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Electron momentum density of graphite from (γ,eγ) spectroscopy
CGJ .__
Nuclear Instruments and Methods in Physics Research B 122 ( 1997) 269-273
__
NOMB
Beam Interactions with Materials&Atoms
@ ELSEVIER
Electron momentum density of graphite from ( y,ey) spectroscopy F.F. Kurp a, A.E. Werner a, J.R. Schneider
a, Th. Tschentscher
b, P. Suortti b, F. Bell ‘, *
’ Hamburger Synchrotronstrahlungslabor
(HASYLAB) at Deutsches Elektronen-Synchrotron (DEW), Notkestr. 85, D-22603 Hamburg, Germany b European Synchrotron Radiation Facility (ESRF), B.P. 220, F-38043 Grenoble, France ’ Sektion Physik, UniuersitZt Miinchen, Am Coulombwall 1, D-85748 Garching, Germany Received 29 March 1996; revised form received 17 May 1996.
Abstract By the coincidental detection of an inelastically scattered hard X-ray photon and its recoil electron (i.e. a so-called (r,er> experiment) cuts through the 3D-electron momentum density (EMD) of graphite have heen measured and are compared with a theoretical EMD based on a pseudopotential calculation. Peak shifts of the EMD as a function of the electron emission angle are discussed by a detailed consideration of the scattering kinematics. Experiments have been performed at the European Synchrotron Radiation Facility (ESRF) at Grenoble, France.
1. Introduction Deep inelastic photon scattering of hard X-rays has been developed as a tool to study the ground state electron momentum density (EMD) of solids [1,2]. In a conventional Compton scattering experiment the energy distribution of the photons scattered at a fixed angle can be related to the so-called Compton profile J(p,) which is a two-dirhensional integration over the EMD p(p): J( pi) = jl&)dp,dp,.
(1)
The P,-component of the electron momentum p is parallel to the momentum transfer vector q. This integration results from the fact that in conventional Compton scattering no information about the momentum distribution of the recoiling electrons is obtained. Since the main interest of solid-state physicists is in the EMD itself, it has been a long standing aim to specify the complete scattering kinematics by measuring the energy of the Compton scattered photon in coincidence with the recoiling electron which makes a direct determination of the EMD possible. The corresponding technique has been named (r,e-y) spectroscopy [3,4]. From the beginning it was recognized that intense elastic scattering of the recoiling electron within the target alters the direction of emission. This effect of multiple scattering introduces uncertainties in the reconstruction of the initial electron momentum p. Therefore,
* Corresponding
author. Fax: + 49.89-289-
0168-583X/97/$17.00 Copyright PII SOl68-583X(96)00828-2
14146.
very thin target foils have to be used so that the mean free path for elastic scattering at the Coulomb field of the nuclei is considerably larger than the target thickness. It is for this reason, together with the small value of the Klein-Nishina cross section, that strong monochromatic photon fluxes at high X-ray energies are needed which can only be delivered by modem synchrotron radiation facilities. In this paper we describe a (r,er) experiment which has been performed at the European Synchrotron Radiation Facility (ESRF). In Section 2 the scattering kinematics are discussed in some detail and in Section 3 results from the experimental EMD of graphite are presented.
2. Kinematics If an X-ray photon is scattered at an electron at rest the scattered photon energy wb will be wb=o/[l+w(l-c&)1,
(2)
where o is the primary photon energy and 8 the scattering angle. We use so-called natural units, i.e. h = m = c = 1. At the same time the recoiling electron will be ejected at an angle (Y (Fig. 1) with respect to the primary photon beam direction where cotcu= (1 + w)tan(Q/2).
(3)
The influence of an initial electron momentum p ;f 0 is twofold: scattered photons have energies w’ # wb and recoil electrons appear at angles different from (Y. Both the
0 1997 Elsevier Science B.V. All rights reserved
F.F. Kurp et al./Nucl.
270
Instr. and Meth. in Phys. Res. B 122 (1997) 269-273
essence given by the Doppler broadening A w’ = w’ - wb and the other components by the angular deviations /?, and p, of the emitted electrons. As indicated in Fig. 1, p, lies in the (&k) scattering plane and p, perpendicular to it.
Fig. 1. Vectors and angles used to describe the experiment: k and k’: momenta of the primary and scattered photon, p’: final electron momentum, qo: momentum transfer to an electron at rest. Note, that in practice q. z=-p.
energetic and angular deviations can be used to reconstruct p in a unique way. In conventional Compton scattering the natural coordinate system for the electron momentum p is that where the p,component is parallel to momentum transfer vector q = k - k’, where k and k’ are the momenta of the primary and scattered photon, see Fig. 1. For our coincident Compton scattering experiment there arises the problem that q changes direction as a function of the scattered photon energy WI. A variation of o’ which corresponds to a change of the p,-component by + 3 a.u. lets the direction of q rotate by A (Y= + 0.35” at a photon energy of 140 keV and a scattering angle 13= 140”. Though this rotation is very small, it changes electron momenta perpendicular to q by qAa = kO.36 a.u. which cannot be tolerated. Therefore, we introduce a space-fixed coordinate system where the p,-component is parallel qo, the momentum transfer to an electron at rest. In this coordinate system the Cartesian initial electron momentum components read (Fig. l>: p, = p’sinflcos
y + w’sinn - wsin ff ,
pY = p’ sin p sin y , pz = p’cos p + o’cosq
- COCOS (Y
(9
,
/3 and y are the polar and azimuthal emission angles of the recoil electron and T = 6 + (Y. The final electron momentum p’ is obtained from energy conservation p’ = [(w
- w’ + EJ2
- I] “2,
3. Experiment The experiment was performed at the high energy X-ray scattering beamline ID15 at the ESRF [5] and is shown schematically in Fig. 2. An asymmetric wiggler with 7 periods and strong poles of 1.9 T (20 mm gap) was used with a critical energy of 45 keV. The white beam was monochromatized by a horizontally focussing bend Si crystal in Bragg geometry [6,7]. At an average storage ring current of 140 mA a photon flux of about 6 X 10” photons/s at o = 135.5 keV in a beam spot of 1 mm (horizontal) X 3 mm (vertical) was obtained. The monochromaticity was A w = 300 eV (FWHM). The monochromatic X-ray beam entered an evacuated target chamber with an externally mounted intrinsic Ge-diode (energy resolution - 600 eV FWHM at 100 keV, solid angle 0.4 msr) at 6 = 140”. The not-needed photon beams are dumped in carefully shielded beam dumps to keep the X-ray background as low as possible. The recoil electrons were measured by a linear array of 7 PIN diodes placed parallel to the (i,&) scattering plane. For our experimental situation electrons initiahy at rest are emitted at (Y= 16.0”. Since the array of diodes cover a total range of transverse momenta of 5 a.u. only, especially the tails of the EMD would not be measured if the array is placed with its center at (Y= 16.0”. We therefore placed it slightly asymmetric with its center at A p, = 15 mrad corresponding roughly to an offset of Ap, = 0.9 a.u. This allows the simultaneous measurement of the center and the tail of the EMD. Each PIN diode was equipped with its own electronic circuit (pre- and main amplifier, discriminator). In a multi-coincidence mode both photon and time correlation spectra for every PIN diode could be obtained. The accuracy by which the initial electron momentum components can be determined depends on the angular and energy uncertainties of
Here, Ei = 1 - E, where E, > 0 is the electron binding energy. However, since the EMD is strongly dominated by the valence electrons it is a very good approximation to neglect E, in Eq. (5) and to discuss the experimental results in a notation which is a good approximation to Eq. (4): px = qo P, + aA J/q,
= qo P, - ( 4,/m)a~,
>
(6a)
py=qoPy.
(6”)
pz=
(6~)
-~AtJ/(qo&).
/3, = p cosy, p, = /3 siny, q. = (CO* + wf 2wwbcose>L/2 and a=osine-(l+o-w;)&. It is immediately seen from Eq. (6) that the pi component is in with
Fig. 2. Experimental set-up: SR: storage ring, W: wiggler, Si(55 1): monochromator, T: target, Ge SSD: photon solid state detector, Si SSD: PIN diode array, BD: beam dump.
F.F. Kurp et al./Nucl. Instr. and Meth. in Phys. Res. B 122 (1997) 269-273 the experiment. They include the monochromaticity of the primary photon beam and the energy resolution of the photon detector on one hand and the uncertainties of 8, p, and /3,, on the other. We estimate a total error of Ap, = 0.55 a.u. (FWHM) and Ap+ = Ap, = 0.75 a.u. (FWHM). For electrons initially at rest the scattered photon energy is wb = 92.3 keV, the kinetic electron recoil energy equals 43.2 keV and the momentum transfer amounts to q. = 58 a.u. The target was a thin carbon foil made by evaporation of pure graphite heated up to 3200 K [8]. The evaporated carbon atoms condensed on a thin Betain film (C,H , ,NO, . H,O) which had a fine crystalline-like structure. This structure acted as a replica for the carbon film guaranteeing a high mechanical stability. After evaporation the Betain film was dissolved in water and the self-supporting foils were mounted on an aluminum frame with 12 mm diameter. Finally, the foil was heat treated by a short pulse from a Nd: YAG laser at temperatures up to 4200 K. Subsequent electron diffraction investigations showed sharp graphite-like diffraction rings from which an average crystalline diameter of 10 nm was estimated [9]. By means of energy loss measurements of energetic ions a foil thickness of 17 nm after the heat treatment was determined. In view of electron multiple scattering discussed above, this thickness should be compared with the mean free path for elastic scattering of 43 keV electrons in carbon of 65 nm [lo]. From Poisson statistics we derive that 88% of the electrons leave the foil unscattered.
4. Results and discussion Fig. 3 shows a photon spectrum coincident with the electrons measured in a pixel of the PIN-diode array placed at /3, = /3, = 0. The scattered photon energy has been converted to a p;scale via Eq. (6~). Since for our experimental situation wb sine= 0.12 the data of Fig. 3 correspond to a cut through the 3D-EMD along p = ( - O.l2p,, 0, p,). The experimental data are compared
P, Fig. 3. Coincident
[a-u-l
photon spectrum for a PIN diode & = p, = 0. Solid line represents theory of Ref. [I 11.
pixel
at
87.5
90
92.5
211
95
97.5
w’[ keV] Fig. 4. Coincident photon spectra for 7 PIN diodes placed at electron emission angles p,. Broken lines are the theory of Ref. [ll] for pY = 0.5 a.“. The arrows indicate the peak positions. Either error bars or the diameter of the symbols reflect the statistical uncertainty of the data.
with an empirical pseudopotential calculation of Yongming et al. [l l] which has been spherically averaged. Theory (solid line in Fig. 3) has been convoluted with the experimental resolution and corrected for multiple electron scattering effects [3]. A (Is)* core contribution [12] has been added to the valence EMD of Ref. [ 111. The most striking feature of graphite is the dip in the EMD at p = 0 which results from the p-character of the T bonds. They are responsible for the weak van der Waals interlayer bonding in graphite and have also been identified by 2D positron annihilation meastuements [ 13,141. It is clearly seen from Fig. 3 that the experimental data show this dip at pz = 0 though it is a little sharper than predicted by theory. Details like this are completely washed out in non-coincident Compton data due to the integration of Eq. (1) and have neither been observed in our singles nor in earlier Compton profile measurements [ 15- 171. In Fig. 4 a series of coincident photon spectra are shown where the array has been placed a little out of the ($,k) scattering plane at p, = 9 mrad which corresponds to p,, = 0.5 a.u., see Eq. (6a). Again, we compare with theory (broken line). We stress that theory has been fitted to the dam by minimizing the squared deviations for all 7 spectra simultaneously. Notice that the number of coincidence events is different for each spectrum, clearly indicating that the EMD falls off for increasing &, i.e. px_ A comparison with Fig. 3 demonstrates that an offset of
F.F. Kurp et al./Nucl.
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Instr. and Meth. in Phys. Res. B 122 (1997) 269-273
pY = 0.5 a.u. suffices to cut the 3D EMD outside the dip from the rr electrons. The agreement between experiment and theory is good though not perfect. A close inspection of the spectra reveals that the maximum of the intensity does not stay at the same o’ value but shifts with p,. The arrows in Fig. 4 indicate the “center-of-gravity” energy of the data where Ni are the number &l = C,w,NJY,,N, of coincidences within a bin at 0:. This shift resembles the displacement of the maximum of the Compton profile from the prediction of the impulse approximation (IA) [ 181, observed in non-coincident Compton spectroscopy [19-211: the so-called Compton defect. In this case it signals the breakdown of the IA, the validity of which must be assumed in order to extract confident Compton profile data. It is caused by a final state interaction of the emerging electron with its nucleus and can be described by a correct first Born approximation [22,23]. Since the validity of the IA is important we want to show in the following that in contrary the shift observed in coincident Compton spectra is of kinematical nature. Due to Eq. (6a) we cut the EMD along pZ for a constant emission angle /?,, i.e. not for pX = const. but for
(7)
P,( P,) = 40 P, - (@b/w)a&.
-0.03
0.00
0.06
0.03
PX[ mrod] Fig. 6. The peak shift A wh as a function of the electron emission angle p,. The lines result from Eq. (11) for c = 0.8 (dash-dotted), 1.O (solid) and 1.2 (dashed).
Eq. (9) shows that the peak shift results not entirely from kinematics but depends on the shape of the EMD also. Assuming
where c is a de-
p = p({w)
formation parameter which for c < 1 yields a prolate and for c > 1 an oblate EMD we derive from Fq. (9)
Thus, in the (p,,p,>
plane the traces of the Doppler broadened photon spectra are not along vertical lines with px = const. but they are inclined to it by a small angle 6 with 6 = arctan( wb a/w),
aP/aPi
dPx p,=p~ = dp,’
(9)
Fig. 5. Contour plot of the EMD p(p) in the (p,,pi) plane. Straight lines indicate the cut made by a photon spectrum coincident with electron pixels at different
Pzl> = &a/w
(10)
or with Eq. (6~)
(8)
which for & = 0 yields 6 = 7” in our case. In Fig. 5 we have plotted contour lines of an isotropic EMD in the (p,,p,) plane. If one follows the coincident photon intensity along lines for p, = const., the maximum will occur at p,” values which depend on p,. Converting pr into w;, via Eq. (6~) explains the shift of Fig. 4. To be more quantitative and independent from the assumption of an isotropic EMD p the maximum count rate will occur if
ap/ap,
CP”/Px(
p,
positions.
AU;, = w;n( &) - o;(O)
= -
4024 c( W/W;)* + a2
(11)
with w&(O) = wb. In Fig. 6 we have plotted the experimental shifts A o& as a function of p,. The lines are from Eq. (11) for c = 0.8, 1.0 and 1.2. We emphasize that the signs for p, and A$,, are correlated: energy and momentum conservation demand, that for an electron emission angle closer to the primary beam direction ( p, > 0) the maximum energy 6~; is smaller than wb, i.e. A oh < 0. It is evident from Fig. 6 that the best fit holds for c = 1.O, i.e. an isotropic EMD. In the first place this means that the influence of electron multiple scattering is negligable: since this effect smears the electron emission angle /3 the measured EMD would dominantly be elongated in px and p,, directions which yields an oblate EMD with c > 1. Evidently this is not observed. From the discussion above it might have become more obvious why we have selected a cut through the EMD at p,, = 0.5 a.u.: the recrystallization of the target foil due to heat treatment may induce a slight texture with a preferential c-axis orientation perpendicular to the foil surface [9]. This would mean an anisotropic dip in the EMD originating from the r-electrons and extending mostly along the pi-direction. To avoid this we have chosen a cut through the EMD where a more isotropic part of it is expected. Finally, we address the question how essential thin targets are and if the
F.F. Kurp et aL/Nucl.
Ins&. andbleth. in Phys. Res. B 122 (1997) 269-273
method is restricted to thin carbon foils. In our case the foil was thin enough to avoid the influence of electron multiple scattering quantitatively. For thicker targets and materials with larger atomic numbers this effect certainly increases but one is able to correct for that by Monte Carlo simulations [3]. In this way, we have been able to measure the EMD from a 80 mn thin Cu target [24]. In a recent extensive Monte Carlo investigation Rollason and Woolf [25] have demonstrated that if one concentrates on the evaluation of 3D Fermi surfaces that there is in principle no limit to the sample thickness and material. The reason for this optimistic view results from the fact that sharp structures like Fermi surface breaks are still visible from the contribution of unscattered electrons originating from the last mean free path for elastic scattering (last layer effect), whereas the scattered electrons contribute to an almost flat, unstructured background only.
5. Summary It is demonstrated that the photon flux from a third generation synchrotron radiation facility (ESRF) is strong enough to measure the 3D electron momentum density of graphite along specific directions. Comparison with theory shows reasonable agreement and especially the contribution of n-bonds is revealed experimentally. Counting statistics are good enough to observe small shifts of the photon energy for which maximum coincidence count rates occur.
Acknowledgements We thank G. Dollinger and P. Maier-Komor from the Target Laboratory of the Technical University of Munich for the carbon foils and V. Honkimski from the University of Helsinki for his help during beam time at the ESRF. This work has been supported by the Bundesministerium fur Bildung, Wissenschaft, Forschung und Technologie under contract Nos. 05 SWMAAI and 05 650 WEA.
273
References
[ll
B. Williams, ed., in: Compton Scattering (McGraw-Hill,
New York, 1977).
Dl M.J. Cooper, Rep. Prog. Phys. 48 (1985) 415. [31 ‘Ph. Tschentscher, J.R. Schneider and F. Bell, Phys. Rev. B
48 (1993) 16965. 141J.R. Schneider, F. Bell, Th. Tschentscher and A.J. Rollason, Rev. Sci. instrum. 63 (1992) 1119. P. Suottti and Th. Tschentscher, Rev. Sci. Instrum. 66 (199.5) I.51 1798. [61 P. Suortti, U. Lienert and C. Schulze, Nucl. Instr. and Meth. A 338 (1994) 27. [71P. Suortti, D. Chapman, J.R. Schneider and Th. Tschentscher, J. Appl. Cryst. 25 (1992) 432. 181P. Maier-Komor, E. Ranzinger and H. Miinzer, Nucl. Instr. and Meth. 200 (1982) 5. [91 G. Dollinger, P. Maier-Komor and A. Mitwalsky, Nucl. Instr. and Meth. A 303 (1991) 79. DO1F. Salvat, J.D. Martinez, R. Mayo1 and J. Parellada, Phys. Rev. A 36 (1987) 467. [ill Lou Yongming, B. Johansson and R.M. Nieminen, J. Phys.: Condens. Matter 3 (1991) 1699. 1121E. Clementi and C. Roetti, At. Data and Nucl. Data Tab. 14 (1974) 177. [131R.R. Lee, EC. von Stetten, M. Hasegawa and S. Berko, Phys. Rev. Lett. 58 (1987) 2863. S. Tanigawa, R. Suzuki, M. Sano and H. [141I. Kanazawa, Inokuchi, Phys. Rev. B 42 (1990) 11583. [15] M. Cooper and J.A. Leak, Philos. Mag. 15 (1967) 1201. [16] A. Issolah, J. Chomillier, G. Loupias, B. Levy and A. Beswick, J. Phys. Coll. C 9 (1987) C9-847. [17] R. Tyk, J. Felsteiner, I. Germer and R. Moreh, Phys. Rev. B 32 (1985) 2625. [18] P. Eisenberger and P.M. Platzman, Phys. Rev. 2 (1970) 415. [19] R.J. Weiss, M.J. Cooper and R.S. Holt, Philos. Mag. 36 (1977) 193. [20] R.S. Holt, J.L. DuBard, M.J. Cooper, T. Paakkari and S. Manninen, Philos. Mag. B 39 (1979) 541. [21] A.D. Barlas, W. Rueckner and H.F. Wellenstein, J. Phys. B 11 (1978) 3381. 1221 P. Holm and R. Ribberfors, Phys. Rev. A 40 (1989) 625 1. [23] F. Bell, J. Chem. Phys. 85 (1986) 303. 1241 F. Bell, Th. Tschentscher, J.R. Schneider and A.J. Rollason, J. Phys. B 24 (1991) L533. 1251 A.J. Rollason and M.B.J. Woolf, J.Phys.: Condens. Matter 7 (1995) 7939.
Nuclear Instruments and Methods in Physics Research B 122 ( 1997) 269-273
__
NOMB
Beam Interactions with Materials&Atoms
@ ELSEVIER
Electron momentum density of graphite from ( y,ey) spectroscopy F.F. Kurp a, A.E. Werner a, J.R. Schneider
a, Th. Tschentscher
b, P. Suortti b, F. Bell ‘, *
’ Hamburger Synchrotronstrahlungslabor
(HASYLAB) at Deutsches Elektronen-Synchrotron (DEW), Notkestr. 85, D-22603 Hamburg, Germany b European Synchrotron Radiation Facility (ESRF), B.P. 220, F-38043 Grenoble, France ’ Sektion Physik, UniuersitZt Miinchen, Am Coulombwall 1, D-85748 Garching, Germany Received 29 March 1996; revised form received 17 May 1996.
Abstract By the coincidental detection of an inelastically scattered hard X-ray photon and its recoil electron (i.e. a so-called (r,er> experiment) cuts through the 3D-electron momentum density (EMD) of graphite have heen measured and are compared with a theoretical EMD based on a pseudopotential calculation. Peak shifts of the EMD as a function of the electron emission angle are discussed by a detailed consideration of the scattering kinematics. Experiments have been performed at the European Synchrotron Radiation Facility (ESRF) at Grenoble, France.
1. Introduction Deep inelastic photon scattering of hard X-rays has been developed as a tool to study the ground state electron momentum density (EMD) of solids [1,2]. In a conventional Compton scattering experiment the energy distribution of the photons scattered at a fixed angle can be related to the so-called Compton profile J(p,) which is a two-dirhensional integration over the EMD p(p): J( pi) = jl&)dp,dp,.
(1)
The P,-component of the electron momentum p is parallel to the momentum transfer vector q. This integration results from the fact that in conventional Compton scattering no information about the momentum distribution of the recoiling electrons is obtained. Since the main interest of solid-state physicists is in the EMD itself, it has been a long standing aim to specify the complete scattering kinematics by measuring the energy of the Compton scattered photon in coincidence with the recoiling electron which makes a direct determination of the EMD possible. The corresponding technique has been named (r,e-y) spectroscopy [3,4]. From the beginning it was recognized that intense elastic scattering of the recoiling electron within the target alters the direction of emission. This effect of multiple scattering introduces uncertainties in the reconstruction of the initial electron momentum p. Therefore,
* Corresponding
author. Fax: + 49.89-289-
0168-583X/97/$17.00 Copyright PII SOl68-583X(96)00828-2
14146.
very thin target foils have to be used so that the mean free path for elastic scattering at the Coulomb field of the nuclei is considerably larger than the target thickness. It is for this reason, together with the small value of the Klein-Nishina cross section, that strong monochromatic photon fluxes at high X-ray energies are needed which can only be delivered by modem synchrotron radiation facilities. In this paper we describe a (r,er) experiment which has been performed at the European Synchrotron Radiation Facility (ESRF). In Section 2 the scattering kinematics are discussed in some detail and in Section 3 results from the experimental EMD of graphite are presented.
2. Kinematics If an X-ray photon is scattered at an electron at rest the scattered photon energy wb will be wb=o/[l+w(l-c&)1,
(2)
where o is the primary photon energy and 8 the scattering angle. We use so-called natural units, i.e. h = m = c = 1. At the same time the recoiling electron will be ejected at an angle (Y (Fig. 1) with respect to the primary photon beam direction where cotcu= (1 + w)tan(Q/2).
(3)
The influence of an initial electron momentum p ;f 0 is twofold: scattered photons have energies w’ # wb and recoil electrons appear at angles different from (Y. Both the
0 1997 Elsevier Science B.V. All rights reserved
F.F. Kurp et al./Nucl.
270
Instr. and Meth. in Phys. Res. B 122 (1997) 269-273
essence given by the Doppler broadening A w’ = w’ - wb and the other components by the angular deviations /?, and p, of the emitted electrons. As indicated in Fig. 1, p, lies in the (&k) scattering plane and p, perpendicular to it.
Fig. 1. Vectors and angles used to describe the experiment: k and k’: momenta of the primary and scattered photon, p’: final electron momentum, qo: momentum transfer to an electron at rest. Note, that in practice q. z=-p.
energetic and angular deviations can be used to reconstruct p in a unique way. In conventional Compton scattering the natural coordinate system for the electron momentum p is that where the p,component is parallel to momentum transfer vector q = k - k’, where k and k’ are the momenta of the primary and scattered photon, see Fig. 1. For our coincident Compton scattering experiment there arises the problem that q changes direction as a function of the scattered photon energy WI. A variation of o’ which corresponds to a change of the p,-component by + 3 a.u. lets the direction of q rotate by A (Y= + 0.35” at a photon energy of 140 keV and a scattering angle 13= 140”. Though this rotation is very small, it changes electron momenta perpendicular to q by qAa = kO.36 a.u. which cannot be tolerated. Therefore, we introduce a space-fixed coordinate system where the p,-component is parallel qo, the momentum transfer to an electron at rest. In this coordinate system the Cartesian initial electron momentum components read (Fig. l>: p, = p’sinflcos
y + w’sinn - wsin ff ,
pY = p’ sin p sin y , pz = p’cos p + o’cosq
- COCOS (Y
(9
,
/3 and y are the polar and azimuthal emission angles of the recoil electron and T = 6 + (Y. The final electron momentum p’ is obtained from energy conservation p’ = [(w
- w’ + EJ2
- I] “2,
3. Experiment The experiment was performed at the high energy X-ray scattering beamline ID15 at the ESRF [5] and is shown schematically in Fig. 2. An asymmetric wiggler with 7 periods and strong poles of 1.9 T (20 mm gap) was used with a critical energy of 45 keV. The white beam was monochromatized by a horizontally focussing bend Si crystal in Bragg geometry [6,7]. At an average storage ring current of 140 mA a photon flux of about 6 X 10” photons/s at o = 135.5 keV in a beam spot of 1 mm (horizontal) X 3 mm (vertical) was obtained. The monochromaticity was A w = 300 eV (FWHM). The monochromatic X-ray beam entered an evacuated target chamber with an externally mounted intrinsic Ge-diode (energy resolution - 600 eV FWHM at 100 keV, solid angle 0.4 msr) at 6 = 140”. The not-needed photon beams are dumped in carefully shielded beam dumps to keep the X-ray background as low as possible. The recoil electrons were measured by a linear array of 7 PIN diodes placed parallel to the (i,&) scattering plane. For our experimental situation electrons initiahy at rest are emitted at (Y= 16.0”. Since the array of diodes cover a total range of transverse momenta of 5 a.u. only, especially the tails of the EMD would not be measured if the array is placed with its center at (Y= 16.0”. We therefore placed it slightly asymmetric with its center at A p, = 15 mrad corresponding roughly to an offset of Ap, = 0.9 a.u. This allows the simultaneous measurement of the center and the tail of the EMD. Each PIN diode was equipped with its own electronic circuit (pre- and main amplifier, discriminator). In a multi-coincidence mode both photon and time correlation spectra for every PIN diode could be obtained. The accuracy by which the initial electron momentum components can be determined depends on the angular and energy uncertainties of
Here, Ei = 1 - E, where E, > 0 is the electron binding energy. However, since the EMD is strongly dominated by the valence electrons it is a very good approximation to neglect E, in Eq. (5) and to discuss the experimental results in a notation which is a good approximation to Eq. (4): px = qo P, + aA J/q,
= qo P, - ( 4,/m)a~,
>
(6a)
py=qoPy.
(6”)
pz=
(6~)
-~AtJ/(qo&).
/3, = p cosy, p, = /3 siny, q. = (CO* + wf 2wwbcose>L/2 and a=osine-(l+o-w;)&. It is immediately seen from Eq. (6) that the pi component is in with
Fig. 2. Experimental set-up: SR: storage ring, W: wiggler, Si(55 1): monochromator, T: target, Ge SSD: photon solid state detector, Si SSD: PIN diode array, BD: beam dump.
F.F. Kurp et al./Nucl. Instr. and Meth. in Phys. Res. B 122 (1997) 269-273 the experiment. They include the monochromaticity of the primary photon beam and the energy resolution of the photon detector on one hand and the uncertainties of 8, p, and /3,, on the other. We estimate a total error of Ap, = 0.55 a.u. (FWHM) and Ap+ = Ap, = 0.75 a.u. (FWHM). For electrons initially at rest the scattered photon energy is wb = 92.3 keV, the kinetic electron recoil energy equals 43.2 keV and the momentum transfer amounts to q. = 58 a.u. The target was a thin carbon foil made by evaporation of pure graphite heated up to 3200 K [8]. The evaporated carbon atoms condensed on a thin Betain film (C,H , ,NO, . H,O) which had a fine crystalline-like structure. This structure acted as a replica for the carbon film guaranteeing a high mechanical stability. After evaporation the Betain film was dissolved in water and the self-supporting foils were mounted on an aluminum frame with 12 mm diameter. Finally, the foil was heat treated by a short pulse from a Nd: YAG laser at temperatures up to 4200 K. Subsequent electron diffraction investigations showed sharp graphite-like diffraction rings from which an average crystalline diameter of 10 nm was estimated [9]. By means of energy loss measurements of energetic ions a foil thickness of 17 nm after the heat treatment was determined. In view of electron multiple scattering discussed above, this thickness should be compared with the mean free path for elastic scattering of 43 keV electrons in carbon of 65 nm [lo]. From Poisson statistics we derive that 88% of the electrons leave the foil unscattered.
4. Results and discussion Fig. 3 shows a photon spectrum coincident with the electrons measured in a pixel of the PIN-diode array placed at /3, = /3, = 0. The scattered photon energy has been converted to a p;scale via Eq. (6~). Since for our experimental situation wb sine= 0.12 the data of Fig. 3 correspond to a cut through the 3D-EMD along p = ( - O.l2p,, 0, p,). The experimental data are compared
P, Fig. 3. Coincident
[a-u-l
photon spectrum for a PIN diode & = p, = 0. Solid line represents theory of Ref. [I 11.
pixel
at
87.5
90
92.5
211
95
97.5
w’[ keV] Fig. 4. Coincident photon spectra for 7 PIN diodes placed at electron emission angles p,. Broken lines are the theory of Ref. [ll] for pY = 0.5 a.“. The arrows indicate the peak positions. Either error bars or the diameter of the symbols reflect the statistical uncertainty of the data.
with an empirical pseudopotential calculation of Yongming et al. [l l] which has been spherically averaged. Theory (solid line in Fig. 3) has been convoluted with the experimental resolution and corrected for multiple electron scattering effects [3]. A (Is)* core contribution [12] has been added to the valence EMD of Ref. [ 111. The most striking feature of graphite is the dip in the EMD at p = 0 which results from the p-character of the T bonds. They are responsible for the weak van der Waals interlayer bonding in graphite and have also been identified by 2D positron annihilation meastuements [ 13,141. It is clearly seen from Fig. 3 that the experimental data show this dip at pz = 0 though it is a little sharper than predicted by theory. Details like this are completely washed out in non-coincident Compton data due to the integration of Eq. (1) and have neither been observed in our singles nor in earlier Compton profile measurements [ 15- 171. In Fig. 4 a series of coincident photon spectra are shown where the array has been placed a little out of the ($,k) scattering plane at p, = 9 mrad which corresponds to p,, = 0.5 a.u., see Eq. (6a). Again, we compare with theory (broken line). We stress that theory has been fitted to the dam by minimizing the squared deviations for all 7 spectra simultaneously. Notice that the number of coincidence events is different for each spectrum, clearly indicating that the EMD falls off for increasing &, i.e. px_ A comparison with Fig. 3 demonstrates that an offset of
F.F. Kurp et al./Nucl.
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Instr. and Meth. in Phys. Res. B 122 (1997) 269-273
pY = 0.5 a.u. suffices to cut the 3D EMD outside the dip from the rr electrons. The agreement between experiment and theory is good though not perfect. A close inspection of the spectra reveals that the maximum of the intensity does not stay at the same o’ value but shifts with p,. The arrows in Fig. 4 indicate the “center-of-gravity” energy of the data where Ni are the number &l = C,w,NJY,,N, of coincidences within a bin at 0:. This shift resembles the displacement of the maximum of the Compton profile from the prediction of the impulse approximation (IA) [ 181, observed in non-coincident Compton spectroscopy [19-211: the so-called Compton defect. In this case it signals the breakdown of the IA, the validity of which must be assumed in order to extract confident Compton profile data. It is caused by a final state interaction of the emerging electron with its nucleus and can be described by a correct first Born approximation [22,23]. Since the validity of the IA is important we want to show in the following that in contrary the shift observed in coincident Compton spectra is of kinematical nature. Due to Eq. (6a) we cut the EMD along pZ for a constant emission angle /?,, i.e. not for pX = const. but for
(7)
P,( P,) = 40 P, - (@b/w)a&.
-0.03
0.00
0.06
0.03
PX[ mrod] Fig. 6. The peak shift A wh as a function of the electron emission angle p,. The lines result from Eq. (11) for c = 0.8 (dash-dotted), 1.O (solid) and 1.2 (dashed).
Eq. (9) shows that the peak shift results not entirely from kinematics but depends on the shape of the EMD also. Assuming
where c is a de-
p = p({w)
formation parameter which for c < 1 yields a prolate and for c > 1 an oblate EMD we derive from Fq. (9)
Thus, in the (p,,p,>
plane the traces of the Doppler broadened photon spectra are not along vertical lines with px = const. but they are inclined to it by a small angle 6 with 6 = arctan( wb a/w),
aP/aPi
dPx p,=p~ = dp,’
(9)
Fig. 5. Contour plot of the EMD p(p) in the (p,,pi) plane. Straight lines indicate the cut made by a photon spectrum coincident with electron pixels at different
Pzl> = &a/w
(10)
or with Eq. (6~)
(8)
which for & = 0 yields 6 = 7” in our case. In Fig. 5 we have plotted contour lines of an isotropic EMD in the (p,,p,) plane. If one follows the coincident photon intensity along lines for p, = const., the maximum will occur at p,” values which depend on p,. Converting pr into w;, via Eq. (6~) explains the shift of Fig. 4. To be more quantitative and independent from the assumption of an isotropic EMD p the maximum count rate will occur if
ap/ap,
CP”/Px(
p,
positions.
AU;, = w;n( &) - o;(O)
= -
4024 c( W/W;)* + a2
(11)
with w&(O) = wb. In Fig. 6 we have plotted the experimental shifts A o& as a function of p,. The lines are from Eq. (11) for c = 0.8, 1.0 and 1.2. We emphasize that the signs for p, and A$,, are correlated: energy and momentum conservation demand, that for an electron emission angle closer to the primary beam direction ( p, > 0) the maximum energy 6~; is smaller than wb, i.e. A oh < 0. It is evident from Fig. 6 that the best fit holds for c = 1.O, i.e. an isotropic EMD. In the first place this means that the influence of electron multiple scattering is negligable: since this effect smears the electron emission angle /3 the measured EMD would dominantly be elongated in px and p,, directions which yields an oblate EMD with c > 1. Evidently this is not observed. From the discussion above it might have become more obvious why we have selected a cut through the EMD at p,, = 0.5 a.u.: the recrystallization of the target foil due to heat treatment may induce a slight texture with a preferential c-axis orientation perpendicular to the foil surface [9]. This would mean an anisotropic dip in the EMD originating from the r-electrons and extending mostly along the pi-direction. To avoid this we have chosen a cut through the EMD where a more isotropic part of it is expected. Finally, we address the question how essential thin targets are and if the
F.F. Kurp et aL/Nucl.
Ins&. andbleth. in Phys. Res. B 122 (1997) 269-273
method is restricted to thin carbon foils. In our case the foil was thin enough to avoid the influence of electron multiple scattering quantitatively. For thicker targets and materials with larger atomic numbers this effect certainly increases but one is able to correct for that by Monte Carlo simulations [3]. In this way, we have been able to measure the EMD from a 80 mn thin Cu target [24]. In a recent extensive Monte Carlo investigation Rollason and Woolf [25] have demonstrated that if one concentrates on the evaluation of 3D Fermi surfaces that there is in principle no limit to the sample thickness and material. The reason for this optimistic view results from the fact that sharp structures like Fermi surface breaks are still visible from the contribution of unscattered electrons originating from the last mean free path for elastic scattering (last layer effect), whereas the scattered electrons contribute to an almost flat, unstructured background only.
5. Summary It is demonstrated that the photon flux from a third generation synchrotron radiation facility (ESRF) is strong enough to measure the 3D electron momentum density of graphite along specific directions. Comparison with theory shows reasonable agreement and especially the contribution of n-bonds is revealed experimentally. Counting statistics are good enough to observe small shifts of the photon energy for which maximum coincidence count rates occur.
Acknowledgements We thank G. Dollinger and P. Maier-Komor from the Target Laboratory of the Technical University of Munich for the carbon foils and V. Honkimski from the University of Helsinki for his help during beam time at the ESRF. This work has been supported by the Bundesministerium fur Bildung, Wissenschaft, Forschung und Technologie under contract Nos. 05 SWMAAI and 05 650 WEA.
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