- Email: [email protected]
Photodetachment of H- near a metal surface
Surface Science 603 (2009) 632–635
Contents lists available at ScienceDirect
Surface Science journal homepage: www.elsevier.com/locate/susc
Photodetachment of H near a metal surface Kuikui Rui, Guangcan Yang * School of Physics and Electronic Information, Wenzhou University, Wenzhou 325027, China
a r t i c l e
i n f o
Article history: Received 19 November 2008 Accepted for publication 7 January 2009 Available online 13 January 2009 Keywords: Photodetachment Image state Semiclassical method
a b s t r a c t The photodetachment cross section of H near metal surface is investigated based on the semiclassical closed orbit theory. It is found that the metal surface has significant influence on the photodetachment process. It is found that the cross section of photodetachment undergoes a transition from a staircase structure to a smooth oscillation when the incoming photon energy increases, When the photodetached electron is trapped in image potential well, the detachment spectrum displays an irregular staircase structure which corresponds to Coulomb-like image states. While the photon energy is higher than the image state potential, the spectrum becomes a smooth oscillation. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction It is well known that atoms or ions adsorbed on a surface show quite different properties from those in free space. An electron at a distance z in front of a metal surface experiences an attractive force e2 FðzÞ ¼ ð2zÞ 2 identical to that produced by a positive charge at a distance z inside the metal. The electron will be trapped in the potential well which formed by the Coulomb-like attractive image e2 [1,2]. Recently, time-resolved twopotential barrier VðzÞ ¼ 4z photon photoemission was used to study the electron dynamics of image potential states on metal surfaces [3].The results reveal the dynamical evolution of excited electrons in real time. Moreover, The binding energies of the image states have been extensively measured by two-photon photoemission (2PPE) [4,5] and time-resolved two-photon photoemission (TR2PPE) [6–8] techniques. These measurements have provided highly accurate data of image states binding energies at surfaces of many noble and transition metals. And the lifetime of some image states have also been evaluated [9–12]. On the other hard, Rous shows theoretically that negative ion resonance can be controlled by adsorption of the molecule onto an epitaxial metallic film [13]. Actually, the presence of a surface alters the resonance lifetime of an adsorbed ion since the barrier penetrability of the detached electron is modified [14,15]. In our former study, the photodetachment of H near an elastic wall has been investigated [16]. In that case, the cross section consisted of a smooth background and a sinusoidal oscillation related with photon energy and distance to surface, which was quite similar to that of electric field [17–19]. In the present work, we investigate the photodetachment cross section of H near the metal
* Corresponding author. Tel.: +86 577 86689033; fax: +86 577 86689010. E-mail address: [email protected] (G. Yang). 0039-6028/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.susc.2009.01.004
surface. The difference of both cases is that the electron near metal surface will produce image potential barrier which significant influence on the photodetachment process. We use semiclassical closed orbit theory to explore the photodetachment cross section of H near the metal surface [20,21]. The closed orbit theory emphasizes the role of closed orbits and their repetitions. The closed orbits are those classical trajectories which go out from the origin where the nucleus or atomic core is located and turned back by external fields and/or other potential barriers to the vicinity of origin. In the semiclassical framework, electronic waves travel along the closed orbits and interfere with the outgoing wave source. We obtain an analytical expression for the cross section as the sum of the contributions of the closed orbits and their repetitions. We hope that our results will be useful in understanding the photodetachment processes of ions in vicinity of interfaces, cavities, and ion traps. The paper is organized as follows. In Section 2 we describe the classical motion of the valence electron of H- and give all the closed orbits. An analytical formula of the photodetachment cross section of the hydrogen ion H- near metal surface is derived in Section 3 based on the closed orbit theory. And Section 4 gives some numerical results and conclusive remarks. Unless indicated otherwise, we use atomic units throughout this work. 2. Closed orbits To investigate the process of photodetachment of H- near metal surface, we consider the following system: In the cylindrical coordinates (q; z; /), a hydrogen negative ion H sits at the origin and a z-polarized laser is applied for the photodetachment. A metal surface perpendicular to z-axis is put at z0 where z0 is positive. The photodetached electron experiences an attractive force e2 FðzÞ ¼ ð2zÞ 2 identical to that produced by a positive charge at z.
K. Rui, G. Yang / Surface Science 603 (2009) 632–635
And it was trapped in the image potential well formed by the Cou2 lomb-like attractive image potential VðzÞ ¼ e4z. Thus when the energy E of detached electron is lower than image potential, the photodetached electrons are oscillated in the well. In all the classical trajectories of the photodetached electron emanating out the origin, those bounced back by image potential to the starting point are called closed orbits. The following are fundamental. (i) The electron goes up along the +z direction, reaches its maximum and then bounced back by the potential and returns to the origin. We call this orbit the up orbit. (ii) The electron goes down in –z direction and hits the metal surface and bounces back and finally return to the origin. We call this orbit the down orbit. (iii) The electron completes the up orbit first and then passes through the origin, and continues to complete the down orbit. (iv) The electron completes the down orbit first and then up orbit. This orbit is similar to the one of (iii) but in reverse order. To classify the system of orbits, indices j and n are used to label the closed orbits, where j = 1,2,3,4 and n = 0,1,2,3,. Here n = 0 means that the orbit is a fundamental closed orbit (j = 1,2,3,4 for the fundamental closed orbits above, respectively). When n > 0, the orbit (j,n) has two par pz ts, the beginning part and the later part. The beginning part is always the jth fundamental closed orbit. The later part consists of n repetitions of periodic orbits j = 3 or 4. For the case of j = 1, it is n repetitions of j = 3, and for j = 2, it corresponds to n repetitions of j = 4. For the periodic orbit j = 3 or j = 4, it repeats itself n times. The classical quantities needed for constructing semiclassical wave function can be obtained according to the systematic procedure of closed orbit theory [20,21]. In the current case, if the escaping energy of photodetached electron is E, we have p2 E ¼ 2z 4ðz01þzÞ þ 4z10 where is the momentum in z-direction. The travaling times for classical orbits can be obtained by the integral Rz Rt of z-motion. For the fundamental closed identity t ¼ 0 dt ¼ 0 dz pz orbits, their returning time T j can be written as
pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi 4z20 2E ð2z0 Þ3=2 ½p=2arcsinð 14z0 EÞ T 1 ¼ 14z þ 3=2 E ð14z0 EÞ 0 pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi 4z20 2E ð2z0 Þ3=2 arcsin 14z0 E T 2 ¼ 14z0 E þ ð14z EÞ3=2 0
3=2
ð1Þ
0
3=2
pð2z0 Þ T 4 ¼ T 1 þ T 2 ¼ 2ð14z EÞ3=2 0
Clearly, we have T 3 ¼ T 4 T. The classical action of a trajectory Rt is defined as Sðq; z; /Þ ¼ 0 pdq, where t is the traveling time, p and q are coordinate and momentum vectors respectively. It is acumulated along a classical trajectory and appears in the phase of the semiclassical wave function. For the fundamental closed orbits, the returning times in Eq. (1) are used in the calculation of the classical actions. The classical actions along the fundamental closed orbits are
S2 ¼
pffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi 14z0 EÞ 2z0 2E;
2z0 ½p=2arcsinð
14z0 E
pffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi14z0 EÞ þ 2z0 2E; 14z0 E
S3 ¼ S1 þ S2 ¼ p2 S4 ¼ S1 þ S2 ¼ p2
qffiffiffiffiffiffiffiffiffiffiffi
Sjn ¼ Sj þ nS:
ð2Þ
2z0 ; 14z0 E
qffiffiffiffiffiffiffiffiffiffiffi
2z0 : 14z0 E
ð4Þ
The returning time of any closed orbit is
T jn ¼ T j þ nT
ð5Þ
with Maslov index
ljn ¼ lj þ 3n:
ð6Þ
3. Photodetachment cross section For hydrogen ion H- near metal surface, the photodetachement can be regarded as a one-electron process if we neglect the influence of the short-range potential Vb(r) when the electron is far 2 from the core. The binding energy Eb ¼ kb =2 is approximately 0.754 eV, where kb is related with the initial wave function Wi = C exp (- kbr)/r, C is a ‘‘normalization function” constant and is equal to 0.31552. When an incident laser beam is applied to the ion, the valence electron absorbs a photon energy Ep = Eb + E and the steady outgoing electron wave propagates outward in all directions. The electron move in a straight line at constant speed before it hits the image potential or metal interface. We propagate the electron wave semiclassically outside the core. The semiclassical wave is reflected by the potential wall or metal interface. Finally, some of the waves return to the vicinity of the core and interfere with the outgoing wave to produce the spectral pattern of the photodetachment. In this physical picture, the photodetachment cross section can be expressed in the form [23]
4Ep ^ þ jDWi i ImhDWi jG c
ð7Þ
^ þ . The dipole The outgoing Green’s function is denoted by G operator D is equal to the projection of the electron coordinate onto the direction of polarization of the laser field. Eq. (7) has the following physical meaning: the initial state is modified by the dipole operator related with the incident laser field to become the source wave function; the Green’s function propagates these waves outward to become the outgoing waves; and finally the waves overlap with the source wave to give the absorption spectrum. The outgoing wave can be divided into two parts. The first part never goes far from the core and it is called the direct wave. The second part is known as the returning wave that propagates outward into the external region first, then is reflected by the potential wall or metal interface, and finally returns to the vicinity of the core to interfere with the outgoing wave. Accordingly, the cross section has two parts,
rðEÞ ¼ r0 ðEÞ þ rret
2z0 ½arcsinð
ð3Þ
The action of any closed orbit consists of the ones of fundamental orbits and can be written as
rðEÞ ¼
pð2z0 Þ T 3 ¼ T 1 þ T 2 ¼ 2ð14z EÞ3=2
S1 ¼
l1 ¼ 1; l2 ¼ 2; l3 ¼ l4 ¼ 3:
633
ð8Þ
The first part is the contribution of the direct wave interfering with the source and is the field-free cross section [24,25]
r0 ðEÞ ¼
pffiffiffi 16 2C 2 p2 E3=2 ; 3c ðEb þ EÞ3
ð9Þ
The second part is the contribution of the returning wave, Clearly, we have S3 ¼ S4 S. Maslov index is related with the topological structure of trjectory manifold and can be obtained simply by counting the singular points such as caustics and foci along the trajectory [22]. For the current case, Maslov index can be easily found by counting the returning points where a p phase loss of electronic wave occurs, thus we have
rret ðEÞ ¼
4Ep ImhDWi jWret i; c
ð10Þ
where Wret is the returning wave, which overlaps with the outgoing source wave to give the interference pattern in the absorption spectrum.
634
K. Rui, G. Yang / Surface Science 603 (2009) 632–635
The construction of semiclassical wave function and the calculation of the overlapping integral to obtain the cross section are central subjects of closed theory. The derivation can be found in references [23,26] in detail. For the current system, we describe the procedure briefly in the following. To construct the semiclassical wave function we start from a spherical surface centered at the origin with radius R 10a0. When a beam of z-polarized laser is applied to the initial state of electron, the outging electron wave is produced. On the surface of the virtual sphere, the outgoing wave is denoted by [27]
W0 ðR; h; /Þ ¼ i
4Ck 2 ðkb
2
cosðhÞ
2 2
þk Þ
eiðkRpÞ : kR
ð11Þ
In the semiclassical approximation, the wave outside this sphere can be expressed as
Wðq; z; /Þ ¼
X
W0 ðR; h; /ÞAi ei½Si li p=2
ð12Þ
i
where Si is the action along the ith trajectory, li is the Maslov index characterizing the geometrical properties of the ith trajectory and its neighboring orbits. The amplitude of the wave function, Ai is given by [16]
J ðq; z; 0Þ1=2 ¼ R : Ai ðq; z; /Þ ¼ i J i ðq; z; TÞ R þ kT
ð13Þ
pffiffiffi X 16 2C 2 p2 E3=2 rðEÞ ¼ C jn sinðSjn ljn p=2Þ: 3c ðEb þ EÞ3 jn
ð19Þ
where c is light speed and
C jn ¼ ð1Þ½ðlj 1Þ=2
pffiffiffi 2p2 4 2C 2 E1=2 : c T jn ðEb þ EÞ3
ð20Þ
4. Numerical results and conclusions The photodetachment cross section can be calculated for different values of the distances between the ion and the metal surface. The results are shown Fig. 1 and Fig. 2. In Fig. 1, we keep the distance z0 fixed at 50 a.u., when we only consider the contribution of the up orbit, i.e. the electron goes up along the +z direction, then bounced back by the potential and returns to the origin, the calculated cross section is shown in Fig. 1a. The detachment spectrums consist of two parts and reveal the characteristic of image potential. When the photon energy is lower than the image potential, the cross section consists of a smooth background and sinusoidal oscillations. The former is the field-free cross section which attribute to the contribution of the direct wave interfering with the source. The latter derive from the contribution of the returning wave. While the photon energy is higher than the potential well,
where
@t
@z @h : @ q @h
ð14Þ
1.0
In Eqs. (13) and (14), T represents the travaling time of the photodetached electron from the origin. Only the waves associated with the closed orbits return to the vicinity of the core and interfere with the steadily-producing outgoing spherical waves and therefore contribute the cross section. For the case without the potential wall, the returning waves near the core must be cylindrically symmetric, and thus are approximated by incoming Bessel functions. The returning wave function related with closed orbit (j,n) must match the Bessel function and can be written as
1
1
2p
2p
ret
ret ikz z Wret ; jn ¼ N jn pffiffiffiffiffiffiffi J 0 ðkq qÞ pffiffiffiffiffiffiffi e
T jn k
By a cumbersome but straightforward analytical derivation, the proportional constant in Eq. (15) now becomes 2
Njn ¼ ð1Þ½ðlj 1Þ=2
i p exp½ ðSjn ljn Þ h 2 þk Þ
4iCk 2 T jn kðkb
2 2
ð17Þ
The oscillatory contribution to the cross section can now be calculated by overlapping the returning waves with the outgoing wave functions, then taking the imaginary part [16,23]. Therefore, we have
4E rret ðEÞ ¼ p Im c
Z
C expðkb rÞ cos hNjn e
sin hdhd/;
0.6 0.4
0.0
0.80
r dr ð18Þ
Integrating out the expression and adding the field-free part, we have the total photodetachment cross section
0.90
0.95
1.00
Photon energy (eV) 1.6
z 0=50a.u.
1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0.8
ikr cos hret jn 2
0.85
b
ð16Þ
:
0.8
0.2
Cross section (a 20 )
Ajn ¼
z 0=50a.u.
ð15Þ
where the normalization factor Ni can be determined by matching Eqs. (12) and (15). For closed orbit (j,n), the modified amplitude in Eq. (13) of the returning function is
1
a Cross section (a 20 )
@z @t Jðq; z; /Þ ¼ qðTÞ @ q
0.9
1.0
1.1
1.2
1.3
Photon energy (eV)
1.4
1.5
Fig. 1. The distance between the ion and the metal surface is 50 a.u. (a) The photodetachment cross section attribute to the contribution of the up orbit and (b) the cross section derive from the superposition of contribution from fundamental closed orbits and their repetitions.
K. Rui, G. Yang / Surface Science 603 (2009) 632–635
a
0.6
Cross section (a 20 )
0.5
z0=100a.u.
0.4 0.3 0.2 0.1 0.0
0.76
0.78
0.80
0.82
0.84
0.86
0.88
Photon energy (eV)
b
1.4
Cross section (a 20 )
1.2
z0=100a.u.
1.0 0.8
635
times vary in a quite range and depend on the type of metals, crystal orientation and the atom adsorption of surface [28]. For a finite lifetime system, the contribution from those orbits whose periods are longer than the lifetime vanishes in Eq. (19), but the shorter orbits still contribute to the summation. In the case, there still exists the staircase structure in the cross section but with smoother steps due to the lack of high harmonics unless the shortest period of closed orbits is longer than the lifetime. For example, the lifetime of n = 1 image state on Cu(1 0 0) is about 40 fs [29], while the shortest period of closed orbit is 22.84 fs. If there is further dephasing such as surface atom adsorption, the staircase structure may disappear. Further investigation is needed to clarify the subject in detail. In summary, we have analyzed the photodetachment of H- near metal surface and obtained an analytical formula of the cross section. It is found that with increasing of the photon energy, the configuration of spectrum evolved from a staircase structure to a smooth oscillation. The phenomena can be interpreted as follows: When the photon energy is lower than the potential, the detachment electron was trapped in the image potential well and bounced back and forth between the metal surface and image potential, corresponding spectrum was a staircase, while the photon energy is higher than the potential, the detachment electron escapes from the image potential well and just reflected by metal surface, so the cross section is a smooth oscillation. Acknowledgement
0.6 0.4
This work is partially supported by a National Key Basic Research Project of China (2007CB310405).
0.2
References
0.0 0.80
0.85
0.90
0.95
1.00
1.05
1.10
1.15
Photon energy (eV) Fig. 2. The distance is 100 a.u. (a) The photodetachment cross section attribute to the contribution of the up orbit and (b) the cross section the superposition of contribution from four fundamental closed orbits and their repetitions.
the detached electrons traverse the potential completely, there is no contribution of returning wave, which is similar to the case of free space, so the spectrum is a smooth line. In Fig. 1b, the structure results from the superposition of contribution from four fundamental closed orbits and their repetitions. We can see that when the photon energy is lower than the image potential, the detachment electron was trapped in the image potential well and the electronic waves are bounced back and forth between the metal surface and image potential, the spectrum displays a staircase structure, contrasting with the regular smooth oscillating curve when only a potential wall exists. While the energy is higher than potential, the cross section is a smooth oscillation. In this case, the detachment electron escapes from the image potential well, the returning wave is just reflected by metal surface, i.e. the cross section only attribute to the contribution of the down orbit. If the metal surface approaches the ion further, taking z0 =100 a.u., we can see the same configuration of spectrum. In the present investigation, we did’nt consider the influence of lifetimes of the image states and assumed that they are infinite long. This is certainly not correct for real systems. Actually, the life-
[1] P.M. Echenique, J.B. Pendry, J. Phys. C 11 (1978) 2065. [2] P.M. Echenique, J.B. Pendry, Prog. Surf. Sci. 32 (1989) 111. [3] U. Höfer, I.L. Shumay, Ch. Reuß, U. Thomann, W. Wallauer, Th. Fauster, 277 (1997) 1480. [4] K. Giesen, F. Hage, F.J. Himpsel, H.J. Riess, W. Steinmann, Phys. Rev. Lett. 55 (1985) 300. [5] W. Merry, R.E. Jordan, D.E. Padowitz, C.B. Harris, Surf. Sci. 295 (1993) 393. [6] R.W. Schoenlein, J.G. Fujimoto, G.L. Eesley, T.W. Capehart, Phys. Rev. Lett. 61 (1988) 2596. [7] M. Wolf, E. Knoesel, T. Hertel, Phys. Rev. B 54 (1996) R5295. [8] R.L. Lingle Jr., N.H. Ge, R.E. Jordan, J.D. McNeil, C.B. Harris, J. Chem. Phys. 205 (1996) 191. [9] E.V. Chulkov, V.M. Silkin, P.M. Echenique, Surf. Sci. 391 (1997) L1217. [10] P.M. Echenique, F. Flores, F. Sols, Phys. Rev. Lett. 55 (1985) 2348. [11] P.L. de Andres, P.M. Echenique, F. Flores, Phys. Rev. B 35 (1987) 4529. [12] P.L. de Andres, P.M. Echenique, F. Flores, Phys. Rev. B 39 (1989) 10 356. [13] P.J. Rous, Phys. Rev. Lett. 83 (1999) 5086. [14] P.J. Rous, D.M. Hartley, Chem. Phys. 236 (1995) 299. [15] D.M. Hartley, P.J. Rous, Surf. Sci. 341 (1995) 213. [16] Guangcan Yang, Yizhuang Zheng, Xianxing Chi, J. Phys. B: At. Mol. Opt. Phys. 39 (2006) 1855. [17] I.I. Fabrikant, Zh. Eksp. Teor. Fiz. 79 (1980) 2070. [18] A.R.P. Rau, H.Y. Wong, Phys. Rev. A 37 (1988) 2393. [19] M.L. Du, Phys. Rev. A 70 (2004) 055402. [20] M.L. Du, J.B. Delos, Phys. Rev. Lett. 58 (1987) 1731. [21] M.L. Du, J.B. Delos, Phys. Rev. A 38 (1988) 1896. [22] S.K. Knudson, J.B. Delos, B. Bloom, J. Chem. Phys. 83 (1985) 5703. [23] A.D. Peters, C. Jaffé, J.B. Delos, Phys. Rev. A 56 (1997) 331. [24] T. Ohmura, H. Ohmrua, Phys. Rev. 118 (1960) 154. [25] M.L. Du, J.B. Delos, Phys. Rev. A 38 (1988) 5609. [26] G.C. Yang, J.M. Mao, M.L. Du, Phys. Rev. A 59 (1999) 2053. [27] M.L. Du, Phys. Rev. A 40 (1989) 4983. [28] E.V. Chulkov, A.G. Borisov, J.P. Gauyacq, D. Sánchez-Portal, V.M. Silkin, V.P. Zhukov, P.M. Echenique, Chem. Rev. 106 (2006) 4160. [29] E.V. Chulkov, I. Sarría, V.M. Sikin, J.M. Pitarke, P.M. Echenique, Phys. Rev. Lett. 80 (1998) 4947.
Contents lists available at ScienceDirect
Surface Science journal homepage: www.elsevier.com/locate/susc
Photodetachment of H near a metal surface Kuikui Rui, Guangcan Yang * School of Physics and Electronic Information, Wenzhou University, Wenzhou 325027, China
a r t i c l e
i n f o
Article history: Received 19 November 2008 Accepted for publication 7 January 2009 Available online 13 January 2009 Keywords: Photodetachment Image state Semiclassical method
a b s t r a c t The photodetachment cross section of H near metal surface is investigated based on the semiclassical closed orbit theory. It is found that the metal surface has significant influence on the photodetachment process. It is found that the cross section of photodetachment undergoes a transition from a staircase structure to a smooth oscillation when the incoming photon energy increases, When the photodetached electron is trapped in image potential well, the detachment spectrum displays an irregular staircase structure which corresponds to Coulomb-like image states. While the photon energy is higher than the image state potential, the spectrum becomes a smooth oscillation. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction It is well known that atoms or ions adsorbed on a surface show quite different properties from those in free space. An electron at a distance z in front of a metal surface experiences an attractive force e2 FðzÞ ¼ ð2zÞ 2 identical to that produced by a positive charge at a distance z inside the metal. The electron will be trapped in the potential well which formed by the Coulomb-like attractive image e2 [1,2]. Recently, time-resolved twopotential barrier VðzÞ ¼ 4z photon photoemission was used to study the electron dynamics of image potential states on metal surfaces [3].The results reveal the dynamical evolution of excited electrons in real time. Moreover, The binding energies of the image states have been extensively measured by two-photon photoemission (2PPE) [4,5] and time-resolved two-photon photoemission (TR2PPE) [6–8] techniques. These measurements have provided highly accurate data of image states binding energies at surfaces of many noble and transition metals. And the lifetime of some image states have also been evaluated [9–12]. On the other hard, Rous shows theoretically that negative ion resonance can be controlled by adsorption of the molecule onto an epitaxial metallic film [13]. Actually, the presence of a surface alters the resonance lifetime of an adsorbed ion since the barrier penetrability of the detached electron is modified [14,15]. In our former study, the photodetachment of H near an elastic wall has been investigated [16]. In that case, the cross section consisted of a smooth background and a sinusoidal oscillation related with photon energy and distance to surface, which was quite similar to that of electric field [17–19]. In the present work, we investigate the photodetachment cross section of H near the metal
* Corresponding author. Tel.: +86 577 86689033; fax: +86 577 86689010. E-mail address: [email protected] (G. Yang). 0039-6028/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.susc.2009.01.004
surface. The difference of both cases is that the electron near metal surface will produce image potential barrier which significant influence on the photodetachment process. We use semiclassical closed orbit theory to explore the photodetachment cross section of H near the metal surface [20,21]. The closed orbit theory emphasizes the role of closed orbits and their repetitions. The closed orbits are those classical trajectories which go out from the origin where the nucleus or atomic core is located and turned back by external fields and/or other potential barriers to the vicinity of origin. In the semiclassical framework, electronic waves travel along the closed orbits and interfere with the outgoing wave source. We obtain an analytical expression for the cross section as the sum of the contributions of the closed orbits and their repetitions. We hope that our results will be useful in understanding the photodetachment processes of ions in vicinity of interfaces, cavities, and ion traps. The paper is organized as follows. In Section 2 we describe the classical motion of the valence electron of H- and give all the closed orbits. An analytical formula of the photodetachment cross section of the hydrogen ion H- near metal surface is derived in Section 3 based on the closed orbit theory. And Section 4 gives some numerical results and conclusive remarks. Unless indicated otherwise, we use atomic units throughout this work. 2. Closed orbits To investigate the process of photodetachment of H- near metal surface, we consider the following system: In the cylindrical coordinates (q; z; /), a hydrogen negative ion H sits at the origin and a z-polarized laser is applied for the photodetachment. A metal surface perpendicular to z-axis is put at z0 where z0 is positive. The photodetached electron experiences an attractive force e2 FðzÞ ¼ ð2zÞ 2 identical to that produced by a positive charge at z.
K. Rui, G. Yang / Surface Science 603 (2009) 632–635
And it was trapped in the image potential well formed by the Cou2 lomb-like attractive image potential VðzÞ ¼ e4z. Thus when the energy E of detached electron is lower than image potential, the photodetached electrons are oscillated in the well. In all the classical trajectories of the photodetached electron emanating out the origin, those bounced back by image potential to the starting point are called closed orbits. The following are fundamental. (i) The electron goes up along the +z direction, reaches its maximum and then bounced back by the potential and returns to the origin. We call this orbit the up orbit. (ii) The electron goes down in –z direction and hits the metal surface and bounces back and finally return to the origin. We call this orbit the down orbit. (iii) The electron completes the up orbit first and then passes through the origin, and continues to complete the down orbit. (iv) The electron completes the down orbit first and then up orbit. This orbit is similar to the one of (iii) but in reverse order. To classify the system of orbits, indices j and n are used to label the closed orbits, where j = 1,2,3,4 and n = 0,1,2,3,. Here n = 0 means that the orbit is a fundamental closed orbit (j = 1,2,3,4 for the fundamental closed orbits above, respectively). When n > 0, the orbit (j,n) has two par pz ts, the beginning part and the later part. The beginning part is always the jth fundamental closed orbit. The later part consists of n repetitions of periodic orbits j = 3 or 4. For the case of j = 1, it is n repetitions of j = 3, and for j = 2, it corresponds to n repetitions of j = 4. For the periodic orbit j = 3 or j = 4, it repeats itself n times. The classical quantities needed for constructing semiclassical wave function can be obtained according to the systematic procedure of closed orbit theory [20,21]. In the current case, if the escaping energy of photodetached electron is E, we have p2 E ¼ 2z 4ðz01þzÞ þ 4z10 where is the momentum in z-direction. The travaling times for classical orbits can be obtained by the integral Rz Rt of z-motion. For the fundamental closed identity t ¼ 0 dt ¼ 0 dz pz orbits, their returning time T j can be written as
pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi 4z20 2E ð2z0 Þ3=2 ½p=2arcsinð 14z0 EÞ T 1 ¼ 14z þ 3=2 E ð14z0 EÞ 0 pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi 4z20 2E ð2z0 Þ3=2 arcsin 14z0 E T 2 ¼ 14z0 E þ ð14z EÞ3=2 0
3=2
ð1Þ
0
3=2
pð2z0 Þ T 4 ¼ T 1 þ T 2 ¼ 2ð14z EÞ3=2 0
Clearly, we have T 3 ¼ T 4 T. The classical action of a trajectory Rt is defined as Sðq; z; /Þ ¼ 0 pdq, where t is the traveling time, p and q are coordinate and momentum vectors respectively. It is acumulated along a classical trajectory and appears in the phase of the semiclassical wave function. For the fundamental closed orbits, the returning times in Eq. (1) are used in the calculation of the classical actions. The classical actions along the fundamental closed orbits are
S2 ¼
pffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi 14z0 EÞ 2z0 2E;
2z0 ½p=2arcsinð
14z0 E
pffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi14z0 EÞ þ 2z0 2E; 14z0 E
S3 ¼ S1 þ S2 ¼ p2 S4 ¼ S1 þ S2 ¼ p2
qffiffiffiffiffiffiffiffiffiffiffi
Sjn ¼ Sj þ nS:
ð2Þ
2z0 ; 14z0 E
qffiffiffiffiffiffiffiffiffiffiffi
2z0 : 14z0 E
ð4Þ
The returning time of any closed orbit is
T jn ¼ T j þ nT
ð5Þ
with Maslov index
ljn ¼ lj þ 3n:
ð6Þ
3. Photodetachment cross section For hydrogen ion H- near metal surface, the photodetachement can be regarded as a one-electron process if we neglect the influence of the short-range potential Vb(r) when the electron is far 2 from the core. The binding energy Eb ¼ kb =2 is approximately 0.754 eV, where kb is related with the initial wave function Wi = C exp (- kbr)/r, C is a ‘‘normalization function” constant and is equal to 0.31552. When an incident laser beam is applied to the ion, the valence electron absorbs a photon energy Ep = Eb + E and the steady outgoing electron wave propagates outward in all directions. The electron move in a straight line at constant speed before it hits the image potential or metal interface. We propagate the electron wave semiclassically outside the core. The semiclassical wave is reflected by the potential wall or metal interface. Finally, some of the waves return to the vicinity of the core and interfere with the outgoing wave to produce the spectral pattern of the photodetachment. In this physical picture, the photodetachment cross section can be expressed in the form [23]
4Ep ^ þ jDWi i ImhDWi jG c
ð7Þ
^ þ . The dipole The outgoing Green’s function is denoted by G operator D is equal to the projection of the electron coordinate onto the direction of polarization of the laser field. Eq. (7) has the following physical meaning: the initial state is modified by the dipole operator related with the incident laser field to become the source wave function; the Green’s function propagates these waves outward to become the outgoing waves; and finally the waves overlap with the source wave to give the absorption spectrum. The outgoing wave can be divided into two parts. The first part never goes far from the core and it is called the direct wave. The second part is known as the returning wave that propagates outward into the external region first, then is reflected by the potential wall or metal interface, and finally returns to the vicinity of the core to interfere with the outgoing wave. Accordingly, the cross section has two parts,
rðEÞ ¼ r0 ðEÞ þ rret
2z0 ½arcsinð
ð3Þ
The action of any closed orbit consists of the ones of fundamental orbits and can be written as
rðEÞ ¼
pð2z0 Þ T 3 ¼ T 1 þ T 2 ¼ 2ð14z EÞ3=2
S1 ¼
l1 ¼ 1; l2 ¼ 2; l3 ¼ l4 ¼ 3:
633
ð8Þ
The first part is the contribution of the direct wave interfering with the source and is the field-free cross section [24,25]
r0 ðEÞ ¼
pffiffiffi 16 2C 2 p2 E3=2 ; 3c ðEb þ EÞ3
ð9Þ
The second part is the contribution of the returning wave, Clearly, we have S3 ¼ S4 S. Maslov index is related with the topological structure of trjectory manifold and can be obtained simply by counting the singular points such as caustics and foci along the trajectory [22]. For the current case, Maslov index can be easily found by counting the returning points where a p phase loss of electronic wave occurs, thus we have
rret ðEÞ ¼
4Ep ImhDWi jWret i; c
ð10Þ
where Wret is the returning wave, which overlaps with the outgoing source wave to give the interference pattern in the absorption spectrum.
634
K. Rui, G. Yang / Surface Science 603 (2009) 632–635
The construction of semiclassical wave function and the calculation of the overlapping integral to obtain the cross section are central subjects of closed theory. The derivation can be found in references [23,26] in detail. For the current system, we describe the procedure briefly in the following. To construct the semiclassical wave function we start from a spherical surface centered at the origin with radius R 10a0. When a beam of z-polarized laser is applied to the initial state of electron, the outging electron wave is produced. On the surface of the virtual sphere, the outgoing wave is denoted by [27]
W0 ðR; h; /Þ ¼ i
4Ck 2 ðkb
2
cosðhÞ
2 2
þk Þ
eiðkRpÞ : kR
ð11Þ
In the semiclassical approximation, the wave outside this sphere can be expressed as
Wðq; z; /Þ ¼
X
W0 ðR; h; /ÞAi ei½Si li p=2
ð12Þ
i
where Si is the action along the ith trajectory, li is the Maslov index characterizing the geometrical properties of the ith trajectory and its neighboring orbits. The amplitude of the wave function, Ai is given by [16]
J ðq; z; 0Þ1=2 ¼ R : Ai ðq; z; /Þ ¼ i J i ðq; z; TÞ R þ kT
ð13Þ
pffiffiffi X 16 2C 2 p2 E3=2 rðEÞ ¼ C jn sinðSjn ljn p=2Þ: 3c ðEb þ EÞ3 jn
ð19Þ
where c is light speed and
C jn ¼ ð1Þ½ðlj 1Þ=2
pffiffiffi 2p2 4 2C 2 E1=2 : c T jn ðEb þ EÞ3
ð20Þ
4. Numerical results and conclusions The photodetachment cross section can be calculated for different values of the distances between the ion and the metal surface. The results are shown Fig. 1 and Fig. 2. In Fig. 1, we keep the distance z0 fixed at 50 a.u., when we only consider the contribution of the up orbit, i.e. the electron goes up along the +z direction, then bounced back by the potential and returns to the origin, the calculated cross section is shown in Fig. 1a. The detachment spectrums consist of two parts and reveal the characteristic of image potential. When the photon energy is lower than the image potential, the cross section consists of a smooth background and sinusoidal oscillations. The former is the field-free cross section which attribute to the contribution of the direct wave interfering with the source. The latter derive from the contribution of the returning wave. While the photon energy is higher than the potential well,
where
@t
@z @h : @ q @h
ð14Þ
1.0
In Eqs. (13) and (14), T represents the travaling time of the photodetached electron from the origin. Only the waves associated with the closed orbits return to the vicinity of the core and interfere with the steadily-producing outgoing spherical waves and therefore contribute the cross section. For the case without the potential wall, the returning waves near the core must be cylindrically symmetric, and thus are approximated by incoming Bessel functions. The returning wave function related with closed orbit (j,n) must match the Bessel function and can be written as
1
1
2p
2p
ret
ret ikz z Wret ; jn ¼ N jn pffiffiffiffiffiffiffi J 0 ðkq qÞ pffiffiffiffiffiffiffi e
T jn k
By a cumbersome but straightforward analytical derivation, the proportional constant in Eq. (15) now becomes 2
Njn ¼ ð1Þ½ðlj 1Þ=2
i p exp½ ðSjn ljn Þ h 2 þk Þ
4iCk 2 T jn kðkb
2 2
ð17Þ
The oscillatory contribution to the cross section can now be calculated by overlapping the returning waves with the outgoing wave functions, then taking the imaginary part [16,23]. Therefore, we have
4E rret ðEÞ ¼ p Im c
Z
C expðkb rÞ cos hNjn e
sin hdhd/;
0.6 0.4
0.0
0.80
r dr ð18Þ
Integrating out the expression and adding the field-free part, we have the total photodetachment cross section
0.90
0.95
1.00
Photon energy (eV) 1.6
z 0=50a.u.
1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0.8
ikr cos hret jn 2
0.85
b
ð16Þ
:
0.8
0.2
Cross section (a 20 )
Ajn ¼
z 0=50a.u.
ð15Þ
where the normalization factor Ni can be determined by matching Eqs. (12) and (15). For closed orbit (j,n), the modified amplitude in Eq. (13) of the returning function is
1
a Cross section (a 20 )
@z @t Jðq; z; /Þ ¼ qðTÞ @ q
0.9
1.0
1.1
1.2
1.3
Photon energy (eV)
1.4
1.5
Fig. 1. The distance between the ion and the metal surface is 50 a.u. (a) The photodetachment cross section attribute to the contribution of the up orbit and (b) the cross section derive from the superposition of contribution from fundamental closed orbits and their repetitions.
K. Rui, G. Yang / Surface Science 603 (2009) 632–635
a
0.6
Cross section (a 20 )
0.5
z0=100a.u.
0.4 0.3 0.2 0.1 0.0
0.76
0.78
0.80
0.82
0.84
0.86
0.88
Photon energy (eV)
b
1.4
Cross section (a 20 )
1.2
z0=100a.u.
1.0 0.8
635
times vary in a quite range and depend on the type of metals, crystal orientation and the atom adsorption of surface [28]. For a finite lifetime system, the contribution from those orbits whose periods are longer than the lifetime vanishes in Eq. (19), but the shorter orbits still contribute to the summation. In the case, there still exists the staircase structure in the cross section but with smoother steps due to the lack of high harmonics unless the shortest period of closed orbits is longer than the lifetime. For example, the lifetime of n = 1 image state on Cu(1 0 0) is about 40 fs [29], while the shortest period of closed orbit is 22.84 fs. If there is further dephasing such as surface atom adsorption, the staircase structure may disappear. Further investigation is needed to clarify the subject in detail. In summary, we have analyzed the photodetachment of H- near metal surface and obtained an analytical formula of the cross section. It is found that with increasing of the photon energy, the configuration of spectrum evolved from a staircase structure to a smooth oscillation. The phenomena can be interpreted as follows: When the photon energy is lower than the potential, the detachment electron was trapped in the image potential well and bounced back and forth between the metal surface and image potential, corresponding spectrum was a staircase, while the photon energy is higher than the potential, the detachment electron escapes from the image potential well and just reflected by metal surface, so the cross section is a smooth oscillation. Acknowledgement
0.6 0.4
This work is partially supported by a National Key Basic Research Project of China (2007CB310405).
0.2
References
0.0 0.80
0.85
0.90
0.95
1.00
1.05
1.10
1.15
Photon energy (eV) Fig. 2. The distance is 100 a.u. (a) The photodetachment cross section attribute to the contribution of the up orbit and (b) the cross section the superposition of contribution from four fundamental closed orbits and their repetitions.
the detached electrons traverse the potential completely, there is no contribution of returning wave, which is similar to the case of free space, so the spectrum is a smooth line. In Fig. 1b, the structure results from the superposition of contribution from four fundamental closed orbits and their repetitions. We can see that when the photon energy is lower than the image potential, the detachment electron was trapped in the image potential well and the electronic waves are bounced back and forth between the metal surface and image potential, the spectrum displays a staircase structure, contrasting with the regular smooth oscillating curve when only a potential wall exists. While the energy is higher than potential, the cross section is a smooth oscillation. In this case, the detachment electron escapes from the image potential well, the returning wave is just reflected by metal surface, i.e. the cross section only attribute to the contribution of the down orbit. If the metal surface approaches the ion further, taking z0 =100 a.u., we can see the same configuration of spectrum. In the present investigation, we did’nt consider the influence of lifetimes of the image states and assumed that they are infinite long. This is certainly not correct for real systems. Actually, the life-
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