1. Introduction
2. Model and method
3. Results and discussion
Figure 1. Effective potential energy curves Veff(r) defined in equation ( |
Table 1. Critical screening parameters for the lowest nine p-wave eigenstates of the ECSCP. |
np | ${\lambda }_{c}^{{np}}$ |
---|---|
2p | 0.148205032642758419285886459123 |
3p | 0.687121436894544378282407881134(−1) |
4p | 0.392634011792194538641491183042(−1) |
5p | 0.253156253177013910985149600317(−1) |
6p | 0.176520702074135587212546717879(−1) |
7p | 0.130010639904740743652758680825(−1) |
8p | 0.997008724443218669747295568387(−2) |
9p | 0.788640558678603012644097745984(−2) |
10p | 0.639311481645614751381777756738(−2) |
Table 2. Convergence of the 2p resonance state for the one-electron system with ECSCP at screening parameters of λ = 0.7, with L = 1000 and θ = 0.785. The results of Nasser et al [42] and Bahlouli et al [43] are included for comparison. |
λ | N | Er | Ei |
---|---|---|---|
0.7 | 1000 | 0.0254564308878703595566369476824 | −0.405997619261475855140293094607 |
2000 | 0.0254564308878410068247375880331 | −0.405997619261439084995429189949 | |
3000 | 0.0254564308878410068247375870520 | −0.405997619261439084995429170708 | |
4000 | 0.0254564308878410068247375870476 | −0.405997619261439084995429170721 | |
5000 | 0.0254564308878410068247375870476 | −0.405997619261439084995429170717 | |
| |||
Nasser et al [45] | 0.025 456 | −0.40599 | |
Bahlouli et al [45] | 0.025 422 | −0.405919 |
Figure 2. Bound state energies (lower) and resonance positions (upper) for the np eigenstates of ECSCP with n = 2 to 10. The inset shows an enlarged view of the high-lying resonances. |
Figure 3. Resonance widths for the np states of ECSCP with n = 2 to 10. The inset shows an enlarged view of the widths for high-lying resonances. |
Table 3. Resonance energies for the np states of ECSCP with n = 2 to 6. Numbers in parentheses represent powers of ten. The results of Nasser et al [42] and Bahlouli et al [43] are included for comparison when they are available. |
np λ | This work | Nasser et al [41] | Bahlouli et al [42] |
---|---|---|---|
2p 0.150 | 8.41463799507136(−4)-6.90016117390905(−5)i | 8.4146380(−4)-6.90016(−5)i | 8.414631(−4)-6.899(−5)i |
0.155 | 3.19835469826691(−3)-4.17271267850758(−4)i | 3.19835(−3)-4.17271(−4)i | 3.19841(−3)-4.17602(−4)i |
0.160 | 5.46594151270155(−3)-8.88396949705082(−4)i | 5.4659415(−3)-8.88396(−4)i | 5.4660(−3)-8.88864(−4)i |
0.165 | 7.63002860494508(−3)-1.49387909741240(−3)i | 7.63002860(−3)-1.4938791(−3)i | 7.6302856(−3)-1.494354(−3)i |
0.170 | 9.70695203299149(−3)-2.23603101512570(−3)i | 9.706952033(−3)-2.23603101(−3)i | 9.70709(−3)-2.23650(−3)i |
0.175 | 1.17112300407746(−2)-3.10935956126523(−3)i | 1.1711230040(−2)-3.109359561(−3)i | 1.171123001(−2)-3.109359(−3)i |
0.178 | 1.28834137420076(−2)-3.69323467301530(−3)i | 1.288341374200(−2)-3.693234673(−3)i | 1.28834137(−2)-3.6932346(−3)i |
0.180 | 1.36533114432199(−2)-4.10642235903903(−3)i | 1.36533114432(−2)-4.10642235903(−3)i | 1.36533114(−2)-4.106422(−3)i |
0.300 | 4.84423224621062(−2)-5.34073390579566(−2)i | 4.844232246(−2)-5.3407339057(−2)i | 4.8442322(−2)-5.34073391(−2)i |
0.400 | 6.29067470676162(−2)-1.19343808546890(−1)i | 6.29067470(−2)-1.193438085(−1)i | 6.290674(−2)-1.1934380(−1)i |
0.500 | 6.44225253344524(−2)-2.01523222601036(−1)i | 6.442253(−2)-2.0152322(−1)i | 6.442259(−2)-2.0152322(−1)i |
0.600 | 5.24634779401062(−2)-2.97544963363380(−1)i | 5.2463(−2)-2.97545(−1)i | 5.246(−2)-2.97543(−1)i |
0.700 | 2.54564308878410(−2)-4.05997619261439(−1)i | 2.5456(−2)-4.0599(−1)i | 2.5422(−2)-4.05919(−1)i |
| |||
3p 0.075 | 2.34087579851323(−3)-3.56157943833318(−4)i | 2.3408757985(−3)-3.5615794383(−4)i | 2.34087603(−3)-3.56158(−4)i |
0.080 | 3.85962887047523(−3)-9.79860400859225(−4)i | ||
0.090 | 6.28423636914734(−3)-2.99789405697670(−3)i | ||
0.100 | 7.99120736008921(−3)-5.82553278005339(−3)i | ||
0.110 | 8.94858397361978(−3)-9.34650665922494(−3)i | ||
0.120 | 9.07430035927160(−3)-1.36013301991694(−2)i | ||
0.130 | 8.39654618124487(−3)-1.88258747777019(−2)i | ||
0.140 | 7.32247886588654(−3)-2.51851313732973(−2)i | ||
0.150 | 6.20578736593378(−3)-3.23345428562951(−2)i | 6.2057(−3)-3.23345(−2)i | 6.2057(−3)-3.23345(−2)i |
| |||
4p 0.050 | 2.82404214024094(−3)-1.22240543251328(−3)i | 1.221(−3)-8.035(−1)i | 1.22(−3)-8.034(−3)i |
0.055 | 3.46040354397717(−3)-2.46379020958530(−3)i | ||
0.060 | 3.69009818410484(−3)-4.07346035370668(−3)i | ||
0.065 | 3.54937316065712(−3)-6.15355044822490(−3)i | ||
0.070 | 3.27423030205280(−3)-8.70856156108195(−3)i | ||
0.075 | 2.94019754414923(−3)-1.14977196676793(−2)i | 2.9401(−3)-1.14977(−2)i | 2.940218(−3)-1.149772(−2)i |
0.080 | 2.44050694521374(−3)-1.44134310547431(−2)i | ||
0.085 | 1.70726275373763(−3)-1.74383483365434(−2)i | ||
| |||
5p 0.027 | 5.81360458456676(−4)-3.46099941466531(−5)i | ||
0.030 | 1.30178012581261(−3)-3.30094559435385(−4)i | 1.0(−6)-8.0(−5)i | |
0.032 | 1.63307990843169(−3)-6.80826923566782(−4)i | ||
0.035 | 1.92936190874223(−3)-1.40295006142746(−3)i | ||
0.037 | 1.99613296099604(−3)-2.02791363535435(−3)i | ||
0.040 | 1.98558854524646(−3)-3.21117430453898(−3)i | ||
0.042 | 1.95597534162070(−3)-4.10351922070500(−3)i | ||
0.045 | 1.82682094908000(−3)-5.50811533399168(−3)i | ||
0.047 | 1.65086235056599(−3)-6.48618317102464(−3)i | ||
0.050 | 1.22197782582441(−3)-8.03490388445078(−3)i | 2.8240421402(−3)-1.2224054325(−3)i | 2.824042(−3)-1.222404(−3)i |
| |||
6p 0.020 | 6.92427903510190(−4)-9.70080858066130(−5)i | ||
0.022 | 1.03624385178502(−3)-3.89514619100446(−4)i | ||
0.025 | 1.25502387376253(−3)-1.16023747385265(−3)i | 4.60(−4)-3.711(−3)i | 1.25502(−3)-1.16023(−3)i |
0.027 | 1.29563016381311(−3)-1.89729540898344(−3)i | ||
0.030 | 1.23313336784715(−3)-3.14185200298618(−3)i | 1.233(−3)-3.141(−3)i | 1.2331(−3)-3.1418(−3)i |
0.032 | 1.04441031869675(−3)-4.06019454769352(−3)i | ||
0.035 | 5.51565165758984(−4)-5.62130286728890(−3)i |
Table 4. Resonance energies for the np states of ECSCP with n = 7 to 10. Numbers in parentheses represent powers of ten. |
np λ | This work |
---|---|
7p 0.014 | 3.28947943545882(−4)-1.34833538327530(−5)i |
0.015 | 5.53459397391246(−4)-9.10187780070191(−5)i |
0.016 | 7.09883391798616(−4)-2.38283903876661(−4)i |
0.017 | 8.08790073247991(−4)-4.49881154538681(−4)i |
0.018 | 8.63918436100496(−4)-7.32402809325174(−4)i |
0.019 | 9.03458649491975(−4)-1.07182076421291(−3)i |
0.020 | 9.22726760736952(−4)-1.43814709485818(−3)i |
0.021 | 9.05005741593023(−4)-1.82787946740459(−3)i |
0.022 | 8.45075214955072(−4)-2.24771277922156(−3)i |
0.023 | 7.46514777847301(−4)-2.70309695051456(−3)i |
0.024 | 6.16566812718687(−4)-3.19311944834636(−3)i |
0.025 | 4.59008576709142(−4)-3.71138911260040(−3)i |
| |
8p 0.010 | 1.15971795975842(−5)-1.16150675741718(−8)i |
0.011 | 3.16061571394108(−4)-2.04410732941511(−5)i |
0.012 | 4.95914048152250(−4)-1.31363610211629(−4)i |
0.013 | 5.96362587422301(−4)-3.35695000604373(−4)i |
0.014 | 6.55082116694703(−4)-6.31336533618891(−4)i |
0.015 | 6.87013975671380(−4)-9.72394483919559(−4)i |
0.016 | 6.67899516270094(−4)-1.35157127594063(−3)i |
0.017 | 6.00387032236425(−4)-1.77788174976687(−3)i |
0.018 | 4.93761733788663(−4)-2.24447283380143(−3)i |
0.019 | 3.44472773060081(−4)-2.74140077060281(−3)i |
0.020 | 1.45869837347042(−4)-3.26903789031866(−3)i |
| |
9p 0.008 | 4.30133228119992(−5)-2.17178914026093(−8)i |
0.009 | 3.09195835539361(−4)-3.70956446968367(−5)i |
0.010 | 4.38721115141968(−4)-1.97993722044232(−4)i |
0.011 | 5.07340097737696(−4)-4.77025398483105(−4)i |
0.012 | 5.30749538523244(−4)-8.14436126949938(−4)i |
0.013 | 4.92705320168512(−4)-1.21079875111701(−3)i |
0.014 | 4.02261644982922(−4)-1.65664265070585(−3)i |
0.015 | 2.52428480947014(−4)-2.14499279279549(−3)i |
| |
10p 0.007 | 1.88888727147462(−4)-7.12117825187625(−6)i |
0.008 | 3.40087820435550(−4)-1.36966956073477(−4)i |
0.009 | 4.11135664655279(−4)-4.12011599015345(−4)i |
0.010 | 4.16386333857880(−4)-7.62517291864373(−4)i |
0.011 | 3.53605788699189(−4)-1.18030350367781(−3)i |
0.012 | 2.18333249353177(−4)-1.65493956518500(−3)i |
Figure 4. n2-scaled bound state energies (lower) and resonance positions (upper) as a function of n2-scaled screening parameters for the 2p, 3p, 5p, and 10p states. |
Figure 5. n2-scaled resonance widths as a function of n2-scaled screening parameters for the 2p, 3p, 5p, and 10p states. The inset shows an enlarged view of the n2Γ at small values of n2λ. |
Figure 6. Variation of the complex momenta of np resonances with n = 2 to 10 as a function of screening parameter. The upper and lower sections show the real and imaginary parts of momentum, respectively. The vertical dashed lines indicate the critical screening parameters. |