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Electron momentum distribution in Al and Al0.97Li0.03
Journal of Physics and Chemistry of Solids 62 (2001) 2223±2231
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Electron momentum distribution in Al and Al0.97Li0.03 P. Suortti a,b,*, T. Buslaps b, V. HonkimaÈki b, A. Shukla b, J. Kwiatkowska c, F. Maniawski c, S. Kaprzyk d,e, A. Bansil e a
Department of Physics, P.O. Box 9, FIN-00014 University of Helsinki, Finland European Synchrotron Radiation Facility, B.P. 220, F-38043 Grenoble, France c Institute of Nuclear Physics, Radzikowskiego 152, 31-342 Krakow, Poland d University of Mining and Metallurgy, al. Mickiewicza 30, 30-059 Krakow, Poland e Department of Physics, Northeastern University, Boston, MA 02115, USA b
Abstract High resolution Compton pro®les from single crystals of Al and Al0.97Li0.03 have been measured along the principal crystallographic directions [100], [110], and [111] using monochromatic synchrotron radiation at two different photon energies. The spectra were recorded by a scanning spectrometer with a momentum resolution of 0.08 a.u. for 29 keV and 0.16 a.u. for 58 keV radiation. Corresponding, highly accurate computations on Al0.97L0. 03 and Al have been carried out within the framework of the selfconsistent KKR±CPA scheme, including the limiting case of a single impurity. A good overall accord is found between theory and experiment with respect to the anisotropies as well as the amplitudes and ®ne structure in the ®rst and second derivatives of the directional pro®les. The Fermi break is found to be the sharpest along the [111] direction in Al and the alloy. The [110] direction shows an anomalously large discrepancy between theory and experiment near the Fermi momentum, suggesting that the present LDA based crystal potential does not reproduce ®ne details of the electron rings of the 3rd band in Al accurately. The effects of alloying are analyzed by taking differences between pro®les of Al and Al0.97Li0. 03, and it is shown that these results can be explained to a ®rst approximation within a free electron model by the depletion of 0.06 electrons/atom from the Fermi surface of Al. q 2001 Elsevier Science Ltd. All rights reserved. PACS: 07.85 Nc; 71.15 Mb; 71.20 Gj
1. Introduction Recent high resolution Compton scattering studies of electron momentum density (EMD) in metals have revealed interesting discrepancies when the experimental results are compared with highly accurate predictions of the local density approximation (LDA) based band theory framework [1±7]. In Li, the size of the discontinuity in the EMD at the Fermi momentum given by the standard parameter ZF has been adduced to be very small, in sharp contrast with the electron gas results of the last several decades [5±7]. Recently, however, it has been shown that ®nal state interactions in Compton scattering provide an additional broadening mechanism which can be signi®cant at low photon energies [8,9]. In Be, the electron correlation effects * Corresponding author. Tel.: 1358-9191-8596; fax: 1358-91918680. E-mail address: pekka.suortti@helsinki.® (P. Suortti).
beyond the LDA appear to be anisotropic which would require the treatment of inhomogeneous electron gas [9]. In short, the Compton scattering experiments have reached a new level of accuracy in providing a unique window for testing the conventional picture of electron correlation effects on the EMD. Al with its three nearly free valence electrons and a Fermi surface extending into the 3rd band has been a traditional touchstone of multivalent metallic systems [10]. For these reasons, there is great motivation for undertaking a state-of-the-art Compton study of Al and Al-alloys, even though this system has been the subject of numerous theoretical and experimental investigations in the literature. The quantity measured in a Compton scattering experiment is the Compton pro®le, which is the one-dimensional projection of the momentum density r (p) on the scattering vector, which de®nes the z-axis of the coordinate system, ZZ J pz r pdpx dpy ; 1
0022-3697/01/$ - see front matter q 2001 Elsevier Science Ltd. All rights reserved. PII: S 0022-369 7(01)00181-0
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where 2 X Z r p 2p23 c rexp ip´rd 3 r
2
The summation extends over all occupied states, and due to the normalization of the ground-state electron wave function c (r), the integral over the Compton pro®le is normalized to the number of the electrons in the scattering unit. Using the Bloch form of the electron wavefunctions, the spindegenerate EMD for band j can be written as X r j p 2 nj kuAj k 1 Gu2 d p 2 k 2 G: 3 Here, p k 1 G, where k denotes the crystal momentum and G the reciprocal lattice vector, nj(k) is the occupation number, and Aj is the amplitude of the Bloch wave component. In this expression, the non-diagonal elements nij, which correspond to P mixing between bands, are neglected, and the sum uAj u2 is normalized to unity. In a nearly free electron system, the main contribution to the EMD comes from the central zone, G 0, and the effects of the crystal potential appear as Umklapp contributions for non-zero G vectors [9,11]. Early Compton experiments on Al were undertaken to demonstrate the free-electron character of the valence electrons [12±14]. A fair agreement with the results of APW calculations was found within the limits of momentum resolution achieved at that time, but it was realized that multiple scattering distorted the pro®les seriously. This led to a series of studies of multiple scattering and the introduction of several Monte Carlo calculations. The approach followed was to extrapolate the results to the zero-thickness limit of the sample. In the case of Al, by invoking these corrections [15], close agreement between an APW calculation [16] and a g-ray measurement on polycrystalline Al [14] was found after the theoretical pro®le was convoluted with the experimental resolution function (FWHM 0.6 a.u.). Nevertheless, the early experiments failed to reveal differences between Compton pro®les along different crystallographic directions. The APW calculation predicted small directional differences of about 0.5% of the Compton peak, while a calculation based on the tight-binding model suggested directional differences up to 1.5% [17]. The anisotropy was reexamined a decade later using two different g-ray energies, 60 and 412 keV [18]. Improved statistical accuracy and momentum resolution revealed directional differences in fair agreement with the APW calculations. A note should be made of the early positron-annihilation (angular correlation of annihilation radiation or ACAR) measurements on Al [19]. Like Compton, ACAR also probes the electron momentum density, except that in ACAR various components in Eq. (3) are weighted by an additional factor which represents the effect of non-uniform positron spatial distribution in the crystal. For this reason, a quantitative comparison between the Compton and ACAR measurements is, of course, not appropriate. Nevertheless, the 1D-ACAR spectra obtained in the long slit geometry involve a double
momentum integral and, thus, parallel the valence electron Compton pro®le. Differences in the 1D-ACAR spectra from the three high symmetry directions in Al [20] have been analyzed and found to be in good accord with the corresponding APW calculations [16]. The 1D-projections of 2D-ACAR measurements further con®rm these ®ndings [21]. The aim of the present study is to test the EMDs obtained theoretically by using the ®rst-principles LDA based framework in Al and in a disordered Al0.97Li0.03 alloy against the results of high resolution Compton experiments. Substantial advances in the last few years have led to a most fruitful interplay between theory and Compton experiments such that it is meaningful now to address discrepancies at the level of a few tenths of a percent of the Compton peak. The availability of crystal spectrometers at high energy synchrotron radiation sources [22,23] has made it possible to measure a Compton pro®le with a momentum resolution of ,0.1 a.u. and a statistical accuracy of ,0.1% in about one day. At the ESRF, much attention has been devoted to the development of relevant experimental methodology and data handling procedures in which Al has played a special role. Our choice of Al in this connection was motivated not only by the fact that Al possesses a relatively complex Fermi surface topology, but also that a study of Al will extend the current high resolution Compton work on Li and Be into the regime of higher electron concentrations. Moreover, many previous Compton and ACAR studies provide subsidiary information and useful reference points for comparisons. Aspects of our Al data have been reported earlier [24]. Concerning Al±Li alloys, note that Al forms a solid solution at room temperature for up to about 4 at.% Li [25]. At higher Li concentrations, the metastable Al3Li phase precipitates and increases the strength of the alloys. For this reason, Al±Li alloys are widely used in applications where a high strength to weight ratio and stiffness are required. Below the 4% limit, there is little change in the unit cell volume (20.022% per 1 at.% of Li), but the concentrations of the core and valence electrons decrease more substantially, by 0.8 and 0.67% per 1 at.% of Li, respectively. We can expect subtle changes in the electronic states and the ®lling of Al bands on alloying, and these changes should be amenable to study by taking differences between the Compton pro®les of Al and Al±Li. Furthermore, these results provide a test of the mean ®eld KKR±CPA (Korringa±Kohn±Rostoker coherent-potential approximation) scheme within which the EMD in the alloy can be computed [26]. More generally, studies of the electronic properties of alloys by high resolution Compton scattering is an emerging ®eld of research, because many of the other methods fail when the translational symmetry of the crystal lattice is broken by the impurities. 2. Experiment The details of the construction and performance of
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Fig. 1. Scattering geometry for a disk-like sample of thickness T. The incident and detected beams are indicated by broken lines and arrowheads, and K is the scattering vector. The normal n of the crystal is 108 off from the [2110] direction towards the [111] direction, so that by rotation about n the principal directions [100], [110], and [111] coincide with K within 28. The scattered intensity as a function of the Bragg angle of the analyzer is sketched below ignoring absorption. The width of the angular distribution is approximately Du A T/p, where p is the distance between the sample and the analyzer.
the focusing monochromator and the scanning spectrometer at Beamline ID15B of the ESRF where the present experiments were carried out are described elsewhere [23]. Two different incident photon energies obtained by using appropriate re¯ections from horizontally focusing Bragg-type bent Si monochromators were employed: 29 keV from the (111) re¯ection and 58 keV from the (311) re¯ection. The photon beam was 0.5 mm wide and 4 mm high with a ¯ux of 1 £ 10 12 photons/sec/100 mA and an energy bandwidth of 10±20 eV. The scanning spectrometer is based on the Rowland circle geometry where the sample, the analyzer crystal and the receiving slit in the front of the detector all stay or move on the Rowland circle during the scan. When the analyzer is translated and rotated, the Rowland circle, in effect, rotates about the sample, with the receiving slit tracking the focus by synchronized rotations and translations. Although our spectrometer requires adequate monitoring of the incident beam, it possesses the advantage of yielding a low background since we energy analyze the scattered radiation at the scintillation detector and use a narrow receiving slit. The resolution of the spectrometer is dominated by the width of the effective volume of the sample as seen by the analyzer. The absorption path length, 1/m, in Al is about 3 mm at 29 keV and 12 mm at 58 keV. Therefore, the effective volume is quite wide if the standard geometry is used where the sample is a disc with its normal along the scattering vector. In our particular case, the scattering angle was Q 1728 at 29 keV and 1608 at 58 keV, so that the effective width, 2T cos(Q /2), due to beam penetration of distance T, would be excessive in the standard geometry except for very thin samples. In order to circumvent this problem, a special geometry was utilized. With the preceding remarks in mind, we brie¯y describe now the preparation of single crystals and the speci®cs of the
sample geometry used. The Al±Li alloy was made by adding Li into molten Al, followed by stirring and casting, all in an argon atmosphere. The single crystals of Al and Al±Li were grown from the melt by the Bridgman±Stockbarger method. In the case of the alloy, the crucible with the starting material was sealed in a stainless steel capsule under about 3 atm pressure of argon to prevent excessive loss of Li. The ®nal composition of the Al±Li crystal was determined by both the atomic absorption spectrometry and the PIGE (proton induced gamma-ray emission) method and was found to be 3.0 at.% Li. The crystals were oriented using Laue X-ray diffraction method and samples in the shape of discs, 9.5 mm in diameter and 0.23 mm thick, were cut along required planes with a diamond saw. The normal to the discs was arranged to be 108 off from the [2110] direction which is perpendicular to all three principal crystallographic directions [001], [110] and [111]. By orienting the normal to the disc approximately perpendicular to the scattered beam, the principal crystallographic directions can then be aligned along the scattering vector with rotations about the normal and small angular adjustments in the scattering plane, as shown in Fig. 1. In this geometry, a large active volume of the sample is achieved together with a small apparent width to obtain good energy resolution without loss of intensity. The resolution function was calculated via ray tracing and the results veri®ed by comparison with the pro®le of the elastically scattered radiation as well as the pro®les of a number of emission lines from ¯uorescent samples. The overall momentum resolution is estimated to be 0.08 a.u. for the 29 keV and 0.16 a.u. for the 58 keV incident radiation. The resolution function is slightly asymmetric due to the re¯ectivity curve of the analyzer crystal. The analyzer crystal was Si(400) in the case of 29 keV and Ge(440) in the case of 58 keV radiation. The bending
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Fig. 2. Experimental differences between various pairs of directional Compton pro®les in Al (shown as histograms) for 29 and 58 keV photons are compared with the corresponding theoretical results which have been broadened to include the effect of experimental momentum resolution (solid lines).
radius was about 6 m in both cases. During the scans, the same area of the analyzer is exposed so that effects of non-uniform analyzer response do not play a signi®cant role. A small fraction of the beam can, nevertheless, be re¯ected by oblique Bragg planes resulting in `glitches' in the recorded spectrum. We emphasize, however, that these effects are cancelled when differences between the directional Compton pro®les are taken. In fact, we ®nd that the differences between normalized, uncorrected pro®les are essentially the same as those between fully corrected pro®les. A scanning cycle involved one long scan from the elastically scattered line symmetrically to the other side of the Compton peak over momentum ranges of ^8 and ^15 a.u. for 29 and 58 keV radiation, respectively; additionally, four short scans were taken over the ^2 a.u. range about the Compton peak. The count rate was more than 3000 cps at the Compton peak, and altogether more than 10 8 counts per pro®le (10 5 ±10 6 counts at the Compton peak) were collected along each crystallographic direction from each sample. Two independent counters were used to monitor and correct for ¯uctuations in the incident ¯ux: a Si diode placed in the incident beam and a Ge detector which recorded scattering from the sample. As noted already, differences between directional pro®les, which are the focus of much of our discussion, are insensitive to systematic errors. Nevertheless, the measured spectra were corrected for the energy-dependent response of
the spectrometer, which includes the effects of absorption in the sample and the air path, and changes of the effective solid angle and re¯ectivity of the analyzer crystal. The background was resolved into components and subtracted on the basis of energy analysis at the detector. Multiple scattering was calculated separately for each dataset in order to account for small differences in the scattering geometry. The relevant Monte Carlo simulations include the effects of beam polarization, sample shape and spectrometer geometry correctly [27]. The recorded energy spectrum was converted to the momentum scale by the relation p z mcr 2 1 1 E2 1 2 cosQ=mc 2 =1 1 r 2 2 2rcosQ 1=2 ; 4 where m is the electron rest mass, c the velocity of light, E1 the energy of the incident photon, E2 that of the scattered photon, r E2/E1, and Q the scattering angle [14]. The Compton pro®le was corrected for the dependence of the scattering cross section on momentum transfer, and ®nally normalized to the theoretical value within a wide momentum range of ^10 a.u.. There is a slight residual asymmetry in the experimental pro®les due to variations in the resolution function, but this effect can be eliminated by taking an average of the pro®le at 1pz and 2pz; the regions affected by the
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Fig. 3. Experimental differences (histograms) between the valence electron Compton pro®les of Al and Al0.97Li0.03, DJAlloy J val(Al) 2 J val(Al0.97Li0.03), as measured with 29 keV radiation, are compared along various high symmetry directions with the corresponding resolution broadened theoretical results (thick solid lines). Thin solid lines give the result of the free electron model discussed in the text.
aforementioned `glitches' are of course excluded from this averaging. 3. Computations The theoretical Compton pro®les of Al and Al0.97Li0.03 have been computed within the all electron charge selfconsistent KKR±CPA framework and are parameter free [28±32]. The exchange-correlation effects were incorporated within the von Barth±Hedin local spin density approximation (LSD) [33]. The experimental lattice constant of Al Ê , and the value for at room temperature is 4.0496 A Al0.97Li0.03 is only 0.022% smaller [25]. All computations Ê (7.6534 a.u.) in this work are based on the value of 4.0496 A for both Al and Al0.97Li0.03. The momentum density r (p) was evaluated over a ®ne mesh of 48 £ 4851 £ 177 p-points covering momenta up to pmax 5 a.u. The Compton pro®le was then obtained by performing the integration in Eq. (1). The numerical accuracy of the computed pro®les is about one part in 10 4. 4. Results and discussion Anisotropy in the Compton pro®le of Al is considered in Fig. 2 which shows differences DJ between various pairs of directional CPs. The experimental results at 29 and 58 keV are seen to be quite similar, some differences within the statistics of the measurements notwithstanding. This speaks to the robustness of these data despite the fact that the maximum amplitude of the anisotropies is only of the order of 1% of the Compton peak. The nearly free electron nature of the electronic states in Al is re¯ected in the fact that DJ is quite
small for pz beyond the Fermi momentum pF 0.9255 a.u. Since the present computations are for a rigid lattice, if anything, we would expect the Umklapp higher momentum components in the experimental pro®les to be suppressed due to ®nite temperature effects [5]. In comparing theory and experiment, Fig. 2 shows a good overall level of accord. The [100]±[111] as well as the [110]±[111] differences are closely the same. In the case of the [100]±[110] anisotropy, however, the computed positive excursion near p 0 and the dip near p 0.6 a.u. are not present in the 29 keV data, although in the 58 keV data, theory and experiment are in somewhat better accord around p 0. Interestingly, there is no signi®cant discrepancy here between theory and experiment with respect to the amplitude of the anisotropies, even though the LDA has been found to predict a larger amplitude compared to measurements in a number of other cases [1,2,4±7]. Ohata et al. [34] have recently reported a high resolution Compton study of Al using a dispersive Cauchois-type analyzer with an imaging plate detector at a photon energy of 60 keV and a momentum resolution of 0.12 a.u.; the [111] Compton pro®le of an Al±Li alloy single crystal with 3 at.% Li is considered in Ref. [35]. The directional anisotropies in the experiments of Ref. [34] are very similar to those given in Fig. 2 here. In particular (see Fig. 4 of Ref. [34]; note, the plotted differences are opposite in sign to those of our Fig. 2), the [100]±[110] difference in Ref. [34] shows a small negative value near p 0 like our measurements. Interestingly, however, the dip around 0.6 a.u. predicted theoretically which, as pointed out above, is missing in our data, is seen in the data of Ref. [34] in good accord with computations. Aside from this, Ref. [34] also ®nds DJ to be quite small for values of pz beyond pF, although the absolute
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Fig. 4. The 3rd band electron rings of the Fermi surface of Al are shown schematically in the ®rst Brillouin zone; images of the rings in a few of the higher zones are also sketched. The section of the Brillouin zone in the (110) plane which includes the three high symmetry directions is shown; the electron `tubes' are perpendicular to this plane at U and K, and the rings project along the lines G ± K, X±U, and U±K. The free-electron sphere is indicated by the dotted curve.
values of DJ in this momentum range are signi®cantly larger than ours. The origin of the aforementioned small, but systematic, differences between data collected using an imaging plate detector versus those obtained on a scanningtype spectrometer is unclear and requires further study. Incidentally, we have compared and found the anisotropy and ®ne structure in the present Compton pro®les to be in qualitative accord with the corresponding positron annihilation (1D-ACAR) results on Al [16,20,21]. These two spectroscopies are, of course, not expected to agree quantitatively
since the positron spatial distribution effects involved in positron annihilation are missing in the case of the Compton scattering process. We turn now to discuss the effects of alloying on the Compton pro®le of Al with reference to Fig. 3 which shows the change, DJAlloy ; J val(Al) 2 J val(Al0.97Li0.03), in the valence electron pro®les along the high symmetry directions for 29 keV radiation. The valence pro®les here have been obtained by subtracting the theoretical core pro®les from the measured CPs. Since DJAlloy is less than 1% of J(0) and the statistical accuracy of data points is about 0.2%, two points were binned and the average of positive and negative sides was taken. The results at 58 keV are quite similar (aside from an increased resolution broadening) and are not shown in the interest of brevity. Theoretical predictions based on highly accurate KKR±CPA computations on Al0.97Li0.03 and the related single impurity limit are given by thick solid lines. The results of a simple free electron model, where the change in J val is assumed to be due to a reduction in the number of valence electrons of 0.06 per atom in going from Al to Al0.97Li0.03, are also shown (thin solid lines) as a useful point of comparison; DJAlloy is then given by the difference between two inverted parabolas, which is essentially a ¯at-topped function extending up to pF. Fig. 3 shows that the KKR±CPA as well as the experimental DJAlloy data is positive and more or less uniformly large below pF Ðmuch like the free electron modelÐindicating that this simple model captures some of the effects of alloying. However, ®ne structure in DJAlloy can be seen clearly. We emphasize that the average value ( < 0.03 units) of DJAlloy below pF is only ,1% of the Compton peak and that the amplitude of the ®ne structure in Fig. 3, which is a fraction thereof, is generally comparable to the size of the error bars on the experimental points. Therefore, caution should be exercised in making any detailed comparisons between the computed and measured ®ne structure in Fig. 3. Nevertheless, it is remarkable that the experimental DJAlloy displays signi®cant anisotropy. For example, there is a prominent dip around 0.4 a.u. along [110], a steplike feature just below the Fermi momentum along [111], and a minimum at p 0 along [100]. Insight into the nature and origin of ®ne structure in DJAlloy in Fig. 3 is obtained by recognizing that DJAlloy may be viewed as being proportional to the concentration derivative (dJ val/dc) of the valence electron pro®le. Recall that the Fermi surface of Al consists of a large hole sheet related to the 2nd band and electron `rings' which appear around the zone edges due to a small number of electrons occupying the 3rd band [36,37]. A schematic plot of the electron rings is shown in Fig. 4 where some of the rings in higher Brillouin zones are also sketched. One may now argue that the ®ne structure in (dJ val/dc) will be dominated by the contribution of states at the Fermi energy which are located around the zone boundaries (mainly the electron rings) where Bloch states deviate most strongly from being free electron like; as noted above, changes in the
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Table 1 Center (ppeak) and width (Dp) of the experimental and theoretical peaks in the second derivative of the Compton pro®le along principal crystallographic directions of Al and Al0.97Li0.03 for the 29 keV data of Fig. 6. As discussed in the text, ppeak does not give the Fermi radius pF; in fact, ppeak is systematically larger than pF due to resolution broadening effects. Atomic units are used [100] Al, pF 0.9255 ppeak,exp 0.930 ppeak,th 0.920 0.167 Dpexp Dpth 0.167 Al0.97Li0.03, pF 0.9195 ppeak, exp 0.924 0.911 ppeak,th Dpexp 0.166 Dpth 0.171
[110]
[111]
0.955 0.942 0.213 0.158
0.947 0.942 0.131 0.133
0.951 0.933 0.186 0.162
0.936 0.933 0.136 0.136
occupation of the free electron states give a roughly uniform contribution to (dJ val/dc). If so, we may expect structure in DJAlloy at momenta where the rings are crossed rapidly. With reference to Fig. 4, it can be shown that this is the case around pz 0 and pz 0.82 a.u. along [100], at
Fig. 6. Same as the caption of Fig. 5, except that this ®gure refers to the second derivatives of the pro®les obtained by taking the derivative of the data of Fig. 5.
Fig. 5. First derivatives of the Compton pro®les of Al and Al0.97Li0.03 along the principal crystallographic directions for the 29 keV data. The experimental values given by open circles have been smoothed by a Gaussian as discussed in the text. The solid lines give the corresponding theoretical derivatives which include resolution broadening. Note that the horizontal scale refers to the left hand side of the Compton peak.
pz 0.71 a.u. along [111], and along [110] the rings are projected on pz at 0.29 and 0.87 a.u. There is, indeed, some qualitative relationship between these numbers and the structure in various theory (KKR±CPA, thick solid) curves for DJAlloy. A more detailed analysis will however be necessary to identify speci®c signatures of the rings in DJAlloy which is considered beyond the scope of this paper. There are several interesting details in the valence electron pro®les near the Fermi momentum which are better studied by taking derivatives of the pro®les. Statistical ¯uctuations are ampli®ed when the derivatives are taken from the differences between neighboring data points. Accordingly, we have smoothed our data by convoluting the ®rst derivative with a Gaussian of 0.06 a.u. FWHM (full-widthat-half-maximum) for the 29 keV data and 0.12 a.u. FWHM for the 58 keV data; this convolution reduces the overall resolution by 0.02 a.u. or less. This is taken into account in the theoretical values of Table 1. Experimental and theoretical ®rst derivatives (dJ val/dpz) are compared in Fig. 5 for the 29 keV data. The corresponding second derivatives (d 2J val/dpz2) are shown in Fig. 6, with the associated peak positions and FWHMs collected in Table 1. It should be emphasized that the peak positions in (d 2J val/dpz2) are systematically higher than pF due to the well-known effect of resolution broadening. We have not attempted to model the related correction since our focus is on delineating changes in the radii with crystallographic directions and/or alloying which will be insensitive to such a correction. Moreover, when pF lies near a zone boundary,
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/dpz2)
the peak in (d J may involve more than one closely placed Fermi surface crossing. Despite these complications, Figs. 5 and 6 and the data of Table 1 provide some insight into the fermiology of Al and Al0.97Li0.03. From Table 1, the experimental peak positions are seen to be lowered upon alloying in all directions; the average estimated lowering of 0.007 a.u. is in close accord with the free electron value of 0.006 a.u. The agreement between theory and experiment with respect to the positions and widths of the peaks in both Al and Al0.97Li0.03 along the [111] direction, where the states are expected to be most free-electron like, is the best. This is less the case along [100] and [110] directions. The [110] direction is clearly anomalous, especially in Al where the shapes of experimental and theoretical peaks in Fig. 6 are quite different, suggesting that the delicate details of the electron rings are not reproduced properly by the LDAbased crystal potential used in this work. Finally, we note that the behavior of (d 2J val/dpz2) has been considered in a number of recent studies in order to obtain a handle on the size of the step ZF in momentum density at pF [5±7,24]. The present results imply that for investigating this fundamental physical parameter for assessing electron±electron correlation effects in Al and Al0.97Li0.03, the [111] direction where crystal potential effects are minimal will be a direction of choice. 5. Summary The ®ndings of the present study may be summarized as follows. In Al, the maximum differences between Compton pro®les along different crystallographic directions are of the order of 1% of the Compton peak. The LDA-based ®rst principles computations predict the amplitudes as well as the momentum dependent structure of these anisotropies rather accurately. An analysis of the ®rst and second derivatives of the pro®les indicates that, among the low-index directions, the states at the Fermi energy are most freeelectron like along the [111] direction. The [110] direction shows an anomalously large discrepancy between theory and experiment, suggesting that some of the delicate details of the 3rd band electron rings are not described correctly by the LDA in Al. We discuss the changes, DJAlloy J val(Al) 2 J val(Al0.97Li0.03), in the valence electron Compton pro®le of Al when it is alloyed with 3 at.% Li. Experimental and theoretical DJAlloy results can be described to a ®rst approximation within the free electron model where DJAlloy is given by a positive constant extending up to the Fermi momentum. Although the measured DJAlloy displays signi®cant anisotropy, the amplitude of such ®ne structure is only slightly larger than the statistical error bars on the data. Therefore, a detailed analysis of the ®ne structure in DJAlloy, and the signatures of deviations of electronic states of Al from the free electron picture contained therein, is not undertaken in this work. On a broader level, the present study taken together with other such recent work establishes the value
of high-resolution Compton data in probing details of electronic structure and fermiology in wide classes of materials. The fact that Compton scattering can be used in imperfect samples and disordered alloys, and the possibility of working at different temperatures and pressures, has the potential to open up new ®elds of applications. Close interaction between theory and experiment in interpreting high resolution Compton data has proven most valuable in identifying fruitful avenues for further theoretical and experimental research. Acknowledgements The authors thank Y. Sakurai and N. Shiotani for making available their directional Compton pro®les of Al [34] prior to publication, and A. Manuel for providing the 1D projections of his 2D-ACAR data. It is a pleasure to acknowledge important conversations with Bernardo Barbiellini and Peter Mijnarends. This work was supported by the Polish Committee for Scienti®c Research Grant 2 P03B 028 and the US Department of Energy contract W-31-109-ENG-38, and bene®ted from the allocation of supercomputing time at the NERSC and the Northeastern University Advanced Scienti®c Computation Center (ASCC). References [1] Y. Sakurai, Y. Tanaka, A. Bansil, S. Kaprzyk, A.T. Stewart, Y. Nagashima, S. Hyodo, H. Nanao, N. Kawata, Phys. Rev. Lett. 74 (1995) 2252. [2] K. HaÈmaÈlaÈinen, S. Manninen, C.-C. Kao, W. Caliebe, J.B. Hastings, A. Bansil, S. Kaprzyk, P.M. Platzman, Phys. Rev. B54 (1996) 5453. [3] M. Itou, Y. Sakurai, T. Ohata, A. Bansil, S. Kaprzyk, Y. Tanaka, H. Kawata, N. Shiotani, J. Phys. Chem. Solids 59 (1998) 99. [4] Y. Sakurai, S. Kaprzyk, A. Bansil, Y. Tanaka, G. Stutz, H. Kawata, N. Shiotani, J. Phys. Chem. Solids 60 (1999) 905. [5] W. SchuÈlke, G. Stutz, F. Wohlert, A. Kaprolat, Phys. Rev. B 54 (1996) 14381. [6] G. Stutz, F. Wohlert, A. Kaprolat, W. SchuÈlke, Y. Sakurai, Y. Tanaka, M. Ito, H. Kawata, N. Shiotani, S. Kaprzyk, A. Bansil, Phys. Rev. B 60 (1999) 7099. [7] Y. Tanaka, Y. Sakurai, A.T. Stewart, N. Shiotani, P.E. Mijnarends, S. Kaprzyk, A. Bansil, Phys. Rev. B 63 (2001) 045120. [8] C. Sternemann, K. HaÈmaÈlaÈinen, A. Kaprolat, A. Soininen, G. DoÈring, C.-C. Kao, S. Manninen, W. SchuÈlke, Phys. Rev. B 62 (2000) R7687. [9] S. Huotari, K. HaÈmaÈlaÈinen, S. Manninen, S. Kaprzyk, A. Bansil, W. Caliebe, T. Buslaps, V. HonkimaÈki, P. Suortti, Phys. Rev. B 62 (2000) 7956. [10] N.W. Ashcroft, N.D. Mermin, Solid State Physics, Holt, Rinehart and Winston, New York, 1976. [11] C. Sternemann, G. DoÈring, C. Wittkop, W. SchuÈlke, A. Shukla, T. Buslaps, P. Suortti, J. Phys. Chem. Solids 61 (2000) 379.
P. Suortti et al. / Journal of Physics and Chemistry of Solids 62 (2001) 2223±2231 [12] W.C. Phillips, R.J. Weiss, Phys. Rev. 171 (1968) 790. [13] M.J. Cooper, P. Pattison, B. Williams, K.C. Pandey, Phil. Mag. 29 (1974) 1237. [14] S. Manninen, T. Paakkari, K. Kajantie, Phil. Mag. 29 (1974) 167. [15] V. Halonen, B. Williams, Phys. Rev. B19 (1990) 1979. [16] Y. Kubo, S. Wakoh, J. Yamashita, J. Phys. Soc Japan 41 (1976) 830. [17] R.A. Tawil, Phys. Rev. B11 (1975) 4891. [18] D.A. Cardwell, M.J. Cooper, Phil. Mag. B 54 (1986) 37. [19] J. Mader, S. Berko, H. Krakauer, A. Bansil, Phys. Rev. Letters 37 (1976) 1232. [20] T. Okada, H. Sekizawa, N. Shiotani, J. Phys. Soc. Japan 41 (1976) 836. [21] A.A. Manuel, S. Samoilov, é. Fischer, M. Peter, Helv. Phys. Acta 52 (1979) 255. [22] Y. Sakurai, M. Ito, T. Urai, Y. Tanaka, N. Sakai, T. Iwazumi, H. Kawata, M. Ando, N. Shiotani, Rev. Sci. Instrum. 63 (1992) 1190. [23] P. Suortti, T. Buslaps, P. Fajardo, V. HonkimaÈki, M. Kretzschmer, U. Lienert, J.E. McCarthy, M. Renier, A. Shukla, Th. Tschentscher, T. Meinander, J. Synchrotron. Rad. 6 (1999) 69. [24] P. Suortti, T. Buslaps, V. HonkimaÈki, C. Metz, A. Shukla, Th. Tschentscher, J. Kwiatkowska, F. Maniawski, A. Bansil, S.
[25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37]
2231
Kaprzyk, A.S. Kheifets, D.R. Lun, T. Sattler, J.R. Schneider, B. Bell, J. Phys. Chem. Solids 61 (2000) 397. Landolt-BoÈrnstein, New Series IV/5a, in: O. Madelung (Ed.), Springer, Berlin, 1991. A. Bansil, Z. Naturforschung 48A (1993) 165. P. Fajardo, V. HonkimaÈki, T. Buslaps, P. Suortti, Nucl. Instrum. Methods B 134 (1998) 337. A. Bansil, S. Kaprzyk, Phys. Rev. B 43 (1991) 10335. A. Bansil, S. Kaprzyk, P.E. Mijnarends, J. Tobola, Phys. Rev. B 60 (1999) 13396. S. Kaprzyk, A. Bansil, Phys. Rev. B 42 (1990) 7358. P.E. Mijnarends, A. Bansil, Phys. Rev. B13 (1976) 2381. A. Bansil, R.S. Rao, P.E. Mijnarends, L. Schwartz, Phys. Rev. B 23 (1981) 3608. U. von Barth, L. Hedin, J. Phys. C 5 (1972) 1629. T. Ohata, M. Itou, I. Matsumoto, Y. Sakurai, H. Kawata, N. Shiotani, S. Kaprzyk, P.E. Mijnarends, A. Bansil, Phys. Rev. B 62 (2000) 16528. I. Matsumoto, J. Kwiatkowska, F. Maniawski, S. Kaprzyk, A. Bansil, M. Itou, H. Kawata, N. Shiotani, J. Phys. Chem Solids 61 (2000) 375. N.W. Ashcroft, Phil. Mag. 8 (1963) 2055. E.P. Volski, Zh. Eksp. Teor. Fiz. 46 (1964) 123 (Sov. Phys. JETP 19, 89).
www.elsevier.com/locate/jpcs
Electron momentum distribution in Al and Al0.97Li0.03 P. Suortti a,b,*, T. Buslaps b, V. HonkimaÈki b, A. Shukla b, J. Kwiatkowska c, F. Maniawski c, S. Kaprzyk d,e, A. Bansil e a
Department of Physics, P.O. Box 9, FIN-00014 University of Helsinki, Finland European Synchrotron Radiation Facility, B.P. 220, F-38043 Grenoble, France c Institute of Nuclear Physics, Radzikowskiego 152, 31-342 Krakow, Poland d University of Mining and Metallurgy, al. Mickiewicza 30, 30-059 Krakow, Poland e Department of Physics, Northeastern University, Boston, MA 02115, USA b
Abstract High resolution Compton pro®les from single crystals of Al and Al0.97Li0.03 have been measured along the principal crystallographic directions [100], [110], and [111] using monochromatic synchrotron radiation at two different photon energies. The spectra were recorded by a scanning spectrometer with a momentum resolution of 0.08 a.u. for 29 keV and 0.16 a.u. for 58 keV radiation. Corresponding, highly accurate computations on Al0.97L0. 03 and Al have been carried out within the framework of the selfconsistent KKR±CPA scheme, including the limiting case of a single impurity. A good overall accord is found between theory and experiment with respect to the anisotropies as well as the amplitudes and ®ne structure in the ®rst and second derivatives of the directional pro®les. The Fermi break is found to be the sharpest along the [111] direction in Al and the alloy. The [110] direction shows an anomalously large discrepancy between theory and experiment near the Fermi momentum, suggesting that the present LDA based crystal potential does not reproduce ®ne details of the electron rings of the 3rd band in Al accurately. The effects of alloying are analyzed by taking differences between pro®les of Al and Al0.97Li0. 03, and it is shown that these results can be explained to a ®rst approximation within a free electron model by the depletion of 0.06 electrons/atom from the Fermi surface of Al. q 2001 Elsevier Science Ltd. All rights reserved. PACS: 07.85 Nc; 71.15 Mb; 71.20 Gj
1. Introduction Recent high resolution Compton scattering studies of electron momentum density (EMD) in metals have revealed interesting discrepancies when the experimental results are compared with highly accurate predictions of the local density approximation (LDA) based band theory framework [1±7]. In Li, the size of the discontinuity in the EMD at the Fermi momentum given by the standard parameter ZF has been adduced to be very small, in sharp contrast with the electron gas results of the last several decades [5±7]. Recently, however, it has been shown that ®nal state interactions in Compton scattering provide an additional broadening mechanism which can be signi®cant at low photon energies [8,9]. In Be, the electron correlation effects * Corresponding author. Tel.: 1358-9191-8596; fax: 1358-91918680. E-mail address: pekka.suortti@helsinki.® (P. Suortti).
beyond the LDA appear to be anisotropic which would require the treatment of inhomogeneous electron gas [9]. In short, the Compton scattering experiments have reached a new level of accuracy in providing a unique window for testing the conventional picture of electron correlation effects on the EMD. Al with its three nearly free valence electrons and a Fermi surface extending into the 3rd band has been a traditional touchstone of multivalent metallic systems [10]. For these reasons, there is great motivation for undertaking a state-of-the-art Compton study of Al and Al-alloys, even though this system has been the subject of numerous theoretical and experimental investigations in the literature. The quantity measured in a Compton scattering experiment is the Compton pro®le, which is the one-dimensional projection of the momentum density r (p) on the scattering vector, which de®nes the z-axis of the coordinate system, ZZ J pz r pdpx dpy ; 1
0022-3697/01/$ - see front matter q 2001 Elsevier Science Ltd. All rights reserved. PII: S 0022-369 7(01)00181-0
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where 2 X Z r p 2p23 c rexp ip´rd 3 r
2
The summation extends over all occupied states, and due to the normalization of the ground-state electron wave function c (r), the integral over the Compton pro®le is normalized to the number of the electrons in the scattering unit. Using the Bloch form of the electron wavefunctions, the spindegenerate EMD for band j can be written as X r j p 2 nj kuAj k 1 Gu2 d p 2 k 2 G: 3 Here, p k 1 G, where k denotes the crystal momentum and G the reciprocal lattice vector, nj(k) is the occupation number, and Aj is the amplitude of the Bloch wave component. In this expression, the non-diagonal elements nij, which correspond to P mixing between bands, are neglected, and the sum uAj u2 is normalized to unity. In a nearly free electron system, the main contribution to the EMD comes from the central zone, G 0, and the effects of the crystal potential appear as Umklapp contributions for non-zero G vectors [9,11]. Early Compton experiments on Al were undertaken to demonstrate the free-electron character of the valence electrons [12±14]. A fair agreement with the results of APW calculations was found within the limits of momentum resolution achieved at that time, but it was realized that multiple scattering distorted the pro®les seriously. This led to a series of studies of multiple scattering and the introduction of several Monte Carlo calculations. The approach followed was to extrapolate the results to the zero-thickness limit of the sample. In the case of Al, by invoking these corrections [15], close agreement between an APW calculation [16] and a g-ray measurement on polycrystalline Al [14] was found after the theoretical pro®le was convoluted with the experimental resolution function (FWHM 0.6 a.u.). Nevertheless, the early experiments failed to reveal differences between Compton pro®les along different crystallographic directions. The APW calculation predicted small directional differences of about 0.5% of the Compton peak, while a calculation based on the tight-binding model suggested directional differences up to 1.5% [17]. The anisotropy was reexamined a decade later using two different g-ray energies, 60 and 412 keV [18]. Improved statistical accuracy and momentum resolution revealed directional differences in fair agreement with the APW calculations. A note should be made of the early positron-annihilation (angular correlation of annihilation radiation or ACAR) measurements on Al [19]. Like Compton, ACAR also probes the electron momentum density, except that in ACAR various components in Eq. (3) are weighted by an additional factor which represents the effect of non-uniform positron spatial distribution in the crystal. For this reason, a quantitative comparison between the Compton and ACAR measurements is, of course, not appropriate. Nevertheless, the 1D-ACAR spectra obtained in the long slit geometry involve a double
momentum integral and, thus, parallel the valence electron Compton pro®le. Differences in the 1D-ACAR spectra from the three high symmetry directions in Al [20] have been analyzed and found to be in good accord with the corresponding APW calculations [16]. The 1D-projections of 2D-ACAR measurements further con®rm these ®ndings [21]. The aim of the present study is to test the EMDs obtained theoretically by using the ®rst-principles LDA based framework in Al and in a disordered Al0.97Li0.03 alloy against the results of high resolution Compton experiments. Substantial advances in the last few years have led to a most fruitful interplay between theory and Compton experiments such that it is meaningful now to address discrepancies at the level of a few tenths of a percent of the Compton peak. The availability of crystal spectrometers at high energy synchrotron radiation sources [22,23] has made it possible to measure a Compton pro®le with a momentum resolution of ,0.1 a.u. and a statistical accuracy of ,0.1% in about one day. At the ESRF, much attention has been devoted to the development of relevant experimental methodology and data handling procedures in which Al has played a special role. Our choice of Al in this connection was motivated not only by the fact that Al possesses a relatively complex Fermi surface topology, but also that a study of Al will extend the current high resolution Compton work on Li and Be into the regime of higher electron concentrations. Moreover, many previous Compton and ACAR studies provide subsidiary information and useful reference points for comparisons. Aspects of our Al data have been reported earlier [24]. Concerning Al±Li alloys, note that Al forms a solid solution at room temperature for up to about 4 at.% Li [25]. At higher Li concentrations, the metastable Al3Li phase precipitates and increases the strength of the alloys. For this reason, Al±Li alloys are widely used in applications where a high strength to weight ratio and stiffness are required. Below the 4% limit, there is little change in the unit cell volume (20.022% per 1 at.% of Li), but the concentrations of the core and valence electrons decrease more substantially, by 0.8 and 0.67% per 1 at.% of Li, respectively. We can expect subtle changes in the electronic states and the ®lling of Al bands on alloying, and these changes should be amenable to study by taking differences between the Compton pro®les of Al and Al±Li. Furthermore, these results provide a test of the mean ®eld KKR±CPA (Korringa±Kohn±Rostoker coherent-potential approximation) scheme within which the EMD in the alloy can be computed [26]. More generally, studies of the electronic properties of alloys by high resolution Compton scattering is an emerging ®eld of research, because many of the other methods fail when the translational symmetry of the crystal lattice is broken by the impurities. 2. Experiment The details of the construction and performance of
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Fig. 1. Scattering geometry for a disk-like sample of thickness T. The incident and detected beams are indicated by broken lines and arrowheads, and K is the scattering vector. The normal n of the crystal is 108 off from the [2110] direction towards the [111] direction, so that by rotation about n the principal directions [100], [110], and [111] coincide with K within 28. The scattered intensity as a function of the Bragg angle of the analyzer is sketched below ignoring absorption. The width of the angular distribution is approximately Du A T/p, where p is the distance between the sample and the analyzer.
the focusing monochromator and the scanning spectrometer at Beamline ID15B of the ESRF where the present experiments were carried out are described elsewhere [23]. Two different incident photon energies obtained by using appropriate re¯ections from horizontally focusing Bragg-type bent Si monochromators were employed: 29 keV from the (111) re¯ection and 58 keV from the (311) re¯ection. The photon beam was 0.5 mm wide and 4 mm high with a ¯ux of 1 £ 10 12 photons/sec/100 mA and an energy bandwidth of 10±20 eV. The scanning spectrometer is based on the Rowland circle geometry where the sample, the analyzer crystal and the receiving slit in the front of the detector all stay or move on the Rowland circle during the scan. When the analyzer is translated and rotated, the Rowland circle, in effect, rotates about the sample, with the receiving slit tracking the focus by synchronized rotations and translations. Although our spectrometer requires adequate monitoring of the incident beam, it possesses the advantage of yielding a low background since we energy analyze the scattered radiation at the scintillation detector and use a narrow receiving slit. The resolution of the spectrometer is dominated by the width of the effective volume of the sample as seen by the analyzer. The absorption path length, 1/m, in Al is about 3 mm at 29 keV and 12 mm at 58 keV. Therefore, the effective volume is quite wide if the standard geometry is used where the sample is a disc with its normal along the scattering vector. In our particular case, the scattering angle was Q 1728 at 29 keV and 1608 at 58 keV, so that the effective width, 2T cos(Q /2), due to beam penetration of distance T, would be excessive in the standard geometry except for very thin samples. In order to circumvent this problem, a special geometry was utilized. With the preceding remarks in mind, we brie¯y describe now the preparation of single crystals and the speci®cs of the
sample geometry used. The Al±Li alloy was made by adding Li into molten Al, followed by stirring and casting, all in an argon atmosphere. The single crystals of Al and Al±Li were grown from the melt by the Bridgman±Stockbarger method. In the case of the alloy, the crucible with the starting material was sealed in a stainless steel capsule under about 3 atm pressure of argon to prevent excessive loss of Li. The ®nal composition of the Al±Li crystal was determined by both the atomic absorption spectrometry and the PIGE (proton induced gamma-ray emission) method and was found to be 3.0 at.% Li. The crystals were oriented using Laue X-ray diffraction method and samples in the shape of discs, 9.5 mm in diameter and 0.23 mm thick, were cut along required planes with a diamond saw. The normal to the discs was arranged to be 108 off from the [2110] direction which is perpendicular to all three principal crystallographic directions [001], [110] and [111]. By orienting the normal to the disc approximately perpendicular to the scattered beam, the principal crystallographic directions can then be aligned along the scattering vector with rotations about the normal and small angular adjustments in the scattering plane, as shown in Fig. 1. In this geometry, a large active volume of the sample is achieved together with a small apparent width to obtain good energy resolution without loss of intensity. The resolution function was calculated via ray tracing and the results veri®ed by comparison with the pro®le of the elastically scattered radiation as well as the pro®les of a number of emission lines from ¯uorescent samples. The overall momentum resolution is estimated to be 0.08 a.u. for the 29 keV and 0.16 a.u. for the 58 keV incident radiation. The resolution function is slightly asymmetric due to the re¯ectivity curve of the analyzer crystal. The analyzer crystal was Si(400) in the case of 29 keV and Ge(440) in the case of 58 keV radiation. The bending
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Fig. 2. Experimental differences between various pairs of directional Compton pro®les in Al (shown as histograms) for 29 and 58 keV photons are compared with the corresponding theoretical results which have been broadened to include the effect of experimental momentum resolution (solid lines).
radius was about 6 m in both cases. During the scans, the same area of the analyzer is exposed so that effects of non-uniform analyzer response do not play a signi®cant role. A small fraction of the beam can, nevertheless, be re¯ected by oblique Bragg planes resulting in `glitches' in the recorded spectrum. We emphasize, however, that these effects are cancelled when differences between the directional Compton pro®les are taken. In fact, we ®nd that the differences between normalized, uncorrected pro®les are essentially the same as those between fully corrected pro®les. A scanning cycle involved one long scan from the elastically scattered line symmetrically to the other side of the Compton peak over momentum ranges of ^8 and ^15 a.u. for 29 and 58 keV radiation, respectively; additionally, four short scans were taken over the ^2 a.u. range about the Compton peak. The count rate was more than 3000 cps at the Compton peak, and altogether more than 10 8 counts per pro®le (10 5 ±10 6 counts at the Compton peak) were collected along each crystallographic direction from each sample. Two independent counters were used to monitor and correct for ¯uctuations in the incident ¯ux: a Si diode placed in the incident beam and a Ge detector which recorded scattering from the sample. As noted already, differences between directional pro®les, which are the focus of much of our discussion, are insensitive to systematic errors. Nevertheless, the measured spectra were corrected for the energy-dependent response of
the spectrometer, which includes the effects of absorption in the sample and the air path, and changes of the effective solid angle and re¯ectivity of the analyzer crystal. The background was resolved into components and subtracted on the basis of energy analysis at the detector. Multiple scattering was calculated separately for each dataset in order to account for small differences in the scattering geometry. The relevant Monte Carlo simulations include the effects of beam polarization, sample shape and spectrometer geometry correctly [27]. The recorded energy spectrum was converted to the momentum scale by the relation p z mcr 2 1 1 E2 1 2 cosQ=mc 2 =1 1 r 2 2 2rcosQ 1=2 ; 4 where m is the electron rest mass, c the velocity of light, E1 the energy of the incident photon, E2 that of the scattered photon, r E2/E1, and Q the scattering angle [14]. The Compton pro®le was corrected for the dependence of the scattering cross section on momentum transfer, and ®nally normalized to the theoretical value within a wide momentum range of ^10 a.u.. There is a slight residual asymmetry in the experimental pro®les due to variations in the resolution function, but this effect can be eliminated by taking an average of the pro®le at 1pz and 2pz; the regions affected by the
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2227
Fig. 3. Experimental differences (histograms) between the valence electron Compton pro®les of Al and Al0.97Li0.03, DJAlloy J val(Al) 2 J val(Al0.97Li0.03), as measured with 29 keV radiation, are compared along various high symmetry directions with the corresponding resolution broadened theoretical results (thick solid lines). Thin solid lines give the result of the free electron model discussed in the text.
aforementioned `glitches' are of course excluded from this averaging. 3. Computations The theoretical Compton pro®les of Al and Al0.97Li0.03 have been computed within the all electron charge selfconsistent KKR±CPA framework and are parameter free [28±32]. The exchange-correlation effects were incorporated within the von Barth±Hedin local spin density approximation (LSD) [33]. The experimental lattice constant of Al Ê , and the value for at room temperature is 4.0496 A Al0.97Li0.03 is only 0.022% smaller [25]. All computations Ê (7.6534 a.u.) in this work are based on the value of 4.0496 A for both Al and Al0.97Li0.03. The momentum density r (p) was evaluated over a ®ne mesh of 48 £ 4851 £ 177 p-points covering momenta up to pmax 5 a.u. The Compton pro®le was then obtained by performing the integration in Eq. (1). The numerical accuracy of the computed pro®les is about one part in 10 4. 4. Results and discussion Anisotropy in the Compton pro®le of Al is considered in Fig. 2 which shows differences DJ between various pairs of directional CPs. The experimental results at 29 and 58 keV are seen to be quite similar, some differences within the statistics of the measurements notwithstanding. This speaks to the robustness of these data despite the fact that the maximum amplitude of the anisotropies is only of the order of 1% of the Compton peak. The nearly free electron nature of the electronic states in Al is re¯ected in the fact that DJ is quite
small for pz beyond the Fermi momentum pF 0.9255 a.u. Since the present computations are for a rigid lattice, if anything, we would expect the Umklapp higher momentum components in the experimental pro®les to be suppressed due to ®nite temperature effects [5]. In comparing theory and experiment, Fig. 2 shows a good overall level of accord. The [100]±[111] as well as the [110]±[111] differences are closely the same. In the case of the [100]±[110] anisotropy, however, the computed positive excursion near p 0 and the dip near p 0.6 a.u. are not present in the 29 keV data, although in the 58 keV data, theory and experiment are in somewhat better accord around p 0. Interestingly, there is no signi®cant discrepancy here between theory and experiment with respect to the amplitude of the anisotropies, even though the LDA has been found to predict a larger amplitude compared to measurements in a number of other cases [1,2,4±7]. Ohata et al. [34] have recently reported a high resolution Compton study of Al using a dispersive Cauchois-type analyzer with an imaging plate detector at a photon energy of 60 keV and a momentum resolution of 0.12 a.u.; the [111] Compton pro®le of an Al±Li alloy single crystal with 3 at.% Li is considered in Ref. [35]. The directional anisotropies in the experiments of Ref. [34] are very similar to those given in Fig. 2 here. In particular (see Fig. 4 of Ref. [34]; note, the plotted differences are opposite in sign to those of our Fig. 2), the [100]±[110] difference in Ref. [34] shows a small negative value near p 0 like our measurements. Interestingly, however, the dip around 0.6 a.u. predicted theoretically which, as pointed out above, is missing in our data, is seen in the data of Ref. [34] in good accord with computations. Aside from this, Ref. [34] also ®nds DJ to be quite small for values of pz beyond pF, although the absolute
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Fig. 4. The 3rd band electron rings of the Fermi surface of Al are shown schematically in the ®rst Brillouin zone; images of the rings in a few of the higher zones are also sketched. The section of the Brillouin zone in the (110) plane which includes the three high symmetry directions is shown; the electron `tubes' are perpendicular to this plane at U and K, and the rings project along the lines G ± K, X±U, and U±K. The free-electron sphere is indicated by the dotted curve.
values of DJ in this momentum range are signi®cantly larger than ours. The origin of the aforementioned small, but systematic, differences between data collected using an imaging plate detector versus those obtained on a scanningtype spectrometer is unclear and requires further study. Incidentally, we have compared and found the anisotropy and ®ne structure in the present Compton pro®les to be in qualitative accord with the corresponding positron annihilation (1D-ACAR) results on Al [16,20,21]. These two spectroscopies are, of course, not expected to agree quantitatively
since the positron spatial distribution effects involved in positron annihilation are missing in the case of the Compton scattering process. We turn now to discuss the effects of alloying on the Compton pro®le of Al with reference to Fig. 3 which shows the change, DJAlloy ; J val(Al) 2 J val(Al0.97Li0.03), in the valence electron pro®les along the high symmetry directions for 29 keV radiation. The valence pro®les here have been obtained by subtracting the theoretical core pro®les from the measured CPs. Since DJAlloy is less than 1% of J(0) and the statistical accuracy of data points is about 0.2%, two points were binned and the average of positive and negative sides was taken. The results at 58 keV are quite similar (aside from an increased resolution broadening) and are not shown in the interest of brevity. Theoretical predictions based on highly accurate KKR±CPA computations on Al0.97Li0.03 and the related single impurity limit are given by thick solid lines. The results of a simple free electron model, where the change in J val is assumed to be due to a reduction in the number of valence electrons of 0.06 per atom in going from Al to Al0.97Li0.03, are also shown (thin solid lines) as a useful point of comparison; DJAlloy is then given by the difference between two inverted parabolas, which is essentially a ¯at-topped function extending up to pF. Fig. 3 shows that the KKR±CPA as well as the experimental DJAlloy data is positive and more or less uniformly large below pF Ðmuch like the free electron modelÐindicating that this simple model captures some of the effects of alloying. However, ®ne structure in DJAlloy can be seen clearly. We emphasize that the average value ( < 0.03 units) of DJAlloy below pF is only ,1% of the Compton peak and that the amplitude of the ®ne structure in Fig. 3, which is a fraction thereof, is generally comparable to the size of the error bars on the experimental points. Therefore, caution should be exercised in making any detailed comparisons between the computed and measured ®ne structure in Fig. 3. Nevertheless, it is remarkable that the experimental DJAlloy displays signi®cant anisotropy. For example, there is a prominent dip around 0.4 a.u. along [110], a steplike feature just below the Fermi momentum along [111], and a minimum at p 0 along [100]. Insight into the nature and origin of ®ne structure in DJAlloy in Fig. 3 is obtained by recognizing that DJAlloy may be viewed as being proportional to the concentration derivative (dJ val/dc) of the valence electron pro®le. Recall that the Fermi surface of Al consists of a large hole sheet related to the 2nd band and electron `rings' which appear around the zone edges due to a small number of electrons occupying the 3rd band [36,37]. A schematic plot of the electron rings is shown in Fig. 4 where some of the rings in higher Brillouin zones are also sketched. One may now argue that the ®ne structure in (dJ val/dc) will be dominated by the contribution of states at the Fermi energy which are located around the zone boundaries (mainly the electron rings) where Bloch states deviate most strongly from being free electron like; as noted above, changes in the
P. Suortti et al. / Journal of Physics and Chemistry of Solids 62 (2001) 2223±2231
2229
Table 1 Center (ppeak) and width (Dp) of the experimental and theoretical peaks in the second derivative of the Compton pro®le along principal crystallographic directions of Al and Al0.97Li0.03 for the 29 keV data of Fig. 6. As discussed in the text, ppeak does not give the Fermi radius pF; in fact, ppeak is systematically larger than pF due to resolution broadening effects. Atomic units are used [100] Al, pF 0.9255 ppeak,exp 0.930 ppeak,th 0.920 0.167 Dpexp Dpth 0.167 Al0.97Li0.03, pF 0.9195 ppeak, exp 0.924 0.911 ppeak,th Dpexp 0.166 Dpth 0.171
[110]
[111]
0.955 0.942 0.213 0.158
0.947 0.942 0.131 0.133
0.951 0.933 0.186 0.162
0.936 0.933 0.136 0.136
occupation of the free electron states give a roughly uniform contribution to (dJ val/dc). If so, we may expect structure in DJAlloy at momenta where the rings are crossed rapidly. With reference to Fig. 4, it can be shown that this is the case around pz 0 and pz 0.82 a.u. along [100], at
Fig. 6. Same as the caption of Fig. 5, except that this ®gure refers to the second derivatives of the pro®les obtained by taking the derivative of the data of Fig. 5.
Fig. 5. First derivatives of the Compton pro®les of Al and Al0.97Li0.03 along the principal crystallographic directions for the 29 keV data. The experimental values given by open circles have been smoothed by a Gaussian as discussed in the text. The solid lines give the corresponding theoretical derivatives which include resolution broadening. Note that the horizontal scale refers to the left hand side of the Compton peak.
pz 0.71 a.u. along [111], and along [110] the rings are projected on pz at 0.29 and 0.87 a.u. There is, indeed, some qualitative relationship between these numbers and the structure in various theory (KKR±CPA, thick solid) curves for DJAlloy. A more detailed analysis will however be necessary to identify speci®c signatures of the rings in DJAlloy which is considered beyond the scope of this paper. There are several interesting details in the valence electron pro®les near the Fermi momentum which are better studied by taking derivatives of the pro®les. Statistical ¯uctuations are ampli®ed when the derivatives are taken from the differences between neighboring data points. Accordingly, we have smoothed our data by convoluting the ®rst derivative with a Gaussian of 0.06 a.u. FWHM (full-widthat-half-maximum) for the 29 keV data and 0.12 a.u. FWHM for the 58 keV data; this convolution reduces the overall resolution by 0.02 a.u. or less. This is taken into account in the theoretical values of Table 1. Experimental and theoretical ®rst derivatives (dJ val/dpz) are compared in Fig. 5 for the 29 keV data. The corresponding second derivatives (d 2J val/dpz2) are shown in Fig. 6, with the associated peak positions and FWHMs collected in Table 1. It should be emphasized that the peak positions in (d 2J val/dpz2) are systematically higher than pF due to the well-known effect of resolution broadening. We have not attempted to model the related correction since our focus is on delineating changes in the radii with crystallographic directions and/or alloying which will be insensitive to such a correction. Moreover, when pF lies near a zone boundary,
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P. Suortti et al. / Journal of Physics and Chemistry of Solids 62 (2001) 2223±2231 2 val
/dpz2)
the peak in (d J may involve more than one closely placed Fermi surface crossing. Despite these complications, Figs. 5 and 6 and the data of Table 1 provide some insight into the fermiology of Al and Al0.97Li0.03. From Table 1, the experimental peak positions are seen to be lowered upon alloying in all directions; the average estimated lowering of 0.007 a.u. is in close accord with the free electron value of 0.006 a.u. The agreement between theory and experiment with respect to the positions and widths of the peaks in both Al and Al0.97Li0.03 along the [111] direction, where the states are expected to be most free-electron like, is the best. This is less the case along [100] and [110] directions. The [110] direction is clearly anomalous, especially in Al where the shapes of experimental and theoretical peaks in Fig. 6 are quite different, suggesting that the delicate details of the electron rings are not reproduced properly by the LDAbased crystal potential used in this work. Finally, we note that the behavior of (d 2J val/dpz2) has been considered in a number of recent studies in order to obtain a handle on the size of the step ZF in momentum density at pF [5±7,24]. The present results imply that for investigating this fundamental physical parameter for assessing electron±electron correlation effects in Al and Al0.97Li0.03, the [111] direction where crystal potential effects are minimal will be a direction of choice. 5. Summary The ®ndings of the present study may be summarized as follows. In Al, the maximum differences between Compton pro®les along different crystallographic directions are of the order of 1% of the Compton peak. The LDA-based ®rst principles computations predict the amplitudes as well as the momentum dependent structure of these anisotropies rather accurately. An analysis of the ®rst and second derivatives of the pro®les indicates that, among the low-index directions, the states at the Fermi energy are most freeelectron like along the [111] direction. The [110] direction shows an anomalously large discrepancy between theory and experiment, suggesting that some of the delicate details of the 3rd band electron rings are not described correctly by the LDA in Al. We discuss the changes, DJAlloy J val(Al) 2 J val(Al0.97Li0.03), in the valence electron Compton pro®le of Al when it is alloyed with 3 at.% Li. Experimental and theoretical DJAlloy results can be described to a ®rst approximation within the free electron model where DJAlloy is given by a positive constant extending up to the Fermi momentum. Although the measured DJAlloy displays signi®cant anisotropy, the amplitude of such ®ne structure is only slightly larger than the statistical error bars on the data. Therefore, a detailed analysis of the ®ne structure in DJAlloy, and the signatures of deviations of electronic states of Al from the free electron picture contained therein, is not undertaken in this work. On a broader level, the present study taken together with other such recent work establishes the value
of high-resolution Compton data in probing details of electronic structure and fermiology in wide classes of materials. The fact that Compton scattering can be used in imperfect samples and disordered alloys, and the possibility of working at different temperatures and pressures, has the potential to open up new ®elds of applications. Close interaction between theory and experiment in interpreting high resolution Compton data has proven most valuable in identifying fruitful avenues for further theoretical and experimental research. Acknowledgements The authors thank Y. Sakurai and N. Shiotani for making available their directional Compton pro®les of Al [34] prior to publication, and A. Manuel for providing the 1D projections of his 2D-ACAR data. It is a pleasure to acknowledge important conversations with Bernardo Barbiellini and Peter Mijnarends. This work was supported by the Polish Committee for Scienti®c Research Grant 2 P03B 028 and the US Department of Energy contract W-31-109-ENG-38, and bene®ted from the allocation of supercomputing time at the NERSC and the Northeastern University Advanced Scienti®c Computation Center (ASCC). References [1] Y. Sakurai, Y. Tanaka, A. Bansil, S. Kaprzyk, A.T. Stewart, Y. Nagashima, S. Hyodo, H. Nanao, N. Kawata, Phys. Rev. Lett. 74 (1995) 2252. [2] K. HaÈmaÈlaÈinen, S. Manninen, C.-C. Kao, W. Caliebe, J.B. Hastings, A. Bansil, S. Kaprzyk, P.M. Platzman, Phys. Rev. B54 (1996) 5453. [3] M. Itou, Y. Sakurai, T. Ohata, A. Bansil, S. Kaprzyk, Y. Tanaka, H. Kawata, N. Shiotani, J. Phys. Chem. Solids 59 (1998) 99. [4] Y. Sakurai, S. Kaprzyk, A. Bansil, Y. Tanaka, G. Stutz, H. Kawata, N. Shiotani, J. Phys. Chem. Solids 60 (1999) 905. [5] W. SchuÈlke, G. Stutz, F. Wohlert, A. Kaprolat, Phys. Rev. B 54 (1996) 14381. [6] G. Stutz, F. Wohlert, A. Kaprolat, W. SchuÈlke, Y. Sakurai, Y. Tanaka, M. Ito, H. Kawata, N. Shiotani, S. Kaprzyk, A. Bansil, Phys. Rev. B 60 (1999) 7099. [7] Y. Tanaka, Y. Sakurai, A.T. Stewart, N. Shiotani, P.E. Mijnarends, S. Kaprzyk, A. Bansil, Phys. Rev. B 63 (2001) 045120. [8] C. Sternemann, K. HaÈmaÈlaÈinen, A. Kaprolat, A. Soininen, G. DoÈring, C.-C. Kao, S. Manninen, W. SchuÈlke, Phys. Rev. B 62 (2000) R7687. [9] S. Huotari, K. HaÈmaÈlaÈinen, S. Manninen, S. Kaprzyk, A. Bansil, W. Caliebe, T. Buslaps, V. HonkimaÈki, P. Suortti, Phys. Rev. B 62 (2000) 7956. [10] N.W. Ashcroft, N.D. Mermin, Solid State Physics, Holt, Rinehart and Winston, New York, 1976. [11] C. Sternemann, G. DoÈring, C. Wittkop, W. SchuÈlke, A. Shukla, T. Buslaps, P. Suortti, J. Phys. Chem. Solids 61 (2000) 379.
P. Suortti et al. / Journal of Physics and Chemistry of Solids 62 (2001) 2223±2231 [12] W.C. Phillips, R.J. Weiss, Phys. Rev. 171 (1968) 790. [13] M.J. Cooper, P. Pattison, B. Williams, K.C. Pandey, Phil. Mag. 29 (1974) 1237. [14] S. Manninen, T. Paakkari, K. Kajantie, Phil. Mag. 29 (1974) 167. [15] V. Halonen, B. Williams, Phys. Rev. B19 (1990) 1979. [16] Y. Kubo, S. Wakoh, J. Yamashita, J. Phys. Soc Japan 41 (1976) 830. [17] R.A. Tawil, Phys. Rev. B11 (1975) 4891. [18] D.A. Cardwell, M.J. Cooper, Phil. Mag. B 54 (1986) 37. [19] J. Mader, S. Berko, H. Krakauer, A. Bansil, Phys. Rev. Letters 37 (1976) 1232. [20] T. Okada, H. Sekizawa, N. Shiotani, J. Phys. Soc. Japan 41 (1976) 836. [21] A.A. Manuel, S. Samoilov, é. Fischer, M. Peter, Helv. Phys. Acta 52 (1979) 255. [22] Y. Sakurai, M. Ito, T. Urai, Y. Tanaka, N. Sakai, T. Iwazumi, H. Kawata, M. Ando, N. Shiotani, Rev. Sci. Instrum. 63 (1992) 1190. [23] P. Suortti, T. Buslaps, P. Fajardo, V. HonkimaÈki, M. Kretzschmer, U. Lienert, J.E. McCarthy, M. Renier, A. Shukla, Th. Tschentscher, T. Meinander, J. Synchrotron. Rad. 6 (1999) 69. [24] P. Suortti, T. Buslaps, V. HonkimaÈki, C. Metz, A. Shukla, Th. Tschentscher, J. Kwiatkowska, F. Maniawski, A. Bansil, S.
[25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37]
2231
Kaprzyk, A.S. Kheifets, D.R. Lun, T. Sattler, J.R. Schneider, B. Bell, J. Phys. Chem. Solids 61 (2000) 397. Landolt-BoÈrnstein, New Series IV/5a, in: O. Madelung (Ed.), Springer, Berlin, 1991. A. Bansil, Z. Naturforschung 48A (1993) 165. P. Fajardo, V. HonkimaÈki, T. Buslaps, P. Suortti, Nucl. Instrum. Methods B 134 (1998) 337. A. Bansil, S. Kaprzyk, Phys. Rev. B 43 (1991) 10335. A. Bansil, S. Kaprzyk, P.E. Mijnarends, J. Tobola, Phys. Rev. B 60 (1999) 13396. S. Kaprzyk, A. Bansil, Phys. Rev. B 42 (1990) 7358. P.E. Mijnarends, A. Bansil, Phys. Rev. B13 (1976) 2381. A. Bansil, R.S. Rao, P.E. Mijnarends, L. Schwartz, Phys. Rev. B 23 (1981) 3608. U. von Barth, L. Hedin, J. Phys. C 5 (1972) 1629. T. Ohata, M. Itou, I. Matsumoto, Y. Sakurai, H. Kawata, N. Shiotani, S. Kaprzyk, P.E. Mijnarends, A. Bansil, Phys. Rev. B 62 (2000) 16528. I. Matsumoto, J. Kwiatkowska, F. Maniawski, S. Kaprzyk, A. Bansil, M. Itou, H. Kawata, N. Shiotani, J. Phys. Chem Solids 61 (2000) 375. N.W. Ashcroft, Phil. Mag. 8 (1963) 2055. E.P. Volski, Zh. Eksp. Teor. Fiz. 46 (1964) 123 (Sov. Phys. JETP 19, 89).