Simple electron-impact excitation cross-sections including plasma density effects
aa r X i v : . [ phy s i c s . a t o m - ph ] J a n Simple electron-impact excitation cross-sectionsincluding plasma density effects
Jean-Christophe Pain a,b, and Djamel Benredjem c a CEA, DAM, DIF, F-91297 Arpajon, France b Universit´e Paris-Saclay, CEA, Laboratoire Mati`ere en Conditions Extrˆemes,91680 Bruy`eres-le-Chˆatel, France c Laboratoire Aim´e Cotton, Universit´e Paris-Saclay, Orsay, France
Abstract
The modeling of non-local-thermodynamic-equilibrium plasmas is crucial for many aspects ofhigh-energy-density physics. It often requires collisional-radiative models coupled with radiative-hydrodynamics simulations. Therefore, there is a strong need for fast and as accurate as possible cal-culations of the cross-sections and rates of the different collisional and radiative processes. We presentan analytical approach for the computation of the electron-impact excitation (EIE) cross-sections inthe Plane Wave Born (PWB) approximation. The formalism relies on the screened hydrogenic model.The EIE cross-section is expressed in terms of integrals, involving spherical Bessel functions, whichcan be calculated analytically. In order to remedy the fact that the PWB approximation is not correctat low energy (near threshold), we consider different correcting factors (Elwert-Sommerfeld, Cowan-Robb, Kilcrease-Brookes). We also investigate the role of plasma density effects such as Coulombscreening and quantum degeneracy on the EIE rate. This requires to integrate the collision strengthmultiplied by the Fermi-Dirac Distribution and the Pauli blocking factor. We show that, using ananalytical fit often used in collisional-radiative models, the EIE rate can be calculated accuratelywithout any numerical integration, and compare our expression with a correction factor presented ina recent work.
Interpretation of spectroscopic measurements and simulations of kinetic and transport processes in non-local-thermodynamic-equilibrium (NLTE) plasmas requires knowledge of many electron-impact excitation(EIE) cross-sections for atoms and ions. This is the case, for instance, for integrated simulations ofhohlraums in inertial confinement fusion, diagnosis of plasma X-ray sources, estimation of radiative powerlosses in magnetic-confinement-fusion devices or photoionized plasmas in astrophysics. The modeling ofatomic physics in plasmas out of equilibrium depends closely on the radiation field and radiation transportand is generally coupled to the hydrodynamic motion of matter, so that NLTE physics must be used inintegrated radiation-hydrodynamics simulations. This requires fast but accurate calculations of cross-sections and rates for the processes involved in collisional-radiative models (see for instance [1–4]). Asan example, we mention the code ATMED CR, which includes relativistic nℓj splitting as well as nonzero ∆ n and elastic ∆ n = 0 collisions with plasma electrons [5–7]. Those codes are helpful for checkingorders of magnitudes and making comparisons between numerical values of rates and other data relevantfor the modeling of NLTE plasmas.Since the measurements of EIE cross-sections in dense plasmas are definitely scarce, it is difficultto provide prescriptions concerning screening charges, near-threshold corrections or energy-level shifts.Bearing in mind this fact, the rates of ATMED CR are really good. In addition, inertial confinement [email protected] et al. [15], that of Chen [16] or theFAC code [17], just to name a few.In this work, we present a simple and rather accurate approach for the computation of the EIE cross-section in the PWB approximation. The formalism relies on the screened hydrogenic model. The EIEcross-section is expressed in terms of integrals involving spherical Bessel functions which can be calculatedanalytically. In order to remedy the fact that the PWB approximation is not correct at low energy (nearthreshold), we also compare different correcting factors (Elwert-Sommerfeld [18–20], Cowan-Robb [21],Kilcrease-Brookes [22]).One of the most significant “standard” contributions to the shift of H-like spectral lines is causedby quenching non-zero ∆ n [23] and elastic ∆ n = 0 [24] collisions with plasma electrons, the so-calledelectronic shift [25, 26]. There is also a so-called Plasma Polarization Shift (PPS), which plays an im-portant role in explaining the observed shifts of the high- n H-like spectral lines [27, 28]. Physically, thePPS is caused by the redistribution of plasma electrons due to interaction with the radiating ion. Whenonly plasma electrons inside the orbit of the bound electron were taken into account, the resulting PPSwas expected to be towards the blue [29]. If the free electrons outside the bound-electron orbit are alsotaken into account, the resulting PPS is towards the red. The theoretical results for red PPS by differentauthors differ by a factor of two [30]. The ionization potential depression (IPD) in a dense plasma issomehow an average quantity characterizing the global effect of the charged particles − perturbers − on agiven ion. Quantum properties, such as the ionization potential are modified due to the interactions of thevalence electron with the perturbers. Two models, namely the Stewart-Pyatt (SP) [31] and Ecker-Kr¨oll(EK) [32] models have been widely used during the past decades to estimate the IPD. A few years ago,their validity has been discussed in the framework of two experiments, one using an X-ray free-electronlaser [33] and the other one using a high-power optical laser [34] to create the dense plasma. It appearedthat neither the SP model nor the EK model were able to explain both experiments. This has initiateda renewed interest for the problem of the IPD in dense plasmas (see for example Refs. [35–40]).We present a model for the computation of EIE cross-sections which has the advantage of beingalmost completely analytical (this was our “requirements specification”). It can be useful for fast “on-line” NLTE calculations in radiative-hydrodynamics simulations for instance. The source code is availableupon request. It relies on the screened hydrogenic approximation and enables one to take into accountthe impact of some plasma density effects on the EIE cross-section, such as electronic shifts or degeneracyeffects. In particular, we suggest to model the effect of electrons (which leads to a blue shift) by extendingthe Li-Rosmej formula [41–44]. The energy shift due to ions (which is in fact a red shift) will not beconsidered here. It can be modeled using an approach similar to the one published recently by Iglesias [43].The range of validity of the model is difficult to establish; owing to the assumptions and approximationsmade, our approach will give reliable results when the screened hydrogenic approximation is reliable. It is2herefore strongly dependent on the quality of the mean charge and subsequently the screening constants.Since it is not simple to determine unambiguously which set of screening constants is the best, we haveinvestigated the impact of different sets (see Refs. [45–47]). The two latter depend on principal n andorbital ℓ quantum numbers. Note that relativistic screening constants (depending on n , ℓ and j ) werederived by several authors (see for instance Ref. [48]), but in the present case we restrict ourselves to thenon-relativistic hydrogenic approximation. It is definitely true that Mayer’s constants [45] depend onlyon n , but they are the ones for which, in that particular case (transition in Li-like carbon), we got the bestagreement with the quantum-mechanical calculations performed using Cowan’s code. This is definitelysurprising, but cannot be generalized to other cases, of course, since it might be due to compensation oferrors due to the approximations of our model.In section 2, we present our model for the EIE cross-section calculation. In section 3, comparisons withDW calculations are performed and discussed. A special care is given to the behaviour near threshold,where several correction factors are compared. In section 4 the impact of energy shifts due to ions andelectrons is studied using analytical formulas. Let us denote respectively a and b the initial and final states of the transition induced by the electronimpact. In the PWB approximation, the EIE cross-section reads (in atomic units): σ ( ǫ ) = πa g a X M,M ′ Z k max k min |h γJM | P j e i k · r j | γ ′ J ′ M ′ i| k dk, (1)where k = √ ǫ is the wavevector and ǫ the energy of the incident electron. k min = √ ǫ − p ǫ − ∆ ǫ )and k max = √ ǫ + p ǫ − ∆ ǫ ), ∆ ǫ representing the excitation energy. The initial and final states of theion are respectively | γJM i and | γ ′ J ′ M ′ i and r j is the position of the j th electron. J is the total atomicangular momentum, M its projection on the z − axis and γ represents all the additional required quantumnumbers in order to define the state in an unique way. The coupling of all the quantum numbers includedin γ leads to J . g a represents the degeneracy of the initial state. In the present study, we restrict ourselvesto dipole-allowed transitions, but the approach can be generalized to non-dipole transitions. This is anadvantage over the Van Regemorter formula [49, 50]. The cross-section can be written as σ ( ǫ ) = πa g a ǫ Ω( ǫ ) , (2)where the collision strength readsΩ( ǫ ) = 8∆ ǫ Z k max k min gf ( k ) d (ln k ) , the quantity gf ( k ) being the generalized oscillator strength. In the hydrogenic approximation, a ≡ n a ℓ a , b ≡ n b ℓ b and the matrix element in Eq. (1) can be written h n a ℓ a m a | e i k . r | n b ℓ b m b i and calculatedanalytically, as will be shown in the next section. Assuming k = k e z , we have : e i k · r = e ikz = e ikr cos θ = ∞ X t =0 i t (2 t + 1) / (4 π ) / j t ( kr ) Y t ( θ )and therefore 3 n a ℓ a m a | e i k · r | n b ℓ b m b i = ∞ X ℓ =0 i ℓ (2 ℓ + 1) / (4 π ) / M r ( n a , ℓ a , t, k, n b , ℓ b ) N a ( ℓ a , m a , t, , ℓ b , m b ) , (3)where M r ( n a , ℓ a , t, k, n b , ℓ b ) = Z ∞ R n a ℓ a ( r ) j t ( kr ) R n b ℓ b ( r ) r dr and N a ( ℓ a , m a , t, , ℓ b , m b ) = Z Y m a ∗ ℓ a Y t Y m b ℓ b d Ω , where Y mℓ ( θ, φ ) are normalized spherical harmonics and the R nℓ ( r ) radial hydrogenic wave functions. d Ω = sin θdθdφ is the infinitesimal solid angle. Angular integrals of the type N a are well-known (“Gauntcoefficients”) and analytical expressions may be found in the current literature. In particular, the Gauntcoefficients can be expressed in terms of 3 j coefficients: Z Y m ℓ Y m ℓ Y m ℓ d Ω = r (2 ℓ + 1)(2 ℓ + 1)(2 ℓ + 1)4 π (cid:18) ℓ ℓ ℓ m m m (cid:19) (cid:18) ℓ ℓ ℓ (cid:19) . Since Z Y m ℓ Y m ℓ Y m ℓ d Ω = ( − m Z Y m ℓ Y − m ∗ ℓ Y m ℓ d Ω , we get Z Y m ℓ Y m ∗ ℓ Y m ℓ d Ω = ( − m r (2 ℓ + 1)(2 ℓ + 1)(2 ℓ + 1)4 π (cid:18) ℓ ℓ ℓ m m m (cid:19) (cid:18) ℓ ℓ ℓ (cid:19) . = s (2 ℓ + 1)(2 ℓ + 1)4 π (2 ℓ + 1) h ℓ | ℓ ℓ ih ℓ m | ℓ m ℓ m i , where h ℓ m | ℓ m ℓ m i is the usual Clebsch-Gordan coefficient, and then N a ( ℓ a , m a , t, , ℓ b , m b ) = Z Y m a ∗ ℓ a Y t Y m b ℓ b d Ω = ( − m a r (2 t + 1)(2 ℓ a + 1)(2 ℓ b + 1)4 π × (cid:18) t ℓ a ℓ b m a m b (cid:19) (cid:18) t ℓ a ℓ b (cid:19) = s (2 ℓ a + 1)(2 ℓ b + 1)4 π (2 t + 1) h ℓ a | t ℓ b ih ℓ a m a | t ℓ b m b i . The latter integral vanishes unless | ℓ a − ℓ b | ≤ t ≤ ℓ a + ℓ b and therefore the sum in Eq. (3) is not infinite.Thus, one can write gf ( k ) = ∆ ǫk (2 ℓ a + 1)(2 ℓ b + 1) X t (2 t + 1) (cid:20)(cid:18) ℓ a t ℓ b (cid:19) Z ∞ P a ( r ) j t ( kr ) P b ( r ) dr (cid:21) , with | ℓ a − ℓ b | ≤ t ≤ ℓ a + ℓ b , mod( ℓ a + t + ℓ b ,
2) = 0, and P a ( r ) = rR n a ℓ a ( r ). The radial wave-functionsare defined as 4 nℓ ( r ) = s Z nℓ ( n − ℓ − n ( n + ℓ )! (cid:18) Z nℓ n (cid:19) ℓ +1 r ℓ exp (cid:20) − rZ nℓ n (cid:21) L ℓ +1 n − ℓ − (cid:18) rZ nℓ n (cid:19) , (4)where Z nℓ is the screened nuclear charge seen by an electron in the nℓ subshell, and L ℓ +1 n − ℓ − ( x ) a gener-alized Laguerre polynomial. The energy of state i is, in atomic units ǫ i = − Z n i ℓ i n i . The generalized Laguerre polynomial can be expanded as L ts ( x ) = s X j =0 (cid:18) t + ss − j (cid:19) ( − x ) j j ! , (5)where (cid:18) mn (cid:19) = m ! n !( m − n )! is the usual binomial coefficient. Eq. (5) yields L ℓ a +1 n a − ℓ a − (cid:18) rZ a n a (cid:19) L ℓ b +1 n b − ℓ b − (cid:18) rZ b n b (cid:19) = n a − ℓ a − X j =0 n b − ℓ b − X u =0 (cid:18) n a + ℓ a n a − ℓ a − − j (cid:19) (cid:18) n b + ℓ b n b − ℓ b − − u (cid:19) × ( − j + u j ! u ! (cid:18) Z a n a (cid:19) j (cid:18) Z b n b (cid:19) u r j + u . The latter formula enables one to obtain the following expression for the generalized oscillator strength: gf ( k ) = ∆ ǫk (2 ℓ a + 1) (2 ℓ b + 1) Z a Z b n a n b ( n a − ℓ a − n b − ℓ b − n a + ℓ a )! ( n b + ℓ b )! × (cid:18) Z a n a (cid:19) ℓ a +1 (cid:18) Z b n b (cid:19) ℓ b +1 n a − ℓ a − X j =0 n b − ℓ b − X u =0 (cid:18) n a + ℓ a n a − ℓ a − − j (cid:19) × (cid:18) n b + ℓ b n b − ℓ b − − u (cid:19) ( − j + u j ! u ! (cid:18) Z a n a (cid:19) j (cid:18) Z b n b (cid:19) u r j + u × X t (2 t + 1) (cid:18) ℓ a t ℓ b (cid:19) I (cid:18) t, j + u + ℓ a + ℓ b + 2 , Z a n a + Z b n b , k (cid:19)) , I being an integral of the type I ( α, β, c, d ) = Z ∞ e − cr j α ( dr ) r β dr, where c and d are real and β is an integer such that β > α . I can be put in the form5 .1 1 10 100 k g f( k ) / ∆ ε CowanZ a =Z b =ZMayer Figure 1: (Color online) gf ( k ) / ∆ ǫ (at. units) for transition 2 s → p in [Ne] Fe XVII. Blue curve: Cowan’scode computation [21], green dot-dashed curve: unscreened case ( Z a = Z b = Z ), red dashed curve: useof Mayer’s screening constants [45]. I ( α, β, c, d ) = 1 c β +1 (2 w ) α (1 + w ) β ( √ w ) β − α − α !( α + β )!(2 α + 1)! " − √ w √ w i × β − α − X i =0 (cid:18) β − α − i (cid:19) (cid:18) α + ii (cid:19) (cid:18) β − i (cid:19)(cid:18) β − i (cid:19) (cid:18) α + 2 i + 12 i (cid:19) , (6)with w = d/c . Expression (6) is simpler than the formulation proposed by Upcraft [51]. In the latterwork, the author expressed the spherical Bessel functions in terms of their sine and cosine terms whichimplied to handle many quantities of the type Z ∞ e − cr sin( dr ) r β dr which is much more tedious than using the compact expression (6). Figure 1 displays gf ( k ) / ∆ ǫ for thetransition 2 s → p in [Ne] Fe XVII. We compare three calculations: one represents the present modelwithout screening constants ( Z a = Z b = Z , green dot-dashed curve), the second one represents thepresent model using the screening constants published by Mayer [45] (green dashed curve) and the thirdone is a quantum-mechanical calculation (but still in the PWB approximation) performed with Cowan’scode (in blue) considered here as the reference. In this specific example, the asymptote for small k isbetter reproduced by the choice Z = Z a = Z b ( i.e. no screening constants), but the intermediate region(for k between 3 and 7) as well as the bump around k =10 reveal that the screening constant improvesthe agreement with Cowan’s code. 6 Corrections near threshold
We now address the issue of the non-validity of the PWB approach near threshold. Since our expressionof the collision strength is obtained within the framework of the screened hydrogenic approximation, weget configuration-averaged collision strengths from the one-electron form by:Ω i → f ( ǫ ) = 2 q i ( g f − q f ) g i g f Ω( ǫ ) ,q i and g i being respectively the population and the degeneracy of the active subshell i . The FAC code [17]gives the collision strength between all the fine-structure levels of two configurations. In order to sumthe fine-structure collision strengths between the γJ levels of the lower configuration C to the γ ′ J ′ levelsof the upper configuration C ′ into the configuration-averaged value, we haveΩ i → f ( ǫ ) = 1 G C X γJ ∈ C X γ ′ J ′ ∈ C ′ g γJ Ω γJ → γ ′ J ′ ( ǫ ) , where g γJ is the degeneracy of the γJ level and G C the degeneracy of the lower configuration, i.e. G C = Y i ∈ C (cid:18) g i q i (cid:19) , the product being taken over all occupied subshells of the lower configuration.The so-called Cowan-Robb correction to the PWB cross-section (see Ref. [21], pp. 568-569) consistsin replacing Ω( ǫ ) by Ω m ( ǫ ) which is defined asΩ m ( ǫ ) = Ω (cid:18) ǫ ∆ ǫ + 31 + ǫ/ ∆ ǫ (cid:19) , in order to extrapolate the collision strength Ω( E ) from above threshold down to the threshold. InRef. [21], Cowan points out that this prescription is tentative, valid only for spin-allowed transitions andwas obtained by comparison with more accurate results relying on close-coupling and DW approaches.Kim proposed a re-scaling of the PWB cross-sections of neutral atoms [52], and of CWB cross-sectionsfor singly charged ions. In the latter case, it consists in multiplying the CWB cross-section withoutexchange by the factor ǫ/ ( ǫ + ∆ ǫ ) [53].Still neglecting exchange, Sommerfeld derived a closed-form solution for the electron-ion Bremsstrahlungprocess (inelastic collision of an electron with a Coulomb point charge [18–20]) and Elwert [20] proposeda correction to the PWB cross-section that approximately reproduced the behaviour of Sommerfeld’s so-lution. The latter correction factor is called the Elwert-Sommerfeld factor (denoted ES in the following)and has been successfully used to correct the PWB cross-section for electron-ion Bremsstrahlung of fullyionized atoms. Invoking the similarity between Bremsstrahlung and electron-ion scattering, Jung [54]suggested to use the ES factor to correct the PWB electron-impact-excitation cross-section. The Elwert-Sommerfeld correction factor f ES is f ES ( ǫ ) = r ǫǫ − ∆ ǫ − e − πZ a / √ ǫ − e − πZ b / √ ǫ − ∆ ǫ ) , where Z a and Z b are the initial and final effective charges, respectively. Kilcrease and Brookes [22] suggestto take Z a = Z b = Z , where Z is the effective ion charge, thus giving f ES ( ǫ ) = r ǫǫ − ∆ ǫ × − e − πZ/ √ ǫ − e − πZ/ √ ǫ − ∆ ǫ ) . The factor f ES slowly tends to 1 as ǫ increases and the latter authors have found that by acceleratingthis behaviour with the use of the heuristic replacement Z → Z ∆ ǫ/ǫ improved the agreement with CWB7 x= ε / ∆ε × -20 × -19 × -19 × -19 × -19 × -19 C r o ss - s ec ti on ( c m ) No correctionCowan-RobbKim (2002)ElwertKilcrease-Brookes (2010)Coulomb Wave Born
Figure 2: (Color online) EIE cross-section with different near-threshold correction factors in [Li] C IV(transition 1 s s → s s p ). Reference: Coulomb Wave Born [11]. The transition energy is ∆ ǫ = 296 eV .cross-sections at higher ǫ values. Figure 2 displays a comparison between the PWB cross-section, thecross-section corrected by different factors: Cowan-Robb [21], Kim [52, 53], Elwert [18–20] and Kilcrease-Brookes [22] and a CWB computation [15] which is our reference here. In the present case, Elwert’sformulation seems to provide the best agreement. In order to take into account plasma density effects on level energies, several analytical formulas wereobtained (see for instance the non-exhaustive list of references [42, 55–57]) to approach ion-sphere po-tentials. Using first-order perturbation theory together with hydrogenic scaled mean ionization yieldsanalytical formulas to estimate energy-level shifts.
Assuming a uniform electron gas in the Wigner-Seitz sphere (of radius R ), Massacrier and Dubau obtainedthe energy-level shift associated with the subshell ( nℓ ) [55]:∆ ǫ nℓ = ǫ c (cid:18) − h r i nℓ R (cid:19) , (7)where h r i nℓ = n Z (cid:2) n + 1 − ℓ ( ℓ + 1) (cid:3) (8)and ǫ c = Z ∗ / (2 R ), Z ∗ being the average ionization (mean charge) of the plasma. Equation (7) is verysimple and easy to handle. For instance, it can be easily implemented in atomic-physics codes [58, 59] or8D Li 2012 Li 2012 Li 2019modified b =2 modified∆ ǫ s ǫ p ǫ nℓ (eV) for EIE channel 1 s s − s s p in Li-like C for different formulations of the energy-levelshift.used to estimate the critical electron density at which pressure ionization occurs, i.e. at which a boundlevel disappears into the continuum (see Appendix A). In 2012, Li and Rosmej proposed an asymptotic expansion of the potential experienced by an ion subjectto free-electron screening in finite-temperature plasmas, in a closed analytical expression [41] (in atomicunits): V f ( r ) = 4 πN e ( R − r √ π (cid:20) Z ∗ k B T e (cid:21) / R / − √ π (cid:20) Z ∗ k B T e (cid:21) / r / ) , (9)where N e is the free-electron density at the Wigner-Seitz radius R = [3 Z ∗ / (4 πN e )] / , k B the Boltzmannconstant and T e the electron temperature. The energy shift of the nℓ subshell is then obtained by∆ ǫ nℓ = h nℓ | V f ( r ) | nℓ i nℓ = Z ∞ V f ( r ) R nℓ ( r ; Z eff ) r dr, (10)where R nℓ ( r ; Z eff ) represents the radial part of the hydrogenic wave-function of the subshell with effectivenuclear charge Z eff . The formula involves expectation values of powers of r . We use the simplified notation h nℓ | f ( r ) | nℓ i = h f ( r ) i nℓ . In 2012, Li and Rosmej, maybe unaware of the fact that an exact formula existsfor h r / i nℓ , derived an alternative analytical fit for V f ( r ), depending only on h r i nℓ (Eq. 8) and h r i nℓ : h r i nℓ = 12 Z eff (cid:2) n − ℓ ( ℓ + 1) (cid:3) . (11)They obtained∆ ǫ nℓ = ǫ c (cid:26) − h r i nℓ R + 8 r ǫ c πk B T e − R / r ǫ c πk B T e (cid:20) √ π h r i nℓ + 110 h r i nℓ (cid:21)(cid:27) , (12)where ǫ c = Z ∗ / (2 R ). It was shown very recently [43] that the fit given in the Li-Rosmej work wasinconsistent with the ion-sphere model, on the contrary to the “original” potential in Eq. (9). Inaddition, the quantity h r / i nℓ can definitely be expressed analytically. This was also pointed out inRef. [43], where the author indicates that such a quantity can be obtained following the proceduregiven in Appendix E of Ref. [60] using the generating-function formalism (see for instance the textbookby Bransden and Joachain [61]) and yielding a complicated expression (in the same paper, a table isprovided with particular values displayed in the form of rational fractions). It was recently pointed outthat a simple expression for h r / i nℓ exists [44], as a particular case of a relation published by Shertzerin 1991 [62], who provided an expression for h nℓ | r β | nℓ ′ i nℓ for arbitrary β : h nℓ | r β | nℓ ′ i nℓ = A n,ℓ,ℓ ′ n − ℓ − X i =0 ( − i Γ( ℓ + ℓ ′ + 3 + i + β ) i !(2 ℓ + 1 + i )!( n − ℓ − i − ℓ − ℓ ′ + 2 + i + β )Γ( ℓ + 3 − n + i + β ) (13)where 9 n,ℓ,ℓ ′ = ( − n − ℓ ′ − n (cid:18) n Z eff (cid:19) β (cid:20) ( n + ℓ )!( n − ℓ − n + ℓ ′ )!( n − ℓ ′ − (cid:21) / , applying therefore also for off-diagonal terms ( ℓ = ℓ ′ ). Γ is the usual Gamma function. It is worthmentioning that a relativistic equivalent of Eq. (13) for ℓ = ℓ ′ was published by Salamin in 1995 [63]. Inthe present case, we have ℓ = ℓ ′ and β = 3 /
2, and we get h r / i nℓ = ( − n − ℓ − n (cid:18) n Z eff (cid:19) / n − ℓ − X i =0 ( − i Γ(2 ℓ + 9 / i ) i !(2 ℓ + 1 + i )!( n − ℓ − i − / i )Γ( ℓ + 9 / − n + i ) . (14)Inserting Eqs. (8) and (14) in Eq. (10) gives∆ ǫ nℓ = ǫ c ( − h r i nℓ R + 8 r ǫ c πk B T e " − n − ℓ √ n RZ eff ) / n − ℓ − X i =0 ( − i Γ(2 ℓ + 9 / i ) i !(2 ℓ + 1 + i )!( n − ℓ − i − × Γ(7 / i )Γ( ℓ + 9 / − n + i ) (cid:21) ) . (15)The potential first published by Rosmej et al. in 2011 [41] and which is consistent with the fundamentalneutrality requirement of the ion-sphere model as shown by Iglesias [43], can therefore be directly usedto derive simple analytical formulas to estimate energy level shifts in dense plasmas. The alternative fitby Li and Rosmej [42], which is not consistent with the ion-sphere model, was motivated by the beliefthat no analytical expression exists for h r / i nℓ , a statement that was invalidated by Iglesias as well. et al. In 2019, Li et al. [64] proposed to use∆ ǫ nℓ = 2 ǫ c (cid:26) x − − x − D rR E x − nℓ (cid:27) , (16)with x = 3 − bπ r ǫ c k B T e ,b = 2 D rR E x − nℓ = ( − n − ℓ − n (cid:18) n RZ eff (cid:19) x − n − ℓ − X i =0 ( − i i !(2 ℓ + 1 + i )! Γ(2 ℓ + 2 + i + x )Γ(1 + i + x )( n − ℓ − i − ℓ + 2 − n + i + x ) . Figure 3 shows the effect of the energy-level shift on the EIE cross-section of the 1 s s → s s p transitionin Li-like C, for the different formulations mentioned above. The numerical values of the energy-level shiftare displayed in table 1. One can see that, in these conditions, the correction of Massacrier and Dubauis rather small and yields a cross-section which is very close to the isolated-atom one. The differencebetween the Li-Rosmej (2012) formula and the modified one indicates that the exact computation of theexpectation value h r / i nℓ (see Eq. (15)) has a significant effect. It tends to reduce the cross-section,compared to the expression approximated by terms proportional to h r i nℓ and h r i nℓ (see Eq. (12).10 x= ε / ∆ε × -20 × -19 E l ec t r on - i m p ac t e x c it a ti on c r o ss - s ec ti on ( c m ) Isolated atomMassacrier-Dubau (1990)Li-Rosmej (2012)Li-Rosmej (2012) modifiedLi et al. (2019) modified [b=2]
Figure 3: (Color online) Effect of different energy-level shifts on EIE cross-section in Li-like C (1 s s → s s p transition): Massacrier-Dubau (Eq. (7)), Li-Rosmej 2012 (Eq. (12)), Li-Rosmej 2012 modified(Eq. (15)), Li et al. modified (Eq. (16)). Many cross-sections were calculated by computer codes or determined experimentally for some values ofincident electron energy. Some of them can be found in available atomic databases. However, publishedcross-sections are often insufficient for detailed simulation of experiments, since data on many cross-sections are missing or do not cover the entire energy range required for calculation of excitation rates,especially for non-Maxwellian plasmas. For this reason, it is desirable to have an easy-to-use formula ofknown accuracy applicable to various classes of transitions (see for instance the non-exhaustive list ofreferences [49, 65–67]) which is usually represented by the following form of the collision strength [68]:Ω (cid:16) ǫ ∆ ǫ (cid:17) = B ln (cid:16) ǫ ∆ ǫ (cid:17) + X i =1 B i (cid:16) ǫ ∆ ǫ (cid:17) − ( i − (17)with i ≥
1. The logarithm is consistent with the high-energy Bethe limit [20]. The form of analyticalformulas such as Eq. (17) is widely used because it can be analytically integrated over a Maxwelliandistribution of electrons. This is important especially for fast codes, which are designed in order to beused “online” in radiative-hydrodynamics simulations. Of course, the values of the parameters B i arespecific of a given transition, i.e. different transition cross-sections have different fitting parameters. Notethat the usual Mewe [69] formulation (see Appendix A) is a particular case of the formalism describedhere, in the case where the summation of the right-hand side of Eq. 17 would end at 3 instead of 5. TheEIE rate for a non-degenerate electron gas, described by a Maxwell-Boltzmann (MB) distribution, is (seefor instance Refs. [70, 71]): 11 MB = 2 N e π Z ∞ ∆ ǫ √ ǫ σ ( ǫ ) e µ − ǫkBTe dǫ, (18)while in the case of a degenerate gas (Fermi-Dirac distribution), it takes the form R FD (∆ ǫ, T, µ ) = 2 N e π Z ∞ ∆ ǫ √ ǫ σ ( ǫ ) f ( ǫ, T e ) [1 − f ( ǫ − ∆ ǫ, T e )] dǫ, (19)where f ( ǫ, T e ) = 11 + e ǫ − µkBTe . The quantity [1 − f ( ǫ − ∆ ǫ, T e )] is the Pauli-blocking factor, which takes into account the fact that allfinal states are not allowed for the free electron. The MB case is recovered if µ/ ( k B T e ) ≪ −
1. Using therelation (2) between the cross-section σ ( ǫ ) and the collision strength Ω( ǫ ) and setting η = µ/ ( k B T e ) < δ = ∆ ǫ/ ( k B T e ) > R MB = C Z ∞ Ω ( x ) e η − δx dx (20)and R FD = C Z ∞ Ω ( x ) 11 + e δx − η (cid:18) − e δ e δx − η (cid:19) dx, (21)where C is a positive constant. Note that the rate of collisional de-excitation from level i to level j wouldbe calculated assuming LTE through detailed balance with collisional excitation R ji = R ij exp [ β ( ǫ j − ǫ i )] . (22)The purpose of the present study is to compare Eqs. (20) and (21). Note that Scott [72] andTallents [73] assume that Ω is nearly constant, since it is a slowly varying function of energy E . In sucha way, they estimate the degeneracy effects by defining the ratio T ( η, δ ) = R ∞ e δx − η (cid:16) − e δ e δx − η (cid:17) dx R ∞ e η − δx dx = e − η − e − δ ln (cid:18) e η e η − δ (cid:19) . Coefficient Value B × − B × − B × − B × − B B B i parameters for the 1 s − p transition in H-like C.In the present work, we would like to go one step further, making a less constraining assumption.12 .2 Maxwell-Boltzmann case We can write R MB = C V , + X i =1 C i V ,i , where C i = C × B i for i ≥ V , = Z ∞ ln x e η − δx dx = e η Γ( δ ) δ V ,i = Z ∞ e η − δx x i − dx = e η E i − ( δ ) for i > , where E n ( x ) = Z ∞ e − xt t n dt. We finally obtain the well-known expression used in many collisional-radiative codes: R MB = e η " C Γ( δ ) δ + X i =1 C i E i − ( δ ) . (23) As in the preceding case, we can write R FD = C V , + X i =1 C i V ,i , with the new integrals V , = Z ∞ ln( x )1 + e δx − η (cid:18) − e δ e δx − η (cid:19) dx V ,i = Z ∞ x i −
11 + e δx − η (cid:18) − e δ e δx − η (cid:19) dx. In order to calculate V , , let us set z ( x ) = e − ( δx − η ) , we then have V , = Z ∞ ln( x ) e − δ [1 + z ( x )] [ e − δ + z ( x )] dx. Expanding the quantity e − δ / (cid:2) (1 + z ( x ))( e − δ + z ( x )) (cid:3) in Taylor series with respect to variable z ( x ) < V , = ∞ X p =1 ( − p +1 (cid:18) − e δp − e δ (cid:19) W ( p )with (see Ref. [74], p. 573, 4.231-1) W ( p ) = Z ∞ ln( x ) e − p ( δx − η ) dx = e pη pδ Γ( pδ ) . We follow the same procedure for the determination of V ,i and get13 ,i = Z ∞ x i − e − δ [1 + z ( x )] [ e − δ + z ( x )] dx, which is equal to V ,i = ∞ X p =1 ( − p +1 (cid:18) − e δp − e δ (cid:19) K i ( p )with K i ( p ) = Z ∞ x i − e − p ( δx − η ) dx = e pη E i − ( pδ ) . Therefore, the final result for the EIE rate taking into account quantum effects is R FD = ∞ X p =1 ( − p +1 (cid:18) − e δp − e δ (cid:19) e pη ( C Γ( pδ ) pδ + X i =1 C i E i − ( pδ ) ) . (24) In the following, we consider an example taken from Ref. [68] (1 s − p transition in H-like C). The valuesof the corresponding B i parameters are listed in table 2. The main issue with this expression (24) is thatit contains an infinite sum. However, it turns out that the truncation of the sum at p max = 4 gives a verygood precision in many cases relevant for our applications. In many cases, only the first two terms aresufficient to achieve good accuracy, as will be illustrated in the next section. The degeneracy effects canbe quantified by the ratio of Eqs. (23) and (24):Λ( η, δ ) = R FD R MB = 1 B δ ) δ + P i =1 B i E i − ( δ ) × p max X p =1 ( − p +1 (cid:18) − e δp − e δ (cid:19) e ( p − η ( B Γ( pδ ) pδ + X i =1 B i E i − ( pδ ) ) , (25)where p max is the truncation order of the sum. Note that the denominator is in fact the first term of thesummation in the numerator.As can be seen in Fig. 4 for δ = ∆ ǫ/ ( k B T e )=0.01, as well as in tables 3 and 4 for δ =0.001, 0.01, 0.1and 0.5, a very good precision is achieved for p max =4. We have checked that this is true over the wholerange of relevant values of ( η , δ ) in warm dense plasmas. δ / p max β ( η, δ, p max ) for η = − δ and p max .As can be seen in Figs. 5, 6, 7 and 8, the new formula (24) can depart significantly from the onederived by Tallents [73], especially for δ =1. For small values of δ , the range of reduced chemical potential η for which the two approaches differ notably gets smaller and smaller (see Figs. 5 ( δ =0.01), 6 ( δ =0.1)and 7 ( δ =0.5)), but the discrepancy is very important for δ =1 (see Fig. 8).14 η R F D / R M B order 2order 4order 6order 8 Figure 4: (Color online) Values of Λ = R FD / R MB (see Eq. (25)) as a function of η for δ =0.01 withdifferent truncations of the sum over index p : p max =2, 4, 6, 8 and 20. δ / p max β ( η, δ, p max ) for η = − . δ and p max .It is worth mentioning that relativistic effects were studied by Beesley and Rose [75] using the Maxwell-J¨uttner distribution [76, 77]: F ( ǫ ) = γ βθK (1 /θ ) e − γ/θ , where θ = k B T e /mc , β = v/c , γ = 1 / p − β and K n is the modified Bessel function of the secondkind: K n ( z ) = √ π Γ( n + 1 / (cid:16) z (cid:17) n Z ∞ e − zx (cid:0) x − (cid:1) n − / dx. Such a formula neglects interactions and quantum effects, which is reasonable since relativistic effectsbecome important at high temperature. Beesley and Rose found the correcting factor R rel ( θ, η T e ) = r π √ θe − /θ K (1 /θ ) (cid:20) θ + η T e ) (cid:21) , where η T e = ǫ/ ( k B T e ). Within the Maxwell-J¨uttner distribution, the kinetic energy is given by E = Z ∞ γf ( γ ) dγ = 1 θK (cid:0) θ (cid:1) Z ∞ γ r − γ e − γ/θ dγ (26)15 η R F D / R M B This workTallents (2016)
Figure 5: (Color online) Values of Λ = R FD / R MB (see Eq. (25)) as a function of η for δ =0.01. Comparisonwith the approach of Tallents [73].and one has E = K (cid:0) θ (cid:1) K (cid:0) θ (cid:1) + 3 . (27)At temperatures T ≈ mc , pair production will become relevant; while this might not change thevelocity distribution, it makes predictions for other quantities based on the velocity distribution andoriginal particle number wrong [78]. The description of NLTE plasmas encountered in different fields of high-energy-density science usuallyinvolves collisional-radiative models coupled to radiative-hydrodynamics simulations. Therefore, onerequires formulas for the cross-sections and rates of the different atomic processes presenting a goodcompromise between accuracy and computational cost. In this work, we presented an analytical modelfor the calculation of the EIE cross-section based on hydrogenic formulas. The source code is availableupon request. We obtained a complete analytical expression of the generalized oscillator strength andinvestigated the sensitivity to the near-threshold corrections as well as to the choice of the screenedcharges. In addition, we studied the impact of different modelings of the plasma density effects basedon recently published formulas. Since the measurements of EIE cross-sections in dense plasmas aredefinitely scarce, it is difficult to provide clear prescriptions, concerning screening charges, near-thresholdcorrections of electronic level shifts. However the present model enables one to get a qualitative (if notquantitative) insight into the cross-section changes induced by such effects and corrections. We alsodeveloped a simple and efficient method to study the impact of degeneracy effects on the EIE rate,following the approach of Tallents et al. but removing an approximation (we do not assume that thecollision strength can be considered as constant). The main difficulty stems from the integration ofthe collision strength multiplied by the Fermi-Dirac distribution and the Pauli blocking factor. Wefound that, using an analytical fit often used in collisional-radiative models, the rate can be calculated16 η R F D / R M B This workTallents (2016)
Figure 6: (Color online) Values of Λ = R FD / R MB (see Eq. (25)) as a function of η for δ =0.1. Comparisonwith the approach of Tallents [73].accurately without any numerical integration. The ratio between classical and quantum-mechanical EIErates can be expressed in terms of Gamma and E n functions, which are widely used and can be easilycomputed. In the future, we plan to make comparisons with experimental EIE cross-sections and toinvestigate the degeneracy effects in the case of very low temperatures, which is important for the startof radiative-hydrodynamics simulations [79]. A Mewe approximation
Instead of using the approximate formula (17) for the collision strength, another alternative consists inresorting to the Mewe formula [69] of the cross-section (for consistency reasons, we keep, in this Appendix,the units of Ref. [80]): σ ( ǫ ) = 4 πa π √ (cid:18) Ryd ∆ ǫ (cid:19) f osc g g ( ǫ/ ∆ ǫ ) ǫ/ ∆ ǫ , (28)where Ryd denotes the Rydberg energy, ∆ ǫ the excitation energy, g the degeneracy of the initial state, f osc the oscillator strength and g ( u ) = A + Bu + Cu + D ln( u ) , (29)with B = C = 0, D = 0 .
28 and A = 0 .
15 for ∆ n = 0 transitions and 0 .
60 for ∆ n = 0 transition [69, 80].The electron-impact excitation rate reads R = N e Z ∞ ∆ ǫ σ ( ǫ ) v ( ǫ ) n ( ǫ ) , (30)with v ( ǫ ) = r ǫm (31)17 η R F D / R M B This workTallents (2016)
Figure 7: (Color online) Values of Λ = R FD / R MB (see Eq. (25)) as a function of η for δ =0.5. Comparisonwith the approach of Tallents [73].as well as n ( ǫ ) = 2 √ π N e √ ǫ ( k B T e ) / e − ǫkBTe (32)when the free electrons are assumed to be non-degenerate. In this case, one has [80]: R = 16 Ryd a c (cid:18) π mc (cid:19) / f osc g N e ( k B T e ) / e − ∆ ǫkBTe [∆ ǫ/ ( k B T e )] G (cid:18) ∆ ǫk B T e (cid:19) , (33)where G ( u ) = A + ( Bu − Cu + D ) e u E ( u ) + Cu. (34)For degenerate free electrons, n ( ǫ ) becomes n ( ǫ ) = 2 √ π N e √ ǫ ( k B T e ) /
11 + e ǫ − µkBTe , (35)and the rate can be approximated by the same procedure as the one described in section 5.3. Note that,in the present work, we do not use any degeneracy reduction. However, we have the possibility to includeit in our code, using the prescription of Zimmerman and More [81] consisting in replacing the degeneracy g nℓ of subshell nℓ by ˜ g nℓ = g nℓ (cid:16) c h r i nℓ R (cid:17) c , (36)where c and c are free parameters. The latter parameters are determined by consistency with thezero-temperature Thomas-Fermi model at solid density and at very high density using the numerical fitsprovided by More [82]. R is the Wigner-Seitz radius and h r i nℓ is given by Eq. (11).18
10 -8 -6 -4 -2 0 η R F D / R M B This workTallents (2016)
Figure 8: (Color online) Values of Λ = R FD / R MB (see Eq. (25)) as a function of η for δ =1. Comparisonwith the approach of Tallents [73]. B A simple model for pressure ionization
Using the simple approach of Massacrier and Dubau (see Sec. 4.1), the one-electron hydrogenic energywith plasma effects reads: ǫ nℓ = − Z n + Z R (cid:18) − h r i nℓ R (cid:19) , with h r i nℓ given by Eq. (8). Using 4 πR N e / Z and setting x = n Z (cid:18) πN e Z (cid:19) / , the equation giving the last bound level, ǫ nℓ = 0, yields x − β nℓ x + β nℓ = 0 , with β nℓ = n Z h r i nℓ = 2 n n + 1 − ℓ ( ℓ + 1) . We use Cardan’s method for cubic equations of the type x + px + q = 0 , where p = − β nℓ , q = β nℓ . In the case where the discriminant ∆ = − (4 p + 27 q ) is positive, we havethree real roots. In the present case β nℓ has the minimum β min = 2 n n + 1 . Therefore the discriminant ∆ reads 19 β − n − n + 1and we are indeed in the case ∆ ≥
0. We find it particularly convenient to use the so-called “trigonomet-ric” form of the solutions of the cubic equation x i = 2 r − p (cid:20)
13 arccos (cid:18) − q p r − p (cid:19) + 2 iπ (cid:21) for i =0, 1 or 2, i.e. x i = 2 p β cos (cid:20)
13 arccos (cid:18) − √ β (cid:19) + 2 iπ (cid:21) , and the critical density of pressure ionization is N e,c = Zn (cid:18) Z πx (cid:19) / . (37)A comparison between the critical density given by formula (37) and the one obtained from a self-consistent-field calculation (SCO-RCG code [83]) for several subshells in different conditions is providedin table 5. Of course, since we assume Z ∗ = Z , the estimate gives poor results when there are fewremaining bound electrons, but the formalism presented here may be improved using screened nuclearcharges.Element Z ∗ Subshell T (eV) N e , SCO-RCG (cm − ) N e , Eq. (37) (cm − )Al 12.9 3 d × × Fe 25.7 5 f × × Fe 25.95 4 d × × Fe 19.5 4 d
500 3.22 × × Table 5: Critical electron density of pressure ionization for several subshells in different conditions.
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