Introduction to neutron scatteringLaboratoire Leon Brillouin

CEA Saclay

Phonons and spin waves in a nickel-iron alloy

Table des Matieres

1 Using neutrons as a probe for exploring matter 21.1 Why using neutrons ? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 How to produce neutrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Introduction to neutron scattering : some theoretical background 42.1 Interaction with matter : the partial differential cross-section . . . . . . . . . . . . . . . . . . . . 4

2.1.1 Nuclear scattering : scattering by a single nucleus . . . . . . . . . . . . . . . . . . . . . . 52.1.2 Array of N nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1.3 Coherent partial differential cross-section for a crystal . . . . . . . . . . . . . . . . . . . . 82.1.4 Magnetic scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Studying lattice and magnetic excitations by inelastic neutron scattering 113.1 Lattice vibration modes - Phonons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 Spin-waves - Magnons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4 Neutron scattering : experimental set up 15

5 Lattice and magnetic excitations in a nickel-iron alloy Ni40Fe60 17

1

Chapter 1

Using neutrons as a probe for exploring matter

1.1 Why using neutrons ?

Neutrons are an interesting probe to explore matter for several reasons :

• They are neutral particles so they do not interact with the electric charges of the target atoms, whichallows neutrons to enter deeply into matter (to probe the bulk)

• They carry a magnetic moment and have a spin S=1/2 which makes neutrons sensitive to magnetism

1.2 How to produce neutrons

A neutron beam can be produced using:

• a fission reactor: a thermal neutron is absorbed by a 235U nucleus, which splits into fission fragments andevaporates off a few neutrons with energy of 1 to 2 MeV.

• spallation: A high energy proton (∼1eV) chops pieces off a heavy nucleus. Twenty to forty neutrons areevaporated with energies of typically a few MeV.

Neutrons then slow down by loosing energy through inelastic collisions with light atoms (H,D,Be).Neutrons are classified depending on their energy given in meV :

• 0.1 - 10 meV : cold neutrons

• 10 - 100 meV : thermal neutrons

2

Using neutrons as a probe for exploring matter 3

• 100 - 500 meV : hot neutrons

• > 500 meV : epithermal neutrons

A neutron of 25 meV has a wave length of ∼ 1.48 A. Its wave length is of the same order of magnitude as thetypical inter-atomic distance in a solid. The neutron energy is of the same order of magnitude as the character-istic excitations in a solid such as the phonons or the magnons. Neutron spectroscopy can be used to probe thenuclear and magnetic structures of a sample and the related nuclear and magnetic excitations. This is a bulkand non destructive probe for measurements.

There are several neutron sources in Europe :

• ILL - Grenoble (Reactor - France)

• LLB - Saclay (Reactor - France)

• FRMII - Munich (Reactor - Germany)

• HMI - Berlin (Reactor - Germany)

• ISIS - Didcot (Spallation source- UK)

• PSI - Villigen (Spallation source - Switzerland)

But also in Russia, Japan, USA, Australia, Canada...

Chapter 2

Introduction to neutron scattering : sometheoretical background

2.1 Interaction with matter : the partial differential cross-section

During a neutron scattering experiment, the scattered intensity measured by the detector is proportional to thepartial differential cross-section which expresses the interaction between neutrons and matter. Formally, thepartial differential cross-section dσ

dΩdE′ gives the fraction of neutrons that can be scattered by a target in a solid

angle dΩ with a final energy between E′ and E′ + dE′. dσdΩdE′ has the dimension of an area.

Within spherical coordinates (r, θ,Φ), the solid angle dΩ = sin(θ)dθdΦ.The state of the neutron is given by the ket |k, σ〉, which incorporates the following properties:

• The neutron is described as a plane wave ψk(~r) = 1√Vei~k~r.

• The neutron has an energy Ek = ~2k2

2M and an impusion ~pk = ~~k.

• The neutron has a spin σ = 12 .

V is the volume of a box where the physics under interest takes place (usually the sample volume) and M is theneutron mass.During a scattering process, the sample is excited by the incident neutron beam and relaxes from an initial state|λ〉 with energy Eλ to a final state |λ′〉 with energy Eλ′ . The conservation law applies to energy and impulsionduring the scattering event and gives :

4

Introduction to neutron scattering : some theoretical background 5

~Q = ~ki − ~kfEλ + k2

i = Eλ′ + k2f

(2.1)

And the energy transfer during the scattering process is defined as :

~ω = k2i − k2

f = Eλ′ − Eλ (2.2)

The general expression of the partial differential cross-section is hence :

∂2σ

∂Ω∂Ef=kikf

(Vmn

2π~2

)∑σ,σ′

pσ∑λ,λ′

pλ|〈kfσ′λ′|V (~r)|kiσλ〉|2δ(~ω + Eλ − Eλ′) (2.3)

with :

V : The sample volume

pσ : The probability for the neutron to be in spin state |σ〉.

pλ : The thermodynamic probability for the target to be in initial state |λ〉.

V (~r) : The interaction potential between neutron and the considered system

The two last terms come from the Fermi Golden Rule which describes the transition from an initial state|kiσiλi〉 to final states |kfσfλf 〉. Depending on the expression of the interaction potential, one candistinguish between nuclear scattering and magnetic scattering.

2.1.1 Nuclear scattering : scattering by a single nucleus

Point-like scattering potential

The nuclear scattering is due to strong interaction between neutron and a nucleus. In the Born approximation,the only potential that gives rise to an isotropic scattering is a δ-function. For a single nucleus at position ~R,the Fermi’s pseudo potential reads:

Vn(~r) =2π~2

Mbδ(~r − ~R). (2.4)

b corresponds to the scattering length. It is determined experimentally and can be either positive or negative.The sign of b is defined such that b is positive for the scattering of an impenetrable sphere of radius b. For agiven nucleus, b varies with the type of isotope under consideration. .

The scattering potential of a nucleus is localized on a single point in real space and its Fourier transformVn(~q) is constant as a function of |q|.For X-ray scattering, the magnitude of the scattering potential grows with the electronic density of atoms. Forneutron scattering, b depends only on the type of nucleus (Fig. 2.1). For instance, light atoms such as Hydrogencan possess a rather large scattering length. This scattering length can be significantly changed by changingthe isotope (Hydrogen → Deuterium, for instance). This isotopic substitution provides a very useful contrastproperty, currently used in soft condensed matter. In addition, atoms which are very close in the periodic tableof classification can have scattering lengths with different signs (Co and Mn for instance).

Nuclear partial differential cross-section

Since the Vn does not depend on spin anymore, one can eliminate all spin dependent terms in the expression ofthe partial differential cross-section:∑

σ,σ′

pσ < σ|σ′ >< σ′|σ >=∑σ,σ′

pσδσ,σ′ = 1 (2.5)

Let us now evaluate the term < k′|Vn(~r)|k >.

VM

2π~2< k′|Vn(~r)|k >=

∫d~rψ†k′(~r)Vn(~r)ψk(~r) = bei~q

~R (2.6)

Introduction to neutron scattering : some theoretical background 6

Figure 2.1: Comparison of the efficiencies of the scattering potentials for X-ray and neutron scatterings. Formore information, see http://www.ncnr.nist.gov/resources/n-lengths.

and one gets the following expression for the partial differential cross-section:

dσ

dΩdE′=

k

k′

∑λ,λ′

pλ < λ|b?e−i~q ~R|λ′ >< λ′|bei~q ~R|λ > δ(~ω + Eλ − Eλ′) (2.7)

Then, one use the general properties of the δ-function:

δ(x) =1

2π

∫ +∞

−∞dteixt (2.8)

δ(ax) =1

|a|δ(x) (2.9)

Furthermore, for a given operator A, the time dependent operator A(t) is defined in quantum mechanics by:

A(t) = e−iHt~ A(t = 0)ei

Ht~ (2.10)

with H the hamiltonian of the system.

Using these properties, one obtains:

dσ

dΩdE′=

k

k′1

2π~

∫ +∞

−∞dteiωt

∑λ,λ′

pλ < λ|b?e−i~q ~R(0)|λ′ >< λ′|bei~q ~R(t)|λ > (2.11)

Then, one can eliminate the projectors |λ′ >< λ′| since∑λ |λ′ >< λ′| = 1 and one introduces the thermody-

namic average: ∑λ

pλ < λ|...|λ >=< ... >T (2.12)

The final expression for the partial differential cross-section is then:

dσ

dΩdE′=

k

k′b2

1

2π~

∫ +∞

−∞dteiωt < e−i~q

~R(0)ei~q~R(t) >T (2.13)

Scattering function, structure factor and correlation function

First, for sake of simplicity, one still consider the case where number of nuclei N is equal to 1.

One can introduce the so-called scattering function (or structure factor) S(~q, ω):

dσ

dΩdE′=

k

k′Nb2S(~q, ω) (2.14)

S(~q, ω) =1

N

1

2π~

∫ +∞

−∞dteiωt < e−i~q

~R(0)ei~q~R(t) >T (2.15)

Introduction to neutron scattering : some theoretical background 7

One further defines Gs(~r, t) the self-correlation function that gives the probability for a nucleus at position~R(0) at t = 0 in a box of volume V to be at ~R(t) at time t. The general expresion for Gs(~r, t) is:

Gs(~r, t) =1

N

∫d~r′

N∑l=1

< δ(~r′ −~(r) + ~Rl(0))δ(~r′ − ~Rl(t)) >T (2.16)

l is the index which is used to identify each nucleus. Here, we consider the case N = 1, only. Then, using theproperties of the δ-function, one obtains:

Gs(~r, t) =1

(2π)3

∫d~qe−i~q~r < e−i~q

~R(0)ei~q~R(t) >T

=1

(2π)3

∫d~qe−i~q~rGs(~q, t)

Gs(~q, t) =

∫d~r′ei~q

~r′Gs(~r, t) (2.17)

One can immediately see that the scattering function is related to the Fourier transform in space and time ofthe self-correlation function.

2.1.2 Array of N nuclei

Let us consider now the case where the target is made of an array of N nuclei. The nuclear scattering potentialV is then gives by:

Vn(~r) =2π~2

M

N∑l=1

blδ(~r − ~Rl) (2.18)

It is usually quite convenient to introduce the density function ρ(~r) and its Fourier transform ρ(~q) :

ρ(~r) =

N∑l=1

δ(~r − ~Rl)

=1

V

∑~q

ρ(~q)ei~q~r (2.19)

ρ(~q) =

N∑l=1

e−i~q~Rl (2.20)

The partial differential scattering cross-section takes now the form:

dσ

dΩdE′=

k

k′1

2π~

∫ +∞

−∞dteiωt <

∑l,l′

blbl′e−i~q ~Rl(0)ei~q

~Rl′ (t) >T (2.21)

The scattering lenght bl depends on the isotopes of each nucleus, but there is no correlation between the isotopedistribution and the position of nuclei in the array: blbl′ = b2 for l 6= l′ and b corresponds to the mean scatteringlength. In general:

blbl′ = b2 + (b2 − b2)δl,l′ (2.22)

The partial differential cross-section splits into two independent terms:

dσ

dΩdE′=

(dσ

dΩdE′

)coh

+

(dσ

dΩdE′

)incoh

(2.23)

The coherent term takes into account the interferences phenomena between nuclei at different sites. It is relatedto Fourier transform in space and time of the density-density correlation function:(

dσ

dΩdE′

)coh

=k′

kNb2Scoh(~q, ω) (2.24)

Scoh(~q, ω) =1

N

1

2π~

∫ +∞

−∞dteiωt <

N∑l=1

e−i~q~Rl(0)

N∑l′=1

ei~q~Rl′ (t) >T

=1

N

1

2π~

∫ +∞

−∞dteiωt < ρ( ~−q, 0)ρ(~q, t) >T (2.25)

Introduction to neutron scattering : some theoretical background 8

At variance, the incoherent term deals with each nucleus, independently from the other: it is related to theFourier transform of the self-correlation function Gs(~r, t).

(dσ

dΩdE′

)incoh

=k′

kN(b2 − b2)Sincoh(~q, ω) (2.26)

Sincoh(~q, ω) =1

N

1

2π~

∫ +∞

−∞dteiωt

N∑l=1

< e−i~q~Rl(0)ei~q

~Rl(t) >T (2.27)

Figure 2.2 shows examples of incoherent scatterings and the related incoherent structure factor.

Figure 2.2: Different types of incoherent scattering: a) isotope distribution, b) local excitations, c) diffuserelation.

2.1.3 Coherent partial differential cross-section for a crystal

A crystal is made of a periodic arrangement of nulei. The buiding block of a crystal is the unit cell whichcontains a specific arrangement of n nuclei. For a crystal with a periodic repetition of N unit cells, the totalvolume is V = Nv0, where v0 is the volum of the unit cell. Each type of nuclei of the unit cell is identified bythe index l, its scattering length is bl and its mass is ml. The location of a nucleus in the crystal is given by asum of three terms: ~Rd + ~rl + ~ud,l(t) :

• ~Rd is a vector of the Bravais lattice and indicates the unit cell the nucleus belongs to.

• ~rl is its mean position in the unit cell.

• ~ud,l(t) is its displacement with respect to its mean position.

~ud,l(t) is induced by the collective vibrations of nuclei, the phonon modes. One uses the label s to identify agiven phononic mode and es~q describes the polarisation of the mode with respect to the propagation wave vector~q. ~ωs~q is the phonon energy. Notice that ~q belongs to the first Brillouin Zone.

We use now the notation ~Q = ~k − ~k′ for the transfered momentum. ~Q is expressed as a combination of avector ~τ of the reciprocal lattice and ~q of the first Brillouin Zone: ~Q = ~τ + ~q as is illustrated in figure 2.3.

The coherent partial differential cross-section has an elastic term and an inelastic term:

dσ

dΩdE′=

(dσ

dΩdE′

)elas

+

(dσ

dΩdE′

)inelas

(2.28)

For the elastic scattering k = k′.(dσ

dΩdE′

)elas

= N(2π)3

v0

∑~τ

δ( ~Q− ~τ)

∣∣∣∣∣∑l

ble−Wle−i

~Q~rl

∣∣∣∣∣2

δ(ω) (2.29)

Introduction to neutron scattering : some theoretical background 9

Figure 2.3: Representation of the scattering vector ~Q in the reciprocal space.

Wl = e−12<(~Q~ud,l)

2> is Debye-Waller factor. The inelastic term reads:

(dσ

dΩdE′

)inelas

=k′

kN

(2π)3

v0

∑~τ

δ( ~Q− ~τ − ~q)

∣∣∣∣∣∣∑l

ble−Wle−i

~Q~rl

(~

2mlωs~q

) 12

es~q~Q

∣∣∣∣∣∣2

×((1 + nB(ωs~q))δ(ω − ωs~q) + nB(ωs~q)δ(ω + ωs~q)

)(2.30)

Note that a specific choice of ~Q provides detailed information on the polarization vector es~q, as

the scattered intensity will be proportional to∣∣∣es~q ~Q∣∣∣2.

The last term of the expression is related to the condition of detailed balance which describes the probability forthe neutron to create and annihilate an excitation. To get a better understanding of the unbalanced processes,one can tune the temperature to 0. As T → 0, nB goes to 0. Thus, at T=0, when the target is in its groundstate, the neutron cannot gain energy because there is no way to lower the energy of the target. At variance,the neutron can give energy to excite the target.

2.1.4 Magnetic scattering

Let us start from the general expression of the partial differential cross-section given in Eq.(2.3). The magneticscattering potential Vm is:

Vm = −µ. ~H (2.31)

µ = γµN σ is the magnetic moment operator for a neutron. It is expressed as a function of γ the gyromagneticfactor, µN the Bohr magneton for a neutron and σ the Pauli matrix. The neutron magnetic moment coupleswith ~H the distribution of magnetic fields inside the target. This field can be due to:

• spins of unpaired electrons

• electronic orbital moments

• nuclear spins

Spin-only scattering

Let us focus on the magnetic field from spins of unpaired electrons, which usually gives the dominant contribu-tion. For a dipolar magnetic interaction between spins of umpaired electrons and the neutron magnetic moment,~H has the following form:

~H = ~curl

(2µBS × ~R

R3

)(2.32)

The incident and final neutron spins can be either ↑ or ↓. Thus one can average over all spin states |σ >and |σ′ >. Thus, ∑

σ

pσ < σ|σασβ |σ >= δα,β (2.33)

Introduction to neutron scattering : some theoretical background 10

The partial differential cross-section can be written as:

dσ

dΩdE′=k′

kr20

∑α,β

(δα,β −

QαQβQ2

)×

∑l′,l′

Fl( ~Q)Fl′( ~Q)ei~Q(~Rl−~Rl′ )

×∑λ,λ′

pλ < λ|Sl,d,α|λ′ >< λ′|Sl′,d′,β |λ > δ(~ω + Eλ − Eλ′) (2.34)

Owing to the specific nature of the interaction potential, one can notice several important differences withrespect to the nuclear scattering.

• r0 is a constant and represent the classical electron radius : r0 = 4π( M2π~2 )2γµNµB = 4π( M

2π~2 )2γ e~2Mc

~e2mec

=γe2

mec2. me and e are the mass and the charge of an electron, and c the speed of light.

• the term∑α,β

(δα,β − QαQβ

Q2

)is due to the dipolar magnetic interaction and indicates that one probes

only the spin components which are perpendicular to ~Q. α and β are the usual Cartesian coordi-nates.

• while the nucleus can be viewed as a point, there is a distribution of electronic spins around the atoms.One therefore needs to introduce magnetic form factor Fl( ~Q) which is the Fourier transform of Sl(~r) thenormalized density of spins for the atoms at position ~rl in the unit cell. At variance with bl which isQ-independent, Fl( ~Q) decreases at large Q.

A single type of magnetic site

When there is only one type of magnetic site (magnetic ion), the partial differential cross section can be sim-plified:

dσ

dΩdE′=

k′

kF ( ~Q)2

∑α,β

(δα,β −

QαQβQ2

)Sα,β(~q, ω) (2.35)

Sα,β( ~Q, ω) =1

N

1

2π~

∫ +∞

−∞dteiωt

∑l,l′

ei~Q(~Rl−~Rl′ ) < Sα,l(0)Sβ,l′(t) >T (2.36)

Sα,β( ~Q, ω) is the Fourier transform in space and time of the spin-spin correlation function.In the case of a magnetic order, the elastic structure factor describes the ordered magnetic pattern and isproportional to the square of the ordered magnetic moment. As in the case of phonons, the dynamical structurefactor is related to the creation or the anihilation of (collective) magnetic excitations, i.e magnons of energy ~ω~q(in general). Note that, in agreement with spin wave theory, it has to be proportional to the ordered magneticmoment, whereas the static structure factor is proportional to the square of the ordered magnetic moment.

Chapter 3

Studying lattice and magnetic excitations byinelastic neutron scattering

Among the many topics addressed in condensed matter physics, we are particularly concerned with these twospecific issues :

• Studying the crystallographic and magnetic structure of the considered system : this helps establishingthe grounds for understanding properties observed experimentally (macroscopic physical properties areoften related to the symmetry of the system)

• Studying the excitation states in the system : this allows in particular to determine the parameters in-volved in the observed structure (lattice constants, magnetic interactions...). Determining such parametersprovides further understanding of the evolution of the system (temperature, electrical or magnetic fielddependence...).

3.1 Lattice vibration modes - Phonons

In order to illustrate the collective vibration modes of atoms in solids, we consider the simple case of a diatomicchain. The chain is a periodic arrangement of N atoms of type 1 (•). The lattice constant between them isa. On the right side of each atom of type 1, there is another atom of type (2) (×) at a distance d(< a). Wefurther consider that the atoms of types 1 and 2 have the same mass M . Atoms are connected to their nearestneighbors by springs K and G(< K) (see Fig. 3.1). The label n stands for the unit cell the atoms belong to.We note u1 and u2 the displacements of the atoms around their equilibrium position.

Figure 3.1: Diatomic chain.

Atoms oscillate in a harmonic potential:

Uharm =K

2

∑n

[u1(na)− u2(na)]2 +G

2

∑n

[u2(na)− u1((n+ 1)a)]2 (3.1)

11

Studying lattice and magnetic excitations by inelastic neutron scattering 12

Their displacement are given by the equation of motion:

Mu1(na) = − dUharm

du1(an)

= −K(u1(na)− u2(na))−G(u1(na)− u2((n+ 1)a) (3.2)

Mu2(na) = − dUharm

du2(an)

= −K(u2(na)− u1(na))−G(u2(na)− u1((n− 1)a) (3.3)

One then looks for solutions of the form:

u1(an) = e1ei(kna−ωt)

u2(an) = e2ei(kna−ωt)

Normal modes are given by : Dej = Mω2ej , where D is 2× 2 Matrix:

D =

(K +G −(K +Geika)

−(K +Ge−ika) K +G

)(3.4)

Once the matrix diagonalized, one gets the frequencies of the normal vibration modes and the relative displace-ments of atoms of types 1 and 2.

Mω2 = (K +G)±√K2 +G2 + 2KGcos(ka) (3.5)

e1

e2= ∓ K +Geika

|K +Geika|(3.6)

There are two modes (Fig.3.2):

• The mode of lower energy is called acoustic (A) and displacements of atoms are in-phase.

• The mode of higher energy is called optical (O) and displacements of atoms are out-of-phase.

Figure 3.2: Dispersion relation for a diatomic chain: the acoustic and optical modes are shown in blue and red,respectively.

For a 3D crystal there are now 3 degrees of freedom for the atomic displacements and p different atoms perunit cell. As a result, D becomes a 3p× 3p matrix and one obtains 3p modes ( 3 accoutic modes and 3(p− 1)optical modes).One can distinguish two types of modes (see Fig. 3.3). Modes for which the direction of displacement ~ej is parallel

to propogation vector ~k are called ”longitudinal” modes (L). Modes for which the direction of displacement ~ej

is perpendicular to propogation vector ~k are called ”transverse” modes (T).Figure 3.4 shows the dispersion curves of phonons in Germanium along the [111] direction. Along this

direction the transverse modes are degenerated.

Studying lattice and magnetic excitations by inelastic neutron scattering 13

Figure 3.3: Schematic description of longitudinal and transverse modes.

Figure 3.4: Germanium :dispersion relation relation along [111] direction.

3.2 Spin-waves - Magnons

Here, we want to illustrate the existence of collective magnetic excitations in solids. We consider the case offerromagnetic chain. In the ground state, spins ~Si are all aligned along z : ~Si = Sz. The Hamiltonian for aferromagnet is:

H = −∑i,j

Ji,j ~Si~Sj (3.7)

Ji,j = J(~Ri− ~Rj) = Jj,i stands for the the magnetic interaction that couples two spins on nearest neighbor sites(i, j).

When spins start oscillating around equilibrium, they acquire a small component δS in (x, y) plane:

~Si ' Sz + δSxx+ δSy y

The equation of motion gives the evolution ~Si as a function of space and time:

d~Sidt

= ~Si × gµB~ho (3.8)

~ho stands for the external magnetic field which acts on ~Si.

In the mean field approximation, ~ho is generated by all other spins, so that: gµB~ho = 2∑j Jij

~Sj .

The equation of motion can be further linearized around equilibrium position:

dδ~Sidt

= 2S∑j

Ji,j(δ~Si − δ~Sj)× z (3.9)

Studying lattice and magnetic excitations by inelastic neutron scattering 14

One introduces the Fourier components ~S(q) =∑l e−i~q ~Rl and J(q).

dδ~S(q)

dt= 2S[J(0)− J(q)]δ~Sj(q)× z (3.10)

dδSx(q)

dt= +2S[J(0)− J(q)]δSy(q) (3.11)

dδSy(q)

dt= −2S[J(0)− J(q)]δSx(q) (3.12)

One looks for solutions of the form: S−(q) = Sx(q)− iSy(q) = Ns(q)eiωt.

One can then deduce the dispersion relation that characterizes the propagation of the collective spin excitations.

ωq = 2S[J(0)− J(q)] (3.13)

When the magnetic interaction is between next-nearest neighbors in a ferromagnetic chain : J(q) = 2Jcos(qa)with a the lattice constant. Figure 3.5 shows the dispersion relation for collective spin excitation in a linearchain. Note that in the limit q → 0, one finds: ω ∼ q2.

Figure 3.5: Spin wave dispersion curve for a ferromagnetic chain.

? Remark : For an antiferromagnet or a helimagnet, ω ∼ q as q → 0.The collective fluctuations of spins can be viewed as a spin wave, given by:

δ~Si = s(q)[cos(~q ~Ri + ωt)x+ sin(~q ~Ri + ωt)y] (3.14)

Figure 3.6 illustrates the propagation of a spin wave in ferromagnetic chain.

Figure 3.6: Propogation of a spin wave along a ferromagnetic chain.

Chapter 4

Neutron scattering : experimental set up

One way to study the excitations in a crystal by inelastic neutron scattering is to use a triple-axis spectrometer(TAS). During a scattering process, the sample is excited by a monochromatic incident neutron beam. Thefirst step is hence to select the energy of the incoming beam. This is performed by using a monochromatorcomprising several co-aligned crystals with high neutron reflectivity (for instance copper crystals). The neutronsare then diffracted by the monochromator and according to the Bragg condition 2dsin(θ) = λ, the neutronsincident energy can be tuned by rotating the monochromator.

Figure 4.1: From left to right: (a) the 3-monochromators of thermal triple-axis spectrometer IN8 at ILL: Si111,PG002,Cu200. b)the analyzer of the cold spectrometer 4F1 at LLB. Monochromator (or analyzer) can be benthorizontally and vertically in order to focus the neutron beam on the sample (gain of flux, but decrease of theresolution).

After hitting the sample, the scattered neutrons can either gain or loose energy, or stay unchanged. Thefinal step is then to define the energy transfer one wants to analyse for a given scattering vector ~Q. This processis exactly the same as selecting the incident energy except that the monochromator used for selecting the finalenergy of the outcoming neutron beam is called an analyser.Finally, the scattered neutron beam with final energy is measured by a detector.

15

Neutron scattering : experimental set up 16

Figure 4.2: Triple-axis spectrometer

Chapter 5

Lattice and magnetic excitations in a nickel-ironalloy Ni40Fe60

Here are some elements to prepare before the practicalWe are interested in measuring the lattice and magnetic excitations in a nickel-iron alloy (Ni40Fe60), for

which the crystallographic structure and the macroscopic magnetic properties are known. Ni40Fe60 crystallizesin a face centered cubic structure (the lattice parameter is a= 3.58A) and is it ferromagnetic below TC ∼ 800 K.

Figure 5.1: Crystalline structure of Ni40Fe60. Ni and Fe are randomly distributed on the FCC lattice positions.

We first need to define which part of the reciprocal lattice we want to explore : ‘

• Determine the atomic positions in the unit cell

• Calculate the structure factor as a function of the (h, k, l) coordinates of a reciprocal lattice vector

17