Theoretical study of X-ray photoelectron diffraction for fixed-in-space CO molecules

Chemical Physics 373 (2010) 261–266

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Theoretical study of X-ray photoelectron diffraction for fixed-in-space CO molecules Misato Kazama a,*, Jun-ichi Adachi b, Hiroshi Shinotsuka c, Masakazu Yamazaki b,1, Yusuke Ohori a, Akira Yagishita b, Takashi Fujikawa a a

Graduate School of Advanced Integration Science, Chiba University, 1-33 Yayoi-cho, Inage, Chiba 263-8522, Japan Photon Factory, Institute of Materials Structure Science, KEK, 1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan c National Institute for Materials Science, 1-2-1 Sengen, Tsukuba, Ibaraki 305-0047, Japan b

a r t i c l e

i n f o

Article history: Received 28 January 2010 In final form 1 April 2010 Available online 26 May 2010 Keywords: XPD EXAFS Multiple-scattering theory Fixed-in-space molecule

a b s t r a c t Diffraction patterns of 1s photoelectrons from fixed-in-space CO molecules have been calculated using full multiple-scattering formula in photoelectron kinetic energy region of k ¼ 50—500 eV. The forward scattering intensities gradually increase as a function of k , whereas the backward scattering intensities oscillate. It has been shown that the backward oscillating structures in calculations taking into up to double scattering converge into those of full multiple-scattering calculations. The present analyses of the oscillating structures show the similarity and difference with those of EXAFS. Ó 2010 Elsevier B.V. All rights reserved.

1. Introduction Extended X-ray absorption fine structure (EXAFS) is widely used as a powerful tool to obtain structural information on complex molecules, solids and their surfaces. EXAFS oscillation is essentially caused by backward scattering of excited photoelectrons from surrounding atoms. In contrast to EXAFS, X-ray photoelectron diffraction (XPD) is controlled by multiple scatterings with small scattering angles [1]. Nowadays XPD is employed to study ordered surface structures. Two different modes of XPD are used in experiments; in one mode, energy-scan XPD is measured ^ which is considerably similar to EXAFS and at a fixed direction k, is often referred to as angle-resolved photoemission extended fine structure (ARPEFS) [2–5]. In the other mode, angular dependence of XPD is measured at a fixed photon energy [1]. Angular distributions of photoelectrons for randomly oriented systems are simply described in the electric dipole approximation by [6]

IðkÞ ¼

r 4p

½1 þ bP2 ðcos hÞ;

ð1Þ

where r is the total cross section, h is the emission angle measured from the electric vector of the incident X-ray, and b is the asymmetry parameter ð1 6 b 6 2Þ. In contrast to these simple angular

* Corresponding author. Tel.: +81 043 290 3699. E-mail address: [email protected] (M. Kazama). 1 Present address: IMRAM, Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai 980-8577, Japan. 0301-0104/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.chemphys.2010.04.038

distributions, the XPD patterns for oriented systems show rich structures. The XPD patterns from solid surfaces are dominated by forward scatterings, whereas those from fixed-in-space molecules provide us with unique information on backward scatterings. Recent experimental developments enable us to measure XPD spectra from fixed-in-space molecules [7–9]. This study reports XPD results of core electrons from fixed-inspace CO molecules using the full multiple-scattering theory [10,11]. We discuss the energy dependence of forward and backward scattering intensities of XPD, and the latter is directly compared with EXAFS. Lee pointed out that the EXAFS expression is obtained by integrating the angle-resolved photoemission expression over 4p steradian, and that the single scattering EXAFS includes second-order scattering processes in angle-resolved photoemission [12]. Fujikawa explicitly derived EXAFS formula by taking into account up to double scattering terms in plane wave XPD formula and by integrating it over 4p steradian [13]. The multiple-scattering method allows us to directly compare XPD with EXAFS in terms of the scattering order. Detailed comparison between backward scattering XPD and EXAFS provides new physical insights into the photoemission dynamics and the widely used EXAFS formula. Useful information on both XPD and EXAFS has been extracted from present sophisticated spherical wave full multiplescattering calculations. To discuss photoelectron diffraction in the low-energy region ðk K 50 eVÞ, some additional effects like nonmuffin-tin effects and electron–phonon interactions should be cared; thus we focus on the energy range k ¼ 50—500 eV. The angular distributions of XPD over the energy region will be also discussed.

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The continuum multiple scattering (CMS) method is often employed to study electron-molecule scattering [14]. In the CMS method, the space is partitioned into three regions. Region I comprises atomic spheres, region II is molecular field encircled by outer sphere, and region III is the outside of the outer sphere. This type of potential is adequate to deal with scattering problems with the long-range Coulomb tail, which has appreciable contribution in low-energy region. However, this paper is concerned with XPD in EXAFS region ðk J 50 eVÞ. Our method employs muffin-tin potential assuming that the Coulomb tail is negligible in this energy region. 2. Theory First principle many-body photoemission theory gives an XPD formula to study no-loss band excited by X-ray photons with the energy of x, in terms of the damped photoelectron wave function w k under the influence of the optical potential [15,16],

  IðkÞ ¼ 2pjhwk jDj/c ij2 d E0 þ x  E0  k ;

h/0k jt a g A Dj/c i

að–AÞ

X

þ

X

h/0k jt b g 0 t a g A Dj/c i

þ 

ð3Þ

b–að–AÞ

where h/0k j is the decaying plane wave under the influence of the imaginary part of the optical potential, h/ Ak j is the photoelectron wave function of a photoelectron emitted from an X-ray absorbing atom A; g 0 is the decaying free Green’s function. g A is the Green’s function influenced by the potential only on absorbing atom A; g A ¼ g 0 þ g 0 t A g 0 . ta is the site-t matrix at the site a. The first term in Eq. (3) describes the direct photoemission amplitude without scatterings from surrounding atoms, which is hereafter referred to as Z 1 . The second term is the single scattering amplitude Z 2 and the third term is the double scattering amplitude Z 3 . The terms Z 1 ; Z 2 and Z 3 are explicitly written as

X

Z 1 ¼ h/Ak jDj/c i ¼

^ LL ; Y L ðkÞM c

where the matrix X is labeled by a set of atomic site a; b; . . ., and angular momentum L ¼ ðl; mÞ. The full multiple-scattering is taken into account by the inverse matrix ð1  XÞ1 . The dipole matrix element M LLc weakly depends on the photoelectron energy k [1]. 3. Results and discussion In this section we discuss the C 1s XPD in the photoelectron energy region of k ¼ 50—500 eV. Geometrical setup for the XPD measurements is shown in Fig. 1; X-rays are linearly polarized parallel to the molecular axis. The bond length of CO is 1.1282 Å. Phase shift calculations are carried out with sufficiently accurate non-local optical atomic potential, which allows taking many-body effects into account [18]. 3.1. Energy dependence of XPD

ð2Þ

where k is the photoelectron momentum, /c is the core wave function localized on the site A. The ground state energies with and without core hole are E0 and E0 , and D is the electron–photon interaction operator. Using site-t matrix expansion, the amplitude is written as [1,10,11]

hwk jDj/c i ¼ h/Ak jDj/c i þ

X bL0aL ¼ ð1  dba Þt bl0 ðkÞGL0 L ðkRb  kRa Þ;

The modulation function F n ðha Þ is defined as the following equation to examine scattering effects on XPD.

F n1 ðha Þ ¼

jZ 1 þ Z 2 þ Z 3 þ    þ Z n j2  jZ 1 j2 jZ 1 j2

where ha is the scattering angle from the molecular axis (see Fig. 1). Fig. 2 schematically shows several scattering paths. The function F n ðha Þ corresponds to the XAFS spectrum vðxÞ ¼ ½lðxÞ  l0 ðxÞ= l0 ðxÞ, where lðxÞ and l0 ðxÞ are the X-ray absorption intensities with and without surrounding atoms at the photon energy x. We focus on XPD intensities of forward scatterings ðha ¼ 0Þ, and those of backward scatterings ðha ¼ pÞ. The energy dependence of the calculated F n ð0Þ and F n ðpÞ spectra are shown in Figs. 3 and 4, respectively. F n ð0Þðn 6 2Þ does not have distinct structures as shown in Fig. 3. In particular, F 1 ð0Þ monotonically increases. This monotonic behavior can be understood by considering Glauber’s approximation, which is highly reliable in the case of high energy and smallangle scatterings [19]. It is helpful to use a simple plane wave approximation for understanding the specific features observed in Figs. 3 and 4. In

ð4Þ

X

h/0k jta g A Dj/c i

að–AÞ

¼

X

eikRa

X

að–AÞ

Z3 ¼

LL

X X

^ a0 ðkÞG 0 ðkR ÞM ; Y L0 ðkÞt a LLc LL l

ð5Þ

C

0

eikRb

bð–aÞ

X X að–AÞ LL0 L00

ð6Þ

where Ra is the position vector of scatterer a measured from the X-ray absorbing atom A. The propagator GL0 L ðkRa Þ describes electron propagation from the site A with angular momentum L to the site a with L0 . By applying multiple-scattering renormalization, we obtain the full multiple-scattering formula [10,11,17];

hwk jDj/c i1 ¼

X a

eikRa

X

^ Y L0 ðkÞ½ð1  XÞ1 La0AL MLLc ;

0

θα

Fig. 1. Geometry for the XPD measurements. X-rays are linearly polarized parallel to the molecular axis of CO.

^ b00 ðkÞ Y L00 ðkÞt l

 GL00 L0 ðkRb  kRa Þt al0 ðkÞGL0 L ðkRa ÞMLLc ;

O 1.1282 Å

h/0k jt b g 0 t a g A Dj/c i

b–að–AÞ

¼

Photoelectron detector

X-ray

L

Z2 ¼

;

ð7Þ

(a)

(c)

(b)

(d)

LL

ð1  XÞ1 ¼ 1 þ X þ X 2 þ   

Fig. 2. Schematic representation of several scattering paths. Shaded circles are the X-ray absorbing C atoms: (a) forward single scattering, (b) forward double scattering, (c) backward single scattering, and (d) backward double scattering.

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M. Kazama et al. / Chemical Physics 373 (2010) 261–266

where 2ReðZ 1 Z 2 Þ is the interference term between the direct and the single scattering waves. Eqs. (8)–(10) give the interference terms as follows:

1.5 1

F (0)

0.5 0

ReðZ 1 Z 2 Þ / jqð1Þc j2

Fn (0)

-0.5

F2 (0)

ReðZ 1 Z 3 Þ / jqð1Þc j2

cos ha

ReðZ 2 Z 3 Þ / jqð1Þc j2

1

-0.5 1

F1 (0)

0.5 0 -0.5 100

200

300

400

500

kinetic energy εk (eV) Fig. 3. Calculated forward photoemission intensities Fn(0) for different scattering orders. F1(0) and F2(0) are the single and double scattering modulation function at ha = 0. Full multiple-scattering one F1(0) is also shown.

Fn (π)

ð11Þ

1 0.5 0

1.5 1 0.5 0 -0.5 -1 1 0.5 0 -0.5 -1 1 0.5 0 -0.5 -1

F (π)

F2 (π)

F1 (π)

100

200

300

400

500

kinetic energy εk (eV) Fig. 4. Calculated backward photoemission intensities Fn(p) for different scattering orders. The arrows indicate the minimum or maximum of Fn(p).

the plane wave expression, Z 1 ; Z 2 and Z 3 are explicitly given for the photoemission from a deep s orbital A

Z 1 / cos hk eid1 qð1Þc ; A

Z 2 / eid1 qð1Þc

X

Z 3 / eid1 qð1Þc

ð8Þ

cos ^ha

a A

 cos ha  Re fa ðha ÞeikRa ð1cos ha Þ ; Ra

X X bð–aÞ að–AÞ

fa ðha Þ ikRa ð1cos ha Þ e ; Ra

cos ^ha

fb ðhkba Þfa ðhbaA Þ ikðRba þRa ÞikRb e ; Rba Ra

ð9Þ

ð10Þ

where the radial dipole integral qð1Þc weakly depends on k . ^ ha is ^ a , where ^ different from the scattering angle ha ; cos ^ ha ¼ ^ eR e is the X-ray polarization vector. fa and fb express the scattering amplitudes at site a and b with scattering angle hbaA ðA ! a ! bÞ and hkba ða ! b ! kÞ. In the present simple case, cos ^ ha ¼ 1; hbaA ¼ hAaA ¼ p; hkba ¼ p  ha and hk ¼ ha . The XPD intensity is written as

Iðhk Þ / jZ 1 þ Z 2 þ    j2 ¼ jZ 1 j2 þ jZ 2 j2 þ 2ReðZ 1 Z 2 Þ þ   

2

Ra

3

Ra

  Re fA ðp  ha Þfa ðpÞe2ikRa ;

  Re fa ðha ÞfA ðp  ha Þfa ðpÞeikRa ð1þcos ha Þ :

ð12Þ

ð13Þ

The interference term ReðZ 1 Z 2 Þ is responsible for the oscillation of F n ðha Þ through the factor exp ½ikRa ð1  cos ha Þ. Then, under the forward scattering condition of ha ¼ 0, the factor reduces to unity, so that the ReðZ 1 Z 2 Þ does not oscillate. In contrast, ReðZ 2 Z 3 Þ oscillates due to the factor of exp ½ikRa ð1 þ cos ha Þ at ha ¼ 0. The single scattering modulation function F 1 ð0Þ results in monotonic behavior because F 1 ð0Þ contains only jZ 2 j2 and ReðZ 1 Z 2 Þ. The double scattering modulation function F 2 ð0Þ shows small oscillation due to ReðZ 2 Z 3 Þ and ReðZ 1 Z 3 Þ. In contrast to F n ð0Þ, the backward photoemission intensities F n ðpÞ exhibit fully different energy dependences; they have EXAFS-like oscillations as shown in Fig. 4. A good convergence is obtained at n ¼ 2 (double scattering). The arrows indicating tops and bottom of the oscillation for F 2 ðpÞ are on the same positions as those for F 1 ðpÞ. Only the single scattering path shown in Fig. 2(c) plays an important role in F 1 ðpÞ, whereas both the paths (c) and (d) in Fig. 2 contribute to F 2 ðpÞ. The path (d) is the important path, because forward scattering by the C atom predominantly contributes to the backward photoemission due to forward-focusing effect. Next, we compare the calculated energy dependences with the experimental results [20]. The experimental data cannot be directly compared with F n ð0Þ or F n ðpÞ, because the contribution of Z 1 is removed from F n . Then, the intensities of experimental and calculated Ið0Þ and IðpÞ are normalized to the angular integrated intensity at each k . The normalized experimental and calculated Ið0Þ and IðpÞ are shown in Fig. 5; the calculated results show good agreements with the experimental ones in relative variations for both forward and backward XPD. The difference in the absolute intensities between the calculated and the experimental results can be due to the finite acceptance angle (in-plane:  5 , outof-plane:  20 for both photoelectrons and photoions) in experiments. The shapes of calculated XPD patterns are much sharper than the experimental ones. In these calculations, k is measured from the muffin-tin constant, whereas the experimental photoelectron kinetic energy exp is measured from the vacuum level. k Thus the calculational photoelectron kinetic energy k is related to the experimental photoelectron kinetic energy exp by k k ¼ exp þ V 0 (V 0 : interstitial energy). As will be shown in Section k 3.3, the calculated XPD patterns for C 1s photoelectron most agree with the experimental angular distribution for exp ¼ 150 eV when k V 0 ¼ 10 eV (see Fig. 9) and thus the same value is used for V 0 in Fig. 5. 3.2. Comparison between XPD and EXAFS The single scattering EXAFS and F 2 ðpÞ are not exactly the same, because only the angular integrated XPD can contribute to the Xray absorption intensity. The K-edge single scattering EXAFS formula for the spatially fixed diatomic molecule is given as follows in the case of parallel polarization as shown in Fig. 1;

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M. Kazama et al. / Chemical Physics 373 (2010) 261–266

Calculational kinetic energy εk (eV) 60

0.008

80

100

120

140

160

where g A ¼ g 0 þ g 0 t A g 0 is the renormalized Green’s function at the X-ray absorbing site A. In the expansions in Eq. (16), both of the second terms describe the ‘‘single” scatterings. The atomic Green’s function g A fully includes the scattering effects on the site A, and the phase shift 2dA1 results from the ‘‘single scattering” term P Imh/c jD g A a ta g A Dj/c i. On the other hand, the first expansion has no contribution to the central atom, and gives no phase shift. The phase part of F 1 ðpÞ is 2kRa þ wa ðpÞ within the plane wave approximation. The information on the phase shift dA1 is lost there. The interference term ReðZ 1 Z 3 Þ has the phase of 2kRa þ wa ðpÞþ wA ð0Þ which partly has information on dA1 in wA ð0Þ. The angular integration of jZ 2 j2 þ ReðZ 1 Z 2 Þ þ ReðZ 1 Z 3 Þ projects out dA1 from fA ð0Þ as discussed before [10,12], where wA ð0Þ is the phase of the forward scattering amplitude fA ð0Þ from the X-ray absorbing atom A (see Fig. 6). The calculated single scattering EXAFS function (Eq. (14)) and the backscattering photoemission intensity F 2 ðpÞ are shown in Fig. 7. The phases of F 2 ðpÞ and vðkÞ are different from

180 0.012

0.01 0.007 0.008

0.0065 0.006

0.006

0.0055 0.004

(a)

0.005 0.0045

Calc. normalized intensity

Exp. normalized intensity

0.0075

40

60

80

100

120

140

Experimental kinetic energy ε

exp k

160

0.002

(eV)

Calculational kinetic energy εk (eV) 60

0.008

80

100

120

140

160

180 0.014

0.006

0.008 0.006

0.005

0.004

0.003

(b) 40

60

80

100

120

140

160

1

0.5

intensity

Exp. normalized intensity

0.01

Calc. normalized intensity

0.007

0.002

-0.5

Fig. 5. The experimental and calculated energy dependence of the forward (a) and backward (b) photoemission intensities. Solid curves show calculated result, and dots are the experimental data. Now we set V0 at 10 eV ðk ¼ exp þ 10 eVÞ. Both the k calculated and the experimental results are normalized to the angular integrated intensities.

ð15Þ

where  is photoelectron kinetic energy, jS0 j is the intrinsic intensity for no-loss channel [21,22]. Two different site-t matrix expansions can be applied to g;

gðÞ ¼ g 0 ðÞ þ

100

200

300

400

500

kinetic energy ε k (eV)

ð14Þ

2

X

χ ×20 F2 (π)

Fig. 7. Calculated C K-edge single scattering EXAFS function v and backward scattering photoemission intensity F2(p) including up to double scatterings.

where fa ðpÞ ¼ jfa ðpÞj expðiwa ðpÞÞ is the backward scattering amplitude from the scattering atom a (oxygen atom, see Fig. 1). The reason why the central atom phase shift should be doubly counted in the EXAFS formula (14) is not quite obvious. According to widelyaccepted heuristic explanation, the photoelectron receives a phase shift of expðidA1 Þ on its outgoing trip and expðidA1 Þ on its incoming trip. When extrinsic loss is neglected, X-ray absorption intensity is written in terms of damping Green’s function g,

IðÞ / Imh/c jD gðÞDj/c ijS0 j2 ;

-1

Fourier Magnitude (arb. unit)

  vðkÞ ¼  2 jfa ðpÞj sin 2kRa þ 2dA1 þ wa ðpÞ ; kRa

0

0

Experimental kinetic energy εkexp (eV)

3

f (π)

Fig. 6. The scattering process of backward double scattering. ‘‘A” and ‘‘a” represent the absorbing atom and the scattering atom.

0.012

0.004

α

A fA (0)

| χ (r) | × 15 | F2 (r) |

0.9 0.95 1 1.05 1.1 1.15 1.2

0

0.5

1

1.5

2

distance (Å)

g 0 ðÞt a g 0 ðÞ þ   

a

¼ g A ð Þ þ

X að–AÞ

g A ðÞt a g A ðÞ þ   

ð16Þ

Fig. 8. The magnitudes of the Fourier transformations of F2(p; k) (jF2(r)j) and single scattering EXAFS v(k) given by Eq. (14) (jv(r)j). The arrow indicates the local maximum of jF2(r)j and the broken line indicates that of jv(r)j.

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M. Kazama et al. / Chemical Physics 373 (2010) 261–266

C

O

O

C

O 1s

C 1s

Fig. 9. Angular distributions of C 1s and O 1s photoelectrons. Solid curves; theoretical results, and dots; experimental results. exp ¼ 150 eV for C 1s and exp ¼ 145 eV for O 1s k k photoelectrons.

C

C

O

O

(b)

(a) Fig. 10. Calculated photoelectron angular distributions for the C 1s core level. (a) scattering intensity.

k = 110–170 eV, and (b) k = 170–270 eV. The intensities are normalized to the forward

each other. Moreover, the intensity of F 2 ðpÞ is twenty times stronger than that of the single scattering EXAFS oscillation vðkÞ, which can be attributed to the forward-focusing factor fA ð0Þ in Eqs. (12) and (13). Fourier transformations of vðkÞ and F 2 ðpÞ are now defined as

sity. Backward scattering intensity increases from k ¼ 110 eV to k ¼ 170 eV, and then decreases from k ¼ 170 eV to k ¼ 270 eV.

vðrÞ ¼

Z

kmax

k

3

vðkÞe2ikr dk;

kmin

F 2 ðrÞ ¼

Z

kmax

ð17Þ 3

k F 2 ðp; kÞe2ikr dk;

kmin 2

This oscillating behavior of the backward scattering intensity is explained by the energy dependence of F 2 ðpÞ shown in Fig. 4; F 2 ðpÞ has local minimums at 110 eV and 266 eV, and a local maximum at k ¼ 169 eV.

2

where kmin =2 ¼ 2:4 a.u., kmax =2 ¼ 6:0 a.u. Fig. 8 shows the absolute values of vðrÞ and F 2 ðrÞ. The difference of the phases between vðrÞ and F 2 ðrÞ causes the differences of peak positions. The side peaks around 1.7 Å should be due to truncated integral range, kmin < k < kmax [23]. 3.3. Angular distribution of XPD The calculated angular distributions of C 1s and O 1s photoelectrons are now compared with the experimental results. The angular distributions of XPD are shown in Fig. 9. The inner potential V 0 is set at 10 eV for C 1s and at 9 eV for O 1s photoionization. The both calculated results of C 1s and O 1s photoemissions agree well with the experimental results, which imply that the present calculations are reliable. We have calculated XPD patterns at every 20 eV intervals to study angle-dependent EXAFS-like oscillations in XPD spectra. Fig. 10 compiles the calculated C 1s photoelectron angular distributions normalized at the forward scattering inten-

4. Concluding remarks We have demonstrated the remarkable differences in the energy dependence of forward and backward scattering photoemission intensities. The forward scattering XPD intensity F n ð0Þ is much stronger than the backward XPD intensity F n ðpÞ in the intermediate- and high-energy region. The former is monotonically and slowly increases with k . The latter is related to EXAFS; both EXAFS and backward XPD show oscillation controlled by expð2ikRÞ. The important role of the double scatterings in XPD analyses appears through the phase factor of 2dA1 , which emerges in the ‘‘single scattering” EXAFS formula.

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