Chapter 9. The excitation process

Transcription

1 Chapter 9 The exitation proess qualitative explanation of the formation of negative ion states Ne and He in He-Ne ollisions an be given by using a state orrelation diagram. state orrelation diagram is drawn in order to analyse the behaviour of the potential urves of the diatomi He-Ne moleule, when the internulear distane varies. It exhibits the qualitative features of the relative energies of the orbitals as a funtion of. elevant exitation mehanisms, leading to speifi exit hannels, an be indiated in this diagram. efore the orrelation diagram is introdued in setion 9.2, the various types of oupling mehanisms will be desribed in a semilassial approah: radial oupling, leading to transitions between states with the same Λ (Λ is the projetion of the total eletron angular momentum on the internulear axis) and rotational oupling, leading to transitions between different Λ. 9.1 adial and rotational oupling Coupling mehanisms are treated in many textbooks. Here we will only give a brief outline, for further information the reader is referred to e.g.. H. ransden [57] The diatomi moleules an be desribed properly in a frame of referene fixed in spae if an atomi basis is used (see e.g. hapter 4). However, moleular orbitals are usually alulated in a frame of referene fixed with respet to the internulear axis, whih is taken as the axis of quantization as shown in figure 9.1. The heavy partile motion an be treated lassially beause the De roglie wavelengths of the nulei are short. In ontrast, the 65

2 The exitation proess eletroni motion must be treated quantum mehanially. The time dependent Shrödinger equation for the eletroni moleular wave funtion reads: ψ([s i ],(t)) t = H e ([s i ],t)ψ([s i ],(t)) (9.1) [s i ] with H e = T e + V (9.2) T e = i V = Z Z 1 2m 2 s i (9.3) i with the redued mass ( Z x i + Z ) + r i i j 1 2 x i x j (9.4) m = M + M M + M +1 1 (9.5) with all quantities in atomi units (a.u.). The square brakets [s i ]imply position vetors s 1, s 2,...,s N as illustrated in figure 9.1. Expanding the - e 1 r x 1 1 s x2 1 s2 - e 2 r2 z v C - - e e N-1 N Figure 9.1 The ollision system and its oordinates. The internulear line is defined as the z-axis. C indiates the entre of masses of and. wave funtion ψ in some omplete set of eletroni wave funtions ϕ k yields ψ([s i ],)= k k (t)ϕ k ([s i ],t). (9.6) 66

3 9.1 adial and rotational oupling The oeffiients follow from the Shrödinger equation 9.1 : i k(t) = ( ϕ k H e ϕ l i ϕ k ) t t ϕ l l (t) (9.7) l The time differentiation of the wave funtions ϕ l, whih depend only impliitly on t via (t), an be separated as ( ) ( ) d d dϑ d t = dt d + (9.8) dt dϑ with (t) expressed in its polar oordinates (t) andϑ(t) (see figure 9.2). The quantum mehanial operators P and L an be introdued: v(t) b (t) (t) Figure 9.2 System of polar oordinates (t) andϑ(t) in whih the rotation of the internulear axis is desribed. P = i d d L = i d (9.9) dϑ with L the omponent of L perpendiular to the plane of sattering. s a result the differential equation beomes i k(t) t = l ( ϕk H e ϕ l + d dt ϕ k P ϕ l + dϑ dt ϕ k L ϕ l ) l (t) (9.10) We distinguish the adiabati basis φ k and the diabati basis χ k. 67

4 The exitation proess The adiabati basis φ k : The wave funtions φ k are the eigenfuntions of the eletroni Hamiltonian H e for fixed internulear distane. Sine the line is an axis of symmetry for H e,theφ k are also eigenfuntions of the omponent L z of the eletroni orbital angular momentum in the diretion of the z-axis: L z φ k = ±Λ k φ k (9.11) If only one eletron is onsidered then the projetion is expressed by the small letter λ. The eigenvalues of Λ and λ areexpressedasσ,π,,φ... or σ, π, δ, φ..., orresponding to the eigenvalues 0, 1, 2, 3,... The matrix φ k H e φ l in equation 9.10 beomes diagonal on this adiabati basis set. This basis set is onvenient when the relative veloity is low, so that the eletrons an adapt adiabatially. However, at internulear distanes where the energy differene E kl between the adiabati wave funtions is small, transitions may take plae via the d dt ϕ k P ϕ l or dϑ dt ϕ k L ϕ l oupling terms. The first oupling term desribes radial oupling beause of the weight fator d/dt. SineP is a salar operator, the eletron angular momentum projetion Λ is onserved. s a result radial oupling desribes only transitions between states with the same Λ. The seond oupling term desribes rotational oupling beause of the weight fator dϑ/dt. L an be expressed in terms of the lowering and raising operator L ± with respet to the internulear axis, so that the matrix elements of L an only be non-zero if Λ k Λ l = ±1. Therefore rotational oupling ours only between states whih differ in Λ by one. adial oupling will in general our at relatively large beause of a large value of d/dt, in ontrast to rotational oupling, whih beomes inreasingly important for 0wheredϑ/dt is dominant. The adiabati potential urves are desribed by W k () = φ k H e φ k, (9.12) with W k () the eigenvalue of the eletroni Hamiltonian H e in whih enters parametrially. In general two adiabati potential urves of the same Λ do not ross aording to the Wigner non-rossing rule as illustrated in figure

5 9.1 adial and rotational oupling Λ energy Λ Figure 9.3 urves. The adiabati (solid line) and diabati (dashed line) potential The diabati basis χ k : Near an avoided rossing the radial oupling term beomes large and hanges very rapidly. eause of these features, it is often better to make an unitary transformation of the adiabati basis funtions to a new basis, alled a diabati basis, in whih the radial oupling term is smoothly varying or zero. However, the transformation will be hosen so that symmetries are preserved; a σ orbital transforms to a mixture of σ orbitals, a π orbital to amixtureofπ orbitals and so on. In the new basis the eletroni Hamiltonian is not diagonal, but it is possible to ensure that it beomes diagonal in the limit of, with elements equal to the energies of the asymptoti atomi levels. suitable diabati basis χ k is obtained when the diabati potential urves are interpolations of the adiabati potential urves in the onsidered rossing region. Therefore diabati potential urves of the same Λ do ross as indiated in figure 9.3. It should be mentioned that φ k and χ k only differ in the oupling region. n example of a set of diabati wave funtions is the set of atomi eigenstates desribed in the body-fixed frame. diabati presentation is useful in ase of fast ollisions, in whih the eletrons are too slow with respet to d/dt, so that the eletrons have no time to adapt to the hanging internulear vetor. 69

6 The exitation proess 9.2 The use of orrelation diagrams The population of exit hannels in atom-atom ollisions an be understood by onsidering the possible transitions between the involved hannels. Transitions our at internulear distanes where potential urves beome nearly degenerate with or ross the inident potential urve. y orrelating the moleular states in the separated-atom limit with moleular states in the united-atom limit a state orrelation diagram is obtained. Suh a state orrelation diagram exhibits the exitation mehanisms of the onsidered exit hannels. state orrelation diagram an be onstruted from a one-eletron orrelation diagram. In a one-eletron orrelation diagram the one-eletron moleular orbitals are skethed as a funtion of. Diabati potential urves with the same λ, i.e. the same symmetry, an ross eah other, while in ase of adiabati potential urves the non-rossing rule is valid. Fano and Lihten [58] disussed a model of the behaviour of moleular one-eletron orbitals in the H + 2 system based on diabati potential energy urves. For asymmetrial systems the model was modified by arat and Lihten [59]. For hetero-nulear systems, as He-Ne, the one-eletron diabati orrelation diagram is shown in figure 9.4. This diabati one-eletron orrelation diagram is onstruted by relating the most strongly bound state in one limit with the strongest bound state in the other limit with the following restritions: 1. The angular momentum projetion λ is onserved. 2. The number of radial nodes n = n l 1 of the radial wave funtions is in both limits the same. 3. In ase of homonulear ollision systems a third restrition holds: the parity must be onserved. The state orrelation diagram an be onstruted by filling the onsidered moleular states with the relevant one-eletron orbitals for = 0or, taking into aount the Pauli exlusion priniple. This means that eah σ orbital may ontain maximum two eletrons, and eah π, δ, φ,... orbital maximum four eletrons. Consider for example the inident He(1s 2 )- Ne(2p 6 ) hannel. ording to the diagram this hannel orresponds to the one-eletron diabati state onfiguration: He[(1sσ) 2 ]+Ne[(1sσ) 2 (2sσ) 2 (2pσ) 2 (2pπ) 4 ]. This states orrelates with the united atom state: Mg (1s 2-2s 2 2p 6 3d 2 ) orresponding to the onfiguration Mg[(1sσ) 2 (2sσ) 2 (2pσ) 2 (2pπ) 4 70

7 9.2 The use of orrelation diagrams n n 4f 4d 4p 4s 3d 3p 3s 2p 2s 3d 3d 3p 3p 3s 3s 2p 2p 2s 2s 1s 1s 0 1s Figure 9.4 The diabati one-eletron orrelation diagram for a hetero-nulear system aording to arat and Lihten [59]. The nulei in the separated atom limit are labelled as and, whih beome nuleus C in the united atom limit. The nulear harge Z is supposed to be larger than Z. The irled rossings indiate avoided rossings in ase of adiabati potential urves. 0 0 (3dσ) 2 ]. In order to explain the formation of negative ion states by using a state orrelation diagram, it is important to know whih other proesses play a role. In the energy loss spetra measured in the time of flight experiments of renot et al [60] the following main proesses are distinguished: a. Elasti sattering: He + Ne. b. One-eletron exitations of helium or neon: He +NeorHe+Ne. 71

8 The exitation proess. Simultaneous exitations of helium and neon: He +Ne. d. Simultaneous ionization of helium and neon: He + +Ne +. e. Single helium ionization (weakly): He + +Ne. utoionizing states are not observed in the eletron spetra, as onfirmed by Gerber et al [42]. However, the hannels, orresponding to the autoionizing neon states in the separated-atom limit, play a vital role in the exitation of the negative ion exit hannels and are therefore onsidered. For the various possible exit hannels the orresponding relevant diabati state orrelation lines, that desribe the possible evolution of the ollision system, an be obtained. These state orrelation lines omprise the so alled state orrelation diagram. For the He-Ne system it is shown in figure 9.5. The -series dissoiate into He + Ne ; the -series into He + Ne; the C series into He + Ne ;thed series into He +Ne. The exit hannels ontaining the negative ion states He (1s2s 2 )+Ne + (2p 5 )andhe + (1s) + Ne (2p 5 3s 2 ) are energetially loated in the range of simultaneously exited states He +Ne. The negative ion exit hannels are nearly degenerate in the asymptoti region with E = 0.19 ev. The rossings between the inident hannel and the two rossed bands of the and C series indue a double eletron promotion for two 2pσ neon eletrons. renot et al estimated that the multirossing pattern ranges between 1.2 <<1.5 a.u. etween 3 <<4a.u. thec and D series ross eah other, leading to simultaneously exited He +Ne states. In the same region the diabati orbitals leading to the negative ion states, ross the C series. ording to the one-eletron orrelation diagram both exit hannels [He (1s2s 2 ) ( 2 S)+Ne + (2p 5 )] and [He + (1s)( 2 S)+Ne (2p 5 3s 2 )] an be haraterized as Σ or Π states, depending on the onfiguration of the neon 2p 5 subshell. The Σ states possess a (2pσ)(2pπ 4 ) onfiguration, and ouple with Mg(2p 5 3l3l 3l ) in the united atom limit. The Π states possess a (2pσ 2 )(2pπ 3 ) onfiguration and unite to Mg(2p 4 3l3l 3l 3l )when 0. If rotational oupling is assumed to be small for >3 a.u. two mehanisms are oneivable in forming the negative ion states: The inident He-Ne Σ hannel ouples radially with the Σ hannels in the C-series between = a.u. seond radial oupling ours for >3 a.u., leading to the onsidered Σ exit hannels.

9 9.2 The use of orrelation diagrams Mg (2p 3p 3d ) Mg (2p s 3p ) ++ 5 Mg (2p 3d) Mg (2p 3d ) + 5 Mg (2p 3s 3d ) 5 2 Mg (2p 3p 3d ) 5 2 Mg (2p 3p 3d) Mg (2p 5 3s3p3d) 5 2 Mg (2p 3s 3p) + Mg (2p 6 3d) 6 2 Mg (2p 3d ) + 6 Mg (2p 3s) 6 Mg (2p 3s3d) C series D series series series 2 He (1s ) + Ne + (2p 5 3s) He (1s ) + Ne (2p 3s ) He (1s) + Ne (2p ) + 5 He (1s) + Ne (2p 3s) He (1s 2s ) + Ne (2p ) He + - (1s) + Ne (2p 53s 2) He (1s2s) + Ne (2p 5 3s) + 6 He (1s) + Ne (2p ) He (1s ) + Ne (2p ) 6 He (1s2s) + Ne (2p ) 2 5 He (1s ) + Ne (2p 3s ) 6 2 Mg (2p 3s ) He (1s ) + Ne (2p ) Figure 9.5 diabati state orrelation diagram for the He-Ne system. The energy levels are presented shematially. The irled rossings indiate radial oupling transitions. The,, C and D-series dissoiate into He + Ne ;He +Ne;He+Ne ;He +Ne, respetively. The exit hannels ontaining the negative ion states He (1s2s 2 )+Ne + (2p 5 )andhe + (1s) + Ne (2p 5 3s 2 ) are energetially loated in the range of simultaneously exited states He +Ne. In the determination of the relative positions of some energy levels the studies of N. ndersen and J. Østgaard Olsen [56] and S. Mengali and. Moia [61] are used. 73

10 The exitation proess 2. nother possible mehanism is that the Σ inident hannel ouples rotationally with a Π state in the C series, after whih the Π negative ion exit hannels an be populated by radial oupling for >3a.u. In both mehanisms the promotion of two 2pσ neon eletrons is ruial. 74