In this study we have to calculate the binding energy of atomic ion and the adiabatic potential of the ionic system . system is a two center system, whereas the ion is a single center system. Many quantum chemical calculations yield only a poor accuracy in adiabatic potential at the large distance limit , and the two center potential does not converges to the separated atomic limit (or the result of a single center system). We check the consistency between the single and double center calculation by comparing the binding energy of ions and the potentials in the asymptotic limit. Table 4-1 shows the calculated values of the electronic adiabatic potentials for the system at the large diatance ( a.u.) and the binding energy of for atoms from to
(He to Ar). Two calculations agree within a.u., which shows that two calculations are consistent with a satisfactory accuracy.
Element Adiabatic potential of in a.u. at
The accuracy in the calculated adiabatic potential itself should be checked by the
comparison with some literature values. The adiabatic potential of for (A= He, Li) have been studied in Refs. [31, 32]. The comparison of our results with the values in the literature is shown in Figure 4-1. For the atoms He, and Li, our computational results agree well with the results obtained by others. It should be noted here that the adiabatic potential for
with A=H is the Coulomb potential, and the exact potential is easily obtained.
Table 4-1 The electronic adiabatic energy of ( a.u.) and the binding energy of for He-Ar targets calculated by CCSD(T) method with Gaussian09.
For other atomic targets than H, He, or Li, we only know the reliable values of the
ionization potential of atoms/ions. Thus we can check the accuracy of our potential only in the united/separated atom limits, namely and of the system. For
example in the case of , should equal the binding energy of the ion Ne+, and should equal the binding energy of the atom F. In Table 4-2, the absolute errors in and are listed, together with the error in the first ionization potential of atoms. Although the error in the first ionization potential is small, the error in or
is of the order of several a.u. for large elements, such as Ar. We consider that these relatively large errors are acceptable in our model, since only the shape of the adiabatic
Figure 4-1 (a) The adiabatic potential of obtained by Ref. [30] (green) and G08 (red)
(b) The adiabatic potential of obtained by Ref.
[30] (green) and G08 (red) (a)
(b)
energy curves determines the cross section, and the absolute value of energy is not important.
As is seen in table 4-2, the errors in is relatively small. This is because the relatively large error is mainly attributed to the effect of inner core electrons, and that the shape of the adiabatic potential may have small error. In the case of Ar target, for example,
, and the error in this is 0.3490834 (see Figure 4-2). We consider the error in the shape is sufficiently small, and does not affect the cross section much.
In order to confirm this, we employed a modified potential function,
. (4.1)
The second term of this function represents the modeled error in our calculated potential. We have calculated the parameters , using this function with a= 0 , 1 , , and . The results are shown in Table 4-3. The parameters , , and do not depend on the error significantly. Thus, we conclude that the present results of the adiabatic potentials calculated by G09 are good enough for our model.
Figure 4-2 The Schematic diagram of the error in the adiabatic energy potentials of the experiment (solid line) and calculated by G09 (dot line) in the case of Ar target.
Element The error in
He 0.0008391 -0.0000011 -0.0002338 0.0002328
Li 0.0017996 0.0434119 0.0008380 0.0425739
Be 0.0010336 0.0484181 0.0452115 0.0032066
B 0.0024159 0.0547662 0.0494517 0.0053146
C 0.0019631 0.0669338 0.0571821 0.0097517
N 0.0012390 0.0849209 0.0688969 0.0160240
O 0.0038193 0.1088765 0.0861600 0.0227165
F 0.0027908 0.1502719 0.1126958 0.0375762
Ne 0.0012008 0.2014630 0.1530627 0.0484003
Na 0.0068393 0.5649223 0.2026638 0.3622585
Mg 0.0042948 0.6723393 0.5717616 0.1005777
Al 0.0007362 0.7938338 0.6766341 0.1171997
Si 0.0006039 0.9495980 0.7945700 0.1550279
P -0.0005149 1.1553597 0.9502018 0.2051579
S 0.0039148 1.4105631 1.1548449 0.2557183
Cl 0.0028012 1.6839647 1.4144779 0.2694868
Ar 0.0008087 2.0358493 1.6867659 0.3490834
a = 0 0.7786534 5.0356663 0.5105075
Table 4-4 The parameters , (approximated by Eq. (2-22)), (given in Eq.
(2-10) and calculated by G09), the polarizability [4-1], and . The values in the last three rows are calculated with the approximated adiabatic potential given by Eq. (3-1).
We have calculated the adiabatic potentials for various atomic targets to
calculate the parameters , , and . The parameters, , (approximated by Eq.
(2-22)), (given in Eq. (2-10) and calculated by G09), the polarizability taken from the literature [32], and , are listed in Table 4-4. The difference of of approximated and calculated by G09 is negligible. and reflect the properties of targets. For example, represents the mean radius of the electron wavefunction of the neutral target atom. The values in the last three rows are calculated with the approximated adiabatic potential given by Eq. (3-1).
and show the periodicity representing the properties of atoms. In Figures 4-3, 4-4, and 4-5, we plotted , , and the first ionization potential as functions of the atomic number of the target. behaves very similar to (see Figure 4-4 and Figure 4-5).
The first ionization potential represents the strength of the electron binding. The electron binding is weakened as the antiproton approaches, and the electron is emitted at . The atoms having large ionization potentials (such as rare gas) cannot be ionized until the
antiproton approaches close to the atomic nucleus. Thus the target atoms with large ionization potentials should have small . This is the reason why the behavior of the is so similar to that of .
Figure 4-3 for
The capture/ ionization cross sections are calculated using the parameters in Table 4-3 for the atoms from H to Ar. The capture/ionization cross sections for the target atoms in the second period in the periodic table (Li-Ne) are plotted in Figure 4-6. The rare gas atom (Ne) has the smallest cross section in this figure. This is mainly because of the large ionization potential.
Roughly speaking atoms having the larger atomic number have the smaller cross section in a common period.
In Figure 4-7, the cross sections for the atoms in the third period (Na-Ar) are plotted.
Reflecting the periodicity of the elements, Figure 4-6 and Figure 4-7 show the similar
tendency. Since our model can be applied to various atomic targets, it can provide the general tendencies in the cross sections, and clear physical picture of the processes.
Figure 4-6 The capture/ionization cross section of Li-Ne.
Figure 4-7 The capture/ionization cross section of Na-Ar.
The validity of our model is discussed by comparing to the results obtained by some reliable theories, i.e. the semi-classical method, the result of K. Sakimoto [5, 26] and X. M.
Tong et al. [25]. Figure 4-8 shows the capture/ionization cross sections obtained by our model together with the results in Refs. [5, 26, 25], for the atomic targets of H, He, and Li. Although our model does not require any heavy computations, it can reproduce general tendencies in the results obtained by sophisticated computations. Generally speaking our model yields relatively larger cross sections. This may be explained from our assumptions of the
probability function given by Eq. (2-32). We assume that the probability is unity if the impact parameter is smaller than . The nonadiabatic probability, however, can be , thus our assumption of the unit probability may be the upper limit of the cross section.
Figure 4-8 The capture/ionization cross section in -atom collision. Li target obtained by Sakimoto [26] (sky blue with solid square), and our model (red with open triangle). He target obtained by Tong et al. [25] (purple with open square) and our model (black with solid circle).
H target obtained by Sakimoto [5] (green with cross), Tong et al. [25] (blue with star), and our model (yellow with open circle).
In Figure 4-9, the cross section for H target is solely plotted. Our model reproduces nicely the result of Sakimoto and Tong et al in the collision energy between 0.1 a.u. and 0.3 a.u. In low collision energy (<0.1 a.u.), on the other hand, the result of our model is little bit larger than the result of Sakimoto. Our cross section clearly shows the sudden change of tendency at . The energy region lower than shows the Langevin cross section ( ), whereas the higher energy regions shows the behavior. It is also seen from Sakimoto’s results (red line) that the behavior in the low energy region has smaller slope, and that the high energy region has steeper slope. The point of behavior change, however, is not clear in Sakimoto’s results. This may be because the number of points is small to see the sudden change.
Figure 4-9 The capture/ionization cross of by Sakimoto (red), Tong et al. (green), and our model (blue)
Figure 4-10 shows the comparison with Tong’s results for the He target. Their results are relatively smaller than ours. At the point of the highest energy in their results a strange
behavior is seen. Except this point, Tong’s results are coincident with our model. In the higher energy region there results shows the steeper slope (roughly parallel to our results of ), and the low energy rigion shows a smaller slope (roughly parallel to our results of ).
Figure 4-10 The capture/ionization cross of derived by X. M. Tong et al. (red), and our model (green)
In the case of Li target, Sakimoto provided the results at many energy points. Therefore the point of behavior change is clearly seen, and agrees with in our model (see Figure 4-11).
Figure 4-11 The capture/ionization cross of derived by Li sakimoto (red), and our model (green)
Our model basically shows similar tendency of cross section with the accurate
calculation results except the fact that our cross section is slightly larger. This may be because of our assumption of the unit probability. If we employ a non-unity reaction probability constant into our model, the results of our model reproduce the accurate calculation much better (see Figures 4-12 to 4-14). We have chosen the reaction probabilities so as to fit the reliable results best. The reaction probabilities obtained are 1, 0.8, and 0.67 for H, He, and Li respectively. It should be noted that the Figures 4-12 to 4-14 are in the linear scale. Our model reproduces the reliable results even in the linear scale plot. Thus further developments in our theory can be expected. A simple theory to find the reaction probability improves our model significantly. Since the reaction probability represents the nonadiabaticity in the region
, we believe that sophisticated theories of nonadiabatic transition can be utilized to find the way to discuss the reaction probability that can improve our model.
Figure 4-12 The capture/ionization cross of obtained by Sakimoto (red), Tong el al (green), and our model (blue) with reaction constant = 1
Figure 4-13 The capture/ionization cross of obtained by Tong et al (red) and our model (green) with reaction constant = 0.8
Figure 4-14 The capture/ionization cross of derived by Sakimoto (red) and our model (green) with reaction constant = 0.67