Electric-field effects on the N - APS Link Manager

PHYSICAL REVIEW A

DECEMBER 1994

VOLUME 50, NUMBER 6

Electric-field efFects on the N

=3 'P'(1) resonance

of H

Y. K. Ho Institute

of Atomic and Molecular Sciences, Academia Sinica, P. O. Box 23-166, Taipei,

Taiwan, Republic

of China

J. Callaway* Department

of Physics and

Louisiana State Uniuersity, Baton Rouge, Louisiana 70803 (Received 16 August 1994)

Astronomy,

Electric-field effects on the N =3 'P'(1) resonance of H are investigated theoretically using a method rotation. Products of Slater orbitals are used to represent the two-electron wave functions with „=5 employed for individual electrons. Block matrices with up to „=6are used. Convergence behavior for the resonance parameters is examined by using different values of „and „. Results for the energy position and width of the N =3 'P'(1) resonance under the influence of external electric fields are compared with experimental observations.

of complex-coordinate

1,

L,

l,

L,

PACS number(s): 32.80.Dz, 32.60. + i

I. INTRODUCTION

H

of This work presents a theoretical investigation electric-field effects on the N =3 'P'(1) state of H . The method of complex-coordinate rotation [1,2] is used. The field effects on the 'P' resonances in H have been subjected to continuous experimental studies. The Stark effect on the lowest 'P' Feshbach resonance of H below the N = 2 hydrogen threshold and on the shape resonance above the N =2 hydrogen threshold have been investigated [3]. Since the 'P'(1) Feshbach resonance lies at a position very close to the second member of the 'S' states, even a weak external electric field will cause a strong mixing of these two states and result in splitting the 'P'(1) state into two components. A third weak component, which was also observed in the experiment, is a result of the mixing of the 'P'(1) state with a nearby 'D'(1) state. A theoretical study [4] using the stabilization method has found qualitative agreement with the experimental results for the resonance positions. On the experimental side, the field effects on the N =2 Feshbach and shape resonances were performed with field strengths of up to 1.2 MV/cm [5]. In addition, field effects on the N=3'P' state were observed [6], as well as an investigation of the field effects on the 'P' states associated with other high excitation thresholds [7]. On the theoretical side, the field effects on the N =2 'P' shape resonance were studied by Wendoloski and Reinhardt [8], and recently by Du, Fabrikant, and Starace [9]. However, theoretical studies of the N=3'P' resonance have not been done. Here we present such an investigation by using the method of complex-coordinate rotation [1,2].

II. THEORY

AND CALCULATIONS

The Stark effect of atoms and ions can be investigated by using a method of complex-coordinate rotation [1,2]. The Hamiltonian of an atom in an external field is 'Deceased. 1050-2947/94/50(6)/4941(4)/$06. 00

50

=T+ V+F r,

where F is the external field, and T and V are the usual kinetic and potential operators, respectively. Under the inhuence of the field the bound states of the atoms and ions become quasibound states, and the electrons will tunnel through the potential barrier formed by the combined Coulomb and external electric fields. As a result, the energies of such states become complex. The real part of a complex energy represents the shifted resonance position, and the imaginary part can be related to the lifetime of the quasibound state by the usual uncertainty principle. In the method of complex-coordinate rotation, the radial coordinates are transformed by

r

re'

(2)

can be written as

and the Hamiltonian

H(g)=Te

's+ ye '8+F re'e

.

Complex eigenvalues are obtained by diagonalizing transformed Hamiltonian,

E=&c IH(8)lc

&/&a le

),

(3)

the (4)

and the complex resonance energy is given by

—

E„,=E„iI /2,

where E, gives the shifted energy position, and I the resonance width with which the Stark broadening can be studied. The method is valid for isolated resonances and if the interaction with the background for such resonances is not too strong. This method was used by Chu and Reinhardt to examine the Stark effect of hydrogen. They have examined the effect on the hydrogen ground state [10] and the N=2 excited states [11]. The field effects on the 'P shape resonance in H lying above the N=2 hydrogen threshold were studied by using the method of complex-coordinate rotation [8]. Recently, we carried out an investigation of electricfield effects on the N =3 'P'(1) resonance of H by using 4941

1994

The American Physical Society

Y. K. HO AND J. CALLA%AY

4942

TABLE I. Convergence behavior of the N F =3.073 X 10 Ry.

=3 'P'

resonance

for

(~ max

&

~max

4

4

5

5 5

6

E

Terms

)

1005 1379

1516

TABLE II. Convergence behavior of the N F =9. 1804X 10 Ry. max

~

I max )

0.002 273 0.002 274 0.002 274

E

Terms

the method of complex-coordinate rotation [12]. The details of such an investigation are reported here. We use products of Slater orbitals to represent the two-electron „=5 employed for individual wave functions, with electrons. The products of Slater orbitals are the following:

1,

b

g, (r, )g&

(rz ) YI, &&(1, 2)S (o»o z),

(6)

la, Ib ij

where

g, (r)=r

'exp(

— g, r) .

In Eq. (6), A is the antisymmetrizing operator, S is a two-particle spin eigenfunction, and the g s are individual Slater orbitals. Y is the eigenfunction of the total angular momentum L,

Y„g(1,2)=g QC(la, lb, L;m„, m, i„M) (1)Yi,

X YI,

(2),

(8)

where C is the Clebsch-Gordan coefficient. For M=O, „=6 are used. The block block matrices with up to matrices of 'S', 'P', 'O', 'F', 'G', 'H', and 'I' are coupled by the external dc fields. For the individual electrons, we use up to a total of nine s-type, eight p-type, seven d-type, six f-type, five g-type, and four h-type orbitals. The highest 1 value for the individual electrons is

L,

Convergence behavior for the resonance parameters is examined by using different values of „and „. For „=4 and „=3, the orbitals would example, when

L,

1,

L,

I,

4.124

)[XX

XX

"x„

h

-0.126-

.

igloo

127-

1/2 I'

E,

%.125-

-2

X

ooocooo 4 044

%.128

o4

I

I

6 4 F(10 'Ry)

FIG. I. Electric-field effects on the resonance = 3 'P'(1) resonance.

width of the N

resonance

energy

and

I (Ry)

(Ry)

—0. 127 657 —0. 127 847 —0. 127 774 —0. 127 79

745 1005 1379

1516

4= A g gC,

= 3 'P'

for

I (Ry)

(Ry)

—0. 125 784 —0. 125 786 —0. 125 786

50

0.004 853 0.005 021 0.005 185 0.005 096

couple to a two-electron wave function that consists of 130 terms for 'S' states, 170 terms for 'P' states, 196 terms for 'D' states, 152 terms for 'F' states, and 97 terms for 'G' states. With the coupling by the external field, they would add up to a total of 745 terms. Similar„=5 and „=5, we have a total of 1397 ly, for terms. The largest basis set used in this work is 1516 „=6 and „=5. Table I terms obtained by using shows the convergence behavior for a field strength of F=3.073X10 " Ry. In the Rydberg units for field Ry. Table II shows strengths, 1 MV/em=3. 89X10 the highest electric field used in this work, i.e. , F =9. 1804X10 Ry. Our results for the energy and width of the N =3 'P'(I) resonance are summarized in Fig. 1 and Table III as a function of ext;nal electric fields. In Fig. 1, it can be seen that initially when the field is increased the width decreases slightly. As the field is increased to a value of about 3 X 10 Ry, the width then starts to increase, and in a more rapid fashion. Also in Fig. 1, the resonance energy is shown shifted downward when the field is increased. It is further noted that at low field, the energy position changes only slightly. But when the field is higher than approximately 2. 7X10 Ry, the

L,

1,

L,

l,

TABLE III. Electric-field effect on the N=3 'P' resonance ofH

F(10 '

Ry)

0.0 2.5 5.0 9.336 12.0 14.0 16.3 18.0 20.0 23.0 26.84 30.73 32.5 35.00 37.5 40.00 45.902 SS.OO 6S.OO 76.63 85.00

91.804

E

(Ry)

—0. 125 429 —0. 125 430 —0. 125 435 —0. 125 452 —0. 125 47 —0. 125 482 —0. 125 502 —0. 125 521 —0. 125 546 —0. 125 592 —0. 125 672 —0. 125 786 —0. 125 848 —0. 125 938 —0.126 023 —0. 126 097 —0. 126 228 —0. 126 422 —0. 126 708 —0. 127 130 —0. 127 481 —0. 127 791

I (Ry)

0.002 362 0.002 361 0.002 360 0.002 353 0.002 346 0.002 340 0.002 332 0.002 324 0.002 313 0.002 294 0.002 271 0.002 274 0.002 297 0.002 36S 0.002 468 0.002 594 0.002 893 0.003 249 0.003 628 0.004 168 0.004 646 0.005 096

ELECTRIC-FIELD EFFECTS ON THE N = 3 'P'(1). . .

50

4943

TABLE IV. Comparison of energy and width of the N =3 'P'(1) state of H under different electric fields. The numbers inside the parentheses under the experimental results are standard deviations.

E, Present theory

0.0 0.24 0.69 0.79

Present theory

Experiment [6]

12.653 44 12.653 13 12.650 14 12.648 58 12.642 57 12.642 57 12.63023 12.621 30 12.621 30 12.621 30

1.18 1.18 1.97 2.36 2.36 2.36

r

(eV)

12.65 12.6457 12.6463 12.6328 12.6502 12.6365 12.6455 12.6203 12.6206 12.5887

Experiment

[6]

0.0321 0.0320 0.0309 0.0309 0.0393 0.0393 0.0567 0.0693 0.0693 0.0693

(0.001) (0.0023) (0.0032) (0.0041) (0.0052)' (0.0046)b (0.0162) (0.0100)' (0.02666) (0.0284)'

(eV)

0.0390 0.0323 0.0357 0 OAAA 0.0581 0.0191 0.0854 0.0568 0.1118 0.0905

(0.0020) (0.0054) (0.0059) (0.0082) (0.0081)' (0.0095) (0.0480) (0.0221)' (0 0377) (0.0416)'

'Run No. 1. "Run No. 8. 'Run No. 2. dRun No. 3. 'Run No. 9.

shifts become more pronounced. The initial decrease in width for the N=3'P' state is consistent with the analysis made by Lin [13] when he investigated the field effects on the N=2'P' shape resonance. By solving the eff'ective potential curve in the hyperspherical coordinates, he found that the height of the inner part of the potential curve actually increases at low field, before it is suppressed by the external electric field and decreased in height at higher field. As a result, the potential barrier turns out to be thickened somewhat as the field starts to increase from zero. It will hence take a longer time for the tunneling effect to take place, and the When the applied field bewidth becomes narrower. thickness of the potential the comes sufficiently large, barrier becomes smaller, and the resonance width broader. However, there is a difference between our finding for the N = 3 'P' Feshbach resonance and that of Lin's for the N =2 shape resonance. In his finding the downward

resonance energy was found to be blueshifted initially before it was shifted downward at higher field. Our finding shows that the energy level is shifted down all the way, although at lower field it is shifted only slightly. Detailed comparisons with experiment [6] are given in Table IV and Figs. 2 and 3. In converting our resonance energies into eV, measured from the ground state of the H ion, we use 1 Ry =13.605 804 eV, the same conversion factor used in the experimental analysis [6]. Furthermore, in a consistent manner, the binding energy of 0.7542 eV for the same value used in Ref. [6], is also used in our present work. In Fig. 2 we show a comparison for our widths with the experimental results, and qualitative agreement is found. It is noted, however, that their standard deviations are somewhat large at the high end of the electric fields. We should also mention that the uncertainties in their electric-field strengths are of 17% (Ref. [5]). Figure 3 and Table IV show comparisons

H,

1248

0.16

a:

t t Q Q Q QQQQQ

x: expelnent

0.12—

12.64—

Q Q

o: theory

12.60—

x ~~"IIa IlSAl ~

084)y

lloo

QQ~

Q Q

0$

&~

coQQ

I

I

1.0

1.5

20

F (MV/ci&g

FIG. 2. Comparison of the field effects with experiment [6] on the width of the N =3 'P'(1) resonance. The bars represent experimental standard deviations.

0$

I

I

lA) 1$ F 44V/ci eg

2A)

28

FIG. 3. Comparison of the field effects with experiment [6] on the resonance energy of the N =3 'P'(1) resonance. The bars represent experimental standard deviations.

Y. K. HO AND J. CAI.LA%'AY between our results and the experimental measurements resonance. Again for the energy position of the X = 3 within the quite large experimental uncertainties at high end of the fields, an "eyeball" comparison can be considered as qualitatively satisfactory. However, quantitatively our finding of the downward shift in energy cannot conclusively be supported by the experimental data. In fact, in Ref. [6] the authors interpreted their results in such a way that they found "no shift of the resonance enIt seems ergy with increasing external field strengths. that more work is needed to shed light on such an interesting phenomenon. In summary, we have carried out an extensive theoretiof the electric-field e6'ects on the cal investigation N=3'P'(I) resonance state of H . The method of complex-coordinate rotation is used in this work. Our results agree qualitatively with experimental observations.

However, due to the relatively large experimental uncertainties at high field, the available data in the literature cannot provide a critical assessment of merit for our finding of the downward shift in resonance energy when the field is increased. It is hoped, therefore, that our elaborate calculations will stimulate other theoretical investigations, and encourage further experimental activities which, in turn, would provide a stringent test of theoretical calculations.

[1] E. Balslev and J. M. Combs, Commun. Math. Phys. 22, 296 (1971); B. Simon, ibid. 27, 1 (1972). [2] Y. K. Ho, Phys. Rep. 99, 1 (1983); W. P. Reinhardt, Ann. Rev. Phys. Chem. 33, 223 (1982); B. R. Junker, Adv. At. Mol. Phys. 18, 208 (1982). [3] H. C. Bryant, David A. Clark, Kenneth B. Butterfield, C. A. Frost, H. Sharifian, H. Tootoonchi, J. B. Donahue, P. A. M. Gram, M. E. Hamm, R. W. Hamm, J. C. Pratt, M. A. Yates, and W. W. Smith, Phys. Rev. A 27, 2889 (1983). [4] J. Callaway and A. R. P. Rau, J. Phys. B 11, L289 (1978). [5] G. Comtet, C. J. Harvey, J. E. Stewart, H. C. Bryant, K. B. Butterfield, D. A. Clark, J. B. Donahue, P. A. M. Gram, D. W. MacArthur, V. Yuan, W. W. Smith, and Stanley Cohen, Phys. Rev. A 35, 1547 (1987). [6] S. Cohen, H. C. Bryant, C. J. Harvey, J. E. Steward, K. B. D. A. Clark, J. B. Donahue, D. Butterfield, MacArthur, G. Comtet, and W. W. Smith, Phys. Rev. A 36, 4728 (1987). [7] M. Halka, H. C. Bryant, C. Johnstone, B. Marchini, W. Miler, A. H. Mohagheghi, C. Y. Tang, K. B. Butterfield,

D. A. Clark, S. Cohen, J. B. Donahue, P. A. M. Gram, R. W. Hamm, A. Hsu, D. W. MacArthur, E. P. MacKerrow, C. R. Quick, J. Tice, and K. Rozsa, Phys. Rev. A 46, 6942

'I"

"

ACKNOWLEDGMENTS

Y.K.H. has been supported by the National Science Council under Grant No. NSC83-0208-M-001-105. He also thanks Professor A. R.P. Rau and Professor S.-I. Chu for interesting discussions.

(1992). [8] John J. Wendoloski and W. P. Reinhardt, Phys. Rev. A 17, 195 (1978). [9] N. Y. Du, I. I. Fabrikant, and A. F. Starace, Phys. Rev. A 48, 2968 (1993). [10] A. Maquet, S. I. Chu, and W. P. Reinhardt, Phys. Rev. A 27, 2946 (1983). [11]C. Cerjan, R. Hedges, C. Holt, W. P. Reinhardt, K. Scheibner, and J. J. Wendoloski, Int. J. Quantum Chem. 14, 393 (1978). [12] Y. K. Ho and J. Callaway, in Abstracts of Contributed Pa pers, XVII International Conference on the Physics of Eiec tronic and Atomic Collisions, edited by I. E. McCarthy, W. R. MacGillivray, and M. C. Standge (GriSth University, GriSth, Australia, 1991), p. 590. [13] C. D. Lin, Phys. Rev. A 2$, 1876 (1983).