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Integral properties of channel interference in resonant X-ray scattering
5 February 1996
PHYSICS LETTERS A ELSFWIER
Physics Letters A 211 (1996)
101-108
Integral properties of channel interference in resonant X-ray scattering Faris Gel’mukhanov a, Hans Agren a, Matthias Neeb b, Jan-Erik Rubensson c, Andreas Bringer ’ a Institute of Physics and Measurement Technology, Linkiiping Universiry, S-58183 Linkiiping. Sweden b Fritz-Haber-lnsritut der Max-Planck-Gesellschaft, Faradayweg 4-6, D-141 95 Berlin, Germany c IFE Forschungszentrum Jiilich, D-52425 Jiilich, Germany Received 14 September 1995; revised manuscript received 13 November
1995; accepted for publication 21 November
1995
Communicated by B. Fricke
Abstract The integral properties of channel interference in resonant X-ray scattering (RXS) are formally investigated. The qualitative difference between vibrational and electronic state lifetime interference is elucidated. It is shown that the integral interference term in the RXS cross section summed over all intermediate and final states is exactly equal to zero. A few experimental cases are given as illustration. As a result of numerical simulations it is found that the integral interference term is also close to zero if only certain finite sets of intermediate and final states are taken into account.
1. Introduction The development of tunable, narrow-band synchrotron radiation sources has stimulated studies of resonant Xray scattering (RXS) by atoms, molecules, and solids. Resonances are accomplished via virtual core excitations that decay by emission of X-ray photons (radiative RXS [ l-3] ) or Auger electrons (nonradiative RXS [ 4-61). There are involved physical and mathematical relationships between these two qualitatively different phenomena [ 71. Common to both scattering processes are that the core excited states constitute the intermediate states and that by virtue of the short lifetime of these intermediate states, the RXS channels can interfere strongly. It has been demonstrated on many occasions that interference can play a crucial role in the formation of the X-ray fluorescence [7-141 and Auger spectra [ 12-141 leading to both distortions and shifts of the emission bands. Many of the predictions have been confirmed experimentally [ 6,15-171. In spite of numerous investigations of interference effects in RXS processes an investigation of integral properties of the interference contribution to the RXS cross section is still lacking. Intuition prompts that integrated over all channels the interference contribution to the total intensity of the RXS process should be equal to zero. Formally, this statement is associated with certain sum rules of a nontrivial nature. Experimentally, such sum rules are of importance when analyzing data with prominent interference effects. The main aim of the present study is to investigate the integral properties of the interference term in the RXS cross section, and to explore under what conditions this term integrated over final particle energies equals zero. 0375-9601/96/$12.00
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SSDIO375-9601(95)00919-l
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reserved
F. Gel’mukhanov
102
et d/Physics
Letters
A 211 (19%)
101-108
The paper is organized as follows. A general investigation of the interference contribution to the RXS cross section is given in Section 2. Here we show that only the total interference term summed over all radiative and nonradiative X-ray scattering channels is equal to zero. This result, albeit being important from a theoretical point of view, is of little practical value because of the difficulty summing the RXS cross section over all radiative and nonradiative scattering channels. In Section 3 we investigate the partial integral interference term for interference between scattering channels through the vibronic levels of an intermediate electronic state. This term is equal to zero as it is shown here. In Section 4 we investigate the integral properties of the partial interference terms under X-ray scattering through electronic states. We show that the partial interference term taking into account an incomplete set of intermediate and final electronic states is not necessarily equal to zero. The interference effects and the integral behavior of the interference term are demonstrated for some realistic cases in Section 5.
2. Structure of the resonant X-ray scattering cross sections An X-ray photon with frequency or solid) radiatively,
w and polarization
vector e can be scattered by target A (atom, molecule
w+A-+A;,+A*+w’,
(1)
or nonradiatively w+A--+A;,-+A++ek.
(2)
The states involved in these scattering processes are the ground Im), and the final state In). The frequency and polarization vector by e’ and w’, the energy of the emitted Auger electron is given F,(z) (w) and nonradiative RXS amplitude F,,,(N)(w> are the sum FcN)(w) O”lll ’ respectively [7,13,18],
state lo), the intermediate core-excited state of the emitted final X-ray photon are denoted by ek. The resonant radiative RXS amplitude of partial scattering amplitudes F,(z,j (w) and
The absorption transition o ---t m is caused by the dipole interaction V = e +D between the X-ray photon and the molecule. The operator Q is the Coulomb interaction responsible for the Auger decay [7,14] in the nonradiative case and Qnln = e’ *D,, for radiative RXS, D,, and w,, are the dipole matrix element and the resonant frequency of the transition n -+ m. The double differential cross section of radiative ( 1) or nonradiative (2) RXS is defined by the square of the RXS amplitude (3), &+R,N’ -=
F,',R,N'12S(w Cl n
dEd0
The lifetime broadening r,,,cc
E -
w,,) .
r,, of the intermediate
(4) core excited state Im) is
J cn lQmn12. d0
Here E is the energy of the final particle scattered into the solid angle d0 in the case of nonradiative RXS. In the radiative case E is replaced by the frequency w’ of the final X-ray photon. The Dirac S-function reflects the energy conservation law for the RXS process. For simplicity we neglected in Eq. (4) the lifetime broadening
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Physks Letters A 211 (1996) 101-108
103
r,, of the final state In). Both the radiative and nonradiative RXS cross sections (4) are the sum of direct and interference terms d2&N)
dir
=
dg d0
c
flddir(@,n)&m
- e - %o),
n
d2a!R.N) m =
de d0
flint(m9n)6(w
c
-
E -
Wno).
(6)
”
Here we introduced the partial integral direct and interference cross sections
(7) integrated over the final particle energy E and corresponding to the scattering channel o --j n to a certain final state n. The interference effects discussed in Section 2 are associated with the second cross section in Eq. (6). In the following we will investigate the integral properties of the interference contribution and elucidate the conditions under which it sums to zero. 2. I. Optical theorem Let us begin to discuss this question from a strict result connected with an optical theorem for the scattering amplitude [ 191. As is well known the total RXS cross section coincides with the X-ray absorption cross section, do’dOde’$&
CT0 =
J
+Jdedo$$
(8)
where the summation is over all Auger electron spins, s. The optical theorem reflects the particle conservation law or, more formally, the unitarity of the scattering matrix [ 191 expressed as aa
= 4lr _ P
Im
F,‘,R’
(x
le Dom’2rm
1 ’ m (w-wd2+G
(9)
where FJf) is the forward elastic X-ray scattering amplitude, and p is the momentum of the X-ray photon. As one can see from this equation and from expression (5) for the lifetime broadening r,,, the interference between different X-ray scattering channels is absent in the total RXS cross section (9),
Thus, the total interference term (10) in the RXS cross section summed over all radiative and nonradiative scattering channels is exactly equal to zero. In practice it is difficult to sum over all radiative and nonradiative RXS channels and to check this sum rule. In the following sections we derive simpler sum rules for the interference term. 3. Vibrational
interference
Let us consider the radiative or nonradiative RXS in molecules. We consider the coherent radiative excitation of the molecule from the lowest vibrational level (0) of the ground electronic state and a group of close lying
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vibrational states Im) of a single intermediate core excited electronic state, and the radiative or nonradiative decay into vibrational levels In) of one final electronic state. The RXS cross section is in accordance with the Frank-Condon (FC) principle given by the following well known equation [ I l-131, d*a( w, E) ded0
=a0
c
I?,
(olm)(mln) * s(w - E - A, - w,,,). w - A - o,,, + iT,,
(11)
Here E is the energy of the final particle (X-ray photon in the radiative RXS case and Auger electron in the nonradiative RXS case); (mln) are the FC factors; w, = Ev,lb - qib, w,,, = I$’ - Glib are the differences between the vibrational energies of the intermediate and the ground electronic states and between vibrational energies of the final and the intermediate electronic states, respectively; all quantities unessential for our consideration are collected into the constant go. A and A, are the energies of electron transitions between the ground state and intermediate core excited state and between the intermediate core excited state and final electronic states, respectively. Like Eq. (6) the RXS cross section ( 11) integrated over the final particle energy E can be presented as the sum of the direct and the interference terms,
(T(w) =
s
ded2a(w,E) de d0
= gdir(W)
+ %t(m).
(12)
To show that the integral of the interference term (12) over the final particle energy E is equal to zero we integrate the RXS cross section ( 11) over E. Taking into account the condition of completeness, C,, In)@\ = 1, and the orthononnality condition, (mlm’) = Sn,,n,~, for the vibronic wave functions of the intermediate core excited electronic state one arrives at the following expression for the integral RXS cross section,
a(w) =ua(w)
1(4m)l* =u,c 1), (~-A-~w,,o,*+f;~’
(13)
This equation shows that the integral RXS cross section (T(O) coincides with the photoabsorption cross section a,(o) for electron vibrational transitions between ground and core excited states. This result can be compared to the general result (8)) (9) based on the optical theorem. There are, however, essential differences between Eq. ( 13) and the optical theorem (S), (9). Contrary to Eqs. (8), (9), Rq. ( 13) is valid if we sum the RXS cross section only over all vibrational states ]m) and In) of one intermediate core excited electronic state and one final electronic state (thus not summing over all final electronic states nor summing over polarizations of final photons or over spins of the Auger electron). Eq. ( 13) shows indirectly that in the case of vibrational interference, the contribution of the interference term in the integral RXS cross section is equal to zero. This result can also be derived explicitly, 6nt(u)
E
J
dcd2gint(a,e) = u* de d0
c
kWWl~Hml4 a~m, (w-A-o,~,+iT,,~)(w-A--omo-iT,)
This term is equal to zero due to the orthogonality core excited state: (m’lm) = 0 for m’ # m.
of the different vibrational
=‘. wave functions
(14)
of the intermediate
4. Electronic interference Let us now consider the partial RXS cross section connected with the radiative RXS or only with the nonradiative RXS. Starting from Eqs. (3)) (6) and following the method described in the previous section one
F. Gel’mukhanovet al./Physics Letters A 211 (19%) 101-108
can obtain the following expression for the interference (final X-ray photon or Auger electron), a;;*N)(W)
z
J
“‘;:g ‘)
&
K
term integrated
105
over the energy of the final particle
(~lvl~‘)(m’lQQl~)(~IVl~) c mZm,(w - wd, + ird) (u - w,,, - ir,,> + O’
(15)
Here we used the approximate condition of completeness for final electron states In)(n] = 1 (this condition is strict only for the total interference term (+int( o) = gj,,,’ (o) + a$’ (w) ) . The indices o, rn and n now denote the ground, intermediate and final electronic states for radiative or nonradiative RXS, w,,*, is the frequency of the electronic transition 10) + Im). Now the interference term ( 15) is not equal to zero because
b’lQQlm) + 0
(16)
for m' # m. As one can see from Eq. ( 10) only the total interference term ainr( w) = gin:’ (w) + a!“:)(o), summed over final photon polarizations and spins of the the Auger electrons and integrated over the solid angle for the scattered final particles, is equal to zero. In the next section we show that the integral interference term ( 15) is close to zero also if only a certain finite set of scattering channels is taken into account.
5. Examples
of vibronic and electronic state lifetime interference
The importance of interference effects in the nonradiative RXS around the Is-‘rs excitation in molecular 02 has been demonstrated earlier [ 171. In this resonance the spacings between the vibrational excitations are comparable to the lifetime broadening, and the vibrational fine structure cannot be resolved in the excitation spectrum. For clarity we have plotted the vibrational profile with a reduced lifetime broadening in Fig. la, assuming a perfectly monochromatized incoming photon beam with an energy (arbitrarily) set to o = 7. The resulting partial integral cross sections
c+(w, n)
= adir(w,
n)
+ @iint(w,
(
n)
17)
and gdtr( w, n) (7) of the nonradiative RXS to the vibrational states n of the final lowest ionic state are shown in Fig. 1b. We note that the widths of the peaks are determined by the monochromator function and the widths of the final states only. The FC factors (elm)and (mln)were calculated using Rydberg-Klein-Rees potentials with experimentally determined spectroscopic constants. The grey bold lines in Fig. lb show the integral cross section a( w, n) ( 17) as a function of energy position of different scattering channels o + n. The thin dark line shows the integral decay spectrum rini( 0, n) (7) taking no interference into account. We see a dramatic vibronic lifetime interference effect, which redistributes intensity from the low to the high kinetic energy side of the spectrum. As it was shown in Section 3 only the integral interference term summed over all intermediate and final vibrational states is equal to zero. To check this important statement we calculated the partial sum of c+i,t(w, n) (7) over of the final vibrational states lying below N (n = 0,. . . , N),
(18) In accordance
with the optical theorem
(+int(W]N) N ~i,t(~)
= 0,
(14) we expect to obtain for sufficiently
N>> 1.
large N (19)
The upper part of Fig. lb shows the integral of the interference term as a function of energy position of the scattering channel u -t N with lowest kinetic energy. One can see from this N-dependence that in spite of
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Letters A 211 (19%)
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Photon Energy [eV]
0, decay
spectrum exuted
a,~,=7
~~__-. 513
514
515
516
517
$=0.18
eV)
,
,
516
519
r_I
-510’
520
Kinetic Energy [eV] Fig. I. (a) Is-’ 1~: vibronic excitation spectrum of 02. The bold line shows the band shape using a realistic lifetime broadening (I’) of 0.18 eV. To make the vibrational substructure obvious, the dotted line gives the result of using a lifetime width of only 0.05 eV. The decay spectrum in (b) has been excited at urn= 7 by a perfectly monochromatized photon beam. (b) The grey bold lines show the X2& final state transitions described by a(w, n) ( 17) using a lifetime broadening of f = 0.18 eV for the intermediate Is-’ ilrs state. The thin dark lines show the decay spectrum given by qdir(U, n) (7) taking no interference into account. The difference between the grey and dark lines gives the pure interference contribution. The integral of the interference contributions Ui”r(O/ N) ( 19) is shown as a full curve in the top panel (b) as the function of energy position of “highest” scattering channel o -+ N.
gtni( w, n) # 0 the integral interference term g’nr( WIN) tends to zero for large N in accordance with the optical theorem. This demonstrates that even in a case with substantial interference effects, the integral of interference term summed over all vibrational excitations is zero. In a partial yield experiment, however, where only part of the excitations are monitored, the intensity would be dependent on the vibronic interference. The situation is very similar for eIectronic state lifetime interference. Around the K edge in neon the absorption spectrum shows resonances assigned to Is-‘np (n = 3,4,5) transitions (Fig. 2a). In the electronic decay to 2p-*n’p states there is a high probability that n f n’, that is all the excitations reach the same final states and one may observe interference effects. In Fig. 2b we have assumed a perfectly monochromatized beam centered on the ls-‘4p resonance. The spectrum consists of 2p-*n’p configurations where n’ = 3,4,5. The 2p electrons couple to ‘D and ‘S parental terms, giving rise to six final states in total. In this case the interference effects are small since the energy spacing is large relative to the lifetime broadening. In the upper part of the panel the integrated intensities m(w,n> (17) and a&,(w,n) (7) are shown. At inter-resonance excitation, however, the interference effects can be emphasized [ 171. In Fig. 2c the monochromator energy is tuned in between the ls-‘3p and ls-‘4p resonances, resulting in large interference effects. Also in this case
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867
Photon
Deexcitation
805
869
Energy
IOI-108
107
871
[eV]
spectra of Neon
810
Kinetic Energy [eV]
Fig. 2. (a) K-shell Rydbeq excitation in neon. The lifetime broadening (r) is set to 0.31 eV. The decay spectra in (b) and (c) have been calculated with a perfectly monochromatized beam tuned in between the ls-‘3~ and Is-‘4~ resonance (inter-resonance) and at the top of the Is-‘4~ resonance (on-resonance), respectively. The decay spectra show the spectator transitions at highest kinetic energy involving two holes in the 2p outer shell. The grey broad lines result (17) when interference is taken into account; the thin dark lines include no interference contributions (see. Eq. (7) for fldir(w,n) ). The dash-dotted curve at the top of each decay spectrum shows the integral of the interference term qint( oJN) ( 19). Note that the interference contribution for sufficiently large N sums to zero both for inter- and for on-resonance excitation.
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find that the interference term integrated over the entire energy region and summed over all intermediate (m) and final (n) close lying electronic states with large oscillatory strengths is zero, and again we note that a partial yield experiment that discriminates against some of the final states gives a result at variance with the absorption probability. The result is dependent on the lifetime of the intermediate states. we
6. Discussion With the increased resolution and efficiency in resonant X-ray scattering (RXS) spectra, it has become progressively more important to understand the role of lifetime interference in such spectra. Quite much is known about the conditions under which interference effects are important, and when simplifying two-step model approaches, excluding interference effects, are appropriate for the analysis of radiative or nonradiative RXS spectra. In the present work we derived for the first time the integral properties of the interference term in RXS spectra, and showed under which conditions the total interference contributions sum up to zero. This was shown to be the general case for vibrational lifetime interference, while for the case of electronic state interference this holds only for the total radiative and nonradiative cross sections, but not for the individual radiative (or nonradiative) RXS cross sections. In practice, the integrated electronic interference contribution to the radiative RXS spectra may be close to zero depending on the precise organization of the intermediate core-excited and final states reached by the resonant X-ray excitation. This was shown to be the case for RXS spectra relating to the resonant K-shell excitations in the neon atom.
References [ I] PL. Cowan, in: Resonant anomalous X-my scattering. Theory and applications, eds. G. Materlik. C.J. Sparks and K. Fisher (NorthHolland, Amsterdam, 1994) p. 449. [ 21 J. Nordgmn, P Glans and N.D. Wassdahl, Phys. Ser. T 34 ( 1991) 100. [3] R. Mayer, D.W. Lindle, S.H. Southworth and P.L. Cowan, Phys. Rev. A 43 ( 1991) 235. [4] A. Kivim%ki,A. Naves de Brito, S. Aksela, O.-P Sairanen, A. Ausmess. S.J. Osborne, L.B. Kantas and S. Svensson, Phys. Rev. Lett. 71 (1993) 4307. [5j H. Wang, J. Woicik, T. Aberg, M.H. Chen, A. Herrena-Gomez. T. Kendelewicz, A. Mantykenttll, K.E. Miyano. S. Southworth and B. Crasemann, Phys. Rev. A 50 ( 1994) 1359. [6] M. Neeb, J-E. Rubensson, M. Biermann and W. Eberhardt. J. Electron Spectrosc. 67 ( 1994) 261. 171 T. Aberg and B. Crasemann. in: Resonant anomalous X-ray scattering. Theory and applications, eds. G. Materlik, C.J. Sparks and K. Fisher (North-Holland, Amsterdam, 1994) p. 449. [ 81 F. Gel’mukhanov and H. Agren, Phys. Rev. A 49 ( 1994) 4378. [9] E Kh. Gel’mukhanov, L.N. Mazalov and A.N. Shlyaeva, Zh. Eksp. Tear. Fiz. 71 ( 1976) 1960 (Sov. Phys. JETP 44 ( 1976) 5041. [ IO] F. Kh. Gel’mukhanov, L.N. Mazalov and A.V. Kondratenko, Chem. Phys. Lett. 46 (1977) 133. [ I1 ] E Kaspar, W. Domcke and L.S. Cederbaum, Chem. Phys. 44 (1977) 33. ( 121 A. Cesar, H. Agnm and V. Carravetta, Phys. Rev. A 40 (1989) 187. [ 131 L.S. Cederbaum and E Tarantelli, J. Chem. Phys. 98 (1993) 9691. [ 141 T.X. Carnal, SE. Anderson. L. Ungier and T.D. Thomas, Phys. Rev. Lett. 58 (1987) 867. [ 151 E.D. Poliakov, L.A. Kelly, L.M. Duffy, B. Space, P Roy, S. Southwotth and M.G. White, J. Chem. Phys. 89 (1988) 4048. [ 161 M. Neeb, J-E. Rubensson, M. Biermann, W. Eberhardt, K.J. Randall, J. Feldhaus, A.L.D. Kilcoyne, A.M. Bradshaw, 2. Xu, PD. Johnson and Y. Ma, Chem. Phys. Lett. 212 ( 1993) 205. [ 171 J.-E. Rubensson, M. Neeb, A. Bringer, M. Biermann and W. Eberhardt, to be published. [ 181 J.J. Sakurai, Advanced quantum mechanics (Addison-Wesley, Reading, MA, 1967) Ch. 2. [ 191 E Gel’mukhanov and H. krgmn, Phys. Rev. A 50 (1994) 1129.
PHYSICS LETTERS A ELSFWIER
Physics Letters A 211 (1996)
101-108
Integral properties of channel interference in resonant X-ray scattering Faris Gel’mukhanov a, Hans Agren a, Matthias Neeb b, Jan-Erik Rubensson c, Andreas Bringer ’ a Institute of Physics and Measurement Technology, Linkiiping Universiry, S-58183 Linkiiping. Sweden b Fritz-Haber-lnsritut der Max-Planck-Gesellschaft, Faradayweg 4-6, D-141 95 Berlin, Germany c IFE Forschungszentrum Jiilich, D-52425 Jiilich, Germany Received 14 September 1995; revised manuscript received 13 November
1995; accepted for publication 21 November
1995
Communicated by B. Fricke
Abstract The integral properties of channel interference in resonant X-ray scattering (RXS) are formally investigated. The qualitative difference between vibrational and electronic state lifetime interference is elucidated. It is shown that the integral interference term in the RXS cross section summed over all intermediate and final states is exactly equal to zero. A few experimental cases are given as illustration. As a result of numerical simulations it is found that the integral interference term is also close to zero if only certain finite sets of intermediate and final states are taken into account.
1. Introduction The development of tunable, narrow-band synchrotron radiation sources has stimulated studies of resonant Xray scattering (RXS) by atoms, molecules, and solids. Resonances are accomplished via virtual core excitations that decay by emission of X-ray photons (radiative RXS [ l-3] ) or Auger electrons (nonradiative RXS [ 4-61). There are involved physical and mathematical relationships between these two qualitatively different phenomena [ 71. Common to both scattering processes are that the core excited states constitute the intermediate states and that by virtue of the short lifetime of these intermediate states, the RXS channels can interfere strongly. It has been demonstrated on many occasions that interference can play a crucial role in the formation of the X-ray fluorescence [7-141 and Auger spectra [ 12-141 leading to both distortions and shifts of the emission bands. Many of the predictions have been confirmed experimentally [ 6,15-171. In spite of numerous investigations of interference effects in RXS processes an investigation of integral properties of the interference contribution to the RXS cross section is still lacking. Intuition prompts that integrated over all channels the interference contribution to the total intensity of the RXS process should be equal to zero. Formally, this statement is associated with certain sum rules of a nontrivial nature. Experimentally, such sum rules are of importance when analyzing data with prominent interference effects. The main aim of the present study is to investigate the integral properties of the interference term in the RXS cross section, and to explore under what conditions this term integrated over final particle energies equals zero. 0375-9601/96/$12.00
@ 1996 Elsevier Science
SSDIO375-9601(95)00919-l
B.V. All rights
reserved
F. Gel’mukhanov
102
et d/Physics
Letters
A 211 (19%)
101-108
The paper is organized as follows. A general investigation of the interference contribution to the RXS cross section is given in Section 2. Here we show that only the total interference term summed over all radiative and nonradiative X-ray scattering channels is equal to zero. This result, albeit being important from a theoretical point of view, is of little practical value because of the difficulty summing the RXS cross section over all radiative and nonradiative scattering channels. In Section 3 we investigate the partial integral interference term for interference between scattering channels through the vibronic levels of an intermediate electronic state. This term is equal to zero as it is shown here. In Section 4 we investigate the integral properties of the partial interference terms under X-ray scattering through electronic states. We show that the partial interference term taking into account an incomplete set of intermediate and final electronic states is not necessarily equal to zero. The interference effects and the integral behavior of the interference term are demonstrated for some realistic cases in Section 5.
2. Structure of the resonant X-ray scattering cross sections An X-ray photon with frequency or solid) radiatively,
w and polarization
vector e can be scattered by target A (atom, molecule
w+A-+A;,+A*+w’,
(1)
or nonradiatively w+A--+A;,-+A++ek.
(2)
The states involved in these scattering processes are the ground Im), and the final state In). The frequency and polarization vector by e’ and w’, the energy of the emitted Auger electron is given F,(z) (w) and nonradiative RXS amplitude F,,,(N)(w> are the sum FcN)(w) O”lll ’ respectively [7,13,18],
state lo), the intermediate core-excited state of the emitted final X-ray photon are denoted by ek. The resonant radiative RXS amplitude of partial scattering amplitudes F,(z,j (w) and
The absorption transition o ---t m is caused by the dipole interaction V = e +D between the X-ray photon and the molecule. The operator Q is the Coulomb interaction responsible for the Auger decay [7,14] in the nonradiative case and Qnln = e’ *D,, for radiative RXS, D,, and w,, are the dipole matrix element and the resonant frequency of the transition n -+ m. The double differential cross section of radiative ( 1) or nonradiative (2) RXS is defined by the square of the RXS amplitude (3), &+R,N’ -=
F,',R,N'12S(w Cl n
dEd0
The lifetime broadening r,,,cc
E -
w,,) .
r,, of the intermediate
(4) core excited state Im) is
J cn lQmn12. d0
Here E is the energy of the final particle scattered into the solid angle d0 in the case of nonradiative RXS. In the radiative case E is replaced by the frequency w’ of the final X-ray photon. The Dirac S-function reflects the energy conservation law for the RXS process. For simplicity we neglected in Eq. (4) the lifetime broadening
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Physks Letters A 211 (1996) 101-108
103
r,, of the final state In). Both the radiative and nonradiative RXS cross sections (4) are the sum of direct and interference terms d2&N)
dir
=
dg d0
c
flddir(@,n)&m
- e - %o),
n
d2a!R.N) m =
de d0
flint(m9n)6(w
c
-
E -
Wno).
(6)
”
Here we introduced the partial integral direct and interference cross sections
(7) integrated over the final particle energy E and corresponding to the scattering channel o --j n to a certain final state n. The interference effects discussed in Section 2 are associated with the second cross section in Eq. (6). In the following we will investigate the integral properties of the interference contribution and elucidate the conditions under which it sums to zero. 2. I. Optical theorem Let us begin to discuss this question from a strict result connected with an optical theorem for the scattering amplitude [ 191. As is well known the total RXS cross section coincides with the X-ray absorption cross section, do’dOde’$&
CT0 =
J
+Jdedo$$
(8)
where the summation is over all Auger electron spins, s. The optical theorem reflects the particle conservation law or, more formally, the unitarity of the scattering matrix [ 191 expressed as aa
= 4lr _ P
Im
F,‘,R’
(x
le Dom’2rm
1 ’ m (w-wd2+G
(9)
where FJf) is the forward elastic X-ray scattering amplitude, and p is the momentum of the X-ray photon. As one can see from this equation and from expression (5) for the lifetime broadening r,,, the interference between different X-ray scattering channels is absent in the total RXS cross section (9),
Thus, the total interference term (10) in the RXS cross section summed over all radiative and nonradiative scattering channels is exactly equal to zero. In practice it is difficult to sum over all radiative and nonradiative RXS channels and to check this sum rule. In the following sections we derive simpler sum rules for the interference term. 3. Vibrational
interference
Let us consider the radiative or nonradiative RXS in molecules. We consider the coherent radiative excitation of the molecule from the lowest vibrational level (0) of the ground electronic state and a group of close lying
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vibrational states Im) of a single intermediate core excited electronic state, and the radiative or nonradiative decay into vibrational levels In) of one final electronic state. The RXS cross section is in accordance with the Frank-Condon (FC) principle given by the following well known equation [ I l-131, d*a( w, E) ded0
=a0
c
I?,
(olm)(mln) * s(w - E - A, - w,,,). w - A - o,,, + iT,,
(11)
Here E is the energy of the final particle (X-ray photon in the radiative RXS case and Auger electron in the nonradiative RXS case); (mln) are the FC factors; w, = Ev,lb - qib, w,,, = I$’ - Glib are the differences between the vibrational energies of the intermediate and the ground electronic states and between vibrational energies of the final and the intermediate electronic states, respectively; all quantities unessential for our consideration are collected into the constant go. A and A, are the energies of electron transitions between the ground state and intermediate core excited state and between the intermediate core excited state and final electronic states, respectively. Like Eq. (6) the RXS cross section ( 11) integrated over the final particle energy E can be presented as the sum of the direct and the interference terms,
(T(w) =
s
ded2a(w,E) de d0
= gdir(W)
+ %t(m).
(12)
To show that the integral of the interference term (12) over the final particle energy E is equal to zero we integrate the RXS cross section ( 11) over E. Taking into account the condition of completeness, C,, In)@\ = 1, and the orthononnality condition, (mlm’) = Sn,,n,~, for the vibronic wave functions of the intermediate core excited electronic state one arrives at the following expression for the integral RXS cross section,
a(w) =ua(w)
1(4m)l* =u,c 1), (~-A-~w,,o,*+f;~’
(13)
This equation shows that the integral RXS cross section (T(O) coincides with the photoabsorption cross section a,(o) for electron vibrational transitions between ground and core excited states. This result can be compared to the general result (8)) (9) based on the optical theorem. There are, however, essential differences between Eq. ( 13) and the optical theorem (S), (9). Contrary to Eqs. (8), (9), Rq. ( 13) is valid if we sum the RXS cross section only over all vibrational states ]m) and In) of one intermediate core excited electronic state and one final electronic state (thus not summing over all final electronic states nor summing over polarizations of final photons or over spins of the Auger electron). Eq. ( 13) shows indirectly that in the case of vibrational interference, the contribution of the interference term in the integral RXS cross section is equal to zero. This result can also be derived explicitly, 6nt(u)
E
J
dcd2gint(a,e) = u* de d0
c
kWWl~Hml4 a~m, (w-A-o,~,+iT,,~)(w-A--omo-iT,)
This term is equal to zero due to the orthogonality core excited state: (m’lm) = 0 for m’ # m.
of the different vibrational
=‘. wave functions
(14)
of the intermediate
4. Electronic interference Let us now consider the partial RXS cross section connected with the radiative RXS or only with the nonradiative RXS. Starting from Eqs. (3)) (6) and following the method described in the previous section one
F. Gel’mukhanovet al./Physics Letters A 211 (19%) 101-108
can obtain the following expression for the interference (final X-ray photon or Auger electron), a;;*N)(W)
z
J
“‘;:g ‘)
&
K
term integrated
105
over the energy of the final particle
(~lvl~‘)(m’lQQl~)(~IVl~) c mZm,(w - wd, + ird) (u - w,,, - ir,,> + O’
(15)
Here we used the approximate condition of completeness for final electron states In)(n] = 1 (this condition is strict only for the total interference term (+int( o) = gj,,,’ (o) + a$’ (w) ) . The indices o, rn and n now denote the ground, intermediate and final electronic states for radiative or nonradiative RXS, w,,*, is the frequency of the electronic transition 10) + Im). Now the interference term ( 15) is not equal to zero because
b’lQQlm) + 0
(16)
for m' # m. As one can see from Eq. ( 10) only the total interference term ainr( w) = gin:’ (w) + a!“:)(o), summed over final photon polarizations and spins of the the Auger electrons and integrated over the solid angle for the scattered final particles, is equal to zero. In the next section we show that the integral interference term ( 15) is close to zero also if only a certain finite set of scattering channels is taken into account.
5. Examples
of vibronic and electronic state lifetime interference
The importance of interference effects in the nonradiative RXS around the Is-‘rs excitation in molecular 02 has been demonstrated earlier [ 171. In this resonance the spacings between the vibrational excitations are comparable to the lifetime broadening, and the vibrational fine structure cannot be resolved in the excitation spectrum. For clarity we have plotted the vibrational profile with a reduced lifetime broadening in Fig. la, assuming a perfectly monochromatized incoming photon beam with an energy (arbitrarily) set to o = 7. The resulting partial integral cross sections
c+(w, n)
= adir(w,
n)
+ @iint(w,
(
n)
17)
and gdtr( w, n) (7) of the nonradiative RXS to the vibrational states n of the final lowest ionic state are shown in Fig. 1b. We note that the widths of the peaks are determined by the monochromator function and the widths of the final states only. The FC factors (elm)and (mln)were calculated using Rydberg-Klein-Rees potentials with experimentally determined spectroscopic constants. The grey bold lines in Fig. lb show the integral cross section a( w, n) ( 17) as a function of energy position of different scattering channels o + n. The thin dark line shows the integral decay spectrum rini( 0, n) (7) taking no interference into account. We see a dramatic vibronic lifetime interference effect, which redistributes intensity from the low to the high kinetic energy side of the spectrum. As it was shown in Section 3 only the integral interference term summed over all intermediate and final vibrational states is equal to zero. To check this important statement we calculated the partial sum of c+i,t(w, n) (7) over of the final vibrational states lying below N (n = 0,. . . , N),
(18) In accordance
with the optical theorem
(+int(W]N) N ~i,t(~)
= 0,
(14) we expect to obtain for sufficiently
N>> 1.
large N (19)
The upper part of Fig. lb shows the integral of the interference term as a function of energy position of the scattering channel u -t N with lowest kinetic energy. One can see from this N-dependence that in spite of
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Photon Energy [eV]
0, decay
spectrum exuted
a,~,=7
~~__-. 513
514
515
516
517
$=0.18
eV)
,
,
516
519
r_I
-510’
520
Kinetic Energy [eV] Fig. I. (a) Is-’ 1~: vibronic excitation spectrum of 02. The bold line shows the band shape using a realistic lifetime broadening (I’) of 0.18 eV. To make the vibrational substructure obvious, the dotted line gives the result of using a lifetime width of only 0.05 eV. The decay spectrum in (b) has been excited at urn= 7 by a perfectly monochromatized photon beam. (b) The grey bold lines show the X2& final state transitions described by a(w, n) ( 17) using a lifetime broadening of f = 0.18 eV for the intermediate Is-’ ilrs state. The thin dark lines show the decay spectrum given by qdir(U, n) (7) taking no interference into account. The difference between the grey and dark lines gives the pure interference contribution. The integral of the interference contributions Ui”r(O/ N) ( 19) is shown as a full curve in the top panel (b) as the function of energy position of “highest” scattering channel o -+ N.
gtni( w, n) # 0 the integral interference term g’nr( WIN) tends to zero for large N in accordance with the optical theorem. This demonstrates that even in a case with substantial interference effects, the integral of interference term summed over all vibrational excitations is zero. In a partial yield experiment, however, where only part of the excitations are monitored, the intensity would be dependent on the vibronic interference. The situation is very similar for eIectronic state lifetime interference. Around the K edge in neon the absorption spectrum shows resonances assigned to Is-‘np (n = 3,4,5) transitions (Fig. 2a). In the electronic decay to 2p-*n’p states there is a high probability that n f n’, that is all the excitations reach the same final states and one may observe interference effects. In Fig. 2b we have assumed a perfectly monochromatized beam centered on the ls-‘4p resonance. The spectrum consists of 2p-*n’p configurations where n’ = 3,4,5. The 2p electrons couple to ‘D and ‘S parental terms, giving rise to six final states in total. In this case the interference effects are small since the energy spacing is large relative to the lifetime broadening. In the upper part of the panel the integrated intensities m(w,n> (17) and a&,(w,n) (7) are shown. At inter-resonance excitation, however, the interference effects can be emphasized [ 171. In Fig. 2c the monochromator energy is tuned in between the ls-‘3p and ls-‘4p resonances, resulting in large interference effects. Also in this case
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et ah/Physics
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Photon
Deexcitation
805
869
Energy
IOI-108
107
871
[eV]
spectra of Neon
810
Kinetic Energy [eV]
Fig. 2. (a) K-shell Rydbeq excitation in neon. The lifetime broadening (r) is set to 0.31 eV. The decay spectra in (b) and (c) have been calculated with a perfectly monochromatized beam tuned in between the ls-‘3~ and Is-‘4~ resonance (inter-resonance) and at the top of the Is-‘4~ resonance (on-resonance), respectively. The decay spectra show the spectator transitions at highest kinetic energy involving two holes in the 2p outer shell. The grey broad lines result (17) when interference is taken into account; the thin dark lines include no interference contributions (see. Eq. (7) for fldir(w,n) ). The dash-dotted curve at the top of each decay spectrum shows the integral of the interference term qint( oJN) ( 19). Note that the interference contribution for sufficiently large N sums to zero both for inter- and for on-resonance excitation.
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find that the interference term integrated over the entire energy region and summed over all intermediate (m) and final (n) close lying electronic states with large oscillatory strengths is zero, and again we note that a partial yield experiment that discriminates against some of the final states gives a result at variance with the absorption probability. The result is dependent on the lifetime of the intermediate states. we
6. Discussion With the increased resolution and efficiency in resonant X-ray scattering (RXS) spectra, it has become progressively more important to understand the role of lifetime interference in such spectra. Quite much is known about the conditions under which interference effects are important, and when simplifying two-step model approaches, excluding interference effects, are appropriate for the analysis of radiative or nonradiative RXS spectra. In the present work we derived for the first time the integral properties of the interference term in RXS spectra, and showed under which conditions the total interference contributions sum up to zero. This was shown to be the general case for vibrational lifetime interference, while for the case of electronic state interference this holds only for the total radiative and nonradiative cross sections, but not for the individual radiative (or nonradiative) RXS cross sections. In practice, the integrated electronic interference contribution to the radiative RXS spectra may be close to zero depending on the precise organization of the intermediate core-excited and final states reached by the resonant X-ray excitation. This was shown to be the case for RXS spectra relating to the resonant K-shell excitations in the neon atom.
References [ I] PL. Cowan, in: Resonant anomalous X-my scattering. Theory and applications, eds. G. Materlik. C.J. Sparks and K. Fisher (NorthHolland, Amsterdam, 1994) p. 449. [ 21 J. Nordgmn, P Glans and N.D. Wassdahl, Phys. Ser. T 34 ( 1991) 100. [3] R. Mayer, D.W. Lindle, S.H. Southworth and P.L. Cowan, Phys. Rev. A 43 ( 1991) 235. [4] A. Kivim%ki,A. Naves de Brito, S. Aksela, O.-P Sairanen, A. Ausmess. S.J. Osborne, L.B. Kantas and S. Svensson, Phys. Rev. Lett. 71 (1993) 4307. [5j H. Wang, J. Woicik, T. Aberg, M.H. Chen, A. Herrena-Gomez. T. Kendelewicz, A. Mantykenttll, K.E. Miyano. S. Southworth and B. Crasemann, Phys. Rev. A 50 ( 1994) 1359. [6] M. Neeb, J-E. Rubensson, M. Biermann and W. Eberhardt. J. Electron Spectrosc. 67 ( 1994) 261. 171 T. Aberg and B. Crasemann. in: Resonant anomalous X-ray scattering. Theory and applications, eds. G. Materlik, C.J. Sparks and K. Fisher (North-Holland, Amsterdam, 1994) p. 449. [ 81 F. Gel’mukhanov and H. Agren, Phys. Rev. A 49 ( 1994) 4378. [9] E Kh. Gel’mukhanov, L.N. Mazalov and A.N. Shlyaeva, Zh. Eksp. Tear. Fiz. 71 ( 1976) 1960 (Sov. Phys. JETP 44 ( 1976) 5041. [ IO] F. Kh. Gel’mukhanov, L.N. Mazalov and A.V. Kondratenko, Chem. Phys. Lett. 46 (1977) 133. [ I1 ] E Kaspar, W. Domcke and L.S. Cederbaum, Chem. Phys. 44 (1977) 33. ( 121 A. Cesar, H. Agnm and V. Carravetta, Phys. Rev. A 40 (1989) 187. [ 131 L.S. Cederbaum and E Tarantelli, J. Chem. Phys. 98 (1993) 9691. [ 141 T.X. Carnal, SE. Anderson. L. Ungier and T.D. Thomas, Phys. Rev. Lett. 58 (1987) 867. [ 151 E.D. Poliakov, L.A. Kelly, L.M. Duffy, B. Space, P Roy, S. Southwotth and M.G. White, J. Chem. Phys. 89 (1988) 4048. [ 161 M. Neeb, J-E. Rubensson, M. Biermann, W. Eberhardt, K.J. Randall, J. Feldhaus, A.L.D. Kilcoyne, A.M. Bradshaw, 2. Xu, PD. Johnson and Y. Ma, Chem. Phys. Lett. 212 ( 1993) 205. [ 171 J.-E. Rubensson, M. Neeb, A. Bringer, M. Biermann and W. Eberhardt, to be published. [ 181 J.J. Sakurai, Advanced quantum mechanics (Addison-Wesley, Reading, MA, 1967) Ch. 2. [ 191 E Gel’mukhanov and H. krgmn, Phys. Rev. A 50 (1994) 1129.