A quantitative study of some sources of uncertainty in opacity measurements
AA quantitative study of some sources of uncertainty inopacity measurements
Jean-Christophe Pain and Franck Gilleron CEA, DAM, DIF, F-91297 Arpajon, France
Abstract
Laboratory (laser and Z-pinch) opacity measurements of well-characterized plasmas provide datato assist inertial confinement fusion, astrophysics and atomic-physics research. In order to test theatomic-physics codes devoted to the calculation of radiative properties of hot plasmas, such experi-ments must fulfill a number of requirements. In this work, we discuss some sources of uncertainty inabsorption-spectroscopy experiments, concerning areal mass, background emission, intensity of thebacklighter and self-emission of the plasma. We also study the impact of spatial non-uniformities ofthe sample.
In the introduction to the article “Opacity calculations: past and future” in 1964 [1], H. Mayer writes“Initial steps for this symposium began a few billion years ago. As soon as the stars were formed, opacitiesbecame one of the basic subjects determining the structure of the physical world in which we live”. Heis also the author of a famous report [2] written in 1947 and presenting many theoretical methods foropacity calculation.Since Mayer’s 1947 report, there have been few comprehensive reviews of methods for opacity cal-culations, although several reviews of limited scope have appeared, focusing on specific aspects [3]. Inearly papers, the authors were mainly concerned with photo-ionization [4–7]. Many interesting summaryarticles about atomic opacities from the astrophysicists viewpoint were published [8–13]. Penner andOlfe discussed atomic and molecular opacities as applied to atmospheric reentry phenomena [14] and theopacity of heated air was the subject of work by Armstrong et al. [15, 16] and Avilova et al. [17, 18].Proceedings of three opacity conferences were published by Mayer [1], Huebner et al. [19] and Adelmanand Wiese [20]. Rickert [21] and Serduke et al. [22] summarized workshops in which opacities were com-pared. Efforts to calculate opacities and the underlying equations of state (EOS) have been renewedby the Los Alamos group, the Livermore group, and the opacity project at University College Londonand the University of Illinois. While the first two groups cover the entire range of atomic opacities, thelast one concentrates on detailed EOS and opacities of light elements for stellar envelopes in the hightemperature 3 10 ≤ T ≤ K and low density regions of the astrophysical plasma domain [23–26].The low temperature limit avoids the presence of molecules and the high density limit is chosen so thatthe isolated atom or ion remains a reasonably good approximation.Seventy-one years have passed, and despite the progress made, many researchers are still working onradiative opacity, due to its applications in the fields of inertial confinement fusion, defense and astro-physics. Several experiments were performed during the past three decades, but the recent experiment onthe Z machine at Sandia National Laboratory (SNL), dedicated to the measurement of the transmissionof iron in conditions close to the ones of the base of the convective zone of the Sun [27], reveals that ourcomputations may be wrong or at least incomplete.The two main facilities used for opacity measurements are lasers and Z-pinches. High-power lasersand Z-pinches can be used to irradiate high- Z targets with intense X-ray fluxes which volumetrically [email protected] a r X i v : . [ phy s i c s . p l a s m - ph ] D ec eat materials in local thermodynamic equilibrium (LTE) to substantial temperatures. These X-rayfluxes produce a state of high-energy density matter that can be studied by the technique of absorptionspectroscopy. In such experiments, also known as pump-probe experiments, the X-ray source creatingthe plasma is expected to be Planckian.The radiative quantities which are measured are the reference I ,ν (unattenuated radiation intensity)and I ν , the radiation attenuated by the plasma. ν is the photon frequency. The transmission of thesample is given by T ν = I ν I ,ν , (1)and the knowledge of T ν and of the areal mass of the sample enables one to deduce the opacity (seesection 2).In laser experiments, the sample is heated by the X rays resulting from the conversion of the energyof laser beams focused inside a gold Hohlraum, and measurement of absorption coefficients in plasmasmay be done for instance using the technique of point-projection spectroscopy, first introduced by Lewisand McGlinchey in 1985 [28]. The technique involves a small plasma produced by tightly focusing a laseron a massive or a fiber target to create a point-like X-ray source with a high continuum emission usedto probe the heated sample. It was first used to probe expanding plasmas [29, 30] and applied to probea radiatively heated plasma for the first time in 1988 [31]. The point-projection spectroscopy techniquecan be used to infer the plasma conditions and/or its spectral absorption.Z-pinch experiments mentioned above proceed as follows [32–36]. The process entails accelerating anannular tungsten Z-pinch plasma radially inward onto a cylindrical low density CH foam, launching aradiating shock propagating toward the cylinder axis. Radiation trapped by the tungsten plasma forms aHohlraum and a sample attached on the top diagnostic aperture is heated during a few nanoseconds whenthe shock is propagating inward and the radiation temperature rises. The radiation at the stagnation isused to probe the sample.For a quantitative X-ray opacity experiment, great care must be taken in the preparation of theplasma, as the latter must be spatially uniform in both temperature and density. Masses and dimensionsof the sample must be well-known. In order to compare with LTE opacity codes (see for instance [37–40]),it is crucial to ensure that the plasma is actually in LTE. Quantitative information on the opacity canbe obtained only if the following requirements are satisfied [41–43]: • (i) The instrumental spectral resolution has to be sufficiently high to resolve key line features andmeasured accurately prior to the experiments. • (ii) Backlight radiation and tamper transmission have to be free of a wavelength-dependent struc-tures. • (iii) Plasma self-emission has to be minimized. • (iv) The tamper-transmission difference has to be minimized. • (v) The sample conditions must be uniform, achieving near-local thermodynamic equilibrium. • (vi) The sample temperature, density, and drive radiation should be independently measured. • (vii) Measurements should be repeated with multiple sample thicknesses. • (viii) Both quantities I ,ν and I ν must be measured during the same experimental “shot” togetherwith the plasma conditions.The lack of simultaneous measurement of plasma conditions and absorption coefficient is a weaknessof most absorption measurements. For example, some experiments rely on radiative-hydrodynamics sim-ulations to infer the plasma temperature and density, while other provide measurements of temperature,density and absorption spectrum, but on different shots. However, even if many experimental teams2evote lots of efforts to perform simultaneous measurements, the inferred quantities are always knownwith a limited accuracy.There are many sources of uncertainty: areal mass of the sample, background radiation, intensityof the backlighter, plasma temperature and density, etc. Strictly speaking, some of error quantificationdepends on platform (such as self-emission). Opacity-measurement uncertainty is challenging because itconsists of three sources of errors that are complicated in different ways: • (i) transmission error, • (ii) areal density error, and • (iii) temperature and density errors.In addition, there are multiple sources for each category. When there is a lateral areal-density non-uniformity, effective areal density (and its error) become transmission dependent. Therefore, areal-densitynon-uniformity must be treated either as transmission error or as a special category.In the present work, we investigate some uncertainty sources in absorption spectroscopy measurements.In section 2, we show how the relative uncertainties on the transmission and on the areal mass ρL arerelated to the uncertainty on the opacity. The question we want to answer is: if we seek a particularvalue of the relative precision on opacity ∆ κ ν /κ ν , knowing the relative uncertainty on the areal mass∆( ρL ) / ( ρL ), which precision ∆ T ν on the transmission do we need? As an example, we impose therequirement ∆ κ ν /κ ν =10 %. The uncertainties on the background emission and on the self-emission ofthe sample are discussed in section 3. We chose to discuss the latter areal-mass non-uniformities ina special category: different sources of uncertainty due to defects in the areal mass are examined insection 4: wedge shape, bulge (concave distortion), hollow (convex distortion), holes in the sample andoscillations. In section 5, we address the issue of the temperature and density uncertainties. Section 6 isthe conclusion. For a homogeneous and optically thin (non-emissive) material, the transmission is related to the opacityby the Beer-Lambert-Bouguer law [44–46]: T ν = e − ρLκ ν , (2)where ρ is the density and L the thickness of the material, κ ν its spectral opacity and T ν its transmission. ρL is the areal mass. Formula (2) is valid for ρLκ ν (cid:46) i.e. T ν (cid:38) . µ = µ meas − µ true , from a single measurement, which can go either positive or negative. On the contrary,uncertainty can be defined as interval (or width) of likelihood where a measured value could fall in.Usually, an uncertainty σ given by an analysis or measurement represents a width of Gaussian probabilitydistribution where the true value can be found. For example, if a measurement found µ meas ± σ meas , thetrue value µ true can be any value but its likelihood follows the Gaussian probability distribution definedas 1 √ πσ meas exp (cid:34) − ( µ − µ meas ) σ (cid:35) . (3)This is why it is usually considered that the true value exists within the measured µ ± σ for 68 % of thetime. 3t is tempting to split the error propagation ∆ T ν → ∆ κ ν in two separate steps: ∆ T ν → ∆ τ ν ( τ ν = ρLκ ν being the optical depth) and then ∆ τ ν → ∆ κ ν . The conversion from ∆ T ν to ∆ τ ν iscomplicated by nature due to their non-linear relation and T ν dependence. There are two main sourcesof ∆ T ν : • (i) miscalibration between unattenuated intensity to attenuated intensity and • (ii) background subtraction error. Plasma self-emission ( i.e. , sample and tamper) can be consideredas a special case of background (see Sec. 3).The second phase is the conversion from ∆ τ ν to ∆ κ ν . This conversion is mathematically much lesscomplicated than the ∆ T ν → ∆ τ ν one, and only areal-density errors ∆( ρL ) have to be quantified. Forexample, if ρL is perfectly known, the percent errors on τ ν and κ ν are the same. If ρL is underestimatedby 10 %, opacity κ ν (= τ ν / ( ρL )) is (additionally) overestimated by 11 % ( i.e. , 1/0.9 ≈ T ν (or ∆ τ ν ). For example, if τ ν is overestimated by 10 % due to transmissionerror (whatever the source is), 10 %-underestimated ρL ends up in giving 1.1/0.9=1.22, which ends upin 22 % overestimate in opacity. So, the impact of areal-density error can be separately computed fromtransmission (or optical-depth) error.The quantity of interest in fine is opacity κ ν , which is considered to be a function of areal mass ρL (measured by specific techniques such as Rutherford back-scattering for instance) and transmission T ν . Actually, T ν is not measured directly; the quantities that are measured are transmitted intensity I ν , backlighter intensity I ,ν , electron density n e , electron temperature T , etc.). Therefore, the properway to study propagation error would be to consider κ ν = f ( n e , L, T, I ν , I ,ν , etc . ). In order to simplifythe problem, we gather all these variables into two ones: areal mass ρL and transmission T ν = I ν /I ,ν .Error propagation must absolutely be performed using κ ν = f ( ρL, T ν ) and not T ν = f ( ρL, κ ν ) (the latterprocedure gives unrealistic uncertainties).Carrying out the same measurement operation many times and calculating the standard deviation ofthe obtained values is one of the most common practices in measurement uncertainty estimation. Eitherthe full measurement or only some parts of it can be repeated. In both cases useful information can beobtained. The obtained standard deviation is then the standard uncertainty estimate. If q = f ( x, y ), thepropagation of uncertainty reads [47]: σ q = (cid:115)(cid:20) ∂f∂x (cid:21) ( σ x ) + (cid:20) ∂f∂y (cid:21) ( σ y ) , (4) i.e. if q = x + y : σ q = (cid:113) ( σ x ) + ( σ y ) . (5)In our case, q = κ ν , x = ρL , y = T ν , f ( x, y ) = − ln( y ) /x and Eq. (4) becomes σ κ ν κ ν = (cid:115)(cid:18) σ ρL ρL (cid:19) + (cid:18) T ν σ T ν T ν (cid:19) . (6)Uncertainty estimates obtained as standard deviations of repeated measurement results are called A-typeuncertainty estimates. If uncertainty is estimated using some means other than statistical treatment ofrepeated measurement results then the obtained estimates are called B-type uncertainty estimates. Thelatter represent upper bounds on the variable of interest and have no statistical meaning. Therefore, ifthe quantity z has the value z with a relative uncertainty of ∆ z/z , it means that z lies between z − ∆ z and z + ∆ z , where ∆ z is the absolute value of errors on any quantity z and thus positive. The othermeans can be e.g. certificates of reference materials, specifications or manuals of instruments, estimatesbased on long-term experience, etc . The propagation of error bounds of a quantity q depending on twoindependent variables x and y is 4 q = (cid:12)(cid:12)(cid:12)(cid:12) ∂f∂x (cid:12)(cid:12)(cid:12)(cid:12) ∆ x + (cid:12)(cid:12)(cid:12)(cid:12) ∂f∂y (cid:12)(cid:12)(cid:12)(cid:12) ∆ y . (7)If q = x + y , then ∆ q = ∆ x + ∆ y . One has therefore∆ κ ν κ ν = − T ν ∆ T ν T ν + ∆( ρL ) ρL . (8)Since Eq. (8) is the simple derivative, this is just a propagation of error while Eq. (6) is the propagationof uncertainty stricto sensu . To differentiate these two better, let us say we want to find f , which is aknown function of x and y , and we measure x and y . Then, there are three cases: • If x and y are known perfectly, f ( x, y ) is perfectly known. • If errors in x and y are known perfectly ( i.e. , ∆ x and ∆ y , respectively), you can correct the errorsin f ( x, y ) perfectly using Eq. (8). • If only uncertainties of x and y are known ( i.e. , σ x and σ y but not actual errors ∆ x and ∆ y ), onecan only compute the likelihood interval (or the probability distribution) of true f using σ f foundwith Eq. (6).Sometimes, it is possible to know perfectly the uncertainty of one of the two variables x or y , and tohave an upper bound for the other. In such a case, neighther Eq. (6) nor Eq. (8) are relevant for thepropagation of errors or uncertainties. In this work, we therefore choose to always use Eq. (8) whichis more restrictive than Eq. (6). Using Eq. (8) one gets, in terms of error bars, the error on spectraltransmission ∆ T ν = − T ν ln( T ν ) (cid:18) ∆ κ ν κ ν − ∆( ρL ) ρL (cid:19) . (9)In that case, errors represent upper bounds on the variables. The same result can be obtained usinginequalities ( i.e. assuming that transmission lies between T ν − ∆ T ν and T ν + ∆ T ν and that areal masslies between ρL − ∆( ρL ) and ρL + ∆( ρL )). The corresponding calculation is provided in Appendix A,since it might be helpful for didactic reasons (the connection between mathematical differentiation andpositive variations ∆ is hidden and not easy to understand). Eq. (9) shows also that the error on theareal mass must not exceed the required precision on the opacity, which can be easily understood.One can also “invert” the formula (9) in order to express the transmission that should be sought inorder to ensure a given value of ∆ T ν : T ν = − ∆ T ν (cid:16) ∆ κ ν κ ν − ∆( ρL ) ρL (cid:17) W (cid:18) − ∆ T ν [ ∆ κνκν − ∆( ρL ) ρL ] (cid:19) , (10)where W is Lambert’s function ( w = W ( z ) is the solution of we w = z , see Fig. 2) [48–54]. TheLambert function is named “ProductLog” in the Mathematica R (cid:13) software. Table 2 contains uncertaintiesmentioned in several publications about absorption-spectroscopy measurements over the past decades(see Refs. [32, 55–66]). For the enigmatic iron experiment on Z [27], as well as for the more recentmeasurements on chromium, iron and nickel [36], the relative uncertainty on the areal mass ∆( ρL ) / ( ρL )was close to 4 % (estimated from Rutherford back-scattering). In a recent paper devoted to the ongoingNational Ignition Facility (NIF) experiment on iron [67], Heeter et al. estimate the relative uncertaintyon the areal mass ∆( ρL ) / ( ρL ) ≈ W in black in Fig. 2) implies that: W − ∆ T ν (cid:104) ∆ κ ν κ ν − ∆( ρL ) ρL (cid:105) ≥ − T ν (cid:104) ∆ κ ν κ ν − ∆( ρL ) ρL (cid:105) ≤ e , (12)which means that the largest acceptable value of ∆ T ν corresponds to T ν = 1 /e ≈ .
37. If we wanta precision ∆ κ ν /κ ν of 10 % on the opacity, assuming a relative uncertainty of 7 % on the areal mass,formula (9) implies that, for a transmission T ν = 0 .
4, a precision of 0.011 is required on the transmission,which corresponds to ≈ et al. was a bit overestimated.If ∆ ρL is perfectly known, in order to achieve ∆ κ ν /κ ν = 10%, Eq. (9) becomes∆ T ν T ν = − . × ln( T ν ) . (13)Assuming ∆ ρL (cid:54) = 0 ends up in tightening the ∆ T ν / T ν measurement. For ∆( ρL ) / ( ρL ) = 7%, Eq. (9)becomes ∆ T ν T ν = − . × ln( T ν ) . (14)Figure 3 represents ∆ T ν / T ν as a function of T ν in both cases (13) and (14).We note that if we had used Eq. (6) for uncertainty propagation, in order to achieve ∆ κ ν /κ ν = 10%with ∆( ρL ) / ( ρL ) = 7%, we would have obtained, for T ν = 0 .
4, ∆ T ν = 0 . i.e. Since Eq. (8) reflects a simple derivative, it is interesting to check the error propagation using MonteCarlo simulations. For given values of the transmission T ν , as well as of the relative uncertainties ∆ T ν / T ν and ∆( ρL ) / ( ρL ), we have generated two sets of random numbers uniformly distributed, namely X i and Y i such as 1 − ∆( ρL ) ρL ≤ X i = ρ i L i ρL ≤ ρL ) ρL (15)and 1 − ∆ T ν T ν ≤ Y i = T i T ν ≤ T ν T ν . (16)Therefore, setting κ i = − ln T i / ( ρ i L i ), one has∆ κ ν κ ν = max random i (cid:18) ∆ κ ν κ ν (cid:19) i (17)with (cid:18) ∆ κ ν κ ν (cid:19) i = | κ i − κ ν | κ ν = (cid:12)(cid:12)(cid:12)(cid:12) − X i (cid:18) ln Y i ln T + 1 (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) , (18)independent of the choice of ρL and κ ν . In the present case, we have chosen T ν =0.4. Fig 2 shows therelative uncertainty on the opacity as a function of the transmission T ν for ∆( ρL ) / ( ρL )=7% and variousvalues of ∆ T ν / T ν . As can be seen, in order to get a precision of 10 % on the opacity for a transmission T ν =0.4, a precision of 2.75 % is required on the transmission. This confirms the value obtained mentionedabove. 6 Background signal, backlighter characteristics and self-emission
Let us consider a wavelength-independent backlight signal I ,ν ≡ I (the effect of wavelength-dependentstructures was studied by Iglesias [68]). Assuming a background radiation of intensity b ν in a restrictedspectral region, we have ˜ T ν = I ν − b ν I − b ν . (19)The quantity ˜ T ν represents the true transmission and T ν is the apparent transmission, i.e. the transmis-sion defined by Eq. (2) without substracting the background δ . Expanding the latter expression up tosecond order yields˜ T ν ≈ T ν (cid:18) − b ν T ν I (cid:19) (cid:18) b ν I (cid:19) = T ν (cid:18) b ν I (cid:20) − T ν (cid:21) − b ν T ν I (cid:19) + O ( δ ) . (20)Therefore, at first order, we have ˜ T ν ≈ T ν (1 + (cid:15) ) − (cid:15), (21)where (cid:15) = b ν /I . As can be seen in figures 5, 6 and 7, for a transmission of T ν ≈ i.e. using Eq.(2) instead of Eq. (19)) would significantly misinfer the sample transmission. This background error isbigger when expected transmission value is low (see Figs. 5 and 7). This is the reason why it is requiredto repeat many experiments with varied sample thicknesses in order to measure strong lines at sufficientlyhigh transmission. It is important to mention that Eq. (19) can also be used to address the question ofhow the uncertainty ∆ b ν on the background b ν would impact the inferred opacity. Indeed, the modifiedtransmission can be rewritten ˜ T ν = I ν − ( b ν + ∆ b ν ) I − ( b ν + ∆ b ν ) = I (cid:48) ν − ∆ b ν I (cid:48) − ∆ b ν (22)with I (cid:48) ν = I ν − b ν and I (cid:48) = I − b ν . The radiative-transfer equation for stationary, homogeneous and non-diffusive material reads dI ν dx = − ρκ ν I ν + j ν , (23)where x represents the position along the line of sight of the spectrometer, I ν is the intensity of theradiation field and j ν the emissivity. For a plasma in LTE, using Kirchhoff’s law j ν = B ν κ ν , where B ν is the Planckian distribution B ν = 2 hν c e hνkBT − , (24)one gets the solution T ν = e − ρLκ ν + B ν I (cid:0) − e − ρLκ ν (cid:1) . (25)The transmission is therefore higher than the one predicted by Beer’s law. However, Eq. (25) holds fora point-like source with a time-independent emission in one direction [69], which is of course not repre-sentative at all of what really happens. The quantities B ν and I are usually given in erg/s/eV/cm /sr.7owever, the measured self-emission and backlight signals integrate B ν and I , respectively, over theiremitting area (observable from each point on the detector) and duration. Comparing these quantitieswithout the integrations has limited applicability. In reality, the measurements also integrate over smallenergy range and solid angle as well, but for many platforms, the integrations over these quantities havesimilar impacts on backlight radiation and self-emission. A point-projection method [75] has a greaterchance of self-emission contamination. For example, if a 2 mm sample foil is heated over 2.5 ns andbacklit by a 200- µ m backlight source over 300 ps, the ratio of sample-self-emission-to-backlight is closeto 1/3 (in terms of expected photons per mm , see Figure 5 of Ref. [70]). In the NIF experiment, there isno dedicated aperture [70], and thus, every point on the detector sees most of the emitting region. As aresult, there is a huge difference in emitting area as well as in duration, which signifies the self-emissioncontamination relative to the backlight radiation. On the other hand, the Z-pinch experiment performedat SNL [27, 36] uses a larger backlight area (800 µ m), and the detector’s view is limited by an apertureand a slit. As a result, the detector sees similar emitting surface areas for backlight radiation and sampleself-emission (see the red rectangle of Fig. 16(a) of Ref. [42]). In that way, the measurements are lesssubject to self-emission issues than the laser experiments. The durations of self-emission and backlightradiation are similar too, because the same source works as heating and backlight radiation. This mightbe the reason why NIF experiment suffers from self-emission and background, on the contrary to theSNL experiment. However, one has to be cautious; the background can also originate from other sources(such as Hohlraum itself, crystal second- (or higher-) order reflection, some sort of fluorescence or hardX rays, etc.). In order to quantify the impact of self-emission on the opacity measurements, one shouldconsider the differences in the emission areas as well as the integration over their time histories (see Sec.IV-C of Ref. [43]).It is worth mentioning that the point-projection method evoked in the introduction is not the bestmethod at high temperature due to this reason. Figure 8 represents microscopy views of several copper samples (before the experiment) used during arecent (2017) experimental campaign in “Laboratoire pour l’Utilisation des Lasers Intenses” (LULI) inFrance [71]. As can be seen, the surface of the sample is far from being perfect. Spatial non-uniformitiesof targets have been widely investigated in the past (see for instance the non-exhaustive list of refer-ences [72–74]). There are various possible modulations. In reality, some modulation that exists in thetarget fabrication is relaxed during the experiment, while some other modulation might be produced byhydrodynamics. In the present work, we focus on five spatial deformations of the target in two dimensions:wedge, bulge (convex deformation), hollow (concave deformation), holes and oscillations (modulationsof the surface), assuming that they are still present at the time of probe. Our goal is to find a simpleanalytical modeling of such non-uniformities in order to get a realistic idea of their respective impact.The areal mass, written ρL , can have defects in both directions x and y (we do not, for simplicity,separate the variations of ρ and L with respect to x and y but ρL is taken as a global quantity) and itsaverage reads (cid:104) ρL ( x, y ) (cid:105) = 1 ab (cid:90) a (cid:90) b ρL ( x, y ) dxdy, (26)where a and b are the dimensions of the sample in directions x and y respectively.Sometimes, the interpretation of an experimental spectrum reveals that the main structures have theright energy and relative intensities which seem consistent with the experiment, but the general level oftransmission is not satisfactory. Even if the areal mass of the sample is guaranteed by the manufacturerwith a good accuracy, it might happen that some variations of the areal mass occur, during the experiment.There might be a difference between the areal density used in the analysis and the true areal density. Inthe following, we assume that area-averaged areal density (cid:104) ρL ( x, y ) (cid:105) during the experiment is known and8efined as Eq. (26) and try to quantify how different types of lateral variations would affect the opacityinferred with (cid:104) ρL ( x, y ) (cid:105) .In the present work, ρL is considered as a function of lateral (or transverse) position x only (we stilldo not separate the dependence of ρ and L with respect to x and take ρL as a global quantity) and inboth cases, we preserve the average areal mass: (cid:104) ρL ( x ) (cid:105) = 1 a (cid:90) a ρL ( x ) dx = ρ L , (27) a being the transverse dimension of the target. For a perfect sample (no defects), we have˜ T ν = T ν = e − κ ν ρ L (28)and for a corrugated sample, we have ˜ T ν = 1 a (cid:90) a e − κ ν ρL ( x ) dx. (29)Equation (29) is correct only when backlight radiation is uniformly filling the observed sample area. ForNIF, it should not be a problem since backlight is only bright over 100 µ m (although it has a self-emissionissue). The concern is modulation over 100 × µ m backlit region. For SNL, it is a bigger concernsince backlighter is bigger (approximately 800 µ m). However, it should not be a serious problem becausethe measurement resolves in one direction and takes lineout only over brightest 300 µ m. Since it doesnot resolve in other direction, the modulation concern for SNL experiment is over 800 (backlight width) ×
300 (lineout width) µ m region. The convexity of the exponential function implies (cid:104) e − κ ν ρL ( x ) (cid:105) ≥ e − κ ν (cid:104) ρL ( x ) (cid:105) (30)and therefore, due to Eq. (27), we have ˜ T ν ≥ T ν . (31)Assuming constant areal mass, the presence of spatial non-uniformities or distortions of the sample tendsto make the foil more transparent. This is correct but may mislead the reader to think effective arealdensity is always lower. This is not true if the sample is tilted somehow by an angle θ , maybe due tomisalignment or some weird hydrodynamics. If tilt happens, the apparent areal density along the line ofsight is elongated by 1 / cos( θ ). We have considered a few distortions: • Wedge: in order to model a wedge-shape distortion of the sample (see Fig. 9), we use the linearform ρL ( x ) = ρ L (cid:16) − (cid:15) xa (cid:17) , (32)where (cid:15) controls the slope of the surface of the target. • Bulge: a concave distortion of the sample (see Fig. 13) is modeled here as ρL ( x ) = ρ L (cid:20) − (cid:15)a x ( x − a ) + 1 (cid:21) . (33) • Hollow: a convex distortion (see Fig. 17) is modeled here as ρL ( x ) = ρ L (cid:20) (cid:15)a x ( x − a ) + 1 (cid:21) . (34)9 Impact of holes: we choose to model the presence of holes (see Fig. 21) in the sample by thereplacement ρ L → ρ L (1 − (cid:15) ) , (35)where 0 < (cid:15) < ρ L to ρ L ).The different expressions of the modified transmission are summarized in table 3 and the derivationsare provided in Appendix B. The effect of the holes is stronger than the effect of the thickness modulations(25 % of holes yield the same result as 100 % of modulations). Nevertheless, to have a visible effect,one needs around 75-100 % of spatial modulations of the areal mass; it seems unrealistic to have such aperturbed hydrodynamic evolution and / or such a bad conception of the targets.The impact of the different defects, as (cid:15) varies, is illustrated respectively in Figs. 10, 11 and 12(wedge), Figs. 14, 15 and 16 (bulge), Figs. 18, 19 and 20 (hollow), Figs. 22, 23 and 24 (holes) and Figs.26, 27 and 28 (modulations). Values of (cid:15) required to obtain an uncertainty of 7 % on the areal mass (asin the NIF experiment [67, 75, 76]) are displayed in table 4.It is difficult for us to clarify what causes each type of non-uniformity (e.g., target fabrication, non1-D expansion during experiment, instabilities during experiments). Of course, if the temperature issufficiently high (which is the case in the laser or Z-pinch experiments mentioned above), the samplebecomes a plasma, and the inhomogeneities will not be the same as the ones before the experiment (inthe solid phase). For instance, one can imagine that the holes will be filled very quickly. In fact, theabove considerations imply that we consider the defects of the sample at the instant of the probe. The temperature and density errors do not contribute to the opacity measurement itself. They areimportant only when comparing with models. Even if temperature and density are off by 50 %, it wouldbe fine if the calculations were identical at the misinferred conditions. In fact, temperature and densityuncertainties are very different from ∆ T and ∆( ρL ) ones, and cannot be discussed in a general way. Thecriteria must depend on the level of model-data discrepancies. The relevant question for this uncertaintyis: can the observed discrepancy between measurement and modeling be explained by temperature anddensity uncertainties? Uncertainties in opacity calculations stem from the fundamental atomic cross-sections, plasma effectscaused by perturbing ions, computational limitations, etc. Measurements of fundamental cross-sectionsare usually carried out on neutral atoms, rather than on charged ions, due to the difficulty in preparinga sample in a specific ion stage and because of the myriad possibilities of excited levels. The problem isthat cross-sections of neutral atoms are more difficult to calculate accurately because of the many-bodyelectron-electron interaction. Thus, comparison of calculations with measured cross-sections for neutralspecies should provide an upper bound on uncertainties. Huebner and Barfield [3] estimate that: • (i) When scattering dominates (high temperature, low density), the uncertainty in the opacity is 5%. • (ii) As the density increases, free-free processes become more important, the uncertainty is less than10 %. 10 (iii) As the temperature decreases and bound-free processes become important, the uncertaintyincreases to 15-20 % and as the temperature decreases still further (photo-excitation can contribute),the uncertainty increases to 30 %.The calculated opacity error purely due to electron temperature and density errors is not just cross-section error, but it involves cross-sections (oscillator strengths) combined with error on quantum-statepopulations as well as on line broadening (in the case of photo-excitation) and edge broadening (in the caseof photo-ionization). Since we are mostly dealing with LTE opacities in the present work, the populationfactor reduces to a Boltzmann factor, and depends on the energies of the configurations (and thereforein particular on the way electron-electron interactions are taken into account). Line shapes depend onmany factors such as the atomic-physics basis used in the computation, the microfield distribution, etc.In the present subsection, we try to find upper bounds on temperature and density uncertainties requiredto ensure a given relative uncertainty on opacity; for that purpose we simplify the problem as much aswe can, and restrict ourselves to one process (the simplest one to model approximately), namely inverseBremsstrahlung. Kramers’ formula for inverse Bremsstrahlung reads: κ ν ≈ (cid:18) N A A (cid:19) π √ (cid:18) e π(cid:15) (cid:19) (cid:126) mc √ πmk B T Z ∗ ρ ( hν ) , (36)which can be put in the form κ ν ≈ CT − / ρ/ν yielding∆ κ ν κ ν ≈
12 ∆ TT + ∆ ρρ . (37)Of course, this is not realistic; in the experiments, our main goal is to study plasmas for which the photo-excitation is important. In general, it is difficult to find a simple scaling with density and temperature.If we require ∆ κ ν /κ ν ≈ .
1, relation (37) implies that∆
TT < × . . (38)and ∆ ρρ <
10 % . (39)Thus, if observed discrepancy is 10 %, T and n e need to be known better than 20 % and 10 %. This is acrude approximation, but since it is reasonable to assume that dκ ν /dT and dκ ν /dn e are similar betweenmodels than κ ν itself, such estimates should be relevant. Recently, Busquet [77] paid attention to the fact that under some circumstances, the transmission of a L-or M-shell weakly inhomogeneous plasma is identical to the transmission of a one-temperature plasma.This is clearly demonstrated in the case of an opacity varying linearly with the temperature. Indeed, if T ν = exp (cid:26) − (cid:90) y y κ ν [ T ( y ) , ρ ( y )] ρ ( y ) dy (cid:27) , (40)where T ( y ) and ρ ( y ) are temperature and density of the sample at depth y , y and y being the limits ofthe sample (see Fig. 29). Defining the average temperature ¯ T as¯ T = (cid:90) y y T ( y ) ρ ( y ) dy/ (cid:90) y y ρ ( y ) dy, (41)Busquet assumed a uniform density ρ (but this is not necessary), and a linear dependence of κ ν withrespect to temperature: 11 ν ( T, ρ ) = κ ν ( ¯ T , ρ ) + ( T − ¯ T ) × d, (42)where d = dκ ν dT (cid:12)(cid:12)(cid:12)(cid:12) T = ¯ T (43)and therefore T ν = exp (cid:20) − (cid:90) y y κ ν ( ¯ T ) ρ ( y ) dy (cid:21) , (44)which means that T ν is identical to the transmission of a sample at the average temperature. This impliesthat, under particular circumstances, one can find a temperature for which the experiment can be inter-preted, although the plasma is subject to gradients... This enforces the need for independent diagnosticsof the plasma conditions (K-shell spectroscopy of a light element, Thomson scattering, shadowgraphy,etc.). It is important to study the effect of relative uncertainties in photo-absorption measurements (areal mass,backlighter, background radiation, self-emission, etc.). In the present work, we discussed, assuming theknowledge of the uncertainty on the areal mass, the required uncertainty on the transmission measure-ment in order to infer the opacity with a given accuracy. The issue of the uncertainty on the backlighteremission was also briefly investigated. We quantified the impact of several spatial non-uniformities of theareal mass on the transmission, considering fives cases: wedge, bulge, hollow, holes and modulations. Thecorresponding formulas can provide an insight on the effect of the amount of corrugations, depending ineach case on a single parameter (cid:15) ( e.g. depth of the hollow relative to the nominal thickness, density ofholes, amplitude of modulations, etc.). Non-uniformities of the sample can be detected by analysis tech-niques (Rutherford back-scattering, scanning electron microscopy, etc.). Such defects always overestimatethe transmission, i.e. make the plasma more transparent, due to convexity of the exponential function.They are more important when the spectral transmission is low. Holes, modulations (oscillations) andhollows (convex distortions) are expected to have the strongest impact. Let us denote κ the opacity, T the transmission and A the areal mass. One has: T ν − ∆ T ν ≤ T ≤ T ν + ∆ T ν (45) κ ν − ∆ κ ν ≤ κ ≤ κ ν + ∆ κ ν (46) ρL − ∆( ρL ) ≤ A ≤ ρL + ∆( ρL ) (47)One has therefore − ln [ T ν − ∆ T ν ] ≥ − ln T ≥ − ln [ T ν + ∆ T ν ] (48)and 12 ρL − ∆( ρL ) ≥ A ≥ ρL + ∆( ρL ) , (49)which implies − ln [ T ν − ∆ T ν ] ρL − ∆( ρL ) ≥ κ ≥ − ln [ T ν + ∆ T ν ] ρL + ∆( ρL ) . (50)Setting κ min = − ln [ T ν + ∆ T ν ] ρL + ∆( ρL ) = κ ν − ∆ κ ν (51)and κ max = − ln [ T ν − ∆ T ν ] ρL − ∆( ρL ) = κ ν + ∆ κ ν (52)where κ ν = − ln T ν / ( ρL ), we have κ max ≈ − ln T ν ρL (cid:18) ρL ) ρL (cid:19) − ρL ln (cid:20) − ∆ T ν T ν (cid:21) (cid:20) ρL ) ρL (cid:21) (53)and taking only the first-order terms in ∆ T ν and ∆( ρL ) and neglecting the second-order terms in∆ T ν ∆( ρL ): κ max = κ ν + ∆ κ ν ≈ κ ν + κ ν ∆( ρL ) ρL + 1 ρL ∆ T ν T ν (54)or ∆ κ ν κ ν ≈ ∆( ρL ) ρL − T ν ∆ T ν T ν , (55)which is exactly Eq. (8). The same result can of course be obtained using κ min (see Eq. (51)). In this appendix, we provide the main steps of the derivations of expressions givven in table 3.
In order to model a wedge-shape distortion of the sample (see Fig. 9), we use the linear form ρL ( x ) = ρ L (cid:16) − (cid:15) xa (cid:17) , (56)where (cid:15) controls the slope of the surface of the target. The preservation of the areal mass reads ρ L a (cid:90) a (cid:16) − (cid:15) xa (cid:17) dx = ρ L , (57)which yields ρ L = ρ L − (cid:15)/ T ν = (1 − (cid:15)/ (cid:15) ln ( T ν ) [ T ν ] − (cid:15)/ (cid:104) − T − (cid:15) − (cid:15)/ ν (cid:105) . (59)13 .2 Bulge (convex distortion) A bulge-shape (concave) distortion of the sample (see Fig. 13) is modeled here as ρL ( x ) = ρ L (cid:20) − (cid:15)a x ( x − a ) + 1 (cid:21) . (60)The quantity ρ L , given by the preservation of the areal mass, is ρ L = ρ L (cid:15)/ . (61)We have therefore ˜ T ν = (cid:115) (cid:15)/ − (cid:15) ln T ν [ T ν ] (cid:15)/ D (cid:32)(cid:115) − (cid:15) ln T ν (cid:15)/ (cid:33) , (62)where D ( x ) represents Dawson’s function D ( x ) = e − x (cid:90) x e t dt. (63) In a similar way, a hollow (convex distortion, see Fig. 17) is modeled here as ρL ( x ) = ρ L (cid:20) (cid:15)a x ( x − a ) + 1 (cid:21) . (64)The thickness L , given by the preservation of the areal mass, is ρ L = ρ L − (cid:15)/ . (65)In the case of a hollow, we have˜ T ν = √ π T ν ] − (cid:15) − (cid:15)/ (cid:115) − (cid:15)/ − (cid:15) ln T ν Erf (cid:34)(cid:115) − (cid:15) ln T ν − (cid:15)/ (cid:35) , (66)where Erf is the usual error function Erf( x ) = 2 √ π (cid:90) x e − t dt. (67)Nevertheless, to have a visible effect, at least 25 % of holes are required in the target (this is veryimportant, but since the considered thicknesses are of the order of a few hundreds of Angstr¨oms, it mightbe realistic). We choose to model the presence of holes (see Fig. 21) in the sample by the replacement ρ L → ρ L (1 − (cid:15) ) , (68)where 0 < (cid:15) < ρ L to ρ L ). The preservation of areal mass implies ρ L = ρ L − (cid:15) (69)14nd the transmission becomes ˜ T ν = (1 − (cid:15) ) [ T ν ] − (cid:15) + (cid:15). (70) We consider the following modulations of the sample (see Fig. 25): ρL ( x ) = ρ L [1 + (cid:15) cos( x )] (71)with 0 ≤ (cid:15) ≤
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20 % ( ρ )Perry et al. (1991) [56] ±
35 % ( ρ ) ± ± et al. (1992) [57] ± ρ ) ± et al. (1992) [58]Eidmann et al. (1994) [59] ± T BL )Perry et al. (1996) [60] ± ±
20 % ( ρ ) ± et al. (1998) [61] ±
35 % ( ρ )Chenais et al. (2000) [62] ±
50 % ( ρ ) ±
20 %Bailey et al. (2003) [32] ±
33 % ( n e )Renaudin et al. (2006) [63] ±
30 % ±
25 % ( ρ ) ± et al. (2009) [64–66]Bailey et al. (2007) [33] ±
25 % ±
25 % ( n e ) ± ± ± T BL represents the effective temperature deduced from the backlighter flux (whichis proportional to the fourth power of T BL according to Stefan’s law).20efect Areal mass Modified transmissionWedge ρL ( x ) = ρ L (cid:0) − (cid:15) xa (cid:1) ˜ T ν = (1 − (cid:15)/ (cid:15) ln( T ν ) [ T ν ] − (cid:15)/ (cid:104) − T − (cid:15) − (cid:15)/ ν (cid:105) Bulge ρL ( x ) = ρ L (cid:2) − (cid:15)a x ( x − a ) + 1 (cid:3) ˜ T ν = (cid:113) (cid:15)/ − (cid:15) ln T ν [ T ν ] (cid:15)/ D (cid:16)(cid:113) − (cid:15) ln T ν (cid:15)/ (cid:17) Hollow ρL ( x ) = ρ L (cid:2) (cid:15)a x ( x − a ) + 1 (cid:3) ˜ T ν = √ π [ T ν ] − (cid:15) − (cid:15)/ (cid:113) − (cid:15)/ − (cid:15) ln T ν Erf (cid:104)(cid:113) − (cid:15) ln T ν − (cid:15)/ (cid:105) Holes ρ L → ρ L (1 − (cid:15) ) ˜ T ν = (1 − (cid:15) ) [ T ν ] − (cid:15) + (cid:15) Modulations ρL ( x ) = ρ L [1 + (cid:15) cos( x )] ˜ T ν = T ν × I ( − (cid:15) ln T ν )Table 3: Modified transmission ˜ T ν as a function of the “unperturbed” transmission T ν in the case of thedifferent areal-mass defects considered in the present paper. D ( x ) = e − x (cid:82) x e t dt represents Dawson’sfunction, Erf( x ) = √ π (cid:82) x e − t dt is the usual error function, and I ( x ) is the Bessel function of the firstkind of order zero. a is the areal size of the sample and (cid:15) quantifies the amplitude of the perturbation.Kind of irregularity Value of (cid:15) (% of distortion)Wedge 0.4 (40 %)Bulge 0.4 (80 %)Hollow 0.4 (60 %)Holes 0.05 (5 %)Modulations 0.2 (20 %)Table 4: Values of (cid:15) required to obtain an uncertainty of 7 % on the areal mass (estimated from Rutherfordback-scattering), as in the NIF experiment [67, 75, 76].21 .4 0.5 0.6 0.7 0.8 0.900.10.20.30.40.50,60.4 0.5 0.6 0.7 0.8 0.900.10.20.30.40.50.6 ∆ κ ν κ ν T ν ∆Τ ν T ν = 1% ∆Τ ν T ν = 2.75% ∆Τ ν T ν = 5% Figure 1: (Color online) Monte Carlo simulation of the propagation of uncertainties. In the three cases,the points correspond to values of ∆ κ ν /κ ν = | κ i − κ | /κ , i=1,10000 and ∆( ρL ) / ( ρL )=0.07. The firstcase (top) corresponds to ∆ T ν / T ν =1 %, the second case (middle) to ∆ T ν / T ν =2.75 % and the last case(bottom) to ∆ T ν / T ν =5 %. As mensioned in the text, the results do not depend on κ .22 z -6-4-20 W ( z ) W W -1 Figure 2: (Color online) The two real branches of Lambert function: W and W − .23 .2 0.4 0.6 0.8 T ν ∆ T ν / Τ ν ( % ) ∆(ρ L)=0 ∆(ρ
L)/ ρ L=7 %
Figure 3: (Color online) Value of ∆ T ν / T ν as a function of T ν for a required precision on opacity ∆ κ ν /κ ν =10% and two cases: ∆( ρL ) = 0 (red curve, see Eq. (13)) and ∆( ρL ) / ( ρL ) = 7% (blue curve, see Eq.(14)). 24 .3 0.4 0.5 0.6 0.7 0.8 0.9 1 T ν ∆ T ν ∆ ( ρ L) / ρ L=7 %
Figure 4: (Color online) Value of ∆ T ν as a function of T ν (see Eq. (9)).25 .2 0.4 0.6 0.8 1 T ν ∆ T ν / T ν ( % ) ε =-0.02 ε =-0.05 ε =-0.1 ε =-0.15 Figure 5: (Color online) Impact of background radiation on ∆ T ν / T ν for different values of (cid:15) = δ/I .26 .2 0.4 0.6 0.8 Transmission T ν -0.03-0.025-0.02-0.015-0.01 P a r a m e t e r ε Background
Figure 6: (Color online) Variation of parameter (cid:15) as a function of transmission in the case of a backgroundperturbation. One has ∆ κ ν /κ ν = 10% and ∆( ρL ) / ( ρL ) = 7%.27
00 950 1000 1050
Energy (eV) T r a n s m i ss i on ε =0 ε =0.02 ε =0.05 ε =0.1 ε =0.15 Figure 7: (Color online) Impact of background radiation on the transmission of copper at T =18 eV, ρ =0.01 g/cm and a resolving power of R = E/ ∆ E =1000 ( E is the photon energy) for different valuesof (cid:15) = δ/I . 28igure 8: (Color online) Microscopy views of a few samples from a recent (2017) absorption-spectroscopycampaign at LULI laboratory [71]. 29igure 9: (Color online) Wedge-shape distortion of the sample. The lowest purple arrows corresponds to I and the highest to I ν . For the sample on the left I ν = T ν I and for the sample on the right I ν = ˜ T ν I .30 .2 0.4 0.6 0.8 T ν ∆ Τ ν / Τ ν ( % ) ε =0.1 ε =0.3 ε =0.5 ε =0.7 ε =0.9 Figure 10: (Color online) Impact of a wedge-shape distortion of the sample on ∆ T ν / T ν for differentvalues of (cid:15) . 31 .2 0.4 0.6 0.8 Transmission T ν P a r a m e t e r ε Wedge
Figure 11: (Color online) Variation of parameter (cid:15) as a function of transmission in the case of a wedge-shape distortion of the sample. One has ∆ κ ν /κ ν = 10% and ∆( ρL ) / ( ρL ) = 7%.32
00 950 1000 1050
Energy (eV) T r a n s m i s i on ε =0 ε =0.1 ε =0.5 ε =0.9
945 950 9550.40.450.50.550.60.650.7
Figure 12: (Color online) Impact of a wedge-shape distortion of the sample on the transmission of copperat T =18 eV, ρ =0.01 g/cm and a resolving power of R = E/ ∆ E =1000 ( E is the photon energy) fordifferent values of (cid:15) . 33igure 13: (Color online) Concave distortion of the sample (bulge). The lowest purple arrows correspondsto I and the highest to I ν . For the sample on the left I ν = T ν I and for the sample on the right I ν = ˜ T ν I .34 .2 0.4 0.6 0.8 T ν ∆ Τ ν / Τ ν ( % ) ε =0.1 ε =0.3 ε =0.5 ε =1.5 ε =2.5 Figure 14: (Color online) Impact of a concave distortion of the sample (bulge) on ∆ T ν / T ν for differentvalues of (cid:15) . 35 .2 0.4 0.6 Transmission T ν P a r a m e t e r ε Bulge
Figure 15: (Color online) Variation of parameter (cid:15) as a function of transmission in the case of a bulge-shape distortion of the sample. One has ∆ κ ν /κ ν = 10% and ∆( ρL ) / ( ρL ) = 7%.36
00 950 1000 1050
Energy (eV) T r a n s m i s i on ε =0 ε =0.1 ε =0.5 ε =2.5
950 960 9700.40.450.50.55
Figure 16: (Color online) Impact of a concave distortion of the sample (bulge) on the transmission ofcopper at T =18 eV, ρ =0.01 g/cm and a resolving power of R = E/ ∆ E =1000 ( E is the photon energy)for different values of (cid:15) . 37igure 17: (Color online) Convex distortion of the sample (hollow). The lowest purple arrows correspondsto I and the highest to I ν . For the sample on the left I ν = T ν I and for the sample on the right I ν = ˜ T ν I .38 .2 0.4 0.6 0.8 T ν ∆ Τ ν / Τ ν ( % ) ε =0.1 ε =0.3 ε =0.5 ε =0.7 ε =0.9 Figure 18: (Color online) Impact of a convex distortion of the sample (hollow) on ∆ T ν / T ν for differentvalues of (cid:15) . 39 .2 0.4 0.6 0.8 Transmission T ν P a r a m e t e r ε Hollow
Figure 19: (Color online) Variation of parameter (cid:15) as a function of transmission in the case of a hollow-shape distortion of the sample. One has ∆ κ ν /κ ν = 10% and ∆( ρL ) / ( ρL ) = 7%.40
00 950 1000 1050
Energy (eV) T r a n s m i s i on ε =0 ε =0.1 ε =0.5 ε =0.9
945 950 9550.40.450.50.550.60.650.7
Figure 20: (Color online) Impact of a convex distortion of the sample (hollow) on the transmission ofcopper at T =18 eV, ρ =0.01 g/cm and a resolving power of R = E/ ∆ E =1000 ( E is the photon energy)for different values of (cid:15) . 41igure 21: (Color online) Presence of holes in the sample. The lowest purple arrows corresponds to I and the highest to I ν . For the sample on the left I ν = T ν I and for the sample on the right I ν = ˜ T ν I .42 .2 0.4 0.6 0.8 T ν ∆ Τ ν / Τ ν ( % ) ε =0.1 ε =0.3 ε =0.5 ε =0.7 ε =0.9 Figure 22: (Color online) Impact of the presence of holes in the sample on ∆ T ν / T ν for different valuesof (cid:15) . 43 .2 0.4 0.6 0.8 Transmission T ν P a r a m e t e r ε Holes
Figure 23: (Color online) Variation of parameter (cid:15) as a function of transmission in the case of a samplehaving holes. One has ∆ κ ν /κ ν = 10% and ∆( ρL ) / ( ρL ) = 7%.44
00 950 1000 1050
Energy (eV) T r a n s m i s i on ε =0 ε =0.1 ε =0.3 ε =0.5 ε =0.7 ε =0.9
945 950 9550.40.450.50.550.60.650.7
Figure 24: (Color online) Impact of the presence of holes in the sample on the transmission of copperat T =18 eV, ρ =0.01 g/cm and a resolving power of R = E/ ∆ E =1000 ( E is the photon energy) fordifferent values of (cid:15) . 45igure 25: (Color online) Presence of oscillations in the sample surface. The lowest purple arrowscorresponds to I and the highest to I ν . For the sample on the left I ν = T ν I and for the sample on theright I ν = ˜ T ν I . 46 .2 0.4 0.6 0.8 T ν ∆ Τ ν / Τ ν ( % ) ε =0.1 ε =0.3 ε =0.5 ε =0.7 ε =0.9 Figure 26: (Color online) Impact of oscillations in the sample surface on ∆ T ν / T ν for different values of (cid:15) . 47 .2 0.4 0.6 0.8 Transmission T ν P a r a m e t e r ε Modulations
Figure 27: (Color online) Variation of parameter (cid:15) as a function of transmission in the case of a samplepresenting surface oscillations. One has ∆ κ ν /κ ν = 10% and ∆( ρL ) / ( ρL ) = 7%.48
00 950 1000 1050
Energy (eV) T r a n s m i s i on ε =0 ε =0.1 ε =0.5 ε =0.9
945 950 9550.40.450.50.550.60.650.7
Figure 28: (Color online) Impact of oscillations of the sample surface on the transmission of copper at T =18 eV, ρ =0.01 g/cm and a resolving power of R = E/ ∆ E =1000 ( E is the photon energy) for differentvalues of (cid:15) . 49igure 29: (Color online) We consider a sample homogeneous along the x and z axis and located between y and y and subject to gradients along the yy