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Vibrationally resolved collisions in cold hydrogen plasma
Nuclear Instruments and Methods in Physics Research B 241 (2005) 58–62 www.elsevier.com/locate/nimb
Vibrationally resolved collisions in cold hydrogen plasma Predrag S. Krstic´
*
Oak Ridge National Laboratory, Physics Division, P.O. Box 2008, Oak Ridge, TN 37831-6372, United States Available online 19 August 2005
Abstract Slow collisions (0.5–100 eV) of hydrogen atoms and ions with vibrationally excited hydrogen molecules and molecular ions are studied using fully quantal (below 10 eV) and semi-classical approaches. Considered transitions include charge transfer, excitation, dissociation, three-body diatomic association, as well as energy and angular spectra of dissociation fragments. Mutual relations of various elastic and inelastic processes, the relevant physical mechanisms governing vibrational dynamics in inelastic channels and comparison of the fully quantal and semi-classical results are also reported. 2005 Published by Elsevier B.V. PACS: 34.70.+e; 82.30.Fi Keywords: Ion–molecule collisions; Vibrationally excited; Vibrationally resolved; Charge transfer; Excitation; Dissociation; Association; Fully-quantal; Semi-classical
The high density–low temperature (1–100 eV) divertor plasma is characterized by similar densities (1015 cm3) of ions and neutrals, consisting predominantly of hydrogen atoms, ions and molecules in various isotopic combinations. The inelastic processes in slow H+ + H2 and H þ Hþ 2 collisions play an important role in the energy and particle redistribution in all hydrogenic astrophysical and fusion divertor plasmas. The role of these processes is particularly pronounced when the hydrogen mole*
Corresponding author. Tel.: +865 574 4701; fax: +865 574 1118. E-mail address: [email protected] 0168-583X/$ - see front matter 2005 Published by Elsevier B.V. doi:10.1016/j.nimb.2005.07.007
cules, or molecular ions, are vibrationally excited. For example, charge transfer and dissociation exhibits a rapid increase of its cross section with increasing initial vibrational excitation and the corresponding decrease or disappearance of their threshold energies. Vibrationally excited H2(v) and Hþ 2 ðvÞ in various isotopic combinations are formed by associative desorption, collisional neutralization and excitation on the plasma-facing surfaces, and possibly, by three-body diatomic association in a H+ + H + H collision. The intensive, often quasi-resonant inelastic processes of the excited molecules with other plasma constituents compete with the elastic scattering in transport of
P.S. Krstic´ / Nucl. Instr. and Meth. in Phys. Res. B 241 (2005) 58–62
the momentum of the molecules. These processes with abundant hydrogen particles can cause dissipation of the divertor plasma momentum. The atomic data for this region which include vibrationally excited molecules have been sparse, and on the experimental side completely missing. An accurate knowledge of the cross sections (and/or of rate coefficients) for all inelastic and elastic processes for the entire spectrum of vibrationally excited molecules is decisively important for further advances in divertor modeling, for formation of the detached plasma regime, and for plasma diagnostics. The divertor plasma detachment layer is desired for the reduction of heat loads on divertor plates, one of the most serious problems in todayÕs fusion energy research. Its formation might critically depend on molecule assisted recombination (MAR), which involves the ion conversion through electron capture by a proton from a vibrationally and rotationally excited hydrogen molecule in cooler plasma regions (1–10 eV), followed by dissociative recombination of the molecular ion with plasma electrons [1,2]. The proton–hydrogen molecule and hydrogen– hydrogen-molecular-ion systems are the most fundamental ion–molecule two-electron collision systems. A typical collision event evolves through dynamically coupled electronic, vibrational and rotational degrees of freedom. These systems have been studied thoroughly only for the processes from the ground states (electronic, vibrational, rotational) as motivated by applications, as well as limited by experimental and theoretical capabilities of the time. Also, these calculations were done within a manifold of bound vibrational states, thus neglecting the possible importance of inelastic processes through ‘‘closed’’ dissociative channels, as well as the dynamic change of the dissociative continuum edge with the position of the projectile. We report on a comprehensive study of scattering of hydrogen ions on vibrationally excited hydrogen molecules as well as of hydrogen atoms on vibrationally excited hydrogen molecular ions in the range of center of mass energies 0.5– 100 eV. Total and partial, initial and final vibrational state resolved cross sections for excitation, charge transfer, dissociation (including dissociative energy spectra), and association have been cal-
59
culated ‘‘on the same footing’’, on the two lowest electronic adiabatic surfaces of the Hþ 3 quasimolecule, using a fully-quantal, coupled-channel (FQCC) approach (for energies below 10 eV, [3– 7]) and semi-classical, split-operator (SCSO) method (above 15 eV). An extensive vibrational basis set, including all bound vibrational states (35 on both H2 and Hþ 2 together) and a several hundred (more than 800) discretized dissociative continua in a large configuration space [3] (of 40 a.u., to include nuclear particle arrangements) was employed, while the rotational dynamics of H2 and Hþ 2 was treated within the sudden approximation [8] (Infinite order sudden approximation, IOSA), implying that the collision center-of-mass energy, ECM, is well above the value of a typical quantum of rotational excitation of the H2 target (0.01 eV). The full calculations were performed for the 12 fixed orientations of the diatomic target and the results averaged at level of the cross sections. The FQCC equations were solved using Johnson logarithmic derivative method [9] for each partial wave. The produced database contains all mentioned + processes for the H þ Hþ 2 and H + H2 collision systems in form of partial and total, initial and final vibrational state resolved cross sections. This represents currently most comprehensive quantum-mechanically obtained set of inelastic data for collisions that involve hydrogen atoms, ions and molecules. All cross sections are available in the tabular form at www-cfadc.phy.ornl.gov, with needed explanations and discussions in [3–7]. The obtained cross sections are in good agreement with the existing quantum-mechanical data for charge transfer and excitation from the ground vibrational state, as well as with classical trajectory surface hoping results of Ichihara et al. [10] from the higher excited states, in agreement with the correspondence principle. It is interesting to note that quantum and the TSH calculations shows similar large contribution of the particle rearrangement channels at low energies, where their role is of particular importance. Our data contain coherent sum of the direct and rearrangement channels. Charge transfers from the first three excited states constitute a separate group. The reason is that CT from the higher states is dominantly exoergic and
60
P.S. Krstic´ / Nucl. Instr. and Meth. in Phys. Res. B 241 (2005) 58–62
often quasi-resonant with the vibrational states of Hþ 2 and therefore are large even at lowest energies. For initially highly excited states (>8) dissociation becomes dominant channel, suppressing other processes. Charge transfer cross sections from excited states of Hþ 2 in collision with H is exoergic from all vibrational states (including the ground one), the characteristic increase of the cross section toward lower energies is expected, and not strongly dependent on the initial state [3]. A dissociative process includes both direct dissociation, into dissociative continuum of H2, as well as charge transfer dissociation, into continuum of Hþ 2 . These two channels are of the same order of magnitude [5]. Concerning the energy spectra of the dissociating fragments, these have the characteristic cusp at the continuum edge, which is more pronounced for the lower collision energies. The calculated total association rates of H2 or + Hþ 2 in the H + H + H collision system are about 32 5 · 10 –6 · 1034 cm6/s for temperatures in the interval 300–20,000 K, and decrease as (approximately) T1 as temperature increases [6]. This indicates that the three-body, diatomic association in a hydrogen plasma may be an important mechanism for H2 and/or Hþ 2 formation in astrophysical environments containing significant populations of protons and hydrogen atoms. In fusion divertor plasmas, however, even with temperatures as low as 1 eV and typical plasma and gas densities of 1015 cm3, the three-body, diatomic association process yields total rate coefficient of 1033 cm6/ s and the probability rate of 103 s1. Apparently, this will not play a significant role in volume plasma recombination in the presence of more powerful competing processes (such as the three-body recombination in e + H+ + e, for instance). Fully quantal approach discussed above, is not practical at higher energies (few tens eV) because of too large number of partial waves involved. An alternative is to replace the partial wave with the impact parameter formalism, assuming a classical motion of the projectile. Electronic and vibrational motions are still treated quantally, on the two lowest adiabatic electronic surfaces of the Hþ 3 , while IOSA and average over diatomic target angles is applied to the diatomic target rotations. The resulting time-dependent Schrodinger equation is
ðH 0 þ VðRðtÞ; rÞÞWðRðtÞ; r; tÞ ¼ i
oWðRðtÞ; r; tÞ ; ot ð1Þ
where R(t) is a classical projectile trajectory defined here inffi the straight line approximation, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R ¼ b2 þ v2 t2 , R is the projectile displacement with respect to diatomic center-of mass, r is vibrational coordinate, V is the coupling matrix, and W is the state vector. The operator H0 defines the unperturbed diatomic vibrational basis u 1 o2 H 0u ¼ I þ V0 ðrÞ uðr; tÞ ¼ eu; 2l or2 V1 0 V0 ðrÞ ¼ ; ð2Þ 0 V2 where indices ‘‘1’’ and ‘‘2’’ correspond to each of the two adiabatic electronic potential curves of H2 and Hþ 2 , respectively, l is the diatomic reduced mass and I is the identity matrix. With a choice of H0 as in Eq. (2), the coupling matrix V(R), with property V(R ! 1) ! 0 is defined as V 11 V 1 V 12 VðR; rÞ ¼ ¼ Vd þ qV 12 ; V 12 V 22 V 2 ð3Þ
1 is the Pauli and Vd is the diag0 onal matrix and V11, V22 are the two lowest diabatic electronic potential surfaces of Hþ 3, constructed from the adiabatic ones by the procedure outlined in [3], and V12 is the coupling between the relevant diabatic states of Hþ 3. Eq. (1) is solved on a numerical mesh, employing the split-operator technique in the energy representation. Iterating the state vector in time from some large ‘‘T’’ with a small step s yields where q ¼
0 1
Wnþ1 ¼ expðiHsÞWn s ¼ Dt
ð4Þ
where Wn = W(tn = T + ns). Applying the well known operator relation exp½iða þ bÞs ¼ expðias=2Þ expðibsÞ expðias=2Þ þ Oðs3 Þ
ð5Þ
P.S. Krstic´ / Nucl. Instr. and Meth. in Phys. Res. B 241 (2005) 58–62
to Eq. (4), using Eqs. (1) and (3), one obtains expðiðH 0 þ V ÞsÞ s s ¼ exp iH 0 exp iV t þ s 2 2 s 3 exp iH 0 þ Oðs Þ; 2 where
one can obtain the total wave function of the system after the collision, which evolved from an initial vibrational state, ui Wn!1 ¼ M n M n1 . . . M 1 ui ¼ Mui
Critical in the applications of Eqs. (6) and (7) are exponents with the differential and non-diagonal operators, H0 and qV12, respectively. These are helped with the exact relations 1 s X s exp iH 0i exp ieki juki ðrÞi ¼ 2 2 k¼0 huki ðrÞj
expðiqV 12 sÞ ¼
ð8Þ
cosðV 12 sÞI
i sinðV 12 sÞI
i sinðV 12 sÞI
cosðV 12 sÞI
; ð9Þ
where i = 1,2 in Eq. (8), similarly as in Eq. (2), correspond to H2 and Hþ 2 vibrational bases juk,i(r)i, respectively. The summation over ‘‘k’’ in Eq. (8) assumes a discrete complete basis, which is here obtained limiting vibrational motion in r to the interval (0.4, 20), which yields a discretized dissociative continuum. We truncate k to N = 400 for each ‘‘i’’, resulting in the vibrational basis of 400 states for each of H2 and Hþ 2 (35 bound and 765 quasi-continuum). The spatial r-discretization is done on a uniform mesh (0.4, 20) with 400 nodes for each surface, yielding dimension of 2N = 800 for all relevant matrices in the iteration procedure. We note that nuclear rearrangement reactions, present at energies in the eV range are not significant at higher energies, which implies a smaller size of the numerical box, here 20 a.u. It can be shown that our iteration procedure is unconditionally unitary and unconditionally convergent. Applying wnþ1 ¼ M nþ1 ðsÞwn þ Oðs3 Þ
ð10Þ
ð11Þ
as well as an S-matrix element, Sji, for transition from an arbitrary initial (i) to an arbitrary final (j) vibrational state of the system S ji ¼ uTj Mui .
ð12Þ
The possibility to calculate the evolution operator M of the whole system, without specifying an initial state during the numerical procedure is the main advantage of the proposed procedure for treatment of the vibrationally resolved transitions from an arbitrary excited state. The final choice of the time step in calculating M, after checking the convergence, was s = 0.002 a.u. A check of quality of the obtained semi-classical data is matching of the cross sections with the fully-quantal results in the overlapping region of energies. The conclusion, after analysis of all transitions, is that the cross sections from the initially lowest vibrational states are overestimated when using the straight-line approximation for the projectile motion. Transitions which are dominated by non-tunneling mechanisms, like resonant, quasi-resonant or endoergic collisions, seems to match well with the fully quantal results. This is true, for -14
10
Charge transfer cross section (cm2)
ð6Þ
expðiV sÞ s s ¼ exp iV d expðiqV 12 sÞ exp iV d 2 2 þ Oðs3 Þ. ð7Þ
and
61
Total charge transfer 7 6
5 -15
10
SClass (Krstic 04)
6,7
4
+
H +H2( vi)
4
10
10
5,6,7
4
+
H(1s)+H2
3 3
2 3 -16
10
1
2
νi=0
1
2 1 vi=0
-17
10
Fully QM (Krstic 2002)
10
0
vi=0, rec. (Linder et al 1985) TSH (Ichihara et al 2000)
vi=0, Holiday [48]
10
1
2
10
CM Energy (eV)
Fig. 1. Charge transfer in H2(vi) + H+ collisions, using fully quantal (solid lines) and semi-classical approaches (triangles), compared with TSH results of Ichihara et al. [10] (dashed lines), experiment of Holliday et al. [11] (circles) and with estimates of Linder et al. [12] (thick dashed line).
62
P.S. Krstic´ / Nucl. Instr. and Meth. in Phys. Res. B 241 (2005) 58–62
example, for charge transfer from higher excited states of H2 in collision with H+ (Fig. 1) as well as for all states of Hþ 2 in collision with H. The cross sections in Fig. 1 match well across the whole energy region only for v P 2. In conclusion, the vibrationally resolved transitions (charge transfer, excitation, dissociation) were considered on the ‘‘same footing’’, in the collision energy range 0.5–100 eV, with two theoretical approaches, fully quantal (for E < 10 eV) and semi-classical (E > 15 eV), the latter within impact parameter, straight-line approximation. The two sets of cross sections seems to match well for higher excited states, but disperse significantly for transitions from the lowest ones. An additional effort is needed for fully quantal study of the collisions at tens-eV region.
through Oak Ridge National Laboratory, managed by UT-Battelle, LLC under contract DEAC05-00OR22725.
Acknowledgement
[11]
I acknowledge support from the US Department of Energy, Office of Fusion Energy Sciences,
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
[12]
S. Krasheninnikov, Phys. Scr. T96 (2002) 7. A.Yu. Pigarov, Phys. Scr. T96 (2002) 16. P.S. Krstic´, Phys. Rev. A 66 (2002) 042717. D.W. Savin, P.S. Krstic´, Z. Haiman, P.C. Stancil, ApJL 606 (2004) L167. P.S. Krstic´, R.K. Janev, Phys. Rev. A 67 (2003) 022708. P.S. Krstic´, R.K. Janev, D.R. Schultz, J. Phys. B 36 (2003) L249. P.S. Krstic´, D.R. Schultz, J. Phys. B. 36 (2003) 385. M. Baer, H. Nakamura, J. Chem. Phys. 66 (1987) 1363, and references therein. B. Johnson, J. Comp. Phys. 13 (1973) 445. A. Ichihara, O. Iwamoto, R.K. Janev, J. Phys. B 33 (2000) 4747. M.G. Holliday, J.T. Muckerman, L. Friedman, J. Chem. Phys. 54 (1971) 1058. F. Linder, R.K. Janev, J. Botero, in: R.K. Janev (Ed.), Atomic and Molecular Processes in Fusion Edge plasmas, Plenum Press, New York, 1995, p. 397.
Vibrationally resolved collisions in cold hydrogen plasma Predrag S. Krstic´
*
Oak Ridge National Laboratory, Physics Division, P.O. Box 2008, Oak Ridge, TN 37831-6372, United States Available online 19 August 2005
Abstract Slow collisions (0.5–100 eV) of hydrogen atoms and ions with vibrationally excited hydrogen molecules and molecular ions are studied using fully quantal (below 10 eV) and semi-classical approaches. Considered transitions include charge transfer, excitation, dissociation, three-body diatomic association, as well as energy and angular spectra of dissociation fragments. Mutual relations of various elastic and inelastic processes, the relevant physical mechanisms governing vibrational dynamics in inelastic channels and comparison of the fully quantal and semi-classical results are also reported. 2005 Published by Elsevier B.V. PACS: 34.70.+e; 82.30.Fi Keywords: Ion–molecule collisions; Vibrationally excited; Vibrationally resolved; Charge transfer; Excitation; Dissociation; Association; Fully-quantal; Semi-classical
The high density–low temperature (1–100 eV) divertor plasma is characterized by similar densities (1015 cm3) of ions and neutrals, consisting predominantly of hydrogen atoms, ions and molecules in various isotopic combinations. The inelastic processes in slow H+ + H2 and H þ Hþ 2 collisions play an important role in the energy and particle redistribution in all hydrogenic astrophysical and fusion divertor plasmas. The role of these processes is particularly pronounced when the hydrogen mole*
Corresponding author. Tel.: +865 574 4701; fax: +865 574 1118. E-mail address: [email protected] 0168-583X/$ - see front matter 2005 Published by Elsevier B.V. doi:10.1016/j.nimb.2005.07.007
cules, or molecular ions, are vibrationally excited. For example, charge transfer and dissociation exhibits a rapid increase of its cross section with increasing initial vibrational excitation and the corresponding decrease or disappearance of their threshold energies. Vibrationally excited H2(v) and Hþ 2 ðvÞ in various isotopic combinations are formed by associative desorption, collisional neutralization and excitation on the plasma-facing surfaces, and possibly, by three-body diatomic association in a H+ + H + H collision. The intensive, often quasi-resonant inelastic processes of the excited molecules with other plasma constituents compete with the elastic scattering in transport of
P.S. Krstic´ / Nucl. Instr. and Meth. in Phys. Res. B 241 (2005) 58–62
the momentum of the molecules. These processes with abundant hydrogen particles can cause dissipation of the divertor plasma momentum. The atomic data for this region which include vibrationally excited molecules have been sparse, and on the experimental side completely missing. An accurate knowledge of the cross sections (and/or of rate coefficients) for all inelastic and elastic processes for the entire spectrum of vibrationally excited molecules is decisively important for further advances in divertor modeling, for formation of the detached plasma regime, and for plasma diagnostics. The divertor plasma detachment layer is desired for the reduction of heat loads on divertor plates, one of the most serious problems in todayÕs fusion energy research. Its formation might critically depend on molecule assisted recombination (MAR), which involves the ion conversion through electron capture by a proton from a vibrationally and rotationally excited hydrogen molecule in cooler plasma regions (1–10 eV), followed by dissociative recombination of the molecular ion with plasma electrons [1,2]. The proton–hydrogen molecule and hydrogen– hydrogen-molecular-ion systems are the most fundamental ion–molecule two-electron collision systems. A typical collision event evolves through dynamically coupled electronic, vibrational and rotational degrees of freedom. These systems have been studied thoroughly only for the processes from the ground states (electronic, vibrational, rotational) as motivated by applications, as well as limited by experimental and theoretical capabilities of the time. Also, these calculations were done within a manifold of bound vibrational states, thus neglecting the possible importance of inelastic processes through ‘‘closed’’ dissociative channels, as well as the dynamic change of the dissociative continuum edge with the position of the projectile. We report on a comprehensive study of scattering of hydrogen ions on vibrationally excited hydrogen molecules as well as of hydrogen atoms on vibrationally excited hydrogen molecular ions in the range of center of mass energies 0.5– 100 eV. Total and partial, initial and final vibrational state resolved cross sections for excitation, charge transfer, dissociation (including dissociative energy spectra), and association have been cal-
59
culated ‘‘on the same footing’’, on the two lowest electronic adiabatic surfaces of the Hþ 3 quasimolecule, using a fully-quantal, coupled-channel (FQCC) approach (for energies below 10 eV, [3– 7]) and semi-classical, split-operator (SCSO) method (above 15 eV). An extensive vibrational basis set, including all bound vibrational states (35 on both H2 and Hþ 2 together) and a several hundred (more than 800) discretized dissociative continua in a large configuration space [3] (of 40 a.u., to include nuclear particle arrangements) was employed, while the rotational dynamics of H2 and Hþ 2 was treated within the sudden approximation [8] (Infinite order sudden approximation, IOSA), implying that the collision center-of-mass energy, ECM, is well above the value of a typical quantum of rotational excitation of the H2 target (0.01 eV). The full calculations were performed for the 12 fixed orientations of the diatomic target and the results averaged at level of the cross sections. The FQCC equations were solved using Johnson logarithmic derivative method [9] for each partial wave. The produced database contains all mentioned + processes for the H þ Hþ 2 and H + H2 collision systems in form of partial and total, initial and final vibrational state resolved cross sections. This represents currently most comprehensive quantum-mechanically obtained set of inelastic data for collisions that involve hydrogen atoms, ions and molecules. All cross sections are available in the tabular form at www-cfadc.phy.ornl.gov, with needed explanations and discussions in [3–7]. The obtained cross sections are in good agreement with the existing quantum-mechanical data for charge transfer and excitation from the ground vibrational state, as well as with classical trajectory surface hoping results of Ichihara et al. [10] from the higher excited states, in agreement with the correspondence principle. It is interesting to note that quantum and the TSH calculations shows similar large contribution of the particle rearrangement channels at low energies, where their role is of particular importance. Our data contain coherent sum of the direct and rearrangement channels. Charge transfers from the first three excited states constitute a separate group. The reason is that CT from the higher states is dominantly exoergic and
60
P.S. Krstic´ / Nucl. Instr. and Meth. in Phys. Res. B 241 (2005) 58–62
often quasi-resonant with the vibrational states of Hþ 2 and therefore are large even at lowest energies. For initially highly excited states (>8) dissociation becomes dominant channel, suppressing other processes. Charge transfer cross sections from excited states of Hþ 2 in collision with H is exoergic from all vibrational states (including the ground one), the characteristic increase of the cross section toward lower energies is expected, and not strongly dependent on the initial state [3]. A dissociative process includes both direct dissociation, into dissociative continuum of H2, as well as charge transfer dissociation, into continuum of Hþ 2 . These two channels are of the same order of magnitude [5]. Concerning the energy spectra of the dissociating fragments, these have the characteristic cusp at the continuum edge, which is more pronounced for the lower collision energies. The calculated total association rates of H2 or + Hþ 2 in the H + H + H collision system are about 32 5 · 10 –6 · 1034 cm6/s for temperatures in the interval 300–20,000 K, and decrease as (approximately) T1 as temperature increases [6]. This indicates that the three-body, diatomic association in a hydrogen plasma may be an important mechanism for H2 and/or Hþ 2 formation in astrophysical environments containing significant populations of protons and hydrogen atoms. In fusion divertor plasmas, however, even with temperatures as low as 1 eV and typical plasma and gas densities of 1015 cm3, the three-body, diatomic association process yields total rate coefficient of 1033 cm6/ s and the probability rate of 103 s1. Apparently, this will not play a significant role in volume plasma recombination in the presence of more powerful competing processes (such as the three-body recombination in e + H+ + e, for instance). Fully quantal approach discussed above, is not practical at higher energies (few tens eV) because of too large number of partial waves involved. An alternative is to replace the partial wave with the impact parameter formalism, assuming a classical motion of the projectile. Electronic and vibrational motions are still treated quantally, on the two lowest adiabatic electronic surfaces of the Hþ 3 , while IOSA and average over diatomic target angles is applied to the diatomic target rotations. The resulting time-dependent Schrodinger equation is
ðH 0 þ VðRðtÞ; rÞÞWðRðtÞ; r; tÞ ¼ i
oWðRðtÞ; r; tÞ ; ot ð1Þ
where R(t) is a classical projectile trajectory defined here inffi the straight line approximation, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R ¼ b2 þ v2 t2 , R is the projectile displacement with respect to diatomic center-of mass, r is vibrational coordinate, V is the coupling matrix, and W is the state vector. The operator H0 defines the unperturbed diatomic vibrational basis u 1 o2 H 0u ¼ I þ V0 ðrÞ uðr; tÞ ¼ eu; 2l or2 V1 0 V0 ðrÞ ¼ ; ð2Þ 0 V2 where indices ‘‘1’’ and ‘‘2’’ correspond to each of the two adiabatic electronic potential curves of H2 and Hþ 2 , respectively, l is the diatomic reduced mass and I is the identity matrix. With a choice of H0 as in Eq. (2), the coupling matrix V(R), with property V(R ! 1) ! 0 is defined as V 11 V 1 V 12 VðR; rÞ ¼ ¼ Vd þ qV 12 ; V 12 V 22 V 2 ð3Þ
1 is the Pauli and Vd is the diag0 onal matrix and V11, V22 are the two lowest diabatic electronic potential surfaces of Hþ 3, constructed from the adiabatic ones by the procedure outlined in [3], and V12 is the coupling between the relevant diabatic states of Hþ 3. Eq. (1) is solved on a numerical mesh, employing the split-operator technique in the energy representation. Iterating the state vector in time from some large ‘‘T’’ with a small step s yields where q ¼
0 1
Wnþ1 ¼ expðiHsÞWn s ¼ Dt
ð4Þ
where Wn = W(tn = T + ns). Applying the well known operator relation exp½iða þ bÞs ¼ expðias=2Þ expðibsÞ expðias=2Þ þ Oðs3 Þ
ð5Þ
P.S. Krstic´ / Nucl. Instr. and Meth. in Phys. Res. B 241 (2005) 58–62
to Eq. (4), using Eqs. (1) and (3), one obtains expðiðH 0 þ V ÞsÞ s s ¼ exp iH 0 exp iV t þ s 2 2 s 3 exp iH 0 þ Oðs Þ; 2 where
one can obtain the total wave function of the system after the collision, which evolved from an initial vibrational state, ui Wn!1 ¼ M n M n1 . . . M 1 ui ¼ Mui
Critical in the applications of Eqs. (6) and (7) are exponents with the differential and non-diagonal operators, H0 and qV12, respectively. These are helped with the exact relations 1 s X s exp iH 0i exp ieki juki ðrÞi ¼ 2 2 k¼0 huki ðrÞj
expðiqV 12 sÞ ¼
ð8Þ
cosðV 12 sÞI
i sinðV 12 sÞI
i sinðV 12 sÞI
cosðV 12 sÞI
; ð9Þ
where i = 1,2 in Eq. (8), similarly as in Eq. (2), correspond to H2 and Hþ 2 vibrational bases juk,i(r)i, respectively. The summation over ‘‘k’’ in Eq. (8) assumes a discrete complete basis, which is here obtained limiting vibrational motion in r to the interval (0.4, 20), which yields a discretized dissociative continuum. We truncate k to N = 400 for each ‘‘i’’, resulting in the vibrational basis of 400 states for each of H2 and Hþ 2 (35 bound and 765 quasi-continuum). The spatial r-discretization is done on a uniform mesh (0.4, 20) with 400 nodes for each surface, yielding dimension of 2N = 800 for all relevant matrices in the iteration procedure. We note that nuclear rearrangement reactions, present at energies in the eV range are not significant at higher energies, which implies a smaller size of the numerical box, here 20 a.u. It can be shown that our iteration procedure is unconditionally unitary and unconditionally convergent. Applying wnþ1 ¼ M nþ1 ðsÞwn þ Oðs3 Þ
ð10Þ
ð11Þ
as well as an S-matrix element, Sji, for transition from an arbitrary initial (i) to an arbitrary final (j) vibrational state of the system S ji ¼ uTj Mui .
ð12Þ
The possibility to calculate the evolution operator M of the whole system, without specifying an initial state during the numerical procedure is the main advantage of the proposed procedure for treatment of the vibrationally resolved transitions from an arbitrary excited state. The final choice of the time step in calculating M, after checking the convergence, was s = 0.002 a.u. A check of quality of the obtained semi-classical data is matching of the cross sections with the fully-quantal results in the overlapping region of energies. The conclusion, after analysis of all transitions, is that the cross sections from the initially lowest vibrational states are overestimated when using the straight-line approximation for the projectile motion. Transitions which are dominated by non-tunneling mechanisms, like resonant, quasi-resonant or endoergic collisions, seems to match well with the fully quantal results. This is true, for -14
10
Charge transfer cross section (cm2)
ð6Þ
expðiV sÞ s s ¼ exp iV d expðiqV 12 sÞ exp iV d 2 2 þ Oðs3 Þ. ð7Þ
and
61
Total charge transfer 7 6
5 -15
10
SClass (Krstic 04)
6,7
4
+
H +H2( vi)
4
10
10
5,6,7
4
+
H(1s)+H2
3 3
2 3 -16
10
1
2
νi=0
1
2 1 vi=0
-17
10
Fully QM (Krstic 2002)
10
0
vi=0, rec. (Linder et al 1985) TSH (Ichihara et al 2000)
vi=0, Holiday [48]
10
1
2
10
CM Energy (eV)
Fig. 1. Charge transfer in H2(vi) + H+ collisions, using fully quantal (solid lines) and semi-classical approaches (triangles), compared with TSH results of Ichihara et al. [10] (dashed lines), experiment of Holliday et al. [11] (circles) and with estimates of Linder et al. [12] (thick dashed line).
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P.S. Krstic´ / Nucl. Instr. and Meth. in Phys. Res. B 241 (2005) 58–62
example, for charge transfer from higher excited states of H2 in collision with H+ (Fig. 1) as well as for all states of Hþ 2 in collision with H. The cross sections in Fig. 1 match well across the whole energy region only for v P 2. In conclusion, the vibrationally resolved transitions (charge transfer, excitation, dissociation) were considered on the ‘‘same footing’’, in the collision energy range 0.5–100 eV, with two theoretical approaches, fully quantal (for E < 10 eV) and semi-classical (E > 15 eV), the latter within impact parameter, straight-line approximation. The two sets of cross sections seems to match well for higher excited states, but disperse significantly for transitions from the lowest ones. An additional effort is needed for fully quantal study of the collisions at tens-eV region.
through Oak Ridge National Laboratory, managed by UT-Battelle, LLC under contract DEAC05-00OR22725.
Acknowledgement
[11]
I acknowledge support from the US Department of Energy, Office of Fusion Energy Sciences,
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