Photodetachment in a cavity: From rectangles to parallel plates

Physica B: Physics of Condensed Matter 530 (2017) 121–126

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Photodetachment in a cavity: From rectangles to parallel plates H.J. Zhao a , M.L. Du b,* a b

School of Physics and Information Science and Center for Molecules Research, Shanxi Normal University, Linfen 041004, China State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China

A R T I C L E

I N F O

Keywords: Photodetachment Closed-orbits theory Quantum well

A B S T R A C T

Simple analytic formulas are presented for photodetachment cross sections of H− inside a rectangular cavity using semiclassical closed-orbit theory. The formulas predict oscillations in the spectrum and correlate the oscillations with closed orbits of the system. Explicit fully-quantum-mechanical formulas are also derived. We compare the results from closed orbit theory and quantum-mechanical calculations. We also show how the photodetachment cross section formulas for a rectangular cavity can be reduced to those for a cavity consisting of two plates.

1. Introduction It is well known that external fields can considerably affect the photodetachment process of negative ions. Many theoretical studies [1–10] and experimental observations [11–14] have revealed the effect in the photodetachment spectra. There are three dominant phenomena in these studies and observations: oscillatory structures above the zero-field threshold, a finite cross-section value at the threshold, and a quantum tunneling effect below threshold. Yang et al. [15,16] applied closed-orbit theory (COT) [17] to study the photodetachment of H− near a reflecting surface, which was closely related to photodetachment in external electric and magnetic fields. Subsequently Afaq et al. [18] developed a theoretical imaging method and applied it to study the same system. Then there has been considerable interest in the photodetachment of H− in different surface environment in recent years, such as in a parallel plate cavity [19,20], in a wedge [21], and in a circular microcavity [22]. The dominant phenomenon which is found in the photodetchment spectra of negative ion in the above cavities is the oscillatory structures above the zero-field threshold. In this paper we consider the photodetachment in a rectangular cavity with an arbitrary aspect ratio. We apply both closed-orbit theory and quantum approach to calculate the photodetachment cross sections. This system is interesting in that it provides an example that the cross section formulas can be explicitly derived by both the closed-orbit theory and the quantum approach. Furthermore, the rectangular cavity we consider here has an arbitrary aspect ratio. By changing the value of * Corresponding author. E-mail address: [email protected] (M.L. Du).

https://doi.org/10.1016/j.physb.2017.10.109 Received 23 June 2017; Received in revised form 24 October 2017; Accepted 25 October 2017 Available online 31 October 2017 0921-4526/© 2017 Elsevier B.V. All rights reserved.

aspect ratio, we can transform the photodetachment cross section formulas in a rectangular cavity into the cross section formulas in a cavity consisting of two parallel plates. Closed-orbit theory provides a clear physical picture and a quantitative tool for analysing the oscillations in the photodetachment. The complete quantum solution can provide all possible knowledge of a system and a different physical picture. Those two approaches are complementary. Here, ions can be placed at any location inside the rectangular cavity, which makes it possible to show the dependence of photodetachment cross sections on the ion position. The paper is organized as follows. In Section 2, we present the COT formulas of the photodetachment cross sections in a rectangular cavity. In Section 3, we present the quantum derivations of the photodetachment cross sections in a rectangular cavity. In Section 4, we compare the two results and show their connections by extracting closed orbits from the quantum formulas by using the Fourier transform. In Section 5, we demonstrate that the cross section formulas in a rectangular cavity are deduced to those in a cavity consisting two plates. 2. Closed-orbit theory formulas for photodetachment cross sections We consider the photodetachment of H− in a tube like cavity with a rectangular intersection. The tube is assumed to be perpendicular to the page. The inner dimension of the rectangular intersection is a0 × b0 as shown schematically in Fig. 1. We use a coordinate system such that the negative ion is at the origin. The distance between the negative ion and the left surface is a, and the distance between the negative ion and the lower surface is b. It is convenient to choose the coordinate system

H.J. Zhao and M.L. Du

Physica B: Physics of Condensed Matter 530 (2017) 121–126

𝜋, corresponding to the “hard” wall case [18]. To use the COT formula in Eq. (1), it is necessary to obtain all the closed orbits and their associated properties such as outgoing angle, length, etc. Because the trajectories are reflected by the inner surfaces of the cavity, it is clear that all the closed-orbits which go out from the position of the negative ion and later return to the position of negative ion must be in the x − y plane. To find closed-orbits in a cavity, we can launch a large number of trajectories going out from the origin in the x − y plane and keep track of the trajectories as they propagate and get reflected inside the cavity. If a group of nearby trajectories come back to the region of the negative ion, the search can be refined to find the closed-orbit. This procedure has been previously used to find the closed-orbits for an atom in a magnetic field [17]. In Fig. 1 we show a group of trajectories near a closed-orbit (heavy solid line) returning to the region of the negative ion. For a rectangular cavity, we can apply the image method as demonstrated in Ref. [21] to find all the closed-orbits and get their associated properties. When an object is placed in a rectangular mirror cavity, there are infinite images outside the cavity. These images can be labeled by the two integers (n, m), both n and m counts from negative infinity to positive infinity, but the case that n and m both are zero at the same time should be excluded. Image (n, m) is located at (xn , yn ), where ( a ) a xn = (−1)n a − 0 + na0 + 0 − a, 2 2 ) ( b b ym = (−1)m b − 0 + mb0 + 0 − b. (2) 2 2

Fig. 1. Schematic illustration of H− photodetachment inside a tube like cavity with a rectangular intersection. We show a group of trajectories propagating away from the H atom and finally returning to the region of the atom after being reflected three times by j the cavity. The center of the group of trajectories is a closed-orbit j (heavy solid line). 𝜙out j

and 𝜙ret denote respectively the azimuthal angles of the outgoing and returning momenta. The quantum interference of the returning detached-electron wave associated with the closed-orbit j leads to an oscillation in the cross section.

such that the z-axis is perpendicular to the intersection. The x-axis is horizontal and parallel to the lower surface in Fig. 1, and the y-axis is vertical and parallel to the left surface. According to the physical picture of closed-orbit theory (COT) [17], when the active electron is detached by a laser, the active electron and the associated wave propagates out from the negative ion in all directions. The electron trajectories and wave fronts follow straight lines inside the cavity until they are reflected by the surfaces of the cavity. The electron may return to the region of the negative ion after several reflections. Then, the returning electron wave will interfere with the initial outgoing electron and induce oscillations in the total photodetachment cross sections. Closed-orbit theory relates the cross sections to all the closed-orbits of detached-electron and provides a recipe to calculate the cross sections based on the closed-orbits and their associated properties. The COT formula for photodetachment cross section of H− in a cavity can be written in the following form [21]

Each image corresponds to a closed orbit, and each closed orbit corresponds to an image. So the index j that specified the closed orbits earlier is replaced by the pair (n, m). The first segment of the closed-orbit before being reflected by the cavity surfaces is given by the straight line connecting the corresponding image and the source. In what follows, we will use these interpretations to derive the required properties of the closed orbits (n, m). It is conspicuous that the geometrical length of the closed-orbit is equal to the distance between the atom and the image: √ 2. Ln,m = xn2 + ym (3) The reflection numbers are 𝜇n,m = |n∣ +|m∣.

All the closed-orbits have polar angles 𝜃out = 𝜃ret = 𝜋∕2. Their outgoing azimuthal angles are given by

3 𝜎(E)COT = 𝜎0 (E) + 𝜎0 (E) √ 2E ×

∑

× sin

𝜙n,m out

j j j j f (𝜃out , 𝜙out ; 𝜃L , 𝜙L )f (𝜃ret , 𝜙ret ; 𝜃L , 𝜙L )

Lj

j

(√

)

2ELj − 𝜇j Δ ,

⎧ arccos xn , ⎪ Ln,m =⎨ xn , ⎪ 2𝜋 − arccos L ⎩ n,m

(m ≥ 0); (m < 0).

(5)

The corresponding azimuthal angles of the returning momentum are given by 𝜋 n (n+m) n,m 𝜙out . (6) 𝜙n,m ret = [(−1) − 1] 2 + (−1)

(1)

√ where 𝜎0 (E) = 16 2B2 𝜋 2 E3∕2 ∕3c(Eb + E)3 represents the photodetachment cross section of H− in free space, B = 0.31522 is related to the normalization of the initial bound state Ψi of H− , c is the speed of light and its value is approximately 137 a.u., E denotes the energy of the escaping electron, Eb = 0.754 eV denotes binding energy of the electron to the negative ion and the photon energy is Ep = E + Eb . The summation is over all closed-orbits going out from and returning to the position of the negative ion. (𝜃L , 𝜙L ) denotes the laser polarization direcj j j j tion. (𝜃out , 𝜙out ) and (𝜃ret , 𝜙ret ) denote the spherical angles of the outj

(4)

Using the above properties of the closed orbits, the total cross section for an arbitrary linear laser polarization direction can be written down straightforwardly following Eq. (1). The formulas for an arbitrary laser polarization direction is very efficient for numerical calculations, but it is too long to write down here. If linear laser polarization direction is taken along axes, the formula becomes sufficiently simple. The COT cross section for laser polarization direction along x-axis can be written as ) (√ 3 ∑ (−1)m 2 x (7) (E) = 𝜎0 (E) + 𝜎0 (E) √ xn sin 2ELn,m , 𝜎cot 3 L 2E n,m n,m

j

going momentum vector ℏ𝐤out and returning momentum vector ℏ𝐤ret of the closed orbit j respectively. f (𝜃, 𝜙; 𝜃L , 𝜙L ) is defined as the form (cos 𝜃 cos 𝜃L + sin 𝜃 sin 𝜃L cos(𝜙 − 𝜙L )). Lj and 𝜇j are, respectively, the length and the number of reflections of the closed-orbit j. Δ denotes the phase loss of the wave function accompanying each reflection. In our COT and quantum calculations throughout this paper, we set Δ to

and the COT cross section for y linear polarization can be written as ) (√ 3 ∑ (−1)n 2 y (8) ym sin 2ELn,m . 𝜎cot (E) = 𝜎0 (E) + 𝜎0 (E) √ 3 2E n,m Ln,m 122

H.J. Zhao and M.L. Du

Physica B: Physics of Condensed Matter 530 (2017) 121–126

where p can be either x, y, or z. Because of the presence of discrete states in x and y directions, the integral in Eq. (10) has collapsed to the expression in Eq. (18). Substituting the dipole matrix elements in Eqs. (15)–(18) and integrating over pz , the quantum cross sections are expressed as ( ( ) ) ∑∑ n𝜋 m𝜋 3𝜋 3 2 2 2 𝜎qx = 𝜎0 3 n cos a sin b a0 b0 2a0 b0 E3∕2 n m

The sums in above equations are over all integer pairs (n, m) except (0, 0). With z-polarized light, there is no outgoing wave in the x − y plane, so the closed orbits have no oscillatory contributions [21]. Therefore the cross section for the z-polarization is z (E) = 𝜎0 (E). 𝜎cot

(9)

At this point we note Wang et al al [23] studied a special case that is related to this work using COT. They considered photodetachment of H − in a square cavity when a = b, and when the ion is at one of the corners or in the middle of the square. Here, ions can be placed at any location in a rectangular cavity with arbitrary aspect ratio.

×√ E− y

𝜎q = 𝜎0

1 2

(

n𝜋 a0

⟩|2 2𝜋 2 |⟨ df 2Ep | Ψf ||D|| Ψi | 𝛿(Ef − E), | | c ∫

𝜋 1∕2

sin(px x + px a)

𝜋 1∕2

sin(py y + py b)

1 eipz z , (2𝜋)1∕2

m𝜋 b0

)2

(20)

∑∑

(21) 1 2

(

n𝜋 a0

)2 −

0

0

tion, when the energy of the detached electron is above each level, we have Δ𝜎 ∝ (E − En,m )−1∕2 . But for z polarization, Δ𝜎 ∝ (E − En,m )1∕2 . The COT results are in good agreement with the quantum results for the x and y polarization. According to COT, the peaks in the cross sections are superpositions of sinusoidal oscillations, and each oscillation corresponds to a closed orbit. The amplitude of the oscillation is proportional to the cosine value of the angle between the laser polarization direction and the outgoing or returning direction of the closed orbit. In this system, because all closed orbits are in x − y plane, thus the COT formulae of cross section for the z-polarization is the same as in free space. However oscillations are still noticeable from the quantum calculations for the z-polarization, although the amplitudes of the oscillations are quite small. If we subtract out the free field cross section for the zpolarization, the weak oscillations become more obvious as shown in Fig. 3. This is an interesting discrepancy between closed orbit theory at the present level approximation and quantum theory. The oscillations of COT can be extracted from the quantum formulae using Fourier transforms defined as

(13)

The cross section for x, y, or z linear polarization in a tube like cavity with a rectangular intersection is given by the expression ⟩|2 2𝜋 2 ∑ ∑ 𝜋 2 |⟨ dpz 2Ep | Ψf ||p|| Ψi | 𝛿(Ef − E), | | c n m a0 b0 ∫

(

At first glance, the cross section formulas from COT and from quantum method are quite different. In the quantum formulae, the number of required terms increases with the energy of escaping electron. However, in the COT formulae, the sum is over infinite closed orbits. In this section, we numerically calculate and compare them for different polarizations. The cross sections in both quantum method and COT can be calculated for different parameters of cavity and the position of the negative ion. We compare the quantum results and the COT results for a set of parameters reflecting the general case in Fig. 2. The most noticeable feature are the presence of many peaks in Fig. 2(a) and (b), while the peaks vanish in Fig. 2(c). The positions of the peaks actually correspond ( )2 ( )2 + 12 m𝜋 . For x and y polarizato the energy levels En,m = 12 n𝜋 a b

Substituting the initial wave function (12) and the final wave function (13) into the dipole matrix element in Eq. (10) and restricting to x, y and z polarizations, we obtain ( ) ( ) ⟩ ⟨ 8B n𝜋 m𝜋 n𝜋 1 cos a sin b , (15) Ψf ||x|| Ψi = √ 2 a0 b0 a0 2𝜋 (k2 + kb )2 ( ) ( ) ⟨ ⟩ 8B n𝜋 m𝜋 m𝜋 1 Ψf ||y|| Ψi = √ sin a cos b , (16) 2 a0 b0 b0 2𝜋 (k2 + kb )2 ( ) ( ) ⟨ ⟩ 8B n𝜋 m𝜋 i Ψf ||z|| Ψi = √ sin a sin b pz . (17) 2 a0 b0 2𝜋 (k2 + kb )2

p

−

1 2

4. Numerical results

where px = n𝜋∕a0 (py = m𝜋∕b0 ) and n(m) = 1, 2, 3 …. The energy of the final state is ( )2 ( )2 1 n𝜋 1 m𝜋 1 + + p2z . (14) Ef = En,m,pz = 2 a0 2 b0 2

𝜎q =

n𝜋 a0

( ) ) n𝜋 m𝜋 a cos2 b a0 b0

0

(12)

1

1 2

(

(19)

where the sum indexes, n and m, are for all integers with E − ( )2 1 m𝜋 ≥ 0. 2 b

where kb is related to the binding energy mentioned above as Eb = k2b ∕2. After the electron absorbs a photon, it quickly propagates out of the atomic region, where the binding potential Vb (r) dominates. Vb (r) is a short-range potential, so we neglect it in the final state. Then the final state can be approximated as a wave function of an electron in the cavity: 1

)2 ,

( ( ) ) n𝜋 m𝜋 3𝜋 2 2 sin a sin b a0 b0 E3∕2 n m a0 b0 √ ( )2 ( )2 1 n𝜋 1 m𝜋 − , × E− 2 a0 2 b0

(11) D = r cos(𝜃) cos(𝜃L ) + r sin(𝜃) sin(𝜃L ) cos(𝜙 − 𝜙L ); ⟨ Ψf | is the final state wave function of the electron after detachment. ) ( The final state is normalized according to ⟨Ψf |Ψf ′ ⟩ = 𝛿 f − f ′ . The integral is over all final states of the system. Similar to previous treatments, we assume that the cavity has no influence on the H− before the electron absorbing a photon. Therefore the initial wave function of the electron is given by Ref. [3]

Ψf (𝐫) =

m𝜋 b0

,

𝜎qz = 𝜎0

(10)

e−kb r , r

(

1 )2

E−

where ∣ Ψi ⟩ is the initial state wave function of H− ; D is a dipole operator, which can be written as

Ψi (𝐫) = B

1 2

(

×√

𝜎quant =

−

∑∑ 3𝜋 3 m2 sin2 2a0 b30 E3∕2 n m

3. Quantum formulas for photodetachment cross sections In this section, we present a quantum formula for the photodetachment cross section for H− in the cavity. We follow an approach in Ref. [24] and derive formulas valid for an arbitrary polarization direction. The photodetachment cross section is given in terms of the dipole matrix elements by the following expression [3]

1 )2

𝜎 ̃(L) ≡

√ 2E2

∫√2E1

𝜎 − 𝜎0 i√2EL (√ ) e d 2E . 𝜎0

(22) y

Fourier transforms of quantum calculated 𝜎qx and 𝜎q are presented in Fig. 4(a) and (b) respectively. There are many peaks: peak B and H

(18) 123

H.J. Zhao and M.L. Du

Physica B: Physics of Condensed Matter 530 (2017) 121–126 10

1.5

Fourier transforms (arb. units)

(a)

1

x

σ (a. u.)

2

0.5 0.8

0.9

(b)

1.5

1

1.1

1.2

1.3

1

y

σ (a. u.)

0

5

B D E

0

D

G

E

C

−5

H

G

F

A

−10

0.5 0

0.8

0.9

(c)

1.5

1

1.1

1.2

−15

1.3

(b) 100

150

200

250

300

L(a. u.) Fig. 4. Fourier transforms of the quantum cross sections for x polarization (a) and y polarization (b). The closed orbits are also displayed in the insets near the corresponding peaks.

1

z

σ (a. u.)

(a)

0.5 0.8

0.9

1

1.1

Photon energy (eV)

1.2

1.3

10

Fourier transforms (arb. units)

0

√ Fig. 2. Photodetachment cross sections for H− inside a tube like cavity with a 97 5 a.u. × 124 a.u. rectangular intersection. The position of the negative ion is fixed by the distances a = 70a. u. and b = 47 a.u. The laser is linearly polarized in the x-axis direction (a), the y-axis direction (b) and the z-axis direction (c). The red and thin lines correspond to quantum results, while the green and thick lines correspond to COT results. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

0.1

Δσ

z

0.05 0

−0.1

0.8

0.9

1

1.1

1.2

B C D

E

F

GH

6

4

2

0

−0.05

A

8

0

50

100

150

200

250

300

L(a ) 0

1.3

Photon energy (eV)

Fig. 5. Fourier transforms of the cross section in Fig. 3.

Fig. 3. The oscillations in Fig. 2(c) are redisplayed with the background removed Δ𝜎 z = 𝜎 z − 𝜎0 .

tions, and then show closed orbits corresponding to peaks. The directions of these two closed orbits corresponding to closed orbit Peak B and H respectively as they leave and return the emitting atom are perpendicular to y-axis. So peak B and H are present only for x polarization. For similar reason, peak A, C, and F are present only for y polarization. In Table 1 we list all the closed orbits appearing in Fig. 4. It should be noted that peak F corresponds to two closed orbits, and these two closed orbits have the same path, but their propagation directions are opposite. There are also two closed-orbits corresponding to peak G propagating out in opposite directions. It should be noted that the oscillations in the quantum calculated cross section for z polarization also can be extracted using Fourier transforms. The absolute value of the Fourier transforms of the cross section in Fig. 3 is presented in Fig. 5. We find that all the peaks appear at the peak positions of x and y polarizations combined. Now we consider the dependence of the photodetachment cross section on the negative ion position. As shown in Fig. 6, we observe that the cross sections display oscillations as the negative ion position is varied. For x or y polarization, the cross section is suppressed when the negative ion approaches the surfaces parallel to the laser polarization, but it is enhanced when the negative ion moves closer to the surfaces perpendicular to the laser polarization. However, for z polar-

appear in x polarization; peak A, C and F appear in y polarization; peak D, peak E and peak G appear in both x and y polarizations. The positions of peaks are listed in Table 1. If we substitute the COT formulae into the above equation, |̃ 𝜎 (L)| should be composed of peaks in L = Ln,m . In Table 1, we compare the length of closed orbit, Ln,m , and the extracted values of the peaks posi-

Table 1 Length of the closed-orbits and peaks positions. Peak Label

Orbit index

Length

Position

A B C D E F G H

(0, −1) (−1, 0) (0, 1) (−1, −1) (−1, 1) (0, 2) and (0, −2) (−1, 2) and (−1, −2) (1, 0)

94.00 140.00 154.00 168.63 208.13 248.00 284.79 293.80

94.0 140.0 154.0 168.5 208.0 248.0 285.0 and 284.0 294.0

124

H.J. Zhao and M.L. Du

Physica B: Physics of Condensed Matter 530 (2017) 121–126

Fig. 6. Dependence of photodetachment cross sections on the position of the negative √ ion. (a), (b) and (c) The pattern of the quantum calculated cross section as a function of the position of the emitting atom. The position is varied inside a tube cavity with a 97 5 a.u. × 124 a.u. rectangular section. The laser is linearly polarized in the x−axis direction (a), y−axis direction (b), and z−axis direction (c). (e), (f) and (g) Comparisons of the COT and quantum results. Green and thick lines represent the COT results and the red and thin lines represent the quantum results. (e) Variations with a for x polarization and setting b = 50 a.u.; (f) Variations with b for x polarization and setting a = 100 a.u.; (g) Variations with b for z polarization and setting a = 100 a.u. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

ization, the cross section is always suppressed when the negative ion moves closer to any surface. It is noticed that quantum formulas will give significantly different results as the ion moves closer enough to the surfaces of the COT.

and ) (√ 2 3 ∑ ym y sin 2EL0,m . 𝜎cot = 𝜎0 + 𝜎0 √ 3 L 2E m 0,m

In the above two functions, the sum index m takes all non-zero integer. Obviously, when a0 → ∞, the rectangular cavity is transformed into two parallel walls, and the normal vector of the walls is parallel to yz x ⊥ of Ref [25] and both equal to and 𝜎cot are just the 𝜎cot axis. Thus, 𝜎cot 𝜎0 . After some manipulations, Eq. (24) can be further rewritten as

5. Reduction to the parallel plates case We should expect that a rectangular cavity characterized by a very large width should mimic the behavior of two parallel plates. We have checked numerically that our expressions coincide with the known results for parallel plates [25] in the limit a0 → ∞, a → ∞ and a∕a0 → const. Here, We show the cross section as a function of the aspect ratio a0 ∕b0 in Fig. 7. The figures demonstrate how the cross sections vary from square cavity to parallel plates. For better display effects, we fix the ion at the center of the cavity. As a0 increases, 𝜎 x pattern varies from a curve consisting many peaks to a smooth curve, while 𝜎 y varies from a curve consisting many peaks to a curve resembling a staircase pattern. Both results are consistent with those we shown in Ref. [25] for the parallel plates. In fact, under the above circumstances, the our general formulas for the rectangular case can be deduced to those in Ref. [25] for parallel plates case. First, we consider closed-orbit theory formulas. As a0 → ∞ and a → ∞, xn in equation (3) will tend to infinity except for the case n = 0. When n is zero, xn equals to zero. If xn tends to infinity, Ln,m will be such that the amplitude of the oscillation in equations (7) and (8) will tend to zero too. Therefore, as long as the sum index n in equations (7) and (8) is set to zero, the photodetachment cross section equations for parallel plate case can be deduced as follows: ) (√ 3 ∑ (−1)m 2 x = 𝜎0 + 𝜎0 √ x0 sin 2EL0,m , 𝜎cot 3 2E m L0,m = 𝜎0 ;

(24)

y

𝜎cot

[ √ ] ⎧ ∞ 3 ∑ ⎪ sin 2 2E(na0 + a) = 𝜎0 + 𝜎0 √ ⎨ na0 + a 2 2E m=0 ⎪ ⎩

+

[ √ ] sin 2 2E(na0 + a0 − a) na0 + a0 − a

[ √ ]⎫ sin 2 2E(n + 1)a0 ⎪ +2 ⎬, n + 1a0 ⎪ ⎭ (25)

which has an identical form to the expression of Eq. (18) in Ref. [25]. So far we have only considered the COT formulas. We now turn ∑ our attention to the behavior of the quantum result. As a0 → ∞, n n𝜋 in functions (19)–(21) should be rewritten as ∫ d( a ), where the range 0 √ √ of integration is from − 2E − (m𝜋∕b0 )2 to 2E − (m𝜋∕b0 )2 . For convenience of notation, we set c = 2E − (m𝜋∕b0 )2 and f = n𝜋∕a0 . Then, we have √ √ ) ( c f2 3𝜋 ∑ 2 2 m𝜋 x sin b 𝜎q = 𝜎0 3∕2 cos2 (af ) √ df , (26) ∫−√c 2E b0 m 𝜋 b c − f2 where m is for all integers satisfying E − 12 (m𝜋∕b0 )2 ≥ 0. As a → ∞, √ c √ − c

the integration in above equation,∫

(23) 125

cos2 (af ) √ f

2

c−f 2

df , tends to

c𝜋 . 4

H.J. Zhao and M.L. Du

Physica B: Physics of Condensed Matter 530 (2017) 121–126

6. Conclusions In summary, we have studied the photodetachment process of H− inside a cavity with a rectangular intersection and calculated the photodetachment cross sections using COT. The cross section can be expressed as a sum of a smooth part and an oscillating part. The smooth part describes the direct escape of the electrons from the negative ion in the absence of the cavity, additionally, the quantum interference effect of the cavity on the cross section is embodied in the oscillating part. We have also derived the explicit fully-quantum-mechanical formulas. The quantum formulas show the peak positions correspond to the energy levels in the rectangle. We have compared the two results and demonstrated numerically that they agree. We have also extracted closed orbits from the quantum formulas using Fourier transforms. Finally, we have shown analytically both our COT formulas and quantal formulas can be deduced to those of H− in a cavity consisting of two parallel plates studied earlier. Acknowledgments This work was supported by NSFC grant Nos. 11421063 and 11474079, SXNSF grant No. 2009011004, 2017YZKC-35 and TYAL. References [1] [2] [3] [4] [5] [6] [7] [8]

[9] [10] [11]

Fig. 7. Photodetachment cross sections in a cavity with a rectangular intersection with varying aspect ratio. We set b0 = 150 a.u. and fix the ion at the cavity center. The ratio a0 ∕b0 is increased from 1 to 1000. Note the ratio is on a logarithmic scale. When the ratio is increased to 1000, the patterns are just like those of parallel plates. The parallel plates cases (red and thick lines) are also shown at lg(a0 ∕b0 ) = 3 for comparison. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Thereby, 𝜎qx = 𝜎0

3𝜋 (2E)3∕2 b0

[12]

[13]

[14]

[ ∑ m

E−

1 2

(

m𝜋 b0

)2 ] sin2

(

) m𝜋 b . b

[15] [16] [17]

(27)

After similar manipulations, we obtain 𝜎qz , which has the same expression as 𝜎qx and gives good agreement with function (13) in Ref. [25]. In the manipulations, the integral formula, √ √ c lim ∫ √ sin2 (af ) c − f 2 df = c𝜋 , has been employed. 4 a→∞

[18] [19] [20] [21] [22] [23] [24] [25]

− c

y

𝜎q can be also given by using a similar calculation. It is as expected y that 𝜎q has the same expression as function (14) in Ref. [25]: y

𝜎q = 𝜎0

) ( 3𝜋 3 ∑ 2 m𝜋 b . m cos2 (2E)3∕2 b0 m b √ c sin2 (af ) df √ √ c−f 2 − c

where the integral formula, lim ∫ a→∞

(28) =

𝜋 , 2

has been used.

126

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