J Theor Appl Phys DOI 10.1007/s40094-017-0252-1
RESEARCH
Photodetachment cross-section evaluation using asymptotic considerations
1,2 1,3 Philippe Babilotte • Mickael Vandevraye
Received: 20 April 2017 / Accepted: 8 May 2017 Ó The Author(s) 2017. This article is an open access publication
Abstract Mathematical calculations are given concerning Keywords Other topics in mathematical methods in the evaluation of the negative ions photodetachment cross- physics Á Metrology Á Determination of fundamental section r, into a so-called saturation regime. The interac- constants tion between a negative ion particle beam and a laser beam is examined under theoretical aspects. A quantitative cri- terion S is proposed to define the saturation threshold Introduction between the linear and the saturated domains, which are both present in this saturation regime. The asymptotic Sustainable non-carbon energy systems, renewable energy behaviours extracted at the low and high energy limits are systems and energy systems that reduce production of used to determine this threshold quantitative criterion S and waste have been identified as domains of interest in the to evaluate also the photodetachment cross-section r. The transition of energy [1]. Hydrogen is the most abundant case of a symmetric gaussian photodetachment laser beam element in the universe, and can be typically considered as shape is examined according to the proposed formalism, the fuel for any potential fusion reactor. This non-hydro- which can be used either for the photo-detachment or carbon energy source requires nevertheless to solve many photo-ionization processes, and could be potentially used scientific and engineering challenges. into technological solutions for negative ion neutralisation Several dozens of fusion reactors are now scattered processes (such as neutral beam injector) in the future around the globe at various stages of completion. The fusion energy devices. Estimations onto the errors related biggest system is the International Thermonuclear Experi- to the use of this methodology are given. mental Reactor (ITER). It is being built in France by an international consortium at an estimated cost of $ 20 billion and should be completed by 2027. Other smaller entities Experimental and theoritical works performed in: Laboratoire Aime´ work also on the fusion concept, such as General Fusion Cotton, UPR 3321 CNRS, LAC, Universite´ Paris Sud XI, 91405 (Vancouver, Canada), Helion Energy (Redmont, Wash- Orsay Cedex, France ington, USA), or Tri Alpha Energy (California, USA) [2]. The nuclear fission involves splitting 235U atoms into Electronic supplementary material The online version of this article (doi:10.1007/s40094-017-0252-1) contains supplementary smaller atoms, thus releasing a large amount of energy. On material, which is available to authorized users. the negative side, it requires a finite resource (235U), expensive (initially and under run-condition) and poten- & Philippe Babilotte [email protected] tially hazardous power plants (Three Mile Island, Cher- nobyl, Fukushima accidents), and it produces significant 1 Laboratoire Aime´ Cotton, UPR 3321 CNRS, LAC, Universite´ quantities of toxic waste that stay hazardously radioactive Paris Sud XI, 91405 Orsay Cedex, France for centuries. At the opposite, the fusion has several sig- 2 UMR 6303 CNRS, ICB, Universite´ de Bourgogne Franche- nificant advantages; first, it runs on common standard Comte´, UBFC, 21078 Dijon Cedex, France elements as hydrogen, which is very abundant, non toxic 3 French Ministry of Education, Paris, France 123 J Theor Appl Phys and non-fossil. In addition, the fusion reactors cannot melt spinning vortex of liquid metal is created, and in which down, and can be stopped immediately (no possible plasma is injected into the empty center (squeezing) [5–9]. uncontrollable chain reaction). Lower quantities of The future fusion reactor requires therefore to manage a radioactive wastes are produced, and fusion could be seen, space charge neutralization of negative ions (negative ions therefore, as a cleaner power source than the existing ones. DÀ species neutralized into D species, as proposed by Schematically, this technology requires the creation of a CYBELE project [10], Eq. 1), which occurs in the Neutral plasma to force the atomic nuclei to fuse. This plasma is Beam Injector (NBI) [4] of the fusion reactor [12]. created under extreme heating and extreme pressure con- As a first option, negative ion beam neutralization could ditions to induce the cloud of free-range electrons and be realized by injecting a gas into the neutralizer cell, but nuclei (T 100 Â 106 C). Confinement and control of the under a low efficiency level around 55% (corresponding to plasma stability remain nevertheless points of weakness of an electric efficiency of the heating system limited to 34%) the fusion process. [13]. Another possible option is the photoneutralization The first method for fusion is the one used by the process [12, 14, 15] that can be seen as an attractive alter- National Ignition Facility (NIF). The NIF is located in San native [16], as no gas is injected into the medium to neu- Francisco (USA, Lawrence National Laboratory, $ 5 billion tralize the negative ions (less parasitic particles produced, no project cost) and integrates around 192 UV laser beams plasma formation in the cell). The neutralization rate is focused onto a gold cylinder full of a D þ T mix, during a estimated around 95%, [17], enhancing in this case the 20 ns, inducing a 500 trillion Watt single shot release. It electrical efficiency of the heating system up to 60% [16]. corresponds to a specific Inertial Confinement Fusion (ICF DÀ þ hm ! D þ eÀ ð1Þ process), as the cylinder (containing D þ T) target simul- taneously explodes and implodes. Neutralization using photodetachment enhanced by several The second method to get fusion is the most common Fabry–Perot cavities have been also invoked [18]. For all method, and consists of magnetically controlled plasma, these photoneutralization processes, it requires to have contained in a metallic doughnut, wrapped in electromag- reliable absolute photo-detachment cross-section r values, netic coils, which contains and compresses the plasma. In as this parameter is used afterwards in the different 1957, in the ZETA fusion reactor (UK), tests were con- numerical models, for instance, in the Space Charge ducted for the first time concerning a magnetic field that Models (SCM) [19]. Mathematical models, behaviour laws was used to spatially confine atoms. In 1968, doughnut- and physics development in the photodetachment processes structured systems appeared (Tokamak Russian process). [12, 14, 20] are also greatly required. Afterwards, in the 1970s, three large-scale tokamaks were Into the present work, benchmark calculations are then developed in Princeton (USA), in Japan, and in the UK. In given concerning the determination of the absolute cross- 2008, the ITER ‘‘proof of concept’’ (and afterwards, the section r into a photo-neutralization process, considering DEMO technological project) [3, 4] was started (300 m the overlap between a laser beam and a negative ion beam. tall, 23,000 tons weight, 105 km of niobium-tin wire for It could be seen as an implementation to the existing magnets, 840 m3 expected plasma volume, $ 20 billions mathematical methodology. This photodetachment config- cost). Nevertheless, the existence of instabilities in the uration used into the present work reproduces a gas-free plasma due to the spiral orbits of particles around the line photoneutralization configuration (space charge neutral- of electric current (particles in the plasma moving in tight ization), probably involved in the future energy generator spiral orbits) induced loss of heat, and alternative techno- demonstrator DEMO [3, 21]. The overlap of the laser sheet logical solutions emerged [5–9, 11]. and the DÀ negative ion beam takes place outside the Another (third) method could be, therefore proposed, nuclear island, in the same configuration that is given in the consisting of creating a single collisional plasma, created present work (overlap between the laser and the negative using two nose-to-nose cannons (‘‘firing’’ around ion outside any ionising radiation conditions). 106 km hÀ1), firing plasmas straight to each other into a The absolute photodetachment cross-section r of nega- central chamber (collisional regime). Positioned around the tive ions quantifies the equivalent surface exposed and central chamber, six Neutral Beam Injector (NBI) intro- available to interact with the photons of the laser beam, duces H atoms at the edges of the spinning cloud (stabi- corresponding therefore roughly to an interaction proba- lization keeping the medium hot). In this type of bility of the particle with the beam. technology, the plasma auto-generates the magnetic field The simplest methodology to determine r into pho- that confines it (‘‘Field Reverse Configuration’’). todetachment experiments (2) consists of using a direct Another (fourth) prototype of fusion reactor is the mag- calculation (3), introducing the number N of neutral atoms À netized target fusion (General Fusion project [2]), in which a A, the number N0 of initially illuminated negative ions A ,
123 J Theor Appl Phys and the number of photons per surface unit and per time The photo-ionisation cross-section can be also determined unit (flux) U, for a total duration of illumination T, con- using this saturation methodology, since the pioneering sidering the probability P of interaction: experiments and estimations proposed by Ambartzumian AÀ þ hm ! A þ eÀ ð2Þ et al. [24], or Heinzman et al. [25]. Optical transitions can be efficiently characterized using this process [36–41], as well as N P ¼ r  U  T ) r ¼  ðÞT  U À1 ð3Þ ions densities and plasma temperature [39, 42]. Nevertheless, N0 this methodology requires a comparison between a reference The absolute cross-section r can be also evaluated using a so- saturation curve (typically the OÀ reference) and the mea- called ‘‘saturation methodology’’, which is generally more sured datapoints, motivating to propose new approaches precise than the direct evaluation methodology [22]. It con- avoiding any comparison processes. sists of measuring the evolution of the number N of photo- Specifically, in the present work, from two asymptotic induced events, changing the light energy E of the laser considerations, at high and low photodetachment laser irra- overlapping the negative ions beam. When this energy is high diation energy E, and using the relevant physical hypothesis, enough (above the saturation energy ES, that is to say if we a scalar parameter S is introduced as a quantitative criteria consider the case in which E [ ES), the photodetachment characterizing the threshold between unsaturated and satu- probability [23] or the photo-ionization probability [24, 25] rated regimes present in the photo-excitation process. The converge on 1 (saturation event, Fig. 1). The pioneering expression of the photodetachment cross-section r will be experiments concerning the photodetachment of negative given, without considering any fitting iterative process ions, and involving this saturation process are proposed by between a reference curve and the measured datapoints. Hall et al. [26] and by Hotop and Lineberger [27]. This satu- In the commonly used experimental configuration ration method has been extensively used afterwards (Fig. 1), a single-mode high power (continuous or pulsed) [22, 27–30]. The mathematical model initially developed in laser beam is used to irradiate a negative ion particle beam, 1973 [27] has been afterwards improved by Balling et al. [30]. to avoid a possible ‘‘bar-code’’ interaction effect poten- This saturation methodology has been also used to evaluate tially induced by the use of a multimode laser (irregular the absolute photo-detachment cross-section of trapped ions temporal profile) [14, 27, 43]. [31, 32], or into the case of a three-photon set-up [33]. The modelized situation considered in the present work Several mathematical models have been proposed to consists of two-actors interaction, e.g. between a photon hm modelize the saturation effect and to extract r from the and a negative ion particle AÀ. The elementary excitation measurements, as Cervenan and Isenor [34], considering probability dP to photodetach one electron from a negative immobile neutral species interacting with laser beam, or ion AÀ (Fig. 1) during the elementary duration dt of illu- Boulassier [35], considering a gaussian profile interacting mination is given by: homogeneously distributed negative ions, or Blondel et al. dP ¼ rðkÞÂU  dt ð4Þ [33]. in which rðkÞ corresponds to the photodetachment cross- section at the laser irradiation wavelength k. The neutral species density variation dn, found in an elementary volumic element dV centered onto a point M moving with the negative ions beam is given by:
dn ¼ rðkÞÂUðM; tÞÂðn0 À nðtÞÞ Â dt: ð5Þ We assume for (5) the uniformity of the ion volumic dis- tribution. We assume a constant value for the initial density À n0 of ions A in the (yOz) plane (Fig. 3). Considering the two normalized spatio-temporal trans- z verse gaussian laser intensity f(x, y) and gðt À cÞ, which are supposed to be identical along the pulse, we can express the flux U: E z UðM; tÞ¼Uðx; y; z; tÞ Â f ðx; yÞÂgtÀ : ð6Þ Fig. 1 Typical photodetachment experimental set-up. Basic pho- hx c toneutralization of negative ions corresponding to an experimental proof. Laser source (LS). Focalisation lenses (Lf). Electrostatic Two conditions of normalization are used, in accordance Deviation Plates (EDP). Fluence Tuning Block (FTB). Rayleigh with considerations proposed by [30], related to these two distance (zR) of the laser beam. Energy E measured after the distribution functions f(x, y) and g(t): interaction of hm with AÀ [14] 123 J Theor Appl Phys Z Z x;y x;y;z f ðx; yÞdxdy ¼ 1 ð7Þ N ¼ nðx; y; zÞ dx dy dz: ð12Þ Z t It gives therefore: gðtÞdt ¼ 1: ð8Þ Zþ1 Zþ1
The flux expression given in (6) allows to rewrite the N ¼ n0L expression (5) in: 0 À1 8À1 91 < Zþ1 = E z E dnðtÞ¼rðkÞÂ Â f ðx; yÞÂgtÀ Âðn0 À nðtÞÞ @ r 0 0 0 A 1 À exp À f ðx þ v À t ; yÞgðt Þ dt dx dy: hx c : hx A ; Â dt: À1 ð9Þ ð13Þ Temporally integrating (9), we find that: The integration according to the z variable is direct as the 0 8 91 pw2 Zþ1 0 < = Rayleigh area zR k (Fig. 2) of the photodetachment @ E z A n ¼ n0 Â 1 À exp ÀrðkÞ Â f ðx; yÞÂgtÀ dt : laser beam (waist w ) is supposed much larger than the : hx c ; 0 À1 diameter L ¼ 2RAÀ of the negative ion beam, if RAÀ is the ð10Þ radius of the negative ions beam profile (supposed cylin- drical), as we have zR L. If vAÀ is the negative ion velocity along the x-axis (Fig. 2) 2 Consequently, from L ¼ y2 þðLðyÞÞ2 (Fig. 2), the length and if we consider the modification x ! x þ vAÀ t and 4 2 t0 ¼ t À z, the density of neutral species produced in a L(y) of the negative ion beam in the propagation direction z c can be considered constant for any altitude y (LðyÞL, given volume of the negative ion beam, located at t0 ¼ 0in Fig. 3), and we assume that a direct integration in (11)is (x, y, z) after its passage inside the laser pulse, can be possible according to the z coordinate. expressed by: This approximation concerning the geometry of the n ¼ n0 0 8 91 interaction area (Fig. 2) is true for the usual photodetach- < Zþ1 = ment experiments performed [14], considering a typical @ E 0 0 A Â 1 À exp:ÀrðkÞ f ðx þ vAÀ t; yÞgðt Þ dt; : waist w0 70 lm, L 2RAÀ 2 mm (typical case of a hx À À1 1nA H ion beam), for a typical laser beam at 1064 nm ð11Þ for instance (20 Hz, laser pulse width s 30 ns), inducing zR 14 mm L 2 mm, and considering no particular The ratio n=n0 corresponds to a photodetachment proba- diffraction phenomenum of the laser beam. bility. The total number N of neutral species photo-pro- In this direct integration consideration, using the nor- duced during the interaction between the laser and the malization conditions (7) and (8), a direct proportionality particle beam is afterwards obtained integrating the density onto the whole space, for a given k (single radiation), that is to say considering r rðkÞ, and considering t as a dummy-variable:
Fig. 3 Typical saturation curve and interception value ES. Typical data for HÀ: k ¼ 1064 nm, v ¼ 479;210 km hÀ1 (typical HÀ particle Fig. 2 Lateral view of the interaction area beam), w0 ¼ 80 lm, s ¼ 20 ns (typical experimental values) 123 J Theor Appl Phys behaviour between N and E appears at the low energy limit, Zþ1 rE using a Taylor expansion (for E variable): 2 2 2 Gðx; yÞÀQ exp À 2 ½ðx þ vtÞ þ y hx w0 Zþ1 Zþ1 Zþ1 À1 rE 0 0 0 t2 N ¼ n0L  f ðx þ vAÀ t ; yÞgðt Þ dt dx dy 2 hx  exp À 2 dt S: ð19Þ À1 À1 À1 s rE ! n0L : ð14Þ Considering the calculations into the Appendix 1, and hx v2s2 introducing the non-dimensional parameter q ¼ 2 0, w0 In other words, 8E ! 0, we have N ! p1E, introducing and replacing Q by its value, we obtain from (19): the following quantity: rE 1 1 2y2 Lr ffiffiffiffiffiffiffiffiffiffiffi ÀGðx; y; tÞ¼  2  p  exp À 2 p1 ¼ n0  : ð15Þ hx pw 1 þ q w hx 0 0 2x2 1 : At the other energy limit (E !1), the total population of  exp À 2  w0 1 þ q neutral species photo-produced (Fig. 1) can be expressed: ð20Þ 0 N ¼ n0VS þ a À a : ð16Þ Above the saturation threshold, from (19) and (20), a In this expression, a corresponds to the number of neutral geometrical condition (21) can then be extracted, intro- species produced into a collisional thermal regime (inde- ducing the real function Y(E) and the parameter pendently from any laser-matter interaction). a0 corre- 2 2 2 2 2 d ¼ w0  ðÞ¼1 þ q w0 þ v s : sponds to the residual number of negative ions undetached 2 2 2r E into the saturated volume itself (traducing the non-effi- x2 þ y2 Ln  YðEÞ: ð21Þ 2 w2 Spw d hx ciency of the laser-particle interaction). VS corresponds to d 0 0 the saturation volume. In a first approximation, we assume All the (x, y) couples of geometrical points concerned by 0 a n0VS and a n0VS, inducing then N n0VS. the saturation regime into the overlapping volume are At high energy (E !1), the saturation starts into the defined by the surface (21). Introducingqffiffiffiffiffiffiffi the two parameters volume (VS) in which the flux is high. Other zones of the a and b, defined by a ¼ d  YðEÞ (along the x axis) and overlapping area, in which the saturation is not effective, qffiffiffiffiffiffiffi 2 stay locally into the linear regime described in (15), but a b ¼ w  YðEÞ (along the y axis), V can be therefore global convergence to the total saturation is nevertheless 0 2 S expressed in: considered in this case. VS is determined by geometrical considerations from the condition Gðx; yÞS,ifS corre- L 2r VS ¼ pabL w0pd  LnE þ Ln : ð22Þ sponds to a scalar number. G(x, y) corresponds to the 2 Spw0hxd argument of (10), for an ion located at the (x, y) position. It traduces that above a given value of the argument, the N n0VS is therefore seen as a function depending from saturation exists. the LnE variable, and (22) can be rewritten under NðLnEÞ¼p  LnE þ d, if: We introduce then Q ¼ Qt  Qxy, and we consider the 2 two normalization conditions (7) and (8) related to the p p ¼ n Lw d ð23Þ transverse laser intensity profile given in (6). The negative 2 0 2 0 ion velocity field is considered constant along the x geo- p 2r 1 metrical coordinate (supposed mono-axis velocity field d ¼ n0 Lw0d  Ln  : ð24Þ 2 Spw0d hx according to x), and we consider for simplification vAÀ ¼ v: rffiffiffi VS should be positive and real, and the expression given in 2 2 1 2 À2t À t gðtÞ¼   e s2 Q  e 2s2 ð17Þ (22) induces then: s p t Spw0d À 2 ÂfðxþvtÞ2þy2g À 2 fðxþvtÞ2þy2g E E ¼  hx: ð25Þ 2 w2 w2 S f ðx; y; tÞ¼  e 0 Q  e 0 : 2r pw2 xy 0 As a consequence of (23) and (24), (22) can be rewritten in: ð18Þ p E The condition onto the argument G(x, y)of(10) can be VS ¼ L   w0d  Ln : ð26Þ 2 ES given by the inequality Gðx; yÞS, developed in:
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d The ratio is independent from the initial density n0 of p2 At the saturation threshold, the laser energy E equalizes negative ion, which is unprecisely known: the critical energy ES, and it comes: d 2 pffiffiffiffiffiffiffiffiffiffiffi 2r 1 p w0 ¼ Ln  : ð27Þ LnðESÞÀLn   1 þ q  hx þ c ¼ 0: ð34Þ p2 Spw0d hx 2 r According to (25) and (27), the saturation energy Using (30), the threshold criteria S of saturation is finally threshold ES is expressed by: given by: d S ¼ eÀc: ð35Þ LnES À : ð28Þ p2 It can be possible then to quantitatively define the saturation The p2 quantity, obtained from (15) and from (23), p1 threshold, into a photo-ionisation or a photodetachment permits to define the cross-section r: process, for a gaussian shape. Therefore, the observation of p1 p the saturated regime allows to determine the photodetach- r ¼   hx  w0d: ð29Þ p2 2 ment cross-section, avoiding to use a comparison method- ology [22, 24–34, 36–42, 44] between two saturation curves. It exhibits the possibility to extract the photodetachment The Eq. (29) and the reading of E (Fig. 3), through the cross-section value from the two asymptotic behaviours, S asymptotic E !1behaviour, allows to a direct evaluation characterized by p (corresponding to E ! 0) and p 1 2 of r [14]. For instance, for the negative ion HÀ, using (corresponding to E !1), and to define quantitatively a k ¼ 1064 nm, v ¼ 479;210 m sÀ1,1 w ¼ 80 lm, s ¼ 20 ns, transition limit between the linear domain and the satura- 0 E ¼ 0:025 mJ, and c ¼ 0:577215, the obtained value of r tion domain, through the parameter S: S is in accordance with all the previous evaluations for HÀ p1 pw0d hx given in the literature [14, 45]. S ¼  ES ) r ¼ S  : ð30Þ p2 2 ES The error onto the final result can be quantified using the Fig. 4. The error onto the E value is therefore given by The mathematical expression for S should nevertheless be S DES 2 2 Dr ¼ r  . In Fig. 4, previous parameters are used, v s ES specified. From (19), introducing q ¼ w2 , integrating only 0 introducing ES ¼ 0:025 mJ, and estimating the impact of a over the temporal term, using x ! x0 ¼ pffiffiffiffiffiffiffix (symmetriza- 1þq Æ0:5 mJ error onto the cross-section value (value of ES 2r pffiffiffiffiffiffiffi1 E À5 À5 tion), introducing the parameter B ¼ 2   hx, the varying from 1:5  10 to 3:0  10 onto the abscissa pw0 1þq neutral species total number is then evaluated by: axis, corresponding to this uncertainty). ÆDr is then pffiffiffiffiffiffiffiffiffiffiffi directly read onto the vertical axis in Fig. 4. N ¼ n0L  1 þ q Zþ1 Zþ1 x02 þ y2 1 exp B exp 2 dx0 dy:  À À  À 2 Conclusion w0 À1 À1 ð31Þ In the present work, a new approach proposes to measure the photo-excitation cross-section r, which is a basic physical At the limit E !1, considering the asymptotic consid- quantity independent from any experimental conditions, and eration N ! n0VS at high energy, replacing B by its value, and considering the detailed calculations given in Appen- characterizing an interaction process between two entities. dix 2, we obtain: The cross section is basically linked to the collision proba- bility P per unit time t for an incident flux U of particles to pffiffiffiffiffiffiffiffiffiffiffi 2 dP pw0 impact a target, and is basically defined using ¼ r  U. N ! n0L  1 þ q  dt 2 The first (old) approach uses directly r ¼ N  1  1, 2r 1 1 N0 T U ÂþLnðEÞþLn  pffiffiffiffiffiffiffiffiffiffiffi  þ c requiring the knowledge of the number N of illuminated pw2 1 þ q hx 0 0 atoms (resp. N for the excited atoms), during the total ð32Þ illumination time T, for an incident flux U. This approach Using (22) and (23), we find from (32) that: could be unprecise, considering possible underlying non- linear effects, targets potentially exposed to different illu- pffiffiffiffiffiffiffiffiffiffiffi 2 pw0 U T d ¼ n0L  1 þ q  minations , or considering a time of illumination that 2 could be different according to the spatial coordinates, or 2r 1 1  Ln  pffiffiffiffiffiffiffiffiffiffiffi  þ c : ð33Þ pw2 1 þ q hx 0 1 Typical orders of magnitude are found from 5  105 msÀ1 (accel- erated ions) to 1 km sÀ1 (trapped ions) [32]. 123 J Theor Appl Phys
Using two temporal and spatial profiles, it comes R QrE þ1 2 2 2 2t2 hx  exp À 2 ÂððxþvtÞ þy Þ Â exp À s2 dt À1 w0 S. It conduces to the intrinsic criteria S ¼ eÀc, which quantitatively defines the saturation threshold, if c is the Euler–Mascheroni constant. This proposed new approach has the strong advantage to avoid comparison between laser-matter interaction curves.
Acknowledgements The authors would like to thank Dr. Ch. Blondel Fig. 4 Numerical estimation of the error induced onto the r final and Dr. Cyril Drag for fruitful discussions, and their precious help value (ÆDr). Parameters used for the simulation: w0 ¼ 80 lm, during experiments onto HÀ, and for overall project guidance. À1 s ¼ 20 ns, k ¼ 1064 nm, v ¼ 479;210 m s , ES ¼ 25 lJ asymptoti- cally determined (at ÆDES) in Fig. 3, ÆDES estimated on Fig. 3 at Open Access This article is distributed under the terms of the Æ0:5 lJ Creative Commons Attribution 4.0 International License (http://crea tivecommons.org/licenses/by/4.0/), which permits unrestricted use, considering possible errors onto counting efficiencies itself distribution, and reproduction in any medium, provided you give (N and N). appropriate credit to the original author(s) and the source, provide a 0 link to the Creative Commons license, and indicate if changes were The second approach to evaluate r use the criteria r  made. UT ¼ 1 at 63% for a saturation process defined under P ¼ 1 À expðÀrUTÞ. This approach (Fig. 5) does not Appendix 1 require to evaluate N and N0, but U and T remain to be measured anyway. We simplify the following expression to evaluate the The new approach developed into the present work integral present into (19): proposes to use the asymptotic behaviours present intrin- Zþ1 sically into the saturation process and to develop a calcu-R 2 2 2 2 2t lation based onto the density n ¼ n0 Âð1 À expfÀr exp À 2  ðÞx þ vt þy  exp À 2  dt w0 s UðtÞdtgÞ of photo-excited ions, considering a laser beam À1 based onto a f(x, y) spatial profile and a g(t) temporal ð36Þ profile, and considering the ion velocity v, conducing to R This infinite integral (36) contains the exponential function n ¼ n Âð1ÀexpfÀr þ1 E Âf ðxþvt;yÞÂgðtÞdtgÞ. Con- 0 À1 hx of a second order polynomial form, which can be given by sidering a standard Gaussian beam, after volume integra- 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi expfÀa2t þ a1ðxÞt þ a0ðxÞg. The three following a0ðxÞ, tion, using d ¼ w2 þv2s2, it comes the expression of the 0 a1ðxÞ and a2 expressions can be then considered: p total number of particles N, given by N ¼ n0 2 v2 1 4vx 2 Lw2dÂfLnE ÀLnðpw0dhxÞþcÀEiðÀBÞg. Asymptotically, a ¼ 2  þ ; a ðxÞ¼À ; a ðxÞ¼À  x2 0 2r 2 w2 s2 1 w2 0 w2 this total number of ions converges to 0 0 0 p 2 pw0dhx ð37Þ N ¼ n0 2 Lw0dÂfLnE ÀLnð 2r Þþcg. Defining the satu- pw0dhx The following integral identity will be then used, consid- ration energy LnES ¼ Lnð 2r Þþc, it conduces then to peÀc hx rEQ the expression of the cross-section r ¼  Âw0d, ering k ¼ : 2 ES hx introducing the two asymptotic parameters p1 (for E ! 0) Zþ1 rffiffiffiffiffi and p (for E !1). In the present study, the saturation 2 p 2 k  expfÀa2t þ a1t þ a0gdt¼ k  volume VS is defined cylindric, exhibiting an elliptic base. a2 À1 Using a composite profile defined by a2 UðM; tÞ¼ E  gðt À zÞÂf ðx; yÞ, and a Gaussian form of  exp 1 þ a hx c 4a 0 these two (temporal and spatial) profiles gðt À zÞ and f( 2 c 38 x, y), it is possible to define a criteria S, which corresponds ð Þ to a quantitative threshold criteria between the saturated Appendix 2 regime and the unsaturated regime (linear-saturated tran- sition limit). This is achieved consideringR the argument of Due to the symmetry considerations, using a radial function rE þ1 0 2 2 02 02 the exponential, through hx Âf À1 f ðx þ vt; yÞÂgðtÞ R of x and y, defined by R w0 ¼ 2 Âðx þ y Þ,(31)is ÂdtgS. changed in:
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As B [ 1, and considering B / E, we can consider the following inequality between integral behaviours, into the high energy E asymptotic domain:
ZÀB ZÀB et dt\ etdt eÀB ð43Þ t À1 À1
ÀB Therefore, if E !1, we have limB!1fe g¼ ÀE limE!1fe g¼0, and consequently: pffiffiffiffiffiffiffiffiffiffiffi pw2 N ! n L  1 þ q  0 ÂfþLnB þ cgð44Þ 0 2
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