Chapter 3. Theory of measurement

Transcription

1 Chapter. Introduction An energetic He + -ion beam is incident on thermal sodium atoms. Figure. shows the configuration in which the interaction one is determined b the crossing of the laser-, sodium- and ion-beam, which are directed along the -, - and-ais, respectivel. The XUV photons, emitted from helium singlet states, are detected at a certain angle. In kev ion-atom collisions the interaction affects mainl the electron orbital angular momenta, while interactions with nuclear spin and electron spin can be neglected. Therefore the shape and orientation of the charge cloud of the active electron are important parameters in electron transfer processes. The isotropic charge cloud of the sodium s electron becomes anisotropic if polaried laser light ecites the electron to the p orbital as sketched in figure.. The p electron can be oriented and aligned in a certain wa depending on the direction of the laser polariation vector. This gives us a powerful tool to prepare specific p electron clouds and investigate the effects on the electron transfer probabilit. For eample, a charge cloud can be aligned along the ion beam or perpendicular to it. However, the actuall realised alignment might be limited b the presence of fine and hperfine structures. As a consequence, the sodium state cannot be described b a pure p,p or p state in case of continuous laser ecitation. This means that the maimum alignment is reduced because the angular momentum of an initial state, sa p, is redistributed over the p,p and p orbitals b couplings between the orbital angular momentum and the electron and nuclear spin. However, if the stead state after redistribution is known, cross

2 XUV photon θ=54.7 o He + beam φ=45 o laser beam sodium beam Figure. Schematic view of the configuration, defining the -,- and- ais and the spherical co-ordinates θ and φ. In the eperiment as plotted here, linearl polaried laser light along the -ais is applied, providing a netto alignment of a sodium p electron orbital in the -direction. Photons are detected at the so-called magic angle sections for electron capture can be determined as if onl a single p,p or p orbital was populated. We carr out five sub-eperiments, b varing the preparation of the sodium atoms in a certain wa. In one sub-eperiment onl ground state sodium atoms are involved. In the other four also ecited sodium atoms are involved, characteried b a specific laser polariation: a linear polariation along the ion beam, a linear polariation perpendicular to the ion beam, a left circular polariation and a right circular polariation. Each sub-eperiment results in a characteristic miture of s, p, p and p states as can be shown b considering the smmetries in the laser-sodium sstem. The s state is alwas present because onl a part of the sodium atoms can be ecited to the p orbital. The laser preparation of sodium atoms is described in section.. To determine electron transfer cross sections in collisions between the pre- 4

3 . The sstem of He atoms pared sodium atoms and incident energetic He + ions, the population rate of the relevant helium neutral states must be measured. In our case we select transfer into the XUV emitting singlet helium states. The He electron distribution is in general not isotropic but contains a certain alignment and orientation, depending on the different smmetries and dnamics of the collision sstem. These smmetries depend on the specific sodium preparations in the several sub-eperiments. Therefore the XUV photon intensit, measured in a certain direction, is in general not simpl proportional to the population rate of corresponding He states but contains also parameters caused b the anisotrop. But when the photon intensit is detected at the magic angle the influence of these parameters is ecluded. In the following the polariation matri is introduced to epress the relation between the emitted XUV photons and the initial state of a singlet helium atom. B using a spherical tensor epansion it is shown how the emitted radiation depends on the population, the orientation and alignment of the considered helium state. Afterwards the magic angle is introduced in section..4. The preparation of the sodium atoms is discussed in section.. Finall, epressions for relative total cross sections are presented in section.4.. The sstem of He atoms.. The polariation matri We will focus on the production rate of He singlet states as a function of impact energ when Na(p) atoms are prepared in different was. The question arises how the population of these states is linked to the intensit of the emitted radiation. Consider an ensemble of helium atoms in a certain singlet He state with principal quantum number n and total electron angular momentum L. This ensemble can be completel described as an epansion in the magnetic substates nlm. The spin function is the same for all singlet states (S =) and will be omitted in the following. The densit operator of the state with particular n and L can be written as: ρ he = MM nlm nlm (.) 5

4 so that nlm ρ he nlm is equal to the densit of helium atoms in the magnetic sub-state M. Consider a radiating transition between an upper level n u L u M u and a lower level n l L l M l due to the interaction Hamiltonian V = µ E(r) (.) with µ the electric dipole moment of the atom and E the electric field operator. The transition probabilit per unit time for deca from state n u L u M u to state n l L l M l, emitting a photon with polariation ε in a solid angle dω can be deducted from the Golden Rule of Fermi. The result reads [8] ω a(m l M u ; ε)dω = 8π ε hc n ul u M u µ ε n l L l M l dω (.) with hω the transition energ and ε the permittivit in vacuum. In the absence of eternal magnetic and electric fields, the states n u L u M u and n l L l M l are degenerate in their magnetic sub-states and the emitted photons do not reveal the values of M l and M u. The total intensit can be determined b summing over the possible values M l and taking the average at the different M u values with n u L u M u ρ he n u L u M u as weighting factors. Off-diagonal elements n u L u M u ρ he n u L u M u give rise to interference terms so that the total power of the emitted radiaton per unit volume and unit solid angle with a detected polariation ε becomes I(ε) = ω 4 8π ε c M l M u,m u n l L l M l µ ε n u L u M u n u L u M u ρ he n u L u M u n u L u M u µ ε n l L l M l This epression can be rewritten in a compact form I(ε) =ε C ε (.4) in which the Cartesian matri C is called the polariation matri. Herein C is defined as C ω4 8π ε c Tr l µ lu ρ he µ ul (.5)

5 . The sstem of He atoms with Tr l the trace over the sub-states n l L l M l and µ ul the electric-dipole matri with (L u +) (L l + ) electric dipole elements between the substates. The distribution of the photon emission is determined b the polariation matri C []. In the present eperimental setup the photons are detected irrespective of their polariation. The intensit angular distribution can be deduced from equation.4 as I =TrC u C u (.) in which the unit vector u =(sinθ cos φ, sin θ sin φ, cos θ) specifies the direction of detection (see also fig..). The anisotropic distribution can var as a function of the collision energ and target preparation, so that the photon intensit in a certain direction depends not onl on the population rate but also on anisotrop terms as presented in the net section... Epansion in multipoles In general an anisotrop of the polariation matri arises from the anisotrop of a densit matri ρ he according to equation.5. B epanding the densit matri in spherical tensors T kq (L ul u ) (see also appendi A.) ρ he = kq ρ kq T kq (L ul u ) q = k, k +,...,k (.7) k =,,...,L u important phsical features as population, orientation and alignment are epressed. The monopole ρ is proportional to the total population of the ecited He state, while the dipole components ρ σ are proportional to the epectation values of the spherical components of angular momentum operator L u, defining the orientation of the ensemble. The five quadrupole moments ρ q are equal to the epectation values of the elements of a smmetrical Cartesian matri with ero trace, for instance [] L u L u L u I (.8) epressing the alignment of the state. B substituting the epansion of a densit matri.7 in equation.5 and 7

6 appling the Wigner-Eckhart theorem and some standard recoupling techniques the polariation matri can also be epanded as [] C = kq c kq S kq (.9) with k restricted to, and. The operators S kq canbeepressedas matrices on a Cartesian basis as given in table A. in appendi A. It turns out that a component c kq is proportional to ρ kq with a factor independent of q. Therefore, the c parameter is a direct measure of the total population of the He magnetic sub-states, while the c q parameters determine the orientation and the c q parameters epress the alignment. The intensit distribution is completel determined b the components ρ kq for k =, and, thus multipoles with k>donot influence the features of spontaneous emission and do not appear in the polariation matri. The polariation matri can be epanded in monopole, dipole and quadrupole terms C = C + C + C (.) which on a Cartesian basis are given b c C = c (.) C = C = c i c i c (c + c ) i (c + c )+ c (c + c ) i (c c ) (c c ) (.) i (c c ) (c c ) i (c c ) (c + c )+ c i (c + c ) c (c c ) i (c + c ) (.) in which C is proportional to the unit matri determined b one parameter. C represents the antismmetrical part of C and is determined b three parameters. C represents the smmetrical part with ero trace and is determined b five parameters... Sstem smmetries As far as the determination of cross sections is concerned onl the population of neutralied helium states is of interest, i.e. the determination of the 8

7 . The sstem of He atoms parameter c. However, the anisotrop parameters influence the photon intensit distribution. The question arises which anisotrop terms influence the photon intensit distribution. For our collision sstem a number of multipoles c kq, appearing in the polariation matri, is interdependent or ero as a consequence of the sstem smmetries. The detector is mounted at such a distance from the collision one that each created He neutral is observed equall well, regardless of its scattering angle. In addition the sodium atoms ma be assumed homogeneousl prepared in the collision volume. As a result the total collision sstem He + -Na(s, p) contains certain smmetries which are conserved during the collision due to certain rotational and parit invariancies of the interaction Hamiltonian. As a result the densit operator of the ecited helium states ρ he contains at least the same smmetries as the initial total sstem. In the determination of the total sstem smmetries it is important how the sodium charge cloud is aligned and oriented in each sub-eperiment. The initial situations of the several sub-eperiments are sketched in figure. in which the following total sstem smmetries can be distinguished: I II III IV laser switched off, isotropic sodium charge cloud. Rotation smmetr around -ais, reflection smmetr in arbitrar plane containing the -ais. linearl polaried laser light along -ais, sodium charge cloud aligned along -ais. Smmetries as in I linearl polaried laser light along -ais, sodium charge cloud aligned along -ais. Reflection smmetr in - plane and - plane. right circularl polaried laser light, sodium charge cloud rotating right. Reflection smmetr in - plane. V left circularl polaried laser light, sodium charge cloud rotating left. Smmetries as in IV The sstem smmetries characteriing the sub-eperiments can be epressed b the invariance of the densit matri in the corresponding smmetr operations Re (α)ρ he R e (α) = ρ he rotation α around -ais (.4) 9

8 I II III IV/ V Figure. Collision sstems for the five different sodium preparations: I. no laser, Na(s) II. linearl polaried laser light parallel to -ais, Na(p) III. linearl polaried laser light parallel to -ais, Na(p) IV +V. right and left circularl polaried laser light, Na(p) PRe (π)ρ he R e (π)p = ρ he reflection in - plane (.5) PRe (π)ρ he R e (π)p = ρ he reflection in - plane (.) in which R and P are the rotation and parit operator. If these smmetries are taken into account the polariation matrices are simplified (see appendi A.): C I,II = c + c c + c c c (.7)

9 . The sstem of He atoms C III = c c + c c + c + c (.8) c c C IV,V = c c + c i(c + c ) c + c + c i(c c ) c c (.9) in which all coefficients are real ecept the pure imaginar c coefficient in situation IV and V. It must be emphasied that a c kq coefficient can differ for each sub-eperiment and must actuall be labelled to the considered subeperiment. Onl the polariation matri of situation IV is transformed into the matri of situation V b reflection in the - plane resulting in a change of sign for the value of c and of c, while the other coefficients remain the same...4 The magic angle Smmetr considerations limit the number of independent and non-ero multipoles. Nevertheless, still certain coefficients are present and influence the measured intensities. According to equation. and matrices.7-.9, the intensities for each situation become: I I,II = c c ( cos θ) I III = c c ( cos θ) c sin θ( cos φ) (.) I IV,V = c c ( cos θ) c sin θ( cos φ)+ ic sin θ sin φ If the detector is placed such that radiation is observed at one of the magic angles defined b cos θ = ± and cos φ = ±, then the intensities are proportional to ρ in sub-eperiment I, II and III, i.e. proportional to the population of the ecited helium atoms. In the eperimental setup the intensit is measured at one of these so-called magic angles in the direction (,, ). When intensities in situation IV or V are detected at this magic angle, the quadrupole parameter is still present in the term ic /.

10 . The sstem of prepared sodium atoms.. The fine- and hperfine structure Before discussing the effective densit matri of the ensemble of partl ecited sodium atoms, we consider the relevant energ levels and transition rules in a sodium atom. Because of electron spin-orbit and electron-nucleus coupling the energ levels ehibit fine and hperfine structures as shown in figure. for the transitions S / P / and S / P /. When electric dipole transitions are P/ P/ 55 GH F= 59 MH F= F= F= F= 89 MH linear polariation - + F= nm F= S/ 77 MH circular polariation - + F= Figure. Na. The hperfine splitting of the S /, P / and P / levels of

11 . The sstem of prepared sodium atoms considered the following rules must be obeed: F =, ± but F g = F e = is forbidden (.) M F = absorption & stimulated emission linearl polaried light M F = ± M F =, ± absorption & stimulated emission circularl polaried light spontaneous emission In. the quantiation ais is defined along the polariation vector in case of linear radiation and parallel to the wave vector in case of circular radiation. Using laser light from a single frequenc de laser with a tpical bandwidth of MH, the ecitation is hperfine sensitive. It turns out that if linearl or circularl polaried light is used, at least three polaried beams are needed in order to achieve a stationar ecited-state population without losing electrons to levels which are not involved in the ecitation process. This applies to all transitions ecept when the laser frequenc is tuned to the transition S /,F g = P /,F e =. In this case no electrons will be trapped and a final stationar state is achieved as indicated in figure.. Remarkable is the circular case which leads to a pure two level sstem: onl left or right circularl polaried light is absorbed and emitted. The densit matri can be described in the F -picture on the basis of hperfine sub-states FM F of the P /, P / and S / states. However, the fine and hperfine structure character of the sodium target will not affect the electron transfer since the interactions with the nucleus and electron spin are relativel small in the ion-atom interaction Hamiltonian. Besides, the collision time ( 5 s) is much shorter than a precession period in electron-nucleus coupling, characteried b the inverse of the frequenc splittings between the hperfine structure levels ( 9 s). It is even shorter than a precession period in the electron spin-orbit coupling, characteried b the inverse of the frequenc splitting between the fine structure levels ( s), so that nuclear and electron spin can be assumed stationar during the short interaction time. Therefore the densit matri of the sodium targets is projected on the angular momentum states of the electron orbit LM L in the L-picture. This can be done b averaging over the angular momentum states of the nuclear spin I and electron spin S, i.e. taking the trace of the densit matri over the electron and nuclear spin states, resulting in a reduction of the polariation. The result of this projection procedure can be clarified b considering the

12 smmetries in the photon-atom sstem. Due to the interaction Hamiltonian V = µ E(r) the photon states are coupled with the free atomic sodium states. This leads to a complicated ecitation mechanism with absorption, stimulated emission and spontaneous emission of radiation with transition probabilities proportional to F i M Fi µ ε l F j M Fj in which ε l the polariation vector of the laser beam [4,5]. After a short time (.µs) stationar conditions are achieved with maimum population of the p orbital before a sodium atom enters the collision one. If we epand the reduced sodium densit matri in spherical tensors with multipole coefficients ρ kq ρ na = ρ kq T kq (LL), (.) as we did before in case of ρ he, then smmetr considerations reduce here also the number of independent and non-ero components. Figure.4 shows the initial smmetries of the photon-atom sstem in case of linear ecitation and circular ecitation. Smmetries of the initial total densit matri of the photon-atom sstem will be conserved in time and hold in the reduced sodium densit matri in the L-picture. If the ecited part of sodium linear circular Figure.4 The initial smmetries of the laser-sodium sstem in case of linear and circular polariation. atoms is considered, i.e. L =, then the spherical tensors T kq () on the Cartesian set of basis functions have eactl the same form as the matrices S kq as presented in table A. (see also equation A. in appendi A). The Cartesian basis functions are given as 4

13 . The sstem of prepared sodium atoms θ, φ p =sinθ cos φ θ, φ p =sinθ sin φ (.) θ, φ p =cosθ The squared functions represent the directional probabilit densit of the sodium valence electron as pictured in figure.5. p* p p* p p* p Figure.5 The directional probabilit densities of the outer sodium electron for pure p, p or p states... Linear polariation In case of linear polariation along the -ais (sub-eperiment II) the rotation smmetr around the -ais and reflection smmetr in the - and - plane impl a reduced densit matri for Na(p) atoms (see table A. in appendi A) ρ na;ii = ρ + ρ ρ + ρ ρ ρ Normaliing this densit matri to the fraction of ecited atoms n p /(n p + n s )=α l ields ρ na;ii = α l r r r (.4) (.5) 5

14 with r a real parameter. A simple rotation transformation gives the matri for the situation in which the linear polariation is oriented along the -ais, ielding ρ na;iii = α l r r r (.) In case of optimal pumping on transition S / (F =) P / (F =) stead state conditions are evaluated for a linear polariation vector along the -ais b Fischer and Hertel []: p : p : p = 9 : 9 : 5 9 This implies a theoretical value r = 9. optimal and 9 r. (.7) In practice the alignment is not.. Circular polariation In case of a circular polariation, rotation smmetr around the -ais (ρ =, ρ = ρ ) and reflection smmetr in the - plane result in a densit matri ρ na;iv,v = ρ + 4 ρ ρ ρ iρ iρ ρ ρ (.8) B pumping with circularl polaried light, mainl transitions between S / M F = ±to P / M F = ± occur, once the stead state is reached. The absorption and emission of photons with circular polariation in the -direction implies a densit matri of the form ρ na;iv,v = α c i ± i (.9) where the matri is normalied to the fraction of ecited sodium atoms α c. Computer simulations confirm the epression, even for such laser intensities that onl a small fraction of atoms is ecited.

15 .4 Phsical quantities..4 Fluorescence measurements The fluorescence of the ecited atoms is detected b a photodiode, placed in the - plane (φ =9 ), perpendicular to the laser beam. If linearl polaried light is used the intensit is proportional to α l ( r sin θ ( r)cos θ)=α l ( + f sin θ) (.) according to equation.. Herein θ is the angle between the linear polariation vector and the detection direction and f ( r)/r. B variation of θ the maimum and minimum intensities can be determined and b division I ma =+f = r I min r (.) the parameter r can be derived. In the case of optimal r = 9 the theoretical value for f becomes 4. Measurements showed a parameter f =.9 ±., slightl lower than the theoretical value. The disadvantage of using a photodiode is that also light of the surrounding region is measured. Therefore another method can be emploed as described in the net section..4 Phsical quantities On the Cartesian basis, the ecited sodium densit matrices are diagonal in situation I, II and III. The absence of off-diagonal interference terms implies that the sub-sstem can be considered as an incoherent superposition of the p,p and p orbitals, with the diagonal terms as weight factors. The remainig fraction of ( α) sodium atoms occupies the ground state s. Therefore, the quantities σ,σ,σ and σ are defined as the total cross sections for electron capture in the considered helium states after collision with Na in a pure s, p,p or p state, respectivel. However, no distinction between σ and σ can be epected because the are phsicall the same in the eperimental setup so that σ = σ σ,. Due to the incoherent epansion of sodium p orbitals and the XUV detection at the magic angle the measured intensities are related to the cross section as follows: I I = Nσ (.) I II = N (α l ( r)σ + α l rσ, +( α l )σ ) (.) 7

16 I III = N (α l rσ + α l ( r)σ, +( α l )σ ) (.4) The factor N depends on the flu of helium ions, densit of sodium atoms, the interaction volume and the efficienc of the XUV detector. In situation IV and V off-diagonal elements are present in the reduced sodium densit matrices and as a consequence interference terms between the p and p states are involved in the transfer process. If we consider the situation in which both left and right circularl polaried light ecite the sodium ensemble simultaneousl, the reduced sodium densit matri is equal to the addition of the densit matrices in case of sub-eperiment IV and V according to the superposition principle. This densit matri contains no off-diagonal elements and describes the sodium state as an incoherent superposition of the p and p state. At the same time the added photon intensities depend onl on the population of He states because the anisotrop c term vanishes, according to equation.. Therefore I IV + I V = N (α c σ + α c σ, +( α c )σ ) (.5) B measuring the different situations in succession for short periods ( seconds) in several series (-4) at a specific ion energ, the factor N can be supposed to be equal for each sodium preparation. From these equations an important energ-independent epression can be derived I II + I III r(i II +I III ) I IV + I V = α l α c ( r) (.) with the condition that α l /α c. This condition is satisfied in case of simultaneous pumping of the two hperfine ground state levels (F g =and F g =) as shown in the net chapter. The relationship makes it possible to fit r without using the photodiode. The advantage is that the fitted r is onl related to the interaction region and not to the surrounding area. Also the ratio α l /α c can be etracted, so that α l is estimated b calculating α c. Furthermore, the epressions of the relative cross sections can be deduced: σ = (r )I II +ri III +( r)( α l )I I (.7) σ (r )α l I I σ, σ = ri II +(r )I III +( r)( α l )I I (r )α l I I (.8) 8

17 .4 Phsical quantities If the intensities I IV and I V are subtracted, onl the c quadrupole term is left I IV I V = 4ic. (.9) The phsical interpretation of c follows from the epression (.4) for I(ε) where the polariation vector ε is epressed in Cartesian coordinates. The c quadrupole moment is similar to the difference in intensit of emitted radiation, each with its specific linear polariation, i.e. I [ ] (,, ) I [ ] (,, ) = ic. (.4) Consequentl, a non-ero c indicates a certain average tilt of the helium electron cloud after collision. The tilt angle can be epressed b multipole coefficients if the total densit matri can be determined. When the He(sd) D eit channel is neglected, the helium densit matrices possess tensors with maimum rank k =. Thus, if the complete polariation matri is measured, then the densit matri of each P helium eit channel is determined (see.9): ρ P = ρ ρ + ρ ρ + ρ + ρ i(ρ ρ ) i(ρ + ρ ) ρ ρ The directional probabilit densit of the p-electron becomes P (θ, φ) = ψ θ, φ θ, φ ψ = A p + A p + A p + A p p (.4) with A = ρ ρ + ρ A = ρ + ρ + ρ A = ρ ρ A = iρ (.4) keeping in mind that ρ is an imaginar coefficient. A possible ecitedelectron distribution in the He atom is sketched in figure.. The cloud is 9

18 + He He* v θ v t before interaction after interaction Figure. An eample of the directional probabilit densit of the helium p electron in the He(sp) or He(sp) states just after collision with sodium atoms, which are prepared with circularl polaried laser light. In the - plane the tilt angle θ t is indicated. tilted in the plane of reflection smmetr, defined b φ =9. The tilt angle can be derived b P(θ, φ =9 ) θ =(A A )sinθ t + A cos θ t = (.4) θt so that iρ tan θ t = ρ +. (.44) ρ However, the quadrupoles c and c are not determined at the magic angle, onl the ratio ic = ( ) IIV I V (.45) c I IV + I V is measurable as a function of the collision energ.