- Email: [email protected]
Collision induced broadening and shifting of the H and K lines of Ca+ at low temperature
Accepted Manuscript
Collision induced broadening and shifting of the H and K lines of Ca+ at low temperature Xin Wang , Rui Zhang , Yong Shen , Qu Liu , Hongxin Zou , Bing Yan PII: DOI: Reference:
S0022-4073(16)30623-9 10.1016/j.jqsrt.2017.04.022 JQSRT 5678
To appear in:
xxx xxx xxx
Received date: Revised date: Accepted date:
26 September 2016 21 March 2017 19 April 2017
Please cite this article as: Xin Wang , Rui Zhang , Yong Shen , Qu Liu , Hongxin Zou , Bing Yan , Collision induced broadening and shifting of the H and K lines of Ca+ at low temperature, xxx xxx xxx (2017), doi: 10.1016/j.jqsrt.2017.04.022
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Highlights •The PECs of the Λ-S states correlating with lowest three atomic limits in Ca+-He molecular system have been calculated. •The SOC have been considered and the PECs of Ω states are obtained.
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• Collision induced broadening and shifting of the H and K lines of Ca+ by He atoms at low temperature have been computed with
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Anderson-Talman theory.
*Email:[email protected] †Email: [email protected]
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Collision induced broadening and shifting of the H and K lines of Ca+ at low temperature Xin Wang, Rui Zhang, Yong Shen, Qu Liu, Hongxin Zou* Interdisciplinary Center of Quantum Information, National University of Defense Technology, Changsha 410073, China
Bing Yan†
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Institute of Atomic and Molecular Physics, Jilin University, Changchun 130012, China
Abstract
Multireference configuration interaction method was used to compute the potential energy
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curves of -S states correlating with lowest three atomic limits in Ca+-He molecular collision system. The potential energy curves of nine states were obtained with inclusion of spin-orbit coupling. And the electric dipole and quadrupole moment matrix elements between excited states and ground state were also computed. Furthermore, with aid of the Anderson- Talman theory we calculated the broadening and shifting coefficients for Ca+-He spectral lines in the low temperature regime. For H line, α=0.303×10-20cm-1/cm-3, β=-0.0527×10-20cm-1/cm-3; For K line, α=0.233×10-20cm-1/cm-3, β=-0.0402×10-20cm-1/cm-3 These results are helpful to understand the collision effects induced by He atom in further spectra investigations of cold Ca+ ions.
Ca+-He molecular system, potential energy curves, spin-orbit coupling, collision
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induced broadening and shifting
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1. Introduction
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In an ion trap, the ions can be cooled to very low temperature in the magnitude of mK, even µK through the diverse mechanisms [1]. When the ions are in the Lame-Dicke regime, the first-order Doppler frequency shift can be eliminated [2]. However, since tiny amount of background gas and rare gas atoms still exist in the traps, they can interact with ions which will result in broadening and shifting of spectral lines. In this work, we have investigated the impact on the broadening and shifting of the H and K lines of Ca+ colliding with He atom. So far, not much work has been done on the experiments and theoretical calculations of the broadening and shifting effects of Ca+-He spectral lines, especially in the case of low temperature. Baur and Cooper [3] measured the collisional broadening coefficients of the 397nm line for Ca+-He over a temperature range of 6100-8300K. Bowman and Lewis [4] have detected the broadening coefficients and shifting coefficients of H and K lines for Ca+-He and Ca+-Ar, respectively, at the temperature of 655K and 5200K. Recently Allard and Alekseev [5] used ab
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2 Theoretical Methods 2.1 Electronic Structure Computations
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initio potential energy curves (PECs), which were calculated with complete active space second-order perturbation CASPT2 for Ca+-He and density expansion, to examine the range of validity of the one-perturber approximation and determine the required order of the density expansion. In this paper, we use the semiclassical Anderson Talman [6] (AT) theory of spectral lines in combination with ab initio PECs to calculate the broadening and shifting coefficients for Ca+-He spectral lines. The PECs for these calculation were computed using multireference configuration interaction (MRCI) method [7,8].
All the ab initio calculations on the electronic structure for Ca+-He were computed with the quantum chemistry MOLPRO 2010 program package [9].
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Firstly, we calculated the low-lying -S states of Ca+-He with MRCI method. The small core Stuttgart relativistic pseudopotentials (PPs) and corresponding basis sets [10] are used for Ca+ ion in the calculation, which consist of 1s22s22p6 ten inner shell electrons. The Def2-TZVP all-electron segmented contracted Gaussian basis sets [11] are used for the He atom. Then the spin
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orbital coupling (SOC) calculation was performed based on the energy eigenvalue of the -S states with the MRCI method. In the subsequent SOC computations, the ECP10MDF pseudopotential [10] was chosen for Ca+, and the Def2-TZVP basis set was used for He atom. In order to obtain the PECs of low-lying state of Ca+-He system, the potential energies were calculated with the open-shell restricted Hartree-Fock (RHF) method to generate the single-configuration wavefunction of the ground state firstly; then the multiconfigurational wavefunction was calculated by the complete active space self-consistent field (CASSCF) method [12,13]; finally, the MRCI approach is used to calculate the correlation energies of the electronic states on the basis of the optimized reference wavefunction obtained from the CASSCF calculation. The Davidson correction (+Q) [14] is taken into account to balance the size-consistency error of the MRCI method. The corresponding SOC results calculated in the asymptotic limit of large internuclear distance, R, are summarized in Table 1.
2.2 Anderson-Talman Theory
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The nondegenerate AT theory are reviewed by Allard and Kielkopf [6] and the degenerate theory is developed by Allard et al. [16, 17]. To begin with, we only review the nondegenerate case briefly. The spectrum I(ω,T) is proportional to the Fourier transform of the autocorrelation function Φ(s,T), where ω is the angular frequency measured from the unshifted line center, T is the temperature, and s is time.
I , T s, T eis ds
(1)
Under some useful approximation, we can write Φ(s,T) like the following expression,
s, T exp{ng s, T }
(2)
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g s, T f v, T g s, v dv
(3)
0
And n is the number density of the perturbers, f(v,T) is the Maxwell speed distribution. s g s, v 2 bdb dx0 1 exp i 1V Rt dt 0 0
(4)
Rt b 2 x0 vt
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Here ΔV(R) is the difference potential (DP) which is the function of internuclear separation R and given by ΔV(R)=[Vi(R)-Vf(R)-(Ei-Ef)] where Vi(R) and Vf(R) are the Ca+-He PECs. Ei and Ef are the initial and final atomic energies of the transition we concerned about. The perturber is assumed to follow a classical rectilinear trajectory with constant speed v in the positive x direction, where x=x0+vt. The emitter is stationary at the origin. b is the impact parameter and the internuclear separation is given by
2 1/ 2
(5)
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To obtain g(s,T) as a function of temperature, the average over the Maxwell speed distribution f(v,T) is performed in Eq.(3). We take the approximation that g(s,v) is evaluated at the average atomic speed
v T 8kT /
1/ 2
(6)
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where k is Boltzmann’s constant, so that the integral in Eq.(3) is omitted. Impact approximation is taken into account for the low number density of perturbers and g(s,v) must be computed for sufficiently large s. For the limit s The g(s,v) can be written as
g s, v v i v s 0 v i0 v
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And with the Eq. (1) and (2) we obtain
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n cos n 0 n sin n 0 I , T 2 exp n 0 2 2 n n
(8)
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When the constant (ɑ0+iβ0) is taken as zero, the profile of Eq. (8) is a Lorentzian lineshape with a half-width nɑ and a shift nβ, which are the real and the imaginary parts of the slope of g(s,v) , respectively. In the impact limit, the broadening coefficient α and shifting coefficient β can be written as
v 2v bdb1 cos v, b 0
(9)
v 2v bdb sin v, b 0
(10)
v, b v
1
V b
2
x2
1/ 2
dx (11)
All calculations of the broadening and shifting coefficients in this paper are performed using Eq.(4) where a linear fit to g(s,v) is performed in the limit s .
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For the degenerate case, the correction formulation of Eq.(2) is
s, T exp n i g i s, T i
(12)
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We make the approximation that the transition dipole moments are constant to determine the weights πi. Allard et al. [17] have demonstrated that the approximation of constant dipole moment primarily influences the line wing and is not expected to significantly affect the broadening and shifting coefficients of the line core.
3. Results and Discussion
The PECs of three lowest energy levels of Ca+ colliding with He are computed. Figure 1
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shows the X2Σ+, A2Π, B2Δ, C2Σ+, D2Π and E2Σ+ PECs of six doublet -S states associated with three lowest dissociation limits. In the atom limits, Ca+ and He atom have no interactions, the X2Σ+ curve corresponds to the ground state Ca+(2S), A2Π, B2Δ and C2Σ+ converge into the Ca+ (2D) state, and the D2Π and E2Σ+ converge into the Ca+(2Po) state. As the internuclear R decreases, the PECs associated with 2D and 2P states become nondegenerate. Figure 2-4 exhibit the PECs of 9 Ω states correlated with lowest 5 atomic limits. Our computed values of equilibrium internuclear distance Re and dissociation energy De are summarized in the Table 2 associated with the theoretical values of Czuchaj et al[18] for comparison. In Figure 2, the X2Σ1/2+ curve corresponds to the Ca+(2S1/2) ground level, with a shallow well of 14.35cm-1 in the region of R=4.64Å. Comparing with previous computational work[18], our computed results give a smaller dissociation energy while a larger equilibrium internuclear distance. In Figure 3, the A2Π1/2 and A2Π3/2 states correspond to the Ca+(2D3/2) energy level, B2Δ3/2, B2Δ5/2 and C2Σ1/2+ correspond to the Ca+(2D5/2) energy level. In contrast to the repulsive nature of A and B states, our computed PECs of A and B states are bound with local minima at 2.6~2.8Å and dissociation energies of ~250cm-1. The values of SOC splitting for A2Π
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and B2 states at equilibrium internuclear distance are 11.3 and 16.9 cm-1, respectively. The SOC splitting values are significant in the regions near local minima of PECs. For the A2Π1/2 and A2Π3/2 states, though the two states are both correlated with the 2D3/2 state, our computations show that the A2Π1/2 and A2Π3/2 curves diverge significantly at approximately R<5Å where the SOC splitting is 1.01cm-1. When the spin-orbit coupling is introduced in the computations, avoided crossings
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may occur between the spin-orbit coupled states with the same values. In Figure 3, there is an avoided crossing point between =3/2 level of A2Π state (A2Π3/2) and =3/2 component in B2Δ state (B2Δ3/2) existing in the region of R=3Å. Figure 4 shows that the D2Π1/2 state corresponds to the Ca+(2P1/2) energy level, while D2Π3/2 and E2Σ1/2+ states correspond to the Ca+(2P3/2) energy level. The potential wells of D2Π1/2 and D2Π3/2 curves are deeper than that of the X2Σ1/2+, while the equilibrium internuclear distance Re and the dissociation energies De are almost same. As shown in Table 2, our computed calculated well depths of D2Π1/2 and D2Π3/2 states are shallower than those from Czuchaj’ work[18]. The relations between the internuclear reparation R and transition dipole moment(TDM) matrix elements with origin at center-of-mass of CaHe+ for the transitions to ground state X2Σ1/2+ are shown in Figure 5. The dipole matrix elements for the transitions between the ground state X2Σ1/2+ and the electronic states associated with Ca+(2D) state are almost zero, which shows these
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transitions in Ca+ are dipole forbidden; while for the transitions from states correlated with Ca+(2Po) to ground state, nonzero dipole matrix elements are found in Figure 5. These computed result is consistent with electric dipole selection rules. The functions of TDM matrix elements in the range of small R differ greatly from those in the range of large R, indicating the effects of He atom on the Ca+ cation. As the internuclear distance R increasing, these dipole moment matrix elements become constant, respectively; the transition properties of the asymptote in Ca+-He indeed become those in atomic Ca+ . Figure 6 shows the absolute values of quadrupole moment (QM) matrix elements for the transitions between the X2Σ1/2+ and excited electronic states. The origin of QM matrix elements is located at Ca+ atom. In the asymptote region, the QM matrix elements of these transitions tend to zero, while in the range of R=2~5Å, the computed values for C2Σ1/2+-X2Σ1/2+ and E2Σ1/2+-X2Σ1/2+ have significant values as large as few Debye. This maybe arises from the external He-atom electrostatic field effect on Ca+. The two potentials, our computed and that from Ref.[18], were used to compute the broadening and shifting coefficients(α and β) determined by the AT theory. In the ion trap, the velocity of the cooled ions is very low, so we assume the ions are static compared to the He atoms which are at the room temperature. We use the formula of average speed v T 8kT /
1/ 2
to
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replace the Maxwell speed distribution. The broadening and shifting β coefficients are calculated by numerically evaluating g(s,v) in Eq.(4). A linear fit to g(s,v) is performed in the impact approximation. Our computational results at low-temperature region are α=0.303, β=-0.0527 for H line and α=0.233, β=-0.0402 for K line with the unit in 10-20cm-1/cm-3. To examine our computational program, firstly, we used the analytic expression of potentials
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for Cs-Xe system in the previous work of Takeo[19] to compute the andfor Cs atom. We summarize these calculated results in the Table 3, and it can be seen that our computed values are in good agreement with those in Ref.[19], which demonstrates the validity of our calculation on broadening and shifting coefficients. In Table 4, we summarize our computed broadening and shifting coefficients under the temperature of 655K. Both our calculated and the reported [18] potentials give similar values for of both H and K lines, and similar value for H line, but give the values with opposite sign of value for K line. In contrast with previous slightly overestimated theoretical value[5], our
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broadening coefficient is underestimated comparing with experimental result[4]. Our computed for H line is only one third of previous theoretical value, while for K line, the two theoretical values are more close. The difference between the two theoretical values should arise from the differences of potentials adopted. Figure 7 shows the relation between the HWHM and shifting and the pressure of the He atoms. It can be seen that both HWHM and shifting are linear to the pressure of perturbers. So in the ion trap, the low concentration of He atoms has little effect on the broadening and shifting of the spectral lines of Ca+. We examine the integrands αint and βint of Eqs.(9) and (10) for H line of Ca+-He.
int , b b1 cos v, b int v, b b sin v, b As illustrated in the Figure 8 and Figure 9, both the integrands oscillate
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Figure
10
and
Figure
11
show
the
integrals
(b) 2v db' int b'
and
0 b
(b) 2v db' int b' . In the region of R less than b0, ɑ(b) increases as b increases and β(b) is 0
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approximately zero. For b>b0,ɑ(b) gradually equals to a constant; β(b) decreases rapidly, and then closes to a certain constant. It is obvious that the shifting coefficient is more sensitive to the long-range PECs than the broadening coefficient.
4. Conclusion
The relativistic MRCI+Q calculation were performed on the ground state and several
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low-lying excited states of Ca+-He. We compute the PECs of -S states associated with three lowest dissociation limits. Using the SOC calculation, the 9 Ω states are obtained. Also, we calculate the transition dipole moment matrix elements and transition quadrupole moment matrix elements. With the AT theory, we obtained the broadening and shifting coefficients for Ca+-He spectral lines in the low temperature regime. For H line, α=0.303×10-20cm-1/cm-3, β=-0.0527×10-20cm-1/cm-3; For K line, α=0.233×10-20cm-1/cm-3, β=-0.0402×10-20cm-1/cm-3 These results can be useful in further understanding of collision effects of the cold Ca+ ions with the background He atoms.
Acknowledgement
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Xin Wang is very grateful to Professor Allard Nicole for the discussions about the Anderson Talman theory. This work was supported by the National Science Foundation of China (Grant No.91436103, 11574114), Research Programme of National University of Defense Technology, China (Grant No.JC15-0203).
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References
[1]Hänsch T W, Schawlow A L. Cooling of gases by laser radiation. Opt Commun 1975;13:68-69. [2]Dicke R H. The effect of collisions upon the Doppler width of spectral lines. Phys Rev 1953;89:472. [3] Baur J F, Cooper J. A shock tube study of line broadening in a temperature range of 6100 to 8300 K. J Quant Spectrosc Radiat Transf 1977;17:311-322. [4] Bowman N J, Lewis E L. Collision broadening and shifts in the spectra of neutral and singly ionised calcium. J Phys B: Atom Molec Phys 1978;11:1703. [5]Allard N F, Alekseev V A. Collisional profiles of ionized calcium perturbed by helium. Adv Space Res 2014;54:1248-1253.
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[6]Allard N, Kielkopf J. The effect of neutral nonresonant collisions on atomic spectral lines. Rev Mod Phys 1982;54:1103. [7]Werner H J, Knowles P J. An efficient internally contracted multiconfiguration–reference configuration interaction method. J Chem Phys 1988;89:5803-5814. [8]Knowles P J, Werner H J. An efficient method for the evaluation of coupling coefficients in configuration interaction calculations. Chem Phys Lett 1988;145:514-522. [9] Werner H-J, Knowles PJ, Knizia G, Manby FR, Schütz M, Celani P, et al. Molpro, version 2010.1, a package of ab initio programs; 2010. [10]Lim I S, Stoll H, Schwerdtfeger P. Relativistic small-core energy-consistent pseudopotentials for the alkaline-earth elements from Ca to Ra. J Chem Phys 2006;124:034107. [11]Weigend F, Ahlrichs R. Balanced basis sets of split valence, triple zeta valence and quadruple zeta valence quality for H to Rn: design and assessment of accuracy. Phys Chem Chem Phys 2005;7:3297-3305. [12]Knowles P J, Werner H J. An efficient second-order MCSCF method for long configuration expansions. Chem Phys Lett 1985;115:259-267. [13]Werner H J, Knowles P J. A second order multiconfiguration SCF procedure with optimum convergence. J Chem Phys 1985;82:5053-5063. [14]Langhoff S R, Davidson E R. Configuration interaction calculations on the nitrogen molecule. Int J Quantum Chem 1974;8:61-72. [15]41Ralchenko Y, Kramida A E, Reader J, and NIST ASD Team, NIST Atomic Spectral Database, version 4.1.0, National Institute of Standards and Technology, Gaithersburg, MD, 2011, see http://physics.nist.gov/asd3. [16]Allard N F, Royer A, Kielkopf J F, et al. Effect of the variation of electric-dipole moments on the shape of pressure-broadened atomic spectral lines. Phys Rev A 1999;60:1021. [17]Allard N F, Spiegelman F. Collisional line profiles of rubidium and cesium perturbed by helium and molecular hydrogen. Astron Astrophys 2006;452:351-356. [18]Czuchaj E, Rebentrost F, Stoll H, et al. Pseudopotential calculations for the potential energies of Ca+He and Ca+Ne. Chem Phys 1996;207:51-62. [19]Takeo M. Theoretical Calculation of Pressure-Broadened Atomic Line Profiles. Phys Rev A 1970;1:1143-1149.
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Figure caption: Figure 1. The PECs of -S states associated with three lowest dissociation limits Figure 2. The PEC of state associated with 2S1/2 ground state Figure 3. The PECs of states associated with 2D3/2 and 2D3/2 state Figure 4. The PECs of states associated with 2Po state Figure 5. Transition dipole moment matrix elements with origin at center-of-mass of CaHe+ Figure 6. The absolute values of Transition quadrupole moment matrix elements with origin at center of Ca+ atom Figure 7. Relation between the HWHM, shifting and the pressure of the He atoms. n represents the density of He
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atoms, H represents H line, and K represents K line. Figure 8. The integrands αint as a function of b Figure 9. The integrands βint as a function of b Figure 10. α(b) as a function of b Figure 11. β(b) as a function of b
spin-orbit splitting. Energies are in cm-1. 2D 5/2
Ca+He
13663
13700
NIST
13650
13711
Δ1
2P
38
25306
25411
105
61
25192
25414
223
1/2
2P
Δ2
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2D 3/2
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Table 1 The potential energies at R=100Å. The experimental NIST values [15] are given, Δ1 and Δ2 are the
Table 2. The equilibrium position (Å), and dissociate energy (cm-1) are showed under the Re and De columns.
Re
De
R e’
De’
4.64
14
4.4
30
A Π1/2
2.60
260
...
...
A2Π3/2
2.62
249
...
...
B2Δ3/2
2.73
249
...
...
B Δ5/2
2.78
232
...
...
C2Σ
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X2Σ1/2+
2
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The pseudopotential calculations by Czuchaj (Ref.18) are listed under the R e’ and De’ columns for comparison.
+ 1/2
3.56
73
...
...
D2Π1/2
2.49
563
2.38
710
D Π3/2
2.49
598
2.38
710
E Σ1/2
5.31
8.9
7.0
7.5
2
2
+
Table 3. Comparisons with broadening and shifting coefficients in Cs-Xe system
References this work Takeo[19]
broadening(10-8cm3/s) 0.207 0.228
shifting(10-8cm3/s) -0.192 -0.156
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H line
References a
this work
b
this work
-0.059
0.374
-0.078
0.297
-0.047
0.263
0.0267
0.635
[4]
0.55
Bowman and Lewis
0.79 -0.19
Computed with our calculated potentials
b
0.78
-0.059
Computed with potentials reported in ref.[18]
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a
0.347 [5]
Allard and Alekseev
K line
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FIG.1. The PECs of -S states associated with three lowest dissociation limits
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FIG.2.The PEC of state associated with 2S1/2 ground state
FIG.3.The PECs of states associated with 2D3/2 and 2D3/2 state
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FIG.4.The PECs of states associated with 2Po state
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FIG.5. Transition dipole moment matrix elements with origin at center-of-mass of CaHe+
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FIG.6. The absolute values of Transition quadrupole moment matrix elements with origin at center of Ca+ atom
FIG.7. Relation between the HWHM, shifting and the pressure of the He atoms. n represents the density of He atoms, H represents H line, and K represents K line.
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FIG.8. The integrands αint as a function of b
FIG.9 The integrands βint as a function of b
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FIG.10 α(b) as a function of b
FIG.11 β(b) as a function of b
Collision induced broadening and shifting of the H and K lines of Ca+ at low temperature Xin Wang , Rui Zhang , Yong Shen , Qu Liu , Hongxin Zou , Bing Yan PII: DOI: Reference:
S0022-4073(16)30623-9 10.1016/j.jqsrt.2017.04.022 JQSRT 5678
To appear in:
xxx xxx xxx
Received date: Revised date: Accepted date:
26 September 2016 21 March 2017 19 April 2017
Please cite this article as: Xin Wang , Rui Zhang , Yong Shen , Qu Liu , Hongxin Zou , Bing Yan , Collision induced broadening and shifting of the H and K lines of Ca+ at low temperature, xxx xxx xxx (2017), doi: 10.1016/j.jqsrt.2017.04.022
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT
Highlights •The PECs of the Λ-S states correlating with lowest three atomic limits in Ca+-He molecular system have been calculated. •The SOC have been considered and the PECs of Ω states are obtained.
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• Collision induced broadening and shifting of the H and K lines of Ca+ by He atoms at low temperature have been computed with
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Anderson-Talman theory.
*Email:[email protected] †Email: [email protected]
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Collision induced broadening and shifting of the H and K lines of Ca+ at low temperature Xin Wang, Rui Zhang, Yong Shen, Qu Liu, Hongxin Zou* Interdisciplinary Center of Quantum Information, National University of Defense Technology, Changsha 410073, China
Bing Yan†
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Institute of Atomic and Molecular Physics, Jilin University, Changchun 130012, China
Abstract
Multireference configuration interaction method was used to compute the potential energy
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curves of -S states correlating with lowest three atomic limits in Ca+-He molecular collision system. The potential energy curves of nine states were obtained with inclusion of spin-orbit coupling. And the electric dipole and quadrupole moment matrix elements between excited states and ground state were also computed. Furthermore, with aid of the Anderson- Talman theory we calculated the broadening and shifting coefficients for Ca+-He spectral lines in the low temperature regime. For H line, α=0.303×10-20cm-1/cm-3, β=-0.0527×10-20cm-1/cm-3; For K line, α=0.233×10-20cm-1/cm-3, β=-0.0402×10-20cm-1/cm-3 These results are helpful to understand the collision effects induced by He atom in further spectra investigations of cold Ca+ ions.
Ca+-He molecular system, potential energy curves, spin-orbit coupling, collision
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induced broadening and shifting
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1. Introduction
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In an ion trap, the ions can be cooled to very low temperature in the magnitude of mK, even µK through the diverse mechanisms [1]. When the ions are in the Lame-Dicke regime, the first-order Doppler frequency shift can be eliminated [2]. However, since tiny amount of background gas and rare gas atoms still exist in the traps, they can interact with ions which will result in broadening and shifting of spectral lines. In this work, we have investigated the impact on the broadening and shifting of the H and K lines of Ca+ colliding with He atom. So far, not much work has been done on the experiments and theoretical calculations of the broadening and shifting effects of Ca+-He spectral lines, especially in the case of low temperature. Baur and Cooper [3] measured the collisional broadening coefficients of the 397nm line for Ca+-He over a temperature range of 6100-8300K. Bowman and Lewis [4] have detected the broadening coefficients and shifting coefficients of H and K lines for Ca+-He and Ca+-Ar, respectively, at the temperature of 655K and 5200K. Recently Allard and Alekseev [5] used ab
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2 Theoretical Methods 2.1 Electronic Structure Computations
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initio potential energy curves (PECs), which were calculated with complete active space second-order perturbation CASPT2 for Ca+-He and density expansion, to examine the range of validity of the one-perturber approximation and determine the required order of the density expansion. In this paper, we use the semiclassical Anderson Talman [6] (AT) theory of spectral lines in combination with ab initio PECs to calculate the broadening and shifting coefficients for Ca+-He spectral lines. The PECs for these calculation were computed using multireference configuration interaction (MRCI) method [7,8].
All the ab initio calculations on the electronic structure for Ca+-He were computed with the quantum chemistry MOLPRO 2010 program package [9].
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Firstly, we calculated the low-lying -S states of Ca+-He with MRCI method. The small core Stuttgart relativistic pseudopotentials (PPs) and corresponding basis sets [10] are used for Ca+ ion in the calculation, which consist of 1s22s22p6 ten inner shell electrons. The Def2-TZVP all-electron segmented contracted Gaussian basis sets [11] are used for the He atom. Then the spin
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orbital coupling (SOC) calculation was performed based on the energy eigenvalue of the -S states with the MRCI method. In the subsequent SOC computations, the ECP10MDF pseudopotential [10] was chosen for Ca+, and the Def2-TZVP basis set was used for He atom. In order to obtain the PECs of low-lying state of Ca+-He system, the potential energies were calculated with the open-shell restricted Hartree-Fock (RHF) method to generate the single-configuration wavefunction of the ground state firstly; then the multiconfigurational wavefunction was calculated by the complete active space self-consistent field (CASSCF) method [12,13]; finally, the MRCI approach is used to calculate the correlation energies of the electronic states on the basis of the optimized reference wavefunction obtained from the CASSCF calculation. The Davidson correction (+Q) [14] is taken into account to balance the size-consistency error of the MRCI method. The corresponding SOC results calculated in the asymptotic limit of large internuclear distance, R, are summarized in Table 1.
2.2 Anderson-Talman Theory
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The nondegenerate AT theory are reviewed by Allard and Kielkopf [6] and the degenerate theory is developed by Allard et al. [16, 17]. To begin with, we only review the nondegenerate case briefly. The spectrum I(ω,T) is proportional to the Fourier transform of the autocorrelation function Φ(s,T), where ω is the angular frequency measured from the unshifted line center, T is the temperature, and s is time.
I , T s, T eis ds
(1)
Under some useful approximation, we can write Φ(s,T) like the following expression,
s, T exp{ng s, T }
(2)
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g s, T f v, T g s, v dv
(3)
0
And n is the number density of the perturbers, f(v,T) is the Maxwell speed distribution. s g s, v 2 bdb dx0 1 exp i 1V Rt dt 0 0
(4)
Rt b 2 x0 vt
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Here ΔV(R) is the difference potential (DP) which is the function of internuclear separation R and given by ΔV(R)=[Vi(R)-Vf(R)-(Ei-Ef)] where Vi(R) and Vf(R) are the Ca+-He PECs. Ei and Ef are the initial and final atomic energies of the transition we concerned about. The perturber is assumed to follow a classical rectilinear trajectory with constant speed v in the positive x direction, where x=x0+vt. The emitter is stationary at the origin. b is the impact parameter and the internuclear separation is given by
2 1/ 2
(5)
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To obtain g(s,T) as a function of temperature, the average over the Maxwell speed distribution f(v,T) is performed in Eq.(3). We take the approximation that g(s,v) is evaluated at the average atomic speed
v T 8kT /
1/ 2
(6)
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where k is Boltzmann’s constant, so that the integral in Eq.(3) is omitted. Impact approximation is taken into account for the low number density of perturbers and g(s,v) must be computed for sufficiently large s. For the limit s The g(s,v) can be written as
g s, v v i v s 0 v i0 v
(7)
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And with the Eq. (1) and (2) we obtain
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n cos n 0 n sin n 0 I , T 2 exp n 0 2 2 n n
(8)
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When the constant (ɑ0+iβ0) is taken as zero, the profile of Eq. (8) is a Lorentzian lineshape with a half-width nɑ and a shift nβ, which are the real and the imaginary parts of the slope of g(s,v) , respectively. In the impact limit, the broadening coefficient α and shifting coefficient β can be written as
v 2v bdb1 cos v, b 0
(9)
v 2v bdb sin v, b 0
(10)
v, b v
1
V b
2
x2
1/ 2
dx (11)
All calculations of the broadening and shifting coefficients in this paper are performed using Eq.(4) where a linear fit to g(s,v) is performed in the limit s .
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For the degenerate case, the correction formulation of Eq.(2) is
s, T exp n i g i s, T i
(12)
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We make the approximation that the transition dipole moments are constant to determine the weights πi. Allard et al. [17] have demonstrated that the approximation of constant dipole moment primarily influences the line wing and is not expected to significantly affect the broadening and shifting coefficients of the line core.
3. Results and Discussion
The PECs of three lowest energy levels of Ca+ colliding with He are computed. Figure 1
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shows the X2Σ+, A2Π, B2Δ, C2Σ+, D2Π and E2Σ+ PECs of six doublet -S states associated with three lowest dissociation limits. In the atom limits, Ca+ and He atom have no interactions, the X2Σ+ curve corresponds to the ground state Ca+(2S), A2Π, B2Δ and C2Σ+ converge into the Ca+ (2D) state, and the D2Π and E2Σ+ converge into the Ca+(2Po) state. As the internuclear R decreases, the PECs associated with 2D and 2P states become nondegenerate. Figure 2-4 exhibit the PECs of 9 Ω states correlated with lowest 5 atomic limits. Our computed values of equilibrium internuclear distance Re and dissociation energy De are summarized in the Table 2 associated with the theoretical values of Czuchaj et al[18] for comparison. In Figure 2, the X2Σ1/2+ curve corresponds to the Ca+(2S1/2) ground level, with a shallow well of 14.35cm-1 in the region of R=4.64Å. Comparing with previous computational work[18], our computed results give a smaller dissociation energy while a larger equilibrium internuclear distance. In Figure 3, the A2Π1/2 and A2Π3/2 states correspond to the Ca+(2D3/2) energy level, B2Δ3/2, B2Δ5/2 and C2Σ1/2+ correspond to the Ca+(2D5/2) energy level. In contrast to the repulsive nature of A and B states, our computed PECs of A and B states are bound with local minima at 2.6~2.8Å and dissociation energies of ~250cm-1. The values of SOC splitting for A2Π
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and B2 states at equilibrium internuclear distance are 11.3 and 16.9 cm-1, respectively. The SOC splitting values are significant in the regions near local minima of PECs. For the A2Π1/2 and A2Π3/2 states, though the two states are both correlated with the 2D3/2 state, our computations show that the A2Π1/2 and A2Π3/2 curves diverge significantly at approximately R<5Å where the SOC splitting is 1.01cm-1. When the spin-orbit coupling is introduced in the computations, avoided crossings
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may occur between the spin-orbit coupled states with the same values. In Figure 3, there is an avoided crossing point between =3/2 level of A2Π state (A2Π3/2) and =3/2 component in B2Δ state (B2Δ3/2) existing in the region of R=3Å. Figure 4 shows that the D2Π1/2 state corresponds to the Ca+(2P1/2) energy level, while D2Π3/2 and E2Σ1/2+ states correspond to the Ca+(2P3/2) energy level. The potential wells of D2Π1/2 and D2Π3/2 curves are deeper than that of the X2Σ1/2+, while the equilibrium internuclear distance Re and the dissociation energies De are almost same. As shown in Table 2, our computed calculated well depths of D2Π1/2 and D2Π3/2 states are shallower than those from Czuchaj’ work[18]. The relations between the internuclear reparation R and transition dipole moment(TDM) matrix elements with origin at center-of-mass of CaHe+ for the transitions to ground state X2Σ1/2+ are shown in Figure 5. The dipole matrix elements for the transitions between the ground state X2Σ1/2+ and the electronic states associated with Ca+(2D) state are almost zero, which shows these
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transitions in Ca+ are dipole forbidden; while for the transitions from states correlated with Ca+(2Po) to ground state, nonzero dipole matrix elements are found in Figure 5. These computed result is consistent with electric dipole selection rules. The functions of TDM matrix elements in the range of small R differ greatly from those in the range of large R, indicating the effects of He atom on the Ca+ cation. As the internuclear distance R increasing, these dipole moment matrix elements become constant, respectively; the transition properties of the asymptote in Ca+-He indeed become those in atomic Ca+ . Figure 6 shows the absolute values of quadrupole moment (QM) matrix elements for the transitions between the X2Σ1/2+ and excited electronic states. The origin of QM matrix elements is located at Ca+ atom. In the asymptote region, the QM matrix elements of these transitions tend to zero, while in the range of R=2~5Å, the computed values for C2Σ1/2+-X2Σ1/2+ and E2Σ1/2+-X2Σ1/2+ have significant values as large as few Debye. This maybe arises from the external He-atom electrostatic field effect on Ca+. The two potentials, our computed and that from Ref.[18], were used to compute the broadening and shifting coefficients(α and β) determined by the AT theory. In the ion trap, the velocity of the cooled ions is very low, so we assume the ions are static compared to the He atoms which are at the room temperature. We use the formula of average speed v T 8kT /
1/ 2
to
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replace the Maxwell speed distribution. The broadening and shifting β coefficients are calculated by numerically evaluating g(s,v) in Eq.(4). A linear fit to g(s,v) is performed in the impact approximation. Our computational results at low-temperature region are α=0.303, β=-0.0527 for H line and α=0.233, β=-0.0402 for K line with the unit in 10-20cm-1/cm-3. To examine our computational program, firstly, we used the analytic expression of potentials
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for Cs-Xe system in the previous work of Takeo[19] to compute the andfor Cs atom. We summarize these calculated results in the Table 3, and it can be seen that our computed values are in good agreement with those in Ref.[19], which demonstrates the validity of our calculation on broadening and shifting coefficients. In Table 4, we summarize our computed broadening and shifting coefficients under the temperature of 655K. Both our calculated and the reported [18] potentials give similar values for of both H and K lines, and similar value for H line, but give the values with opposite sign of value for K line. In contrast with previous slightly overestimated theoretical value[5], our
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broadening coefficient is underestimated comparing with experimental result[4]. Our computed for H line is only one third of previous theoretical value, while for K line, the two theoretical values are more close. The difference between the two theoretical values should arise from the differences of potentials adopted. Figure 7 shows the relation between the HWHM and shifting and the pressure of the He atoms. It can be seen that both HWHM and shifting are linear to the pressure of perturbers. So in the ion trap, the low concentration of He atoms has little effect on the broadening and shifting of the spectral lines of Ca+. We examine the integrands αint and βint of Eqs.(9) and (10) for H line of Ca+-He.
int , b b1 cos v, b int v, b b sin v, b As illustrated in the Figure 8 and Figure 9, both the integrands oscillate
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Figure
10
and
Figure
11
show
the
integrals
(b) 2v db' int b'
and
0 b
(b) 2v db' int b' . In the region of R less than b0, ɑ(b) increases as b increases and β(b) is 0
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approximately zero. For b>b0,ɑ(b) gradually equals to a constant; β(b) decreases rapidly, and then closes to a certain constant. It is obvious that the shifting coefficient is more sensitive to the long-range PECs than the broadening coefficient.
4. Conclusion
The relativistic MRCI+Q calculation were performed on the ground state and several
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low-lying excited states of Ca+-He. We compute the PECs of -S states associated with three lowest dissociation limits. Using the SOC calculation, the 9 Ω states are obtained. Also, we calculate the transition dipole moment matrix elements and transition quadrupole moment matrix elements. With the AT theory, we obtained the broadening and shifting coefficients for Ca+-He spectral lines in the low temperature regime. For H line, α=0.303×10-20cm-1/cm-3, β=-0.0527×10-20cm-1/cm-3; For K line, α=0.233×10-20cm-1/cm-3, β=-0.0402×10-20cm-1/cm-3 These results can be useful in further understanding of collision effects of the cold Ca+ ions with the background He atoms.
Acknowledgement
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Xin Wang is very grateful to Professor Allard Nicole for the discussions about the Anderson Talman theory. This work was supported by the National Science Foundation of China (Grant No.91436103, 11574114), Research Programme of National University of Defense Technology, China (Grant No.JC15-0203).
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References
[1]Hänsch T W, Schawlow A L. Cooling of gases by laser radiation. Opt Commun 1975;13:68-69. [2]Dicke R H. The effect of collisions upon the Doppler width of spectral lines. Phys Rev 1953;89:472. [3] Baur J F, Cooper J. A shock tube study of line broadening in a temperature range of 6100 to 8300 K. J Quant Spectrosc Radiat Transf 1977;17:311-322. [4] Bowman N J, Lewis E L. Collision broadening and shifts in the spectra of neutral and singly ionised calcium. J Phys B: Atom Molec Phys 1978;11:1703. [5]Allard N F, Alekseev V A. Collisional profiles of ionized calcium perturbed by helium. Adv Space Res 2014;54:1248-1253.
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[6]Allard N, Kielkopf J. The effect of neutral nonresonant collisions on atomic spectral lines. Rev Mod Phys 1982;54:1103. [7]Werner H J, Knowles P J. An efficient internally contracted multiconfiguration–reference configuration interaction method. J Chem Phys 1988;89:5803-5814. [8]Knowles P J, Werner H J. An efficient method for the evaluation of coupling coefficients in configuration interaction calculations. Chem Phys Lett 1988;145:514-522. [9] Werner H-J, Knowles PJ, Knizia G, Manby FR, Schütz M, Celani P, et al. Molpro, version 2010.1, a package of ab initio programs; 2010. [10]Lim I S, Stoll H, Schwerdtfeger P. Relativistic small-core energy-consistent pseudopotentials for the alkaline-earth elements from Ca to Ra. J Chem Phys 2006;124:034107. [11]Weigend F, Ahlrichs R. Balanced basis sets of split valence, triple zeta valence and quadruple zeta valence quality for H to Rn: design and assessment of accuracy. Phys Chem Chem Phys 2005;7:3297-3305. [12]Knowles P J, Werner H J. An efficient second-order MCSCF method for long configuration expansions. Chem Phys Lett 1985;115:259-267. [13]Werner H J, Knowles P J. A second order multiconfiguration SCF procedure with optimum convergence. J Chem Phys 1985;82:5053-5063. [14]Langhoff S R, Davidson E R. Configuration interaction calculations on the nitrogen molecule. Int J Quantum Chem 1974;8:61-72. [15]41Ralchenko Y, Kramida A E, Reader J, and NIST ASD Team, NIST Atomic Spectral Database, version 4.1.0, National Institute of Standards and Technology, Gaithersburg, MD, 2011, see http://physics.nist.gov/asd3. [16]Allard N F, Royer A, Kielkopf J F, et al. Effect of the variation of electric-dipole moments on the shape of pressure-broadened atomic spectral lines. Phys Rev A 1999;60:1021. [17]Allard N F, Spiegelman F. Collisional line profiles of rubidium and cesium perturbed by helium and molecular hydrogen. Astron Astrophys 2006;452:351-356. [18]Czuchaj E, Rebentrost F, Stoll H, et al. Pseudopotential calculations for the potential energies of Ca+He and Ca+Ne. Chem Phys 1996;207:51-62. [19]Takeo M. Theoretical Calculation of Pressure-Broadened Atomic Line Profiles. Phys Rev A 1970;1:1143-1149.
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Figure caption: Figure 1. The PECs of -S states associated with three lowest dissociation limits Figure 2. The PEC of state associated with 2S1/2 ground state Figure 3. The PECs of states associated with 2D3/2 and 2D3/2 state Figure 4. The PECs of states associated with 2Po state Figure 5. Transition dipole moment matrix elements with origin at center-of-mass of CaHe+ Figure 6. The absolute values of Transition quadrupole moment matrix elements with origin at center of Ca+ atom Figure 7. Relation between the HWHM, shifting and the pressure of the He atoms. n represents the density of He
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atoms, H represents H line, and K represents K line. Figure 8. The integrands αint as a function of b Figure 9. The integrands βint as a function of b Figure 10. α(b) as a function of b Figure 11. β(b) as a function of b
spin-orbit splitting. Energies are in cm-1. 2D 5/2
Ca+He
13663
13700
NIST
13650
13711
Δ1
2P
38
25306
25411
105
61
25192
25414
223
1/2
2P
Δ2
3/2
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2D 3/2
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Table 1 The potential energies at R=100Å. The experimental NIST values [15] are given, Δ1 and Δ2 are the
Table 2. The equilibrium position (Å), and dissociate energy (cm-1) are showed under the Re and De columns.
Re
De
R e’
De’
4.64
14
4.4
30
A Π1/2
2.60
260
...
...
A2Π3/2
2.62
249
...
...
B2Δ3/2
2.73
249
...
...
B Δ5/2
2.78
232
...
...
C2Σ
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X2Σ1/2+
2
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The pseudopotential calculations by Czuchaj (Ref.18) are listed under the R e’ and De’ columns for comparison.
+ 1/2
3.56
73
...
...
D2Π1/2
2.49
563
2.38
710
D Π3/2
2.49
598
2.38
710
E Σ1/2
5.31
8.9
7.0
7.5
2
2
+
Table 3. Comparisons with broadening and shifting coefficients in Cs-Xe system
References this work Takeo[19]
broadening(10-8cm3/s) 0.207 0.228
shifting(10-8cm3/s) -0.192 -0.156
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H line
References a
this work
b
this work
-0.059
0.374
-0.078
0.297
-0.047
0.263
0.0267
0.635
[4]
0.55
Bowman and Lewis
0.79 -0.19
Computed with our calculated potentials
b
0.78
-0.059
Computed with potentials reported in ref.[18]
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a
0.347 [5]
Allard and Alekseev
K line
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FIG.1. The PECs of -S states associated with three lowest dissociation limits
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FIG.2.The PEC of state associated with 2S1/2 ground state
FIG.3.The PECs of states associated with 2D3/2 and 2D3/2 state
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FIG.4.The PECs of states associated with 2Po state
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FIG.5. Transition dipole moment matrix elements with origin at center-of-mass of CaHe+
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FIG.6. The absolute values of Transition quadrupole moment matrix elements with origin at center of Ca+ atom
FIG.7. Relation between the HWHM, shifting and the pressure of the He atoms. n represents the density of He atoms, H represents H line, and K represents K line.
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FIG.8. The integrands αint as a function of b
FIG.9 The integrands βint as a function of b
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FIG.10 α(b) as a function of b
FIG.11 β(b) as a function of b