B Merit

G Lemarquand
email: guy.lemarquand@ieee.org

V Lemarquand

Performances and Design of Ironless Loudspeaker Motor Structures

Keywords: Permanent magnets, loudspeaker motor, loudspeaker performance, motor design, analytical magnetic field calculation, . I

This paper presents several kinds of structures of ironless loudspeaker motors. The proposed devices go from simple structures, in terms of manufacturing process, to structures in which the magnet rings can be stacked, with different magnetization directions or values. The structures are studied and compared according to their created induction level and uniformity across the coil displacement range, which are directly linked to the loudspeaker performances (force factor). The model used is the coulombian model of magnets to calculate the magnetic field created, therefore all the calculations are analytical. The paper shows that no structure is universal when it comes to the loudspeaker design: some are well adapted to micro displacements of the coil while other structures are adapted to large displacements.

The power is linked to the acoustical flow, thus to the product of the axial displacement times the section of the emissive surface. If this surface is diminished and the power remains the same, then the displacement of the moving part has to be increased. In classical structures of electrodynamic loudspeakers, the increase of this displacement often means the increase of nonlinear effects and distortions.

Numerous studies of the loudspeakers highlight that their non-linearities have four major causes [START_REF] Cunningham | Nonlinear distortion in dynamic loudspeakers due to magnetic effects[END_REF][2]: the defects of the motor, the acoustical propagation, the viscoelastic behavior of the suspension, and the vibrating modes of the emissive surface and of the moving part. Many solutions have been found in order to reduce these imperfections or their effects. Especially the drawbacks linked to the iron in the classical motor structures have already been described (reluctant force, Eddy currents) [START_REF] Vanderkooy | A model of loudspeaker driver impedance incorporating eddy currents in the pole structure[END_REF] and we presented a concept to get rid of them: structures of ironless loudspeaker motors [START_REF] Berkouk | Analytical calculation of ironless loudspeaker motors[END_REF]- [START_REF] Ravaud | Modelling an ironless loudspeaker by using three-dimensional analytical approaches[END_REF].

The present paper goes further in this direction, describing different types of existing structures of ironless motors and comparing their performances. Predominantly we pay our attention on their ability to create a uniform induction over the axial trajectory of the coil. When this induction is uniform, the force that makes the coil oscillate, and consequently the coil support and the membrane, does not depend on the position of the coil anymore. Then all the non-linearities due to the motor are suppressed in such ironless loudspeakers.

In the same way, we evaluate structures on the space they need, especially in the axial direction of the loudspeaker. The more reduced the structure height is, the closer the coil can be to the cone diaphragm and the shorter the moving part can be: this is an advantage in terms of vibration modes.

II. CONSIDERATIONS FOR THE STUDY

The way the structures are presented in the following study, taking into account the various geometrical possibilities, is consistent with the way electrical machines are presented by Wang [START_REF] Wang | Design optimisation and comparison of permanent magnet machines topologies[END_REF]. Many solutions have been imagined and studied to create homogeneous fields in closed cavities or open spaces and we meet here the same problematic [START_REF] Halbach | Design of permanent multiple magnets with oriented rec material[END_REF]- [START_REF] Ravaud | Discussion about the magnetic field produced by cylindrical halbach structures[END_REF]. Some of these structures were also applied to other purposes in electrical machines [START_REF] Maosan | Magnetic field analysis of rec devices with gradually varied magnetizations[END_REF]- [START_REF] Zhu | Design considerations for permanent magnet polarised electromagnetically actuated brakes[END_REF]. The reasoning used to associate magnets to create the intended magnetic field is the important and common point.

The structures presented are ring-shaped with rectangular sections. Then, the important point is to calculate the magnetic field created by such magnets or magnet assembly. The model chosen for the magnets is the coulombian one, which uses fictitious surface and volume charge densities to represent the magnet and thus to calculate the field they create in the space in and out of the magnet [START_REF] Furlani | A three-dimensonal field solution for radially polarized cylinders[END_REF]- [START_REF] Ravaud | Discussion about the analytical calculation of the magnetic field created by permanent magnets[END_REF]. It has to be noted that the formulations obtained for the three components of the field are totally analytical. This enables easy optimization of the parameters of a structure, once a criterion is chosen.

Two kinds of applications are considered: the high frequency loudspeakers, for which the needed coil displacements are small, and the low frequency loudspeakers, which need large coil displacements. Typically, high frequency loudspeakers require peak-to-peak coil displacements smaller than 1mm (tweeters) or reaching 3mm (high medium loudspeakers); lower frequency loudspeakers can require peak-to-peak coil displacements of 10mm (bass medium loudspeakers) and even larger than 20mm (woofers).

The reference model usually describing the functioning of loudspeakers uses the Thiele&Small parameters (lumped parameters) Re, Le, B r l, M ms, Rms, and Kms. This useful model leads to a linear and stationary system of two differential equations:

     Re.i(t) + Le di(t) dt + B r l dz(t) dt = u(t) M ms d 2 z(t) dt 2 + Rms dz(t) dt + Kms.z(t) = B r l.i(t) (1)
However it is known that loudspeakers are highly nonlinear devices [START_REF] Ravaud | Time-varying non linear modeling of electrodynamic loudspeakers[END_REF]. And the dependence of the force factor B r l on the displacement x(t) of the coil is among the major sources of non-linearities [START_REF] Klippel | Tutorial: Loudspeaker nonlinearities -causes, parameters, symptoms[END_REF].

This B r l product, with B r the radial induction flowing through the coil of length l, determines not only the driving force B r l.i(t) for a given current i(t), but also an electrical damping B r l. dz(t) dt = B r l.v(t)

of the loudspeaker connected to an amplifier with low impedance output. Thus, variation of B r l versus displacement will produce two nonlinear terms in the differential equations. In both terms, time signals are multiplied. This multiplication produces new spectral components in the output signal measured and eared as harmonic and intermodulation distortions. Fig. 9 of [START_REF] Klippel | Speaker auralization subjective evaluation of nonlinear distortion[END_REF] shows how critical is the non-linearity due to the force factor B r l, leading to a distorted signal with high amplitudes of the harmonic components.

A symmetrical variation of B r l around the rest position of the coil leads to odd-order distortions; an asymmetry leads to even-order distortions and causes 40% of intermodulation distortions in a loudspeaker [START_REF] Klippel | Prediction of speaker performance at high amplitudes[END_REF].

A. Definition of a linearity criterion related to the force factor 1) Quantification of the distortions induced by a non-uniform force factor:

In this section, the simple linear model given by the system (1) is considered. In this system, a displacement-varying model of the force factor is introduced in order to visualize its effect on the calculated pressure response of a related loudspeaker. The considered lumped parameters are those of a classical loudspeaker whose force factor has been measured. Table I gives these parameters, used to solve the system [START_REF] Cunningham | Nonlinear distortion in dynamic loudspeakers due to magnetic effects[END_REF].

The value of the force factor B r l given by Table I is only available at the rest position of the voice-coil of the loudspeaker. In reality, the force factor depends on the displacement of the voice-coil, and this variation is not symmetrical with the rest position, i.e. z = 0, of the voice-coil (continuous line in Fig. 1).

Its model of dependence is found using a 4 th order polynomial function. It well fits the values of the force factor across the entire displacement range of the loudspeaker, as well as its derivative (dashed line in Fig. 1) This model of dependence on the displacement is introduced in the system [START_REF] Cunningham | Nonlinear distortion in dynamic loudspeakers due to magnetic effects[END_REF]. The solving of this system in the time domain allows the calculation of the displacement z(t) of the voice-coil and of the a.

b. electric current i(t) running through the voice-coil. These later variables are used to estimate the pressure response of the loudspeaker, in time and frequency domains.

The sound-pressure spectrum of the loudspeaker in response to a 5V sinusoidal voltage at 43Hz has been measured (Fig. 2.a). This response is compared with the calculated spectrum thanks to the model (Fig. 2.b). Despite this simple model only takes into account the nonlinearities created by the force factor, it well fits the real sound-pressure spectrum of the loudspeaker: the levels of the created harmonics are quite corresponding, even if even-order harmonics are higher in the model than in reality. Thus, this model can be used to understand the contribution of a non-uniform force factor on the distorted response of a loudspeaker. Fig. 3 shows the time response of the voice-coil displacement to the previous excitation. The maximal reached displacement is only 1.2mm while the voice-coil can move with a ±4mm displacement. To clearly see the effects of a non-uniform force factor, the excitation voltage is increased until the maximal displacement is 4mm. When the frequency is chosen equal to 43Hz, the 4mm displacement is obtained with a 16V excitation voltage. Fig. 4 shows the calculated time responses of the voice-coil displacement and of the current intensity to this increased excitation. The time signals are clearly distorted, and obviously lead to distortions in the frequency domain (see Fig. 5). This figure 4 points out another problem related to a non-uniform force factor: the loudspeaker consumes more current but its voice-coil displaces less than in the linear case, which is the proof of a waste of energy. Fig. 5 is aimed to point out the harmonic and intermodulation distortions created by the loudspeaker whose force factor still varies like in Fig. 1. Thus, the excitation signal is a two-tone excitation voltage:

a 16V one at 43Hz, to obtain the 4mm displacement of the voice-coil, and a 20V one at the arbitrary chosen frequency 500Hz.

When the voice-coil displaces across its entire displacement range, high harmonic distortion and intermodulation distortions are produced, because of the displacement-varying force factor. The higher harmonic component at 129Hz is only 18dB lower than the fundamental at 43Hz, and 13dB lower than the fundamental at 500Hz. The higher intermodulation component at 414Hz is only 19dB lower than the fundamental at 500Hz and 24dB lower than the fundamental at 43Hz. With only two tones, the non-uniform force factor B r l induces a lot of distortions, most of them being potentially audible or disturbing for some applications [START_REF] Blondel | Electropneumatic transducers as secondary actuators for active noise control part i: Theoretical analysis[END_REF], [START_REF] Lane | Improving loudspeaker performance for active noise control applications[END_REF]. These effects are reinforced when talking about low frequency drivers, for which the displacements of the voice-coil are large, and for which the design of the motor becomes very difficult. But the distortions caused by a non-uniform force factor are also discussed in high frequency drivers [START_REF] Hwang | Analysis of harmonic distortion due to uneven magnetic field in a microspeaker used for mobile phones[END_REF].

These are the reasons why it is important to search for loudspeaker motors able to induce force factors B r l that do not depend on the displacement of the coil. The first step to the decrease of the nonlinearities due to the force factor can be to make its dependence on the displacement symmetrical with the rest position of the voice-coil. Thus, from the measured force factor, a model of force factor wih symmetrical dependence on the displacement we call symmetrical-made force factor is obtained (Fig. 6).

This symmetrical-made force factor has a 48% maximal peak-peak variation across a ±4mm range. As expected, Fig. 7 shows that each even-order harmonic component is removed. In classical loudspeaker motors, making a symmetrical force factor is very difficult: it means creating a symmetrical magnetic induction in the airgap around the rest position of the voice-coil. The presence of iron makes this operation very difficult, and designers use tricks to evade this problem [START_REF] Mazin | Non-uniform voice coil winding for electrodynamic loudspeaker[END_REF].

When the magnetic circuit is only made of magnets, the symmetry of the force factor is simply obtained thanks to the symmetry of the magnets disposition. In the sequel, we consider that the modeled loudspeaker creates a symmetrical magnetic induction in the airgap. To study the influence of the variation percentage of the force factor on the creation of distortions, we make its maximal variation vary from 48% (Fig. 6 and Fig. 7) to 1% across a ±4mm range. In Fig. 8, the measured symmetrical-made force factor of the real loudspeaker (thick line) has a 48% peak-peak variation, and is made even more uniform until 1% peak-peak variation.

Fig. 9 to Fig. 13 show the sound-pressure spectra of the modeled loudspeaker whose force factor maximal variation goes respectively from 30% to 1%, in response to a two-tone excitation voltage.

From 30% to 1%, the improvement in the reduction of the harmonic and intermodulation components is as much logical than necessary. When the B r l maximum variation reaches 1%, the higher harmonic and intermodulation components are 50dB lower than the fundamental components. The intermodulation components are known to be more disturbing than the harmonic components, as much in the signal generated by the loudspeaker [START_REF] Czerwinski | Multitone testing of sound system components -some results and conclusions, part 1: History and theory[END_REF]- [START_REF] Klippel | Assessment of voice-coil peak displacement x-max[END_REF] than in the auditive electric signal created by the ear itself [START_REF] Abuelma'atti | Prediction of the two-tone suppression and intermodulation performance of auditory systems[END_REF].

That is why it is important to decrease these components as much as possible. Thus, searching for an Fig. 8. Models of force factor as a function of the voice-coil displacement, and whose maximal peak-peak variation varies from 48% (thick line) to 1% (continuous line) across a ±4mm range. Dashed lines represent force factors with 30% to 5% maximal variation across the ±4mm range. accurate loudspeaker motor is searching for a perfectly uniform force factor, whose maximal variation is 0% along the displacement range of the voice-coil. The constraints in the dimensioning of a traditional magnetic circuit imply that the designers would be largely satisfied with a variation of 1%, given the low levels of the intermodulation components.

2) Definition of the linearity criterion z uni :

The best way to the ideal 0%-variation force factor is to search for a symmetrical ironless motor structure, which is able to create a magnetic field whose radial component is very uniform along the trajectory of the coil. Ideally, for an accurate and space-saving loudspeaker motor, a magnetic structure should create a perfectly uniform induction -i.e. with 0% variation -across a distance as large as the total heigth of the structure. This ideal structure is not realistic and we present in this paper existing structures that may approach the ideal case by optimizing their characteristics and by suffering a 1% peak-to-peak variation. It is not necessary to reach smaller variations than 1%: on the one hand the magnetization of the currently available magnets is not really uniform itself. And on the other hand, especially in the case of underhung loudspeakers, the force that makes the voice-coil oscillate is related to the mean value of the magnetic induction over the coil height (the coil height is the axial length of the coil). Therefore, if the induction variation across a range of displacement that interests us does not exceed 1%, then the maximal force factor variation may not exceed 1%. We have seen in the previous section that such a displacementvarying force factor is acceptable when looking at the level of the intermodulation components. That is why we consider that an induction with a 1% peak-to-peak maximal variation is sufficient to design an acceptable loudspeaker motor.

Thus, for the study, let us consider a parameter we call z uni . Given a structure, this parameter represents the axial distance across which the created magnetic induction seen by the coil has a peak-to-peak variation smaller than 1%. Thus, across this distance, the induction and then the force factor are considered uniform.

In classical loudspeakers, the B r l variation and the induction variation easily exceed 10% across a distance that corresponds to the trajectory of the coil; that is why significant nonlinear effects are produced and it is important to search for structures that lead to more uniform induction.

In the considered applications, the coil moves in a direction, Oz, parallel to the revolution axis of Fig. 14. General composition of an ironless loudspeaker; here the magnet structure is made with two sets of one magnet ring.

the loudspeaker (see Fig. 14): the induction has to be uniform along this direction. Oz is the observation axis, placed in the airgap at a location that corresponds to the mean value of the coil thickness (the coil thickness is the radial length of the coil).

III. SMALL DISPLACEMENTS LOUDSPEAKERS

The coil of a loudspeaker generally moves with small amplitudes at high frequencies. The following sections present the structures which are well adapted to these displacements. As the devices are often intended to be small themselves [START_REF] Yu | Design of a voice coil motor used in the focusing system of a digital video camera[END_REF], we consider structures with only one outer set of magnet ring(s) and an inner coil (see Fig. 15). The origin O of the observation axis Oz corresponds to the center of the coil when this latter is in its equilibrium position. The induction is observed at a radial distance a/2 = 0.3mm from the magnet ring(s). We compare structures that have approximately the same magnet volume with regard to their ability to create a uniform induction.

A. Two axially magnetized permanent magnets: the simplest structure

Structure A of Fig. 15 shows an ironless structure constituted by two axially magnetized rings. These rings are concentric, placed one above the other, and separated with an airgap of thickness e. Their respective magnetization are in repulsion -the facing poles are of the same kind-. The coil is centered in front of the airgap e and the magnetic flux in the coil is then expected to be radial.

This kind of structure has been patented by the Harman society [START_REF] House | Transducer motor assembly[END_REF]. The main advantage of this structure is its simple manufacturing process, because it only requires classical axially magnetized permanent magnets. Fig. 16 shows the global induction behavior when the magnets deviate. The induction intensity and the possible linear coil displacement z uni decrease when e decreases. For example, when the magnets are close to each other (see the solid bold line in Fig. 16), the induction is intense (up to 0.6T ) but the distance z uni is so small that it is not convenient to represent it on the curve.

Thus, this structure is simple but, to prevent nonlinear phenomena, it should only be dedicated to tweeters for which the needed coil displacements are small.

B. One permanent magnet ring radially magnetized.

Let us consider the structure B of Fig. 15. It is constituted by one ring facing the moving coil. Ideally the ring is made in a radially magnetized permanent magnet. For a simpler design, it can also be made up of permanent magnet tiles -the same that are used in electrical machines-which are radially magnetized.

Previous studies [START_REF] Lemarquand | Ironless loudspeakers[END_REF] on the induction created by a rectangular magnet show that a good combination between its dimensions can always lead to a quite uniform induction in front of the magnets. In the best case, the axial distance z uni is about 60% of the considered magnet height.

Fig. 17 shows that, for a given length, the induction can be quite constant over an axial range if the magnet height is sufficient (dot-dashed line in Fig. 17). This figure can also be interpreted alternately: it shows that small height magnets produce an important induction over short axial areas (see bold line in Fig. 17). This localized action will be useful in further structures.

A structure B of 9mm long and 9mm high is now compared to the structure A proposed in the previous section: the magnet volume is the same in both structures. For the structure B, the induction is uniform over a range of ±2.7mm along the Oz axis around the middle of the magnet with a variation of 1% -so z uni = 5.4mm -. The induction intensity in front of the middle of the magnet is 0.33T with a magnet magnetization of 1T and with a magnet section of 81mm 2 . The behavior difference compared to the structure A is obvious: the induction level is smaller but is uniform over a significant range. If the length of the magnet decreases, the uniformity distance z uni decreases. z uni equals about 4mm for a 7mm length, and 1.6mm for a 5mm length. We conclude that the magnet length cannot go below a given value when the variation has to remain to one percent. If smaller, the magnet will not be well used.

In fact, the set L = H = 9mm is the optimum in this case.

C. Combination of magnet rings

The Sony society patented the structure C of Fig.15 [START_REF] Ohashi | Magnetic circuit and speaker[END_REF]. It is constituted by a stack of three magnet rings of equal length. It is notable that such stacked structures were proposed to enhance the performances of other applications such as passive magnetic bearings [START_REF] Yonnet | Stacked structures of passive magnetic bearings[END_REF]. The central ring is radially magnetized while the external rings are axially magnetized and their magnetization directions help the total magnetic flux to close. According to the patent, the three magnetizations of the central ring, J, and of the external ones, j, are equal.

Let us consider the dimensions of the structure given in the patent:

H = 5mm, h = 2mm, L = 8mm,
with a magnetization of 1T . Fig. 18 shows the induction calculated along the observation axis created by the whole structure (bold line) and by the separated elements. This induction is not really uniform. Its central value is 0.5T . The induction variation along the Oz axis over a range corresponding to the height of the central magnet represents 19% of the central value. Because of this significant variation, z uni is only 1.28mm, that is 14% of the total height of the structure. Note that the induction variation is 10% across the distance of 4.2mm that separates the two maxima of this induction.

The structure C, which is 11% more compact than the previous optimal structure B, produces a more intense induction but with a larger induction variation. The induction increases by 34% while the distance z uni decreases by 76% compared to the optimal stucture B .

The principal advantage of the structure C is then its ability to create really high induction levels while using relatively small magnet volumes. Despite its induction variation, the structure C has on this point a substantial lead over the structure A. Nevertheless, because of its small z uni , and taking into account the behavior of its created induction, we conclude that the structure C should be used for tweeters or high medium loudspeakers only.

In addition, Fig. 19 shows that the total induction created by the structure C can be more uniform if the magnetization of the external rings is smaller than the magnetization of the central ring. Fig. 18 shows the contribution of each ring to the total induction and especially that the external rings (see dashed line in Fig. 18) provide a really high magnetic induction compared to the induction provided by the central ring. Fig. 19 shows the result when the external magnetizations are lowered to 0.52T . In this case z uni equals 2.1mm, and the induction variation is 1.8% across the 3.4mm distance that separates the two maxima of this induction. The uniformity improvement is significant but leads to a 8% decrease of the total induction.

D. Several magnetizations

The structure D of Fig. 15 goes further in the idea of using magnets whose magnetizations differ in order to give the intended flat shape to the observed induction. Like the structure C, it is constituted by one set of three stacked magnet rings, but their magnetizations are all radial. The purpose of the induction created by the external rings is to compensate the bell shape of the induction created by the central ring.

This is achieved by combining the higher magnetization of the external rings and their small height to use their very localized action (as described in Fig. 17).

Fig. 20 shows the contribution of each ring and the resulting total induction (bold line). The set of dimensions is chosen in order to have the same magnet section as the one of the previous optimal structure B (81mm 2 ). The resulting induction has a z uni of 6.4mm, which represents 67% of the total height of the structure. In fact, z uni is larger than the height of the central magnet because the structure is calculated in order to create the most uniform possible induction across the height of the central magnet;

across H, the induction variation is only 0.1%.

In this case, the performance difference between both structures B and D is not significant, but it is important to note that the structure D can be optimized. Indeed, the distance z uni can always correspond to the dimension H, so the starting point of the optimization is facilitated by this occurence: the intended z uni fixes the height of the central ring. The other dimensions and magnetizations are calculated in order to optimize the volume of the whole structure and the uniformity and the intensity of the magnetic field.

The previous geometry has been optimized in order to obtain an induction of equal intensity and having about the same z uni of 5.4mm. The optimal dimensions we find correspond to a more spacesaving structure whose total height is 7.2mm. The total section of this structure D is 53.4mm 2 (34% decrease compared to the previous section). The created induction reaches 0.33T and the distance z uni is 5.4mm (see Fig. 21).

The structure D is now compared to the structure C: the magnet volume is kept constant for both structures (Fig. 22). The dimension H = 4mm of the structure D is arbitrary chosen so that z uni approximately corresponds to the distance between the two maxima of the induction created by the structure C.

While the total height of the structure D is 7mm, that is 2mm less than the structure C, z uni equals 4.2mm. This represents a significant improvement compared to the distance z uni of 1.28mm obtained with the structure C. The induction created by the structure D reaches 0.37T , which represents a 37% decrease compared to the induction created by the structure C. Of course, the volume of the structure D can be optimized. But, whatever the configuration is, it is not possible to reach an induction as intense as the one created by the structure C without using big magnets. The reason is that the external rings of the structure D are strictly compensation rings.

As a result, the structures D, with several magnetizations, lead to very compact devices that have a very uniform induction over a range that can be chosen. They allow the miniaturization of the device as well as a linear functioning, since a small coil easily remains in a uniform induction. The structure D and the structure B are well adapted to small displacements loudspeakers, but, thanks to their large z uni , they are not limited to these loudspeakers.

IV. LARGE DISPLACEMENTS LOUDSPEAKERS

When fed with low frequencies, the coil of a loudspeaker moves with large amplitudes. So, the motor has to have good performances for large coil displacements. The requirements are more demanding than in the previous structures, since nonlinear effects increase with large displacements. The ironless structures which are well adapted to these requirements, that is displacements larger than 20mm, are presented in the Fig. 23. Ironless motor structures for large displacements. The coil has an axial oscillating movement centered on the point 0. BB: two radially magnetized rings DD: two sets of three radially magnetized rings following section. Given the conclusions of the first part concerning small displacements, and according to our calculations, the only structures that are well adapted to large and linear displacements are those inspired from the structures B and D, as shown on Fig. 23. Structures inspired from the structures A et C cannot be adapted for large displacements. Indeed the axially magnetized rings do not have effect anymore on the total induction when these two rings are too far from each other.

Hence, this section considers motor structures which are constituted by two sets of magnet ring(s) separated by an airgap where the moving coil is located. Having one or two sets of ring(s) does not change the behavior of the device, especially the uniformity of the field. However the induction intensity is increased. The calculation method is the same, as the superimposition theorem applies. The induction is calculated at a radial distance a/2 = 2mm from the magnets.

A. Two permanent magnet rings radially magnetized.

Let us consider the structure BB of Fig. 23. This structure derives from the structure B. Of course the behavior of this structure is the same as the one of the structure B presented for small displacements. Fig. 24 illustrates the dimensioning of the device for large displacements.

For an arbitrary chosen magnet height of 40mm, and a length of 26mm, the induction is uniform over a range of ±12.5mm along the Oz axis around the point O with a variation of 1%: z uni = 25mm, which represents 63% of the height of the structure. The induction reaches 0.54T in front of the middle of the rings for a magnet magnetization of 1T . The section of one magnet ring is 10.4cm 2 . Note that the length L = 26mm is the optimal length for this configuration: with other lengths, z uni decreases.

B. Several magnetizations.

As it is natural to try to improve the performance of a device, the structure DD of Fig. 23 is considered now. It is derived from the structure D by concentrically placing two sets of three radially magnetized rings. Its behavior is the same as described in section III-D.

The previous structure BB of 40mm high and 26mm long is compared with the structure DD. First, the magnet volume is kept constant for both structures (Fig. 25). The chosen parameters for the structure DD are: H = 25mm, L = 30mm and J = 1T for the central rings, and h = 6mm, l = 24mm, and j = 1.07T for the external rings. With a smaller height, then with a larger length, the structure DD provides a more constant induction that reaches 0.58T . This is 7.5% higher than the induction intensity reached by the structure BB. z uni is larger than 25mm: it represents 71% of the total height of the structure. Thus, this structure DD is more competitve but it is important to note that its external rings need a magnetization j = 1.07T .

The geometry of the structure DD is now optimized in order to reduce its volume. The new parameters for the structure are: H = 25mm, L = 25mm and J = 1T for the central rings, and h = 5mm, l = 20mm, and j = 1.09T for the external rings. Fig. 26 compares this structure DD to the same previous structure BB. The created induction reaches 0.55T and z uni is still larger than 25mm. The total magnet section of one set of the structure DD equals 8.25cm 2 . This represents a 21% decrease compared to the structure BB. While the total volume of the structure DD is considerably reduced, its created induction is similar to the induction created by the structure BB, as much for its intensity as for its uniformity.

As for small displacements, the methodology for dimensioning the structure DD is rather simple: the maximal displacement of the coil gives the height of the central rings, if the induction is intended to be uniform over the whole displacement. Even in the case of large displacements, the optimization of the structure volume leads to compact devices that create very uniform inductions.

V. CONCLUSION

This paper presents several kinds of ironless structures of loudspeakers. The principal point of comparison between them is their ability to create a uniform induction. Based on this point, their study distinguishes the structures dedicated to small coil displacements and the ones that can also be dedicated to large coil displacements. While structures A and C should only be integrated in high frequency loudspeakers, it appears that structures B and D are more universal: their dimensions can be adjusted to correspond to small displacements (structures B and D) as well as large displacements (structures BB and DD).

Then, compromises are to be done between the difficulties in manufacturing the structure, its total magnet volume and of course the intensity and the uniformity of its created induction. The structure A is the simplest structure to realize. But the technologies advancement makes radial magnetizations become more and more common, so that structures B, C and D are easy to realize. The volume of the magnet structure is an obstruction criterion as well as a price criterion; the structure D well answers to both criterions. For efficient loudspeakers, the important point is the intensity of the induction. Both structures A and C create intense inductions in their reduced application domain. But the important point to obtain a good sound quality is to have a uniform induction over the whole excursion range of the coil. Whatever the range is, structures B and D, and then BB and DD, permit to obtain more accurate loudspeakers.

Fig. 1 .

 1 Fig. 1. Displacement-varying force factor Bl measured on the modeled loudspeaker (continuous line) and its model (dashed line).

Fig. 2 .

 2 Fig. 2. a. Measured sound-pressure spectrum of the loudspeaker supplied with a 5V sinusoidal voltage at 43Hz and b. calculated sound-pressure spectrum of the nonlinear model in response to the same excitation signal.

Fig. 3 .

 3 Fig. 3. Time variations of the voice-coil displacement in response to a 5V sinusoidal voltage at 43Hz, calculated with the linear model (dashed line) and with the model including displacement-varying force factor (continuous line).

Fig. 4 .

 4 Fig. 4. Time variations a. of the current running through the voice-coil and b. of the voice-coil displacement, in response to a 16V sinusoidal voltage at 43Hz, calculated with the linear model (dashed line) and with the model including displacement-varying force factor (continuous line).

Fig. 5 .

 5 Fig. 5. Sound-pressure spectrum of the nonlinear loudspeaker model to a two-tone excitation voltage at 43Hz (16V ) and 500Hz (20V ).

Fig. 6 .Fig. 7 .

 67 Fig.6. Asymmetrical measured force factor of the loudspeaker (dashed line) and symmetrical-made force factor (continuous line) as a function of the voice-coil displacement.

Fig. 9 .

 9 Fig.9. Sound-pressure spectrum of the nonlinear loudspeaker model whose force factor has a maximal variation of 30% in response to a two-tone excitation voltage at 43Hz (16V ) and 500Hz (20V ).

Fig. 10 .

 10 Fig.10. Sound-pressure spectrum of the nonlinear loudspeaker model whose force factor has a maximal variation of 20% in response to a two-tone excitation voltage at 43Hz (16V ) and 500Hz (20V ).

Fig. 11 .

 11 Fig.11. Sound-pressure spectrum of the nonlinear loudspeaker model whose force factor has a maximal variation of 10% in response to a two-tone excitation voltage at 43Hz (16V ) and 500Hz (20V ).

Fig. 12 .

 12 Fig.12. Sound-pressure spectrum of the nonlinear loudspeaker model whose force factor has a maximal variation of 5% in response to a two-tone excitation voltage at 43Hz (16V ) and 500Hz (20V ).

Fig. 13 .

 13 Fig.[START_REF] Jensen | Generation of highly uniform magnetic fields with magnetized wedges[END_REF]. Sound-pressure spectrum of the nonlinear loudspeaker model whose force factor has a maximal variation of 1% in response to a two-tone excitation voltage at 43Hz (16V ) and 500Hz (20V ).

Fig. 15 .Fig. 16 .

 1516 Fig.[START_REF] Maosan | Magnetic field analysis of rec devices with gradually varied magnetizations[END_REF]. Ironless motor structures for small displacements. The coil is located on the observation axis Oz and has an oscillating movement centered on the 0 point. A: two axially magnetized rings B: one radially magnetized ring C: one radially magnetized ring and two axially magnetized rings D: three radially magnetized rings

Fig. 17 .

 17 Fig.17. Radial magnetic induction, Br (T), along the observation axis at a distance a/2 = 0.3mm, created by the structure B for L = 9mm and for various values of the height: H = 1mm (bold line), then from top to bottom H = 3mm, 5mm, 7mm, 9mm (dot-dashed line) and H = 18mm (dashed line).

Fig. 18 .

 18 Fig.[START_REF] Furlani | A three-dimensonal field solution for radially polarized cylinders[END_REF]. Radial magnetic induction, Br (T), along the observation axis at a distance a/2 = 0.3mm, created by the structure C (bold line) with its central magnet with H = 5mm, L = 8mm, J = 1T (full line) and its two external magnets with h = 2mm, l = 8mm, j = 1T (dashed line).

Fig. 19 .

 19 Fig.[START_REF] Yonnet | Magnetomechanical devices[END_REF]. Radial magnetic induction, Br (T), along the observation axis at a distance a/2 = 0.3mm, created by the structure C as patented (dotted line) and by the structure C with lowered magnetizations (bold line); this structure uses a central magnet with L = 8mm, H = 5mm and J = 1T (full line) and two external magnets with l = 8mm, h = 2mm and j = 0.52T (dashed line).

Fig. 20 .

 20 Fig.[START_REF] Furlani | Field analysis and optimization of ndfeb axial field permanent magnet motors[END_REF]. Radial magnetic induction, Br (T), along the observation axis at a distance a/2 = 0.3mm, created by the structure D (bold line) with its central magnet with H = 5mm, L = 9mm, J = 1T (full line) and its two external magnets with h = 2mm, l = 9mm, j = 1.01T (dashed line).

Fig. 21 .

 21 Fig.[START_REF] Zhilichev | Calculation of magnetic field of tubular permanent magnet assemblies in cylindrical bipolar coordinates[END_REF]. Radial magnetic induction, Br (T), along the observation axis at a distance a/2 = 0.3mm, created by the structure D (bold line) with its central magnet with H = 5.2mm, L = 7mm, J = 1T (full line) and its two external magnets with h = 1mm, l = 8.5mm, j = 1.09T (dashed line).

Fig. 22 .

 22 Fig.[START_REF] Rakotoarison | Using coulombian approach for modelling scalar potential and magnetic field of a permanent magnet with radial polarization[END_REF]. Radial magnetic induction, Br (T), along the observation axis at a distance a/2 = 0.3mm, created by the structure D (bold line) with its central magnet with H = 4mm, L = 10.5mm, J = 1T and its two external magnets with h = 1.5mm, l = 10mm, j = 1.05T , and by the structure C as patented (dashed line).

Fig. 24 .

 24 Fig.[START_REF] Ravaud | Analytical calculation of the magnetic field created by Permanent-Magnet rings[END_REF]. Radial magnetic induction, Br (T), along the observation axis at a distance a = 2mm, created by the structure BB for H = 40mm and for various values of the length, from bottom to top, L = 6mm (bold line), 16mm, 26mm (dot-dashed line), and 36mm, and by the nearest face from the coil (dashed line).

Fig. 25 .

 25 Fig.[START_REF] Ravaud | Discussion about the analytical calculation of the magnetic field created by permanent magnets[END_REF]. Radial magnetic induction, Br (T), along the observation axis at a distance a = 2mm, created by the structure DD (bold line) with its central rings with H = 25mm, L = 30mm, J = 1T and its four external rings with h = 6mm, l = 24mm, j = 1.07T , and by the structure BB with H = 40mm, L = 26mm, J = 1T (dashed line).

Fig. 26 .

 26 Fig.[START_REF] Ravaud | Time-varying non linear modeling of electrodynamic loudspeakers[END_REF]. Radial magnetic induction, Br (T), along the observation axis at a distance a = 2mm, created by the structure DD (bold line) with its central rings with H = 25mm, L = 25mm, J = 1T and its four external rings with h = 5mm, l = 20mm, j = 1.09T , and by the structure BB with H = 40mm, L = 26mm, J = 1T (dashed line).

TABLE I THIELE&

 I SMALL PARAMETERS OF THE MEASURED AND MODELED LOUDSPEAKER.


S Farizon

T D Ouaskit

Märk

J Tabet

S Eden

S Feil

H Abdoul-Carime

B Farizon

M Farizon

S Ouaskit

T D Märk

Absolute molecular flux and angular distribution measurements to characterize DNA / RNA vapor jets

Absolute molecular flux and angular distribution measurements to characterize DNA / RNA vapor jets

Introduction

The experimental study of biomolecular ionization can provide insights to further our understanding of radiation damage mechanisms in living material. Indeed a significant goal in radiobiology is to characterize the ionization events which can damage key biomolecules (e.g. DNA, RNA, and their constituents) [1]. This represents an important step towards an elucidation of the physical interactions which can initiate the sequences of chemical, biochemical, and cellular processes associated with harmful physiological effects. From this perspective, Monte Carlo track structure codes provide a valuable framework to study initial radiation damage mechanisms on the molecular scale [2]. Such programs require a complete set of cross sections for particle-biomolecule collisions in order to simulate localized energy deposition and the transport of primary (incident energetic species) and secondary particles (products of ionization and fragmentation events) in a representative cellular environment. However, absolute cross section measurements are rare for biomolecules which require heating (or another technique such as laser desorption or MALDI) to produce suitable gas-phase targets. The scarcity of reliable absolute cross sections for interactions with sublimated biomolecules is primarily due to the difficulty in determining the target thickness traversed by a projectile beam intersecting a vapor jet from an oven. This parameter depends on (i) the respective positions of the oven and the incident beam, (ii) the total flux of target molecules as a function of the oven temperature, and (iii) the angular distribution of molecules in the jet. Accordingly the present work provides new experimental measurements of the total flux and angular profiles of DNA and RNA base jets (adenine, cytosine, thymine, and uracil) from a Knudsen-type oven. Sublimation enthalpies calculated using the total flux measurements are in good agreement with the literature and the observed angular profiles are consistent with theoretical calculations based on the dimensions of the oven's capillary exit. The Lyon oven design has been used in a wide range of crossed-beam experiments on sublimated DNA / RNA bases [3][4][5] so the present jet characterization may be applied to normalize previous relative measurements. More generally, a full description is provided of the procedure to calculate the thickness of a DNA / RNA target jet intersected by an incident projectile beam.

Theoretical description of a DNA / RNA base vapor jet

2.1. Equilibrium vapor pressure and total flux

The oven system comprises a cylindrical reservoir with a capillary exit to produce a jet with high directivity (see section 3). As the orifice cross section is ~1% of the reservoir cross section, the rate at which molecules escape through the capillary is very low compared to the frequency of collisions with the oven walls. Therefore, at a temperature T, the evaporative system reaches an equilibrium state characterized by a vapor pressure P e which is close to the saturation pressure P s . In these conditions, the thermodynamic properties of the vapor flow can be described using the Knudsen [6] method based on the kinetic theory of rarefied gas valid at equilibrium pressures < 10 Pa. Accordingly, in a closed container with a gas pressure P s , the flow of molecules per unit area can be determined using the equation:

(eq. 1)
where n is the number of molecules per unit volume and the velocity of the evaporated molecules is given by: (eq. 2)

where k B is the Boltzmann constant and m is the molecular mass. The vapor flow emerging from an ideal orifice of area A 0 can then be calculated as follows:

(eq. 3) 2 1 8         = m T k v B π ( ) e B o P T k m A dt N d 2 1 2 / - = π 4 v n dt dN =
If we consider the capillary effect, the expression for the vapor flow must take into account the transmission probability given by the Clausing factor C of the tube capillary [7,8]. At a given temperature, the equilibrium vapor pressure can then be determined indirectly by measuring the total mass flow from the oven.

Vapor pressure and consequently temperature are the key parameters for the study of jet properties. Accordingly, we can distinguish three flow regimes characterized by the Knudsen number K n = λ 0 /L which is related to the capillary tube dimensions (L, d) and the molecular mean free path λ 0 as follows:

(eq. 4)

where σ is the effective diameter of the molecule (~ 6 -7 Å for a DNA / RNA base [9]); πσ The transition from molecular to viscous flow is defined as the intermediate regime with a mean free path which is similar to or smaller than the capillary dimensions: λ 0 ≤ d < L. In this case, the flow is characterized by the competition between intermolecular collisions and collisions with the oven walls. The description of this regime is based on the introduction of an effective length:

(eq. 5)

e B P T k . 2 0 2 1 σ π λ = L L eff λ 2 =
Thus it is possible to take into account both the capillary dimensions and the molecular density. Following Troitskii's formula [9,10], the angular distribution is given by: (eq. 6) (eq. 7)

Where f(u) and f(τ) are error functions:

(eq. 8)

(eq. 9)

The original theoretical contribution of this work is to combine Troïtskii's [10] description of the angular distribution with the Knudsen [6] method for calculating the total flow.

Experimental methods

Total flow measurements

The gas-phase DNA / RNA bases which have been studied in the present experiments are uracil, adenine, thymine, and cytosine. Previous studies indicate that minimal thermal decomposition or isomerization of these DNA / RNA bases occurs between 375 and 475 K [11], thus we expected negligible effects in the temperature range 400-500 studied in the present work. Guanine, the remaining DNA base, was not studied in this work due to the reported difficulty of sublimating to a high vapor pressure without significant isomerization and / or thermal decomposition [12]. The DNA / RNA base powder samples were purchased from Sigma-Aldrich (minimum purity 99%).

) ( cos ) 0 ( ² ² 2 3 u f dt d N d dt d N d θ Ω = Ω 0 θ θ ≤ 0 θ θ ≥ ) ( cos ) 0 ( ² ² 2 3 τ θ f dt d N d dt d N d Ω = Ω θ cos eff L L u = θ θ τ sin cos eff L d =
The vapor jets studied were obtained by heating the DNA / RNA base powder in a custom-made oven. As mentioned above, this oven consists of a closed cylindrical powder reservoir (copper; internal diameter 10 mm) with a capillary exit (copper; length L = 35 mm, orifice diameter d = 1 mm).

The reservoir cylinder was heated by a resistive filament ribbon and mounted in a ceramic outer casing for insulation. A thermocouple was inserted tightly in a hole in the base of the insulated reservoir cylinder. The DNA / RNA powder was heated progressively, with fine tuning of the heating power supply, until a stable sublimation temperature T was reached. The stabilization of the temperature was necessary to work in equilibrium state evaporation conditions. The DNA / RNA base vapor pressure was sufficiently high to form an intense neutral molecular beam at an oven temperature of 400 -500 K.

The total vapor flux emerging from the capillary was determined indirectly by measuring the weight loss Δm of the oven for a given evaporated base powder during a time interval Δt. Figure 1 shows the system used to measure the mass flow from the oven at equilibrium temperature. The evaporated DNA bases were condensed onto a cold plate target during an exposure time of several hours. The evaporative system and the collecting target were mounted in a vacuum chamber maintained at approximately 10 -6 mbar. Each deposit was carried out in stable temperature and steadystate evaporation conditions. The weight of deposit was deduced from the difference between the weights of the collecting plate target before and after vapor exposure.

Because of the limited acceptance of the collecting system, we have measured systematically the percentage of the collected weight relative to the initial powder weight in the oven. The mean value of this percentage was Ψ = 69.4 ± 1.6 % (the uncertainty is the standard deviation of all the measurements).

Angular flux measurements

Knowledge of the angular distribution of the jet is essential in order to determine the target thickness in a crossed-beam experiment. In our previous study of a gas jet target [13] we have demonstrated that in the viscous flow regime the angular jet profile is independent of the pressure.

Similarly, the angular distribution depends only on the geometry of the capillary in the molecular flow regime [8,10,[14][15][16]. In the case of intermediate flow (equations 6-9), however, the angular dependence is included in the error functions which depends on the effective length (L eff ). Therefore, it is necessary to study the effect of this angular dependence as a function of the oven temperature.

Figure 2 shows photographs corresponding to a thin and thick deposit of uracil obtained at the same sublimation temperature and for two exposure times, respectively 1h30 and 8 mn. The wavelength and order of interference rings visible in the photograph of the thin deposit are directly related to the thickness of condensed DNA / RNA bases. Thus the angular vapor flux pattern can be extracted from the optical profile. Photographs were taken with using CCD camera in order to store the interference pattern digitally. Using an image processing program developed in MATLAB, each tint was decomposed into its red, green, and blue (RGB) components so that pixels could be identified by a virtual matrix in the RGB space. (eq. 10) (eq. 11)

where n represents the optical index and e(x) represents the local thickness.

In order to determine the wavelength of the G component used in the camera, the well-known black cross calibration method was applied. An interference pattern was generated using a spath plate mounted between two crossed polarizers. The precise wavelength of the camera G component could then be calculated by plotting the optical path difference induced by the spath plate against the interference order k. The linear fit gives a wavelength λ(G) of 550.15 ± 0.15 nm.

Results and discussion

Total flow as a function of temperature

λ δ k x e n = = ) ( 2 2 1 2 2 / ) ( ) ( λ δ + = = k x e n
The measured mass deposition rates are plotted in Figure 4 as a function of the sublimation temperature T for the four DNA/or RNA bases studied. Two distinct regimes are identified for lower and higher temperature corresponding to molecular and viscous flow, respectively (see Table 1).

According to Table 1, we can verify that a large part of our measurements are in the intermediate regime except the case of uracil molecules where the measurements concern the three flow regimes.

Figure 5 presents the same data on a log-log scale. In the present conditions the thermo-dynamical properties of vapor flow are determined by the Knudsen method based on the kinetic theory of rarefied gas and applicable for equilibrium pressure P e < 0.1 mbar.

2 / 1 2         = b o t M T R A C D P π (eq. 12)
Where A 0 is the surface area of the effusion orifice, C the Clausing coefficient of the capillary, T the temperature of the measurements, R the gas constant, M b the molar mass of the nucleobase, and D t = m/t (mass evaporation rate where m the mass of nucleobase sublimated over time t). Thus the present D t (T) data has been used to determine the nucleobase temperatures which correspond to the vapor pressure ranges associated with molecular and viscous flow (Table 1). The present total mass flow measurements were carried out in the following temperature ranges: 390-454 K for thymine, 400-493 K for uracil, 414-452 K for adenine, and 428-478 K for cytosine. The Table 1 shows that these ranges correspond principally to the intermediate flow regime. However the uracil and thymine data also overlap with the viscous regime (above 471 and 446 K, respectively).

The molar enthalpy of sublimation ∆H sub can be extracted from the vapor pressure and temperature using the Clausius-Clapeyron equations.

        ∆ - = T R H A T P sub exp ) ( (eq. 13)
where A is a constant.

Thus ∆H sub was determined by generating a least-squares-fit to the Clausius-Clapeyron relation expressed in terms of the quantities measured in the present experiments:

( ) T R H R M A C Ln T D Ln sub b o t ∆ -         = π 2 2 / 1 2 / 1 (eq. 14)
Table 2 compares the present ∆H sub (based on the absolute mass measurements, AMM) for uracil, cytosine, thymine, and adenine with previous values determined using caloric (C), gas saturation (GS), Langmuir vaporization (LE), Mass Effusion (ME), Quartz Resonator (QR), Head Space Analysis (HSA), and Torsion Effusion (TE) methods. The uncertainties are also derived from the least-squares-fit and are mainly due to the experimental method of determination of the evaporated mass rate D t . The agreement of our measurements with available data is a validation test for our method of extracting vapor pressure from the mass loss rate D t using the Knudsen method.

Target thickness determination from theoretical profiles

In order to determine the absolute target thickness and thus derive absolute cross sections in a crossed beam experiment, both total and local vapor flux measurements are required. The target thickness ε at temperature T is given by the integral density n(x,y,z) over the incident beam -vapor jet overlap distance (x):

(eq. 15) where I(x,y,z) is the normalized beam profile which is assumed to be Gaussian, or uniform when the width of the incident beam is smaller than that of the vapor jet. Figure 6 shows a schematic view of the geometrical parameters used to define the vapor jet and thence derive its thickness along the trajectory of an incident beam. The flux density is the number of molecules entering the elementary surface dS per second at a given position.

∫ = dx z y x I y x n y T ) , , ( ) , , ( ) , ( 0 ε
Ω = d r dS 2 v r n r r dt dS N d r j ) , , ( ² ) , , ( ϕ θ ϕ θ = = (eq. 16)
is the density of vapor molecules (number of molecules per unit volume dr.dS) and is the molecular velocity (equation 2).

In order to obtain a high target thickness in recent proton impact experiments using the present oven system [3,[START_REF] Tabet | [END_REF], the incident beam axis was located at a similar distance (y ~ 1 mm) above the capillary to the diameter of the orifice. Taking into account the results of the total flux measurements and the angular distributions, the target density can be approximated by the combination of a uniform density directly above the capillary orifice (cross sectional area A 0 ) with the angular distribution law of intermediate regime flow (equation 7).

Target thickness determination from the optical interference profile

As mentioned above, the present measurements concern two complementary parameters: the total flux (dN/dt) total and the angular flux (dN/dΩ dt). The total flux is determined by measuring the oven weight loss ∆m during the time ∆t by weighing the target plate before and after exposure to vapor. The angular flux distribution and consequently the density n(R,ϕ,z) are expressed in cylindrical coordinates. Consider an elementary surface (dS = dRdϕ) of the interference pattern (Figure 6), where j(R) denotes the vapor flux density. The number of molecules deposited on this surface element, dN d , is given by: (eq. 17)

This can be also expressed as a function of the deposit thickness e d (R): (eq. 20) where η is the geometrical factor related to the fraction of collected vapor compared to the total flowing from the capillary: η = 0.694 ± 0.014 (see section 3.1). The relation between the measured deposit D t and the local thickness deposit e(R) for a given position R is given by: (eq. 21)

(eq. 18) v ) , , ( ϕ θ r n ϕ d dR R t R j dN d ∆ = ) ( ϕ ρ d dR R R e M N dN
Thus equations 16 -20 provide a complete description of the total flux deduced from D t and the angular flux related to the optically measured local thickness e(R). Figure 7(a) shows the profile of the deposit thickness e(R) for uracil evaporated at ~425 K obtained by interference rings analysis (Section 3.2). Figure 7(b) shows the same profiles obtained by varying the sublimation temperature and normalizing to the maximum thickness. It should be noted that the width of the profile increases with the sublimation temperature. Indeed this behavior has been observed previously [14,16,[START_REF] Pollard | [END_REF] in the intermediate regime. The comparison with the theoretical profiles demonstrates that this effect is related to the L eff parameter and the competition between molecular and viscous flow.

Figures 8 and9 include also theoretical profiles calculated using equations 6 and 7 for the present experimental conditions at various sublimation temperatures. In the upper part of Figure 8, both and

M N K A d ρ = ) ( ) ( R e K R j = t d dt dN dt dN t N       =       =       ∆ ∆ η t A s S D M N RdR R Ke dR R R j η π π = = ∫ ∫ ) ( ) ( 2
2
experimental and theoretical profiles are compared to the theoretical upper limit where intermolecular collisions in the capillary are neglected (f(u) = 1 and f(τ) = 1). These profiles have been obtained when the capillary is at 26 mm from the deposit surface (see Figure 1).

The lower part of Figure 8 shows the experimental and the theoretical profiles F 0 c when the error functions f(u) and f(τ) are taken into account. The error function depends on the length and diameter of the capillary. The theoretical profiles IF 0 c reported in the lower part of Figure 8 are obtained by integrating F 0 c over the capillary dimensions in Cartesian coordinates. Naturally, the calculated upper limits of F 0 and IF 0 are equal in the upper and lower parts of the figure since F 0 does not depend on the capillary geometry. Figure 9 shows the same results plotted for six uracil sublimation temperatures corresponding to the intermediate flow regime.

The experimental results and the theoretical model are in good agreement both quantitatively and qualitatively. The dependence on the temperature is also described well. This indicates that the analytical expression of the theoretical model used in this study can be applied for absolute determination of both the total flux and angular distribution of DNA / RNA base jets with good accuracy.

Conclusion.

The DNA -base vapor-jet has been studied in the intermediate flow regime, corresponding to Knudsen number 0, 1 < K n <10. The heat of the oven containing DNA -base powder has been adjusted in such manner to obtain stationary flow for all the studied temperature intervals. At a given temperature the corresponding pressure is determined assuming Knudsen hypothesis and knowing the transmission probability of associated capillary Clausing factor. This allows to relate temperature and pressure to total flux which is directly measured by a weighing method of the evaporated base powder quantity. The absolute flux density depending on the angular distribution of vapor molecules has been also studied experimentally using a method of optical interference. The measured profiles are in good agreement with analytical law given by Troïtskii in the intermediate regime, taking into account both (cosθ) 3/2 angular dependence and the error function dependence on the free path of vapor molecule collisions. The absolute target thickness of the jet could be extracted with an accuracy of 10% resulting mainly from errors in the relative positions of the beam axes.



  2 represents the collision cross section for the given DNA/RNA base molecule. The molecular regime for K n > 10 or P e ≤ 0.26 Pa, dominated by molecules traversing the capillary without intermolecular collisions  The intermediate regime for K n > 0.1 or P e ≤ 100 Pa  The viscous regime for K n > 0.001 or 100 Pa ≤ P e ≤ 2600 Pa, dominated by intermolecular collisions along the capillary

Figure 3

 3 Figure 3 represents the intensity variation of the green (G) component along the interference ring

  Equations 17 and 18 express the linear dependence between the angular flux density and the deposit local thickness. The following expression can thus be deduced: (eq.19) and e(R) = e d (R) / ∆t with The angular flux distribution is derived directly from the optical analysis of the interference rings. The total flow (dN/dt) total is determined from the measured mass deposition rate during steady-state evaporation.

Figure 1 :Figure 3 :

 13 Figure 1: Schematic view of the DNA / RNA base vapor deposit set-up

Figure 4 :

 4 Figure 4: Mass deposition rates of DNA / RNA bases as a function of sublimation temperature

Figure 5 :Figure 6 :Figure 8 :

 568 Figure 5: DNA / RNA base mass deposition rates plotted against sublimation temperature on a log-log

  Thickness of the deposit and calculated functionDistance from the center (mm)

Table 1 :

 1 The vapor pressure ranges for molecular, intermediate, and viscous regimes presented with the corresponding sublimation temperature ranges of DNA / RNA bases in a Knudsen-type oven

	T= 411K
	T= 425 K
	T= 430 K
	T= 439 K
	T= 454 K
	T= 468 K
	F 0
	I F 0

Table 2 :

 2 Enthalpy of sublimation for nucleobases: our results are compared to data compiled by Chikos and Acree[17].

Thin deposit

Thick deposit