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Surface plasmons and interaction of charges with their images in a metal–atom system
Nuclear Instruments and Methods in Physics Research B 193 (2002) 408–413 www.elsevier.com/locate/nimb
Surface plasmons and interaction of charges with their images in a metal–atom system V.G. Drobnich *, I.S. Sharodi Department of Physics, Uzhgorod National University, 88000, Uzhgorod, Ukraine
Abstract The relation between surface plasmons (SPs) and image charges in a ‘‘metal–atom’’ system is investigated. It is shown, that the interaction of external charge with image of another charge should be treated as a pair interaction of these external charges which should be added to their Coulomb interaction. This circumstance allows to choose definitely within the frames of the known image approach the form of the perturbation operator which is responsible for the shifts of atomic levels near a metal surface. The analogous but more precise operator is obtained in the case of description of the long-range interaction between metal and atom in SPs terms. Ó 2002 Published by Elsevier Science B.V. Keywords: Metal; Charges; Image approach; Surface plasmons; Atomic levels
1. Introduction As it is known, surface plasmons (SPs) are quasiparticles, bosons which correspond to normal modes of charge fluctuations on a metal surface, and which are responsible, from the quantummechanical point of view, for long-range interaction between metal and the rest of the world [1]. SPs in many respects determine evolution of ‘‘metal–moving atom’’ system, thus participating in different emission phenomena, which accompany bombardment of metals with accelerated ions. One kind of SP manifestation which occur in indicated system, namely, the long-range interaction between metal and atom, is considered below.
*
Corresponding author. E-mail address: [email protected] (V.G. Drobnich).
It was stated [2,3], that by its nature this interaction is identical to the interaction between the charges and their electric images in metal which is well known from the electrostatics of conductors [4]. So it could be simulated both from quantummechanical point of view, as a manifestation of SPs, and within the frames of image approach from classical point of view. As a manifestation of SPs, this interaction is explored enough for the system ‘‘metal–one charged particle’’. As to the system ‘‘metal–atom’’ it was investigated mainly within the frames of image approach. Using this approach in different models of ion emission and emission of excited atoms the shifts of atomic levels near the metal surface were calculated. Recently authors of works [5,6] questioned the correctness of these calculations offering (within the frames of image approach) the new expression for the perturbation operator, which is responsible for the shifts of atomic levels near the metal surface.
0168-583X/02/$ - see front matter Ó 2002 Published by Elsevier Science B.V. PII: S 0 1 6 8 - 5 8 3 X ( 0 2 ) 0 0 8 1 3 - 3
V.G. Drobnich, I.S. Sharodi / Nucl. Instr. and Meth. in Phys. Res. B 193 (2002) 408–413
This work is devoted to the question of choice of the mentioned operator both within the frames of image approach, and for the case of description of the long-range interaction between metal and atom in SP terms.
details in works [5,6]. Authors started from the assumption, that X Ui ; ð2aÞ Eint ¼ i
where Ui is potential energy of interaction of charge ei with metal. Taking into account (1a), it is naturally to put
2. Setting the problem Ui ¼ V ðRi Þ þ Let us consider interaction between metal and system fei g of classic charges ei , which are placed in points Ri ¼ ðXi ; Yi ; Zi Þ. It is known, that in approximation, which is given by electrostatics of conductors, this interaction can formally be considered as Coulomb interaction between fei g and system fei g of fictitious charges ei ¼ ei , which are placed in points (Xi , Yi , Zi ) (we consider, that the beginning of a frame of reference is on a surface of metal, and the OZ-axis is directed along the normal to this surface) [4]. In given approximation the energy of this interaction (or taken with an inverse sign work on infinitely slow removal of system fei g to infinity) Eint ¼
X
V ðRi Þ þ
i
1X W ðRi ; Rj Þ; 2 i;j
ð1aÞ
i6¼j
where
1X W ðRi ; Rj Þ: 2 j
ð1bÞ
ei ej ffi: W ðRi ; Rj Þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ri Rj 2 þ 4Zi Zj
ð1cÞ
It is possible to define V ðRi Þ value as energy of interaction of charge ei with its image ei , and W ðRi ; Rj Þ as the sum of energies of interactions ei with ej and ej with ei [4–6]. The indicated approximation becomes valid when Zi becomes large enough i.e. when there is no effect of macroscopic character of boundary conditions of an electrostatics of conductors. A question arises: how it is possible to construct on the basis (1a)–(1c) the perturbation operator which is responsible for shifts of energy levels of atom, which is placed near (but not too close to) the metal surface? This question was considered in
ð2bÞ
i6¼j
From here for the system fei g, which is classic analog of an one-electron atom, we have [5,6]: Ue ¼ V ðRe Þ þ 12W ðRe ; Rn Þ;
ð3aÞ
Un ¼ V ðRn Þ þ 12W ðRe ; Rn Þ;
ð3bÞ
where Re and Rn are radius vectors of an electron and core, correspondingly, and Ue and Un are potential energies of these particles, conditioned by presence of metal. Considering a core to be a classical particle, authors of works [5,6] on the base of expression (3a) have constructed the perturbation operator which is responsible for shifts of energy levels of an one-electron atom near the metal surface, and have calculated these shifts. It is underlined in works [5,6], that earlier in similar calculations the energy Ue ¼ V ðRe Þ þ W ðRe ; Rn Þ
e2 V ðRi Þ ¼ i ; 4Zi
409
ð4Þ
was inaccurately assigned to an atomic electron. Let us consider the question of expression (4) correctness. As it is clear from the stated above, it is not correct, if the assumption (2a) is valid. But it is possible to make the alternate assumption. In fact, the expressions (1a)–(1c), which were found in approximation of the electrostatics of conductors, are consent to assumption, that only V ðRi Þ is a potential energy of interaction of a charge ei with metal, and W ðRi ; Rj Þ is energy of pair interaction between ei and ej , which at the presence of metal is added to energy of a Coulomb interaction of charges. Such treatment of W ðRi ; Rj Þ value authorizes the use of expression (4) for calculation of atomic shifts and makes the expression (3a) to be senseless. To be convinced of correctness of the proposed treatment, it is enough to show, considering the metal as system of SPs, that, for example,
410
V.G. Drobnich, I.S. Sharodi / Nucl. Instr. and Meth. in Phys. Res. B 193 (2002) 408–413
in case of interaction of electrons with metal the following operator: Z Z 1 3 b þ ðr0 Þ b þ ðrÞ C b d r d3 r0 C W ¼ 2 b ðr0 Þ C b ðrÞ ð5Þ W ðr; r0 Þ C (here C þ ðrÞ and CðrÞ are operators of birth and annihilation of an electron in a point r), i.e. the operator of two-particle interaction [7], corresponds to W ðRi ; Rj Þ value. Let us show it. Besides this let us find out expressions which are analogous to (1a) and (1b), and which in difference from the last will be valid for every given distances from metal to external charges.
3. Relation between SPs and electric images of the system of external charges Let us write down a Hamiltonian of SPs which not interact with the rest of the world. Using known ‘‘jellium’’ model for metal, we have [3,8– 10] X 0 b sp ¼ ^q ; H eq a^þ ð6Þ qa q
where aþ q , aq are operators of creation and the annihilation of SP with wave vector q (all q are parallel to metal surface); eq is energy of plasmon, which depends on the vector q module; the prime on the summation implies jqj < qc where qc is the plasmon cutoff wave vector module. The second time quantized potential of an electric field generated by these plasmons is [3,8,9] i X h 0 ^ ðrÞ ¼ U Cq cðq; rÞ^ aþ aq ; ð7aÞ q þ c ðq; rÞ^
b ðrÞ is the operator of b þ ðrÞ C where q^ðrÞ ¼ e C electron charge density in the point r. The analogous operator for the SP–fei g system X ^ ðRi Þ b sp–fei g ¼ H ei U ð8bÞ i
(we use the former notations for values and radius vectors of classic charges). On the base of (6)–(8a) and (8b), introducing the convenient notations which are presented in Table 1, we can write X 0 bþ b b sp þ H b sp–el ¼ H eq B q B q þ h^sp–el ; ð9aÞ q
b sp þ H b sp–fei g ¼ H
q
where
X
0
^ eq b^þ q bq þ hsp–fei g :
ð9bÞ
q
cðq; rÞ ¼ exp ð qjzj iqqÞ; Cq ¼ ðpeq =qSÞ
regions (here xp is characteristic frequency of plasma oscillation in metal, vF is Fermi velocity, c is light velocity). In these regions dielectric function of metal acquires significant space dispersion. Also it is necessary to note that Eq. (7c) for Cq is required to be corrected in the region of small q (c=xp 102 a.u. [9]) which are not of practical interest. Let us consider two systems: ‘‘SP-electrons’’ (SP–el system) and ‘‘SP-electrons, as classical charges’’ (SP–fei g system). Hamiltonian of interaction between SPs and electrons for SP–el system Z ^ ðrÞ; b sp–el ¼ d3 r^ H qðrÞU ð8aÞ
1=2
:
ð7bÞ
It is easy to become convinced, that at any q, q0 the bq , X b þ commute with X bq0 . Correoperators X q
ð7cÞ
Here q is parallel to the metal surface component of a radius-vector r; z is projection of r on the surface normal, S is surface area. The Eqs. (7a)– (7c) are valid in wide range of possible q, eq values. The exceptions are q > xp =vF and eq < hxp vF =c
Table 1 SP–el system R bq e1 Cq d3 r^ X qðrÞcðq; rÞ, q b b B q a^q þP Xq, bqþ X bq , h^sp–el q 0 eq X
SP–fei g system P vq e1 q Cq i ei cðq; Ri Þ, ^ bq a^q þ vqP , hsp–fei g q 0 eq vq vq
V.G. Drobnich, I.S. Sharodi / Nucl. Instr. and Meth. in Phys. Res. B 193 (2002) 408–413
b þ, B b q0 obey to the same spondingly, the operators B q þ commutation relations, as a^q , a^q0 operators, i.e. to the standard commutation relations [7] for the operators of creation and annihilation of bosons. ^ The latter is valid also in relation to the b^þ q , bq0 operators because vq values are not the operators, but usual complex numbers. Let us clarify physical sense of items in the right part of expressions (9a) and (9b). For this let us find at first hsp–fei g . Making the standard replacement Z X 2 0 ! S=ð2pÞ d2 q ; ð10Þ jqj
q
we find (see Appendix A) X 1X e hsp–fei g ¼ Ve ðRi Þ þ W ðRi ; Rj Þ; 2 i;j i
ð11aÞ
i6¼j
where e2 Ve ðRi Þ ¼ i ½1 expð2qc jZi jÞ; ð11bÞ 4jZi j Z qc e ðRi ; Rj Þ ¼ ei ej W dq exp½qðjZi j þ jZj jÞ 0
J0 ðqjqi qj jÞ:
ð11cÞ
Here qi and qj are parallel to the metal surface components of Ri and Rj vectors, J0 ðxÞ is Bessel function of I kind and of zero order. For large enough Zi (Zi q1 c ) the value hsp–fei g coincides with energy of interaction Eint , because in this area expressions (11b) and (11c) are transformed to expressions (1b) and (1c). Taking into account this result and mentioned above commutation rela^ it is possible to tions for the operators b^þ q , bq 0 , P 0 ^þ ^ come to the conclusion, that q eq bq bq is the Hamiltonian of system of new quasi-particles SPs, properties of which are changed by the presence of classical charges ei . This modified surface plasmons (mSPs) have the same energies, as SPs, but differ from last by ground and exited states. In particular, the ground state jGi of the system mSP is determined by the equation b^q jGi ¼ 0
or
a^q jGi ¼ vq jGi;
where q transverses all possible values. When all Zi ! 1, then vq ! 0, i.e. aq jGi ¼ 0 at all q and jGi
411
state transfers in a ground state of the SP system. At infinitely slow removal of charges ei to infinity, the mSP system, if initially it was in a state jGi, will still stay in this state (this state is in parametric dependence from Zi ), i.e. the energy of the plasmon system will not change. Therefore the hsp–fei g value as well as Eint , is equal to the work (with an inverse sign) on infinitely slow removal of system fei g to infinity, i.e. quantities hsp–fei g and Eint have the same physical sense. Now physical sense of items in a right of expression (9a) becomes eviP0 part bþ b dent: q eq B q B q is the Hamiltonian of system of SPs, the properties of which are changed by presence of electrons over the metal surface, and operator h^sp–el , classical analog of which is hsp–fei g value, is operator which is responsible for shifts of energy levels of electrons near the of metal surface. The last task is to obtain physically transparent equation for h^sp–el . Using the known commutation relations for the b þ ðrÞ; C b ðrÞ and the results of fermion operators C calculating of hsp–fei g value, we find (see Appendix A) Z b þ ðrÞ Ve ðrÞ C b ðrÞ h^sp–el ¼ d3 ðrÞ C Z Z 1 b þ ðr0 Þ b þ ðrÞ C þ d3 r d3 r0 C 2 b ðr0 Þ C b ðrÞ: e ðr; r0 Þ C ð12Þ W The classical analogs of the first and the second operator items in (12), when the electrons are placed, mainly, on large distances from a metal surface (q1 c ), are values V ðRi Þ and W ðRi ; Rj Þ correspondingly. Thus, the operator of a twoparticle interaction (5) really corresponds to the W ðRi ; Rj Þ energy. It proves validity of the assumption, made by us, that W ðRi ; Rj Þ is the energy of the pair interaction of charges, which is caused by presence of metal. 4. ‘‘SP–hydrogen atom’’ system Hamiltonian of this system is b ¼H b sp þ H bA þ H b sp–A ; H
ð13aÞ
b A is Hamiltonian of free hydrogen atom, where H b sp–A is Hamiltonian of interaction between SP H
412
V.G. Drobnich, I.S. Sharodi / Nucl. Instr. and Meth. in Phys. Res. B 193 (2002) 408–413
b , using the mentioned and atom. Let us rewrite H above mSP idea (in this case mSPs are SPs which properties are changed by presence of atom above the metal surface). Let us consider a proton to be a classical particle. Let R to be its radius vector, and r to be the shift of atomic electron relatively to the proton. Taking into account, that in our case there is only one classical charge, we have vq ¼ vq ðRÞ ¼ e1 q Cq ecðq; RÞ; bq ¼ X
Z
b þ ðrÞvq ðR þ rÞ C b ðrÞ: d3 r C
b sp–A operator is Taking into account also that H equal to the sum of Hamiltonians (8a) and (8b) and introducing the bq Ybq ¼ vq þ X
h^A ¼
Z
h b þ ðrÞ H b A ðrÞ þ Ve ðR þ rÞ d3 r C i b ðrÞ; e ðR; R þ rÞ C þW
ð14bÞ
This expression justifies the use of Eq. (4) for calculation of shifts of atomic levels in the region e of large Zð q1 c Þ, where energies V ðR þ rÞ and e ðR; R þ rÞ acquire the sense of energies of inW teraction of atomic electron with its electric image and the image of proton correspondingly. Within the framework of chosen SP model the obtained e values are exact for any expressions for Ve and W given distance between atom and metal. It allows to solve the problem of the shifts of atomic levels near a metal surface more precisely.
5. Conclusion
and b q ¼ a^q þ Ybq ; A we have b ¼ h^mSP þ h^A ; H
ð13bÞ
where h^mSP ¼
X
0
bþA b eq A q q;
ð13cÞ
q
bA h^A ¼ H
X
0
eq Ybqþ Ybq :
ð13dÞ
q
Here h^mSP is Hamiltonian of mSPs, h^A is Hamiltonian of atom in an electric field of mSPs, and the sum in h^A is the perturbation operator, which describes action of this field on atom. Let us find out this sum. Doing calculations similar to ones given in Appendix A, we receive (omitting the non operational item Ve ðRÞ and the two-electron (5)-kind operator which action on any state of one-electron atom results in 0): Z h X 0 bþ b b þ ðrÞ Ve ðR þ rÞ eq Y q Y q ¼ d3 r C q
i.e.
i b ðrÞ; e ðR; R þ rÞ C þW
ð14aÞ
Long-range interaction between metal and system of charges is investigated. It is shown that interaction of one charge with the electric image of another charge which appears in the known image approach should be treated as a pair interaction of these charges, which is added to their Coulomb interaction. This circumstance allows to choose definitely within the frames of the mentioned approximation the form of the perturbation operator which is responsible for the shifts of atomic levels near a metal surface. The analogous but more precise operator is obtained in the case of description of the long-range interaction between metal and atom in SP terms.
Acknowledgements This work was performed in the framework of the PICS France-Ukraine collaborative project. Appendix A. Calculation of hsp–fei g and ^hsp–el values Coming out from the definition of hsp–fei g we can write hsp–fei g ¼
X i
1 Ve ðRi Þ þ 2
X i;j i6¼j
e ðRi ; Rj Þ; W
ðA:1Þ
V.G. Drobnich, I.S. Sharodi / Nucl. Instr. and Meth. in Phys. Res. B 193 (2002) 408–413
where Ve ðRi Þ ¼ e2i
X
0 1 2 eq Cq jcðq; Ri Þj2 ;
e ðRi ; Rj Þ W ðA:2Þ
¼
jZi jþjZi jq1 c
ei ej ffi: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Ri Rj þ 2Zi Zj þ 2Zi Zj
q
e ðRi ; Rj Þ ¼ 2ei ej W
X
413
0 1 2 eq Cq c ðq; Ri Þcðq; Rj Þ:
ðA:9Þ
q
ðA:3Þ Sums in (A.2) and (A.3) after replacement of them with integrals (10) appear in a following manner: Z 2p Z qc X 1 0 1 2 eq Cq ¼ dq du ; ðA:4Þ 4p 0 0 q where q, u are polar coordinates of vector q. This replacement results in Z qc e2i e V ðRi Þ ¼ dq exp ð 2qjZi jÞ; ðA:5Þ 2 0 Z qc
e ðRi ; Rj Þ ¼ ei ej W dq exp qðjZi j þ jZj jÞ 2p 0 Z 2p du expðiqjqi qj jÞ; ðA:6Þ 0
where qi , qj are parallel to the metal surface components of Ri and Rj vectors. Calculating the integral in (A.5) we get e2 Ve ðRi Þ ¼ i ½1 expð2qc jZi jÞ: 4jZi j
ðA:7Þ
Let us notice that in region of large positive Zi ðZi q1 c Þ, where (1a)–(1c) equations are valid, the expression (A.7) transforms into expression (1b) for the potential energy V ðRi Þ. Let us simplify the (A.6) equation. We have [11] Z p expðix cos uÞdu ¼ pJ0 ðxÞ; 0
where J0 ðxÞ is Bessel function of I kind and of zero order. Thus Z qc e W ðRi ; Rj Þ ¼ ei ej dq exp q jZi j þ Zj 0 J0 qjqi qj j : ðA:8Þ The integral in (A.8) is calculated analytically, when jZi j þ jZj j q1 c . In this case we have [11]
Let us notice that in region of large positive Zi , Zj , where (1a)–(1c) equations are valid, the expression (A.9) transforms into expression (1c) for the energy W ðRi ; Rj Þ. Let us find out now the h^sp–el operator. Coming out from its definition we can write h^sp–el ¼
Z
d3 r (
Z
e
b ðrÞ b þ ðrÞ C d3 r0 C 2
X
) 0 1 2 eq Cq c ðq; rÞcðq; r0 Þ
b ðr0 Þ: b þ ðr0 Þ C C
q
ðA:10Þ bþ
b ðrÞ; C ðr Þ operators and Commuting first the C b ðrÞ; C b ðr0 Þ operators in (A.10) and taking the then C integral on r0 in the item, which arise during these operations and which contains delta function d3 ðr r0 Þ, finally we get, taking into account (A.2) and (A.3), Z ^ b þ ðrÞ Ve ðrÞ C b ðrÞ hsp–el ¼ d3 r C Z Z 1 b þ ðr0 Þ b þ ðrÞ C d3 r d3 r0 C þ 2 b ðr0 Þ C b ðrÞ: e ðr; r0 Þ C ðA:11Þ W 0
References [1] P.J. Feibelman, Surf. Sci. 282 (1993) 129. [2] R.H. Ritchie, Surf. Sci. 34 (1973) 1. [3] E.N. Economou, K.L. Ngai, Advan. Chem. Phys. 27 (1974) 265. [4] L.D. Landau, E.M. Lifshitz, Electrodynamics of Continuous Media, Pergamon Press, Oxford, New York, 1984. [5] M.I. Chibisov, A.V. Rojtman, ZhETF 116 (1999) 236. [6] M.I. Chibisov, A.V. Rojtman, Materials of XIV Int. Conf. Interaction of Ions with a Surface, Moscow, 1999, p. 235. [7] R.P. Feynman, Statistical Mechanics, W.A. Benjamin Inc., Advanced Book Program Reading, Massachusetts, 1972. [8] J.M. Elson, R.H. Ritchie, Phys. Rev. B. 4 (1971) 4129. [9] R.H. Ritchie, Phys. Lett. 38A (1972) 189. [10] A.A. Almulhem, M.D. Girardeau, Surf. Sci. 210 (1989) 138. [11] I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series, and Products, Academic Press Inc., 1980, p. 1160.
Surface plasmons and interaction of charges with their images in a metal–atom system V.G. Drobnich *, I.S. Sharodi Department of Physics, Uzhgorod National University, 88000, Uzhgorod, Ukraine
Abstract The relation between surface plasmons (SPs) and image charges in a ‘‘metal–atom’’ system is investigated. It is shown, that the interaction of external charge with image of another charge should be treated as a pair interaction of these external charges which should be added to their Coulomb interaction. This circumstance allows to choose definitely within the frames of the known image approach the form of the perturbation operator which is responsible for the shifts of atomic levels near a metal surface. The analogous but more precise operator is obtained in the case of description of the long-range interaction between metal and atom in SPs terms. Ó 2002 Published by Elsevier Science B.V. Keywords: Metal; Charges; Image approach; Surface plasmons; Atomic levels
1. Introduction As it is known, surface plasmons (SPs) are quasiparticles, bosons which correspond to normal modes of charge fluctuations on a metal surface, and which are responsible, from the quantummechanical point of view, for long-range interaction between metal and the rest of the world [1]. SPs in many respects determine evolution of ‘‘metal–moving atom’’ system, thus participating in different emission phenomena, which accompany bombardment of metals with accelerated ions. One kind of SP manifestation which occur in indicated system, namely, the long-range interaction between metal and atom, is considered below.
*
Corresponding author. E-mail address: [email protected] (V.G. Drobnich).
It was stated [2,3], that by its nature this interaction is identical to the interaction between the charges and their electric images in metal which is well known from the electrostatics of conductors [4]. So it could be simulated both from quantummechanical point of view, as a manifestation of SPs, and within the frames of image approach from classical point of view. As a manifestation of SPs, this interaction is explored enough for the system ‘‘metal–one charged particle’’. As to the system ‘‘metal–atom’’ it was investigated mainly within the frames of image approach. Using this approach in different models of ion emission and emission of excited atoms the shifts of atomic levels near the metal surface were calculated. Recently authors of works [5,6] questioned the correctness of these calculations offering (within the frames of image approach) the new expression for the perturbation operator, which is responsible for the shifts of atomic levels near the metal surface.
0168-583X/02/$ - see front matter Ó 2002 Published by Elsevier Science B.V. PII: S 0 1 6 8 - 5 8 3 X ( 0 2 ) 0 0 8 1 3 - 3
V.G. Drobnich, I.S. Sharodi / Nucl. Instr. and Meth. in Phys. Res. B 193 (2002) 408–413
This work is devoted to the question of choice of the mentioned operator both within the frames of image approach, and for the case of description of the long-range interaction between metal and atom in SP terms.
details in works [5,6]. Authors started from the assumption, that X Ui ; ð2aÞ Eint ¼ i
where Ui is potential energy of interaction of charge ei with metal. Taking into account (1a), it is naturally to put
2. Setting the problem Ui ¼ V ðRi Þ þ Let us consider interaction between metal and system fei g of classic charges ei , which are placed in points Ri ¼ ðXi ; Yi ; Zi Þ. It is known, that in approximation, which is given by electrostatics of conductors, this interaction can formally be considered as Coulomb interaction between fei g and system fei g of fictitious charges ei ¼ ei , which are placed in points (Xi , Yi , Zi ) (we consider, that the beginning of a frame of reference is on a surface of metal, and the OZ-axis is directed along the normal to this surface) [4]. In given approximation the energy of this interaction (or taken with an inverse sign work on infinitely slow removal of system fei g to infinity) Eint ¼
X
V ðRi Þ þ
i
1X W ðRi ; Rj Þ; 2 i;j
ð1aÞ
i6¼j
where
1X W ðRi ; Rj Þ: 2 j
ð1bÞ
ei ej ffi: W ðRi ; Rj Þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ri Rj 2 þ 4Zi Zj
ð1cÞ
It is possible to define V ðRi Þ value as energy of interaction of charge ei with its image ei , and W ðRi ; Rj Þ as the sum of energies of interactions ei with ej and ej with ei [4–6]. The indicated approximation becomes valid when Zi becomes large enough i.e. when there is no effect of macroscopic character of boundary conditions of an electrostatics of conductors. A question arises: how it is possible to construct on the basis (1a)–(1c) the perturbation operator which is responsible for shifts of energy levels of atom, which is placed near (but not too close to) the metal surface? This question was considered in
ð2bÞ
i6¼j
From here for the system fei g, which is classic analog of an one-electron atom, we have [5,6]: Ue ¼ V ðRe Þ þ 12W ðRe ; Rn Þ;
ð3aÞ
Un ¼ V ðRn Þ þ 12W ðRe ; Rn Þ;
ð3bÞ
where Re and Rn are radius vectors of an electron and core, correspondingly, and Ue and Un are potential energies of these particles, conditioned by presence of metal. Considering a core to be a classical particle, authors of works [5,6] on the base of expression (3a) have constructed the perturbation operator which is responsible for shifts of energy levels of an one-electron atom near the metal surface, and have calculated these shifts. It is underlined in works [5,6], that earlier in similar calculations the energy Ue ¼ V ðRe Þ þ W ðRe ; Rn Þ
e2 V ðRi Þ ¼ i ; 4Zi
409
ð4Þ
was inaccurately assigned to an atomic electron. Let us consider the question of expression (4) correctness. As it is clear from the stated above, it is not correct, if the assumption (2a) is valid. But it is possible to make the alternate assumption. In fact, the expressions (1a)–(1c), which were found in approximation of the electrostatics of conductors, are consent to assumption, that only V ðRi Þ is a potential energy of interaction of a charge ei with metal, and W ðRi ; Rj Þ is energy of pair interaction between ei and ej , which at the presence of metal is added to energy of a Coulomb interaction of charges. Such treatment of W ðRi ; Rj Þ value authorizes the use of expression (4) for calculation of atomic shifts and makes the expression (3a) to be senseless. To be convinced of correctness of the proposed treatment, it is enough to show, considering the metal as system of SPs, that, for example,
410
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in case of interaction of electrons with metal the following operator: Z Z 1 3 b þ ðr0 Þ b þ ðrÞ C b d r d3 r0 C W ¼ 2 b ðr0 Þ C b ðrÞ ð5Þ W ðr; r0 Þ C (here C þ ðrÞ and CðrÞ are operators of birth and annihilation of an electron in a point r), i.e. the operator of two-particle interaction [7], corresponds to W ðRi ; Rj Þ value. Let us show it. Besides this let us find out expressions which are analogous to (1a) and (1b), and which in difference from the last will be valid for every given distances from metal to external charges.
3. Relation between SPs and electric images of the system of external charges Let us write down a Hamiltonian of SPs which not interact with the rest of the world. Using known ‘‘jellium’’ model for metal, we have [3,8– 10] X 0 b sp ¼ ^q ; H eq a^þ ð6Þ qa q
where aþ q , aq are operators of creation and the annihilation of SP with wave vector q (all q are parallel to metal surface); eq is energy of plasmon, which depends on the vector q module; the prime on the summation implies jqj < qc where qc is the plasmon cutoff wave vector module. The second time quantized potential of an electric field generated by these plasmons is [3,8,9] i X h 0 ^ ðrÞ ¼ U Cq cðq; rÞ^ aþ aq ; ð7aÞ q þ c ðq; rÞ^
b ðrÞ is the operator of b þ ðrÞ C where q^ðrÞ ¼ e C electron charge density in the point r. The analogous operator for the SP–fei g system X ^ ðRi Þ b sp–fei g ¼ H ei U ð8bÞ i
(we use the former notations for values and radius vectors of classic charges). On the base of (6)–(8a) and (8b), introducing the convenient notations which are presented in Table 1, we can write X 0 bþ b b sp þ H b sp–el ¼ H eq B q B q þ h^sp–el ; ð9aÞ q
b sp þ H b sp–fei g ¼ H
q
where
X
0
^ eq b^þ q bq þ hsp–fei g :
ð9bÞ
q
cðq; rÞ ¼ exp ð qjzj iqqÞ; Cq ¼ ðpeq =qSÞ
regions (here xp is characteristic frequency of plasma oscillation in metal, vF is Fermi velocity, c is light velocity). In these regions dielectric function of metal acquires significant space dispersion. Also it is necessary to note that Eq. (7c) for Cq is required to be corrected in the region of small q (
1=2
:
ð7bÞ
It is easy to become convinced, that at any q, q0 the bq , X b þ commute with X bq0 . Correoperators X q
ð7cÞ
Here q is parallel to the metal surface component of a radius-vector r; z is projection of r on the surface normal, S is surface area. The Eqs. (7a)– (7c) are valid in wide range of possible q, eq values. The exceptions are q > xp =vF and eq < hxp vF =c
Table 1 SP–el system R bq e1 Cq d3 r^ X qðrÞcðq; rÞ, q b b B q a^q þP Xq, bqþ X bq , h^sp–el q 0 eq X
SP–fei g system P vq e1 q Cq i ei cðq; Ri Þ, ^ bq a^q þ vqP , hsp–fei g q 0 eq vq vq
V.G. Drobnich, I.S. Sharodi / Nucl. Instr. and Meth. in Phys. Res. B 193 (2002) 408–413
b þ, B b q0 obey to the same spondingly, the operators B q þ commutation relations, as a^q , a^q0 operators, i.e. to the standard commutation relations [7] for the operators of creation and annihilation of bosons. ^ The latter is valid also in relation to the b^þ q , bq0 operators because vq values are not the operators, but usual complex numbers. Let us clarify physical sense of items in the right part of expressions (9a) and (9b). For this let us find at first hsp–fei g . Making the standard replacement Z X 2 0 ! S=ð2pÞ d2 q ; ð10Þ jqj
q
we find (see Appendix A) X 1X e hsp–fei g ¼ Ve ðRi Þ þ W ðRi ; Rj Þ; 2 i;j i
ð11aÞ
i6¼j
where e2 Ve ðRi Þ ¼ i ½1 expð2qc jZi jÞ; ð11bÞ 4jZi j Z qc e ðRi ; Rj Þ ¼ ei ej W dq exp½qðjZi j þ jZj jÞ 0
J0 ðqjqi qj jÞ:
ð11cÞ
Here qi and qj are parallel to the metal surface components of Ri and Rj vectors, J0 ðxÞ is Bessel function of I kind and of zero order. For large enough Zi (Zi q1 c ) the value hsp–fei g coincides with energy of interaction Eint , because in this area expressions (11b) and (11c) are transformed to expressions (1b) and (1c). Taking into account this result and mentioned above commutation rela^ it is possible to tions for the operators b^þ q , bq 0 , P 0 ^þ ^ come to the conclusion, that q eq bq bq is the Hamiltonian of system of new quasi-particles SPs, properties of which are changed by the presence of classical charges ei . This modified surface plasmons (mSPs) have the same energies, as SPs, but differ from last by ground and exited states. In particular, the ground state jGi of the system mSP is determined by the equation b^q jGi ¼ 0
or
a^q jGi ¼ vq jGi;
where q transverses all possible values. When all Zi ! 1, then vq ! 0, i.e. aq jGi ¼ 0 at all q and jGi
411
state transfers in a ground state of the SP system. At infinitely slow removal of charges ei to infinity, the mSP system, if initially it was in a state jGi, will still stay in this state (this state is in parametric dependence from Zi ), i.e. the energy of the plasmon system will not change. Therefore the hsp–fei g value as well as Eint , is equal to the work (with an inverse sign) on infinitely slow removal of system fei g to infinity, i.e. quantities hsp–fei g and Eint have the same physical sense. Now physical sense of items in a right of expression (9a) becomes eviP0 part bþ b dent: q eq B q B q is the Hamiltonian of system of SPs, the properties of which are changed by presence of electrons over the metal surface, and operator h^sp–el , classical analog of which is hsp–fei g value, is operator which is responsible for shifts of energy levels of electrons near the of metal surface. The last task is to obtain physically transparent equation for h^sp–el . Using the known commutation relations for the b þ ðrÞ; C b ðrÞ and the results of fermion operators C calculating of hsp–fei g value, we find (see Appendix A) Z b þ ðrÞ Ve ðrÞ C b ðrÞ h^sp–el ¼ d3 ðrÞ C Z Z 1 b þ ðr0 Þ b þ ðrÞ C þ d3 r d3 r0 C 2 b ðr0 Þ C b ðrÞ: e ðr; r0 Þ C ð12Þ W The classical analogs of the first and the second operator items in (12), when the electrons are placed, mainly, on large distances from a metal surface (q1 c ), are values V ðRi Þ and W ðRi ; Rj Þ correspondingly. Thus, the operator of a twoparticle interaction (5) really corresponds to the W ðRi ; Rj Þ energy. It proves validity of the assumption, made by us, that W ðRi ; Rj Þ is the energy of the pair interaction of charges, which is caused by presence of metal. 4. ‘‘SP–hydrogen atom’’ system Hamiltonian of this system is b ¼H b sp þ H bA þ H b sp–A ; H
ð13aÞ
b A is Hamiltonian of free hydrogen atom, where H b sp–A is Hamiltonian of interaction between SP H
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b , using the mentioned and atom. Let us rewrite H above mSP idea (in this case mSPs are SPs which properties are changed by presence of atom above the metal surface). Let us consider a proton to be a classical particle. Let R to be its radius vector, and r to be the shift of atomic electron relatively to the proton. Taking into account, that in our case there is only one classical charge, we have vq ¼ vq ðRÞ ¼ e1 q Cq ecðq; RÞ; bq ¼ X
Z
b þ ðrÞvq ðR þ rÞ C b ðrÞ: d3 r C
b sp–A operator is Taking into account also that H equal to the sum of Hamiltonians (8a) and (8b) and introducing the bq Ybq ¼ vq þ X
h^A ¼
Z
h b þ ðrÞ H b A ðrÞ þ Ve ðR þ rÞ d3 r C i b ðrÞ; e ðR; R þ rÞ C þW
ð14bÞ
This expression justifies the use of Eq. (4) for calculation of shifts of atomic levels in the region e of large Zð q1 c Þ, where energies V ðR þ rÞ and e ðR; R þ rÞ acquire the sense of energies of inW teraction of atomic electron with its electric image and the image of proton correspondingly. Within the framework of chosen SP model the obtained e values are exact for any expressions for Ve and W given distance between atom and metal. It allows to solve the problem of the shifts of atomic levels near a metal surface more precisely.
5. Conclusion
and b q ¼ a^q þ Ybq ; A we have b ¼ h^mSP þ h^A ; H
ð13bÞ
where h^mSP ¼
X
0
bþA b eq A q q;
ð13cÞ
q
bA h^A ¼ H
X
0
eq Ybqþ Ybq :
ð13dÞ
q
Here h^mSP is Hamiltonian of mSPs, h^A is Hamiltonian of atom in an electric field of mSPs, and the sum in h^A is the perturbation operator, which describes action of this field on atom. Let us find out this sum. Doing calculations similar to ones given in Appendix A, we receive (omitting the non operational item Ve ðRÞ and the two-electron (5)-kind operator which action on any state of one-electron atom results in 0): Z h X 0 bþ b b þ ðrÞ Ve ðR þ rÞ eq Y q Y q ¼ d3 r C q
i.e.
i b ðrÞ; e ðR; R þ rÞ C þW
ð14aÞ
Long-range interaction between metal and system of charges is investigated. It is shown that interaction of one charge with the electric image of another charge which appears in the known image approach should be treated as a pair interaction of these charges, which is added to their Coulomb interaction. This circumstance allows to choose definitely within the frames of the mentioned approximation the form of the perturbation operator which is responsible for the shifts of atomic levels near a metal surface. The analogous but more precise operator is obtained in the case of description of the long-range interaction between metal and atom in SP terms.
Acknowledgements This work was performed in the framework of the PICS France-Ukraine collaborative project. Appendix A. Calculation of hsp–fei g and ^hsp–el values Coming out from the definition of hsp–fei g we can write hsp–fei g ¼
X i
1 Ve ðRi Þ þ 2
X i;j i6¼j
e ðRi ; Rj Þ; W
ðA:1Þ
V.G. Drobnich, I.S. Sharodi / Nucl. Instr. and Meth. in Phys. Res. B 193 (2002) 408–413
where Ve ðRi Þ ¼ e2i
X
0 1 2 eq Cq jcðq; Ri Þj2 ;
e ðRi ; Rj Þ W ðA:2Þ
¼
jZi jþjZi jq1 c
ei ej ffi: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Ri Rj þ 2Zi Zj þ 2Zi Zj
q
e ðRi ; Rj Þ ¼ 2ei ej W
X
413
0 1 2 eq Cq c ðq; Ri Þcðq; Rj Þ:
ðA:9Þ
q
ðA:3Þ Sums in (A.2) and (A.3) after replacement of them with integrals (10) appear in a following manner: Z 2p Z qc X 1 0 1 2 eq Cq ¼ dq du ; ðA:4Þ 4p 0 0 q where q, u are polar coordinates of vector q. This replacement results in Z qc e2i e V ðRi Þ ¼ dq exp ð 2qjZi jÞ; ðA:5Þ 2 0 Z qc
e ðRi ; Rj Þ ¼ ei ej W dq exp qðjZi j þ jZj jÞ 2p 0 Z 2p du expðiqjqi qj jÞ; ðA:6Þ 0
where qi , qj are parallel to the metal surface components of Ri and Rj vectors. Calculating the integral in (A.5) we get e2 Ve ðRi Þ ¼ i ½1 expð2qc jZi jÞ: 4jZi j
ðA:7Þ
Let us notice that in region of large positive Zi ðZi q1 c Þ, where (1a)–(1c) equations are valid, the expression (A.7) transforms into expression (1b) for the potential energy V ðRi Þ. Let us simplify the (A.6) equation. We have [11] Z p expðix cos uÞdu ¼ pJ0 ðxÞ; 0
where J0 ðxÞ is Bessel function of I kind and of zero order. Thus Z qc e W ðRi ; Rj Þ ¼ ei ej dq exp q jZi j þ Zj 0 J0 qjqi qj j : ðA:8Þ The integral in (A.8) is calculated analytically, when jZi j þ jZj j q1 c . In this case we have [11]
Let us notice that in region of large positive Zi , Zj , where (1a)–(1c) equations are valid, the expression (A.9) transforms into expression (1c) for the energy W ðRi ; Rj Þ. Let us find out now the h^sp–el operator. Coming out from its definition we can write h^sp–el ¼
Z
d3 r (
Z
e
b ðrÞ b þ ðrÞ C d3 r0 C 2
X
) 0 1 2 eq Cq c ðq; rÞcðq; r0 Þ
b ðr0 Þ: b þ ðr0 Þ C C
q
ðA:10Þ bþ
b ðrÞ; C ðr Þ operators and Commuting first the C b ðrÞ; C b ðr0 Þ operators in (A.10) and taking the then C integral on r0 in the item, which arise during these operations and which contains delta function d3 ðr r0 Þ, finally we get, taking into account (A.2) and (A.3), Z ^ b þ ðrÞ Ve ðrÞ C b ðrÞ hsp–el ¼ d3 r C Z Z 1 b þ ðr0 Þ b þ ðrÞ C d3 r d3 r0 C þ 2 b ðr0 Þ C b ðrÞ: e ðr; r0 Þ C ðA:11Þ W 0
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