Athanasios Laliotis

Isabelle Maurin

Michèle Fichet

Daniel Bloch
email: bloch@lpl.univ-paris13.fr

Martial Ducloy

Nikolai Balasanyan

Apetnak Sarkisyan

David Sarkisyan

SELECTIVE REFLECTION SPECTROSCOPY AT THE INTERFACE BETWEEN A CALCIUM FLUORIDE WINDOW AND Cs VAPOUR

A special vapour cell has been built, that allows the measurement of the atom-surface van der Waals interaction exerted by a CaF 2 window at the interface with Cs vapour.

Mechanical and thermal fragility of fluoride windows make common designs of vapour cells unpractical, so that we have developed an all-sapphire sealed cell with an internal CaF 2 window. Although impurities were accidentally introduced when filling-up the prototype cell, leading to a line-broadening and shift, the selective reflection spectrum on the Cs D 1 line (894 nm) makes apparent the weak van der Waals surface interaction. The uncertainties introduced by the effects of these impurities in the van der Waals measurement are nearly eliminated when comparing the selective reflection signal at the CaF 2 interface of interest, and at a sapphire window of the same cell. The ratio of the interaction respectively exerted by a sapphire interface and a CaF 2 interface is found to be 0.55 ± 0.25, in good agreement with the theoretical evaluation of ~0.67.

Introduction

The long-range van der Waals (vW) atom-surface attraction [START_REF] Bloch | Atom-Wall Interaction[END_REF] tackles various fundamental problems, and imposes various constraints in the development of nanotechnologies, like the deposition of atoms in nano-lithography. With regards to its universality and to the possibilities to achieve exotic situations [2][START_REF] Dutier | Laser Spectroscopy. Proceedings of the XVI International Conference[END_REF][START_REF] Gorza | [END_REF], the experimental investigations have remained relatively limited, and in particular, the list of materials submitted to experimental investigation has remained short. Most of the experiments have employed an optical detection, hence requiring transparency of the surface (occasionally, a thin metal film provides a sufficient transparency [START_REF] Bloch | Atom-Wall Interaction[END_REF]5]). To investigate these vW effects, we have developed for a long time Selective Reflection (SR) spectroscopy at a vapour interface [START_REF] Bloch | Atom-Wall Interaction[END_REF]2,[6][7][8][9], a technique that typically probes the vapour at the interface on a depth λ/2π, with λ the optical irradiation wavelength, and more recently extended it to spectroscopy in a vapour nanocell [10]. Materials were limited to glass -fused silica-, sapphire and YAG (for nanocells, to YAG and sapphire only). Alternate measurements were performed with cold atoms, at a glass interface or at an interface of a glass piece coated with a thin layer (see e.g. [10] for references). Extending the number of materials for the measurement of the vW interaction is of interest, especially when exotic effects such as vW repulsion [2], occur for specific interfaces [2][START_REF] Dutier | Laser Spectroscopy. Proceedings of the XVI International Conference[END_REF][START_REF] Gorza | [END_REF]11].

A particular interest of fluoride crystals, is that their surface resonances are located in the thermal IR range. This makes them good candidates to investigate how the thermal emissivity of the surface influences the vW interaction exerted onto a resonantly absorbing atom [START_REF] Dutier | Laser Spectroscopy. Proceedings of the XVI International Conference[END_REF][START_REF] Gorza | [END_REF]11]. To implement the corresponding SR experiments, that require both a relatively dense (hot) vapour of alkali (Cs in our case), and hot windows of a controllable temperature, a specially designed vapour cell must be built-up. Indeed, fluoride materials are mechanically fragile and exhibit a large thermal expansion coefficient, making unpractical most of the standard designs for glass or sapphire cells. In the present work, a prototype Cs vapour cell with a CaF 2 window was constructed (section 2). A SR experiment on the D 1 resonance line (894 nm) was implemented (section 3), in order to test the cell via the evaluation (section 4) of the vW atom-surface interaction. Impurities having appeared when filling-up the cell make the measurement more complex, but we show that the vW interaction exerted by the CaF 2 window can be reliably compared to the one exerted by a sapphire interface when recording and fitting simultaneously the two spectra.

Design of a cell with a CaF 2 window

The vapour cell must obey the following requirements: it must resist to a high temperature (to warrant a sufficient density of Cs and for future experiments involving thermal emissivity), it must be evacuated to avoid unwanted collision broadening and shift, and it has to be compatible with a CaF 2 window. We discuss below the available technologies before describing our cell.

The current technology with glass-blowing allows the convenient build-up of temperatureresistant (~ 200 °C) sealed glass cells, but it cannot be extended to the sealing of special crystalline windows. Sealed cells with a metallic housing and various windows such as sapphire or even fluoride windows, can be prepared with commercial systems of windows soldered to a metallic mount; however, it is advised for the metal/window region not to exceed a maximal temperature of ~200°C. Alternately, a metallic housing cell, with windows mechanically hold by metal (copper) rings, is conceivable for a system under permanent evacuation but this makes difficult to monitor the vapour pressure. Such a design, operative with materials of a sufficient mechanical resistance, is at a high risk with fragile fluoride windows.

All-sapphire cells [12] can be realized with a special mineral glue connecting a sapphire tube and sapphire windows, the cell being sealed after evacuation and introduction of the alkali-metal. Such cells resist to very high temperature (up to ~1000 °C with an appropriate cell design). If sapphire windows can easily be replaced by nonbirefringent YAG windows, such a replacement is not permitted for fluoride materials as due to their mechanical fragility and to very different thermal expansion coefficients. In a similar manner, the mineral gluing technology cannot be extended to an all-fluoride cell. This led us to design our special cell on the basis of an all-sapphire sealed cell, with a fluoride window inserted inside the tubing, mechanically maintained close to one of the window. An issue for the observation of a SR signal at the CaF 2 interface is to make negligible the absorption in this interstitial region.

The cell (fig. 1) has a T-shaped geometry, in order to allow for an independent control of the vapour density (through the temperature of the alkali-metal reservoir), and of the temperature of the window (controlled through the temperature of the cell body). It has two opposite windows, one of which is in YAG, and the other one in sapphire (⊥ c-axis, to avoid birefringence). The CaF 2 window consists of a long (~ 7 cm in our prototype cell) cylindrical piece. This allows the CaF 2 tube to accommodate a temperature gradient: the YAG window is maintained at room temperature outside the oven, so that the corresponding extremity of the CaF 2 tube remains cold, while the other extremity of the CaF 2 tube reaches an adjustable temperature (up to ~350 °C) (fig. 1). This makes the free-space between the CaF 2 window and the YAG window non critical: indeed, at room temperature, the very low alkali-metal vapour density yields negligible effects on the transmission (conversely, in the absence of a temperature gradient, the SR signal at the CaF 2 interface is contaminated by the transmission signal of the interstitial volume for separation distances as short as 100 nm). To reduce still this vapour density, and to avoid Cs condensation in the interstitial region, a cooled (~ 5°C) metal ring is usually set around the sapphire tubing.

As shown in fig. 1, the oven has a complex structure and is made of independent sections, each of them contains heating wires (structured in doubled coils to minimize the induced magnetic field). These wires are surrounded by thermally isolating bricks of alumina powder.

Note that, as usual in the construction of a T-shaped all-sapphire cell, a convenient filling-up of the cell reservoir imposes the bottom part of the reservoir to be in glass (and hence at T < 200°C) but the Cs density is governed by the temperature of the upper free-surface, i.e. the temperature of the oven T res . Also, the T-shape of the cell divides the heating wire in the top part of the oven (T windows ) in two independent subsections. This is the reason for a temperature gradient inside the top part of the oven (the temperature of the top part is monitored through two thermal probes located close to the two respective windows), that can be suppressed if sending independent currents in the heating wires, or if specifically adjusting the lengths of the wire. In view of experiments where the surface temperature could be varied while keeping constant the Cs density (in principle governed by the temperature in the Cs reservoir), the independence of the top and bottom sections of the oven was also tested. An adjustable temperature difference 30-120°C is currently obtained for a bottom part at 150-200°C (we operate at a minimal ~30°C nominal difference because of the uncertainty in the temperature measurement). With an improved isolation between the oven sections, it should be not difficult to increase this relative independence to a larger scale, and the upper limit in temperature is essentially a matter of the outgassing conditions.

Although we could not find any data on the chemical resistance of fluoride windows to hot alkali vapour, and in spite of the possibility of a migration of fluor atoms from heated fluoride materials, no special problems have been detected when repeatedly heating the cell up to ~350 °C. Unfortunately, in spite of a standard procedure for the cell outgassing (350°C, Laliotis et al.

Selective Reflection spectroscopy at the interface between a calcium fluoride window... 10 hours), an incident has apparently occurred when filling-up the cell with Cs, that brought impurities in the only cell prototype available for a long duration. These impurities are responsible for a major (collision) broadening of the Saturated Absorption (SA) signal observed on the D 1 line of Cs (894 nm), turning it into a Doppler-broadened signal instead of a Doppler-free one. The expected sub-Doppler signal, that is nevertheless tiny and appearing on a broad background (figure 2), is recovered only with a fast modulation [13], that minors the influence of velocity-changing collision processes. As shown below, these impurities have a less dramatic effects in SR spectroscopy.

Experimental set-up and main spectroscopic features

The smaller sensitivity of SR spectroscopy to impurities, as compared to the SA technique, originates in its linear nature, and its fast response time (solely governed by the optical transition, and not by the build-up of an optical population). However, one still clearly observes a residual broadening in our SR experiments.

Several SR spectroscopy experiments were already performed on the Cs resonance lines, allowing to measure the vW interaction. The present one, schematized in figure 3, does not differ essentially from these previous experiments. Rather, a double SR set-up has been implemented in order to monitor simultaneously the SR spectrum at the CaF 2 interface, and at the opposite sapphire window. The laser is here a tuneable narrow-linewidth DBR laser at 894 nm, and its frequency scan linearity is monitored by analyzing the transmission through a low-finesse very stable Fabry-Perot (FP). The laser is also frequency modulated (FM), in order to obtain purely Doppler-free SR spectra [14] through demodulation of the reflected beam. A frequency marker for the atomic resonances in volume is usually required for the analysis of the vW interaction in the SR spectrum: this is why a simultaneous SA experiment is implemented in an auxiliary cell, with an amplitude modulation (AM) applied to the pump beam. Note that to compare the atomic resonances in volume, and close to the surface, the pressure conditions should be the same as in the SR cell. This is why the tiny and narrow SA signal available in the special cell when the pump undergoes a fast modulation, was compared to the SA signal of the auxiliary experiment. The SA signal in the special cell appears red shifted by ~ -5.2 ± 0.5 MHz for all h.f.s. components (experiments performed at room temperature so that the pressure shift induced by Cs-Cs collisions is entirely negligible).

To evaluate the possible competitive presence of a spectroscopic signal originating in the thin interstitial region between the CaF 2 window and the YAG window, we have attempted to measure the (FM) signal reflected by the CaF 2 -YAG interface. The SR signal, typically integrating the atomic response over λ/2π, is hence negligible relatively to the Doppler-broadened linear absorption signal resulting from the back and forth propagation through the interstitial region. Comparing this signal with the absorption through the whole cell kept at room temperature (vapour column ~8 cm), we extrapolate that the interstitial spacing is on the order of 50 µm (the back and forth absorption in this interstitial region is below 10 -3 , and is observable only through a large amplitude FM). This ensures that when recording the SR spectrum with a hot vapour, the signal of the transmission through the interstitial region remains largely negligible: the contribution of the interstitial region is still reduced by the cooling close to the YAG window, and by the FM technique that specifically enhances the narrow signals.

We have recorded the SR spectra on the Cs D 1 line 6S 1/2 (F=3,4)→6P 1/2 (F'=3,4) across the four (entirely resolved) h.f.s. components. The signals from the sapphire and from the Selective Reflection spectroscopy at the interface between a calcium fluoride window... negligible, the minimal SR width is found to be on the order of 27 MHz, largely exceeding the natural width of the transition. The SR spectra exhibit a notable asymmetry, along with a red frequency shift, typical of the vW interaction (see fig. 4). The apparent shift, relatively to the SA frequency marker, varies slightly with the Cs pressure conditions and with the considered h.f.s. components. It is actually much larger than the red shift currently found for the vW interaction, on the order of 2 -3 MHz, and hence originates in the residual impurities in the cell, and possibly in a Cs-pressure shift for the higher Cs temperatures. As shown in the next section, it remains possible to extract an estimate of the vW interaction, although the vW effects are largely hindered by volume collisions effects.

Analysis of the vW interaction

The vW interaction simultaneously shifts and distorts (asymmetry) the optical transition.

The vW interaction on the D 1 resonance line is known to perturb only weakly the (FM) SR spectrum, and is responsible for an approximate spectral shift not exceeding a few MHz, smaller than the minimal width of the transition (~ 5 MHz in the absence of any collisions). It had however been demonstrated that, even when collisions (buffer gas, or self-broadening) impose the dominant broadening, the vW interaction, although of a weaker influence, had to be approximately taken into account to allow for a correct evaluation of the influence of collisions [8]. Here, the novelty is that we aim at evaluating as precisely as possible the vW interaction exerted by the CaF 2 window, or at least to compare it with the one exerted onto the sapphire window, in spite of the large impurities broadening.

We recall here the general method for extracting the vW interaction from a (FM) SR spectrum. When the Doppler width largely exceeds the optical width, and in the linear regime, (FM) SR experimental lineshapes are amenable to dimensionless lineshapes [START_REF] Bloch | Atom-Wall Interaction[END_REF][6][7][8][9], dependent on a single dimensionless parameter A characterizing the vW interaction (A = 2C 3 k 3 /γ, with k=2π/λ, γ the optical width of the relevant transition [6], and C 3 the coefficient of the spatial dependence of the transition frequency, evolving according to ω(z) = ω ∞ -C 3 z -3 , with z the atom-surface distance). In numerous realistic cases, the finite ratio between the Doppler width (ku) and the optical width (γ) imposes a slight correction to the dimensionless theoretical lineshapes [START_REF] Bloch | Atom-Wall Interaction[END_REF], but no additional parameter is introduced as the Doppler width is intrinsically fixed by the experimental conditions. One hence attempts to fit an experimental lineshape with theoretical dimensionless lineshapes for different A values.

The adjustable parameters are the amplitude, width, location of the resonance (when not imposed by the free-space reference), and possibly a vertical offset. C 3 is hence determined from the limited range of A values allowing an acceptable fitting. The consistency of the fittings is usually checked by establishing that a single C 3 value is found in spite of phenomenological changes imposed to the lineshapes when varying the pressure (see e.g. [START_REF] Bloch | Atom-Wall Interaction[END_REF]2,[7][8][9]).

In the "weak vW regime" (A << 1) of our experiments, the optimal fitting width γ(A) is nearly independent of the A value, but there is a strong interplay between the location of the resonance (i.e. frequency shift as possibly resulting from collisions) and the A value (fig. 4).

This makes it very difficult to extract unambiguously the VW interaction from a single SR spectrum, notably when the spurious shift originating in the residual impurities is unknown. This is why we have attempted to benefit from the simultaneous recording of the SR signal at the two different interfaces, assuming that the Cs vapour (density, and collisions) is homogeneous in the top part of the cell. Hence, only a single parameter for the width, and another one for the shift, are to be adjusted when simultaneously fitting the experiments with sapphire and with CaF 2 . The tighter constraint imposed in the fitting method bring a serious help to the consistency of the fittings. In addition, the coupled fitting of two independent signals recorded simultaneously with the same laser eliminates most of the effects related to a possible tiny laser frequency drift.

In our situation, the vW coefficients for CaF 2 and sapphire are expected to differ notably, owing to an approximate (ε-1)/(ε+1) dependence [START_REF] Bloch | Atom-Wall Interaction[END_REF], yielding a theoretical ratio ~ 0.67 (as predicted from the respective refractive indices of CaF 2 (n=1.430) and sapphire (n=1.758), and ε = n²). Hence, in analyzing the data, the objective is to find the optimal range of coefficients (A 1 ,A 2 ) characterizing the vW coupling. Figure 5 shows that high quality fittings of the experimental curves can be obtained in spite of the restricted number of adjustable parameters. Significantly, for each experimental curve to be fitted, the domain of acceptable (A 1 ,A 2 ) values is strongly limited, and the cloud of fitting (A 1 ,A 2 ) parameters is notably characterized by the ratio A 1 /A 2 between the interaction exerted by the CaF 2 window and by the sapphire window. In addition, the assumption that the Cs vapour density is the same at the two interfaces should impose the relative amplitudes of the two SR signals. This is indeed verified, within the limits of the experimental uncertainty, mostly affecting the non resonant reflection, which usually departs from the predicted value (i.e. Fresnel formula) owing to scattering losses at the interface. As illustrated in figure 6 to plot a Cs pressure self-broadening and shift (figure 7). These variations of the pressure effects, partly dependent upon the investigated hyperfine component, are compatible with previous estimates of self-broadening and shift [15,16], although the accuracy, affected as usual by the pressure scale uncertainty (~ 40 % originating in a ~5°C temperature uncertainty), suffers here from the offset introduced by the impurities broadening. Also, the residual shift and broadening attributed to the impurities are independent of the cell temperature, showing that the impurities that have appeared when filling-up the cell are of a nearly constant density, and not stored in a dense phase with a temperature-controlled gas equilibrium; moreover, the residual shift corroborates the SA experiment (section 3). As a further evidence of the consistency of our fittings and extrapolation from a series of spectra, fig. 8 shows the ability to fit a series of spectra with a single C 3 value (for respectively sapphire, and CaF 2 ), and the optical width and shift values deduced from fig. 7.

Conclusion

In summary, we have designed an alkali vapour cell with a fluoride window susceptible to be heated up to a high temperature. The principle of construction, allowing the contact of high temperature vapour with the window of interest, is actually applicable to any interface made of a crystal (or material) resistant to hot alkali vapour. In spite of spurious impurities that has degraded the vacuum conditions in the prototype cell, we show that the atom-surface vW interaction can be measured at the interface with the special window, hence validating the design of the construction. Such a cell is an important step to study the specific influence of the thermal emittance of fluoride windows on suitable atomic transitions, such as the Cs 6S 1/2 →8P 3/2 line [9]. In addition, it appears that the coupled analysis of the vW interaction exerted on two different windows, as extracted from a simultaneous recording, is a particularly robust method to estimate reliably the strength of the C 3 coefficient of the vW interaction, notably when stray effects can hinder the specific vW contribution. The zoom on the left wing shows that it may be not easy to decide if A =0.02 is better than A = 0.006 Selective Reflection spectroscopy at the interface between a calcium fluoride window...

Figure captions

CaF 2

 2 windows were recorded simultaneously, and for various temperatures of the Cs reservoir. The irradiation intensity is kept low ( ≤ 0.1 mW/cm² ) to ensure the linearity of the SR technique. The temperature of the top part of the cell has no significant influence over the SR signal amplitude. At low temperatures (~ 100°C), when the Cs self-broadening is Laliotis et al.

  constant (~ 0.55 ± 0.25) when varying the temperature conditions, in very good agreement with the theoretical value (~0.67). The accuracy for the individual values for C 3 is lower than for the relative interaction exerted by sapphire and CaF 2 respectively; the obtained values C 3 sapphire =1.4 kHz.µm 3 , C 3 CaF2 = 1.0 kHz.µm3 are in an acceptable agreement with the predicted ones (respectively 1.0 kHz.µm 3 , and 0.67 kHz.µm 3 , following a 2.0 kHz.µm 3 value predicted for the D 1 line at an ideal interface, 2 ) = 2.42 kHz.µm 3[7]). At last, an optical width and collision shift are associated to each fitting of an experimental recording, allowing

Figure 1 :

 1 Figure 1 : Scheme of the Cs vapour cell. In the top oven, the heating source is located in the left and right parts surrounding the connection to the Cs reservoir.

Figure 2 :

 2 Figure 2 : Saturated Absorption (SA) spectrum recorded on the prototype cell with impurities (a) AM (500Hz) applied on the pump the signal, (b) In FM, the sub-Doppler signal emerges more clearly from the broad background for a fast FM.

Figure 3 :

 3 Figure 3 : Scheme of the experimental set-up.

Figure 4 :

 4 Figure 4 : An experimental SR spectrum (T Cs = 140 °C) recorded on the CaF 2 window. For A = 0.006 , A= 0.02, A = 0.04, the optimal fittings lead to the respective shifts -7.1 MHz, -5.1 MHz, and -3.3 MHz, and widths 32.0 MHz, 32.6 MHz, and 32.9 MHz. The vertical dashed line is the SA reference. The dotted line is a fit neglecting the vW interaction (A=0).

Figure 5 :Figure 6 :

 56 Figure 5 : Optimal fitting (black) of two simultaneously recorded spectra (grey), under the constraint of a single parameter, for the shift (-4.5 MHz), and respectively for the width (27.8 MHz). one has T Cs = 120 °C. The vertical dashed line is the SA reference. One has A(CaF 2 )= 0.024 (C 3 =0.96 kHz.µm 3 ), A(sapphire) = 0.042 (C 3 =1.68 kHz.µm 3 ). The ratio of fitting amplitudes is 0.57 in the fitting, 0.516 in theory

Figure 7 :

 7 Figure 7 : Widths (a) and shifts (b) as a function of Cs pressure for the various hyperfine transitions, s extrapolated from the fitting of SR spectra.

Figure 8 :

 8 Figure 8 : A series of experimental (grey) spectra fitted (black) with a unique C 3 value (1.4 kHz/µm 3 for sapphire, 1 kHz/µm 3 for CaF 2 ) and only adjustable amplitudes ; shifts and widths are also imposed as a result of the linear extrapolation of fig.7.

  Figure 1

  Figure 5

Figure 6 20

 6 Figure 6

  Figure 7

Acknowledgments

This work was partially supported by FASTNet (European contract HPRN-CT-2002-00304) and by the INTAS South-Caucasus Project (grant: 06-1000017-9001).

Selective Reflection spectroscopy at interface between a calcium fluoride window...


Jean-Guillaume Dumas
email: jean-guillaume.dumas@imag.frwww-lmc.imag.fr

Aude Rondepierre
email: aude.rondepierre@imag.frwww-lmc.imag.fr

Modeling the electrical activity of a neuron by a continuous and piecewise affine hybrid system

A hybrid system is proposed to model the electrical potential emitted by a neuron as a response to an externally applied DC current. Experimentally, Hodgkin and Huxley built a four-dimensional and nonlinear dynamical system to simulate this activity. Our idea is to use a new continuous and piecewise affine approximation as a hybrid model of the Hodgkin-Huxley dynamic. Our new model reproduces the Hodgkin-Huxley features with good accuracy (e.g. including the fact that the incoming current intensity is a bifurcation parameter), and, moreover, still allows an analytic computation of its solutions.

Introduction

Neurons communicate with each other by generating electrical signals, called action potentials. Action potentials are the result of currents that pass through ion channels in the cell membrane. Each neuron integrates the incoming signals and, when the stimulation reaches a certain threshold, an output signal is generated and delivered. This paper deals with the dynamics of the membrane potential of a single neuron under a certain stimulus.

As a result of their experiments on a giant squid nerve fiber, Hodgkin and Huxley proposed a first model describing the dynamics of the action potentials [START_REF] Hodgkin | A quantitative description of membrane current and its application to conduction and excitation in nerve[END_REF]. The main idea is that because of active ion transport (mainly sodium, potassium and chloride) through the cell membrane, this membrane acts as a capacitor. From then their measures tended to show that potassium ions can cross the membrane when four similar molecules occupy a certain region of this membrane (so that potassium conductance is considered proportional to the number of sites on the inside of the membrane which are occupied by the four molecules, g K ≈ ḡK n 4 ). Similarly, the sodium conductance is supposed to be proportional to the number of sites on the inside of the membrane which are occupied simultaneously by three activating molecules and not blocked by an inactivating molecule (g N a ≈ ḡNa m 3 h). The chloride and other ions conductance is supposed constant. From this assumptions, Hodgkin and Huxley then proposed the following dynamical system as a simplified neuron model:

(H-H)                    C M dV dt = I(t) -ḡK n 4 (V -V K ) -ḡNa m 3 h(V -V N a ) -ḡl (V -V l ) dn dt = (1 -n).α n (V ) -n.β n (V ) dm dt = (1 -m).α m (V ) -m.β m (V ) dh dt = (1 -h).α h (V ) -h.β h (V )
• V is the displacement of the membrane potential from its resting value • n is the proportion of the K + activating molecules inside the membrane at time t ; 0 ≤ n ≤ 1 • m is the proportion of the N a + activating molecules inside the membrane at time t ; 0 ≤ m ≤ 1 • 1 -h is the proportion of the N a + inactivating molecules inside the membrane at time t ; 0 ≤ h ≤ 1 • I(t) is the intensity of the current applied to the neuron cell.

The α and β being respectively the transfer rate constants from outside to inside and from inside to outside, the ḡX and V X being respectively the conductances and resting potentials of potassium, sodium and chloride ions [START_REF] Frank | Modeling and Simulation in Medicine and Life Sciences[END_REF].

Despite the small dimension of the (H-H) system, the mathematical analysis is still quite complex. Therefore only numerical simulations are used and simplified models are required for a better understanding of the neuronal dynamics.

Some simplifications have been proposed [START_REF] Abbott | Model neurons: From hodgkin-huxley to hopfield[END_REF], models like integrate-and-fire, resonate-and-fire ( [START_REF] Izhikevich | Resonate-and-fire neurons[END_REF]) or even Hopfield or FitzHugh-Nagumo (see [15, §6.5] or [7, §3.1] for further references) provide different level of relevant informations on the neuronal dynamics. For instance, these models are used to show the existence of a spike (excitable state), or of a threshold potential for the emission of a periodical state (see e.g. [7, §3.3] for more details). Unfortunately, these simplifications give only a few fully analytical results or low quality numerical approximations. Therefore some piecewise linear simplifications have been proposed by Tonnelier [START_REF] Tonnelier | Dynamique non-linéaire et bifurcations en neurosciences mathématiques[END_REF] to combine analytic results and good numerical approximations. Still there is no complete analytic analysis of the dynamics and Tonnelier's approximations suffer from a non-continuous behavior.

Our approach is to use a hybrid system modeling and to refine Tonnelier's approximation. Indeed the main advantage of an hybrid system is that it combines fully symbolic and multi-scale resolutions. On the one hand, by using continuous and linear approximations, any refinement is possible: the original system is replaced by piecewise linear parts at as many points as needed to ensure any quality of approximation. On the other hand in each resulting "cell" (corresponding to a state of the hybrid system), the system is fully linear and therefore analytically solvable. These two aspects together enable an analytic analysis of most of any system properties, whatever its complexity is.

There are many possible linearization by parts. For instance, one can build a mesh of the phase space and use multi-dimensional interpolation to define a linear approximation of the system in each cell (simplex) of the mesh, see [START_REF] Della | Hybrid computation[END_REF][START_REF] Girard | Approximate solutions of ordinary differential equations using piecewise linear vector fields[END_REF][START_REF] Asarin | Reachability of non-linear systems using conservative approximations[END_REF] for more details. We present a few simultations using this approximation in section 5.3. In this paper, we will instead use another idea: we linearize each equation separately by implicit representation and one-dimensional linearization on each variable, see sections 2.2 and 2.3 (using the latter on the Hodgkin-Huxley model, we actually need fewer hybrid states to prove the desired analytic properties).

We therefore propose a new continuous and piecewise affine approximation of the Hodgkin-Huxley model. We implicitly build a hybrid automaton defined by the pieces of the approximation within the phase plane of the dynamical system (see figure 6). Now, in each state, the solutions of our model are analytically computable. Moreover, simulations of this automaton show the good quality of our approximations.

In section 2, we give a reduction from the four-dimensional Hodgkin-Huxley model to a two-dimensional continuous and piecewise affine model. We next linearize both equations separately by way of an implicit representation in sections 2.2 and 2.3 and propose the associated hybrid automaton in section 3. We then fully analyze the model dynamics in section 4 and produce some simulations and comparisons between the Hodgkin-Huxley model and our approximation in section 5.

Reduction of the Hodgkin-Huxley model

In this section we propose a reduction from the Hodgkin-Huxley model to our piecewise affine and continuous model. We combine both FitzHugh-Nagumo [7, §3.1.1] and Tonnelier's approaches [16, §1.4.3] and refine them. The first step is to reduce the dimension. We first rewrite the activating or inactivating molecule equations in the form X = -1 τ X (V ) X -X(V ) . Then for Fig. 2. Simulation of n + h variation with time for a high intensity a fixed potential V , the variable X approaches the value X(V ) (its asymptotic value) with a time constant τ X (V ). Experimental data shows that the variable m always reaches its asymptotic value m(V ) much faster than the variables n and h do [7, §2.2.1]. The first hypothesis is then that the m-gate of sodium are instantaneously activated, so that we can assume m ≈ m(V ). Then, FitzHugh noticed that whatever incoming current intensity, the n+h quantity remains close to 0.8 as shown on figures 1 and 2 from Jeff Moehlis 1 . We now have a system with only two dimensions : V and n. In the following, we will consider the functions and constants given e.g. in [START_REF] Fukai | Hopf bifurcations in multiple-parameter space of the hodgkin-huxley equations i. global organization of bistable periodic solutions[END_REF]:

From four dimensions to two dimensions

V N a = 115mV ; V K = -12mV ; V l = 10.599mV ḡNa = 120mΩ -1 cm -2 ; ḡK = 36mΩ -1 cm -2 ; ḡl = 0.3mΩ -1 cm -2 C M = 1µF cm -2 ; m(V ) = 25 -V 25 -V + 40 e -7 4 V + 5 2 -e -1 18 V α n (V ) = 10 -V 100 e 10-V 10 -1 ; β n (V ) = e -V 80 8

The first equation approximation

Let ϕ be the right hand side of the first equation, leaving out the intensity:

ϕ : (V, n) → -ḡ K n 4 (V -V K ) -ḡNa m(V ) 3 (0.8 -n)(V -V N a ) -ḡl (V -V l ) Figure 3 shows an example of the value of ϕ(V, n) as an implicit curve ϕ(V, n) = c for some constant c. The implicit curve representing ϕ(V, n) = c (of the form n = δ(V )
) has clearly a cubic shape. FitzHugh and Nagumo then proposed to approximate δ by a degree three polynomial p (scaled and shifted to maintain its roots between 0 and 1). They obtained the following equation with w corresponding to a scaling and shifting of n, v is a scaling of C M V and I is modified consequently:

dv dt = v(1 -v)(v -a) -w + I = p(v) -w + I, for 0 < a < 1. ( 1
)
Unfortunately this approximation is still too complex for a fully analytical approach. Consequently we decided to linearize ϕ(V, n) by pieces and care to preserve the continuity of the curve. We also used the previous scalings and shiftings and choose to preserve (v -,p(v -)) and (v + ,p(v + )), the respective coordinates of the local minimum and maximum of p (v

-= a+1- √ a 2 -a+1 3 , v + = a+1+ √ a 2 -a+1
3

), as well as the two extreme roots. As shown on figure 4, p is now approached by

p(v)=                p(v -) v -v if v < v - p(v + )-p(v -) v+-v- v + p(v + ) -p(v + )-p(v -) v+-v- v + if v -≤ v ≤ v + p(v+) 1-v+ (1 -v) if v > v + (2)
Finally the first equation of the system becomes: dv dt = p(v) -w + I.

The second equation approximation

Unlike FitzHugh-Nagumo, we propose to also approach the second equation the same way. Let ψ be the second right hand side, before scaling and shifting:

ψ : (V, n) → (1 -n)α n (v) -nβ n (v), for 0 ≤ n ≤ 1.
Figure 5 shows the implicit curve ψ(V, n) = c, here for c = 0. The graph V = χ(n) of this implicit curve has a logoïde shape. Therefore, we linearize again by pieces and use the scales and shifts of the first approximation. This introduces extra constant factors in this second equation, so that ψ(V, n) is replaced by bv -χ(w), with χ(w) the affine approximation chosen: linear for most of its range (indeed equation 1 shows that for no incoming current and with no displacement potential, the neuron must remain inactive, i.e. at v = 0 and I = 0, dv dt must be zero and so is w) and with a small slope outside two extreme thresholds w 0 and w 1 , as follows:

χ(w) =            λ 0 (w -w 0 ) + γw 0 if w ≤ w 0 γw if w 0 ≤ w ≤ w 1 λ 1 (w -w 1 ) + γw 1 if w ≥ w 1 (3)
Finally the second equation of the system becomes: dw dt = bv -χ(w).

Hybrid automaton

Now, we have a continuous and piecewise affine system modeling the neuron activity. To study the dynamics of this model (M), we consider it as a hybrid As there are 3 different pieces for each equation, this system is a hybrid automaton with nine contiguous states. Inside each state (rectangular cell), the trajectory evolves continuously according to its affine system (defined in sections 2.2, and 2.3). Figure 6 shows these states together with the affine system for a positive incoming current. Next section will study the different possible configurations, and therefore the transitions between the states, depending on the applied DC current I.

Analysis of the (M) model dynamics

We assume the current intensity I to be constant and positive (we are in the case where the neuron is under an external stimulation). We begin the analysis with the study of the equilibrium points of the system. This enables us to study I as a bifurcation parameter of the system. The equilibrium point of the system (M) is an intersection of the two curves induced by the differential equations: (w = p(v)+I) and (bv = χ(w)).

In concordance with experimental data, we assume that the linear part of χ is neither nearly horizontal, nor nearly vertical. Therefore, first, we assume that γ (resp. λ) is small enough (resp. big enough), which means that the middle part of p is is more gentle than the linear part of χ and steeper than the nearly horizontal parts of χ. Second, as the linear part of χ is not extremely steep, γ b w 1 > v + . Lastly, we consider that w 0 is negative so that (v = 0, w = 0), the resting point for I = 0, is reachable. This together gives us the following conditions:

b λ i < p(v + ) -p(v -) v + -v - < b γ (4) b γ v + < w 1 ( 5
)
w 0 < 0 (6)
By looking at the original system and the graphs of the functions, we see that the experimental data of e.g. [9, Table 3], [10, §3.2.3] or [5, §2] largely fulfill these conditions. Therefore we can consider that they are non-restrictive. Consequently, we actually only have to consider the four possible configurations shown on figure 7. Now, we will look for equilibrium points in each cell of the automaton. This together with the stability of the points, will enable us to describe the transitions between states and the phase portrait of the approximation.

Stability and bifurcation parameter

The Hodgkin-Huxley system is actually parameterized by a number of parameters and shows a variety of qualitatively different behaviors depending on these parameter values. [START_REF] Fukai | Hopf bifurcations in multiple-parameter space of the hodgkin-huxley equations i. global organization of bistable periodic solutions[END_REF][START_REF] Fukai | Hopf bifurcations in multiple-parameter space of the hodgkin-huxley equations ii. singularity theoretic approach and highly degenerate bifurcations[END_REF], for instance, have explored numerically the dynamics of the HH equations for a wide range of parameter values in the multiple-parameter space, that is, they examined the global structure of bifurcations, by way of numerical simulations. In particular, they considered the externally applied DC current as the basic bifurcation parameter [5, §3.3] and used numerical simulations with the AUTO software [START_REF] Doedel | Auto 2000: Continuation and bifurcation software for ordinary differential equations (with homcont)[END_REF] to detect the bifurcations.

Our idea in this section is to show that our linearized approximation models also this behavior. The method is to first search for equilibrium points corresponding to the system associated to the cell. For instance, if the point is stable and belongs to cell, many trajectories entering this cell will converge to this point and the automaton will remain in this state. If the point is unstable, there is a possibility of periodical behavior, etc. The trick is that the necessary conditions so that an equilibrium point is located in a certain cell, will actually induce some equivalent conditions on the incoming intensity:

First, for any intensity and under the conditions (4), ( 5), ( 6), there are no fixed point in states 1, 2, 3, 7 and 8. In state 4, the equilibrium point verifies the following: w * = p(v -) vv * + I ; bv * = γw * . Then a simple computation shows that belonging to cell 4 (i.e. v * < v -and w 0 < w < w 1 )) is equivalent to I < I 1 and I 0 < I < I 1 , where

I 1 = b γ v --p(v -), I 0 = γ b b γ -p(v-) v-
w 0 and

I 1 = γ b b γ -p(v -)
v - w 1 . Now, as w 0 < 0, I 0 is negative. Also, using condition 5, we have I 1 < I 1 . Moreover, when computing the jacobian of the system at the fixed points one can show that this point is actually stable. So we have the nice condition on I:

-whenever 0 ≤ I < I 1 , there exist a stable fixed point inside cell 4 and we are in case (a).

In the same manner, we can show that there exists two other threshold intensities

I 2 = b γ v + -p(v + ) and I 3 = w 1 + p(v + ) 1-v + γ b w 1 -1 , so that:
-whenever I 1 < I < I 2 , there exists an unstable fixed point inside cell 5 and we are in case (b). -whenever I 2 < I < I 3 , there exists a stable fixed point inside cell 6 and we are the case (c).

And, the last possible fixed points are located in cells 8 and 9 with the condition:

-whenever I > I 3 there exists a stable fixed point inside cell 9 and we are in case (d).

In conclusion, we analytically proved that the neuronal stimulus I (equivalently the externally applied DC current) is a bifurcation parameter of our linearized system. In the following section we actually prove that when a stable fixed point exists in a cell, the automaton tends to switch to the state containing this fixed point. Therefore the neuron emits one or more spike and then stabilizes at a given potential. On the contrary, in case (b), we will see that the configuration leads to a periodic behavior.

Phase portrait

The phase space associated to our model is divided into nine cells defined by the automaton [START_REF] Fukai | Hopf bifurcations in multiple-parameter space of the hodgkin-huxley equations ii. singularity theoretic approach and highly degenerate bifurcations[END_REF]. In the previous section, we proved that in cases (a), (c) and (d) the linearized system has only one fixed point. This point is located in a certain cell, so that whenever a trajectory arrives in this cell, it is attracted to the fixed point. Then, the question is: do we have global convergence towards the fixed point ? The idea is to show that for every initial condition (v i , w i ), the associated trajectory tends to cross the cell containing the fixed point.

Well, as shown on figure 8, the two nullclines of our system divide the plan into four parts. Each one of those four parts has then a different combination of v and w derivative signs. On the smae figure, we show an example trajectory simulated with CONTENT (an environment for continuation and bifurcation analysis of dynamical systems [START_REF] Yu | Content: integrated environment for analysis of dynamical system[END_REF]). For instance, in the part defined by and w > 0), the trajectory evolves in concordance, i.e. v and w are increasing. Now, since the third affine part of the v-nullcline decreases in v, the trajectory must cross it at some point. When this happens, w keeps increasing while v now starts to decrease as the trajectory is in the (v < 0 and (w > 0) part. The key point is that a trajectory will always turn anti-clockwise in the phase space. In conclusion, this together with the analysis of section 4.2 proves the following:

(v > 0 v'>0 w'>0 v'>0 w'>0 w'<0 w'<0 v'<0 v'<0
-whenever 0 ≤ I < I 1 , all the trajectories collapse to the stable fixed point of cell 4. This correspond to a spike emission followed by a return back to the resting potential. -whenever I 1 < I < I 2 , all the trajectories keep turning around the unstable fixed point of cell 5. This correspond to oscillations of the potential. -whenever I 2 < I, all the trajectories expand to a stable fixed point inside cell 6, 8 or 9. This correspond to a stimulation by a very high incoming intensity with a spike emission and a stabilization at a high potential.

Quality of the Approximation

We now confront the theoretical results of the previous section with the numerical simulations of the Hodgkin-Huxley model. The simulations are done with CONTENT or Matlab. We first produce simulations of the original Hodgkin-Huxley model and compare them with simulations of our hybrid model. Then we present an alternative way of linearizing which gives an even better quality of simulations but no fully analytical analysis yet. The simulations of the Hodgkin-Huxley model show that there are two main characteristic for the neuronal behavior: a spike solution and a periodic solution shown respectively on figures 9 and 10. Here we used Matlab as the system is too complex for CONTENT (CONTENT had some troubles to draw the trajectory before finding the limit cycle and the trajectory even crossed itself !). Next section will produce the simulations of our linearized model for the two

Hodgkin-Huxley model

The continuous and piecewise affine model

The analysis of section 4 has shown that our model also produces the two characteristic behaviors: an excitable state and a periodic state. We here show the good numerical properties of these two states. Indeed, on figure 11, the left simulation is the convergence towards the stable fixed point and the right simulation shows the quality of our approximation of the spike solution. Analytically, within this excitable state (e.g. one of the configurations (a), (c) or (d) holds), the neuron emits actually a finite number of spikes, each one corresponding to a full loop around the fixed point. This does not really shows up in the Hodgkin-Huxley model as the other spikes are very small. On the contrary, in our model, the scales are much smaller and the other spikes are more visible. Moreover our When comparing with the simulations of the Hodgkin-Huxley model of the previous subsection, we clearly see the good quality of the approximation. Indeed, the continuity and the accuracy of our approximation ensures a good regularity of the new solution. Fig. 13. Matlab Simulations of the periodic state using linear interpolation with 30 and 200 points on the mesh build a mesh of the phase space (this can be done dynamically as the trajectories evolve). For the Hodgkin-Huxley model we can use either the 4-dimensional or the 2-dimensional version. For the sake of visualization, we produce here figures linearizing in dimension 2. The system of A. Girard [START_REF] Girard | Approximate solutions of ordinary differential equations using piecewise linear vector fields[END_REF][START_REF] Asarin | Reachability of non-linear systems using conservative approximations[END_REF] uses interpolation at each node of the mesh to define a linear approximation of the system in each cell (simplex). Figure 13, shows the periodic trajectory of figure 12 for different mesh sizes. One can check the convergence of the trajectory when the mesh is refining. Indeed, linearization can always be close enough to the original model so that the phase portrait of the linearization reproduces the full original behavior [START_REF] Girard | Approximate solutions of ordinary differential equations using piecewise linear vector fields[END_REF]. Also, we are able to automatically compute fixed points or stable and unstable invariants and we can semi-automatically compute e.g. the limit cycles (i.e. a point close enough to the cycle must be provided manually) or even the local attracting sets. Still, this method does not allow us to analyse any property of the system. The study of the Intensity as a bifurcation parameter, for instance, seems still pretty difficult with this approach.

Conclusion

The Hodgkin-Huxley model only considers two different ionic species (K+, Na+) and a leakage current. However, even for this four dimensional model, the mathematical analysis is complex and most of the results are numerical. We therefore need simplified models to understand the neuronal dynamics better. Our approach was first to reduce the dimension of the Hodgkin-Huxley model (using FitzHugh-Nagumo approach) and then to approximate each equation of the reduced system by a continuous and piecewise affine function. Indeed the continuity of our approximation ensures the regularity of the new solution and the piecewise affine approximation allows us to compute the solutions analytically. Both simulations and analytical study of our model bring to the fore the two characteristic features of the neuronal behavior:

-The spike solution → correspond to the generation of a finite number of action potentials. → is a transient state that characterize the property of neuronal excitability. -The periodic solution → correspond to the generation of a infinite number of action potentials.

→ is an asymptotic state that shows the presence of a limit cycle.

We also formally proved that the applied current intensity is a bifurcation parameter of the linearized system.

With this case-study, we showed how continuous and piecewise affine hybrid systems can be used to analytically compute phase portraits that approach complex dynamical systems with no loss of intrinsic properties. Next step would then be to provide an automatic or quasi-automatic symbolic computation of the phase portrait of piecewise affine systems. Indeed, we are able to automatically compute fixed points or stable and unstable invariants. Still the full automatic analysis is not yet possible as we can only compute e.g. the limit cycles with some manual inputs. Also, a combination of different kind of linearizations would allow multi-resolution and therefore an even higher quality of simulation together with the semi-automatical analysis.

Fig. 1 .

 1 Fig. 1. Simulation of n + h variation with time for a low intensity

Fig. 3 .

 3 Fig. 3. Implicit representation of the first right hand side, here c=-3

Fig. 4 .

 4 Fig. 4. The piecewise affine p and the degree 3 polynomial approximations of the first right hand side

Fig. 5 .

 5 Fig. 5. Implicit representation of the second right hand side

1 Fig. 6 .

 16 Fig. 6. Hybrid automaton of the continuous and piecewise affine model

Fig. 7 .

 7 Fig. 7. Some equilibrium points of the system (M) when I > 0

Fig. 8 .

 8 Fig. 8. Trajectory behavior using the sign of the derivatives (here for I > 0)

Fig. 9 .

 9 Fig. 9. Excitable state (low incoming intensity) in a simulation of Hodgkin-Huxley model: fixed point in the phase space corresponding to the Spike solution for the displacement current

Fig. 10 .

 10 Fig. 10. Periodic state (high incoming intensity) in a simulation of Hodgkin-Huxley model: limit cycle in the phase space corresponding to Oscillations of the displacement current

Fig. 11 .

 11 Fig. 11. Excitable state within the hybrid system

Fig. 12 .

 12 Fig. 12. Periodical state within the hybrid system

http://www.math.princeton.edu/˜jmoehlis/APC591/tutorials/tutorial2


M Fifield

E Lewitowicz

Y Plagnol

Zhan Schutz

Belozyorov

N A Orr

W Mittig

L K Fifield

M Lewitowicz

E Piagnol

Zhan Y Schutz

Wen Long

L Bianchi

A Gillibert

A V Belozyorov

S M Lukyanov

Yxxe Pcnionhkevich

A C C Villari

A Cunsolo

A Foti

G Audi

N A Oit

E Plagnol

Yu E Penionhkevich

NEW MASS MEASUREMENTS OF NEUTRON-RICH NUCLEI NEAR N=20*

New mass measurements of neutron-rich nuclei near N

= 20

The Z=l 1 and 12, N»20 nuclei have long been known to constitute a region of stable deformation corresponding to a collapse in the conventional N=20 shell closure [1], [2]. This behaviour was originally deduced from the much stronger than expected binding energies of the sodium nuclei, 31 Na and 32 Na, as measured via direct on-line mass spectrometry [3]. The overbinding corresponded to an increase in the two-neutron separation energies at the crossing of the neutron sd-shell closure, where a decrease would be expected. Subsequently similar anomalies were observed in the masses of 31 Mg and 32 Mg [4]. Additionally a much lower than expected excitation energy was determined for the first 2 + state in 32 Mg [5], [6], Since these original observations there has been a paucity of experimental information forthcoming concerning these and neighbouring nuclei -although transfer reaction studies of some less exotic nuclei have been performed ( 33 A1 [7] and 34 « 35 Si [8], [9]) which have aided in determining the extent of the effect to nuclei of higher Z. This has been due to the inherent difficulty in producing such nuclei very far from stability with reasonable yields. The advent, however, in recent years of the techniques of projectile and target fragmentation coupled with a new generation of spectrometers, notably SPEC [10] at GANIL and TOFI [11] at Los Alamos, have facilitated the continued investigation of these nuclei. The availability of the very neutron-rich beam 4 **Ca at GANIL has further enhanced the prospects for studying the very neutron-rich light nuclei [12] - [15].

In this paper we report on the results of one such study which undertook direct time-offlight mass determinations of light neutron-rich nuclei produced in the fragmentation of a 48 Ca beam. The objectives of this investigation were twofold -the remeasurement with improved precision of the masses of the neutron-rich isotopes of Ne to Al and the extension of the measurements to nuclei further from stability. The techniques employed in this experiment were similar to those used in earlier studies [16], [START_REF] Mittig | Nuclei far from Stability[END_REF], [START_REF] Gillibert | [END_REF] and the details are only briefly described here.

The principle of the technique requires only the determination of die magnetic rigidity and velocity of an ion. That is, for a particle of mass m and charge q traversing an achromatic system, the magnetic rigidity Bp is related to the velocity as, Bp = mv/q

(1) Thus, a precise measurement of the rigidity and velocity will allow the mass of the ion to be deduced. The determination of the magnetic rigidity was provided in the present arrangement by use of the SPEC spectrometer [10]. A precise measurement of the velocity of each ion was achieved by measuring the flight-time over an 82m path length.

In the present study a beam of 55 MeV/nucleon 48 Ca ions delivered by the GANIL cyclotrons was used to bombard a 330 mg/cm 2 thick Tantalum target. Beam intensities on target were typically 100 nA. The projectile like fragments emitted from the target were analysed by an alpha shaped spectrometer. A time-of-flig.it start signal was provided by a microchannel plate detector located at the exit of the o,-specttometer. The fragments were then transported along an achromatically tuned beam line to the focal plane of the spectrometer SPEG. The time-of-flight stop signal was derived from a two dimensionally position sensitive microchannel plate detector positioned immediately before the focal plane. The intrinsic resolution of the start and stop detectors was 150 ps, while the overall resolution of the measured time-of-flight was 250 ps for 48 Ca ions.

The identification of the ions arriving at the focal plane of SPEG was derived from the measured flight-time (T) together with energy loss and total energy signals from a detector telescope. The telescope consisted of three cooled silicon detectors of thicknesses 50, 300 and 60CO urn. The energy loss signal (AE) was obtained by summiug the signals from the first two elements of the telescope, while the total energy signal (E) was obtained by summing the signals from all three detectors. Standard particle identification (A,Z) was thus possible via the application of the following relations in the off-line analysis,

A/qccT (2)
Z a AE^/T

A a ET2 (4)

All the fragments observed at the present beam energy were fully stripped (q=Z).

As noted above both the time-of-flight stop detector and the second element of the telescope were position sensitive in two dimensions, thus allowing the angle-of-entry of particles arriving at the focal plane to be determined. Consequently small corrections necessary to compensate for second order aberrations in the system could be made in the offline analysis.

Measurement of the rigidity of each ion was performed using a position sensitive parallel plate avalanche counter located in the conventional target chamber of SPEG, where the dispersion of the system was 13 cm per percent in momentum. A mass resolution of 3x10-4 was observed with the experimental arrangement described. This is consistent with the resolutions of 10" 4 in the measurement of the rigidity and 2.5x10" 4 for the time-of-flight.

Two field settings of the spectrometer and transport system were employed. The first, Bp=2.46 Tm with the production target oriented at 45° to the beam direction, provided broad elemental and isotopic distributions -120 nuclei up to 46 C1 were identified. The second setting of 2.7 Tm, with the target oriented normal to the beam axis, gave rise to more restricted distributions (90 nuclei up to ^S) with higher yields for nuclei in the region of the neutron-rich sodium and magnesium isotopes. Many nuclei with well known trasses were also observed at each field setting and those with adequate yields were employed in the final calibration of the data. The masses of these nuclei were taken from an updated version of the 1986-87 midstream mass evaluations [START_REF] Haustein | 1986 -1987 Atomic Mass Predictions[END_REF]. A paucity of heavy mass calibrants, A=38-46, prevented reliable extraction of masses beyond A=37.

Table 1 lists the results obtained for 39 nuclei along with the results of earlier investigations for comparison. The uncertainties quoted, which range from 70 keV ( 20 N) to 1600 keV ( 35 Mg), comprise three components: the statistical uncertainty, an uncertainty arising from the calibration and an uncertainty which represents the limiting precision of the experimental arrangement. The latter was determined, as outlined in [START_REF] Gillibert | [END_REF], by obtaining a reasonable fit of the data to the high yield nuclei of known mass. In this experiment it was determined to be 40 keV. In the case of the uncertainty ascribed to the calibration, it should be noted that the further a nucleus lies from nuclei with known masses, the larger this quantity will be. For example, for the sodium nuclei this uncertainty evolves from 70 keV for 28 Na to 420 keV for 33 Na. Similarly an increase occurs in the statistical uncertainty as the yields decrease along isotopic chains -90 keV for 29 Na, the most prolifically produced sodium isotope (5500 counts), to 1400 keV for 33 Na (21 counts) at the 2.7 Tm setting.

The present results constitute the first measurements of the masses of 29 » 30 Ne, 34 ' 35 Mg and 36 > 37 A1, while the majority of the other determinations are the most precise yet made. In comparison with previous measurements good agreement is generally found with the present results with the conspicuous exception of the very neutron-rich sodium isotopes. It should be noted that the masses deduced in the present work for 33 A1 and 37 P are in excellent accord with the transfer reaction studies of [7], [START_REF] Fifield | [END_REF], [26]. Significant disagreement arises, as noted above, for the very neutron-rich sodium nuclei 31 ~33 Na. The present results indicate that 31 Na and 32 Na are 2 MeV less bound than originally reported [3], while 33 Na is 4 MeV less bound. The present mass for 31 Na is, however, in good agreement with the only previous direct time-of-flight measurement [16].

These results and the discrepancies with the measurements which provided the original evidence for deformation in the N~20 region [3], are of considerable interest in the light of recent shell model calculations directed at understanding the structure of the neutronrich nuclei in the vicinity of this so called "island of inversion" [31]. Such calculations have become feasible through the development of shell model codes which can handle large matrices and of effective interactions which encompass the sd-fp model space [27]- [30], of which the most comprehensive is the "WBMB" interaction [30].

In figure 1, comparison is made between the binding energies measured in the present work with WBMB calculations [31] which employed complete Oi Tco bases. Excellent agreement between experiment and the predictions is evident for all the isotopes of silicon and aluminium and for the lighter isotopes of magnesium, sodium and neon up to N=18 or 19. However, for the latter three elements, the measured values begin to depart significantly from the predictions at N=20 and the trend continues to at least N=21 for sodium and N=22 for magnesium. The overbinding of these nuclei is less dramatic than originally proposed, but nevertheless the effect persists and is evidence of some shortcoming of the Otlto calculations in this region. Notably the present results provide the first evidence that the effect extends below Z=ll and 12 to Z=10.

The failure of the Ot Tco calculations has prompted Warburton et al [31] to extend their calculations in order to investigate, within the framework of the shell model, the suggestion (originally proposed through constrained Hartree-Fock calculations [1]) that the overbinding may be attributed to a transition from a spherical shape to a prolate deformation driven by npnh neutron excitations across the sd-fp shell gap. Using the WBMB interaction and introducing weak-coupling approximations and truncation schemes in the cases where full basis calculations were intractable, they have estimated the binding energies of states with 1, 2 and 3 neutrons promoted from the sd to fp shell orbitals. These they refer to as 1,2 and 31to states. These calculations indicate that for the Z=10-12, N=20-22 nuclei, where the discrepancies between the Otto predictions and experiment are found, but not elsewhere, the lowest 2iTco state (and to a similar extent for the N=21 isotones the lowest ttlo state also) is more bound than the lowest Otfco state. A quantitative comparison between their estimates and the present results are presented in figure 2. The discrepancies, although substatially reduced, are evidently not eliminated entirely. As noted by the authors a further reduction could be anticipated due to repulsion between the Oho and 2fito ground states. Hence, in the light of the measurements reported in this paper, we conclude that the WBMB interaction contains the ingredients necessary to understand the existence of the "island of inversion" in this region of the chart of nuclides. Particularly noteworthy is that these calculations were performed before the results of the present study became available.

Other mass predictions based on empirical and semi-empirical mass relations may be found in [START_REF] Haustein | 1986 -1987 Atomic Mass Predictions[END_REF]. These models in general have very poor predictive abilities in the region of interest -probably due to the sudden change in structure (a smooth behaviour of the nuclear mass surface as a function of A and Z is usually assumed). For example as noted by [31], in the case of the microscopic-macroscopic model of Moeller et al [START_REF] Haustein | 1986 -1987 Atomic Mass Predictions[END_REF] an rms deviation of 1.4 MeV is found between experimentally determined masses and those predicted for nuclei of Z=13-20, N>20. By comparison the WBMB calculations yield an rms deviation of only 0.3 MeV.

The present results are also displayed in figure 3 in terms of two-neutron separation energies (thereby avoiding the odd-even neutron pairing effect). Such a plot, as noted earlier, graphically illustrates the region of stable deformation whereby the monotonic decrease with increasing neutron number is arrested abruptly for the Z=10-12 nuclei at or near N=20. Given that 31 Ne is unbound [13] (implying an upper limit of 3.9±1.0 MeV on the twoneutron separation energy) and the apparent resumption in the decline of the separation energies at 33 Na and 35 Mg, it would be interesting to extend the measurements further along the Z=9-12 isotopic chains. In particular, the masses of 29 F and 32 Ne, the last known particle stable nuclei of fluorine and neon [13] would be of considerable interest.

In summary, the masses of 39 neutron-rich nuclei in the region of the N=20 shell closure have been measured using a direct time-of-flight method following the fragmentation of an intermediate energy 48 Ca beam. Comparison of the results with shell model calculations have demonstrated that the so called "island of inversion" persists into the Z=10 nuclei, becoming apparent at 30 Ne (N=20). The effect does not appear to intrude into the Z=13 nuclei. Its extent along the Z=10-12 isotopic chains, while giving indications of terminating at N=22 and 23 for Z=l 1 and 12 requires further investigation. The magnitude of the effect on the binding energies of the sodium isotopes 3l-33N a has been found to be substantially less (2-4 MeV) than indicated by earlier measurements.

This experiment was performed at the GANIL laboratories and the authors wish to acknowledge the assistance and support of the staff. NAO also wishes to thank Dr W. Mittig and the staff of GANIL for their generous hospitality during his sejour.

FIGURE CAPTIONS

FIGUREl: Comparison between the present experimental results and binding energies predicted by Ot Tco WBMB calculations [31].

FIGURE 2: Comparison between the present experimental results and 2froc» weak coupling binding energy estimates of [31] for the N=20-22 Ne, Na and Mg nuclei. For the nuclei with N<20 and N=23, for which E x (OtTco) < E x (2ttco), the comparison is with the Ot Tco WBMB calculations.

FIGURE 3: Plot of two-neutron separation energy versus neutron number. For nuclei not appearing in table 1 the data of [START_REF] Haustein | 1986 -1987 Atomic Mass Predictions[END_REF] have been used.

NOTE ADDED IN PROOF:

Mass measurements of neutron-rich isotopes between Neon and Argon have very recently been performed using the TOFI spectrometer [34]. While not extending to 30 Ne, 33Na and 35Mg, these data support the conclusions of the present work. The authors are grateful to D. Vieira for bringing this work to their attention.

[34] X.G. Zhou et ah, Bull. Am. Phys. Soc. 35 (1990) 1648 and D.J. Vieira, priv. comm. Neutron Number

TABLE 1 :

 1 Mass excesses of present and earlier work (A (±o) MeV). MeV derived for32 Na(|3")32 Mg[4], coupled to the present32 Mg mass excess implies a32 Na mass excess of 17.4±1.4 MeV. e) S n = -1.0711,66 MeV. The particle stability of35 Mg and the present 34 Mg mass excess would imply a 35 Mg mass excess of less than 16.3210.44 MeV. f) Observed only at Bp = 2.46 Tm.

	Nucleus	Present	[19]	[16]	[20]	[32]	Other
	17 B	43.90(0.23)	43.62(0.17)	42.82(0.80)
	19 C	33.07(0.24)	32.77(0.12)	32.95(0.42)	32.30(0.24)
	20 C	37.41(0.46)	37.60(0.22)	37.20(1.13)
	20 N	21.79(0.07)	21.78(0.12)	21.62(0.14)	21.64(0.26)		22.20(0.36)I 21 1
	21 N	25.30(0.12)	25.09(0.20)	25.21(0.18)	24.76(0.19)		25.58(0.51)1 21 1
	22 N	32.31(0.36)	31.99(0.23)	30.73(0.74)
	22 0	9.17(0.08)				9.44(0.09)	9.48(0.10)[ 22 '
	23 0	14.55(0.17)	14.62(0.14)	14.77(0.30)	14.6 (0.3)		14.61(0.66)t 21 l
	24 0	19.24(0.41)	18.6 (0.5)	18.70(1.00)
	24 F	7.58(0.08)		7.87(0.22)	7.52(0.16)		7.86(0.45)l 21 '
	25 F	11.29(0.14)	11.37(0.14)	11.19(0.27)	11.18(0.20)		10.51(0.57)[ 21 '
	26p	18.21(0.28)	18.46(0.20)	19.51(0.60)	18.4 (0.9)		17.55(1.27)[ 21 1
	27 F	24.22(0.72)	25.6 (0.7)
	26Ne	0.42(0.08)				0.44(0.07)	0.44(0.07)I 23 1
	27Ne 2 «Ne	7.05(0.16) 11.14(0.22)		6.96(0.28) 11.43(0.52)	5.6 (0.6) 1 10.7 (0.4)
	29fj e 3 °Ne	18.10(0.51) 22.24(0.82)
	28 Na	-1.02(0.09)			-1.14(0.18)	-1.14(0.14)	-1.14(0.08)I 3 1
	29Na 30^3 31 Na 33 Na c 32j\f a	2.64(0.13) 8.50(0.20) 12.63(0.31) 25.5 (1.5) 18.37(0.59)		8.57(0.34) 12.52(0.93)	2.63(0.21) 7.1 (0.5) 1	2.65(0.15) 8.21(0.25) 11.83(0.58) b 21.47(1. 14) b 16.55(0.74) b	2.65(0.10)t 3 J 8.37(0.20)t 3 ) 10.60(0.80)I 3 1 16.4 (l.l)t 3 '
	34 Na					26.65(3.57) b
	30 Mg	-8.94(0.09)			-9.04(0.21)	-9.10(0.21)
	31 Mg	-3.22(0.14)		-3.28(0.17)	-3.56(0.20)		-5.0 (0.7) W
	32 Mg	-0.86(0.20) d		-0.83(0.25)	-0.75(0.24)	-1.75(1.58)	-1.9 (1.5) W
	33 Mg	4.85(0.30)		5.09(0.84)
	34 Mg	8.25(0.44)
	35 Mg c	1 7.4(1. 6) e
	32 A1 f	-11.06(0.10)		-11.06(0.19)	-11.33(0.20)
	33 Al	-8.54(0.13)		-8.62(0.15)	-8.84(0.23)		-8.53(0.14)' ? J
	34 Al	-3.04(0.20)		-3.17(0.23)	-3.5 (0.4)
	35 A1	-0.28(0.28)		-0.32(0.43)
	36 A1	5.76(0.39)
	57 Al	9.60(0.54)
	36 Si f	-12.65(0.18)		-12.57(0.30)	-12.9 (0.6)
	3 7 S i	-6.81(0.28)		-7.03(1.31)
	37pf	-19.04(0.19)				-18.99(0.04)1 26 J

13 14 15 16 17 IB 19 20 21 22 23 24

	2	-
												Si
	2	t	i	i	i	i	i	i	t	i	i	i


Rafik Absi

Revisiting the pressure-area relation for the flow in elastic tubes: Application to arterial vessels

Keywords: Pressure-area relation, Transmural pressure, Elastic tubes, Arterial vessels, Compliance

For the description of the flow behaviour in elastic tubes as arterial vessels, we need a relationship between the transmural (internal minus external) pressure and the variation in the cross-sectional area A (or diameter), i.e., the pressure-areaconstitutive relation. However, a literature review shows different relations. In this study, the method based on the linear theory of elasticity is revisited. A new pressure-arearelation is proposed.

Results for the variation of cross-sectional area, arterial compliance and distensibility are presented. To define a unique threshold value for the applicability of the former equations, all results are presented in dimensionless form using the parameter = ℎ / (where E is Young's Modulus, ℎ and are respectively the vessel wall thickness and the internal radius at =0). Comparisons with the so-called linear and non-linearequations show that all results are similar for / < 0.05. Our results indicate that the former equations could be used with an accepted gap until / =0.1. However, the inaccuracy increases with and at / =0.2, the difference is of 26.7% and 24.6% respectively for the linear and non-linear relations. Proposed equations were applied to arterial vessels with =150mmHg for radius from 0.8 to 6 mm. Results show an increase in the diameter of 4% for =0.8mm while it is of 30% for =6mm.

Introduction

Physiological and cardiovascular fluid mechanics provide an understanding of advanced concepts in fluid mechanics to study blood flow in the cardiovascular system. The knowledge of the fluid mechanics of the circulatory system is indispensable for well understanding of many cardiovascular diseases [START_REF] Korsakova | Model of blood circulation in the cerebral cortex on the theory of fluid flow in heterogeneous medium[END_REF][START_REF] Liepsch | Study of wall shear stress in an idealized 90-bifurcation with Newtonian and non-Newtonian fluid models[END_REF]. Principles of conservation of mass and momentum provide the main equations of fluid flow which are non-linear, partial differential equations and need numerical solutions. For some cases, simplification of these equations allows analytical solutions [START_REF] Wang | Exact solution for laminar flow in partially collapsed tubes[END_REF][START_REF] Absi | Analytical solutions for the modeled k equation[END_REF]. Computational fluid dynamics (CFD) modelling provides detailed pressure and flow fields [START_REF] El Gharbi | An improved near-wall treatment for turbulent channel flows[END_REF] and the quantification of some parameters which cannot be obtained experimentally as wall shear stress [START_REF] Morris | Computational fluid dynamics modelling in cardiovascular medicine[END_REF][START_REF] Antonova | Numerical analysis of 3D blood flow and common carotid artery hemodynamics in the carotid artery bifurcation without stenoses[END_REF]. Fluid dynamics in elastic tubes is of high interest in different industrial and biological applications [START_REF] Hoekstra | Unsteady flow in a 2D elastic tube with the LBGK method[END_REF][START_REF] Riahi | Modeling unsteady two-phase blood flow in catheterized elastic artery with stenosis[END_REF][START_REF] Ghigo | A 2D nonlinear multiring model for blood flow in large elastic arteries[END_REF]. The description of the flow behaviour in elastic tubes as arterial vessels needs three independent variables namely the pressure ( , ), the fluid velocity ( , ) (or equivalently the flow rate ( , )) and the crosssectional area ( , ). The main governing equations are the conservation of mass and momentum (i.e. the continuity and the momentum equations). In this problem, we only have two equations and three variables, namely, , , and . Therefore, we need a third relation which describes the deformation of the vessel walls due to a variation in the pressure. A third equation could be obtained from the energy conservation which is related to the interaction between the fluid and the tube wall or by analytical equations which provide a relationship between the transmural (internal minus external) pressure and the variation in the crosssectional area (or diameter), i.e., the so-called state equation or pressure-areaconstitutive relations.

A literature review shows different pressure-area relations [START_REF] Rammos | A computer model for the prediction of left epicardial coronary blood flow in normal, stenotic and bypassed coronary arteries, by single or sequential grafting[END_REF][START_REF] Olufsen | Structured tree outflow condition for blood flow in larger systemic arteries[END_REF][START_REF] Formaggia | A. One-dimensional models for blood flow in arteries[END_REF][START_REF] Sherwin | One-dimensional modelling of a vascular network in spacetime variables[END_REF][START_REF] Urquiza | Multidimensional modelling for the carotid artery blood flow[END_REF][START_REF] Canic | Blood flow through compliant vessels after endovascular repair: wall deformations induced by the discontinuous wall properties[END_REF][START_REF] Smith | An anatomically based model of transient coronary blood flow in the heart[END_REF][START_REF] Mynard | A numerical model of neonatal pulmonary atresia with intact ventricular septum and RV-dependent coronary flow[END_REF][START_REF] Valdez-Jasso | Modeling and Identification of Vascular Biomechanical Properties in Large Arteries[END_REF][START_REF] Mynard | A simple versatile valve model for use in lumped parameter and one-dimensional cardiovascular models[END_REF][START_REF] Sochi | The flow of Newtonian and power law fluids inelastic tubes[END_REF]. In this study, we will consider the relations which are derived from the linear theory of elasticity without considering the viscoelastic behaviour. The more used equations which show interest in practical purposes are the so-called linear and [START_REF] Urquiza | Multidimensional modelling for the carotid artery blood flow[END_REF]). The aim of this study is to provide an answer about these different pressure-area relations. It is important to understand why there are different relations, and which one should be used. To provide an appropriate answer to these questions, it is important to re-examine the method which provides these relations.

In the following sections we will first review the linear and non-linearrelations, then present the main assumptions of the study. The method of obtaining arelation will be revisited and finally our results will be compared to former relations and we will discuss the implications for arterial vessels.

Review of pressure-area relations

The vessel is represented as a cylindrical tube (Fig. 1) of length , wall thickness ℎ, inner (or internal) radius = , outer (or external) radius and circular cross-sectional area ( , ) =

. Pressurearea equations provide relation between the transmural pressure and the variation in the cross-sectional area (or the diameter). The transmural pressure is defined as

= -

, where = is the internal fluid (blood) pressure and is the external pressure (from surrounding tissue). The variation in the area is between the tube section = (at internal fluid pressure p) and the area 0 = 0 2 when there is zero transmural pressure (i.e. =

), where is the radius at =0.

In this study, we will consider the more usedrelations, i.e., linear and non-linear equations (presented in Table 1).

Fig. 1: A diagram of a vessel represented as a cylindrical tube, = is the internal radius; h is the wall thickness; L is the length of the tube; and σ is the circumferential stress.

The linear pressure-area relation

We write the linearrelation in the following form

(1)

= ( -)

where is the proportionality factor which is a measure for the stiffness of the tube wall, the value from Rammos equation is therefore =

, where the coefficient of proportionality is related to Young's Modulus , the vessel wall thickness ℎ, = 2

the diameter and the cross-sectional area when =0. = + ℎ -1

= + √ ℎ (1 -) (√ -√ )

Non-linear pressure-area relations

We write the non-linearrelation in a single general form as

= (√ -√ ) [START_REF] Urquiza | Multidimensional modelling for the carotid artery blood flow[END_REF] ℎ 1

4 3 ℎ 1 √ Sherwin et al. (2003) [14] √ ℎ (1 -) Urquiza et al. (2006)
In the two first non-linear relations of [START_REF] Olufsen | Structured tree outflow condition for blood flow in larger systemic arteries[END_REF] [START_REF] Olufsen | Structured tree outflow condition for blood flow in larger systemic arteries[END_REF] and [START_REF] Sherwin | One-dimensional modelling of a vascular network in spacetime variables[END_REF] [START_REF] Sherwin | One-dimensional modelling of a vascular network in spacetime variables[END_REF] (Table 2) the variation of the thickness ℎ (equal to ℎ at =0) due to the deformation is introduced through the Poisson ratio . In Olufsen's relation =0.5 and therefore 1 -2 = 3/4 and it contains ℎ instead of ℎ .

In the last nonlinear elastic relation of Table [START_REF] Liepsch | Study of wall shear stress in an idealized 90-bifurcation with Newtonian and non-Newtonian fluid models[END_REF], [START_REF] Urquiza | Multidimensional modelling for the carotid artery blood flow[END_REF] [START_REF] Urquiza | Multidimensional modelling for the carotid artery blood flow[END_REF] didn't consider the Poisson ratio. It is important to understand why there are linear and non-linear relations and why different relations for β, and which relation should be used? An answer requires a re-examination of the method related to these relations.

Pressure-area relation revisited

Main assumptions of flow in elastic tubes (arterial vessels)

Several approaches can be taken to write the relation between the pressure and the cross-sectional area. The arterial wall shows a time-delay in the response from a change in pressure to the corresponding change in cross-sectional area i.e., the viscoelastic behaviour [START_REF] Caro | The Mechanics of the Circulation[END_REF][START_REF] Mcdonald | Blood Flow in Arteries[END_REF][START_REF] Rockwell | Model studies of the pressure and flow pulses in a viscoelastic arterial conduit[END_REF]. However, these viscoelastic effects seem to be small within the physiological range of the flow and pressure [START_REF] Tardy | Non-invasive estimates of the mechanical properties of peripheral arteries from ultrasonic and photoplethysmographic measurements[END_REF]. Therefore, many studies use relations derived from the linear theory of elasticity and disregard the viscoelastic behaviour [START_REF] Rammos | A computer model for the prediction of left epicardial coronary blood flow in normal, stenotic and bypassed coronary arteries, by single or sequential grafting[END_REF][START_REF] Olufsen | Structured tree outflow condition for blood flow in larger systemic arteries[END_REF][START_REF] Formaggia | A. One-dimensional models for blood flow in arteries[END_REF][START_REF] Sherwin | One-dimensional modelling of a vascular network in spacetime variables[END_REF][START_REF] Urquiza | Multidimensional modelling for the carotid artery blood flow[END_REF][START_REF] Canic | Blood flow through compliant vessels after endovascular repair: wall deformations induced by the discontinuous wall properties[END_REF][START_REF] Smith | An anatomically based model of transient coronary blood flow in the heart[END_REF][START_REF] Mynard | A numerical model of neonatal pulmonary atresia with intact ventricular septum and RV-dependent coronary flow[END_REF][START_REF] Valdez-Jasso | Modeling and Identification of Vascular Biomechanical Properties in Large Arteries[END_REF][START_REF] Mynard | A simple versatile valve model for use in lumped parameter and one-dimensional cardiovascular models[END_REF][START_REF] Sochi | The flow of Newtonian and power law fluids inelastic tubes[END_REF][START_REF] Caro | The Mechanics of the Circulation[END_REF]. Our study is based therefore on the following assumptions: • the flow is axisymmetric • the arterial vessels walls are thin, i.e., ℎ<< , that the loading and deformation are axisymmetric • the structural arterial properties are constant • the vessel is tethered in the longitudinal direction To obtain the relationship between the pressure and the cross-sectional area, we need to examine the equilibrium of the internal and external forces acting on a unit element of the wall.

Pressure force

The elementary force due to the pressure differences is given by

(3) = - = ( - )
For the half cylinder (Figure 1), the vertical component is

= ( -) sin

After integrating from 0 to π, we obtain the vertical force due to the pressure differences

(5) = 2 ( - )
If the vessel is thin-walled (ℎ ≪ ), then ≈ = and

(6) = 2 ( - ) = 2 3.3. LaPlace'ś Law
The aim is to link the transmural pressure to the tension in the walls related to the wall stress (force per unit area) σ. The force pulling the half cylinder down is

(7) = 2 ℎ
In equilibrium, Fσ is balanced by the vertical force due to the transmural pressure (Eq. 6)

ℎ =

Equation ( 8) is an expression of LaPlace'ś law for a thin-walled cylinder. Note that for a given transmural pressure, the wall tension ( = ℎ) per unit length increases as the radius increases and vice-versa. The elasticity of the tube involves a relation between stress and strain (proportional deformation) as = ∆ / . By considering the variation of the wall thickness through the Poisson ratio and Eq. ( 8), we write

(9) = = ( ) ∆
If the radius varies from the initial value related to =0 (zero transmural pressure) to a given value , the strain is equal to ( -)/ , Eq. ( 9) becomes [START_REF] Ghigo | A 2D nonlinear multiring model for blood flow in large elastic arteries[END_REF] = ( ) -1

The former linearrelation (Eq. 1) is based therefore on the assumption / = / which is not realistic and could be valid only for small deformations. The accurate relation is / = √ / and therefore involves a non-linearrelation.

(

) = ( ) √ (√ -√ ) 11
With Eq. ( 2), the coefficient is therefore equal to

= (

) √ With = 0.5 in Eq. ( 12), we obtain the same coefficient of Olufsen et al. (1999) [START_REF] Olufsen | Structured tree outflow condition for blood flow in larger systemic arteries[END_REF] (Table 2). However, in Eq. ( 12), ℎ and are not constants since they change with

. Therefore, we need a relation for with ℎ and instead of ℎ and . To write this equation, we need to know the variation of the wall thickness. Based on the assumption related to this variation, we present the following formulations.

A newrelation

A first approximation

If we assume that the variation of the wall thickness is negligible (ℎ = ℎ and = 0) Eq. ( 12) becomes = √ and we write Eq. ( 2) in the following dimensionless form

= 1 -where =

Coefficient has the unit of pressure, it could be interpreted as the initial value (at =0) of the coefficient of proportionality between and the strain. From Eq. ( 13), we have the following explicit equation for ( )

=

In Eq. ( 14), / ≠1, this equation will be used for < .

Proposedrelation

If we consider the variation of the wall thickness ℎ through the Poisson ratio , it is possible to write a relation between ℎ and ℎ based on the area conservation equation 2 ℎ = 2 ℎ, Eq. ( 12) becomes

= √ (

) and we write Eq. ( 2) as In this equation, is known through input data , ℎ and . However, it is possible to write Eq. ( 15) in dimensionless form using and , since = /(1 -), as

= ( )

-

In equation ( 16), the dimensionless coefficient of proportionality

( )
is related to the assumption based on the area conservation equation. When the variation of the wall thickness is negligible ℎ = ℎ , this coefficient becomes equal to 1 since for this case = 0 and = in this dimensionless coefficient and Eq. ( 16) reverts to Eq. ( 13) of the first approximation (Appendix 1).

We write equations ( 13) and ( 16) in a single form as:

(17) = 1 - where = 1 = 0 ( ) ≠ 0
However unlike in the first approximation, Eq. ( 16) doesn't allow one to write an explicit equation for (

). In allrelations (Table 3), the different coefficients of proportionality ( , , or ) play an important role which impacts strongly the results. This coefficient depends on the biomechanical behaviour of the blood vessels which are determined by the physical properties of the individual wall constituents (mostly elastin, collagen, and smooth muscle), and their relative content [START_REF] Mynard | A simple versatile valve model for use in lumped parameter and one-dimensional cardiovascular models[END_REF].

Table 3 Summary of former and proposedrelations.

Coefficient

/

Linear [START_REF] Rammos | A computer model for the prediction of left epicardial coronary blood flow in normal, stenotic and bypassed coronary arteries, by single or sequential grafting[END_REF] ( - 2√ 2 2

) = ℎ + 1 2 -1 Non-linear [15] (√ -√ ) = ℎ + -1 1sr Approximation √ - √ = ℎ 1 - 1 - Proposed (√ - ) = √ ℎ (1 -) 1 (1 -) 1 -
1 st Approximation 2 / 2 / 2 Proposed ( -0.5 √ ) (1 -) 1 -0.5
(1 -) 1 -0.5

Arterial compliance or capacitance and the distensibility

The compliance or capacitance describes how volume changes in response to a change in pressure [START_REF] Spencer | Pulsatile blood flow in the vascular system[END_REF], it is inversely proportional to elasticity. The arterial capacitance per unit length or cross-sectional compliance [START_REF] Reneman | Age-related changes in carotid artery wall properties in men[END_REF] may be calculated assuming that vessel length does not vary with transmural pressure.

(

) 18

=

The capacitance is a key parameter involved in many calculations such as:

• The flow rate is related to the cross-sectional compliance through =

• The distensibility is defined by

= =

Our two proposed relations for compliance and distensibility are summarized in table [START_REF] Absi | Analytical solutions for the modeled k equation[END_REF] and presented together with the former linear and non-linear equations. with the coefficient (Table 2) given by Urquiza et al. [START_REF] Urquiza | Multidimensional modelling for the carotid artery blood flow[END_REF]) (dash-dotted line) and our approximation given by Eq. ( 14) (solid line). with coefficients (Table 2) given rby Sherwin et al. [START_REF] Sherwin | One-dimensional modelling of a vascular network in spacetime variables[END_REF] (dashed line) and our result which is obtained from the resolution of the ordinary differential equation (ODE) given by the cross-sectional compliance (Table 4) (solid line). All other equations remain quite similar to the result from our proposed equations until a value of / equal to about 0.05. However, from / =0.1, the gap increases significantly with

Results and discussion

Variation of the cross-sectional area and diameter

. Former linear and non-linear relations underestimate the variation of cross-sectional area for a given transmural pressure. At / =0.2, the difference is of 26.7% and 24.6% respectively for linear (Rammos) and non-linear (Urquiza) relations. 4).

Variation of cross-sectional compliance and distensibility

Figure [START_REF] Wang | Exact solution for laminar flow in partially collapsed tubes[END_REF] shows a comparison between cross-sectional compliance obtained from the four equations presented in table (4) namely, the linear with the coefficient of Rammos [START_REF] Rammos | A computer model for the prediction of left epicardial coronary blood flow in normal, stenotic and bypassed coronary arteries, by single or sequential grafting[END_REF] (dashed line), the non-linear with the coefficient of Urquiza et al. [START_REF] Urquiza | Multidimensional modelling for the carotid artery blood flow[END_REF]) (dash-dotted line), Eq. ( 13) (thin solid line) and our proposed Eq. ( 16) with =0.5 (thick solid line). At the opposite of figure [START_REF] Liepsch | Study of wall shear stress in an idealized 90-bifurcation with Newtonian and non-Newtonian fluid models[END_REF] where results were similar for lower values of / , results for compliance (figure 3) show more scatter. The difference increases with / and therefore with (figure 3.b). For / =1.5, the gap of the non-linearrelation (with the coefficient of Urquiza et al.) is of 43% with respect to our Eq. [START_REF] Canic | Blood flow through compliant vessels after endovascular repair: wall deformations induced by the discontinuous wall properties[END_REF]. Figure [START_REF] Absi | Analytical solutions for the modeled k equation[END_REF] presents results for A/A 0 distensibility . Even if the difference between the four curves seems more important, the difference (of the non-linearrelation) is the same as for cross-sectional compliance namely 43%.

In figures (3) and ( 4), all curves begin at a value of dimensionless cross-sectional compliance equal to 2 except our Eq. ( 16) which starts from a value of 1.5. This value is due to the term of the Poisson ratio, i.e., (1 -) which is equal to 3/4 (for =0.5) and therefore at / =1 a value equal to 1.5. The approximation of ≈ ( 3)/(2 ℎ ) (for << ) [START_REF] Olufsen | Structured tree outflow condition for blood flow in larger systemic arteries[END_REF] provides the same value / = 1.5

and confirms that this approximation is valid only for very small values of / . For larger values, the dependency on (or ) should be considered (figure 3). 4).

Application to arterial vessels

In order to understand the implications for arterial vessels, we will consider the empirical relation for the parameter β1 as a function of [START_REF] Olufsen | Structured tree outflow condition for blood flow in larger systemic arteries[END_REF] namely ( 18)

1 ( 0 ) = 1 exp( 2 0 ) + 3
The constants =2.00 × 10 7 g.s -2 .cm -1 , =-22.53 cm -1 and =8.65 × 10 5 g.s - [START_REF] Westerhof | Analog studies of the human systemic arterial tree[END_REF]. Values of β1 obtained from Eq. ( 18) are therefore in g.s -2 .cm -1 . The application of the proposed equation to an arterial vessel with =150mmHg (Table 5) show an increase in the diameter of 4 % for =0.8mm while it is of 30 % for =6mm. Figure [START_REF] El Gharbi | An improved near-wall treatment for turbulent channel flows[END_REF] presents the variation of the dimensionless cross-sectional area and diameter for different arterial vessels with initial radius equal to 0.8, 2 and 6 mm. Results show that at =150mmHg, the dimensionless cross-sectional area is more than 1.6 for =6mm, about 1.4 for =2mm and very small 1.08 for =0.8mm.

Conclusions

In this study, the pressure-area relation was revisited and a new relation was proposed. For the variation of the cross-sectional area, the proposed result was obtained from the resolution of the ordinary differential equation (ODE) given by the cross-sectional compliance or capacitance . The parameter = ℎ / allowed us to write all results in dimensionless form and therefore to define a unique threshold value for the applicability of the former equations. Comparisons with the so-called linear and non-linearequations show that results of cross-sectional area are similar until a value of dimensionless transmural pressure / equal to about 0.05. Former equations could be used with an accepted gap until / =0.1. However, the inaccuracy increases with and at / =0.2, the difference is of 26.7% and 24.6% respectively for the linear and non-linear relations. Results for the variation of arterial compliance and distensibility were presented. These results showed non-negligible differences between proposed and former equations. The proposed equation was applied to arterial vessels with =150mmHg for radius from 0.8 to 6 mm. Results show an increase in the diameter of 4 % for =0.8mm while the increase is of 30 % for =6mm. This study was about the linear theory of elasticity. In future studies, for better understanding of the flow behaviour in arterial vessels, we will consider fluid-structure interaction phenomena [START_REF] Pielhop | Experimental analysis of the fluid-structure interaction in finitelength straight elastic vessels[END_REF] and the influence of transmural pressures on the change in tube shapes [START_REF] Nahar | Influence of elastic tube deformation on flow behavior of a shear thinning fluid[END_REF].

Figure ( 2 )

 2 Figure (2) shows a comparison betweencurves obtained from the methods presented in tables (2) and (3). For =0, fig. (2.a) presents comparisons between three explicit equations of namely, the linear relation = + with the coefficient of Rammos [11] (dashed line), the non-linearrelation = +

Figure ( 2

 2 .a) shows that all equations present practically same results for / <0.05. This could explain the applicability of the linear equation for small values of / . For =0.5, fig. (2.b) presents comparisons between two non-linearrelations = +

Figure ( 2

 2 .b) shows that Sherwin's equation underestimates the area compared to our equation. They present similar results for / <0.05.

Fig. 2 :

 2 Fig. 2: Comparison ofrelations, / ( / ), (a) =0, from linear (Rammos, dashed line), nonlinear (Urquiza et al., dash-dotted line) equations and results from Eq. (14) solid line, (b) =0.5 from nonlinear Sherwin et al. (dashed line) and our solution (solid line) obtained from the resolution of ODE of compliance (Table4).

Fig. 3 :

 3 Comparison of dimensionless cross-sectional compliance obtained by the four equations (Table4).

4 :

 4 Comparison of dimensionless distensibility obtained by the four equations (Table

Fig. 5 :

 5 Fig. 5: Variation of dimensionless arterial vessel area and diameter for from 0 to 150 mmHg.

Table 1

 1 Linear and non-linearrelations used in the literature.

	Author		-Relation
	Rammos et al. (1998) [11]	=	1 +	ℎ	( -	)
	Olufsen et al. (1999) [12]	-	=	4 3	ℎ	1 -
	Sherwin et al. (2003) [14]

Table 2

 2 Coefficient for the non-linearrelations.

Author

Olufsen et al. (1999)

[START_REF] Olufsen | Structured tree outflow condition for blood flow in larger systemic arteries[END_REF]

  2 .cm -1 were obtained by fitting the data from Segers et al. (1998)[START_REF] Segers | Role and relevancy of a cardiovascular simulator[END_REF],[START_REF] Stergiopulos | Computer simulation of arterial flow with applications to arterial and aortic stenosis[END_REF] [START_REF] Stergiopulos | Computer simulation of arterial flow with applications to arterial and aortic stenosis[END_REF] and[START_REF] Westerhof | Analog studies of the human systemic arterial tree[END_REF]

Table 5

 5

	Parameters for	=150mmHg (0.2×10 5 Pa).
	(cm)	×10 5 (Pa)	/
	0.08	4.16	0.048
	0.2	1.08	0.18
	0.6	0.86	0.23