Anisotropic Compton scattering in LiF using synchrotron radiation

Anisotropic Compton scattering ^{in LiF} using synchrotron ^radiation

G. Loupias ^{and J. Petiau} (*)

Laboratoire de Minéralogie-Cristallographie (**), Université Pierre-et-Marie-Curie, Paris, ^France

and Laboratoire _pour l’Utilisation du Rayonnement Electromagnétique (***), ^Bât. 209C, Université Paris-Sud, Orsay, ^France (Reçu ^le 31 juillet 1979, accepte ^{le 19 novembre} 1979)

Résumé.

L’utilisation du rayonnement synchrotron ^a permis ^{la mesure} de la distribution des moments élec-

troniques ^{dans un} monocristal de LiF pour les directions 100 > ^et 111 > ^{avec une} résolution de 0,15 ^unité atomique. ^Cette résolution est environ 4 fois meilleure que celle des ^mesures existantes en diffusion _03B3. Le spectro- mètre utilise le rayonnement ^de LURE-DCI, ^un monochromateur channel-cut, ^{un cristal} analyseur ^focalisant

et un détecteur à localisation spatiale. ^Les résultats obtenus confirment l’anisotropie importante ^{de la} distribution de moments dans LiF. La comparaison ^{avec les} anisotropies calculées dans la zone 0 à 2 u.a. montre un très bon accord avec les calculs LCAO de Berggren ^et al. pour q 0,6 ^{u.a. et avec} les calculs H.F. de Euwema et al. pour q > 0,7 ^u.a.

Abstract.

Using synchrotron radiation, ^the electronic momentum distribution is measured with a 0.15 atomic unit resolution in a LiF single crystal ^{for the} 100 > and 111 > directions. This resolution is about four times better than obtained in _03B3 experiments.

The spectrometer ^uses the LURE-DCI radiation with a channel-cut monochromator, ^a focusing crystal analyser

and a position sensitive detector. The measurements performed confirm the significant anisotropy of the electron distribution in LiF. The present ^{results are} compared with calculated anisotropies ^{in the range 0 to 2 a.u. ;} they

agree quite well with the Berggren et ^al. calculations (LCAO) for q ^{0.6 a.u.} and with the Euwema et al. calcula- tions (H.F.) for q > 0.7 a.u.

Classification

Physics ^Abstracts

71.25T - 07.85

1. Introduction.

Measurements of the electron momentum distribution have recently [1] proved ^to

be a particularly ^sensitive probe ^{of the behaviour}

of the slowly moving electrons. In condensed matter, such electrons are precisely ^{those which are taking}

part ⁱⁿ bonding ^{and are thus of a} particular ^interest

for the physicist and the chemist.

Measurements of the momentum distribution for the target ^{electrons are} principally ^{obtained by two}

methods :

-

the spectral analysis ^{of the} inelastically ^scattered

X or y photons, the so-called Compton profile (CP) ;

-

the long-slit angular correlation profile ^{of the}

two photons ^emitted during ^the annihilation of

positron-electron pairs (ACP).

The experimental profiles usually ^{lead to the total}

electron momentum density n(p) integrated ^{in momen-}

tum space ^over planes perpendicular ^{to the} scattering

vector k, ^{i.e. :}

with

However this relation is only ^{valid if :}

-

in the ACP measurement, the effects of the

positive charge (repulsion by ^{ion core,} affinity ^for

defects and vacancies) ^{can be} neglected ^{and the} position wave-function can be considered as a

constant. These assumptions ^are usually ^{far from} being ^true and, ^{as a} consequence, the interpretation

of the ACP in terms of electron momentum is some-

times difficult. On the other hand, ^this technique ^has

hitherto enjoyed ^{a better} resolution than usual CP

techniques ^{and is then to} be used for Fermi surface (*) ^Et E.N.S.J.F., Paris.

(**) Laboratoire associe au C.N.R.S. (LA 09).

(***) Laboratoire du C.N.R.S. associ6 a l’Universit6 Paris-Sud.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01980004103026500

266

studies, ^when high quality single ^{crystals are avai-} lable ;

-

in the CP measurements, ^the Impulse Approxi-

mation [3] ^can be assumed i.e. the _energy transferred to the recoil electron greatly exceeds its binding

energy. This assumption ^is always ^{true in the case}

of a y-ray ^source whereas in the case of a classical

X-ray ^tube it may be different ; ^{then a} fundamental advantage ^{of a} synchrotron ^{radiation source} (SRF)

is the possibility ^of picking ^{out a} suitable incident

photon energy.

The first measurements of Compton scattering

with synchrotron ^{radiation were made} by ^M. Cooper

et al. [4] using radiation from the synchrotron ^NINA,

at three energies ^{in the} range 40-80 keV ; they ^showed

that the count rates were very favourable while the resolution was severely ^limited by ^{the use of a solid}

state detector.

At LURE-DCI the available energies range from 4 keV to 15 keV.

Compton scattering experiments ^{have been per-}

formed to check the _accuracy of wave-function calculations [1]. ^To improve ^the sensitivity ^{of these} checks, ^{it is} necessary ^to improve ^{both the} statistical accuracy and the resolution of the data. In order to solve the former problem ^several highly ^efficient

y-ray Compton spectrometers [5] ^{have been} designed

in the last two years. However the momentum reso-

lution _{is poor} (0.55 ^{to 0.4 atomic} unit), being ^limited by ^the energy resolution of the analysing ^{solid state}

detector.

On the other hand, ^{the momentum} resolution of the X-ray crystal analysing spectrometers (typically

0.2 a.u.) ^is actually ^limited by statistics considerations for two main reasons : first the inelastic cross section is weak due to the severe competition ^{with the} photo-

electron absorption (except ^{for low} z) ^and secondly

the detection is usually performed point by point [6-7].

These considerations led us to choose a high ^flux X-ray ^source (SRF), ^{a curved} crystal analyser ^and

a position ^{sensitive detector} hoping ^{in this} way ^to

improve ^{the momentum} resolution while keeping

a satisfactory statistical _accuracy.

The measurements of directional Compton pro- files of LiF using ^{this new} spectrometer may be justified

for several reasons. First from the theoretical point

of view LiF is a simple system ^and directional Compton profiles may be derived according ^{to two models :}

either by Hartree-Fock calculation using ^{a Gaussian}

basis set by ^{Euwema et al.} [8] ^or by ^a tight binding

calculation using ^Kunz’s crystal ^orbitals by Berggren

et al. [9]. Secondly y-ray Compton profiles ^{are avai-}

lable [9] ^{and offer a} good opportunity ^{for a} comparison

of the assumptions ^{and data} corrections involved in both X and _y-ray techniques.

2. 1’he X-ray spectrometer.

2.1 ^APPARATUS

DESCRIPTION. - The SRF photons being mostly horizontally polarized [10] ^the spectrometer ^{has been}

built in the vertical plane. A schematic diagram ^is

shown in figure ^{I and a} detailed set-up in figure ^2.

Fig. ^1. - Schematic representation ^{of the} Compton spectrometer

at LURE-DCI.

The white synchrotron ^{beam is} monochromatized by ^a double reflection in a channel-cut single crystal.

In the present experiments ²²⁰ reflections in the

symmetrical Bragg ^case have been used in a silicon

crystal but, ^{in order to} increase the monochromatic beam flux (by ^{a factor} roughly equal ^to four) ^a ger- manium single crystal has ^{been cut} [11] ^{in such a} way that the reflections 220 are asymmetric ^with angles of respectively ^{10 and 7} degrees ^{between the} reflecting planes ^{and the surfaces. The incident} wavelength Ao

is selected by ^use of rotation Rl ^{with a} step ^{of two} seconds of arc and the whole spectrometer ^{is matched}

to the position ^{of the} monochromatic beam through

the translation T 1.

The vertical width of the slit F1 ^{has to be limited}

in order to improve ^the resolution (under ^{a width}

of 1.5 mm the incident _energy resolution is controlled

by ^{the source} size). ^The slit, systems (F2, F3) ^{and the}

monochromator and analyser shielding ^{are dictated} by background considerations. Moreover in order to save intensity and reduce the background, ^{the mono-}

chromator as well as the paths ^{of the incident and} analysed ^{beams are} included in an evacuated chamber.

The incident flux is monitored in an air ionization chamber I and irradiates the sample ^{S mounted on a} goniometer head fixed on a rotation axis R2 ^{so that} any hkl > crystallographic ^{axis can} be oriented

along ^the scattering ^{vector k. The value of the scatter-} ing angle T is adjusted by the rotation R3. ^{For this}

configuration ^{of the} spectrometer (Fig. 2) ^{the value} of T ^can be selected at will from 1000 to 1700.

The scattered radiation is energy-analysed by Bragg reflection in a curved crystal A, ^{in the trans-} mission geometry. ^The analysed beam is focused

on the Rowland circle. The coupling ^{of a} position

sensitive detector (PSD) ^{with a} multichannel analyser

allows all the points ^{of the} spectrum ^{to be simul-}

taneously collected. The crystal plate ^{A is cut in such}

Fig. ^2.

Experimental set-up ^{of the} Compton spectrometer ^{at LURE-DCI. M :} channel-cut monochromator (Ge ^or Si). ^{I : ionization} chamber. S : sample. ^{A : curved} analyser (Si 220). ^{D :} position sensitive detector.

a way that the Bragg ^{reflected beam is normal to}

its surface. This is indeed the geometrical ^condition providing ^{both the best} crystal focusing [12] ^and

the smallest parallax ^{in the 5 mm thick detector of the} backgammon type (Jeu ^de Jacquet) [13]. ^{The rotation} R4, actually ^achieved by ^{a vertical} translation of the bracket shown on the right ^{side of} figure 2, ^allows

the analyser adjustment ^for Bragg reflection of the

Compton ^scattered wavelength. To reach the best resolution available with this spectrometer configu- ration, ^the crystal may be bent with a radius of cur-

vature of _up to two meters. The PSD is always posi-

tioned normal to the diffracted beam (Fig. 2). ^All

these motions. are motorized and remote controlled.

Finally and still for background reduction, ^the geometry ^{has been} designed ^{in such a} way that the detector should not be in direct view of _any point

of the sample.

2.2 COMPARISON BETWEEN THE -PERFORMANCES OF THE PRESENT APPARATUS AND A SELECTION OF COMPTON

SPECTROMETERS USING THE DIFFERENT AVAILABLE TECH- NIQUES.

All the data presented ^{in table I are}

concerned with a 0.2 ^mm thick aluminum sample

and have been derived from the information given

in each experiment. For y-ray spectrometers, ^the sample ^{thickness can be increased} up ^{to a} few milli-

meters (usually ¹ mm) ^{with a} corresponding ^increase

in intensity ^{but also in the amount of} multiple Compton scattering ^{events which} requires ^careful

corrections. For the DCI spectrometer, ^{the data} have been recorded with the germanium ^mono-

chromator, ^{the incident} energy being ^{fixed at 10.5 keV}

and the scattering angle ^at 140°. The conditions of the impulse approximation ^are satisfied for all the electrons : the energy transfer at the Compton peak

is 380 eV while the electron binding energies ^{are res-} pectively ^{1 559 eV and 72-87 eV for K and L shells}

in aluminum.

The experimental data listed in table I clearly

demonstrate the large gain in resolution obtained with the curved crystal analyser set-up ^as compared

to the y-ray spectrometers. The resolution function has been evaluated through ^{the width of the fluores-}

cence line Kocl ^of germanium ^{used as a} sample.

No serious drop ^{in the} statistical _accuracy is observed although aluminum is not the most favou- rable case for a low incident _energy source. However

an increase of the analysing crystal quality ^can bring

the resolution down to 0.05 atomic unit without

intensity loss; ^this X-ray spectrometer will then be

an order of magnitude ^{more sensitive than a} y-ray spectrometer ^{with a SSD detector.}

2 . 3 DIFFERENT CONFIGURATIONS ^{OF THE APPARATUS.}

-

Compton scattering involving large transfers of momentum and _energy is studied in the configuration

a or b, ^as described in figure ^3. Figures ^{1 and 2 show}

the a-configuration ^{which was} used for all the experi-

ments described in this paper. The b-configuration

has to be chosen for a better resolution : the analyser

is then positioned ^{in the} reflection geometry ^with

a Bragg angle larger than 50°. On the other hand the

range of _energy analysed ^{in one} measurement is

268

Table I.

Performances of ^various Compton spectrometers.

All the data are concerned with a 0.2 mm thick aluminum sample.

(*) This number results from the superposition ^{of the two} Compton profiles ^{due to} the incident doublet and of a high background.

For the present work, the numbers indicated between brackets represent performances expected ^{in the near future} (autumn 1979) ^when

DCI will be operated ^{at 1.80 GeV, 400 mA} instead of 1.72 GeV, 200 mA.

Fig. ^3.

^{The four} configurations ^{of the} apparatus (a ^{and b corres-} ponding to qJ > 90°, ^{c and d to} 9 90°).

smaller ; recording ^{of an entire} Compton profile

then requires several successive measurements in the 6 cm long ^PSD.

Configurations ^c and d allow the study ^{of scatter-} ing ^within the range of low and intermediate momen-

tum transfer _e.g. from q ^{= 0.2 a.u.} (Eo ^{= 4 keV,}

9

200) to q

^{2.7 a.u.} (Eo

^15.5 keY, ({J

800).

The best resolution is achieved in the c-configuration.

3. Experimental profiles.

^{We have} performed

the measurement of two directional Compton pro- files with the scattering ^{vector k} successively parallel

to the ( 100 ) and 111 > crystallographic ^{axes of}

LiF : the largest difference between directional _pro- files is of course expected for these two directions.

3.1 EXPERIMENTAL CONDTTIONS. - The experimen-

tal arrangement ^has been selected in order to fulfil the impulse approximation. ^{The electron} binding energies ^are respectively ^{55 eV for the K shell of}

lithium and 687 eV for that of fluor. Thus an incident energy equal ^{to 10 870 eV and a} scattering angle equal

to 1300 yield ^an energy transfer of 380 eV at the

Compton peak.

The entrance slit F1 ^{is three} millimeters wide ver-

tically (and ^{2.5 cm wide} horizontally); ^{the mono-}

chromator is the symmetrical ²²⁰ silicon channel-cut.

The incident _energy bandwidth AEO ^{is 2} eV ; ^this is evaluated experimentally by ^{use of the well-known}

structure of the K absorption edges ^of copper and titanium recorded with the ionization chamber.

Using ^a series of such K edges (Ge, ^Cu, Ti) ^{an absolute}

calibration of the incident _energy is obtained.

The monochromator and the slits are covered with silver foils whose characteristic K and L lines

are far from the values of the incident and scattered

energies.

The monochromatic beam with 109 photons ^within

a cross section of 0.75 cm2 irradiates a cleaved (100) single crystal sample ^{of LiF, 2 mm thick.} The scattered radiation is analysed by ^{the 220} reflection in a silicon

crystal, ^{0.180 mm thick and} bent with a radius of

50 cm.

The measured spatial resolution of the position

sensitive detector is 170 ym. Its linearity ^{in both} space and amplitude ^is better than 2 ^x 10-3. The energy

scale of the spectrum ^can be calculated from the

crystal dispersion; ^{moreover it has been calibrated} by ^{use of} the three florescence lines of the germa- nium K-spectrum. ^{The momentum scale is then}

calculated from relation (2).

For each direction approximately ^{105 counts were}

collected in the Compton peak during ^{ten hours}

when LURE-DCI was running ^{with an} averaged

electronic current of 150 mA at 1.80 GeV. In fact two

independent measurements were performed ^{in each}

case; the difference plot ^{for the} two measurements

corresponding ^{to the} direction 111 > is shown in

figure ^4.

Fig. ^4.

Difference between two independent 111 ) ^measure-

ments in LiF.

3.2 DATA ANALYSIS METHOD. - The _very weak and uniform background (less ^{than 1 count for 100}

collected in the central Compton channel, ^{0.027 0 a.u.}

of momentum wide) ^is first subtracted. Then the measured scattering ^cross section is corrected from the wavelength dependent ^{factors : these are the} photoelectric absorption ^{in the} sample ^and analyser,

the reflectivity and Lorentz factor of the analyser

and the efficiency ^{of the} detector. The last term is evaluated from both the photoelectric absorption

and the _energy of a pair creation in the xenon filling

the detector cell. The first harmonic, energy 2 Eo,

is not detected.

Finally ^the profile without the K shell electrons of fluor is normalized to a theoretical profile [9] ^{so that :}

+2

- 2 ^{J(q) dq =} 4.408. The limits of ± ^{2 a.u. ofmomen-}

-2

tum are actually ^{chosen to} eliminate the beginnings

of the K shells Compton Profile from both Li+ and F-. No correction for multiple scattering ^{is made}

since the thickness of the sample is of the same

order [1] ^as g-’

The data have not been corrected for instrumental resolution but the theoretical profiles ^{have been}

convoluted with the resolution function for _compa-

rison.

Table II.

Compton profiles for LiF 100 ).

(*) ^The contribution of the K shell electrons of fluor is subtracted (Clementi ^{atomic orbital} [15]), and the theoric profile ^is convoluted

by ^our resolution function.

(**) ^Ref. [9]. ^The experimental profile ^{corrected for} multiple scattering ^and spectrometer resolution is treated ^as (*).

270

3.3 RESULTS AND DISCUSSION. - The experimental profile ^{for LiF} f 100 > averaged ^over ± q ^{is in} agree- ment with Berggren et ^al.’s theory (see ^Table II) : dJ(0)/J(0) ^{= 1.4 x I O - 3 and} AAq)lAq) , ^{1.3 x 10-2}

for 0 , q 1.5 a.u. while Euwema’s theory ^over-

estimates J(1.0) by 3 %. ^The statistical accuracy is

± ¹ % ^of J(q). ^{However, for the values of} q > ^{1.6 a.u.}

the X-ray experimental profile ^is always ^{above the}

calculated values, ^so corrections for multiple ^scatter- ing ^{have to} be included before deriving any ^accurate conclusion in the range 1.6 q ^{2.0 a.u. of momen-} tum.

However as it has been already ^shown [9], ^the quality ^{of the} crystal ^wave function is more accurately

checked by ^the study ^{of the} anisotropy itself evaluated

by ^the difference J(q) - J(q) ^{where the} multiple

(100> 111 >

scattering correction needs not be considered since both profiles ^are equally ^{affected. A} point by point

subtraction of the ( 111 ) profile ^from the ( 100 ) profile ^{has been} performed ^{and the} uncertainty ^on

the difference value has been estimated as being

inferior or equal ^to + 0.01. The agreement ^{with the}

tight binding theory (and ^Kunz’s crystal orbitals)

has been found satisfactory for q ^{0.6 a.u. whereas} for q > 0.7 a.u. the agreement is better with the H.F. calculation of Euwema.

Fig. ^5.

Difference between 100 > and 111 ) Compton profiles ^{in LiF.}

The poor quality ^{of the} Gaussian basis is well known for the low values of momentum but for an inter- mediate _range of momentum this Gaussian basis is reliable. It is not easy ^to make a direct comparison

with _{the y-ray} anisotropy ^{due to} its broad resolution function. Let us recall that the elastic X-ray ^scatter- ing measurements show that the charge density ^is isotropic, ^within experimental accuracy [16] ^while figure ^{5 shows an} amplitude ^{of the momentum aniso-} tropy approximatively equal ^to ± 2 % ^of J(O).

4. Conclusion. - The X-ray spectrometer ^described here is a reliable apparatus ^{as is} demonstrated by

these first results on the anisotropy ^{of LiF. Moreover}

we have obtain a large gain in resolution without

an important loss of statistical _accuracy for the low Z elements when compared ^{to the best} performant

y-ray spectrometers. However the investigation ^of

the anisotropy ^{of the} Compton profile ^measured along ^different crystallographic directions alone is not yet sufficient since, ^{even at small} q, the Compton profile ^contains high ^momentum contributions due to the integration ^over planes ^{in the} k-space (rela-

tion (I)). ^{Therefore one} should aim at reconstructing

the three dimensional momentum density n(p) ^and

its Fourier transform, the autocorrelation function

B(Z).

In such a study resolution is a critical parameter otherwise this function is drastically damped by

the Fourier transform of a too broad resolution function. The atomic orbitals would then be hardly separated ^for long correlation lengths. ^{On the other}

hand the tunable wavelength ^{allows the conditions}

for the impulse approximation ^or, alternatively, ^the study ^{of its breakdown} when the K shell binding edge ^is expressly located within the Compton profile

to be studied.

Acknowledgments.

^The realization of the _spec- trometer was supported by ^the Délégation ^Générale

a la Recherche Scientifique. ^{We are} very much undebted to J. Frouin and Y. Bernard, ^{from the} Laboratoire de Mineralogie-Cristallographie, ^for taking ^a very

important part ^{in the} design ^and construction of the apparatus. ^We express thanks ^to the Laboratoire de I’Acc6l6rateur Lineaire, ⁱⁿ Orsay, ^and parti- cularly ^{to Dr. P. Marin for} providing ^the synchrotron

beam.

References

[1] ^{For a recent review see} Compton Scattering, ^{ed. B. Williams} (McGraw-Hill, London) ^1977.

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(1978) 115.

PATTISON, ^P. and SCHNEIDER, ^J. R., ^Nucl. Instr. and Meth. 158

(1979) ^145.

[6] Comparison of various Compton spectrometers ^{in use before} 1976 : REED, ^W. A., ^Acta Crystallogr. ^{A 32} (1976) ^676.

[7] ^{However a} focusing X-ray spectrometer ^{with a} photographic

detection has been used to measure Compton profiles ^of

water and ice : the momentum resolution was 0.11 a.u.

LOUPIAS, G., Thesis University ^{Paris 6} (1978).

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DAGNEAUX, P. et al., ^Ann. Phys. ⁹ (1975) ^9.

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COOPER, M. J., PATTISON, P., WILLIAMS, ^{B. and} PANDRY, K. C., Philos. Mag. ²⁹ (1974) ^12-37.

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