experimental and theoretical evidence for the magic angle in transmission electron energy loss...

Ultramicroscopy 96 (2003) 523–534

Experimental and theoretical evidence for the magic angle intransmission electron energy loss spectroscopy

Howard Daniels, Andy Brown, Andrew Scott, Tony Nichells, Brian Rand,Rik Brydson*

Institute for Materials Research, University of Leeds, Leeds LS2 9JT, UK

Received 10 July 2002; accepted 11 November 2002

Abstract

We present experimental measurements of the C K-ELNES of high temperature pyrolysed graphite and related

crystalline materials as a function of collection angle and sample tilt. These results together with a corresponding

theoretical analysis indicate that the so-called ‘‘magic angle’’ for EELS measurements of an anisotropic crystal such as

graphite, where spectra are independent of sample orientation, is approximately two times the characteristic scattering

angle. We briefly discuss the implications of this result for the experimental measurement of anisotropic structures,

including interfaces, as well as for the detailed modelling of ELNES structures using advanced electronic structure

calculations.

r 2003 Elsevier Science B.V. All rights reserved.

PACS: 61.14; 61.72; 71.15; 71.20; 78.90

Keywords: EELS; ELNES; Anisotropy; Modelling

1. Introduction

Transmission electron energy loss spectroscopy(EELS) is an analytical technique in the transmis-sion electron microscope whereby fast incidentelectrons (typically 100–400 keV) impart momen-tum and energy to electrons in a material. Thisinelastic scattering process results in a correspond-ing loss in both momentum and energy of the fastelectrons which forms the detected signal [1,2]. The

direction of momentum transfer measured isdependent on the angular range over which thescattered incident electrons are collected which istypically of the order of milliradians.There are a number of characteristic features in

the EELS spectrum [1,2], of particular interest isthe fine structure on characteristic inner shellionization edges, electron loss near edge structure(ELNES), which reflects transitions of inner shellelectrons to site- and symmetry-projected unoccu-pied electronic states, which in a solid lie above theFermi level [3]. The modelling of such ELNESusing advanced electronic structure calculationsallows the extraction of bonding information from

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*Corresponding author. Tel.: +44-0113-233-2369; fax: +44-

0113-242-2531.

E-mail address: [email protected] (R. Brydson).

0304-3991/03/$ - see front matter r 2003 Elsevier Science B.V. All rights reserved.

doi:10.1016/S0304-3991(03)00113-X

projected specimen areas, which in modern-dayinstruments can approach those of atomic dimen-sions [4].It is usual to resolve the direction of momentum

transfer imparted during the scattering process,into components parallel and perpendicular to theincident beam direction. In a specimen regionwhich is crystallographically anisotropic, the exactdirection of momentum transfer and hence energystates available to the excited electron will dependon the specimen orientation with respect to theincident electron beam [5]. Practically this canpresent problems, especially where quantitativecomparisons of electronic structure between dif-ferent specimen regions at different orientations(e.g. curved specimens such as nanotubes or withina polycrystalline material) or between differentsamples are required. One suggested solution tothis problem has been to perform all measurementsat a so-called ‘‘magic angle’’, where spectra do notvary with sample orientation; this angle is afunction of the semi-angle subtended by the usuallycircular aperture over which the scattered electronsare collected, b; as well as the convergence semi-angle of the incident electrons on the specimenregion, a: Note that these two quantities may becombined in the concept of an effective collectionangle, beff for energy loss measurements [1].The seminal work of Browning et al. [6,7]

followed by the work of Menon and Juan [8],Souche et al. [9] and Nelheibel et al. [10] hassuggested that the value of the magic angle, bmagic;is close to four times the characteristic scatteringsemi-angle, yE ; corresponding to the mean energyloss Eav: yE is given relativistically by yE ¼Eav=ðgm0v

2Þ; where g is the relativistic factor ðg ¼½1� ðv=cÞ2�1=2Þ and m0 is the electron rest mass; forlow beam energies this approximates to yE ¼Eav=2E0: However, recent theoretical calculationsby Paxton et al. [11] have suggested that the valueof bmagic is much smaller than these earlier reports.The majority of these studies have been primarilytheoretical and little direct experimental evidencefor the exact magic angle has been published. It isthe purpose of this brief report to establishexperimentally a value for this magic angle andto support it further with a correspondingtheoretical analysis.

2. Theory

It is common to represent the inelastic scatteringof incident electrons by atoms in the sample interms of a vector diagram as shown in Fig. 1.Before scattering, the incident electrons of energyE0; will have a particular wave vector k0 whereE0 ¼ ðh=2pÞ2k20=ð2meÞ. Experimentally the range ofincident wavevectors will be defined by theconvergence semi-angle, a; in the microscope.After scattering, the wavevector will have changedto kf (the range of wavevectors collected willdepend on the collection angle). The momentumtransfer suffered by the incident electron is givenby ðh=2pÞq where q ¼ k0 � kf (whereas the mo-mentum transfer imparted to the specimen is

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Fig. 1. Vector diagram of momentum transfer during inelastic

scattering.

H. Daniels et al. / Ultramicroscopy 96 (2003) 523–534524

correspondingly �ðh=2pqÞ: For high incident beamenergies, the momentum transfer q is much smallerthan k0: Conservation of momentum and energyleads to the result

q2Ek20ðy2 þ y2EÞ; ð1Þ

where y is the scattering semi-angle and yE is thecharacteristic semi-angle of scattering as defined inSection 1. As we see below, effectively this meansthat for a particular inelastic process, such asionization of inner shell electrons, the angulardistribution of inelastically scattered electrons willfollow a Lorentzian distribution of half-widthequal to the characteristic angle, yE : Since q ismuch smaller than k0; it is possible to decompose q

into a component qjj (parallel to k0) and q>

(perpendicular to k0) where q2 ¼ q2J þ q2>: Forsmall y these are given by qjj ¼ k0yE and q> ¼k0 sin yEk0y in accord with Eq. (1).What is experimentally measured in an EELS

spectrum is related to the total inelastic cross-section for scattering, s; or more exactly itsdependence on energy loss, E; and solid scatteringangle, O; and is known as the double differentialcross-section, d2s=ðdE dOÞ: This quantity representsthe fraction of incident electrons (energy E0) whichare scattered into a solid angle dO ð¼ 2p sin y dy;if we ignore any diffraction effects which maydestroy the cylindrical symmetry of the experi-mental measurement) with an energy between E

and E þ dE:The form of the double differential cross-section

can be derived quantum mechanically using anumber of equivalent approaches. One descrip-tion, known as Fermi’s Golden Rule, relies on time-dependent perturbation theory for the transitionrate between initial and final states of theinteracting electrons.It is assumed that the kinetic energies of the

incident electrons are much greater than theenergies of the excited atomic states—known asthe Born approximation. This allows the totalwavefunction of the system to be written as aproduct of the atomic states and plane wave states.The plane wave states describe the fast incidentelectron before and after scattering and are of thegeneral form expðikr0Þ; where k is the wavevector(either k0 or kf) and r0 the vector coordinate of the

fast electron. The fast electron interacts throughthe Coulomb potential (given by Vcoul ¼ e2=½4pe0jr0 � rj�) with the atomic electron which hascoordinate system r and causes it to make thetransition from the initial atomic state wavefunc-tion, Ci (r) to a final atomic state Cf ðrÞ:Mathematically the result is in the form of adifferential cross section with respect to a solidangle given by theory due to Bethe [12]:

ds=dO ¼ ð2pme=h2Þ2ðkf=k0Þ

Z Z

C�f expðiðk0 � kf Þ � r0Þ

�� VcoulCid

3r0d3r

��2

ð2Þ

integration over the coordinates of the fastelectron gives

ds=dO ¼ 4g2=ða20q4Þðkf=k0Þ

Z

C�f expðiq � rÞCi d

3r

��2

; ð3Þ

where g is the relativistic correction factor for aparticle of velocity, v; and a0 ¼ e0h2=ðpe2meÞ is theBohr radius=0.053 nm. If the final states of theatomic electron are normalized with respect toenergy loss, E; to give Cf ðrÞ; this gives us thedouble differential cross section:

d2s=ðdE dOÞ ¼ 4g2=ða20q4Þðkf=k0Þ /f jexpðiq � rÞjiSj j2;

ð4Þ

where/f jexpðiq � rÞjiS denotesRC0�f expðiq � rÞCi d

2r:Since in a solid there a number of possible band-like final states, the single integral term in Eq. (4)should actually be a sum over all possible finalstates leading to the concept of a density of states(DOS).The magnitude of q ¼ jqj can be written in terms

of the scattering angle, y; since by the cosine rule inFig. 1, q2 ¼ k20 þ k2f � 2k0kf cos y: Differentiatingthis with respect to y allows us to write the solidangle in terms of q; which is

dO ¼Zdf sin y dy ¼

Zdfq dq=ðk0kf Þ;

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H. Daniels et al. / Ultramicroscopy 96 (2003) 523–534 525

where f is the azimuthal angle (varying between 0and 2p) relative to the polar angle y: This gives

d2s=ðdE dqÞ ¼ 4g2R=ðEk20ÞZ

ð1=qÞ df =dE df; ð5Þ

where E is the energy loss and df =dE is an atom-dependent quantity known as the generalisedoscillator strength (GOS) per unit energy loss.This quantity depends both on energy loss, E; andthe change in momentum, q;

df =dE ¼ E=ðRa20q2Þ /f jexpðiq � rÞjiSj j2; ð6Þ

where R is the Rydberg=ðe4me=2Þ=ð2e0hÞ2

¼ 13:61 eV. Since q2 ¼ k20ðy2 þ y2EÞ; where yE is

the characteristic semi-angle and, for high incidentbeam energies, the term kf=k0 can be approxi-mated as unity, it is also possible to express Eq. (4)in terms of a double differential cross section withrespect to energy loss and scattering angle:

d2s=ðdE dyÞ ¼ 4g2R=ðEk20Þ

Z

y=ðy2 þ y2EÞ df =dE df

¼ 4g2=ða20k40ÞZ

ðy=ðy2 þ y2EÞ2Þ

/f j expðiq � rÞjiSj j2 df: ð7Þ

Here we have approximated sin y ¼ y: Alterna-tively, in terms of solid angle, the doubledifferential cross-section can be expressed byEq. (4) as

d2s=ðdE dOÞ ¼ 4g2R=ðEk20Þ1=ðy2 þ y2EÞ df =dE: ð8Þ

As we shall see, for small scattering angles andthus small q; it is usual to assume that the GOS isindependent of q: Effectively this means that for aparticular inelastic process, such as ionization ofinner shell electrons, the angular distribution ofinelastically scattered electrons will follow aLorentzian distribution of half-width equal to thecharacteristic angle, yE :The theory summarized above was originally

formulated by Bethe. As in X-ray scattering, theBorn approximation allows the differential cross-section to be separated into the product of twoterms: one dependent on the incident electron andthe other dependent only on the excited atom.These are known as an amplitude factor and adynamic structure (or inelastic form) factor. The

amplitude factor represents the scattering of theelectron by a free electron given by the Rutherfordcross-section ð¼ 4g2=ða20q

4ÞÞ; while the dynamicstructure factor depends on the properties of theatom and reflects the fact that the electrons areinvolved in bonding. The general scatteringproperties of a particular atomic species aresummarized in the generalized oscillator strength(GOS), f ; which for a solid, tends to be formulatedas an energy-dependent quantity, df =dE:We now consider the form of the GOS. For

small q; we can expand the exponential term inEq. (6) as follows

expðiq � rÞ ¼ 1þ iq � r þ? ð9Þ

this is known as the dipole approximation at smallq and allows us to ignore q2 (quadrupole) termsand above. Since Ci and Cf are orthogonal, thematrix elements of 1 will be zero. Thus we have

/f jexpðiq � rÞjiSj j2¼ /f jq � rjiSj j2: ð10Þ

In a solid, if we are considering an inner shellionization edge, Ci will be an atomic-like corelevel, whereas Cf will be an unoccupied band-likestate above the Fermi energy. As discussed above,in the expression for the double differential cross-section in Eq. (4) we need to sum over all possiblefinal states, which means that the dipole matrixelements are weighted by the unoccupied DOS.Eq. (10) may now be expanded as follows:

/f jqxyz � rxyzjiS�� 2¼ /f jqxx þ qyy þ qzzjiS

�� 2¼ /f jqxx þ qyyjiS��þ/f jqzzjiS

��2: ð11Þ

Now we choose a coordinate system where y isthe polar angle to the z-axis and the azimuthalangle f lies in the (xy) plane (i.e. the microscopecoordinate system with the beam going down the�z direction). Then we can substitute in Eq. (11)for: qx ¼ qxy cos f ¼ q> cos f ¼ ky cos f forsmall y; qy ¼ qxy sin f ¼ q> sin f ¼ ky sin f; andqz ¼ qJ ¼ k0yE where qxy is the magnitude ofvector qxy—the momentum transfer, or morestrictly the change in wavevector, in the ðxyÞ planeperpendicular to the beam direction. Note alsothat x ¼ rxy cos f and y ¼ rxy sin f; where rxy isthe magnitude of vector rxy—the position vector in

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the ðxyÞ plane. This gives:

/f jqxyz � rxyzjiS�� 2

¼ /f jqxycosfrxycosf��þ qxysinfrxysinfjiSþ/f jqzzjiS

��2¼ /f jqxyrxyðcos2fþ sin2fÞjiSþ/f jqzzjiS

�� 2¼ /f jqxyrxyjiSþ/f jqzzjiS

�� 2¼ q2xy/rxyS2 þ q2z/zS2 þ 2qxyqz/rxyS/zS;

ð12Þ

where /zS2 ¼ /f jzjiS2 and, since the initial stateis an atomic core level and the final state is a band-like state, /zS2 is effectively the projected densityof unoccupied band-like states in the z-direction(projected both in terms of site and, by thedipole selection rule, also symmetry). Corres-pondingly /rxyS

2 is the projected unoccupiedDOS in the (xy) plane and may be expressedas /rxyS2 ¼ /ðx2 þ y2Þ1=2S2; note that /rxyS2a/xS2 þ/yS2:This division of ðq�rÞ in Eqs. (11) and (12) is

important since, under normal experimental con-ditions, we are effectively integrating over thecircular collection aperture which we will chooseto lie in the ðxyÞ plane. This integration over theazimuthal angle f (lying in the xy plane) meansthat it is impossible to distinguish between x and y

directions, except in that they are perpendicular toz: Thus the experimental arrangement really onlydefines two distinct directions (as opposed tothree, i.e. x; y and z): one direction parallel tothe beam direction (�z) and another perpendicularto beam direction in the (xy) plane.We believe that the final cross term in Eq. (12) is

zero, since the vector components qxy (equals q>)and qz (equals qjj) are orthogonal

1, therefore weobtain:

/f jqxyz � rxyzjiS�� 2¼ q2xy/rxyS2 þ q2z/zS2

¼ q2>/rxyS2 þ q2J/zS2

¼ q2>/ðx2 þ y2Þ1=2S2 þ q2J/zS2:

ð13Þ

We now use the resulting dipole approximationfor the GOS, given by Eq. (13), in Eq. (7) andintegrate with respect to f (between 0 and 2p) toobtain d2s=ðdE dyÞ; noting that Eq. (13) has nodependence on f:

d2s=ðdE dyÞ ¼ ð8pg2=ða20k40Þðy=y

2 þ y2EÞ2Þ

ðkyÞ2ð/ðx2 þ y2Þ1=2S2Þh

þ ðk0yEÞ2/zS2

i: ð14Þ

In order to calculate the energy differentialcross-section for a given value of scattering angle,y; we have to integrate Eq. (14) with respect to y:As we have defined the direction parallel to theincident beam as the �z direction, then in thisdirection we have integrals of the form:

dsJ=dEp

Zyðk20y

2EÞ=ðy

2 þ y2EÞ2ð/zS2Þ dy; ð15Þ

where the momentum transfer parallel to theincident beam direction, q2J ¼ q2zEk20y

2E from

Fig. 1. Whilst for the perpendicular direction wehave integrals of the form:

ds>=dEp

Zyðk20y

2Þ=ðy2 þ y2EÞ2

ð/ðx2 þ y2Þ1=2S2Þ dy; ð16Þ

where the momentum transfer perpendicularto the incident beam direction, q2> equals k20y

2

for small y:Integrating Eqs. (15) and (16) between a collec-

tion semi-angle of 0 and b; we then have

dsJ=dE pk20=2½m2=ðm2 þ 1Þ�ð/zS2Þ ¼ MJð/zS2Þ;

ð17Þds>=dE pk20=2½lnðm

2 þ 1Þ � m2=ðm2 þ 1Þ�

ð/ðx2 þ y2Þ1=2S2Þ

¼ M>ð/ðx2 þ y2Þ1=2S2Þ; ð18Þ

where we have defined m ¼ b=yE ; and the paralleland perpendicular weighting factors are given byMJ¼ k20=2½m

2=ðm2þ 1Þ� and M>¼ k20=2½lnðm2þ 1Þ

�m2=ðm2 þ 1Þ�; respectively. These terms involvingm ¼ b=yE describe the weighting of the paralleland perpendicular directional components of theunoccupied DOS in the total energy differentialcross-section. Note that the sum of these twoexpressions gives the correct form for the energy

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1We also believe that /qxyrxyS/qzzS ¼ /qzrxyS/qxyzS ¼ 0:


differential cross-section integrated over the mo-mentum transfer, q; as depending on ln ððb2=y2EÞ þ1Þ: This latter expression is one that should beemployed for the case of an isotropic materialwhere /zS2 ¼ /ðx2 þ y2Þ1=2S2; i.e. the projectedDOS along the beam direction (z) is equivalent tothe DOS along any general direction in the (xy)plane perpendicular to the beam direction.The derivation presented here owes much to the

work described in Refs. [5–11]. The form ofEqs. (16) and (17) is identical, albeit simplified,to that obtained in Ref. [7] for a uniaxial material.To understand the form of this expression we

now consider a specific case of the graphite C K-ELNES with the electron beam incident along the[0 0 1] direction. The initial state is a carbon 1slevel and the corresponding dipole allowed finalstates are of p symmetry. Here /zS2 will be the p�DOS, formed from the overlap of carbon 2pz

orbitals which are directed along the c-axis, andð/ðx2 þ y2Þ1=2S2Þ will be the s� DOS formed fromthe 3sp2 hybridized orbitals in the (a; b) plane. Forthis particular orientation, in the C K-ELNESspectrum we can therefore equate dsJ=dE with theintensity in the p� peak at 285 eV and ds>=dE

with the intensity in the s� region commencing atca. 292 eV. However we need to generalize thisexpression for any given sample orientation.Inclusion of sample tilt may be achieved by simplytransforming the coordinate scheme describedabove involving z and rxy ¼ ðx2 þ y2Þ1=2 (with thebeam incident along the �z direction) to theconventional crystal axes as discussed in Refs. [7,9].To do this we realise that the ðrxy; zÞ experi-

mental coordinate system is related to the co-ordinate system of the sample, which we define asðRxy;ZÞ with the p� DOS directed along Z; via thestandard rules governing the rotation of one set ofaxes to another [13]:

z ¼ cos f sin gRxy þ cos gZ; ð19aÞ

rxy ¼ cos f cos gRxy � sin gZ: ð19bÞ

Here, as before, �z is the beam direction, f isthe rotation (between 0 and 2p) around the z-axis(i.e. in the (xy) plane of the collection aperture)and g is the angle of tilt from the z-axis. Using

these expressions for z and rxy in Eq. (7) andintegrating with respect to f between limits 0 and2p; we arrive at an expression for the doubledifferential cross-section, d2s=ðdE dyÞ which wecan then separate into parallel and perpendicularcomponents. However, now we note that, follow-ing integration, instead of the simple, common 2pterm at the front of the right-hand side of Eq. (14),we will now have terms of the form (note that thecross terms vanish in the integration over f):

/zS2 ¼ psin2 g/RxyS2 þ 2pcos2 g/ZS2; ð20aÞ

/rxyS2 ¼ pcos2 g/RxyS2 þ 2psin2 g/ZS2: ð20bÞ

As we have seen, after integration with respectto y (Eqs. (17) and (18)), the DOS componentparallel to the electron beam direction ð�zÞ isweighted by the factor MJ; while the correspond-ing perpendicular component is weighted by thefactor M>: This therefore gives the followingexpressions for the parallel and perpendiculardifferential cross-sections, in terms of the samplecoordinate system as

dsJ=dE ¼ MJp sin2 g/RxyS2

þ MJ2p cos2 g/ZS2; ð21aÞ

ds>=dE ¼ M>pcos2 g/RxyS2

þ M>2psin2 g/ZS2: ð21bÞ

The intensity in the p� peak is given by the termscontaining /ZS2, while the s� intensity is given byboth /RxyS2 components, which results in:

dsp�=dE ¼ 2pðMJcos2 gþ M>sin

2 gÞ/ZS2;

ð22aÞ

dss�=dE ¼ pðM>cos2 gþ MJsin

2 gÞ/RxyS2:

ð22bÞIt is clear that these expressions (as well as the

ratio of these expressions relative to each other or,indeed, to the total differential cross section) areonly independent of tilt angle, when MJ ¼ M>:Equating the two weighting factors in the expres-sion for the energy differential cross-sectionstherefore provides us with a measure of the magicsemi-angle, bmagic; the experimental condition

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where the parallel and perpendicular contributionsare weighted equally. In a planar anisotropicsample such as graphite this is where the ELNESwill be independent of sample and/or beamorientation. Previously published derivations havesuggested various values for the magic angle includ-ing 4yE [8], 3:86yE [9] and most recently 1:36yE [11].A plot of both the parallel and perpendicular

weighting factors and their dependence on collec-tion angle, as described in Eqs. (17) and (18), isshown in Fig. 2. These results suggest that equalweighting of the two components occurs whenm ¼ b=yE ¼ 1:98: Thus our current results provideus with a value for bmagic ¼ 1:98yE : This is almostidentical to the value given by Zhu et al. [14] forthe oxygen K-edge who, using Egerton’s Hydro-genic model for the GOS, empirically determined avalue of the collection angle at which the averagemomentum transfer parallel and perpendicular tothe beam direction was equal. We believe that thediffering value for bmagic with parallel illuminationpredicted by the work of Souche [9], who resolve

the dipole matrix element into separate x; y and z

components, arises due to the fact that they do notresolve the momentum transfer components ontoall three of the axes of the sample coordinatesystem X ; Y and Z:An approximation commonly used in the dipole

regime is that the GOS is independent of q: Thiswill lead to a 1=q2 ¼ 1=ðy2 þ y2EÞ term droppingout of the integrals (15) and (16) and results in apredicted value of bmagic ¼ 1:59yE :The present results have been derived for the

simple case of a parallel incident beam as in aconventional TEM. Inclusion of the incident beamconvergence semi-angle a will lead to expressionscompletely symmetrical in both b and a asdemonstrated in Ref. [9]. The presence of incidentbeam convergence can be summarized in theconcept of an ‘‘effective collection semi-angle’’,beff ; which will be a convolution of both b and theconvergence semi-angle, a; this may, in principle,be calculated using standard methods as outlinedin Egerton [1] such as summing in quadrature.Generally this will have an appreciable effect if band a are of a similar magnitude, leading tobeff > b:Having derived a theoretical value for the EELS

magic collection semi-angle, we now outline theprocedure for the experimental determination ofthis quantity.

3. Experiment

Initial EELS measurements were made onsamples of high temperature (2750 C) pyrolysedgraphite, ground, dispersed and pipetted ontoholey carbon support films. Subsequent measure-ments were also made on samples of crystallinehexagonal boron nitride and magnesium diboride.Thin sample regions (less than 0.3L; where L is themean free path for inelastic scattering) overhan-ging holes in the support film were analysed in aFEI CM200 field emission transmission electronmicroscope fitted with a Supertwin lens, operatingat 200 keV and equipped with a Gatan imagingfilter (GIF) (The TEM actually operates at197 keV with the GIF also operational). AllEELS measurements were performed with the

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0 1 2 3 4 50

0.5

1

1.5

2

2.5

m

Fig. 2. Plot of functions describing the weighting of the parallel

(dashed line) and the perpendicular components (solid line) of

the momentum transfer as a function of m ¼ b=yE :


microscope in diffraction mode (i.e. image cou-pling to the spectrometer), so that the collectionangle was defined by the diffraction patterncamera length and the radius of the spectrometerentrance aperture.Collection semi-angles, b; for various combina-

tions of camera lengths and spectrometer entranceapertures were accurately calibrated by simulta-neously viewing on the GIF a diffraction patternfrom a standard material with a large d-spacing(e.g. Crocodilite—Agar Scientific) together withthe inserted spectrometer entrance aperture. Thisimaging method was also employed for accuratelyaligning the position of the diffraction pattern withrespect to the spectrometer entrance apertureduring the measurements on graphite and theother materials. This is particularly important aswe observed that any deviation from cylindricalsymmetry can significantly affect the resultsobtained—this follows from the derivation ofEq. (5). All collection angles employed lay withinthe dipole regime which is satisfied ifq{ð2m0EavÞ

1=2=ðh=2pÞ [5] and, furthermore, allwere small enough not to include significantcontributions from diffracted beams; at 200 keVthe first order diffraction spots in the [1 1 0] zoneaxis pattern of graphite are at 2yBragg ¼ 7:6mrad.Convergence angles were calibrated as a func-

tion of condenser lens current by observing thewidth of diffraction discs in diffraction patternsfrom reference materials. All measurements wereperformed with a low value of convergence semi-angle, o0.5mrad (i.e. almost parallel illumina-tion). In all cases the collection angles reported areeffective collection angles and include the effects ofthe incident beam convergence—here we havesimply assumed that in the convolution thecollection and convergence angles sum in quad-rature, i.e. beff ¼ ðb2 þ a2Þ1=2:

4. Results

Initial measurements were made on a thin graphitecrystal with the incident beam along /001S.In this orientation the qjj component isolatesthe p� feature at 285 eV, while the q> componentis reflected in the s� feature at ca. 292 eV and

above which corresponds to sigma-type bondingwithin the graphene sheets. Fig. 3 shows theeffect of varying the collection angle on theC K-ELNES recorded close to /0 0 1S zoneaxis. For small collection angles (0.8mradEyE),the p� peak intensity is enhanced relative to thes� region, while for larger collection angles(3.2mradE3.8yE) the opposite is true. Further-more the detailed structure in the s� regionbecomes more well defined (as four main featuresfollowing the p� feature) as the collection angleincreases.We have measured the C K-ELNES as a

function of sample tilt, and collection angle andintegrated the areas under the p� peak and p� þ s�

features (total intensity) using energy windows ofwidth 5 and 20 eV, respectively, both beginning atthe edge threshold. The relative intensity of the p�

peak, IRELp� ; normalized to the total edge intensityeffectively gives a measure of the relative weightingof the qJ component in the spectra. Measurementswere made close to the /0 0 1S zone axis (weavoided exact zone axis conditions so as to removeany possibility of channelling effects) and also atboth 730 tilt.Fig. 4 shows a plot of the difference in

IRELp� ðDIRELp� Þ between 0 and +30 tilt (from/0 0 1S) as a function of collection angle (normal-ized to the relativistically corrected value of yE forthe C K-edge at 197 kV: yE ¼ 0:84mrad). Forsmall collection angles, we should be isolating theqJ component and we observe that for b=yEo1:5

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CK Edge Spectra (Graphite, Beam Along <002>)With Varying Collection Angles

275 285 295 305 315 325 335Energy Loss (eV)

Inte

nsi

ty

1.0 E

2.0 θ

θ

θ

E~magic angle

3.8 E

Fig. 3. C K-ELNES of graphite after background subtraction

measured with incident beam along [0 0 1] as a function of

collection semi-angle.


the difference, IRELp� ; is a negative quantity reflect-ing that we are tilting away from the direction ofthe 2pz orbitals of carbon in the graphene sheetsand thus reducing the relative weight of the p�

feature. For b=yE > 2:5 the difference is a positivequantity, which can only be explained if here theq> component is dominant. The point at which thedifference is zero, should correspond to an equalweighting of the qJ and q> components and isgiven approximately by b=yE ¼ 2:1 This corre-sponds to a value of bmagic of close to 2yE inexcellent agreement with our theoretically pre-dicted value. Note the decrease in the (positive)value of IRELp� at large values of b=yEð> 4Þ which webelieve is real and is related to the contributionfrom the first-order diffraction beams (at 7.6mrad, i.e. b=yE ¼ 9 in this orientation).Using methods identical to those employed in

the production of Fig. 4, additional results oflimited measurements on a /0 0 1S graphitecrystal recorded at an incident beam energy of100 keV are also shown overlaid in Fig. 4 and theseagain indicate a value of bmagic of approximately2yE : Furthermore, similar measurements at200 keV (not shown), on a graphite crystaloriented such that the incident electron beam wasnow parallel to the /1 0 0S direction of graphiteagain suggest a value of bmagic of close to 2yE inaccord with the results obtained in the /0 0 1Scrystal orientation.

Following on from this experimental method, asecond means of determining bmagic; is to measurerelative intensity of, say, the graphite C K-edge p�

peak, IRELp� ; as a function of collection angle fortwo different sample orientations; the point atwhich the relative intensities are equal will thencorrespond to bmagic: Fig. 5 shows the results oftwo independent measurements displayed as a plotof IRELp� versus b=yE for the graphite C K-edge withthe sample in both the /0 0 1S and /1 0 0Sorientations. The two curves intersect at a valueof bmagic which lies slightly below 2yE :Figs. 6 and 7 show similar plots to Fig. 5,

however these have been obtained for the B K-and N K-edges in hexagonal BN (Figs. 6a and b)as well as the B K-edge in MgB2 (Fig. 7)—bothmaterials have a similar hexagonal, anisotropiclayered structure to that of graphite. At 197 keV,yE for the B K-edge is 0.57mrad, whilst for theN K-edge the corresponding value is 1.18mrad;the relative cross-over points in the three graphs alllie close to a value of 2yE which again supports ourprevious analysis for the magic collection angle.To summarize: a number of experimental

methods as well as independent datasets allindicate that the ELNES from anisotropic struc-tures such as graphite, hexagonal BN and MgB2become independent of sample orientation for amagic collection angle of approximately two timesthe characteristic angle for inelastic scattering.

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Plot Showing Difference in [ π∗/(π∗+σ∗) π∗ /(π∗ +σ∗ )] ratio Between 0˚ And +/- 30˚ Tilit Away From c-axis.

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0 1 2 3 4 5 6

Multiples of θE

Dif

fere

nce

in π

∗/(π

∗+σ∗

) π

∗/(π

∗+σ∗

) rat

io r

atio

Fig. 4. Difference in the relative intensity of p� peak at thegraphite C K-edge between 0 and 30 tilt from /0 0 1S as afunction of normalized effective collection angle, b=yE : Twosets of data are shown overlaid: data recorded at both 197 keV

where yE ¼ 0:84mrad (K) and 97 keV where yE ¼ 1:6mrad(m).

C-K in Graphite

0

0.05

0.1

0.15

0.2

0.25

0 1 2 3 4 5

θΕ = 0.84mrads

Multiples of θE

π∗/(

π∗+

σ∗)

π∗/(π

∗+σ∗

)

Fig. 5. Relative intensity of p� peak at the graphite C K-edge asa function of normalized effective collection angle, b=yE : Twosets of data are shown: data recorded with incident electron

beam parallel to /0 0 1S (E) and perpendicular to /0 0 1S(’). The point of intersection of the two curves is the magic

collection angle.


5. Discussion

The determination of the EELS magic angle iscritically important for a number of reasons:

(a) It allows the study of anisotropic samples suchas graphitic nanotubes, filaments, fibres andtapes, without the need to perform all energyloss measurements at the same sample orien-tation if comparisons between the electronicstructure and hence degree of ordering indifferent samples are to be subsequentlymade. One such area of interest to researchersat Leeds is a detailed study of the graphititisa-tion of different carbons and carbon compo-sites as a function of heat treatment orprocessing route. If chemical mapping usingthe relative intensity of ELNES features,either in EFTEM mode or STEM spectrumimaging mode, is desired then clearly orienta-tion effects can be minimized by recordingimages with a collection angle as close apossible to bmagic (subject to the experimentallimitations such as the limited objectiveaperture sizes in EFTEM).

(b) It provides a measure of the experimentalparameters required to perform orientationdependent EELS measurements. Clearly ifbobmagic then the experiment isolates princi-pally transitions involving momentum trans-fer parallel to the incident beam direction.This will be of great use in exploringanisotropic bonding in solids, both in bulkmaterials [15] and at spatially localized,aperiodic features, which are often intrinsi-cally anisotropic (e.g. interfaces or defectsprobed using either EELS spatial differencespectroscopy in the TEM/STEM [16] ordirectly using atomic-column EELS in aSTEM [17]). One major problem for spatiallylocalized studies is that a high degree oflocalization of the probe is required whichleads to low beam currents and low spectralsignals. Our predicted value for the magicangle is, in fact, rather small (compared tothat chosen for routine spectroscopic studies)which will further diminish the collected signaland hence long acquisition times may be

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B-K in MgB2

0

0.02

0.04

0.06

0.08

0.1

0 2 4 6 8 10

θΕ = 0.57mrads

Multiples of θE

π∗/(π

∗+σ∗

)

Fig. 7. Relative intensity of p� peak at the B K-edge of MgB2as a function of normalized effective collection angle, b=yE :Two sets of data are shown: data recorded with incident

electron beam parallel to /0 0 1S (E) and perpendicular to

/0 0 1S (’). The point of intersection of the two curves is themagic collection angle.

(a)

(b)

B-K in BN

00.05

0.10.15

0.20.25

0.30.35

0 2 4 6 8 10

N-K in BN

00.05

0.10.15

0.20.25

0.3

0 1 2 3 4

θΕ = 0.57mrads

Multiples of θE

π∗/(

π∗+

σ∗)

π∗/(π

∗+σ∗

)

θΕ = 1.18mrads

Multiples of θE

π∗/(

π∗+

σ∗)

π∗/(π

∗+σ∗

)

Fig. 6. Relative intensity of p� peak at (a) the B K-edge and (b)the N K-edge of hexagonal BN as a function of normalized

effective collection angle, b=yE (note at 197 kV, yE ¼ 0:57 and1.18mrad for the B K- and N K-edges, respectively). Two sets

of data are shown: data recorded with incident electron beam

parallel to /0 0 1S (E) and perpendicular to /0 0 1S (’). Thepoint of intersection of the two curves is the magic collection

angle.


required in order to achieve an adequatesignal to noise ratio; this may lead toproblems with specimen drift and/or radiationdamage. A solution may to record ELNESspectra at two differing collection angles andthen extract the separate parallel and perpen-dicular components of the momentum trans-fer as discussed in Ref. [6].

(c) Finally knowledge of the EELS magic angle isrequired if accurate comparison betweenexperimental ELNES data from anisotropicstructures and the results of theoreticalmodelling procedures (e.g. either state-of-the-art band structure or full multiple scatter-ing methods [18]) is to be achieved.

To reiterate this final point, in Fig. 8 we show acomparison of the C K-ELNES of graphite,

measured at approximately the magic collectionangle, bE2yE with the results of band structurecalculations for the unoccupied p-like DOS ingraphite using the full linear augmented planewave approach as embodied in the commerciallyavailable code WIEN ’97 [19]. The DOS resultshave been multiplied by a matrix element term andbroadened using a broadening parameter equal tothe experimental resolution (ca. 0.8 eV). It isimportant to note that in Fig. 8 the calculatedDOS is averaged over all possible directions, sinceit is usual not to assume an a priori knowledge ofthe weighting of the various directional DOScomponents (governed by the sample orientationand value of the collection angle). The calculationsinclude an approximation for the carbon 1s corehole created during the excitation process, via theuse of a 2 2 1 supercell. Clearly the calculationwith a core hole matches the magic angle data well,particularly in terms of the relative intensities ofthe p� and s� features.In a forthcoming publication we will discuss the

general applicability of the theory developed herefor all crystal systems. For now we note that thispresent analysis appears to be valid for hexagonaland tetragonal unit cell materials.

6. Conclusions

This work has presented experimental evidencefor the existence and magnitude of the magiccollection angle in transmission electron energyloss spectroscopy. Measurements of the electronenergy loss near-edge structure conducted at thiscollection angle, bmagicE2yE ; are independent ofthe orientation of an anisotropic crystal structuresuch as graphite. We show, theoretically that thisarises since the individual components of themomentum transfer parallel and perpendicular tothe incident electron beam direction integratedover the collection aperture are equal. Finally, wediscuss the implications of this result for thegeneral study of the electronic structure of bothanisotropic materials and inherently anisotropicstructures such as the study of aperiodic systems athigh spatial resolution.

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-5 0 5 10 15 20 25 30 35 40Energy loss (eV)

(b)

(a)

Fig. 8. (a) Experimental C K-ELNES of HOPG graphite after

background subtraction with beam incident down [0 0 1]

measured at magic collection angle, bmagic ¼ 2yE : (b) Theore-tical C K-ELNES calculated using FLAPW band structure

code including a supercell approximation for the core hole (see

text).


Acknowledgements

The authors gratefully acknowledge EPSRC(for a quota studentship to HD) and EPSRC andHEFCE (for equipment grants and postdoctoralresearch assistantships to AB and AS) as well asProf D.V. Edmonds for continued departmentalsupport. The time and patience of an anonymousreferee are also greatly appreciated.

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experimental and theoretical evidence for the magic angle in transmission electron energy loss...

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