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Scaling for state-selective charge exchange due to collisions of multicharged ions with hydrogen

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J. Phys. B: At. Mol. Opt. Phys. 48 (2015) 235201 (10pp) doi:10.1088/0953-4075/48/23/235201

Scaling for state-selective charge exchange due to collisions of multicharged ions with hydrogen

A Jorge1, Clara Illescas1, J E Miraglia2 and M S Gravielle2

1 Laboratorio Asociado al CIEMAT de Física Atómica y Molecular en Plasmas de Fusión. Departamento de Química, módulo 13, Universidad Autónoma de Madrid, Cantoblanco E-28049 Madrid, Spain

2 Instituto de Astronomía y Física del Espacio (IAFE, CONICET-UBA) Casilla de Correo 67, Sucursal 28, (C1428EGA) Buenos Aires, Argentina

E-mail: alba.jorge@uam.es

Received 13 April 2015, revised 1 August 2015 Accepted for publication 1 September 2015 Published 6 October 2015

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Abstract

In this article we evaluate state-resolved charge exchange cross sections for Be4+, B5+, C6+, N7+, and O8+ projectiles colliding with atomic hydrogen employing two different methods: the classical trajectory Monte Carlo and the eikonal impulse approximations. These cross sections are used to extend previously derived scaling laws for n-, ni-, and nim-distributions to highly excited final levels with 4 ^ n ^ 9, covering energies in the range 50 — 2000 keV/ amu. Present total and partial capture cross sections are in agreement with available experimental and theoretical data for these collision systems. Besides, the proposed scaling rules are also verified by other theories, becoming a useful instrument for plasma research.

Keywords: electron capture, scaling, multicharged ions, hydrogen, partial charge-exchange cross sections

(Some figures may appear in colour only in the online journal)

1. Introduction

Charge exchange (CX) in collisions of multiply charged ions with hydrogen atoms has attracted renewed attention in the last years. Such an interest is motivated by the importance of this process, also called electron capture, in both laboratory and astrophysical plasma environments [1]. In the case of laboratory magnetic confined plasmas, the CX between multicharged ions and neutral H plays an important role to estimate the temperature of the plasma and the density and charge of ionic impurities [2, 3], while in astrophysical plasmas it was found to be responsible for x-ray emission from comets due to the scattering of solar wind ions with cometary neutral gases, having been also detected in the heliosphere and in planetary atmospheres [4-6].

When multiply charged ions collide with H(1s) at intermediate and moderated high impact energies, electrons are mainly transferred to excited states of the ion, which then decay via photon emission giving rise to specific emission

lines. These emission spectra, extensively employed as a diagnostic tool in plasmas [7], strongly depend on the nl-distribution of transferred electrons [8, 9], making the state-selective electron capture cross section a key magnitude for the analysis of the experimental data. However, partial CX cross sections into different nl-sublevels are less studied than their corresponding total and partial n values and consequently, several plasma studies assume approximated l-dis-tributions for the different levels [10-12]. In this article we focus precisely on these state-selective CX cross sections, extending previously derived scaling laws for n-, nl-, and nlm-distributions [13] to deal with final capture levels up to the n = 9 shell.

Two different theoretical methods are employed: the classical trajectory Monte Carlo (CTMC) method and the eikonal impulse (EI) approximation. Each of these methods can be applied in a different region of impact energies. The CTMC method is a classical theory that results appropriate for impact velocities v in the range v ~ 1 — 3 a.u. [14], while the

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© 2015 IOP Publishing Ltd Printed in the UK

EI approach is a high-energy method, which can be extended to describe CX in the intermediate velocity range as well, i.e. for velocities v > 2 a.u. [13]. We use both approaches to evaluate state-selective CX cross sections for different ion-H collisions with the aim of providing reliable scaling rules that cover a wide range of incidence energies, from intermediate to high impact velocities. Notice that for highly charged ions colliding with H(1s), many scaling rules for total capture, and even for partial capture to a given n shell, have been proposed [15-18]. But for nl-distributions the studies on scaling laws are scarce [13, 19, 20]. Therefore, present state-selective scaling rules are useful tools to estimate partial contributions coming from specific final capture states, avoiding tedious calculations, specially for capture to highly excited levels.

In this article, the CTMC and EI methods are applied to evaluate CX cross sections for different bare ions—Be4+, B5+, C6+, N7+, and O8+-impinging on neutral hydrogen in its ground state. The range of considered projectiles is of interest in fusion and astrophysical research. Most of these ions are found as impurities in laboratory fusion plasmas, like beryllium and carbon that are expected to be present in ITER [21, 22], or nitrogen that is routinely introduced in the ASDEX tokamak to protect the plasma-facing components [23, 24]. Also in astrophysics, ions of carbon and oxygen are considered the predominant sources of soft x-ray emission in collisions involving the solar wind [4, 25, 26].

This article is organized as follows. The theoretical methods used in this work to investigate scaling rules for partial CX distributions are presented in section 2. In section 2.1 we summarize the parametric dependence of the CX cross section to a specific final state, obtained within the EI approximation, which is here employed to derive scaling laws for highly excited final states. While in section 2.2 we show how total capture cross sections derived within the CTMC and EI approximations merge in the intermediate velocity regime, allowing us to combine both theories to produce a join CTMC-EI approach that provides reliable results for a wide range of impact velocities, from 1 a.u to 6 a.u. Scaling rules for the corresponding partial cross sections are investigated in section 3, where scaled n-, nl- and nlm-distributions are presented and discussed in sections 3.1, 3.2 and 3.3, respectively. Finally, in section 4 we outline our conclusions. Atomic units (a.u.) are used unless otherwise stated.

2. Description of the theoretical methods

2.1. Parametric dependence of the EI capture cross section to a given final state

The prior version of the EI approximation is a distorted wave method that makes use of the exact impulse wave function to describe the final collision channel, while the initial channel is represented by means of the eikonal wave function [27]. It has been successfully applied to describe CX in asymmetric collisions, where the nuclear charge of one of the particles is larger than the other, at high and intermediate impact velocities. In particular, the prior EI approach was found

Figure 1. Total electron capture cross sections for Be4+, C6+ and O8+ ions impinging on H(1 s), as a function of the incidence energy. (a) (upper panel) Calculations derived with :-■-, EI; , CTMC approachs. (b) (lower panel): — , join CTMC-EI method; other

theories: H__я AOCC [32], + _ MOCC [33], A__д FBA

[34], ,__, AOCC [35]; expt.: 0 [18] and [36].

to provide proper electron capture predictions for both, multicharged ions colliding with light atoms (H and He) [13] and proton-atom collisions [28].

Following previous articles [13, 19], we use the prior EI approach to derive a scaling rule for the asymmetric reaction:

aZp + + h^ H+ + a(Zp-i)+(n i, m),

where the charge of the fully stripped projectile, ZP, is larger than the one of the target nucleus, ZT (with ZT = 1 for H), and n, l, m are the quantum numbers of the final electronic state, fn l m, bounded to the projectile with final electronic energy en,l. For this reaction the EI transition matrix reads [28]:

where is the impact velocity and WP = K — ¡PKf is the

projectile momentum transfer, with K (Kf ) the initial (final) momentum of the projectile and ¡p its final reduced mass. The functions L®-, IE+, and JE+ represent the Nordsieck-type integrals defined in [28], and fnlm (if ) is the Fourier transform of the final electronic wave function. From equation (2), using the explicit expressions of the L®-, IE+, and JE+ functions, the EI capture cross section into the n, l, m state, sEIim, is found to satisfy [13]

s Elm - z— I C (Zp/v)\ 2<Jnlm (v , W), (3)

(a) CTMC and EI

(b) Other theories

Figure 2. Scaled cross sections Sn, as a function of the scaled momentum W, for different n values in the range n = 4 — 9. Solid black line, proposed curve given by equation (8). Column (a) (left pannels), results for ZP = 4-8 derived within the considered approaches: # EI, CTMC, each of them in the corresponding velocity range, as indicated in the text. Column (b) (right pannels), results from other theories:

and H AOCC forZP = 4, 6, 8, n = 6, 4, 7, respectively [35]; D and ^ CTMC forZP = 4, n = 6, 7 [38]; <> and A CDW forZP = 6, n = 3, 4 [39]; FBA for ZP = 6, n = 5 [34]; , ^ and o CTMC for ZP = 6, n = 5 and ZP = 8, n = 6,7 [40]; , y, B AOCC for ZP = 5,6,7,8 and n = 5, 5, 4, 3 [41] and recommended ADAS sets for ZP = 6, n = 4, 8 [42].

where Zp = ZP/n represents the mean value of the final electron velocity and

C (a) = exp (pa/2) r (1 — ia) (4)

is the projectile-electron Coulomb factor coming from the asymptotic conditions, with r the gamma function. In equation (3) the function anlm depends on the scaled parameters vZ and WZ , denoted with tilde, which are defined as

v = v/zP and

respectively, with W = WP. ^/v the component of WP parallel to the incidence velocity and zT = Zj/Zp (zT = ZPl for H targets).

For positive W values, the function anim of equation (3) presents a weak dependence on vZ and consequently, it can be approximated by an universal function of WZ , depending only

of the final quantum numbers n, l, m; that is,

Slm (, W) » Unlm (w). (6)

In this article we will use this approximation, together with equation (3), to derive state-selective scaling laws for capture to highly excited levels, with main quantum numbers in the range 4 ^ n ^ 9. As stated in [13], the approximated relation of equation (6) starts to fail for impact velocities lower than 2 a.u. when the condition Zp > 1 is not verified.

2.2. Join CTMC-EI approach: total capture cross sections

In this section we inspect the validity of the used theoretical approaches—CTMC and EI—by comparing total capture cross sections for the studied collision systems with available experimental data and with results obtained from other theories.

tN_ 10-17

6 o 10-18

oB 1019

E (keV/amu)

Figure 3. Partial CX cross sections sn, as a function of the impact energy, for Ne10+ + H(1 s) collisions. Solid black line, present scaling; calculations: __ CTMC [44], one-electron-diatomic-molecule (OEDM) [44], AOCC [35].

The present CTMC calculations were computed using a superposition of ten standard microcanonical distributions, fitting the radial probability density of the atomic hydrogen [29]. Nuclear straight-line trajectories and a statistic of 2 x 105 electrons have been considered. The trajectories have been integrated up to tmax = 500v-1. Notice that capture to low n levels (n < 3) is well characterized by the usual microcanonical distribution, but for higher n levels, like those considered in this article, the hydrogenic density-fitted one provides a better description of the electron capture process [30, 31].

Concerning the EI approach, each electron capture cross section to a given n, l, m state was calculated separately, involving a three-dimensional integral on the momentum space, as given by equation (2), and a further integration on the final angular distribution of the scattered projectiles. For highly excited final levels, such numerical calculations were almost prohibitive twenty years ago, forcing us to use extrapolation rules [13], but now they can be done in a reasonable CPU time (for example, for the state (n, l, m) = (9, 8, 0) of carbon, it takes less than 3' in a PC I7 computer). In this article we evaluated EI total capture results by adding (n, l, m)- cross sections up the shell n = 9, while the contribution of final levels with n ^ 10 was estimated by using an extrapolation rule based on the n-dependence of the Brinkman-Kramer cross section, as given by equation (12) of [13].

In figure 1 (a) we display CTMC and EI total CX cross sections for Be4+, C6+, and O8+ colliding with H(1s), as a function of the impact energy, considering a different velocity range for each method: v ^ 3 a.u. for the CTMC method and v ^ 2 a.u. for the EI one. For the three projectiles, the CTMC and EI curves fairly agree each other in the intermediate energy region, between 100 and 200 keV amu-1, and in the case of beryllium, the accord also extend to lower energies. In order to quantify the level of concordance of both theories, we can mention that for C6+ projectiles the differences between CTMC and EI total capture cross sections at 100 keV amu-1 are around 15%, increasing up to 27% at 200 keV amu-1. This fact allows us to combine both theoretical methods by merging them around 100 keV amu-1 to build a join CTMC-EI approach, which provides realistic cross sections in a wide velocity region, ranging from intermediate to high incidence velocities.

Total cross sections derived within the join CTMC-EI method are shown in figure 1 (b) and compared with results obtained from other theories and with available experimental data. For all the projectiles the curve corresponding to the join CTMC- EI approximation is in good accord with values derived from different theories, like semiclassical close-coupling approaches with molecular (MOCC) [33] or atomic (AOCC) [32, 35] wave functions, or the first born approximation (FBA) [34. Join CTMC-EI results are also in agreement with available experimental data for carbon and oxygen projectiles in the intermediate energy regime [18, 36]. Such a concordance is an indication of the adequate description of the CX process obtained by means of the merge of the CTMC and the EI methods.

The same criterion is followed in the next section to combine the CTMC and EI theories also at the level of state-selective cross sections, where the differences between CTMC and EI n-resolved cross sections at 100 keV amu-1 are 20% at most for Be4+ projectiles, decreasing up to a relative error lower than 10% as ZP augments. At low velocities, the join CTMC-EI results verified that the most populated n-level, nmax, qualitatively agrees with the scaling law [37], but the value of nmax slowly decreases as the velocity increases, tending in all the cases to the ground level at very high velocities.

3. Results

3.1. Scaling law for partial n-resolved cross sections with n ^ 4 In this section we focus on the partial CX cross section into a

given final n level, sn, which is defined as sn — ^

From equations (3) and (6) approximated relation for sn:

snlm■

we obtain the following

Sn —

\C (Zp/v )

- Un (WW) — ^Unlm (W) (7)

where Un is expected to be an universal function of WW, depending on the main quantum number n. In previous articles [13, 19], this scaling rule was successfully tested for

0.2 —

I I If I/ I ¿\ > I' I »\

- If A

4 /// " fit nt

0.5 0.4 0.3 0.2 0.1 0 0.5 0.4 0.3 0.2 0.1 0

-' 1 ж l_r

n = 4/ S>

Jt 1 \V4 n=!

и = 8

J_Г;-г1 . I

I. I I

}%" = 4 l/T

1 = 2 _

0 1 2 3 4 5 6

W (a.u.)

Figure 4. Join CTMC-EI Pnl distributions as a function of W. Solid black line, present scaling, given by equation (9); results for different projectiles: . Be4+, B5+, B C6+, A N7+, ^ O8+. In each panel the curves shift from left to right as n increases.

partial capture into the fundamental and first excited levels, i.e., n = 1—4. In this work the rule is extended to consider final levels in the range n = 4 — 9. In addition, we found that when the scaled cross sections Sn for the different levels are plotted together, all the CTMC-EI results can be gathered in a relatively narrow band, indicating a weak dependence on n of the universal function, i.e. Un (W) — U (W) for 4 ^ n ^ 9, as shown in figure 2(a). In this figure we have split the WZ range in two panels (upper and lower) in order to display clearly the different results.

Furthermore, we provide a simple expression to represent the approximate function U (W), which is valid for final levels with principal quantum numbers in the range n = 4—9. The proposed function (for sn expressed in cm2) reads:

и (WW) =

CiW2)4

C2 WW2)4'

Figure 5. Pnl distributions, as a function of WW, for (a) l = n — 1, and (b) l = n — 2. Lines, scaled nl-populations for different n shells, derived within the join CTMC-EI method considering charges ZP = 4 — 8.

where a1 = 3.46 x 10—14 cm2, b1 = 2.63, c1 = 6.23 (a.u.)-2, a2 = 1.0 x 10—16 cm2, b2 = 2.21, c2 = 1.43 (a.u.)-2. This function, also plotted in figure 2, is valid for positive WZ values and can be used to evaluate capture from H(1s) to a

§ ID"1

, N7+ + H -> Nfi+(n/ = 54) !.....1 . . ___□ -

- 08+ + H -> 01+(nl = 83) .....1 . . 'n, -

E (keV/amu)

for high WW values and highly excited n levels, where theoretical results are particularly scarce.

In order to illustrate the utility of the n-resolved scaling law, we compare in figure 3 partial cross sections for Ne10+ projectiles, available in the literature, with results derived from equations (7) and ((8)), considering final n levels close to the most populated one. Results of figure 3 confirm that the proposed scaling provides reasonable predictions for projectiles with ZP > 8.

3.2. Scaling for nl-distributions

We investigate the nl-distributions, defined as Pnl = uni/sn, with snl the partial CX cross section into the nl subshell, which is obtained as snl = £m snlm. Using equations (3) and (6) we derive the following scaling rule for Pnl:

= ^ ~ Unl(WW) * £

Unlm (W)

Un (WW) '

Figure 6. Partial CX cross sections sni for: (a) ZP = 6, nl = 43; (b) ZP = 7, nl = 54; (c) ZP = 8, nl = 83. Solid black line, present scaling; calculations: # EIA, B CTMC, A AOCC [35], 4 CDW [39], CTMC [40], * FBA [34], □ CDW [43], B AOCC [41].

given n-shell through equation (7) considering velocities v > 2 a.u., as well as lower energies (i.e. E ^ 70 keV/ amu) if the inequality zp > 1 is satisfied. Notice that all the CTMC-EI results displayed in figure 2(a) (more than 500 Sn-values that cover an extended cross section range of almost 9 orders of magnitude) can be approximated by equation (8) with a maximum relative error of 40% at most, decreasing as the impact velocity augments.

Other theoretical methods also verify the universal behavior given by equation ((8), as shown in figure 2 (b), where scaled partial cross sections Sn, defined by equation (7), are plotted as a function of WW for different sets of theoretical data: those of Igenbergs et al [35] and Toshima and Tawara [41] obtained with the AOCC; those of Schultz et al [38] and Olson et al [40] obtained with the CTMC; those of Mandal et al [39] and Saha et al [43] obtained with the continuum distorted wave (CDW); and those of Belkic et al [34] obtained with the FBA. We observe that the different theoretical and experimental data lie again inside a narrow band around the universal curve of equation (8), assuring the validity of the proposed scaling. We have also included in figure 2 (b) results for C6+ projectiles and final capture levels n = 4-8 extracted from the ADAS website [42], considering the energy range E ^ 70 keV/amu. Differences between the recommended ADAS data for the intermediate and high energy regions and the ones derived from the universal curve were found lower than 15% and 10%, respectively. Notice that the scaling rule of equation (8) becomes specially useful

where Unl is assumed as an universal function. In figure 4 we plot CTMC and EI results for the Pnl distributions corresponding to Be4+, B5+, C6+, N7+, and O8+ projectiles, as a function of WW, considering final n levels from n = 4 to n = 9. While the CTMC method is used to obtain the nl-distributions for the lower values of WW, i.e. WW ^ 1.5 a.u., the EI approach is employed to describe the high energy region, which corresponds to WW ^ 2.0 a.u. In all the cases, the Pnl values derived from the CTMC approach match with those obtained with the EI theory in the intermediate region, which corresponds to scaled transferred momenta WW in the range between 1.5 and 2.0 a.u. Therefore, it is possible to derive practically universal Unl functions for the different nl- subshells, which can be used to estimate partial angular momentum distributions for ionic projectiles with larger nuclear charges than the ones considered here. To build these curves we have used the join CTMC-EI method, connecting the results of both theories at an impact velocity v » 2.2 a.u.

As a general tendency we found that for low W values, in the range W < 0.5 a.u., the capture proceeds to the subshell with the highest orbital momentum, i.e. l = n — 1, but the lvalue corresponding to the maximal contribution decreases as the scaled momentum WW augments, as shown in figure 5, where the Pnl distributions for the orbital quantum numbers l = n — 1 and l = n — 2 are displayed as a function of WW. Even though all the curves for l = n — 1 (or l = n — 2) present similar shapes, the population of the nl-subshell also depends on the main quantum number n, since as n increases, the position of the maximum slightly shifts towards lower WW values and its relative contribution diminishes. This behavior is expected to hold also for higher ZP charges.

With the aim of providing a more precise representation of the Pnl functions for the different nl-subshells, numerical values corresponding to the universal distributions obtained from the join CTMC-EI method are tabulated in tables 1-6 in the appendix. These tables can be easily used to compute a

1 2 3 1 2 3 4

W (a.u.)

Figure 7. Pnlm distributions derived with the EI approach, for n = 6 as a function of WW. Symbols: • Be4+, x B5+, q] C6+, A N7+, * O8+.

partial nl cross section for a given electronic transition and impact velocity. For example, <r43 for the C6+ + H ^ C5+(nl = 43) process. First, we have to calculate s4 by applying equation (7) with ZP = 6 and zp = 6/4 as follows:

04 = ^ |C (6/v)|2 U (W), v2 + (4/6)2 - 1

with W =

3.3. Scaling for nlm-distributions

Finally, we study the scaling for the nlm-distributions, defined as Pnim = &nim/snl, which are expected to verify:

Pnim = 2 "Unim (í^),

and U (W) from equation (8). Once we have computed s4, we only need to use equation (9) and table 1 to obtain s43 as s4 x P43. As an illustrative example, in figure 6 we display partial CX cross sections, derived from the proposed scaling functions, as a function of the impact energy, considering: (a) ZP = 6, n = 4, l = 3; (b) ZP = 7, n = 5, l = 4; (c) ZP = 8, n = 8, l = 3. These particular cases were only chosen as a proof of the efficacy of the proposed scaling rule. In all the cases, the scaling- derived curve, displayed with a black line, runs close to the CTMC and EI results, as well as to the theoretical data extracted from [35, 46]. Only for the O8+ example with n = 8, the scaling slightly overestimates the data for the lowest energies, since the condition ZP/n > 1 is not verified.

where Unlm denotes again an universal function.

As an illustration, in figures 7 and 8 we plot the magnetic number distribution for the n = 6 and n = 7 shells, considering the quantization axis along the direction of the incidence velocity. In all the cases electrons are mainly captured to final states oriented parallel to the velocity, which correspond to m = 0, while the contribution of other final state orientations decreases as m increases. This is a general behavior, also observed for other n shells.

4. Conclusions

We have used two distinct approximations, CTMC and EI, in order to describe the CX process between multicharged ions and atomic hydrogen for a wide range of impact energies, encompassing from intermediate to high velocities. For five different bare projectiles, total and state-resolved CX

0,75 0,5 0,25

pf 0,25 0

0,3 0,2 0,1 0

-t * -H?

' ^t^U * * x» xi1' it ■ K-

^ 7f2 7f3

* » «¿WK »AXAH X

7g2 7g3 7g4

....... ......

a" X m A Mm AXA)KX» >3^*

Xifc >»" x ft

1 2 3 1 2 3 4

W (a.u.)

Figure 8. Pn1m distributions derived with the EI approach, for n = 7, as a function of W. Symbols: • Be4+, x B5+, q] C6+, A N7+, * O8+.

cross sections derived with these methods show a good agreement with available experimental and theoretical data, being indicative of the accuracy of the considered descriptions. In addition, the fact that CTMC and EI state-selective cross sections for capture to highly excited levels, from n = 4 to n = 9, match between them in the intermediate energy region allowed us to extend previously derived scaling laws for shell and sub-shells capture distributions [13, 19] along the studied energy range. In the case of the n-resolved scaling, an analytical universal function in terms of the scaled transferred momentum WW is suggested, while for the nl-distributions, values obtained from the proposed scaling are listed in the appendix to facilitate their interpolation. These scaling rules are important tools, not only to display partial cross sections for different multicharged projectiles in a comprehensive way, but also to predict n-, nl-, and even nlm-distributions originated by highly charged ions.

Acknowledgments

MSG and JEM acknowledge financial support from CONICET, UBA, and ANPCyT of Argentina. A J and C I financial support from the projects ENE2011-28200 and ENE2014—52432-R of the Secretaría de Estado de Investigación, Desarrollo e Innovación (Spain).

Appendix A. Appendix: Pnl tables

Numerical values for the distribution corresponding to equation (9), derived from the proposed scaling rule, are listed in the following tables.

Table 1.100 x Un¡ values for n = 4.

wh Ao P41 P42 P43

0.13 1 9 25 64

0.31 2 9 23 66

0.44 2 10 25 64

0.50 2 10 26 62

0.60 2 9 30 57

0.67 2 8 36 52

0.80 3 10 44 42

0.88 4 12 48 36

0.99 3 19 50 28

1.06 2 22 50 25

1.23 2 32 48 17

1.40 4 41 43 12

1.57 7 47 38 8

1.74 11 50 33 6

1.90 15 52 29 5

2.07 19 53 25 3

2.23 23 53 21 3

2.39 28 52 19 2

2.55 31 51 16 2

2.68 34 50 14 1

Table 2.100 x Unl values for n = 5.

P50 P51 P52 P53 P54

0.10 1 5 13 26 55

0.30 1 5 12 25 57

0.38 1 5 12 25 56

0.57 1 6 14 29 49

0.67 1 7 16 32 42

0.74 1 7 17 40 37

0.89 2 6 21 45 26

0.98 2 6 26 44 20

1.14 3 8 35 40 14

1.20 3 10 38 38 11

1.41 2 18 44 29 6

1.62 2 28 45 22 4

1.83 4 37 42 16 2

2.03 6 43 38 12 1

2.23 10 47 34 9 1

2.43 14 49 30 6 0

2.64 18 51 26 5 0

2.84 22 51 23 4 0

3.04 26 50 20 3 0

3.23 30 50 17 2 0

Table 3.100 x Unl values for n = 6.

WP P60 P61 P62 P63 P64 P65

0.08 0 2 7 16 26 48

0.32 1 3 8 14 24 50

0.60 1 4 9 17 29 40

0.78 1 5 11 22 33 28

0.96 1 7 14 27 38 16

1.01 1 7 15 29 36 13

1.16 3 6 20 35 31 8

1.26 3 6 23 39 26 6

1.41 3 7 28 39 19 3

1.51 3 8 34 37 16 2

1.76 2 16 41 30 9 1

2.00 2 26 43 23 6 0

2.24 3 35 41 18 3 0

2.48 6 41 37 13 2 0

2.72 9 46 34 10 1 0

2.96 13 48 30 8 1 0

3.20 18 49 26 6 1 0

3.44 22 50 23 4 0 0

3.67 26 50 20 4 0 0

3.91 30 49 18 3 0 0

Table 4.100 x Unl values for n = 7.

P70 P71 P72 P73 P74 P75 P76

0.30 0 2 5 10 14 23 45

0.52 0 3 6 10 16 26 38

0.81 1 4 8 14 23 31 21

1.00 1 4 10 18 28 28 10

1.16 1 5 12 23 33 23 6

1.24 1 6 13 26 35 19 4

1.41 2 6 18 31 32 12 2

1.50 2 6 20 34 28 10 1

1.73 3 6 28 37 20 5 1

Table 4. (Continued.)

WPz P70 P71 P72 P73 P74 P75 P76

1.81 3 8 31 36 18 4 0

2.09 2 15 39 30 11 2 0

2.37 2 25 41 24 7 1 0

2.65 3 33 40 18 4 0 0

2.92 6 40 37 14 3 0 0

3.20 9 44 34 11 2 0 0

3.48 13 47 30 8 1 0 0

3.75 17 49 27 6 1 0 0

4.03 22 50 24 5 1 0 0

4.31 26 49 21 4 0 0 0

4.58 30 49 18 3 0 0 0

Table 5.100 x Uni values for n = 8.

WPz P80 P81 P82 P83 P84 P85 P86 P87

0.70 0 2 5 9 13 19 26 25

1.00 1 3 7 12 20 27 23 7

1.13 1 4 8 15 23 27 18 3

1.19 1 4 9 17 25 26 16 2

1.39 1 6 12 22 29 23 10 0

1.48 1 6 13 25 30 20 7 1

1.66 2 5 16 31 31 14 3 0

1.79 3 4 19 35 28 11 2 0

1.99 3 6 25 36 22 7 1 0

2.11 3 7 30 35 19 5 1 0

2.42 2 15 38 31 12 3 0 0

2.74 2 24 40 25 7 1 0 0

3.05 3 33 40 19 5 1 0 0

3.36 6 39 37 15 3 0 0 0

3.67 9 44 34 11 2 0 0 0

3.99 13 47 30 9 1 0 0 0

4.31 17 48 27 7 1 0 0 0

4.62 21 49 24 5 1 0 0 0

4.94 25 49 21 4 0 0 0 0

5.25 30 48 18 3 0 0 0 0

Table 6.100 x Unl values for n = 9.

WPz P90 P91 P92 P93 P94 P95 P96 P97 P98

0.68 0 1 4 7 9 12 18 25 23

0.88 0 2 5 8 12 17 23 22 11

1.12 1 3 7 10 17 23 22 14 3

1.29 1 4 8 13 21 25 20 9 1

1.41 1 4 9 16 24 25 18 5 1

1.68 1 4 12 23 29 23 10 2 0

1.91 2 5 16 29 30 16 5 1 0

2.05 3 5 19 33 28 12 3 0 0

2.30 3 6 26 35 22 7 1 0 0

2.40 3 7 29 35 19 6 1 0 0

2.75 2 14 37 31 13 3 0 0 0

3.10 2 24 40 25 8 1 0 0 0

3.45 3 32 39 19 5 1 0 0 0

3.80 6 39 37 15 3 0 0 0 0

4.15 9 43 34 11 2 0 0 0 0

4.51 13 46 30 9 1 0 0 0 0

4.86 17 48 27 7 1 0 0 0 0

5.21 21 49 24 5 1 0 0 0 1

5.57 25 48 21 4 0 0 0 0 1

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