Four-body methods for high-energy ion-atom collisions

The progress in solving problems involving nonrelativistic fast ion (atom)-atom collisions with two actively participating electrons is reviewed. Such processes involve, e.g., (i) scattering between a bare nucleus (projectile) $P$ of charge $Z_P$ and a heliumlike atomic system consisting of two electrons $e_1$ and $e_2$ initially bound to the target nucleus $T$ of charge $Z_T$, i.e., the $Z_P\text{−}{(Z_T;e_1,e_2)}_i$ collisions; (ii) scattering between two hydrogenlike atoms ${(Z_P,e_1)}_{i_1}$ and ${(Z_T,e_2)}_{i_2}$, etc. A proper description of these collisional processes requires solutions of four-body problems with four active particles including two nuclei and two electrons. Among various one- as well as two-electron transitions which can occur in such collisions, special attention will be paid to double-electron capture, simultaneous transfer and ionization, simultaneous transfer and excitation, single-electron detachment and single-electron capture. Working within the four-body framework of scattering theory and imposing the proper Coulomb boundary conditions on the entrance and exit channels, the leading quantum-mechanical theories are analyzed. Both static and dynamic interelectron correlations are thoroughly examined. The correct links between scattering states and perturbation potentials are strongly emphasized. Selection of the present illustrations is dictated by the importance of interdisciplinary applications of energetic ion-atom collisions, ranging from thermonuclear fusion to medical accelerators for hadron radiotherapy.

Figure 1

Total cross sections $Q\left({\mathrm{cm}}^2\right)$ for double-charge exchange in the collisional reaction ${\mathrm{H}}^{+}+\mathrm{He}\left(1s^2\right)\to {\mathrm{H}}^{-}\left(1s^2\right)+{\mathrm{He}}^{2+}$. Theory: solid curve: the CDW-4B method (36) [the wave functions of 145]; dashed curve: the CDW-4B method (36) [the wave functions of 135]; dotted curve: the three-state two-center close coupling approximation of 143, and singly chained curve: the first Born approximation (113). Experimental data: ● (208, ◻ (228, ▿ (94), and ○ (236).

Figure 2

Total cross sections, as a function of the incident energy $E\left(\mathrm{keV}\right)$ for reaction (163). The theoretical curves relate only to the transition $1s^2\to 1s^2$. Doubly-chained curve: the CB1-4B method (18, 19); solid curve: the BCIS-4B method (20); dotted curve: the BDW-4B method (40); singly chained curve: the CDW-4B1 method (40); dashed curve: the CDW-4B2 method (162). All computations are performed by using the one-parameter 135 orbital for the initial and final states of helium. Experimental data: ○ (189), ▵ (166), ▿ (85), ◻ (72), and ● (209).

Figure 3

Total cross sections $Q\left({\mathrm{cm}}^2\right)$ in the CDW-4B method for the reaction ${\mathrm{Li}}^{3+}+\mathrm{He}\left(1s^2\right)\to {\mathrm{Li}}^{+}\left(\Sigma \right)+{\mathrm{He}}^{2+}$ as a function of the incident energy. The solid curve corresponds to double-electron capture into the ground and all singly excited states, whereas the dashed curve corresponds to double-electron capture into the ground state only (106). The upper singly chained curve corresponds to the results from the CDW-IPM for double-electron capture into all states (109). Experimental data: ● (213).

Figure 4

Total cross sections $Q\left({\mathrm{cm}}^2\right)$ in the CDW-4B method for the reaction ${\mathrm{B}}^{5+}+\mathrm{He}\left(1s^2\right)\to {\mathrm{B}}^{3+}\left(\Sigma \right)+{\mathrm{He}}^{2+}$ as a function of the incident energy. The solid curve corresponds to double-electron capture into the ground and all singly excited states, whereas the dashed curve corresponds to double-electron capture into the ground state only (106). The upper singly chained curve represents the results from the CDW-IPM for double-electron capture into all states (109). Experimental data: ● (129).

Figure 5

Differential cross sections $dQ∕d\Omega \equiv {(dQ∕d\Omega )}_{\mathrm{lab}}({\mathrm{cm}}^2∕\mathrm{sr})$ as a function of the scattering angle ${\theta }_{\mathrm{lab}}$ for reaction (163) at incident energy $E=1.5\mathrm{MeV}$. The computations relate only to the $1s^2\to 1s^2$ transition. Theoretical results: (i) solid curve: the BDW-4B method (40); (ii) dashed curve: the BCIS-4B method (20); (iii) dotted curve: the CB1-4B method (18, 19). All computations have been performed using the one-parameter Hylleraas wave function for the initial and final bound states of helium. The computed cross sections are not convolved with the experimental resolution function. Experimental data (including double-electron capture into all bound states of helium): ○ (209).

Figure 6

Total cross sections $Q\left({\mathrm{cm}}^2\right)$ as a function of laboratory incident energy $E\left(\mathrm{keV}\right)$ for double-charge exchange in the reaction ${}^7\mathrm{Li}^{3+}+{}^4\mathrm{He}\left(1s^2\right)\to {}^7\mathrm{Li}^{+}+{}^4\mathrm{He}^{2+}$. The theoretical curves relate only to the $1s^2\to 1s^2$ transition. Both the initial and final heliumlike bound states are described by the one-parameter orbital of 135. The CB1-4B method: the prior $Q_{if}^{-}$ and the post $Q_{if}^{+}$ total cross sections are represented by the dashed and solid curves, respectively (19). Experimental data (including double-electron capture into all bound states of ${\mathrm{Li}}^{+}$): ○ (213).

Figure 7

Total cross sections $Q\left({\mathrm{cm}}^2\right)$ as a function of the laboratory incident energy $E(\mathrm{keV}∕\mathrm{amu})$ for transfer ionization ${}^4\mathrm{He}^{2+}+{}^4\mathrm{He}\left(1s^2\right)\to {}^4\mathrm{He}^{+}\left(1s\right)+{}^4\mathrm{He}^{2+}+e$. The solid curve represents the post total cross section of the CDW-4B method (38). The dashed curve represents the post cross sections of the CDW-IEM (87). Experimental data: ◻ (215) and ○ (213).

Figure 8

Total cross sections $Q\left({\mathrm{cm}}^2\right)$ as a function of laboratory incident energy $E(\mathrm{keV}∕\mathrm{amu})$ for transfer ionization ${}^4\mathrm{He}^{2+}+{}^4\mathrm{He}\left(1s^2\right)\to {}^4\mathrm{He}^{+}\left(1s\right)+{}^4\mathrm{He}^{2+}+e$. The solid and dashed curves correspond, respectively, to the post and prior cross sections of the CDW-4B method (38). Experimental data: ◻ (215) and ○ (213).

Figure 9

Total cross sections $Q$ $\left({\mathrm{cm}}^2\right)$ as a function of the laboratory incident energy $E$ (keV/amu) for the reaction ${\mathrm{Li}}^{3+}+\mathrm{He}\to {\mathrm{Li}}^{2+}+{\mathrm{He}}^{2+}+e$. The solid and dashed curves represent, respectively, the prior $Q_{if}^{-}$ and post $Q_{if}^{+}$ cross sections of the CDW-4B method (151) with the complete perturbation potentials. The dotted curve represents a quantum field method of the second order (48). Experimental data: ◼ (213) and ● (239).

Figure 10

Total cross sections $Q$ (${\mathrm{cm}}^2$) for single-electron detachment from ${\mathrm{H}}^{–}\left(1s^2\right)$ by ${\mathrm{H}}^{+}$. Impact energy $E$ (keV) is in the laboratory reference system. Theories (in the prior versions), with only $\mathrm{H}\left(1s\right)$ in the exit channel: the ECB method (104) and the MCB method (24, 25). Integers $N$ with the acronyms ECB and the MCB represent the numbers of variational parameters in the target CI wave functions, which are taken from 219 for $N$=2 as well as $N$=3 and from 140 for $N$=61. Experimental data, with all the final states H($\Sigma$) in the exit channel: ◻ (186) and ○ (172). The original data of 186 are for electron impact, and here they are scaled to the equivalent proton impact energy. The ECB and MCB methods share the same properly behaving scattering wave functions, but differ sharply in their initial perturbation potentials $V_i^{\left(\mathrm{ECB}\right)}$ and $V_i^{\left(\mathrm{MCB}\right)}$, where the former is incorrect, whereas the latter is correct (see the text). More details are given by 29.

Figure 11

Differential cross sections ${(dQ∕d\Omega )}_{\mathrm{c.m.}}$ at proton energies 0.293, 2, and 7.4 MeV in the post version of the CDW-4B method for single-electron capture from $\mathrm{He}\left(1s^2\right)$ by ${\mathrm{H}}^{+}$ (26, 27). Dotted curves: the results from the full perturbation $V_f$ summing the contributions from (i) $–{\vec{\nabla }}_{s_1}\ln {\phi }_f^{*}\left({\vec{s}}_1\right)\bullet {\vec{\nabla }}_{x_1},$(ii) $\Delta V_{12}=1∕r_{12}–1∕x_1,$ and (iii) $V_P(R,s_2)=1∕R–1∕s_2$. Dashed and solid curves: the separate contributions from (i) and (ii), respectively. To avoid clutter, the contribution from (i) is shown only at 7.4 MeV. Perturbations (i) and (ii) yield the Thomas double scatterings of the 1st and 2nd kind, ${\mathrm{H}}^{+}-e-{\mathrm{He}}^{2+}$ (at sufficiently high energies) and ${\mathrm{H}}^{+}-e-e$ (at all energies). The critical Thomas angle is indicated by the arrow at ${\theta }_{\mathrm{c.m.}}$=0.027 deg. The ${\mathrm{H}}^{+}-e-{\mathrm{He}}^{2+}$ and the ${\mathrm{H}}^{+}-e-e$ peaks always appear with and without splitting, respectively. A dip in the Thomas peak is unphysical, as it has never been detected experimentally.

Figure 12

Total cross sections $Q$ $\left({\mathrm{cm}}^2\right)$ for single-electron capture from helium by protons as a function of laboratory impact energy obtained by means of CDW-4B method (31). The wave function of 219 is used for the initial bound state. The solid and dashed curves correspond to the cases where the potential $\Delta V_{12}$ is included and excluded from the complete perturbation $V_f$, respectively. Experimental data: ▿ (198), ▵ (215), ○ (213), ● (130), ◇ (46), ▴ (237), ▾ (159), and ◆ (234).

Figure 13

Total cross sections $Q$ $\left({\mathrm{cm}}^2\right)$ as a function of laboratory incident energy $E$ (keV) for the reaction ${}^4\mathrm{He}^{2+}+{}^4\mathrm{He}\left(1s^2\right)\to {}^4\mathrm{He}^{+}\left(\Sigma \right)+{}^4\mathrm{He}^{+}\left(1s\right)$. The solid and dotted curves represent the post cross sections $Q_{if}^{+}$ of the CDW-4B method (149) with the complete perturbation potential and without the potential $(1∕r_{12}-1∕x_1)$, respectively. The results of the CDW-4B method are obtained using the wave function of 135 for the ground state of the helium target atom. The dashed curve represents the results of the CDW-IEM of 87 based upon the Pluvinage wave function with the corresponding theoretical binding energy. Experimental data: ○ (85), ▿ (215), ◇ (175), ● (213), ◻ (188), ▵ (134), and ◼ (72).

Figure 14

Total cross sections $Q$ $\left({\mathrm{cm}}^2\right)$ as a function of laboratory incident energy $E$ (keV) for the reaction ${\mathrm{H}}^{+}+{}^7\mathrm{Li}^{+}\left(1s^2\right)\to \mathrm{H}\left(\Sigma \right)+{}^7\mathrm{Li}^{2+}\left(1s\right)$. The solid curve represents the post cross sections $Q_{if}^{+}$ from the CDW-4B method with the complete perturbation potential. The dashed curve represents the results from the CDW-4B method obtained without the term $(1∕r_{12}-1∕x_1)$ in Eq. (197). Experimental data: ● (212).

Figure 15

Total cross sections $Q$ $\left({\mathrm{cm}}^2\right)$ as a function of laboratory incident energy $E$ (keV/amu) for the reaction ${\mathrm{Li}}^{3+}+\mathrm{He}\to {\mathrm{Li}}^{2+}+{\mathrm{He}}^{+}$. The solid and dashed curves represent the post cross sections $Q_{if}^{+}$ of the CDW-4B method (151) with the complete perturbation potential and without the potential $(1∕r_{12}-1∕x_1)$, respectively. The symbol ▵ refers to the post total cross sections $Q_{\Sigma }^{+}$ obtained from Eq. (204), whereas the two curves correspond to capture into the ground state and the contribution from exited states is accounted by 1.202 as an overall multiplying factor. The results are obtained using the orbital of 135 for the ground state of the helium target atom. Experimental data: ◼ (213) and ● (239).

Figure 16

Total cross sections $Q$ $\left({\mathrm{cm}}^2\right)$ as a function of laboratory incident energy $E$ (keV) for the reaction ${}^4\mathrm{He}^{2+}+{}^4\mathrm{He}\left(1s^2\right)\to {}^4\mathrm{He}^{+}\left(\Sigma \right)+{}^4\mathrm{He}^{+}\left(1s\right)$. The solid curve represents the cross sections $Q$ of the CDW-BFS method (152). The dashed curve represents the results of the prior form of the CDW-4B method (149). Experimental data: ○ (85), ▿ (215), ◇ (175), ● (213), ◻ (188), ▵ (134), and ◼ (72).

Figure 17

Total cross sections $Q$ $\left({\mathrm{cm}}^2\right)$ as a function of laboratory energy $E$ (keV) for the reaction ${\mathrm{H}}^{+}+{}^4\mathrm{He}\left(1s^2\right)\to \mathrm{H}\left(\Sigma \right)+{}^4\mathrm{He}^{+}$. The solid curve represents the results from the CDW-BFS method with the full perturbation potential (152). The dashed curve represents the results of the CDW-BFS method obtained without the potential $V_P(R,s_2)=Z_P(1∕R-1∕s_2)$. Experimental data: ▿ (215), ◼ (46), ○ (211), ● (234), ◻ (198), ◇ (213), ▴ (4), ▾ (130), and ▵ (198). Theory (CC): ⊞ (238).

Figure 18

Differential cross sections ${(dQ∕d\Omega )}_{\mathrm{c.m.}}$ $({\mathrm{cm}}^2∕\mathrm{sr})$ as a function of the scattering angle ${\theta }_{\mathrm{c.m.}}$ (rad) for single-electron capture in the ${\mathrm{H}}^{+}$-He collisions at a proton impact energy of $100\mathrm{keV}$. Cross sections and scattering angles are in the center-of-mass system. Solid curve: the results of the CDW-BFS method with the full perturbation potential (156). Dashed curve: the results of the CDW-BFS method obtained without the potential $V_P(R,s_2)=Z_P(1∕R-1∕s_2)$. Experimental data: ● (159).

Figure 19

Differential cross sections ${(dQ∕d\Omega )}_{\mathrm{lab}}$ (a.u./sr) for single-electron capture in the ${\mathrm{H}}^{+}$-He collisions as a function of the scattering angle ${\theta }_{\mathrm{lab}}$ (rad) at a proton impact energy of $150\mathrm{keV}$. Solid curve: the results of the CDW-BFS method with the full perturbation potential (156). Dashed curve: the results of the CDW-BFS method obtained without the potential $V_P(R,s_2)=Z_P(1∕R-1∕s_2)$. Experimental data: ◼ (174).

Figure 20

The same as in Fig. 19, except for $200\mathrm{keV}$. The additional symbols denoted by ○ represent the experimental data of 144.

Figure 21

The same as in Fig. 20, except for $300\mathrm{keV}$.

Figure 22

Differential cross sections ${(dQ∕d\Omega )}_{\mathrm{lab}}$ (a.u./sr) for single-electron capture in the ${\mathrm{H}}^{+}$-He collisions as a function of the scattering angle ${\theta }_{\mathrm{lab}}$ (rad) at $400\mathrm{keV}$. Cross sections and scattering angles are in the laboratory system. Solid curve: the results of the CDW-BIS method (153, 154). Dashed curve: the CDW-BFS method (152; 156). Experimental data: ◼ (174) and ○ (144).

Figure 23

Differential cross sections ${(dQ∕d\Omega )}_{\mathrm{lab}}$ (a.u./sr) for single-electron capture in the ${\mathrm{H}}^{+}$-He collisions as a function of the scattering angle ${\theta }_{\mathrm{lab}}$ (rad) at a proton impact energy of $630\mathrm{keV}$. Cross sections and scattering angles are in the laboratory system. Solid curve: the results of the CDW-BIS method (153, 154). Dashed curve: the results of the CDW-BFS method (152; 156). Experimental data: ◼ (174).

Figure 24

Total cross sections $Q$ $\left({\mathrm{cm}}^2\right)$ for single-electron capture in the collisional reaction $\mathrm{H}+\mathrm{H}\to {\mathrm{H}}^{-}+{\mathrm{H}}^{+}$ as a function of incident energy. Solid curve: the CB1-4B method (147); dashed curve: the first Born approximation (157, 158). Experimental data: ● (165) and ○ (208).

Figure 25

Total cross sections $Q$ $\left({\mathrm{cm}}^2\right)$ for single-charge exchange in the reaction ${}^4\mathrm{He}^{+}+\mathrm{H}\to {}^4\mathrm{He}+{\mathrm{H}}^{+}$ as a function of incident energy. Dashed curve: the CB1-4B method (147). Solid curve: the CDW-4B method (155). Both computations refer to the post forms carried out using the two-parameter wave function of 219 for helium. Experimental data: ● (180), ◼ (133), and ▿ (214).

Figure 26

Total cross sections $Q$ $\left({\mathrm{cm}}^2\right)$ as a function of laboratory incident energy for symmetric single-charge exchange ${}^4\mathrm{He}^{+}+{}^4\mathrm{He}\to {}^4\mathrm{He}+{}^4\mathrm{He}^{+}$. Dashed curve: the CB1-4B method (148, 155). Solid curve: the CDW-4B method (155). Both computations refer to the post forms carried out using the two-parameter wave function of 219 for helium in the exit channel and the RHF wave function for helium in the entrance channel. Experimental data: ◼ (96), ◻ (72) ○ (86), ▿ (5), ▾ (136), and ▵ (188).

Figure 27

Total cross sections $Q$ $\left({\mathrm{cm}}^2\right)$ as a function of laboratory incident energy $E$ (keV) for single-charge exchange $\mathrm{H}+{}^4\mathrm{He}\to {\mathrm{H}}^{-}+{}^4\mathrm{He}^{+}$. The solid curve is due to the post version of the CB1-4B method as computed by 148 using the two-parameter wave function of 219 for the negative hydrogen ion ${\mathrm{H}}^{-}\left(1s^2\right)$ in the exit channel and the RHF wave function for target helium in the entrance channel. Experimental data: ● (208).

Figure 28

Total cross sections $Q$ $\left({\mathrm{cm}}^2\right)$ as a function of laboratory incident energy for asymmetric single-charge exchange ${}^7\mathrm{Li}^{2+}+{}^4\mathrm{He}\to {}^7\mathrm{Li}^{+}+{}^4\mathrm{He}^{+}$. Dashed curve: the CB1-4B method (147). Solid curve: the CDW-4B method (155). Both computations refer to the post forms carried out using the two-parameter wave function of 219 for the heliumlike ion ${\mathrm{Li}}^{+}\left(1s^2\right)$ in the exit channel and the RHF wave function for target helium in the entrance channel. Experimental data: ◼ (239).

Figure 29

The ${\mathrm{Ca}}^{17+}\text{−}\mathrm{He}$ collisional system. Shown are the RTEX peaks for these asymmetric collisions $(Z_P⪢Z_T)$. Theoretical curve (171) and experimental data (circles) (224).

Figure 30

The ${\mathrm{Nb}}^{31+}\text{−}{\mathrm{H}}_2$ collisional system. Shown are the RTEX peaks for these asymmetric collisions $(Z_P⪢Z_T)$. Theoretical curve (171) and experimental data (circles) (224).

Figure 31

The RTEX and NTEX cross sections for the collisional ${\mathrm{S}}^{13+}\text{−}\mathrm{He}$ system. Theoretical curves (125) and experimental data (circles) (223). The results shown by the curve NTEX are multiplied by 3.

Figure 32

Total cross sections for production of the $K\alpha \text{−}K\alpha$ lines from ${\mathrm{S}}^{14+}$. Dashed curve: the DR model (141). Solid curve: the CDW-4B method (8). Solid squares: experimental data (141).

Figure 33

State-selective total excitation cross sections in the CDW-4B method for the ${\mathrm{S}}^{15+}\text{−}\mathrm{H}$ system (8). $\left(2s^2\right){}^{1}S$: singly chained curve; $\left(2p^2\right){}^{1}S$: doubly chained curve; $\left(2p^2\right){}^{1}D_0$: solid curve; $\left(2p^2\right){}^{1}D_{+1}$: dotted curve; $\left(2p^2\right){}^{1}D_{-1}$: dashed curve.

Figure 34

Total cross sections for production of the $K\alpha \text{−}K\beta$ lines from $S^{14+}$. The solid curve represents the CDW-4B method (8), and solid squares are the experimental data (141).

Figure 35

Total cross sections in the CDW-4B method for excitation of the $\left(2p^2\right){}^{1}D_0$ state of ${\mathrm{S}}^{14+}$ in a collision of ${\mathrm{S}}^{15+}$ with atomic hydrogen target $(Z_T=1)$ (8). Dotted curve: $Q_{1D_0}^{\mathrm{NTEX}}$; dashed curve: $Q_{1D_0}^{\mathrm{RTEX}}$; solid curve: $Q_{1D_0}^{\mathrm{TEX}}$; doubly-chained curve: CDW-IPM (103). In this process, the RTEX and TEX total cross cross sections practically coincide with each other, due to a totally negligible influence of the NTEX mode, $Q_{1D_0}^{\mathrm{TEX}}\simeq Q_{1D_0}^{\mathrm{RTEX}}$ (solid and dashed curves indistinguishable). Here we set $1D_0\equiv {}^{1}D_0$.

Figure 36

The same as in Fig. 35, except for $Z_T=10$.

Figure 37

The total cross section in the CDW-4B method for the RTEA, NTEA, and TEA modes of the collision ${\mathrm{He}}^{+}\left(1s\right)+\mathrm{H}\left(1s\right)\to \mathrm{He}\left(2s2p{}^{1}P_0\right)+{\mathrm{H}}^{+}$ (105, 107). Singly-chained curve: $Q_{\mathrm{RTEA}}$. Dashed curve: $Q_{\mathrm{NTEA}}$. Solid curve: $Q_{\mathrm{TEA}}$. The solid circles are the experimental data (241) with an error of 30%. According to (164), the CDW-4B method should be valid above 110 keV/amu, as indicated by the vertical arrow on the abscissa.

Figure 38

The same as in Fig. 37, except for the collision ${\mathrm{He}}^{+}\left(1s\right)+\mathrm{H}\left(1s\right)\to \mathrm{He}\left(2p^2{}^{1}D_0\right)+{\mathrm{H}}^{+}$.

Figure 39

The same as in Fig. 37, except for the collision ${\mathrm{He}}^{+}\left(1s\right)+\mathrm{H}\left(1s\right)\to \mathrm{He}(2p^2{}^{1}D_0+2s2p{}^{1}P_0)+{\mathrm{H}}^{+}$.

Figure 40

Zero-degree electron energy spectra at an impact energy $E$ of $75\mathrm{keV}$ for the collision ${}^3\mathrm{He}^{+}\left(1s\right)+{}^4\mathrm{He}\left(1s^2\right)\to {}^3\mathrm{He}\left(2l2l^{\prime }{}^{2S+1}L\right)+{}^4\mathrm{He}^{+}\left(1s\right)$ (183). The $\left(2l2l^{\prime }\right){}^{2S+1}L$ states are $\left(2s^2\right){}^{1}S$, $\left(2s2p\right){}^{3}P$, $\left(2p^2\right){}^{1}D$, and $\left(2s2p\right){}^{1}P$. The CDW-4B method: solid curve, $Z_{\mathrm{NTE}}=1$; dotted curve, $Z_{\mathrm{NTE}}=1.6875$. Solid circles are the experimental data of 137.

Figure 41

The same as in Fig. 40, except for $100\mathrm{keV}$.

Figure 42

The same as in Fig. 40, except for $500\mathrm{keV}$.

Figure 43

Total cross sections for the collision ${}^3\mathrm{He}^{+}\left(1s\right)+{}^4\mathrm{He}\left(1s^2\right)\to {}^3\mathrm{He}\left(2s2p{}^{1}P_0\right)+{}^4\mathrm{He}^{+}\left(1s\right)$. The results of the CDW-4B method (183): solid curve, $Z_{\mathrm{NTE}}=1$; dashed curve, $Z_{\mathrm{NTE}}=1.6875$. The results of the CDW-4B method (105, 107): singly-chained curve, $Z_{\mathrm{NTE}}=1$; dotted curve, $Z_{\mathrm{NTE}}=1.6875$. Experimental data of 137: solid circles.

Figure 44

The same as in Fig. 43, except for ${}^3\mathrm{He}\left(2p^2{}^{1}D_0\right)$.

Figure 45

The same as in Fig. 43, except for ${}^3\mathrm{He}\left(2s^2{}^{1}S\right)$.

Figure 46

The same as in Fig. 43, except for ${}^3\mathrm{He}\left(2s2p{}^{3}P_0\right)$.