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Two-electron-one-photon transitions in polycenter systems
Volume 104, number 2,3
CHEMICAL PHYSICS LETTERS
3 February 1984
TWO-ELECTRON-ONE-PHOTON TRANSITIONS IN POLYCENTER SYSTEMS Alexander A. BEREZIN * Department of Engineering Physics, McMaster University, Hamilton, Ontario, Canada LSS 4M1 Received 18 October 1983;in final form 22 November 1983
A system of four potential wells with two bound electrons is considered. For specific combinations of spatial locations and depths of the wells, the two-electron-one-photon transitions (TEOPTs) can provide the main channel for the spontaneous relaxation of the excited-state configuration into the ground-state level. This is distinctly different from the case of compact atomic systems where TEOPTs are several orders of magnitude weaker than two-electron-two-photon transitions.
1. Introduction One o f the questions o f principle interest for the theory o f energy transitions in quantum mechanics can be formulated in the following form: Is it possible to suggest a realistic situation in which there is a significant probability that the simultaneous transitions o f two or more spatially separated electrons will result in the emission o f only one " c o m m o n " photon? The existence o f correlated multielectron transitions in atomic spectra has been predicted by Heisenberg as early as in 1925 [1], but the experimental detection of these transitions has been reported only recently [2,3]. Cited papers by W61fli et al. [2] and Salem et al. [3], both entitled "Two-electron--one-photon transitions...", discuss a simultaneous two-electron jump accompanied by the emission o f only one photon. Two points regarding these papers should be mentioned in the context o f the present work: (I) Simultaneous transition o f both electrons occurs within the compact atomic system, i.e. both electrons are spatially very close to each other when they jointly generate one common photon. (II) Alternative de-excitation channels in the form o f either two separate one-electron transitions (when * Supported by the Natural Sciences and Engineering Research Council of Canada. 226
both electrons jump independently and non-synchronously) or two-electron-two-photon transition (jumps o f both electrons are simultaneously and correlated) are also allowed and even with greater (by a factor o f 104-105 ) probability than the correlated two-electron one-photontransition (TEOPT). Here I will construct an example o f a two-electron system in which this entire situation is, in a sense, reversed; namely: (I*) Both jumping electrons remain distinctly spatially separated when their correlated jumps generate one common photon, and moreover, (II*) (1) The corresponding one-electron transitions are absolutely prohibited by the energy conservation law; (2) the TEOPT becomes the main channel for the decay of excited state o f the system because, (3) the two-electron-two-photon transitions have, apparently, lower (and not higher, as in the case o f compact atomic systems) probability than TEOPTs.
2. The model: four-site-two-electron array To serve the purpose, let us consider a four-sitetwo-electron array [4,5]. This is a system o f four potential wells (trapping sites) with two electrons residing on them (fig. 1). Generally, four wells may be spatially arranged in a variety o f ways and need not even lie in the same plane. For simplicity, however, let us assume that all four wells are positioned along the 0 009-2614/84/$ 03.00 @Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 104, n u m b e r 2,3 A
B
?
CHEMICAL PHYSICS LETTERS C
?
D
~--~x
I
Fig. 1. Linear array o f four potential wells for which trapping energies and inter-site distances are such that t h e simultaneous tunneling o f b o t h electrons is the only open channel for s p o n t a n e o u s transition from t h e excited configuration into the g r o u n d state level.
same line (x-axis) with distances between them being a, b, and c. Here, o f course, we are talking about a three-dimensional and not a one-dimensional problem: we have four "spherically symmetrical" potential wells with their centers located on a common line. The site energies associated with these wells (i.e. the binding energies o f an extra electron) are CA, eB, eC, and eD, respectively;the sites A and C have trapped electrons while sites B and D are empty. The consideration which follows has a general quantum mechanical character. However, for the purpose of concreteness, I will mention two cases o f the possible physical realization. One can think o f these wells as being four different atoms capable o f forming negative ionswith four different binding energies. Their relative positions can be stabilized either by the environment (in the case o f solid state-type problems) or could be changing in a negligibly slow (adiabatic) manner in the case o f slow collision problems. The very spread of the site energies may also be associated with the outside potential sources, especially in the case when these wells are inserted into a molecular or crystalline matrix. The very formation o f the four-site-two-electron level-reversed configuration (an example o f which is shown in fig. 1) can occur as a result o f the ionization (photo, thermal, etc.) o f a relatively deeper lying level (in our example site B) if the site D has already been ionized. One can also think o f a simultaneous (consecutive) ionization o f both levels (B and D), e.g. in the field of an intense (laser) electromagnetic wave. Any o f four (geometrically conceivable) one-electron jumps o f the two electrons can change the total energy o f the system since (a) site energies are generally different, and (b) interelectron Coulomb repulsion depends on the inter-site distance, which is different for various pairs o f filled sites. I will show now
3 February 1984
that some combinations o f a, b, c, CA, eB, eC, and e D satisfy the above conditions (I*) and (II*) and, therefore, are able to make the TEOPT the principle channel of energy relaxation. Let us designate the differences o f site energies as A 1 = e A -- e B and A 2 = e D -- e C and assume, for a further simplification, that a = b = c and e A = e C (both electrons initially occupy equi-energetic levels). Due to the Coulomb repulsion between the electrons on sites A and C it is easy to verify that if A 1 > q2/a q2/2a = q2/2a (we are using atomic units) the one-electron tunnel transition A -+ B is energetically allowed and can proceed spontaneously as a radiativetunnel transition [4,5]. Alternatively, if A 2 < q2/2a -- q2/3a = q2/6a the spontaneous upward jump of the second electron (C -+ D) also becomes energetically allowed. Opposite inequalities will energetically prohibit these two jumps (and also the C ~ B jump). The jump A -+ D is energetically prohibited anyway. Therefore, to forbid all four one-electron transitions'let us take q2/6a < A 2 < A 1 < q2/2a,e.g. A 1 = q2/3a and A 2 = q2/5a. At the same time, the simultaneous tunnel jump o f both electrons (A ~ B and C -+ D) does not change the energy o f the coulombic repulsion (the Final distance between the electrons remains unchanged) but, nevertheless, releases the energy AE = A 1 -- A 2. Note that a similar, although more complicated, chain o f unequalities could be established for the general case, when a 4=b 4:c, e A 4=ec, and the geometry o f the four-site array is non-linear or even non-planar. Therefore, we arrived at the situation when all four one-electron transitions (A -+ B, A ~ D, C -> B and C ~ D) are individually energetically prohibited and yet the total energy o f the system can be spontaneously decreased by the correlated transition of both electrons jumping simultaneously. In other words, the initial state o f the system as a whole is, in fact, an excited state which can relax into the genuine ground state (electrons on sites B and D) only through the simultaneous correlated tunnel transition o f both participating electrons. Now, let us verify that this double transition has a non-zero matrix element and, consequently, the emission of just one "joint" photon is allowed in the firstorder perturbation theory. The wavefunction of the initial (excited) state (before the TEOPT occurs) o f two-electron-four227
Volume 104, number 2,3
CHEMICAL PHYSICS LETTERS
site system has the structure •Iti(rl, r2) = ~bA(rl)~C(r2) -+ ~A(r2) ~C(rl)
(1)
and reflects the presence o f electrons on sites A and C. The wavefunction o f the final (ground) state (after the TEOPT is completed) should be taken as orthogonalized to xPi, i.e. q~f(rl,/'2) = ~bB(rl ) ~bD(r2) ± ~bB(r2) ~bD(rl ) + K ~ i ( r l , r2) ,
(2)
where K (K < 1) is the orthogonalization factor. The two-electron dipole matrix element (~ilrl + r 2 Iq~f) with xlt i and xitf has a non-zero value, i.e. the TEOPT is allowed as the first-order transition (cf. ref. [6] for the similar "atomic" case). In our "polycenter" case the order o f magnitude of the probability o f the TEOPT is comparable to the probability o f the radiative tunnel transition [4,5] o f a single electron jumping over the joint distance a + c.
3. Discussion: on the N-electron transitions The probability o f the two-electron-two-photon transition (each o f the two simultaneously jumping electrons emits "its o w n " photon) has, apparently, the same (or, perhaps, even smaller) order o f magnitude as for the TEOPT. The calculation o f their branching ratio is, however, a separate problem for further investigation; we can mention papers [ 6 - 8 ] dealing with the same problem for the case o f "compact" atomic systems. In the present paper we considered the four-site array in which all one-electron transitions are strictly prohibited and the de-excitation o f an excited configuration can relax only through the simultaneous double jump o f both electrons. The non-zero matrix elements for the TEOPT sub-channel implies that the two jumping electrons may parent one common pho- " ton (despite the fact that they remain spatially well separated!). Acting similarly, one can consider a system o f N electrons distributed over 2 N trapping sites. For properly selected geometry and depths o f the potential wells, this polycenter system will possess the specific N-electron configurational excited state. This excited state can be such that from it all 1-, 2-, ..., and N - 1-electron transitions into the genuine ground 228
3 February 1984
state level will be energetically prohibited. The only channel o f the spontaneous energy deactivation will, therefore, consist of the correlated simultaneous jump o f all N electrons with a significant relative probability that this collective jump will produce only one " c o m m o n " photon. According to Goudsmit and Gropper [9] the maximum value o f N is restricted; namely N < 3. In what follows I will, however, offer an example in which N can be any arbitrary number. Consider an array o f 2 N trapping centers placed on the vertices of a regular planar 2N-polygon. Assume now that N electrons are trapped, on sites 1, 3, 5, ..., 2 N - 1, respectively. This configuration (configuration X) corresponds to the ground state o f the system. This ground state, however, will be doubly degenerate since the configuration with electrons on site 2, 4, ..., 2 N (configuration Y) has the same energy. Let us now remove this degeneracy by the radial outward shift o f one unfilled (even) vertex (e.g. vertex with number 2) by the small distance a (a < the side o f polygon). The total energy o f the electrostatic repulsion in the configuration Y is now smaller than in the configuration X, i.e. Y is now the "genuine" ground state configuration, while X constitutes the "first excited state". The transition X + Y is possible only through the simultaneous (clockwise or counterclockwise) jump o f all N electrons from odd to even sites, since all other partial moves (involving 1,2 ..... or N - 1 electrons) would result in the increase o f the total coulombic repulsion. In analogy with section 2, one may suggest that such a correlated N-electron transition could result in the emission of either one " c o m m o n " photon or several separate photons. Similar examples o f N-electron transitions can be based on a variety o f non-planar arrays of trapping sites.
References [1] W. Heisenberg, Z. Physik 32 (1925) 841. [2 ] W. WSlfli, Ch. Stoller, G. Bonani, M. Suter and M. StSckli, Phys. Rev. Letters 35 (1975) 656. [3] S.I. Salem, A. Kumar and B.L. Scott, Phys. Letters 97A (1983) 100. [4] A.A. Berezin, Phys. Rev. Letters 50 (1983) 1520. [5] A.A. Berezin, Phys. Letters 97A (1983) 105. [6] S.I. Salem, A. Kumar, B.L. Scott and R.D. Ayers, Phys. Rev. Letters 49 (1982) 1240. [7] T. Aberg, K.A. Jamison and P. Richard, Phys. Rev. Letters 37 (1976) 63. [8] H.P. Kelly, Phys. Rev. Letters 37 (1976) 386. [9] S. Goudsmit and L. Gropper, Phys. Rev. 38 (1931) 225.
CHEMICAL PHYSICS LETTERS
3 February 1984
TWO-ELECTRON-ONE-PHOTON TRANSITIONS IN POLYCENTER SYSTEMS Alexander A. BEREZIN * Department of Engineering Physics, McMaster University, Hamilton, Ontario, Canada LSS 4M1 Received 18 October 1983;in final form 22 November 1983
A system of four potential wells with two bound electrons is considered. For specific combinations of spatial locations and depths of the wells, the two-electron-one-photon transitions (TEOPTs) can provide the main channel for the spontaneous relaxation of the excited-state configuration into the ground-state level. This is distinctly different from the case of compact atomic systems where TEOPTs are several orders of magnitude weaker than two-electron-two-photon transitions.
1. Introduction One o f the questions o f principle interest for the theory o f energy transitions in quantum mechanics can be formulated in the following form: Is it possible to suggest a realistic situation in which there is a significant probability that the simultaneous transitions o f two or more spatially separated electrons will result in the emission o f only one " c o m m o n " photon? The existence o f correlated multielectron transitions in atomic spectra has been predicted by Heisenberg as early as in 1925 [1], but the experimental detection of these transitions has been reported only recently [2,3]. Cited papers by W61fli et al. [2] and Salem et al. [3], both entitled "Two-electron--one-photon transitions...", discuss a simultaneous two-electron jump accompanied by the emission o f only one photon. Two points regarding these papers should be mentioned in the context o f the present work: (I) Simultaneous transition o f both electrons occurs within the compact atomic system, i.e. both electrons are spatially very close to each other when they jointly generate one common photon. (II) Alternative de-excitation channels in the form o f either two separate one-electron transitions (when * Supported by the Natural Sciences and Engineering Research Council of Canada. 226
both electrons jump independently and non-synchronously) or two-electron-two-photon transition (jumps o f both electrons are simultaneously and correlated) are also allowed and even with greater (by a factor o f 104-105 ) probability than the correlated two-electron one-photontransition (TEOPT). Here I will construct an example o f a two-electron system in which this entire situation is, in a sense, reversed; namely: (I*) Both jumping electrons remain distinctly spatially separated when their correlated jumps generate one common photon, and moreover, (II*) (1) The corresponding one-electron transitions are absolutely prohibited by the energy conservation law; (2) the TEOPT becomes the main channel for the decay of excited state o f the system because, (3) the two-electron-two-photon transitions have, apparently, lower (and not higher, as in the case o f compact atomic systems) probability than TEOPTs.
2. The model: four-site-two-electron array To serve the purpose, let us consider a four-sitetwo-electron array [4,5]. This is a system o f four potential wells (trapping sites) with two electrons residing on them (fig. 1). Generally, four wells may be spatially arranged in a variety o f ways and need not even lie in the same plane. For simplicity, however, let us assume that all four wells are positioned along the 0 009-2614/84/$ 03.00 @Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 104, n u m b e r 2,3 A
B
?
CHEMICAL PHYSICS LETTERS C
?
D
~--~x
I
Fig. 1. Linear array o f four potential wells for which trapping energies and inter-site distances are such that t h e simultaneous tunneling o f b o t h electrons is the only open channel for s p o n t a n e o u s transition from t h e excited configuration into the g r o u n d state level.
same line (x-axis) with distances between them being a, b, and c. Here, o f course, we are talking about a three-dimensional and not a one-dimensional problem: we have four "spherically symmetrical" potential wells with their centers located on a common line. The site energies associated with these wells (i.e. the binding energies o f an extra electron) are CA, eB, eC, and eD, respectively;the sites A and C have trapped electrons while sites B and D are empty. The consideration which follows has a general quantum mechanical character. However, for the purpose of concreteness, I will mention two cases o f the possible physical realization. One can think o f these wells as being four different atoms capable o f forming negative ionswith four different binding energies. Their relative positions can be stabilized either by the environment (in the case o f solid state-type problems) or could be changing in a negligibly slow (adiabatic) manner in the case o f slow collision problems. The very spread of the site energies may also be associated with the outside potential sources, especially in the case when these wells are inserted into a molecular or crystalline matrix. The very formation o f the four-site-two-electron level-reversed configuration (an example o f which is shown in fig. 1) can occur as a result o f the ionization (photo, thermal, etc.) o f a relatively deeper lying level (in our example site B) if the site D has already been ionized. One can also think o f a simultaneous (consecutive) ionization o f both levels (B and D), e.g. in the field of an intense (laser) electromagnetic wave. Any o f four (geometrically conceivable) one-electron jumps o f the two electrons can change the total energy o f the system since (a) site energies are generally different, and (b) interelectron Coulomb repulsion depends on the inter-site distance, which is different for various pairs o f filled sites. I will show now
3 February 1984
that some combinations o f a, b, c, CA, eB, eC, and e D satisfy the above conditions (I*) and (II*) and, therefore, are able to make the TEOPT the principle channel of energy relaxation. Let us designate the differences o f site energies as A 1 = e A -- e B and A 2 = e D -- e C and assume, for a further simplification, that a = b = c and e A = e C (both electrons initially occupy equi-energetic levels). Due to the Coulomb repulsion between the electrons on sites A and C it is easy to verify that if A 1 > q2/a q2/2a = q2/2a (we are using atomic units) the one-electron tunnel transition A -+ B is energetically allowed and can proceed spontaneously as a radiativetunnel transition [4,5]. Alternatively, if A 2 < q2/2a -- q2/3a = q2/6a the spontaneous upward jump of the second electron (C -+ D) also becomes energetically allowed. Opposite inequalities will energetically prohibit these two jumps (and also the C ~ B jump). The jump A -+ D is energetically prohibited anyway. Therefore, to forbid all four one-electron transitions'let us take q2/6a < A 2 < A 1 < q2/2a,e.g. A 1 = q2/3a and A 2 = q2/5a. At the same time, the simultaneous tunnel jump o f both electrons (A ~ B and C -+ D) does not change the energy o f the coulombic repulsion (the Final distance between the electrons remains unchanged) but, nevertheless, releases the energy AE = A 1 -- A 2. Note that a similar, although more complicated, chain o f unequalities could be established for the general case, when a 4=b 4:c, e A 4=ec, and the geometry o f the four-site array is non-linear or even non-planar. Therefore, we arrived at the situation when all four one-electron transitions (A -+ B, A ~ D, C -> B and C ~ D) are individually energetically prohibited and yet the total energy o f the system can be spontaneously decreased by the correlated transition of both electrons jumping simultaneously. In other words, the initial state o f the system as a whole is, in fact, an excited state which can relax into the genuine ground state (electrons on sites B and D) only through the simultaneous correlated tunnel transition o f both participating electrons. Now, let us verify that this double transition has a non-zero matrix element and, consequently, the emission of just one "joint" photon is allowed in the firstorder perturbation theory. The wavefunction of the initial (excited) state (before the TEOPT occurs) o f two-electron-four227
Volume 104, number 2,3
CHEMICAL PHYSICS LETTERS
site system has the structure •Iti(rl, r2) = ~bA(rl)~C(r2) -+ ~A(r2) ~C(rl)
(1)
and reflects the presence o f electrons on sites A and C. The wavefunction o f the final (ground) state (after the TEOPT is completed) should be taken as orthogonalized to xPi, i.e. q~f(rl,/'2) = ~bB(rl ) ~bD(r2) ± ~bB(r2) ~bD(rl ) + K ~ i ( r l , r2) ,
(2)
where K (K < 1) is the orthogonalization factor. The two-electron dipole matrix element (~ilrl + r 2 Iq~f) with xlt i and xitf has a non-zero value, i.e. the TEOPT is allowed as the first-order transition (cf. ref. [6] for the similar "atomic" case). In our "polycenter" case the order o f magnitude of the probability o f the TEOPT is comparable to the probability o f the radiative tunnel transition [4,5] o f a single electron jumping over the joint distance a + c.
3. Discussion: on the N-electron transitions The probability o f the two-electron-two-photon transition (each o f the two simultaneously jumping electrons emits "its o w n " photon) has, apparently, the same (or, perhaps, even smaller) order o f magnitude as for the TEOPT. The calculation o f their branching ratio is, however, a separate problem for further investigation; we can mention papers [ 6 - 8 ] dealing with the same problem for the case o f "compact" atomic systems. In the present paper we considered the four-site array in which all one-electron transitions are strictly prohibited and the de-excitation o f an excited configuration can relax only through the simultaneous double jump o f both electrons. The non-zero matrix elements for the TEOPT sub-channel implies that the two jumping electrons may parent one common pho- " ton (despite the fact that they remain spatially well separated!). Acting similarly, one can consider a system o f N electrons distributed over 2 N trapping sites. For properly selected geometry and depths o f the potential wells, this polycenter system will possess the specific N-electron configurational excited state. This excited state can be such that from it all 1-, 2-, ..., and N - 1-electron transitions into the genuine ground 228
3 February 1984
state level will be energetically prohibited. The only channel o f the spontaneous energy deactivation will, therefore, consist of the correlated simultaneous jump o f all N electrons with a significant relative probability that this collective jump will produce only one " c o m m o n " photon. According to Goudsmit and Gropper [9] the maximum value o f N is restricted; namely N < 3. In what follows I will, however, offer an example in which N can be any arbitrary number. Consider an array o f 2 N trapping centers placed on the vertices of a regular planar 2N-polygon. Assume now that N electrons are trapped, on sites 1, 3, 5, ..., 2 N - 1, respectively. This configuration (configuration X) corresponds to the ground state o f the system. This ground state, however, will be doubly degenerate since the configuration with electrons on site 2, 4, ..., 2 N (configuration Y) has the same energy. Let us now remove this degeneracy by the radial outward shift o f one unfilled (even) vertex (e.g. vertex with number 2) by the small distance a (a < the side o f polygon). The total energy o f the electrostatic repulsion in the configuration Y is now smaller than in the configuration X, i.e. Y is now the "genuine" ground state configuration, while X constitutes the "first excited state". The transition X + Y is possible only through the simultaneous (clockwise or counterclockwise) jump o f all N electrons from odd to even sites, since all other partial moves (involving 1,2 ..... or N - 1 electrons) would result in the increase o f the total coulombic repulsion. In analogy with section 2, one may suggest that such a correlated N-electron transition could result in the emission of either one " c o m m o n " photon or several separate photons. Similar examples o f N-electron transitions can be based on a variety o f non-planar arrays of trapping sites.
References [1] W. Heisenberg, Z. Physik 32 (1925) 841. [2 ] W. WSlfli, Ch. Stoller, G. Bonani, M. Suter and M. StSckli, Phys. Rev. Letters 35 (1975) 656. [3] S.I. Salem, A. Kumar and B.L. Scott, Phys. Letters 97A (1983) 100. [4] A.A. Berezin, Phys. Rev. Letters 50 (1983) 1520. [5] A.A. Berezin, Phys. Letters 97A (1983) 105. [6] S.I. Salem, A. Kumar, B.L. Scott and R.D. Ayers, Phys. Rev. Letters 49 (1982) 1240. [7] T. Aberg, K.A. Jamison and P. Richard, Phys. Rev. Letters 37 (1976) 63. [8] H.P. Kelly, Phys. Rev. Letters 37 (1976) 386. [9] S. Goudsmit and L. Gropper, Phys. Rev. 38 (1931) 225.