Fermi surface parameters from the Fourier analysis of Compton profiles

Solid State Communications, Vol. 20, PP. 585—588, 1976

Pergamon Press.

Printed in Great Britain

FERMI SURFACE PARAMETERS FROM THE FOURIER ANALYSIS OF COMPTON PROFILES P. Pattison * Department of Physics, University of Warwick, Coventry CV4 7AL, England and B. Williams Department of Applied Physics, University of Strathclyde, Glasgow G4 ONG, Scotland (Received 18 June 1976 by C. W. McCombie) A new approach to the analysis and interpretation of Compton profiles, based on the properties of the Fourier transform of the proffles, is presented. The method is applied to the determination of Fermi momenta for some simple metals. The remarkable precision of the results obtained in this way ifiustrates both the power of the technique and also the manner in which this approach enables one to circumvent some major problems inherent in the usual analysis of Compton data. COMPTON scattering studies have undergone a remarkable process of development during the last decade. However, it is only within the last two or three years that workers have begun to feel real confmdence in the accuracy and reliability of their results.’3 Increasing confidence and understanding, together with improvements in experimental techniques4 has meant that measurements of directional profiles with an accuracy of better than 1% are regularly reported. This has led to a search for more sophisticated approaches to the processing and analysis of Compton profile (CP) data. The possibility of obtaining structural information from CP measurements has been considered,5 and attempts have been made to reconstruct 3-dimensional momentum distributions from sets of 1-dimensional CP measurements.6 Despite the improved statistical accuracy of experimental CP’s, the interpretation of CP data is still hampered by the relatively poor resolution’ (particularly rn 7-ray experiments) and the large contribution from multiple scattering which can be present in the data.2 Other problems such as the separation of the core from the conduction or valence electron profiles may give rise to additional uncertainties. In this letter we describe a new approach which is based on Fourier analysis of the CP. We illustrate the power of this technique by obtain-

reconstruction of a 3-I) momentum distribution p (p) since the 1-D transform of a directional profile gives the 3-D transform of the momentum density along the same direction.6’7 IfJ(p~)is the CP measured in the p~direction J(p~)= p(p) dp~,dp~. (1)

JJ

i’~ p~,

We can also define a function B(r) which is the 3-D Fourier transform of the momentum distribution, (i~~ e’~p (p) dp. B = 2irj

JfJ

(2)

-

From (1) and (2) we can obtain7 (1)

=

B(z)

1/2 -

+°~

~

e~z ~

dp~

(3)

so that each direction of the CP, J(p~),yields the function B(r) associated with the z direction. From a set of such functions it is possible to synthesise the momentum distribution p(p) using (2). However, we suggest that rather than attempting to build up the 3-D momentum distribution p(p), attention should instead be focussed on the intermediate function B(r) obtained from the Fourier transform of a single direction profile J(p 2). We ing accurate values for the Fermi momenta of some note that B(r) may alternatively be regarded as the autosimple metallic systems from experimental CP’s. In this correlation function of the position wave function. way we show how consideration of the Fourier transWe now consider B(r) for a free electron gas model form of the profile, rather than the profile itself, enables of a simple metal. Application of (I) yields the characus to circumvent each of the difficulties described above. teristic free electron parabola for the CP, falling to zero The 1-dimensional (1-D) Fourier transform of a at the Fermi momentum ±pF.If the parabola has unit directional CP may be used as a first step in the area then 3 [sin (pFr) cos PFrJ. (4) * Supported by a Science Research Council Fellowship. B(r) ~ PFF 585 —

—

—

586

FOURIER ANALYSIS OF COMPTON PROFILES

Vol. 20, No.6

AL

40

0

I 00

80

I

120

k (r=k.~r)

Fig. I. The real part of the function B(r) for aluminium. Experimental results are shown in curve (a) with open squares (reference 8), and curve (b) with circles (reference 9). The continuous curve is obtained from the free electron model for aluminium. The channel spacing ~ along the r-axis is 0.245 a.u. (0.130 A).

Li

I

I

20

40

,~nO’.~’ O’~~~°•O oo.-~-q

60

8~’~’

100

120

k (r=k.~r)

Fig. 2. The real part of the function B(r) for polycrystalline lithium obtained from experimental Compton profiles given in reference 12 (circles) and reference 13 (squares). The channel spacing LIr is as in Fig. 1. It follows therefore that, for a free electron gas, B(r) is zero whenever pFr = 4.493, 7.725, 10.904, 14.066 The same result for B(r) can, of course, be obtained by making a direct 3-D transformation of the spherical momentum distribution using (2). We will now compare this result for B(r) with the Fourier transforms of experimental CP’s of simple metals

measured by several authors on Al, Li and Na. In each case the behaviour of the conduction electrons should be described reasonably well by the free electron model, although the experimental profiles will be severely affected by the low resolution, the presence of multiple scattering and the large core electron contribution. Figure 1 shows the function B(r) for a free electron

Vol. 20, No.6

FOURIER ANALYSIS OF COMPTON PROFILES

587

Na

20 I

60 I

80 I

I

k (r=k.tir)

Fig. 3. The real part of the function B(r) for sodium obtained from the experimental Compton profile given in reference 12. The channel spacing ~ is as in Fig. 1. model of Al (,pF = 0.926 a.u.) in the region ofr where B(r) oscifiates. Also shown in the figure are two experimental results for Al measured with 412 keV 7-rays8 and 60 keV 7-rays.9 It should be emphasised that no attempt has been made to remove the effects of multiple scattering or finite resolution from the experimental data. Indeed even the large core CP was not subtracted from the experimental profile before performing the Fourier transform. The agreement between the experimental curves and the theory is remarkably good, particularly In the regions where B(r) intersects the r-axis. If the first intersection should occur at pFr = 4.493 [see (4)J then these experimental results yield values for PF of 0.931 a.u.8 and 0.936 a.u Y These values for the Fermi momentum are both within 0.01 a.u. of the accepted value. This can be compared with the widths of the resolution function in the two experiments of 037 a.u.8 and 0.70 a.u.9 respectively. From other recent CP measurements on Al using 60 keV 7-rays at a similar resolution we obtain values of 0.933 a.u.10 and 0.9 17 a.u.11 for the Fermi momentum and once again the calibration of the Fermi surface is achieved to within 1%. The amplitudes of these osdillations in the function B(r) are shown here on an arbitrary linear scale since, in these examples, it is only necessary to determine the values of r for which B(r) = 0. Figures 2 and 3 show values of B(r) obtained from experimental CP’s of polycrystalline Li and Na, respectively. The various B(r) are given only in the region where the functions are oscifiating around the r-axis, and the amplitudes are again shown on an arbitrary scale. In

the case of U~”3the values of r for which B(r) =0 lead to Fermi momenta of 0.56 and 0.57 a.u. compared with the free electron value of 0.588 a.u. For Na~the experimental B(r) curve leads to a PF of 0.47 a.u. in this free electron picture, compared with the theoretical value of 0.483 a.u. At first sight it seems surprising that it is possible to locate the Fermi momentum with such precision when using data obtained at such modest resolution. This behaviour highlights one of the main advantages of working with the position space functionB(r), rather than the momentum profile J(p). In momentum space the observable J(p) is convoluted with the resolution function so that details in the profile (e.g. a discontinuity at the Fermi surface) are hard to resolve. In position space, on the other hand, we must multiply B(r) by the Fourier transform of the resolution function, typically Gaussian. The result of this multiplication will be a reduction in the amplitudes of B(r), but it cannot affect the positions of the zeros and hence our estimate ofthe Fermi momentum. It is easy to see the effects of resolution in Fig. 1 where the amplitude of the oscifiations is reduced by a larger factor in the data measured at lower resolution (curve b). However, in both curves (a) and (b) the positions of the zeros are unaffected, and very accurate Fermi momentum values can be obtained from these zero positions. We also see from these results that the presence of a multiple scattering contribution (as much as 30% of the total curve (a)8 of Fig. 1) does not appear in the region of B(r) where these oscillations occur. This should be expected since multiple scattering makes a

588

FOURIER ANALYSIS OF COMPTON PROFILES

broad and slowly varying contribution to the profile J(p). Hence the multiple scattering will make a sharp, narrow contribution to the position space function B(r), well away from the tail ofB(r) where the function is oscifiating. The same arguments will apply to the smooth flat contribution to the profiles from the tightly bound core electrons which must, of course, be highly localised around the origin in the position space representation. In the regions of B(r) shown in Figs. 1—3 it is immaterial whether either the core electron profile or the multiple scattering contribution is included or removed from J(p) before transformation. In this letter we have examined the behaviour of the function B(r) only for the very simple case of a free electron gas. The power of this approach is clearly illustrated

Vol. 20, No.6

however by the accuracy with which we are able to determine the Fermi momentum even from data measured at a very low resolution. Further results which involve the study of the behaviour ofB(r) for ionic crystals, more complex metals and simple molecules will be published elsewhere.’4 In particular the directional properties of B(r) for such systems will be considered in detail. The importance of the function B(r) is that it provides a complementary view of Compton scattering data and in so doing allows one to avoid many of the problems inherent in the analysis of Compton data. The relationship between B(r) and the charge density distribution should help to provide a more intuitive understanding of the behaviour of Compton profiles.

REFERENCES 1.

WILLIAMS B.G.,Acta. Cryst. A32, 513 (1976).

2.

FELSTEINER J., PATTISON P. & COOPER M., Phil. Mag. 30, 537 (1974); HALONEN V., WILLIAMS B.G. & PAAKKARI T.,Phys. Fenn. 10, 107 (1975). RIBBERFORS R.,Phys. Rev. B12, 2067,3136(1975).

3. 4.

6. 7.

WEISS R.J., REED W.A. & PATTISON P., in Compton Scattering (Edited by WILLIAMS B.G.). to be published by McGraw-Hill (1976). EPSTEIN I .R., Invited talk, 32 Annual Pittsburgh Diffraction Conference (1974); BEARDSLEY G.M., BERKO S., MADER Ji. & SHULMAN M.A., AppL Phys. 5,375 (1975). MIJNARENDS P.E.,Phys. Rev. B4, 2820 (1971). MUELLER F.M., Bull. Am. Phys. Soc. II 19, 200 (1974); and submitted to Phys. Rev.

8. 9.

COOPER M., PATTISON P. & SCHNEIDER J.R. (to be published). PATTISON P. & HOLT R. (unpublished data).

5.

10.

MANNINEN S., PAAKKARI T. & KAJANTIE K., Phil. Mag. 29, 167 (1974).

11.

PATTISON P., MANNINEN S., FELSTEINER J. & COOPER M., Phil. Mag. 30,973 (1974).

12.

EISENBERGERP., LAM L., PLATZMAN P.M. & SCHMIDT P., Phys. Rev. B6, 3671 (1972).

13.

WEISS Ri. & PHILLIPS W.C.,Phil. Mag. 20, 1239 (1969).

14.

PATTISON P., WEYRICH W. & WILLIAMS B.G. (to be published).