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Absorption and resonant scattering of x-rays

Dans le document The DART-Europe E-theses Portal (Page 45-48)

Since the pioneering work from de Bergevin and Brunel on two antiferromagnets (NiO and Fe2O3) in 1981 [111–113], x-ray scattering became a tool to study magnetic materials due to the development of third-generation light sources. They provide high brilliance photon fluxes with a tunable energy, therefore allowing the probing of different elements in materials.

Resonant scattering is a photon-in photon-out process (figure 3.1): following the ab-sorption of a photon, a second photon is emitted with either the same (elastic) or a different energy (inelastic). In both cases, the wavevectork due to a transfer of the momentum to the electronic system. Measurement of the variation of the wavevector,q=k’ - kwithq themomentum transfer, brings information about the structure of the system such as the crystallinity, the shape or the presence of magnetic correlations.

X-rays are chemical sensitive probes which can be tuned to match the incident photon energy with exact difference between two energy levels (absorption edge or resonance).

Such properties are not available for other probes such as neutrons or muons. In 3d magnetic transition metals such as Fe, Co or Ni, the L3 absorption edge is located in the 700 - 800 eV range. Exciting the system with soft x-rays thus allows the study of their magnetic properties (2p → 3d transitions). In addition, the possibility to change the polarization of the photon allows the study of magnetism through the x-ray magnetic circular dichroism (XMCD) [13, 114–116].

Following the discovery of the resonant scattering and, at the same time, of the sensitiv-ity of scattering to magnetism [117–120], soft x-ray resonant scattering became a standard technique to study a broad range of materials: magnetic multilayers [121, 122], coupled nanowires [10], spin-valves [11] or sub-micron patterned films [12].

3.1.1 Absorption and Magnetic Circular Dichroism X-Ray Absorption Spectroscopy

Absorption is the transfer of the energy and momentum of a photon to an electron (figure 3.1 a). This happens at specific energy value, which corresponds to the absorption edge

3.1. ABSORPTION AND RESONANT SCATTERING OF X-RAYS 45 (or resonance) of an element [109]. These edges vary from element to element, making the photon a chemically sensitive probe. These values are known and tabulated [123].

The experiments reported here were carried out in the soft x-ray photon range; this energy range covers the 2p → 3d (L2,3) edge of 3d transition metals Co,Fe,Ni and the 5d→4f (M4,5) of 4f edge rare-earth, e.g. Sm or Gd.

In the following, we will briefly discuss x-ray absorption using a quantum mechanics approach. This discussion follows the approach of the book by Altarelli in ref. 108. In particular, our interest will be the establishment of the Hamiltonian of the system, and the expression of the transition rate within the perturbation theory, which we will compare later with its expression in the case of scattering.

The total Hamiltonian Htot is the sum of two Hamiltonians, Hel which describes the electrons andHrad for the photons:

Htot =Hel+Hrad (3.1) p is the momentum operator and m the electron rest mass. V(ri) is the potential energy which in the case of a system with N electrons:

V(r1, ...,rn) = where Vnuc describes the coulombic interaction of all electron with the nucleus and VC the coulombic repulsion between electrons. This expression can be simplified to a one-electron system by considering the behaviour of one one-electron in an averaged field through theself-consistent field approximation [108]:

In the presence of an electro-magnetic field, the momentum of the electrons changes.

To take this into account, we replacepi by pi−(e/c)A: whereA(ri) is the vector potential [108]:

A(r, t) =X withh the Planck constant, epolarization vector of the modeλ,Ωvolume of the quanti-zation box andwk=c|k|.

a(k, λ) and a(k, λ) are the operators ofannihilation and creation of a photon which are also used to write the Hamiltonian related to the electro-magnetic field [108]:

Hrad =X

k,λ

¯

k(a(k, λ)a(k, λ) + 1/2) (3.7)

The total Hamiltonian is thus given by:

withHint, interaction Hamiltonian between photons and electrons, expressed as:

Hint=−X

The fist term linear in A corresponds to the absorption of a photon. The second one, quadratic in A, corresponds to the Thomson scattering, i.e. the elastic scattering of a photon by the electronic cloud. It also corresponds to the scattering of the photon via an intermediate state, which we will describe in section 3.1.2 as resonant scattering.

In the following, we are only interested in absorption. To simplify the theoretical treatment, it is usual to use theelectric dipole approximation. Let us consider the transition between an initial state|ii and a final state |fi. As the wavelength of the photon is much larger than the size of an atom, the matrix elements which describes the transition in the first and second terms of the equation (3.10) of formhf|ei(q·r)|ii can be rewritten as hf|q·r|ii [109]. Within this approximation, the probability that a photon is absorbed is given by the absorption cross-section σabs; its expression is given in equation (9.59) in ref. 109. Using σabs, we can now write the transition rate up to the first order in the perturbation theory using theFermi’s Golden Rule [109]:

w= 2π

As a consequence of the dipole approximation, one obtains from equation 3.11 the following selection rules, written here in the case of single-electron transition [109]:

∆l=±1; ∆ml= 0,±1; ∆s= 0; ∆ms= 0.

withl the azimuthal quantum number,ml the magnetic quantum number,sthe spin and ms the spin projection quantum number.

X-ray Magnetic Circular Dichroism

Absorption cross-section in ferromagnetic elements does not depend only on the photon energy, but also on the polarization. This was first demonstrated by Schütz and co-workers at the Fe K-edge and was the first demonstration of the XMCD [13]. To describe this effect, we use a two-component model to the 2p → 3dtransition (figure 3.2) following ref. 109 and 124; the XMCD effect stays however a single step process.

In a first part, a circularly polarized photon is absorbed by an electron. Conservation of the momentum imposes the transfer of the photon momentum to the electron via spin-orbit coupling [125]; absorption of a photon alone cannot change the spin. The relative

3.1. ABSORPTION AND RESONANT SCATTERING OF X-RAYS 47

Dans le document The DART-Europe E-theses Portal (Page 45-48)