>10$
keV). Beamline optical elements such as pinholes and slits are then used to
improve coherent fraction to values typically $>80\%$ needed for applications
exploiting coherence, such as X-ray photon correlation spectroscopy, coherent
diffraction imaging, propagation-based phase-contrast imaging, and
ptychography [1]. The coherent beam interaction with the optical elements
produce diffraction, thus modifying the beam characteristics and also
affecting the beam focusing, as discussed here.

Let us consider the case of an ideally focusing system of focal length $f$
(made by a mirror or lens that focuses the source into the image plane). The
position of the focus with respect to the focusing element $q$ is given, in
the framework of geometric optics, by the lens equation
$f^{-1}=p^{-1}+q^{-1}$, with $p$ the source-element distance. If the numerical
aperture (NA) of the beam is reduced (for example by the finite dimension of
the lens or mirror, or by using a slit) the location and dimension of the
focus change as a result of diffraction. It has been shown [2] that the focal
position of a Gaussian beam moves towards the lens position when the NA
decreases. This shifting of the focal position is relevant for synchrotron
beamlines, as demonstrated by Westfahl et al. [3]. These authors noticed a
shift of the horizontal focal position upstream from the position given by
geometrical optics (Fig. 7 ibid.) when the horizontal acceptance is reduced by
a slit.

The diffraction effect (due to the slits and finite size of the optical
elements) not only shifts the focal position (up- or downstream), but also
changes the focal dimensions. These facts must be taken into account in
beamlines in fourth-generation synchrotron sources. They are critical when
designing beamlines with several coupled focusing elements. We studied this
phenomenon in the context of the project for the new “EBSL1” beamline at the
upgraded EBS-ESRF storage ring. This beamline will produce highly coherent
beams of variable cross section at the sample position. Two refractive systems
(transfocators) will be paired to allow a varying focal size, whereas a slit
placed upstream from the transfocators is used to control the coherent
fraction. The optical matching of the transfocators is strongly correlated to
the slit aperture (or coherent fraction). The performances of such systems are
calculated in the framework of the partially coherent optics using a fast
algorithm for decomposition of the undulator radiation in coherent modes that
are propagated along the beamline using the algorithms presented in [4]. The
suitability of this method is discussed (ibid.) and compared with other
partial coherence simulation algorithms. The key point is to treat separately
the horizontal and vertical planes, therefore working with one-dimensional
wavefronts.

##  2 Focal position produced by a single focusing element

Let us consider an aperture of dimension $a$ placed between the source and a
focusing optical element (here we consider a lens), at a distance $p_{a}$ from
the lens ($p_{a}<p$). The lens is set to focus the source into an image plane
at $q$ downstream from the lens. Following the geometrical optics, the
position of the focal plane is defined by the lens equation (which does not
depend on the aperture $a$), and the focal size is $M\times s$ (considering
the aperture size $a$ is larger than the source size $s$). $M=q/p$ is the
optical magnification. If $a$ is smaller than the beam size, the aperture
obscures part of the source reducing the beam intensity. However, the focal
size and focal position (where the beam waist is found) are not modified. When
the beam has a high coherence, these results predicted by geometrical optics
are insufficient [5]. This section presents a numerical study of the position
and size of the beam waist for different apertures taking into account the
partial coherence of undulator emission. We simulated a U18 undulator (period
$\lambda_{u}=$18\text{\,}\mathrm{mm}$$) with $N_{u}=138$ periods. The gap is
tuned to have the first harmonic at $E=7$ keV (deflecting parameter
$K=1.851$). We consider a Be lens with parabolic profile and radius at the
apex $R=$0.2\text{\,}\mathrm{mm}$$ ($f=$14.35\text{\,}\mathrm{m}$$ at 7 keV),
located at a distance $p=$65\text{\,}\mathrm{m}$$ from the source. A slit of
variable aperture $a$ is placed at $p_{a}=$29\text{\,}\mathrm{m}$$ upstream
from the lens.

We first suppose an ideal storage ring with zero emittance, therefore
producing a fully coherent emission. The beam at the slit plane has a full-
width at half-maximum (FWHM)
$a_{\text{FWHM}}=~{}$565\text{\,}\mathrm{\SIUnitSymbolMicro m}$$. The aperture
$a$ takes different values from fully opening (the whole beam passes the slit
therefore the transmittivity is $T=1$) to a very narrow aperture ($T\ll 1$).
The refracted beam is analyzed at different positions from the lens by
calculating the FWHM of the intensity distribution. At the beam waist the FWHM
presents a minimum.

Figure 1: Evolution of the size of an undulator beam cropped by a slit and
focused by a lens of $R=$0.2\text{\,}\mathrm{mm}$$. The plots show the FWHM of
the beam as a function of the distance from the lens for different values of
the aperture $a$ (in $\text{\,}\mathrm{\SIUnitSymbolMicro m}$). a) Zero
emittance case ($\varepsilon=0$, full coherence), b) horizontal emittance
($\varepsilon=130$ pm rad) and c) vertical emittance ($\varepsilon=10$ pm rad)
for the EBS-ESRF storage ring. The focal positions given by geometrical optics
when the source is considered at either the undulator position (blue vertical
line at $q$) or at the slit position (orange vertical line at $q_{a}$) are
marked. Full lines in b), c) correspond to partial coherent (multi-mode) beam,
whereas the dashed lines correspond to the full coherent beam (first mode).

Fig. 1a shows the evolution of the coherent beam size after being focused by
the lens. It also shows markers for the focal positions predicted by the lens
equation: $q=$18.417\text{\,}\mathrm{m}$$ considering the source at the
undulator position (blue line) and $q_{a}=$28.408\text{\,}\mathrm{m}$$ if one
considers the slit as an effective source (orange line). As expected, the beam
waist calculated numerically converges to these respective values when the
slit is opened ($T=1$, Fresnel number $N_{F}=78$, grey line) and almost closed
($T\approx 0$, blue line). For intermediate values of $a$, the beam waist
“moves” from one case to another. For $T=0.93$ ($N_{F}=35$, magenta line) the
cropping by the slit is negligible so the situation does not change with
respect to the open slit. For $T=0.56$ ($N_{F}=8.7$, brown line) the waist
presents a flat depression, increasing its minimum FWHM and also the depth of
focus. Smaller values of $T$ and $N_{F}$ shift the minimum to higher distances
(e.g., $T=0.12$, $N_{F}=0.3$, red line), and the minima become less
pronounced. A small FWHM is found close to $q_{a}$ (orange marker) for
$T=0.05$ ($N_{F}=0.09$, green line), showing a twin minimum due to the
interference fringes found in the intensity distributions. Both minima
converge to this position for $T=0.01$ ($N_{F}=0.003$, orange line), and
$T=0.003\approx 0$ ($N_{F}=0.0009$, blue line). The numeric values of the FWHM
agree with those calculated using geometrical concepts only for the limiting
cases of waist at $q$ and $q_{a}$. However, for intermediate slit apertures,
where the waist position is in between $q$ and $q_{a}$, the waist size is
different to that predicted by geometric optics, and significantly higher than
the limiting values at $q$ and $q_{a}$, showing the envelope in Fig. 1a
(dashed black line). This is an important fact: good focusing, i.e., a very
small beam size is only obtained in the limiting cases of open slit and
almost-closed slit. This means that for a fully coherent beam, the slit
worsens the focusing. The diffraction at the slit creates a spurious
divergence that affects the lens focusing. When the focal distance of the lens
is reduced (using lenses with smaller radius, or piling several lenses), the
$q$ and $q_{a}$ positions shift to shorter distances, and become closer one to
another ($|q-q_{a}|$ also reduces). They converge to a single position: the
lens focal length $q=q_{a}=f$. This happens when both source-lens and slit-
lens distances can be considered infinite.

We used the EBS-ESRF emittance values111Throughout this work we used the
electron beam sizes and divergences at the center of the straight section:
$\sigma_{x}=$29.7\text{\,}\mathrm{\SIUnitSymbolMicro m}$$,
$\sigma_{x^{\prime}}=$4.37\text{\,}\mathrm{\SIUnitSymbolMicro rad}$$,
$\sigma_{y}=$5.29\text{\,}\mathrm{\SIUnitSymbolMicro m}$$,
$\sigma_{y^{\prime}}=$1.89\text{\,}\mathrm{\SIUnitSymbolMicro rad}$$,
corresponding to beam emittances:
$\varepsilon_{x}=$130\text{\,}\mathrm{pm}\text{\,}\mathrm{rad}$$,
$\varepsilon_{y}=$10\text{\,}\mathrm{pm}\text{\,}\mathrm{rad}$$, and beta
functions $\beta_{x}=$6.8\text{\,}\mathrm{m}$$,
$\beta_{y}=$2.8\text{\,}\mathrm{m}$$.  to perform the coherent mode
decomposition of the undulator source. The details for that are presented in
[4]. This is done for the horizontal and vertical directions, yielding
coherent fractions (at 7 keV) $CF_{h}=13\%$ and $CF_{v}=58\%$, respectively.
These values are much higher than for the old ESRF-1 source, but still too low
to apply the full-coherence approximation. Therefore, we propagated a number
of modes large enough to contain more than 99% of the source intensity (36
modes in H and 8 in V). The illumination at the entrance slit plane is
$610\text{\,}\mathrm{\SIUnitSymbolMicro m}$ (H) $\times$
$566\text{\,}\mathrm{\SIUnitSymbolMicro m}$ (V). The slit aperture is
effectively used to tune the coherence of the beam: closing the slit increases
the $CF$. In the limit (zero aperture) the beam after the slit is fully
coherent ($CF=1$), but obviously with zero transmittivity ($T=0$). The choice
of the right slit aperture comes from a compromise between coherence and flux.
Figures 1b and 1c show the focal size along the optical axis for several
apertures in the horizontal and vertical planes. We calculated the case of the
partially coherent beam (multi-mode, solid lines) and also the case of a fully
coherent beam (only the first coherent mode, dashed lines). We observe that
the focal positions (the minima of the plot lines) do not change significantly
when passing from full to partial coherence. However, the focal dimension
changes significantly in the horizontal direction for cases with
$CF_{h}\leq~{}70\%$. In the vertical direction, where the beam at the source
was more coherent, there is not much difference in sizes when going from full
(dashed lines) to partial coherence (solid lines). Looking at the focal
position shift versus $q_{a}$ (orange marker) we find important differences in
the horizontal and vertical directions. Closing the slit produces a gradual
shift of the focal position from the position of the geometrical image of the
undulator source (blue vertical line) to the geometrical image of the slit
(orange vertical line). However, in the vertical plane, the slit crops the
beam only slightly, and closing the slit to go from the source $CF_{v}=58\%$
to values up to 90% does not produce any focal shift from the position of the
geometrical image of the undulator source. Only when the slit is very narrow
(e.g., $a_{v}=$25\text{\,}\mathrm{\SIUnitSymbolMicro m}$$, red curve) the
focal position shifts to the locus of the geometrical image of the slit. This
case is in principle not interesting experimentally as it reduces the
intensity from an already quite coherent beam ($CF_{v}=90\%$). In summary, for
practical values of aperture selected to increase the beam $CF$ to values up
to 90% the behaviour in vertical (V) and horizontal (H) directions is
different: the better source coherence in V permits working with quite open
slits thus the focusing system “sees” the source at the undulator position,
whereas in H one needs a significant crop of the beam thus shifting and
enlarging the beam waist. Notice that the good coherence of the beams emitted
by the EBS-ESRF and other 4${}^{\text{th}}$ generation storage permits
performing coherence experiments with a “conservative” use of slit (open in V
and partially closed in H). By contrast, these experiments at
3${}^{\text{rd}}$ generation sources require a drastic closing of the slits to
a pinhole size.

##  3 Focal lengths and sizes for paired focusing elements

Consider an ideal optical system composed of two focusing elements which have
focal lengths $f_{1}$ and $f_{2}$, and which are separated by a distance $D$.
Following [6], the relationship between object-to-element-1 distance $p_{1}$
and the element-2-to-image distance $q_{2}$ given by the geometrical optics
is:

$D-(f_{1}+f_{2})=\frac{f_{1}^{2}}{p_{1}-f_{1}}+\frac{f_{2}^{2}}{q_{2}-f_{2}},$
(1)

which corresponds to an hyperbola222Expanding eq. (1) one gets
$f_{1}(Dq_{2}+p_{1}q_{2})+f_{2}(Dp_{1}+p_{1}q_{2})-f_{1}f_{2}(D+p_{1}+q_{2})=Dp_{1}q_{2}$,
with horizontal asymptote at $f_{2}=((p_{1}+D)^{-1}+q_{2}^{-1})^{-1}$ and
vertical asymptote at $f_{1}=(p_{1}^{-1}+(D+q_{2})^{-1})^{-1}$.  in the
($f_{1},f_{2}$) plane. The global magnification is the product of the
magnification of the individual focusing elements.

$M=M_{1}M_{2}=\frac{1-q_{2}/f_{2}}{1-p_{1}/f_{1}}.$ (2)

a) b)

Figure 2:  Maps of beam peak intensity in the horizontal plane (column a) and
vertical plane (b) at the sample position obtained by two lenses of variable
focal lengths ($f_{1}$,$f_{2}$). Slit values (in
$\text{\,}\mathrm{\SIUnitSymbolMicro m}$) are $a_{h}=[1000,145.5,85.1,40.3]$
(a) and $a_{v}=[1500,506.7,227.0,25]$ (b), corresponding to transmitted
coherence fraction $CF_{h}=[0.13,0.5,0.7,0.9]$ and
$CF_{v}=[0.58,0.7,0.9,0.999]$, respectively. The analytical hyperbolic
trajectories from geometric optics (eq. (1)) are over-plotted in white (dotted
if the source is at the undulator, and dashed if the source is at the slit).
We represented in red dots some optimum pairs ($f_{1},f_{2}$) that guarantee
that the beam waist is at the focal plane.

a)

b)

Figure 3:  Size of the image in a) horizontal and b) vertical planes for
different slit apertures.

The magnification is not dependent on $D$, however, the optical throw (length
of the optical system $L=p_{1}+D+q_{2}$) will change if one changes the focal
distances of the elements. For a constant $L$ one can change magnification by
changing the inter-element distance $D$ (zoom effect). In a synchrotron
beamline, the transfocators are refractive focusing elements that allow
changing the magnification by varying their focal lengths (by adding or
removing lenses) [7]. Alternatively, two Kirkpatrick-Baez systems with
bendable mirrors or multilayers can be used.

We consider an optical system made by two paired transfocators to get a
variable magnification (variable focal size) in a fixed location (the sample
position). The configuration match the requirements of the EBS-ESRF EBSL1
beamline. The source is a U18 undulator tuned at a photon energy of
$7\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}$ (as used before). A slit of
aperture $a_{h}\times a_{v}$ is placed at $36\text{\,}\mathrm{m}$ from the
undulator. We represent the two transfocators by two real beryllium lenses of
variable curvature radius $R_{1}$ and $R_{2}$ that correspond to focal
distances $f_{1}$ and $f_{2}$, with $f_{1,2}=R_{1,2}/(2\delta)$,
$\delta=6.96\times 10^{-6}$ for Be at
$7\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}$. Lens-1 is placed at
$p_{1}=$65\text{\,}\mathrm{m}$$ from the source, and lens-2 at
$D=$105\text{\,}\mathrm{m}$$ downstream from lens-1. The image plane, where
the sample is placed, is at $q_{2}=$30\text{\,}\mathrm{m}$$ downstream from
the lens-2. The optical throw is $L=p_{1}+D+q_{2}=$200\text{\,}\mathrm{m}$$.
The slit is $p_{a}=$29\text{\,}\mathrm{m}$$ upstream from lens-1 (i.e. at
$36\text{\,}\mathrm{m}$ from the undulator).

For a given value of $f_{1}$ and the the fixed distances ($p_{1}$, $D$ and
$q_{2}$), $f_{2}$ can be calculated analytically with eq. (1). The trajectory
pairs ($f_{1},f_{2}$) obtained in this way guarantee that the focus is at the
sample position ($L=$200\text{\,}\mathrm{m}$$ from source). However, as shown
in the previous section, these results of the geometrical optics are not exact
when partially coherent beams are cropped by a slit. Figure 2 shows the
results of the numerical search of the $f_{2}$ to produce the waist at the
sample plane. It is obtained by performing a map of the on-axis intensity
$\mathcal{I}_{0}$ of the beam at the sample plane versus $(f_{1},f_{2})$. Each
pixel corresponds to a simulation to compute the pattern at the image plane,
considering partial coherence, i.e., propagating many coherent modes. For a
given $f_{1}$ value, one can select the $f_{2}$ value that produces the best
focus at the sample position (picking the maximum of $\mathcal{I}_{0}(f_{2})$,
which also corresponds to the minimum FWHM). The
$\mathcal{I}_{0}(f_{1},f_{2})$ maps depend strongly on the slit aperture. In
the horizontal direction, the low $CF_{h}$ of the source (13%) is strongly
increased when the slit is being closed, producing an appreciable displacement
of the pattern in Fig. 2a, thus requiring a fine tuning of $f_{2}$ to keep the
beam focused at the sample plane. In the vertical direction, the source is
more coherent ($CF_{v}=58\%$), and closing the slit does not shift the waist
position until reaching $CF_{v}\approx 90\%$. However, when the slit is almost
closed ($a_{v}=$25\text{\,}\mathrm{\SIUnitSymbolMicro m}$$) acting as a
pinhole the patterns approach the ($f_{1},f_{2}$) trajectories resulting from
the geometrical optics considering the source at the slit position.

Figure 3 shows the calculated focal sizes. Analytical values (dashed lines),
based on geometrical optics, i.e. eq. (2), are close to numeric values of beam
sizes for the limiting cases of open slit (source at the undulator) or almost-
closed slit (source at the slit). For other useful cases where the slit crops
partially the beam, the numerically calculated values should be used. Here,
the focal sizes cannot be predicted by geometrical optics (also seen in Fig.
1a), and cannot be simply interpolated from the sizes corresponding to the
limiting slit openings. The size values calculated analytically have very
limited applicability for practical cases where the beam is partially cropped
by the slit.

In the horizontal direction, for slit apertures
$a_{h}>$40\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ the sizes change
considerably for small changes in $a_{h}$. The focusing characteristics of the
system highly depend on the diffraction effects produced at the slit. For the
slit $a_{h}=$40.3\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ ($CF_{h}=90\%$) the
focal size can vary from roughly 10 to 50 microns. In the vertical direction
the focal size can be changed from 5 to 100
$\text{\,}\mathrm{\SIUnitSymbolMicro m}$ at $CF_{v}=90\%$.

In summary, it is shown that a slit that crops a partially coherent X-ray beam
induces a diffraction effect in the beam originating an appreciable change in
the focal characteristics (position and dimensions). In the case that two (or
more) focusing elements are used, the basic concepts from geometric optics are
insufficient to predict the conditions to pair them (define the focal lengths)
for creating a focus in a precise position. Numeric methods for partially
coherent optics based on coherent mode decomposition and wavefront propagation
(as described in [4]) are used to predict the focal lengths of the focusing
elements and the resulting image size. The data that support the finding of
this study are openly available333https://github.com/srio/paper-transfocators-
resources.

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We now address the second issue, namely that the quasi-equilibrium state
identified above is in a regime where isochoric modes are stable while
isobaric modes are unstable. The turbulent heating rate we described above has
no explicit dependence on gas density or temperature, but only on position.
Photoheating by the UV background has a negligible effect on gas with $T\lower
2.15277pt\hbox{$\;\buildrel<\over{\sim}\;$}10^{5}\,{\rm K}$. Photoheating by
AGN can affect gas at these temperatures (Vogelsberger et al., 2013), however
the IPM is far away from any massive galaxy or accreting black hole, rendering
AGN heating negligible. We therefore assume a total heating rate which is
independent of density or temperature, which was the case studied by Das,
Choudhury & Sharma (2021). In this case, the stability of isobaric and
isochoric modes depends on the logarithmic derivative of the cooling function
versus temperature, $\Lambda_{\rm T}=d{\rm log}~{}\Lambda/d{\rm log}~{}T$. If
$\Lambda_{\rm T}<0$ then both isobaric and isochoric modes are unstable, if
$0<\Lambda_{\rm T}<2$ then isobaric modes are unstable while isochoric modes
are stable, and if $\Lambda_{\rm T}>2$ then both modes are stable (Das,
Choudhury & Sharma, 2021). In LABEL:dlnL, we show $\Lambda_{\rm T}$ as a
function of temperature for the cooling curve used in our simulations. We
assume a metallicity of $[Z]=-3.0$, though the results are effectively
identical for metallicity as high as $[Z]\lower
2.15277pt\hbox{$\;\buildrel>\over{\sim}\;$}-2.0$ and for primordial gas. While
photoheating by the UVB is expected to be negligible at $T\sim 10^{5}\,{\rm
K}$, photoionization is still important and can alter the cooling curve. We
show results assuming photoionization due to the UVB at $z=3.93$, both with
and without accounting for photoheating as well in blue and red respectively.
We here assumed a density of $n=10^{-3.3}\,{\rm cm}^{-3}$, the median density
in the region of $nT$ space highlighted in LABEL:phase_low. While
photoionization alters the cooling curve at all $T<2\times 10^{5}\,{\rm K}$,
photoheating has hardly any effect at $T\lower
2.15277pt\hbox{$\;\buildrel>\over{\sim}\;$}6\times 10^{4}\,{\rm K}$ as
expected. Regardless, in both cases gas with $T\sim(10^{4.6}-10^{5})\,{\rm K}$
has $0<\Lambda_{\rm T}<2$ so isochoric cooling modes are stable. These results
are unchanged by assuming the UVB at $z\sim 3$ or $z\sim 5$ instead of $z\sim
4$.

We note that the above analysis suggests that linear isochoric perturbations
around the quasi-equilibrium thermal state should be stable. However,
perturbations with initially non-linear amplitudes can cool isochorically even
if linear isochoric modes are stable (Das, Choudhury & Sharma, 2021).
Supersonic turbulence tends to generate a lognormal density PDF (e.g. Vazquez-
Semadeni, 1994; Padoan, Nordlund & Jones, 1997; Scalo et al., 1998; Federrath,
Klessen & Schmidt, 2008; Price, Federrath & Brunt, 2011; Hopkins, Quataert &
Murray, 2012), as indeed seen in the IPM (LABEL:dens_low). If we assume that
the density at the peak of the lognormal distribution, ${\bar{\rho}}$,
corresponds to the equilibrium density, then clouds with
$\delta\rho/{\bar{\rho}}<1$ will be unable to cool isochorically, while those
with $\delta\rho/{\bar{\rho}}>1$ will be unaffected by the arguments presented
above. For isothermal turbulence, the width of the lognormal distribution is
given by

$\sigma_{\rm ln(\rho)}\simeq\left[{\rm ln}\left(1+b_{\rm
turb}^{2}\mathcal{M}_{\rm turb}^{2}\right)\right]^{1/2},$ (A7)

where $\mathcal{M}_{\rm turb}$ is the turbulent Mach number and $b_{\rm turb}$
depends on the ratio of compressive to solenoidal forcing driving the
turbulence. While our system is not isothermal, and eq. (A7) may not be
precisely valid in a gravitationally stratified medium in any case (Mohapatra,
Federrath & Sharma, 2021), we use this as a proxy for the expected width of
the PDF. To obtain an upper limit, we assume purely compressive forcing with
$b=1$ (Federrath, Klessen & Schmidt, 2008). For a turbulent Mach number of
$\mathcal{M}_{\rm turb}\sim(1-2)$ (LABEL:profiles) we obtain $\sigma_{\rm
ln(\rho)}\sim(0.8-1.25)$. The condition $\delta\rho/{\bar{\rho}}<1$
corresponds to ${\rm ln}(\rho/{\bar{\rho}})<0.7$, so $\sim(40-65)\%$ of the
gas mass should have $\delta\rho/{\bar{\rho}}<1$ and be affected by the
isochoric stability criterion discussed above.

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