MuPAD 3D Polarization Analysis in magnetic Neutron Scattering

Physik-Department

MuPAD3D Polarization Analysis in

magnetic Neutron Scattering

Diploma thesissubmitted by

Marc Janoschek

September 2004

Technische Universitat Munchen

Contents

1 Introduction 5

2 Theory of Polarization Analysis 92.1 Neutron scattering cross section . . . . . . . . . . . . . . . . . 92.2 Magnetic scattering . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.1 The magnetic interaction potential . . . . . . . . . . . 112.2.2 The magnetic interaction vector . . . . . . . . . . . . . 122.2.3 Magnetisation of the sample . . . . . . . . . . . . . . . 132.2.4 The geometric selection rule . . . . . . . . . . . . . . . 13

2.3 Polarization of a neutron beam . . . . . . . . . . . . . . . . . 142.4 Polarization in neutron scattering experiments . . . . . . . . . 18

2.4.1 Polarized neutron scattering cross-section . . . . . . . . 182.4.2 Polarization of the scattered beam . . . . . . . . . . . 212.4.3 The polarization tensor . . . . . . . . . . . . . . . . . . 23

3 Polarization Analysis on a Three-Axis-Spectrometer 253.1 Three-Axis-Spectrometer (TAS) . . . . . . . . . . . . . . . . . 253.2 Equipping a TAS for Polarization Analysis . . . . . . . . . . . 28

3.2.1 Polarizer . . . . . . . . . . . . . . . . . . . . . . . . . . 283.2.2 Turning the Polarization Vector . . . . . . . . . . . . . 303.2.3 Polarizing analyzer . . . . . . . . . . . . . . . . . . . . 35

3.3 Setups for Polarization Analysis . . . . . . . . . . . . . . . . . 373.3.1 Classical polarization analysis . . . . . . . . . . . . . . 383.3.2 Three Dimensional Polarization Analysis . . . . . . . . 403.3.3 CryoPAD . . . . . . . . . . . . . . . . . . . . . . . . . 43

4 MuPAD 454.1 The Principle of MuPAD . . . . . . . . . . . . . . . . . . . . . 454.2 Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.2.1 Zero Field Chamber . . . . . . . . . . . . . . . . . . . 53

3

4 CONTENTS

4.2.2 Precession coils . . . . . . . . . . . . . . . . . . . . . . 614.2.3 Coupling Coils . . . . . . . . . . . . . . . . . . . . . . 77

4.3 Calibration of MuPAD . . . . . . . . . . . . . . . . . . . . . . 794.3.1 Mechanical Adjustment . . . . . . . . . . . . . . . . . 794.3.2 Calibration of MuPAD precession coils . . . . . . . . . 83

5 Measurements 855.1 Setup on TAS IN22 . . . . . . . . . . . . . . . . . . . . . . . . 855.2 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 875.3 Final Measurement . . . . . . . . . . . . . . . . . . . . . . . . 90

5.3.1 The sample: MnSi . . . . . . . . . . . . . . . . . . . . 905.3.2 The Satellite Peaks . . . . . . . . . . . . . . . . . . . . 915.3.3 Chirality . . . . . . . . . . . . . . . . . . . . . . . . . . 945.3.4 Full Polarization Analysis . . . . . . . . . . . . . . . . 975.3.5 Inelastic measurements . . . . . . . . . . . . . . . . . . 1025.3.6 Accuracy of MuPAD . . . . . . . . . . . . . . . . . . . 104

5.4 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

6 Conclusion 111

Appendix 113

A Commutation of rotary matrices 113

B pMuPAD 115

C Chiral Magnetic structures 121C.1 Chiral structure in real space . . . . . . . . . . . . . . . . . . 121C.2 Chiral structure in reciprocal space . . . . . . . . . . . . . . . 123C.3 Neutron scattering cross-section of a chiral structure . . . . . . 124C.4 The different magnetic contributions . . . . . . . . . . . . . . 128

Bibliography 129

Acknowledgements 133

Chapter 1

Introduction

The basic properties of thermal neutrons make them a highly suitable probeto measure static and dynamic properties of condensed matter:

• Due to their wavelength being of the same order as interatomic dis-tances in solids and liquids, interference effects occur in the scatteringprocess which yield information on the structure of the scattering sys-tem.

• Being uncharged particles, neutrons only interact with the scatteringsystem via nuclear forces. Therefore there is no Coulomb barrier toovercome. Neutrons can deeply penetrate in the sample. In differenceto X-ray scattering the scattering lengths are no monotonic function ofthe atomic number. They vary strongly for neighbouring cores in thetable of elements.

• Energies of thermal neutrons and of elementary excitations in con-densed matter are of the same order. By analyzing neutron energies inthe scattering process, sample dynamics can be studied easily.

• Due to its magnetic moment the neutron interacts with the magneticfields generated by the electrons of a magnetic sample. On account ofthis the neutron scattering cross-section does not only include nuclearcontributions but also magnetic ones.

By also analyzing the spins of the scattered neutrons, usually regarded interms of neutron polarization, it is possible to gain additional informationabout the scattering sample. In 1963, Blume [Blu63] derived the changeof the polarization vector upon elastic scattering. This clearly showed thatnuclear and magnetic contributions to the scattering cross-section, which are

5

6 CHAPTER 1: INTRODUCTION

superposed in an unpolarized neutron scattering experiment, can be disen-tangled at a single point in (Q, ω)-space by means of polarization analysis.In general, a method which measures the diagonal elements of the polariza-tion tensor only, as realized first by Moon et al [MRK69] in 1969, is usedin polarization dependent measurements. Central in this experimental ar-rangement is the existence of a magnetic guide-field all the way throughthe instrument, from the polarizing monochromator to the analyzer. Thisguide-field is used to adiabatically conserve the projection of the initial neu-tron polarization to the guide field in smooth rotation towards a particulardirection at the sample position. The neutron spin is conserved or flippedthere with regard to the quantization axis, given by the guide field due to themicroscopic scattering process. The guide field conserves only the projectionof of final polarization to the guide field towards the analyzer. Thus onlythe projections of the final polarization on the incident polarization, corre-sponding to the diagonal terms of the polarization tensor, can be measuredin this kind of setup. As the measured final polarization is therefore longi-tudinal on the incident this is also called ’longitudinal polarization analysis’(LPA). Because the neutron polarization transforms as a vector during thescattering process this means a loss of information, namely all off-diagonalterms of the polarization tensor.In 1989, Tasset [Tas89] presented an experimental setup named ’CRYOPAD’which allowed to measure all elements of the polarization tensor in diffractionexperiments. Later on it was adapted for inelastic measurements on a Three-Axis-Spectrometer. Similar setups have already been realized by Rekveldt([RS79], [STRG79]) and Okorokov([ORVG75]) in the seventies for SANS andreflectometry techniques. It is based on a zero field chamber, realized througha double superconducting Meissner-shield, in which neutrons enter througha non-adiabatic field-transition which conserves the polarization. Inside thezero-field region the neutron spin is not precessing. The incident and finalpolarization-vector can be turned in any arbitrary direction using a total offour coils outside and between the Meissner-shields. This method is knownas ’spherical neutron polarimetry’ (SNP) because the polarization vector isturned in terms of spherical coordinates.The technique of spherical neutron polarimetry (SNP) developed since 1989at the Institut Laue Langevin (ILL), Grenoble, has proven to be a powerfultool to solve magnetic structures in elastic scattering, which where intractablebefore. For inelastic scattering one big advantage over conventional analysisconsists in the zero-field sample environment which is important when type-IIsuperconductors should be studied in the superconductive phase. Further-more, nuclear-magnetic interference terms which should exist in principle,could be measured with SNP but not with conventional polarization analy-

7

sis.The main task of this diploma work was to built and test the new non-cryogenic polarization analysis device ’MuPAD’ (Mu-Metal PolarizationAnalysis Device) - an alternative setup to the existing ’CRYOPAD’ - fora Three-Axis-Spectrometer (TAS). ’MuPAD’ relies on two compact preces-sion coils with well defined field geometry and magnetic screens up and downstream of the sample. A split mu-metal shield in the sample area with cou-pling coils for in- and outgoing beam guides the neutron polarization throughthe instrument. All is based on existing components from NRSE-instruments(Neutron Resonance Spin Echo) where the scattering sample is in zero fieldas well ([GG87],[Kli03]). The low cost of ’MUPAD’ and its ease of handlingare attractive.The theory of polarization analysis is introduced and experimental tech-niques for polarization analysis on a TAS are illustrated. The principle ofthe experimental approach ’MuPAD’ is discussed and its construction dur-ing this work is described in detail. Field measurements and calculationsof the central MuPAD precession coils are explained and analyzed. Finallythe successful test measurements performed with MuPAD on a MnSi sam-ple on the thermal TAS IN22 at the ILL are presented. For the first timethe chiral term of a magnetic sample was measured on off-diagonal terms ofthe polarization tensor. The performance of MuPAD will be analyzed andpropositions for further improvement are given.

8 CHAPTER 1: INTRODUCTION

Figure 1.1: The mascot of the MuPAD project. Designed by Oliver Janoschek ([Jan].)

Chapter 2

Theory of Polarization Analysis

2.1 Neutron scattering cross section

Figure 2.1: The geometry of a neutron scattering experiment; picture taken from [Squ78]

In a conventional neutron scattering experiment a beam, with unpolarizedneutrons of a known energy, is aimed on the sample which is to be examined.An unpolarized beam has equal probabilities for the neutron spins being inthe parallel or antiparallel state with respect to any chosen quantization axis.The angular distribution and the energy of neutrons scattered by the sampleare analyzed to reveal information about the sample. In Fig.2.1 the geometryof such an experiment is shown. The measured quantity in such an experi-

9

10 CHAPTER 2: THEORY OF POLARIZATION ANALYSIS

ment is the partial differential cross-section denoted by ([Lov84],[Squ78])

d2σ

dΩdE ′ =

number of neutrons per second with a certain final

energy between E′ and E′ + dE′ that are scattered by the

sample into a small solid angle dΩ in the direction (θ, ϕ)

/ΦdΩdE ′

=k′

k

∑

λ,σ

pλpσ

∑

λ′,σ′

〈λ, σ|V †~Q|λ′, σ′〉〈λ′, σ′|V ~Q|λ, σ〉δ(~ω + Eλ − Eλ′),

(2.1)

where Φ[area−1time−1] is the flux of the incident neutrons, therefore thenumber of neutrons which pass a cross-section of the incident beam per sec-ond divided by the size of this cross-section. k and k′ are the magnitudes ofthe initial and scattered neutron wave vector respectively. ~ ~Q = ~~k − ~~k′

and ~ω are the momentum transfer and the energy transfer in the scatteringprocess. ~Q is called the scattering vector. The states of the scattering sys-tem and neutron spin before the scattering are given through the quantumnumbers λ and σ; the neutron energy is given by Eλ. The primed quantitiesdescribe the same quantities after scattering. pλ and pσ are the probabilitiesfor the scattering system and the neutron spin for being in a specific stateλ and σ. Because the beam is unpolarized the probabilities for the up anddown states of the neutrons spins are equal. Therefore pσ = 1

2for all σ. Av-

eraging over λ and σ takes into account all possible initial states before thescattering process, while summing over λ′ and σ′ considers all possible finalstates after the scattering process. The δ-function assures the conservationof energy during the scattering process. V ~Q is the Fourier transformation ofthe interaction potential between the neutron and the sample multiplied by(m/2π~

2); it has the dimension of a length. All the properties of the sam-ple accessible through a neutron scattering experiment are encoded in V ~Q.The partial differential cross-section is a measure for the probability that aneutron with wave vector ~k will be scattered into the direction (θ, ϕ) due tothe interaction with the sample, represented by V ~Q, undergoing a momentum

transfer ~ ~Q and a energy transfer ~ω) and then being detected by an detectorthat covers a solid angle dΩ and counts neutrons with energies between E ′

and E ′ + dE ′.The scattering potential is basically composed out of two parts1: the neu-trons are scattered by strong interaction with the nuclei and by magneticinteraction with the magnetic moments of electrons in the sample:

V ~Q = N ~Q + r0~σ~⊥ ~Q (2.2)

1Scattering processes due to the electric field produced by the nuclei and atomic elec-trons in a solid won’t be considered in this work.

2.2 MAGNETIC SCATTERING 11

The nuclear part is given through N ~Q =∑

i bi exp(i ~Q~ri), where bi is thescattering length and ~ri the position of the nucleus i in the sample. It containsinformation about the structure of the scattering sample. The magnetic partis composed out of the magnetic scattering length r0 and ~⊥ ~Q, generallycalled the magnetic interaction vector. The magnetic interaction vector holdsinformation about magnetic properties of the sample. The Pauli-matricesdenoted by

σx =

(0 11 0

)

, σy =

(0 −ii 0

)

, σz =

(1 00 −1

)

(2.3)

describe the spin state of the neutron in the magnetic interaction.

2.2 Magnetic scattering

2.2.1 The magnetic interaction potential

As polarization analysis is mainly done to reveal magnetic structures anddynamics of samples, the magnetic interaction which is dependent on theseproperties of the scattering system will be examined. The magnetic interac-tion of the neutron with the sample is the interplay between the magneticdipole moment of the neutron carried by its spin and the magnetic fieldsgenerated by the electrons of the sample. The operator corresponding to themagnetic dipole moment of the neutron is

~µn = −γµn~σ, (2.4)

where µn =e~

2mp

(2.5)

is the nuclear magneton. mp is the mass of the proton and e its charge. γ

is a positive constant whose value is γ = 1.913. ~σ are the Pauli-matrices ofEq.(2.3). The interaction of the magnetic dipole moment of the neutron witha magnetic field is

Vm = −~µn~B. (2.6)

The magnetic field generated by the sample is composed out of two parts.Imagine an electron with spin ~s = 1

2~σ and with momentum ~p:

• Due to the magnetic dipole moment of the electron

~µe = −2µB~s (2.7)


the magnetic field produced at a point ~R from the electron is

~Bs = curl ~A, ~A =µ0

4π

~µe × ~R

R3. (2.8)

Here µB = e~

2meis the Bohr magneton and me is the mass of the electron.

• The electron representing a moving charge of magnitude e also gener-ates the magnetic field

~BL = −µ0

4π

2µB

~

~p × ~R

R3. (2.9)

at the point ~R. Because ~L = ~R × ~p is the angular momentum of theelectron this field is denoted by ~BL.

The geometric relationship between ~R and the positions of the neutron ~r andof the electron ~ri is explained in Fig.2.2.In total the magnetic interaction potential is then

Vm = −µ0

4πγµN2µB~σ( ~Ws + ~WL), (2.10)

where ~Ws = curl(~s × ~R

R3), (2.11)

and ~WL =1

~

~p × ~R

R3. (2.12)

2.2.2 The magnetic interaction vector

The quantity really included in the partial differential cross-section (2.1) isthe Fourier transformation of the interaction potential multiplied by (m/2π~

2).For magnetic scattering it is given by2

(m/2π~2)

∫

Vm exp(i ~Q~r)d3r = r0~σ~⊥ ~Q, (2.13)

where ~⊥ ~Q =∑

i

exp(i ~Q~r)

~Q × (~si × ~Q) +i

~Q(~pi × ~Q)

(2.14)

is the magnetic interaction vector. r0 = 5.391 · 10−15m is a collection of allthe multiplying factors in Eqns. (2.10) and (2.13). It can be referred to as a

magnetic scattering length. ~Q is a unit vector in direction of ~Q.

2This results are taken out of [Lov84]. They include some lengthy algebra.

2.2 MAGNETIC SCATTERING 13

Figure 2.2: Magnetic Scattering: The neutron at position ~r is scattered by the magnetic field in thesample generated by the spin ~s and momentum ~p of electrons at ~ri

2.2.3 Magnetisation of the sample

A relation between the magnetisation operator ~M(~r) of the sample and themagnetic interaction vector can be derived ([Squ78]):

~⊥ ~Q = ~Q × (~ ~Q × ~Q) (2.15)

~ ~Q = − 1

2µB

~M( ~Q) (2.16)

= − 1

2µB

∫

~M(~r) exp(i ~Q~r)d3r (2.17)

~M(~r) describes the local magnetisation of the sample at the point ~r. Themagnetisation operator and therefore also the magnetic interaction vectorcontain vectorial information about the three dimensional distribution of themagnetic moments in a magnetic sample.

2.2.4 The geometric selection rule

From Eq.(2.15) a useful geometric selection rule can be easily derived. Theevaluation of this expression proofs that only components of ~ ~Q perpendicular


(a) The geometric selection rule: Only com-

ponents of ~~Qperpendicular to ~Q contribute

the scattering process. Further on these com-

ponents are denoted as ~⊥~Q

.

(b) The Basic frame of reference for polariza-

tion analysis: ~ki and ~kf are the initial and final

wavevectors of the neutrons; the coordinates are

defined with respect to ~Q.

Figure 2.3: Geometric details in a magnetic scattering process.

to ~Q contribute to the scattering process as shown in Fig. 2.3(a). Becauseof that the frame of reference in polarization analysis experiments is usuallychosen with respect to ~Q:

x ‖ ~Q

y ⊥ ~Q in the scattering plane (2.18)

z ⊥ ~Q out of the scattering plane

The coordinate frame is defined in Fig. 2.3(b). In that frame of reference~⊥ ~Q has only non-zero components along the y- and z-axis. In this work itwill be further denoted as analysis frame.

2.3 Polarization of a neutron beam

In the last section we learned that magnetic scattering of neutrons is dueto the interaction of the neutron spin and the magnetic fields of the in thesample. The neutron spins in a neutron beam are generally described bythe concept of neutron polarization. Therefore we will develop a descriptivepicture of neutron polarization before the theory of neutron polarizationanalysis is introduced. As a neutron spin represents a spin-1

2-system, we

will investigate the properties of such a system. The most general quantum-

2.3 POLARIZATION OF A NEUTRON BEAM 15

mechanical description of such a system is given by

χ = aχ↑ + bχ↓ = a

(10

)

+ b

(01

)

(2.19)

which describes the superposition of eigenstates being parallel or antiparallelto the chosen quantization axis in a two-dimensional Hilbert space H2. Fromnow on they will be call the up and down states respectively. |a|2 and |b|2 arethe probabilities of the system being in the up or the down state. Becausethe probability that the system is in any of these state is 1 the normalizationcondition

χ†χ = |a|2 + |b|2 = 1 (2.20)

has to be fulfilled. This can be generally achieved by the following choice fora and b:

a = cosθ

2ei ϕ

2 (2.21)

b = sinθ

2ei ϕ

2 (2.22)

(2.23)

Then Eq.(2.20) is true for any choice of θ and ϕ. With that choice we willbe able to understand Eq.(2.19) in a more descriptive way.

χ = cosθ

2ei ϕ

2

(10

)

+ sinθ

2ei ϕ

2

(01

)

≡ |θ, ϕ〉 (2.24)

|θ, ϕ〉 is an eigenvector to the operator ~n~σ; where ~σ are again the Pauli-

matrices (s.Eq.2.3) and ~n is a unit vector given through

~n =

sin θ cos ϕsin θ sin ϕ

cos θ

. (2.25)

It is pointing towards a certain point of the surface of the unit sphere S2

fixed by the two angles θ and ϕ. By solving the eigenvalue equation for thisoperator, namely

(~n~σ)χ = λχ (2.26)

⇔(

cos θ − λ sin θe−iϕ

sin θeiϕ −(cos θ + λ)

)(uv

)

= 0, (2.27)


the eigenvalues λ = ±1 with the eigenstates

λ = +1 : χ+ =

(cos θ

2ei ϕ

2

sin θ2ei ϕ

2

)

(2.28)

λ = −1 : χ− =

(sin θ

2ei ϕ

2

cos θ2ei ϕ

2

)

(2.29)

are gained. ~n~σ projects the components of the spin of the considered systemonto the unit vector ~n. Thus, the inserted eigenvalues show that for the twocalculated eigenstates (Eqns.(2.28) and (2.29)) the spin is fully aligned in

direction of ~n:

(~n~σ)χ = ±χ (2.30)

As the eigenstate in Eq.(2.28) is perfectly the same as that in Eq.(2.24),we conclude that the parameter set (θ, ϕ), which originally describes thesuperposition of up and down states to a given general state in the madechoice of eigenfunctions, also denotes the three dimensional orientation ofthe spin of that state in the corresponding frame of reference. Hence, thespin is a vectorial quantity.This geometrical interpretation of a general spin-1

2-state is called the Bloch-

representation of a state. The Bloch-representation is shown in Fig.2.4. Thevector ~n is denoted as the Bloch-vector and the unit sphere S2 as the Bloch-sphere. This more descriptive picture of a spin is usually used in quantumcomputation for qubits represented by a spin-1

2-system.([NC02])

We have to take into account that the spin is a vectorial quantity in theconcept of polarization. Thus, the polarization is defined as a unit vectorpointing in the direction of the neutron spin, given by the expectation valueof the Pauli-matrices ([Lov84]):

~P ≡ < ~σ >= χ†~σχ = Tr(ˆ~σ), (2.31)

where ˆ = χχ† =

(|a|2 ab∗

ba∗ |b|2)

(2.32)

is the density matrix operator which defines the probability of a certain spinstate ([Fan57]). The polarization vector ~P of the spin state defined as asuperposition of up and down states in Eq.(2.19) is then

~P =

2ℜ(a∗b)2ℑ(a∗b)|a|2 − |b|2

=

sin θ cos ϕsin θ sin ϕ

cos θ

= ~n (2.33)

2.3 POLARIZATION OF A NEUTRON BEAM 17

in agreement with the above developed picture of such a state.The polarization of a neutron beam is then

~P =1

N

∑

i

~Pi =<< ~σ >>beam, (2.34)

where N is the number of neutrons in the beam and ~Pi is their correspondingpolarization. Therefore the polarization is defined as a property of a neutronbeam. This statistical quantity will be measured by averaging over all neu-trons in the beam. For an unpolarized beam ~P0 is then zero; for a completelypolarized beam | ~P0| = 1; and for a partially polarized beam 0 < | ~P0| < 1.

Figure 2.4: Bloch-sphere S2 with Bloch-vector ~n


2.4 Polarization in neutron scattering exper-

iments

In conventional neutron scattering experiments only direction and energy ofthe scattered neutrons are analyzed. More information about the sample isgained by additionally analyzing the spins of the neutrons. The additionalinformation serves to separate the nuclear and the magnetic contributionsto the scattering process, which superpose in unpolarized neutron scatter-ing experiments (s. Eqns. (2.1) and (2.2)). The polarization vector of theneutron beam incident on the sample is aligned along a known direction be-fore the scattering process. The change of polarization during the scatteringprocess is analyzed afterwards. Hence, this technique is generally known aspolarization analysis.

2.4.1 Polarized neutron scattering cross-section

To take into account the polarization of the neutron beam incident on thesample the expression for the partial differential cross-section in Eq.(2.1)has to be modified. Only the part of the cross-section that depends on theneutron spin is regarded:

∑

σ,σ′

pσ〈σ|V †~Q|σ′〉〈σ′|V ~Q|σ〉. (2.35)

The density matrix ˆ in Eq.(2.32) can be represented through

ˆ =∑

σ

pσ|σ〉〈σ|, (2.36)

if it is diagonal. With that Eq.(2.35) can be rewritten as

∑

σ,σ′

pσ〈σ′|V †~Q|σ〉〈σ|V ~Q|σ′〉 =

∑

σ′

〈σ′|V †~QV ~Q ˆ|σ′〉 = Tr(V †

~QV ~Q ˆ) = Tr(ˆV †

~QV ~Q)

(2.37)Then the partial differential cross-section changes to

d2σ

dΩdE ′ =k′

k

∑

λ,λ′

pλTr ˆ〈λ|V †~Q|λ′〉〈λ′|V ~Q|λ〉δ(~ω + Eλ − Eλ′), (2.38)

where the trace is only to be taken with respect to neutron spin coordinates.

2.4 POLARIZATION IN NEUTRON SCATTERING EXPERIMENTS 19

If the trace is evaluated (s.[Blu63]) the partial differential cross-sectionfor polarized neutrons is gained:

d2σ

dΩdE ′ =k′

k

∑

λ

pλ

〈λ|N ~QN †~Q|λ〉 + r2

0〈λ|~⊥ ~Q~†⊥ ~Q|λ〉 +

+ r0~P0

[

〈λ|N †~Q~⊥ ~Q|λ〉 + 〈λ|~†⊥ ~Q

N ~Q|λ〉]

−

− ir20~P0〈λ|~⊥ ~Q × ~†⊥ ~Q

|λ〉

δ(~ω + Eλ − Eλ′), (2.39)

where ~P0 is the polarization of the incident beam. The closure relation∑

λ′ |λ′〉〈λ′| = 1 was used to simplify the expression. Eq.(2.39) consists outof four different terms. These are

• the nuclear contribution N ~QN †~Q

which contains the pure nuclear scat-

tering,

• the magnetic contribution ~⊥ ~Q~†⊥ ~Qwhich contains the pure magnetic

scattering,

• the nuclear-magnetic interference term N ~Q~†⊥ ~Q+ ~⊥ ~QN †

~Qwhich only

arises if there is interference between the nuclear and magnetic scatter-ing, e.g. when a magnon is modulated by a passing phonon,

• the chiral term i(

~⊥ ~Q × ~†⊥ ~Q

)

which only arises if there is a chiral

ordering of the magnetic moments in the scattering system.

Eq.(2.39) shows that the partial differential cross-section will be different for

polarized and unpolarized neutrons. If ~P0 = 0 the nuclear-magnetic interfer-ence and the chiral term will not be observed. In that case the cross-sectiondepends only on the square of the nuclear structure and the magnetic scatter-ing vector. Measuring the partial differential cross-section only with unpo-larized neutrons clearly results in a loss of directional and phase information.Therefore it is very hard or sometimes even impossible to disentangle a mag-netic structure by only measuring that quantity. Fortunately this problemcan be solved by also analyzing the final polarization after the scatteringprocess which reveals additional information about the sample.In a last step we formulate Eq.(2.39) in terms of correlation-functions:


d2σ

dΩdE ′ =k′

k

1

2π~

∫

dt exp(−iωt)

〈N ~QN †~Q(t)〉 + r2

0〈~⊥ ~Q~†⊥ ~Q(t)〉 +

+ r0~P0

[

〈N †~Q~⊥ ~Q(t)〉 + 〈~†⊥ ~Q

N ~Q(t)〉]

−

− ir20~P0〈~⊥ ~Q × ~†⊥ ~Q

(t)〉

, (2.40)

This is a more convenient representation in the case of studying sample dy-namics and thus inelastic scattering ([Mal99]). The correlation-functions aredefined by

〈A ~QA†~Q(t)〉

=∑

λ

pλ〈λ|A ~QA†~Q(t)|λ〉

=∑

λ,λ′

pλ〈λ|A ~Q|λ′〉〈λ′| exp(iHt/~)A†~Q

exp(−iHt/~)|λ〉, (2.41)

where H is the Hamiltonian of the scattering system andA†

~Q(t) = exp(iHt/~)A†

~Qexp(−iHt/~) is the operator A†

~Qin the Heisenberg

picture. Using the fact that the states λ of the scattering system are eigen-functions of H with the eigenvalues Eλ

H|λ〉 = Eλ|λ〉 (2.42)

the relationship to the original form of the cross-section can be seen:

1

2π~

∫

dt exp(−iωt)〈A ~QA†~Q(t)〉

=1

2π~

∑

λ,λ′

pλ

∫

dt exp(−iωt)〈λ|A ~Q|λ′〉〈λ′| exp(iHt/~)A†~Q

exp(−iHt/~)|λ〉

=1

2π~

∑

λ,λ′

pλ

∫

dt expi(Eλ′ − Eλ)t/~ exp(−iωt)〈λ|A ~Q|λ′〉〈λ′|A†~Q|λ〉

=∑

λ,λ′

pλ〈λ|A ~Q|λ′〉〈λ′|A†~Q|λ〉δ(~ω + Eλ − Eλ′). (2.43)


In the last step the integral representation for the δ-function for conservationof energy was used:

1

2π~

∫

dt expi(Eλ′ − Eλ)t/~ exp(−iωt) = δ(~ω + Eλ − Eλ′). (2.44)

Correlation-functions are very useful representations of sample properties (s.for example [Sch97], chapter 4 or [Squ78], chapter 4).

2.4.2 Polarization of the scattered beam

If a beam of polarized neutrons is scattered at the sample, the incident po-larization ~P0 = 〈~σ〉 is transformed due to the interaction with the sample

represented by V ~Q. Thus 〈~σ〉 is transformed by V ~Q like

~P ′ ∝ Tr(ˆV †~Q~σV ~Q), (2.45)

where the trace over the density matrix ˆ assures the averaging over allpossible initial spin states of the beam and the summing over all possiblefinal states. The constant of proportionality is determined by normalization:

~P ′ = Tr(ˆV †~Q~σV ~Q)/Tr(ˆV †

~QV ~Q). (2.46)

The full expression is then

~P ′ d2σ

dΩdE ′ =k′

k

∑

λ,λ′

pλTr ˆ〈λ|V †~Q|λ′〉~σ〈λ′|V ~Q|λ〉δ(~ω + Eλ − Eλ′). (2.47)

The evaluation of the trace over the neutron spin states in Eq.(2.47)([Blu63])leads to the expression for the final polarization vector after the scatteringprocess:

~P ′ d2σ

dΩdE ′ =k′

k

∑

λ

pλ

~P0〈λ|N ~QN †~Q|λ〉 − r2

0~P0〈λ|~⊥ ~Q~†⊥ ~Q

|λ〉 +

+ r20〈λ|( ~P0~

†⊥ ~Q

)~⊥ ~Q|λ〉 + r20〈λ|~†⊥ ~Q

( ~P0~⊥ ~Q)|λ〉 +

+ r0

(

〈λ|N †~Q~⊥ ~Q|λ〉 + 〈λ|~†⊥ ~Q

N ~Q|λ〉)

+

+ ir0~P0 ×

(

〈λ|~†⊥ ~QN ~Q|λ〉 − 〈λ|N †

~Q~⊥ ~Q|λ〉

)

+

+ ir20〈λ|~⊥ ~Q × ~†⊥ ~Q

|λ〉

δ(~ω + Eλ − Eλ′). (2.48)


Again the several parts are expressed as correlation functions for the case ofinelastic scattering:

~P ′ d2σ

dΩdE ′ =k′

k

1

2π~

∫

dt exp(−iωt)

~P0〈N ~QN †~Q(t)〉 − r2

0~P0〈~⊥ ~Q~†⊥ ~Q

(t)〉 +

+ r20〈( ~P0~

†⊥ ~Q

(t))~⊥ ~Q〉 + r20〈~†⊥ ~Q

(t)( ~P0~⊥ ~Q)〉 +

+ r0

(

〈N †~Q~⊥ ~Q(t)〉 + 〈~†⊥ ~Q

N ~Q(t)〉)

+

+ ir0~P0 ×

(

〈~†⊥ ~QN ~Q(t)〉 − 〈N †

~Q~⊥ ~Q(t)〉

)

+

+ ir20〈~⊥ ~Q × ~†⊥ ~Q

(t)〉

. (2.49)

Table 2.1: Terms contained in the polarization tensor P ; the upper indices corre-spond to the three directions x, y, z in space.

σN 〈N ~QN †~Q(t)〉 the nuclear contribution

My/z r20〈

y/z

⊥ ~Q†y/z

⊥ ~Q(t)〉 the y- and z-components of

the magnetic contribution.

The x-component is missing

because of the geometric se-

lection rule!

Ry/z r0〈N †~Q

y/z

⊥ ~Q(t)〉 + 〈†y/z

⊥ ~QN ~Q(t)〉 real parts of the nuclear-

magnetic interference term.

Iy/z r0〈N †~Q

y/z

⊥ ~Q(t)〉 − 〈†y/z

⊥ ~QN ~Q(t)〉 imaginary parts of the

nuclear-magnetic interference

term.

Tchiral ir20(〈y

⊥ ~Q†z⊥ ~Q

(t)〉 − 〈z⊥ ~Q

†y⊥ ~Q

(t)〉) chiral contribution

Mmix r20(〈y

⊥ ~Q†z⊥ ~Q

(t)〉 + 〈z⊥ ~Q

†y⊥ ~Q

(t)〉) mixed magnetic contribution

or magnetic-magnetic inter-

ference term, no obvious phys-

ical meaning


2.4.3 The polarization tensor

Before looking at the different terms of the final polarization vector it is moreconvenient to formulate Eq.(2.48) as a tensor equation

~P ′ = P ~P0 + ~P

=

σN−My−Mz

σxiIz

σy − iIy

σz

− iIz

σxσN+My−Mz

σyMmix

σz

iIy

σxMmix

σyσN−My+Mz

σz

P x0

P y0

P z0

+

Tchiral

σx/y/z

Ry

σx/y/z

Rz

σx/y/z

,

(2.50)

where σi is the partial differential cross-section in Eq.(2.39) with the initialpolarization pointing in direction i, where i can be x, y and z. The tensorP is called the polarization tensor. The coordinate frame in which thisexpression is valid, is defined in (2.18). The final polarization vector can beseparated into two parts:

• P ~P0 is dependent upon the initial polarization vector ~P0. The termscontained in this part rotate the initial polarization.

• ~P is completely independent of ~P0. The terms represented by this partproduce polarization even from an unpolarized beams. Samples whichhave properties that result in such terms are actually used to polarizean unpolarized neutron beam.

All terms in Eq.(2.50) are of the form k′

k1

2π~

∫dt exp(−iωt)〈A ~QA†

~Q(t)〉. The

corresponding correlation-function for each term is given in Table (2.1). Animportant thing to recognize is that in all terms containing magnetic contri-bution there’s no x-component. This is because of the geometric selectionrule.The cross-section (2.39) can also be expressed in terms of the given notation:

d2σ

dΩdE ′ = σN + My + M z − P x0 Tchiral + P y

0 Ry + P z0 Rz (2.51)

The information collected by measuring all terms of the polarization ten-sor P in a point in (~Q, ω)-space can be used to separate all the differentcontributions that usually superpose in the cross-section. This can be doneover the whole (~Q, ω)-space accessible by neutron scattering. Out of these

data the microscopic magnetisation density ~M(~r, t) can be reconstructed.Therefore neutron polarization analysis is a powerful tool to characterize themicroscopic magnetic properties of a sample.

Chapter 3

Polarization Analysis on aThree-Axis-Spectrometer

Various types of neutron scattering instruments use polarization analysis toobtain information upon the magnetic properties of matter. As MuPAD wasespecially designed to be installed on a Three-Axis-Spectrometer (TAS) forinelastic measurements, we only deal with polarization analysis on this typeof instrument.

3.1 Three-Axis-Spectrometer (TAS)

The principle of Three-Axis-Spectroscopy consists in directing neutrons witha certain incident wavevector ~ki on the sample and analyzing the scatteredneutrons in dependency of their wavevector ~kf . Here we measure the partialdifferential cross-section (2.1) for a specific momentum- and energy-transfer

on the sample at a specific point in (~Q, ω)-space. This follows from therelationships of momentum and energy conservation during the scatteringprocess:

~ ~Q = ~~ki − ~~kf (3.1)

and ~ω =~

2

2mN

(~k2i − ~k2

f ), (3.2)

where mN is the mass of the neutron. The neutrons coming from the sourcehave a broad wavevector distribution. Thus, the incident wave vector ~ki is

25

26 CHAPTER 3: POLARIZATION ANALYSIS ON A

THREE-AXIS-SPECTROMETER

selected by first order Bragg reflection from a known single crystal monochro-mator:

~ki sin θM =1

2τM

ki =2π

λi

(3.3)

τM =2π

dM

,

where θM is the Bragg angle at the monochromator, dM and τM are the lat-tice spacing and the reciprocal lattice vector of the monochromator crystal,and λi is the wavelength of the neutron. As the beam coming from the sourceis divergent only the momentum transfer ~~τ is defined exactly. Though it ispossible to satisfy Eq.(3.3) for different neighboring ~ki and θM . The use ofcrystals with mosaicity1 as monochromator even relaxes the exact definitionof ~τ and allows much higher neutron fluxes at the sample position. This isdemonstrated in Fig. 3.1. The direction and therefore also the magnitudeof ~ki is often defined exact enough just by the alignment of monochromatorand sample. By use of collimators or diaphragms the divergence of the beamcan be reduced.The neutrons with well-defined wavevector ~ki are scattered at the sample.There they undergo the momentum transfer ~ ~Q and energy transfer ~ω de-fined in Eqns.(3.1) and (3.2). To select a specific final neutron wavevector~kf a second single crystal is used as analyzer:

Figure 3.1: Bragg reflection at the monochromator crystal. Due to the finite beam divergence α0 and

the crystal mosaic spread ηM not only a single wavevector ~ki is selected by the crystal. Instead a volumein reciprocal (denoted by blue shaded surface) is selected around a mean wavevector ~kI which allows muchhigher neutron flux upon the sample.

1A crystal consisting of several small single crystals whose crystal planes are slightlytilted against each other.

3.1 THREE-AXIS-SPECTROMETER (TAS) 27

~kf sin θA =1

2τA

kf =2π

λf

(3.4)

τA =2π

dA

,

where, analog to the monochromator crystal, θA is the Bragg angle at theanalyzer, dA and τA are the lattice spacing and the reciprocal lattice vectorof the analyzer crystal, and λf is the wavelength of the scattered neutron. Tosuppress contamination of the beam through higher order Bragg reflection afilter (e.g. graphite) is used which scatters out a certain bandwidth of wavevectors ([Far00]).After the analyzer a neutron counter detects all neutrons which have the in-cident wavevector ~ki and the final wavevector ~kf after the scattering process.Therefore the intensity for this specific choice corresponding to a point in( ~Q, ω)-space is recorded2.. Hereby different combinations of ~ki and ~kf can

select the same combination of ~Q and ω. The intensity-distribution recordedby varying ~ki and ~kf reveals the interaction potential V ~Q between the sampleand the neutrons. As with a TAS instrument momentum and energy transfercan be varied systematically, it is especially suitable to measure the disper-sion relation of excitations in the sample.The whole principle is shown in Fig.3.2. The setup is called Three-Axis-Spectrometer because of its three axes Monochromator-Sample-Analyzer.

Figure 3.2: The principle of a Three-Axis-Spectrometer.

2Due to the beam divergence and the mosaicity of the monochromator and analyzercrystals the intensity is not only recorded for a point in the 4D (~Q, ω)-space but for a 4Dvolume, called the resolution ellipsoid, by a TAS instrument. (s. [Dor82] and [CN67])



3.2 Equipping a TAS for Polarization Analy-

sis

For polarization analysis the following modifications of a TAS-instrument arerequired.

3.2.1 Polarizer

Because the neutrons produced in a fission reaction do not have a preferreddirection for their spins, a general neutron beam is unpolarized. Hence, theneutron beam has to be polarized first before any polarization measurementscan be performed. There are basically two methods which are commonlyused to polarize the beam in the case of a TAS. In future, a third method,He-3 polarizers will certainly find broad application in TAS (s. for example[TR95]).

Supermirror Polarizer

The angle of total reflection for neutrons from a magnetized ferromagneticthin film is given by

θ± = λ√

N(b ± p)/π, (3.5)

where λ is the neutron wavelength, N is the nuclear density and b and p arethe nuclear and magnetic scattering length, respectively. The + and − casedescribe the reflection angle for neutrons having spin antiparallel or parallelto the direction of magnetization of the film. For the ideal case b = p allreflected neutrons are polarized.Unfortunately the reflection angles are very small for thermal and cold neu-trons and dependent on the wavelength of the neutron (e.g. θ ≈ 0.4 forλ ≈ 4A and Fe50Co48V2 as magnetized film). This problem is usually solvedby producing films out of multiple magnetic and non-magnetic layers, calledsupermirrors. Typical combinations of materials are

• Fe/Si

• Co/Si

• Fe50Co48V2/T iNix

3.2 EQUIPPING A TAS FOR POLARIZATION ANALYSIS 29

Often multilayers are used in devices called Benders. Multiple multilayer-wafers (up to a few hundred) are pressed in a curved shape, e.g. S- orC-form, in order to polarize the beam completely by multiple reflection. Aswafers high transparent substrate material (e.g. Si) is used, on which thesupermirror is deposited by sputtering techniques. Such a device is placeddownstream of the monochromator of the TAS.

Single Crystal Polarizer

The method uses the nuclear-magnetic interference term in Eqns.(2.39) and(2.48). It is non-zero for centrosymmetric ferromagnetic single crystals whichdo have the magnetic property

~⊥ ~Q = ~ †⊥ ~Q

. (3.6)

Such a crystal is then employed as a monochromator crystal. A magneticfield is applied in such a way that all its magnetic moments are saturated andaligned perpendicular to the scattering vector ~Q (remember the geometricselection rule in chapter 2.2.4) which is equal to ~τ in Eq.(3.3) for a Bragg

reflection. By setting ~P0 = 0 and inserting (3.6) in Eq.(2.39) the cross-sectionfor such a crystal is obtained:

d2σ

dΩdE ′ =k′

k

∑

λ

pλ

〈λ|N2~Q|λ〉 + r2

0〈λ|~2⊥ ~Q

|λ〉

δ(~ω + Eλ − Eλ′), (3.7)

as for the nuclear contribution N ~Q = N †~Q

is always true3. The final polariza-

tion in Eq.(2.48) is then:

~P ′ d2σ

dΩdE ′ =k′

k

∑

λ

pλ2〈λ|N ~Q~⊥ ~Q|λ〉δ(~ω + Eλ − Eλ′). (3.8)

In total the polarization of the beam after being scattered on the Bragg Peak(k = k′ and Eλ = Eλ′) is:

3N~Q=

∑

i bi exp(i ~Q~ri); the scattering length bi is only imaginary if the neutron isabsorbed and not scattered.



~P ′ =

∑

λ pλ2r0〈λ|N ~Q~⊥ ~Q|λ〉∑

λ pλ

〈λ|N2~Q|λ〉 + r2

0〈λ|~2⊥ ~Q

|λ〉 . (3.9)

Hence, the diffracted beam from such a single crystal is completely polarizedparallel to the magnetic moments of the crystal, if there is a Bragg reflectionwith the property r0|~⊥ ~Q| = |N ~Q|. For example if we apply the magneticfield on the crystal in direction z (in correspondence with the analysis framewe defined for magnetic scattering in (2.18)) the magnetic interaction vectorwill look like

~⊥ ~Q =

00

(3.10)

and |N ~Q| = r0. (3.11)

This leads to the final polarization

~P ′ =2r2

02

r20

2 + r20

2·

001

=

001

(3.12)

The geometric situation is sketched in Fig. 3.3. There are several crystalsshowing Bragg reflections with the property r0|~⊥ ~Q| = |N ~Q|, examples are

• the (111) reflection of the Heusler Cu2MnAl (d-spacing 3.43A),

• the (200) reflection of the alloy Co0.92Fe0.08 (d-spacing 1.76A).

3.2.2 Turning the Polarization Vector

For both types of polarizers, usually installed on TAS instruments, the di-rection of the created initial polarization is fixed. Eq.(2.49) shows that it isnecessary to turn the polarization vector in several different initial directions


Figure 3.3: Polarizing the beam at a single crystal.

to get the full information about the magnetic properties of a sample fromPolarization Analysis.Because its spin endows the neutron also with a magnetic moment, it willinteract with an applied magnetic field. The spin will precess around theaxis of the magnetic field. As the polarization vector of a neutron beam isa property of the whole ensemble of all neutron spins in the beam, it alsochanges under the influence of a magnetic field. There are two methods toturn the polarization with the help of a magnetic field.

Larmor precession

The change of the polarization vector under the influence of a magnetic fieldis quantum mechanically calculated in the Heisenberg representation wherethe time-dependency is expressed via operators.

OH = exp[i

~H(t − t0)]O exp[− i

~H(t − t0)], (3.13)

andd~σH(t)

dt=

i

~

[

HH , OH

]

+∂

∂tOH , (3.14)



where H is the Hamiltonian of the considered system and O any valid opera-tor in the Schrodinger representation. The Hamiltonian H = −~µn

~B describesthe interaction between the magnetic moment of the neutron ~µn (s. Eq.(2.4))

and a time independent homogenous magnetic field ~B. The change of thepolarization vector ~P = 〈~σ〉(s. Eq.(2.31)) with time is then

d~σH(t)

dt=

i

~

[

HH , ~σH

]

+∂

∂t~σH(t) (3.15)

= −γl(~σH(t) × ~B), (3.16)

where γl = 2π · 2916 radsGs

. By setting the magnetic field ~B =(

0 0 B)4

this differential equation is solved by

~P (t) = 〈~σ(t)〉 =

〈σx(0)〉 cos(γlBt) − 〈σy(0)〉 sin(γlBt)〈σy(0)〉 cos(γlBt) + 〈σx(0)〉 sin(γlBt)〈σz(0)〉

=

P⊥(0) cos(ωlt + α)P⊥(0) sin(ωlt + α)P‖(0)

, (3.17)

where 〈σi(0)〉 are the expectation values of the Pauli-matrices at time t = 0and P⊥(0) =

√

〈σx(0)〉2 + 〈σy(0)〉2 and P‖ = 〈σz(0)〉 are the componentsof the polarization vector perpendicular and parallel to the magnetic fieldaxis at time t = 0 respectively. α = arctan( 〈σx(0)〉

〈σy(0)〉) is the phase angle.

ωl = γlB is called the Larmor frequency. Eq.(3.17) describes the Larmorprecession of the polarization around the axis of the magnetic field. Thecomponent of the polarization parallel to the magnetic field is conserved,the components perpendicular to it precess around the field axis (s. Fig.3.4).Hence a homogenous magnetic field which is perpendicular to the polarizationvector of the beam is suitable to turn the polarization by a specific angle.The turning angle is determined by the magnitude of the field and the timethe neutrons need to pass the magnetic field:

ϕ = γlB[Gs]t[sec]

= 2π · 2916[rad

sGs]B[Gs]

l[m]

v[ msec

], (3.18)

4This already solves the general case, because the frame of reference can be alwaysturned to have the magnetic field along the z-axis.


where l is the length of the field and v is the velocity of the passing neu-trons. There are several devices which use this principle in order to turnpolarization. Some of them will be described in detail further down.

Figure 3.4: The Larmor precession of the Polarization Vector in a magnetic field.

Adiabatic field transitions

Eq.(3.17) shows that if the polarization vector of a polarized neutron beamis perfectly aligned with a magnetic field, its direction will be conserved.Therefore such a magnetic field is also called a guide field. A field whosedirection turns slowly with respect to the Larmor frequency ωl can be usedto guide the polarization vector into another direction. The polarizationvector will just precess around the slowly changing field direction. As thefield direction changes scarcely during one full precession of the polarizationthe polarization vector will adapt to the new direction of the field. This canbe demonstrated by examining the projection of the polarization vector onto

the vector of the magnetic field (~P~b) with ~b =~B

| ~B| . The time derivative of

(~P~b) is

d

dt(~P~b) = (~P

d~b

dt) + (~b

d~P

dt) ≈ 0 (3.19)

The first term can be neglected because of the very slow turning of the fieldwhereas the second is equal to zero in view of Eq.(3.17) as ~P is parallel to ~b.



The principle is shown in Fig.3.5. In this context two different types of fieldtransitions are distinguished:

• ωl = γlB >> ωB: (ωB = d( ~B/| ~B|)dt

is the frequency of the rotation ofthe magnetic field) The change of the field is so small with respect toits magnitude (which of course affects the precession frequency ωl) thatthe polarization vector is able to follow the field rotation. This is calledan adiabatic transition.

• ωl = γlB << ωB: The change of the field with respect to its magnitudeis so big that the polarization vector can’t follow the field rotation. Thepolarization vector is conserved along its original direction and beginsto precess around the new field direction after the transition. This iscalled a non-adiabatic transition.

The quality of an adiabatic transition is described by the adiabaticity pa-rameter

E =γl

ωB

. (3.20)

The bigger E is, the better the polarization vector is guided by the turningfield (s. also [Gut32],[Vla61],[Her98]). Some values are given in Table 3.1.

E 3 4 8 15P [%] 95 96 98 99.5

Table 3.1: Different sets of the adiabaticity parameter E and the corresponding conserved polarizationalong the new field direction after the adiabatic transition. Values are taken from [Her98].

Figure 3.5: An adiabatic field transition: the polarization vector precesses around the slowly turningmagnetic field.


3.2.3 Polarizing analyzer

After the scattering process at the sample, the final polarization can bemeasured with a similar device as for polarizing the beam.

Supermirror Analyzer

In section 3.2.1 it was shown that only neutrons with their spin being inthe state parallel to the magnetization axis of a magnetized thin film willbe reflected. Now imagine a beam of neutrons already having a polarization~P ′ 6= 0 before the reflection at the film. Because the different components ofthe Pauli-Operators do not commute, e.g.

[~σx, ~σy] = 2i~σz and cyclical, (3.21)

only the projection of the polarization vector on the quantization axis canbe measured. The magnetization of the film represents such a quantizationaxis for the neutrons spins: all the spins will collapse to states being parallel(up) or antiparallel (down) with respect to this axis. The distribution of thespins into these two states will fulfill the condition

P ′z =< ~σz >= |a|2 − |b|2, (3.22)

where the z-axis had been chosen in the direction of magnetization, and |a|2and |b|2 are the probabilities for the spins being in the up or down staterespectively (s. also Eq.(2.19)). A detector behind the supermirror countsthe reflected neutrons. In a dedicated time t = tcount it will detect only thosereflected neutrons, which are in the up state. If the direction of magnetizationin the film is reversed, only the down states will be reflected. Now countingfor the same time tcount only the neutrons which are in the down state aredetected. Therefore out of the intensities I+ and I− for the up and downchannel the probabilities for the neutrons being in up and down state arequickly calculated:

|a|2 =I+

I+ + I− (3.23)

|b|2 =I−

I+ + I− (3.24)



Eq.(3.22) then already gives the polarization of the beam along the axis ofmagnetization in the thin film.

P ′z =

I+ − I−

I+ + I− (3.25)

Thus, experimentally, polarization is defined as the projection of the po-larization vector on an axis defined by the applied quantization axis of themeasurement device.For practical reasons the magnetization direction of the supermirrors won’tbe changed to measure the polarization. Instead a device is installed up-stream of the supermirror which rotates the polarization vector by 180

around an axis perpendicular to the quantization axis of the mirror by usinga magnetic field (s. section 3.2.2).This has obviously the same effect. Sucha device is called a ’π-flipper’.

Single Crystal Analyzer

When using this type of polarizer as a polarization analyzer in principleeverything that is true for the supermirror type is also true. But here thedetermination of the polarization can be seen even more easily. Again thepolarization vector before the reflection of the beam on the Bragg peak ofthe single crystal is denoted as ~P ′. Also a π-flipper is installed in frontof the crystal. Remembering the properties for a single crystal polarizerwhich is magnetized along the z-direction (~⊥ ~Q = ~ †

⊥ ~Q, and |N ~Q| = r0 and

~⊥ ~Q =

00

, s. also section 3.2.1) the calculated elastic cross-section on

that Bragg peak looks like (s. Eq.(2.39))

dσ

dΩ

±= 2r2

02 ± 2r2

0~P ′2

001

= 2r20

2 ± 2r20P

′z

2, (3.26)

where dσdΩ

+and dσ

dΩ

−are the cross-sections for flipper on and off respectively.

Measuring those two quantities also gives us the projection of polarizationvector along the quantization axis:

3.3 SETUPS FOR POLARIZATION ANALYSIS 37

I+ − I−

I+ + I− ∝dσdΩ

+ − dσdΩ

−

dσdΩ

++ dσ

dΩ

−

=4r2

0P′z

2

4r20

2

= P ′z (3.27)

3.3 Setups for Polarization Analysis

A general setup for polarization analysis on a TAS would look like this:

1. The beam incident on the sample will be polarized with help of a po-larizer (s. section 3.2.1).

2. The incident polarization vector will be turned in any wanted directionby Larmor precession or adiabatic field transitions (s. section 3.2.2).

3. The neutrons are scattered at the sample, therefore the polarizationvector of the beam is transformed by the interaction with the sample(s. Eq.2.50).

4. The final polarization vector is analyzed by using a polarizing analyzer(s. section 3.2.3).

This actually assumes that the polarization vector of the beam is not changedbetween those four steps. But in fact the situation is different. The harshenvironment of an experimental setup like a TAS adds several magnetic fields:

• the magnetic earth field which is in the order of about 300mGs,

• magnetic fields of step motors, moving the several parts of the instru-ment, or the fields of some magnetic parts of the instrument, add fieldsin the range of several Gs,

• electromagnetic fields produced in different frequency ranges and withdifferent amplitudes produced by power electronics.

• magnetic fields produced by sample environments e.g. like supercon-ducting magnets, which are used to magnetize magnetic samples onother instruments in the surroundings with inner fields of several Tesla.

Any of these magnetic fields (s. section 3.2.2) will turn the polarizationvector. None of these field sources is known in such a way that their influence



could be considered in the experiments. The magnetic moment of a neutronis quite sensitive to these magnetic fields. Even small fields in the mGs-rangemay be enough to disturb the experiment. For example if a neutron beamwith wavelength of 1A corresponding to a velocity of 3956m

sis traveling 0, 5m

(in the range of typical distances between polarizer and sample) through ahomogenous field of 300mGs (magnitude of earth field!) the componentsof the polarization vector perpendicular to the direction of the field will bealready turned by (s. Eq.(3.18))

ϕ = 0, 69rad ≈ 40

around the axis of the field. A setup for polarization analysis has to preventthe change of the polarization vector due to any of those magnetic fields.

3.3.1 Classical polarization analysis

The simplest way to prevent uncontrolled magnetic fields to turn the polar-ization vector is using magnetic guide fields. These are fields which havethe same direction as the initial polarization of the beam generated at thepolarizer. Due to Eq.(3.17) any component parallel to the axis of a homoge-nous magnetic field will be just conserved. Therefore if the magnitude ofthe installed field is well above the disturbing field, the polarization will beconserved in such a field. Typical magnitudes for the fields are between 10and 100Gs. Mostly they are realized with permanent magnets.These guide fields are installed from polarizer to sample and polarizing ana-lyzer to conserve the polarization on the whole way through the instrument(s. Fig.3.7(a)). But there is quite a limitation to that concept: Eq.(2.50)shows that the polarization vector will be eventually turned instantly in thescattering process. As the guide fields point in the direction of the polariza-tion vector incident on the sample, all components of the final polarizationvector perpendicular to that direction start precessing around the axis of theguide field and are therefore lost for the measurement. This is called par-tial depolarization as the magnitude of the polarization vector remains wellabove zero in this process. Fig. 3.6(a) shows this situation, which is in factequal to that of a non-adiabatic field transition, where instead of the polar-ization vector the field is turned instantly. Altogether this signifies that themethod is only suitable to analyze the component of the final polarizationvector parallel to the incident polarization vector.By using adiabatic field transitions (s. section 3.2.2) at least the incidentpolarization vector can be turned upstream of the sample into any wanted


(a) The incident polarization vector ~Pi is conserved by the use of a guide field ~B. The

scattering process transforms the polarization vector into ~Pf . The component P‖ of the final

polarization vector ~Pf which is parallel to the guide field ~B is guided to the polarization

analyzer. The component perpendicular to the guide field P⊥ depolarizes and the information

is lost.

(b) An adiabatic field transition turns the incident polarization vector ~Pi upstream of the

sample to any wanted direction (here from z to y direction). The scattering process transforms

the polarization vector into ~Pf . The component P‖ of the final polarization vector ~Pf which

is parallel to the guide field (here: y-component) is turned back to the analyzer axis(here:

z-axis) by a second adiabatic field transition. The component perpendicular to the guide field

P⊥ depolarizes. (s. also (a))

Figure 3.6: Concepts in classical polarization analysis



direction. But downstream of the sample still only the component of the finalpolarization vector parallel to the guide field direction can be turned backby adiabatic transitions into the direction of the analyzer axis (remembersection 3.2.3, only the projection of the polarization vector onto the quan-tization axis can be measured). This is generally the same direction as theone of the initial polarization. The use of adiabatic field transitions is ex-plained in Fig. 3.6(b). The adiabatic field transitions are usually realizedwith a combination of permanent guide fields between polarizer and sampleand sample, and analyzer, and a special setup of coils around the sampleaxis. These coils are necessary to allow adiabatic transitions in different di-rections. They are denoted as ’Polarization Selector’ in Fig.3.7(a).In all, classical polarization analysis allows only to measure the terms of thepolarization tensor in Eq.(2.50), where the direction of the initial and finalpolarization vector are the same, thus the diagonal terms only:

P =

Pxx

Pyy

Pzz

.

Hence, clearly some of the additional information which can be revealed bydoing polarization analysis is not measured by this kind of setup. It was firstused by Moon et al.([MRK69]). Because it was the first setup for polarizationanalysis it is called ’Classical Polarization Analysis’.

3.3.2 Three Dimensional Polarization Analysis

Only the diagonal terms of the polarization tensor can be measured withclassical polarization analysis. This is due to the presence of guide fields inthe sample region where the initial polarization vector is transformed in thescattering process. The guide field depolarizes any component of the polar-ization vector perpendicular to the direction of the guide field. But if therewere no disturbing fields along the path of the polarized neutron beam noguide fields would be needed to conserve the polarization and therefore thisproblem would be bypassed.The disturbing fields can be removed from the beam area just by buildinga magnetic screen around. Such a ’zero field chamber’ is supposed to shieldthe magnetic field from the concerned area. ’Zero’ signifies that the mag-netic field inside has to be so small that the field inside the chamber doesn’tturn the polarization significantly in the view of the experimental precision.For example if a neutron beam with wavelength of 1A corresponding to a


(a) Classical Polarization Analysis: The polarization vector is guided through the instrument by guide

fields. Only the component of the final polarization vector parallel to the initial polarization vector can be

measured, because the component perpendicular to it will depolarize in the guide field (s. also Fig.3.6(a)).

The polarization selector is a combination of different coils, which allow together with the permanent guide

fields to realize different adiabatic field transitions up- and downstream of the sample. This serves to select

different directions of the initial polarization vector (s. also Fig.3.6(b)).

(b) Three Dimensional Polarization Analysis: The polarization vector is guided into a zero field

chamber where it is not depolarized due to disturbing fields. It can be directed into any initial direction by

Larmor precession devices before and and from any direction to the analyzer axis after the scattering process.

Therefore the whole polarization tensor can be measured.

Figure 3.7: Setups for polarization analysis on a TAS instrument



velocity of 3956ms

travels 2m (in the range of typical TAS instrument length)through a homogenous field of 1mGs, the components of the polarizationvector perpendicular to the direction of the field will be turned by ϕ ≈ 0, 5

around the axis of the field. Therefore for such neutrons zero field would besomething in the order of slightly beyond 1mGs over the whole length of theinstrument. There are basically two methods to screen volumes of magneticfields:

• The Meissner-Ochsenfeld effect of a superconductor can be used: asuperconductor which is cooled beyond its critical temperature Tc -where it enters the superconducting phase - expells any magnetic fieldout of its own volume.

• A closed shielding out of highly permeable material (e.g. mu-metal)tends to guide magnetic field lines in a way that the volume enclosedby it holds only a field which is several orders of magnitude lower thanthe outer field.

Assuming that the TAS instrument is equipped with a zero field chamber,it needs a device capable to turn the polarization vector in any wanted di-rection before and after the scattering process at the sample. These devicesare generally a combination of several coils which produce well-defined ho-mogenous fields in order to turn the polarization vector with high precisionby Larmor precession into any wanted direction. They have to be designedin a way that the zero field condition for the rest of the neutron flight path isnot disturbed. In the next chapter the coils designed for MuPAD will be ex-plained in detail. Because of the missing guide field there’s no risk of loosingany component of the final polarization vector, as they are all conserved inthe zero field chamber until one is selected by the Larmor precession device.Hence, the combination of a zero field chamber and high precision Larmorprecession techniques enables to measure all components of the polarizationtensor P in Eq.(2.50). Each component can be selected by setting the direc-tion of the polarization vector before the scattering at the sample with thefirst Larmor precession device and measure the appropriate component ofthe final polarization vector with the second. Such a setup is shown in Fig.3.7(b). If this technique is properly applied it allows to gain the maximum ofinformation in magnetic neutron scattering. Because the incident and finalpolarization vector can be turned in any direction this technique is called’Three Dimensional Polarization Analysis’ or ’Spherical NeutronPolarimetry’.


3.3.3 CryoPAD

CryoPAD (Cryogenic Polarization Analysis Device) was the first setup forthree dimensional polarization analysis at the ILL. It was presented by Tas-set in 1989 (s. [Tas89], [Tas98], [Tas99]). First it was only used in diffractionexperiments to disentangle complicated magnetic structures. In 2003 a mod-ified CryoPAD setup (called CryoPADUM) for inelastic polarization analysismeasurements on a TAS was presented by Regnault et al([RGF+03] and[RGF+04]).The zero field chamber of CryoPAD consists of a double Meissner screen outof niobium. The polarization vector is manipulated in ’3D’ upstream anddownstream of the sample by the combination of a nutation field (motorizedrotation in the plane perpendicular to the incident or scattered beam) and a

precession field between the two Meissner screens (perpendicular to ~ki or ~kf

in the scattering plane). The action of both coils is decoupled by the outerMeissner screen. The setup is shown Fig.3.8. For more details we refer tothe articles mentioned above.From our point of view the CryoPAD setup suffers from different major draw-backs. But still we want to point out that there’s no unique opinion aboutthat and the topic is still open for discussion. These drawbacks are:

• The use of a Meissner screen as zero field chamber is not perfect. Anyfield, present before the superconducting transition takes place, is ac-tually not removed out of the enclosed volume but trapped inside ofit. The only way to use the superconductor as a shielding anyway is tocool it down inside another magnetic shielding, e.g. out of mu-metal.In that case it will enclose a ’zero field region’, and as no magneticfield can enter in the superconductor, it will keep any magnetic fieldout of the enclosed volume as long as it is still in the superconduct-ing phase. During cooling down, temperature differences between thedifferent metals of the cryostat cause magnetic fields (Seebeck-effect),which increase significantly the field in the screened area.

• A lot of cryogenics are needed to hold the superconducting shieldingbeyond the critical temperature, which makes it inconvenient to operatethe device.

• The device always has to be cooled down in the second shielding half aday before it is ready to be operated. Only after being cooled down itcan be mounted on the TAS. Often during the transportation the fieldinside increases again.



• If the device runs out of liquid helium or nitrogen the superconductingphase breaks down and magnetic fields enter in the chamber. After-wards it needs at least half a day to cool down the chamber inside themu-metal shielding. Therefore it has to be dismounted again and to beput inside the shielding. A lot of measurement time is lost due to that.

We think that a device without these drawbacks can be realized with a zerofield chamber out of a mu-metal screen together with highly accurate Lar-mor precession devices also based on the use of mu-metal. Similar deviceshave already been built in the seventies for use on SANS and reflectometryinstruments ([ORVG75], [STRG79], [RS79]). Therefore the idea was bornto built such an option also for a Three-Axis-Spectrometer. This concept isnamed MuPAD.

(a) Scheme of CryoPAD (picture taken from [Tas99]).

(b) CRYOPADUM

Figure 3.8: CryoPAD: (a) Shows a scheme of the CryoPAD setup. (b) Thenutation devices can be seen on the left and right of the niobium Meissnerscreen in the center.

Chapter 4

MuPAD

MuPAD - the Mu-Metal Polarization Analysis Device - is an option forThree-Axis-Spectrometers equipping them with three dimensional polariza-tion analysis (s. section 3.3.2).

4.1 The Principle of MuPAD

The option MuPAD uses a mu-metal screen to realize a zero field chamberas discussed in section 3.3.2. Its design is described in the section 4.2. Asa polarized beam can not pass a mu-metal screen with a thickness in therange of several mms without being depolarized, it has to be guided throughspecial devices - called the coupling coils - into and out of the zero fieldchamber. These devices are presented later in section 4.2.3. At present weassume a perfect zero field chamber capable of maintaining the polarization.For three dimensional polarization analysis Larmor precession devices areneeded to turn the polarization vector upstream and downstream of thesample arbitrarily in three dimensions. Assuming the polarized beam to en-ter in the zero field chamber with the polarization vector perpendicular tothe scattering plane (this direction will be further on denoted as zi-axis), itwill pass two regions of perfectly homogenous magnetic fields of well definedlength along the incident beam direction (denoted as xi axis, xi ‖ ~ki). Thefirst field is perpendicular to the beam in the scattering plane (denoted asyi-axis) whereas the second points along the zi-axis. Direction and lengthof the two fields are fixed, but their magnitude can be varied. These twofields will be realized by two identical coils, called the precession coils. Theirspecial design assures that the neutron beam only passes through their inner

45

46 CHAPTER 4: MUPAD

homogenous field but not through their outer return fields. The coils arepresented in section 4.2.2. The arrangement of those two coils with respectto the beam is shown in Fig.4.1.

Figure 4.1: Two precession coils with inner homogenous fields perpendicular to each other areshown. The return fields of the coils do not intersect with the beam area. This arrangement of two coilsis sufficient to turn the polarization vector in three dimensions.

From Eq.(3.17) we know that the polarization vector of a neutron beampassing through such a field will be turned by a certain angle around theaxis of the field depending on the magnitude of the field and the velocity ofthe neutrons. These angles will be called δi and ϕi for the first and secondcoil respectively. The action of the first coil on the initial polarization vector~P0 can be expressed as a rotary matrix, which describes a rotation by theangle δi around the yi-axis

Tδi=

cos δi 0 sin δi

0 1 0− sin δi 0 cos δi

. (4.1)

The action of the second coil on the polarization vector ~P ′0 = Tδ · ~P0 is then

described by the rotary matrix representing a rotation around zi by ϕi

Tϕi=

cos ϕi − sin ϕi 0sin ϕi cos ϕi 0

0 0 1

. (4.2)

The action of both coils is just given by the product of the two matrices.The initial polarization vector was assumed to be aligned along the z-axis,

4.1 THE PRINCIPLE OF MUPAD 47

therefore:

~Pi = Tϕi· Tδi

· ~P0

=

cos ϕi cos δi − sin ϕi cos ϕi sin δi

sin ϕi cos δi cos ϕi sin ϕi sin δi

− sin δi 0 cos δi

︸︷︷︸

Ti(δi,ϕi)

·

00P

= P

cos ϕi sin δi

sin ϕi sin δi

cos δi

(4.3)

This is the representation of a vector of magnitude P in spherical coordinates,which proves that such a combination of homogenous fields can turn theinitial polarization vector into any arbitrary direction before the neutronbeam is scattered at the sample (s. also Fig. 4.1).The same setup of precession fields is used for the scattered beam, only theirsequence is exchanged. Apart from that, they have to be considered in acoordinate frame with respect to the direction of the scattered neutron beamwhich is ~kf . Hence, now xf ‖ ~kf , yf ⊥ ~kf in the scattering plane and zf ⊥ ~kf

perpendicular to the scattering plane. Note that the z-direction is conserved(zf = zi = z). We will call this the final frame of reference whereas thecoordinate frame used before the scattering process will be denoted as theinitial frame of reference. Thus, after being scattered at the sample the beamwill pass a homogenous field pointing in the z-direction of the same lengthas the fields before the scattering process. The action of this field on thepolarization vector is then described by the following rotary matrix:

Tϕf=

cos ϕf − sin ϕf 0sin ϕf cos ϕf 0

0 0 1

. (4.4)

Afterwards it passes the second field pointing in the yf direction whose actionon the polarization vector is then:

Tδf=

cos δf 0 sin δf

0 1 0− sin δf 0 cos δf

. (4.5)

The action of the the two precession fields downstream of the

48 CHAPTER 4: MUPAD

Figure 4.2: The principle of MuPAD: The setup of MuPAD is shown schematically. A neutronbeam polarized along the z-direction enters into the zero field chamber denoted by the blue volume throughthe incident coupling coil. The polarization vector is turned by two precession coils with homogenousfields perpendicular to each other in order to be aligned along the x-direction. It is scattered on a non-magnetic Bragg peak of a sample (this case was chosen due to the simplicity of the picture). Thereforethe polarization vector is not changed in the scattering process. Now the second pair of precession coilsdownstream of the sample is used to turn the x-component of the final polarization in the direction of theanalyzer axis (z-axis). This component is guided out of the zero field chamber to the analyzer by the exitcoupling coil. Thus in this configuration the term P xx of the polarization tensor is measured.

Figure 4.3: The relationship of the three different coordinate frames necessary to describe the principleof MuPAD are shown. The one in red is the analysis frame in which the change of the polarization vectoris described. The two others are denoted by indices i and f and are the ones of the beam incident on andscattered by the sample respectively. They are called initial and final frames.


sample is denoted by the matrix

Tf (δf , ϕf ) = Tδf· Tϕf

=

cos ϕf cos δf − sin ϕf cos δf sin δf

sin ϕf cos ϕf 0− cos ϕf sin δf sin ϕf sin δf cos δf

(4.6)

This second pair of fields enables MuPAD to manipulate the polarizationvector after the scattering process. The setup is sketched in Fig. 4.2 and willbe discussed in more detail now.The coordinate frame for polarization analysis is defined in (2.18). It will bedenoted as analysis frame in the following. To measure a single term P ij ofthe polarization tensor the polarization vector incident on the sample has tobe turned in direction i whereas the component j of the polarization vectorafter the scattering process has to be analyzed and therefore to be turned inthe direction of the analyzer axis. Here i and j can be x, y and z respectively.The situation is demonstrated in Fig. 4.3. The picture shows that the initialand final frame of reference are transformed into the analysis frame by beingturned by the angles βi = ∠(xi, x) = ∠(~ki, ~Q) and βf = ∠(x, xf ) = ∠( ~Q, ~kf )around the common z-axis respectively. These angles have to be consideredin the process of turning the polarization vector.The whole process from the polarization vector entering into the chamber tothe analysis will be examined in more detail now. The initial polarizationvector aligned along the z-axis will be put along each of the axis of the analysisframe. This is done in the initial frame because the rotation matrices Tδi

andTϕi

are only valid there.

x-axis: The initial polarization vector has to be turned first by δi = π/2around the yi-axis to orient it in the scattering plane. Then by turningit by ϕi = βi around the z-axis, (s. Fig. 4.3) it is aligned along thex-axis (s. Eq.(4.3)):

~Pi = P

cos βi sin π/2sin βi sin π/2

cos π/2

= P

cos βi

sin βi

0

. (4.7)

To prove that the polarization is really aligned along the x direction inthe analysis frame, ~Pi will be transformed into the analysis frame by

50 CHAPTER 4: MUPAD

turning the coordinate frame by −βi around the z-axis1:

~P ai = P

cos(−βi) − sin(−βi) 0sin(−βi) cos(−βi) 0

0 0 1

cos βi

sin βi

0

= P

cos2 βi + sin2 βi

00

=

P00

. (4.8)

y-axis: The initial polarization vector first has to be turned by δi = π/2around the yi-axis to orient it in the scattering plane. Then by turningit by ϕi = βi + π/2 around the z-axis, it is aligned along the y-axis:

~Pi = P

cos(βi + π/2) sin π/2sin(βi + π/2) sin π/2

cos π/2

= P

− sin βi

cos βi

0

. (4.9)

In the analysis frame this is:

~P ai = P

cos(−βi) − sin(−βi) 0sin(−βi) cos(−βi) 0

0 0 1

− sin βi

cos βi

0

= P

sin βi cos βi − cos βi sin βi

sin2 βi + cos2 βi

0

=

0P0

. (4.10)

z-axis: Nothing at all has to be done as the initial polarization vector is al-ready aligned along that direction when it enters the zero field chamberand the z-axis is the same for all three frames.

Now the scattering process at the sample takes place. Therefore the po-larization incident on the sample ~P a

i will be transformed in the scatteringprocess into a final polarization vector depending on the sample propertiesand the direction of the incident polarization vector. We assume the initialpolarization vector in the analysis frame will be transformed into the finalpolarization

~P af =

abc

, (4.11)

1Note that coordinate frames transform inverse to vectors.


where a, b and c are the components of the final polarization in x, y and zdirection of the analysis frame respectively. Each component of (4.11) has tobe analyzed. Because the rotation matrix Tf (δf , ϕf ) is only valid in the final

frame we express ~P af in terms of it by turning the coordinate frame by −βf

around the z-axis:

~Pf = P

cos(−βf ) − sin(−βf ) 0sin(−βf ) cos(−βf ) 0

0 0 1

abc

=

cos(βf ) · a + sin(βf ) · b− sin(βf ) · a + cos(βf ) · b

c

. (4.12)

Now each component of the final polarization vector is analyzed:

x-component: To analyze the x-component we have to turn ~P af inside the

scattering plane first by ϕf = βf and then turn the component whichis now along the xf axis by δf = −π/2 into the direction of the z-axiswhich is the analyzer axis (s.Figs 4.2 and 4.3) by using (4.6):

~P ′f =

cos βf cos(−π/2) − sin βf cos(−π/2) sin(−π/2)sin βf cos βf 0

− cos βf sin(−π/2) sin βf sin(−π/2) cos(−π/2)

~Pf

=

−c

sin(βf ) cos(βf )a+sin2(βf )b−cos(βf ) sin(βf )a+cos2(βf )b

cos2(βf )a+cos(βf ) sin(βf )b−sin(βf ) cos(βf )b+sin2(βf )a

=

−cba

. (4.13)

As only the component aligned along the z-axis will be guided out of thezero field chamber by the exit coupling coil to the polarization analyzer,that choice of ϕf and δf corresponds to analyzing the x-component of~P a

f .

y-component: To analyze the y-component we first have to turn ~P af inside

the scattering plane by ϕf = βf − π/2 and then turn the componentwhich is now aligned along the xf axis by δf = −π/2 into the direction

52 CHAPTER 4: MUPAD

of the z-axis which is the analyzer axis (s. again Figs.4.2 and 4.3):

~P ′f =

cos(βf−π/2) cos(−π/2) − sin(βf−π/2) cos(−π/2) sin(−π/2)

sin(βf−π/2) cos(βf−π/2) 0

− cos(βf−π/2) sin(−π/2) sin(βf−π/2) sin(−π/2) cos(−π/2)

~Pf

=

−c

− cos2(βf )a−cos(βf ) sin(βf )b−sin2(βf )a+sin(βf ) cos(βf )b

sin(βf ) cos(βf )a+sin2(βf )b−cos(βf ) sin(βf )a+cos2(βf )b

=

−c−ab

. (4.14)

Only the z-component of this is analyzed, thus this choice of ϕf and δf

corresponds to analyzing the y-component of ~P af .

z-component: Nothing at all has to be done , since the z-component of ~P af

is already aligned along the analyzer axis, namely the z-axis.

Hence, we showed that with the combination of two precession coils upstreamand downstream of the sample respectively, all terms P ij of the polarizationtensor (2.50) can be measured. During an experiment the TAS, on which

MuPAD is mounted, will be set to a certain point in (~Q, ω)-space. From

the corresponding relationships of ~Q, ~ki and ~kf the angles βi and βf can becalculated. Then for each of the three components of the polarization vectorincident on the sample (corresponding to the choices derived above for theangles δi and ϕi) the three components of the final polarization vector areanalyzed by adjusting the precession fields to the corresponding angles δf

and ϕf . This results in nine measurements for each point in ( ~Q, ω)-space tobe made.As the precession fields are produced by coils the adjustment of the field isachieved by simply setting the right current in the coils. Therefore the wholeprinciple is very convenient to use.

4.2 Design

All components needed to realize MuPAD were developed within this work.Their design and special features will be described in detail.

4.2 DESIGN 53

4.2.1 Zero Field Chamber

Mu-Metal

-0.7-0.6-0.5-0.4-0.3-0.2-0.1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

-10-5 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

B[T

]

H[A/m]

new curceHysteresis

Figure 4.4: The hysteresis and new curve of mu-metal measured with a sample in form of stampedrings of dimensions 28, 5mm × 20mm − 0, 8mm ([Han]).

The zero field chamber of MuPAD is made out of mu-metal. Mu-metal isan alloy out of 77% Ni, 16% Fe, 5% Cu, 2% Co. It is a material with avery high relative permeability µr. As it is ferromagnetic µr, depends on themagnetic flux through the material. The hysteresis curve is shown in Fig.4.4. For screening magnetic fields from several mGs up to several hundredGs the mu-metal can be regarded as linear with µr ≈ 30.000. After perfectheat treatment, µr can reach values near 70.000. Minor mechanical stresshowever, will reduce µr and the value of 30.000 is more realistic for welltreated mu-metal.For the description of magnetic loops a magnetic resistance - the Reluc-tance - can be defined similar to the electrical resistance ([Rai87])

R =l

µ0µrA, (4.15)

where µ0 = 4π · 10−7 is the permeability of vacuum, l is the length of themagnetic conductor and A is its cross-section. Thus it is energetically ofadvantage for a magnetic field to float through a material with a high relativepermeability. Therefore a volume which is enclosed by a highly permeablematerial, e.g. mu-metal, will be screened from outer fields because the fieldlines are deformed by the presence of the material in a way that the majorityof the field lines will now travel through the high permeable region. This

54 CHAPTER 4: MUPAD

holds as long as the bigger path lengthes of the field lines do not significantlycompensate the effect of the higher µr. Fig. 4.5 demonstrates the shieldingof a homogenous field B0.To describe the quality of a magnetic screen the screening factor is defined:

S =B0

Bi

, (4.16)

where Bi is the magnitude of the field inside the shielding. Typical values forscreening factors are . 100 for a single layer of mu-metal depending on thesize and geometry of the volume to be screened. Apart from that multiplelayers can be used to improve the screening factor up to several thousands.

Figure 4.5: Screening a volume from magnetic fields. The left picture shows the field ~B which shouldbe screened. In the right picture the volume to be screened is enclosed by a highly permeable materialwith µr >> 103. The magnetic field lines are deformed so that they travel through the screen with themuch lower reluctance (s. Eq.(4.15)). Therefore the magnitude of the field Bi inside the volume is nowmuch smaller than outside.

Shielding of the sample region

The main part of the zero field chamber of MuPAD is a cylindrical shield-ing around the sample table of the TAS. It is designed to host a standard’ILL Orange Sample Cryostat’ ([Cry]) which has a diameter of 316mm. Thesample shielding consists of two separate parts: an upper and a lower doublecylinder out of mu-metal with a slit in between them which can be variedfrom 40 to 100mm centered on the beam height. The slit allows the neutronbeam to enter the shielding without loss of polarization. A single doublecylinder with only one entrance and one exit hole is not possible as MuPADis operated on a TAS where the scattering angle varies with the selected pointin ( ~Q, ω)-space. Two provide a good magnetic connection between upper and

4.2 DESIGN 55

lower part despite the slit a 90-section of a cylinder out of mu-metal is fixedbetween them for both - inner and outer - cylinders at a ’blind spot’ wherethe beam is never analyzed. Supplementary ten 15-sections serve as movablelamellas to establish a magnetic connection at all other places apart from theone where the beam exits. They are planned to be moved automatically inand out in the future. Due to lack of time within that work they are oper-ated manually until now. They are fixed on the cylinders by strong rubberbands (s. Fig. 4.7). The analyzer arm of the TAS can be moved around thesample axis by −30 ≤ θS ≤ 110 when MuPAD is mounted. The two lowercylinders have a closed bottom whereas the upper ones are open in order tobe able to insert the sample cryostat. Good magnetic screening is providedin that case just by extending the upper two cylinders to the limit(600mm)allowed by the cryostat. The whole shielding is shown in Figs.4.6 and 5.1(b).The screening factor of such a cylinder construction can be calculated from

the empiric formulas ([VAC88]):

St =µrd

D+ 1, (4.17)

Sl =4N(Sq − 1)

1 + D2L

+ 1, (4.18)

where St and Sl are the screening factors transverse and longitudinal tothe cylinder axis respectively. d is the thickness of the cylinder wall, D itsdiameter and L its length. N is the ’demagnetization factor’ which dependson the fraction L/D (s. also [VAC88]). For a double cylinder there’s amultiplying effect for the transverse screening factor. It is then calculatedlike this:

Si = µrdi

Di

,

SD = S1S2[1 − (D2/D1)2] + S1 + S2 + 1, (4.19)

where Si, Di and di are the screening factor, diameter and thickness of cylin-der i (2 is the inner one!) and SD is the transverse screening factor for thedouble cylinder. All formulas are only valid for the center of the cylinder.All cylinders for MuPAD are made of 2mm thick mu-metal. The diameter ofthe inner cylinders is 360mm whereas the one of outer is 404mm. As a firstapproximation we will treat upper and lower cylinders as one for inner andouter cylinders respectively. This should be possible as we provided a goodmagnetic connection between them with the lamellas and the 90-connectionsegment. In that case they are approximately 850mm long. For that L/Dis N ≈ 0.15. For these values Eqs.(4.17) to (4.19) give SD ≈ 25.000 and

56 CHAPTER 4: MUPAD

Figure 4.6: A technical drawing of the complete zero field chamber. In the center the sample shieldingis shown. Left and right from the arms hosting the precession and coupling coils can be seen.

4.2 DESIGN 57

(a) (b)

Figure 4.7: The magnetic connection of upper and lower cylinders of the sample shielding.(a) Theslit between upper and lower double cylinders can be seen. On the left side the slit between the outercylinders is still open. on the right the movable magnetic connections lamellas of the inner cylinders canbe recognized. (b) Here the slit is completely closed by movable lamellas for inner and outer cylinders.They are fixed by strong rubber bands.

Sl ≈ 15.000. Experimentally we observed screening factors of approximately1.000. The difference to the calculated values is due to several reasons: onone hand the connected upper and lower cylinders are certainly much worsethan one continuous cylinder and on the other hand the sample and the beamare not at the center of the cylinders. Apart from that it is difficult to sayhow big µr - the most important factor in the equations - really is. Mu-metalis very sensitive on mechanical distortion which may appear during manu-facturing, transport and installation.Assuming a screening factor of 1.000 in both direction and the earth fieldof magnitude 0, 3Gs to be screened, we find an inner field of approximately0, 3mGs in the sample shielding. The polarization vector of 4.1A neutronstherefore will be turned inside the shielding (s. Eq.(3.18)):

ϕ = 2π · 2916[Hz

Gs]0, 3[mGs]

0, 4[m]

2586[ msec

]= 0, 04. (4.20)

The Arms

The arms of MuPAD are a compact prolongation of the shielding on themonochromator and analyzer side of the TAS. Each of them hosts a pair ofprecession coils to turn the polarization vector in 3D upstream and down-stream of the sample and one coupling coil which guides the polarization

58 CHAPTER 4: MUPAD

vector into and out of the zero field chamber (s. Fig4.6). Due to intensity2

reasons MuPAD was kept as compact as possible. Therefore the arms arejust as big as needed to host the coils. As we assume the precession coilsdo not have any outer return field in the beam leading to crosstalk betweentwo coils (s. Fig. 4.10), the coils were put together as near as 20mm. Thusthe main part of the arm is a nearly cubic shielding (a ≈ 180mm) whichhosts the precession coils. The cube has a lit which is fixed by screws inorder to be able to insert the precession coils. On the entrance or exit siderespectively there is an opening which joins into a tube in which the cou-pling coil is mounted. The tube has a length of 140mm and a diameter of100mm. Therefore it is nearly 1.5 times as long as its diameter is wide. Thisratio was chosen because field lines only enter approximately as deep as onediameter through a hole in a shielding3. With that design almost no fieldlines will penetrate as deep as the length of the tube in the shielding whichis connected with the cube hosting the arms (Exactly at the end of the cou-pling coil the polarization vector is guided into the zero field chamber by anon-adiabatic field transition. See section 4.2.3 for more detail). The otherside of the cube towards the sample cylinders is closed with a 60-section ofa 404mm diameter cylinder of length 320mm, which is welded to it. Hencegood mechanical and therefore magnetic contact to the cylinders of the sam-ple shielding is assured. In the center the cylinder section has a quadratic(100mm × 100mmm) opening towards the slit of the sample shielding. Allshielding parts of the arms are also out of 2mm thick mu-metal. For detailssee Fig. 4.9.Because the arm on the monochromator side is fixed with respect to the sam-ple shielding it is just attached mechanically to the sample cylinders. Thearm on the analyzer side moves around the cylinder axis of the sample shield-ing corresponding to the choice of the examined point in (~Q, ω)-space. As thesample shielding cylinders can not be manufactured perfectly circular, goodcontact between arms and sample shielding requires a special constructionfor optimal contact. The arm is mounted on a track system4 which allowsradial movement of the arm shielding towards the cylindrical sample shield-ing. The arm is pulled to the nearest possible position towards the cylinderby two strong rubber bands (s. Fig. 4.9(c)).

2Intensity drops quadratically with the length of the flight path of the beam.3This is a simple geometric argument. If they enter as deep as one diameter of the

hole the path length along that way is approximately as long as the path around the holeinside the mu-metal. But for bigger path lengthes the bigger µr of mu-metal makes itmore favorable to go through the mu-metal for the lines. Thus entering more deeply isenergetically worse for the field lines.

4http://www.hepco.fr/

4.2 DESIGN 59

The screening factor for a cube of side length a is defined empirically by([VAC88])

Scube =4

5

µrd

a+ 1. (4.21)

Therefore the screening factor of one arm is S ≈ 268. Experimentally weobserved factors of . 100. The difference is due to the fact that the arm isnot really a closed cube but has fairly big openings on two sides. Screeningthe earth field of 0, 3Gs results then in a mean field inside one arm of 3mGs.The polarization vector of 4.1A neutrons therefore will be turned inside theshielding (s. Eq.(3.18)):

ϕ = 2π · 2916[rad

sGs]3[mGs]

0, 18[m]

2586[ msec

]= 0, 22. (4.22)

Hence, the polarization vector will be turned about (0, 04+2 ·0, 22) = 0, 44

for 4.1A neutrons throughout the whole chamber and the screen of MuPADcan be regarded as a zero field chamber. The zero field chamber is showncompletely in Fig. 4.8. It was manufactured by Sekels 5.

Figure 4.8: The complete zero field chamber of MuPAD is shown.

5http://www.sekels.de/

60 CHAPTER 4: MUPAD

(a) the arm on the monochromator side

(b) Both arms before being mounted

(c) the arm on the analyzer side

Figure 4.9: The MuPAD arms:(a) The arm on the entrance side of the zero field chamber. Thecoupling coil in the entrance tube and the inserted precession coils in the arms can be seen. (b) Thecontact plate for magnetic contact to the sample shielding with the beam opening are visible at the leftside. The right arm shows the entrance tube for the coupling coil. (c) The closed exit arm mounted on atrack system. The rubber bands assuring the contact can be seen on the left. The inset shows a detailedview of the track.

4.2 DESIGN 61

Figure 4.10: Cross talk between two PCs: Shown is the inner and outer field on the beam axis ofeach PC. If the outer field does not decline fast enough within the distance between two coils, their outerfields superpose each other in an uncontrolled way. This is denoted by the red arrows. That would affectthe independence of both coils.

4.2.2 Precession coils

The four MuPAD Precession Coils (PCs) were designed to fulfill the followingrequirements necessary for MuPAD:

1. Homogenous inner field over the whole trajectory neutrons pass andover the cross-section of the beam in order to turn the polarizationvector by any angle around the axis of the inner field with high accuracy.

2. The return field of the coils should be guided in such a way that theneutron beam is only affected by the inner homogenous field. As alwaysa pair of coils is used to turn the polarization vector, leakage of outerfields would cause cross talk effects between those two coils (s. Fig.4.10).

3. The field transitions into and out of the coils should be strictly non-adiabatic in order to preserve the polarization vector outside the coil.

4. The inner field should be high enough to allow rotatation of the po-larization vector by 180 for neutrons with wave number k = 4.1A−1.From Eq.(3.18) we get an inner field integral of 44, 4Gscm for this case.

5. A neutron beam of cross-section 25mm × 30mm (W × H) should beable to pass through the coil.

6. The coils should be as compact as possible.

7. Polarized inelastic measurements on a TAS suffer from low intensitiesand high transmission of neutrons through the coils is a primary goal.

62 CHAPTER 4: MUPAD

Preliminary Tests and Models

The main idea to solve the first three points was to build rectangular cylin-drical coils with fields perpendicular to the beam direction. The rectangularcross-section of the coil allows to have almost the same field integral for thewhole beam. A beam window in the center of the coil body allows the beamto pass through the coil. The windings of the coil then have to be made outof transparent material for neutrons, e.g. Aluminium (s. Fig. 4.16 for thefinal design). A mu-metal-yoke serves to guide the return field of the coil. Asmagnetic field lines are closed loops the field of a solenoid coil has an outerfield in the opposite direction with respect to the inner field like shown inFig. 4.11(a). Thus, a neutron passing through such a coil, has to pass theouter return field regardless of its path through the coil. Apart from that,the inner field of such a coil is not really homogenous. Both effects lead topartial depolarization and inaccuracy of the coils. But by inserting the coilin a yoke out of mu-metal, like shown in Fig. 4.11(b) both problems will besolved.

• Most of the return field will now pass through the mu-metal because itis energetically more favorable for the field to go through the mu-metalthan through the air, as showed in section 4.2.1. Therefore there is nowa direction in which the beam does not have to pass the return field ofthe coil (red arrow in Fig. 4.11(b)). As now the outer field directly atthe surface of the coil is much smaller, the non-adiabaticity of the fieldtransmission into and out of the coils is ameliorated too.

• As mu-metal simulates fairly well a solenoid coil with infinite length,the inner field is much more homogenous than before.

To verify if this method is good enough, simulations on such coil models weredone ([Pro04]) using the software FARADAY([IES]) based on the boundaryelements method (BEM)([YKP91], [CR96]). The results revealed that sucha yoke of 2mm thick mu-metal in combination with a second outer mu-metalshield of 1mm thickness would reduce the outer field integral by a factor 1000with respect to the inner field integral. As the biggest turning angle for onecoil is 180 the outer field integral would correspond to a turning error of0, 18, which is good enough. The calculation further on indicated how theratio between inner field integral and outer field integral can be increased:

4.2 DESIGN 63

(a) Cut through the field of a solenoid coil.

The not perfectly homogenous inner fields

and the return fields of such a coil would

lead to partial depolarization of a neutron

beam.

(b) Solenoid coil equipped with a

yoke out of mu-metal. The outer

field is mostly guided back through

the yoke. A neutron beam can pass

in the direction of the red arrow

without striking the outer return

field.

Figure 4.11: The use of mu-metal for the PCs

• The gap between the yoke and the coil is very crucial. The bigger thedistance between the yoke and the coil at the two points where theyoke is taking over the magnetic flux of the coil (top and bottom of thecoil) the worse is the ratio.

• Increasing the length of the mu-metal tube, which represents the yoke,improves the ratio.

Prototypes have been built and good agreement between experiment andsimulation was obtained ([JSB+02]).These preliminary results also revealed another significant problem. Forthese PCs already very small fields perpendicular to the inner field (in therange of 10mGs over several centimeters) do decrease the accuracy. In thefollowing the direction of the inner field is defined to be along the z-axis.For a standard coil the windings will be wrapped around the body like aspiral, causing a slope with respect to the plane perpendicular to the innerfield direction, namely the x-y-plane. Due to the slope a component of thecurrent in the windings will be along the z-axis. This current will generate amagnetic field in the x-y-plane which will decrease the precision of the coil.This is demonstrated in Fig. 4.12. Another problem linked to this are thefields produced by currents of the connections.

64 CHAPTER 4: MUPAD

Figure 4.12: The slope of the windings of a usual solenoid coil with respect to the x-y-plane will leadto a small component of the current passing along the z-axis. This current leads to a disturbing field inthe x-y-plane.

Final Coil design

Taking into account all these subtle details the following final design for theprecession coil was made. It has the following special properties:

• The coil has the outer dimension 110, 1mm×70mm×20mm (height×width×depth, height along the direction of the inner field). The heightand width of the coil are already big enough, that even without mu-metal yoke the inner field is almost homogenous for the neutron beamof 25mm × 30mm passing in the center (s. Fig. 4.13 and 4.16(a)).The body is out of anodized aluminium for best heat conduction asany deforming of the coil body due to thermal expansion will changeits field.

• In its center the coil body has a beam window of 42mm × 42mm forthe beam. On the surfaces of that window cadmium stripes were gluedin order to prevent scattering of neutrons at the coil body.

• The coil windings are made out of pure aluminium wire of 1mm di-ameter without insulation because this kind of material has the besttransmission of neutrons (s. next section). One coil has exactly 100windings.

• Grooves were milled in the coil body to guide the wires of the coils.On one side this enables a very well defined field, because the positionof the wires is exactly known, which is also very useful for simulationsof the coils. Also all produced coils are to high extend similar to each

4.2 DESIGN 65

Figure 4.13: Technical drawing of the coil bodies of the PCs. The window in the center is for thebeam. The grooves guide the wires in order to have very well defined fields and to suppress unwantedfields in x-y-plane in combination with a current sheet (s. also Figs. 4.12 and 4.16(e)). Apart from thatthe not-insulated pure Al-wires are air-insulate from each other by putting 0, 1mm distance between twogrooves. The threads are for fixing the yoke. The holes serve to enter a magnetic probe inside the coiland for air cooling.

Figure 4.14: Technical drawing of the mu-metal yoke: It’s manufactured to host exactly the coilbody. This is marked by the red region, which has a tolerance of +0 − 0, 2mm only. Apart from that itis one cm deeper than the coil body. Both features improve the screening of the outer field.

66 CHAPTER 4: MUPAD

other.

(a) The field vectors of the precession coil with-

out the current sheet projected on the plane

z = 0. The main component inside the coil is

along the x-axis.

(b) The field vectors of the precession coil with

the current sheet projected on the plane z = 0.

The main component inside the coil is along

the x-axis. The current sheet can be seen on

the right side.

-0.23-0.22-0.21-0.2

-0.19-0.18-0.17-0.16-0.15-0.14-0.13-0.12-0.11-0.1

-0.09-0.08-0.07-0.06

-20 -15 -10 -5 0 5 10 15 20

Bx[

Gs]

y[mm]

Precession Coil without current plate

x=5mmx=0mm

x=10mm

(c) The x-component of the field along the y-axis

for different values for x for the coil without current

sheet.

-0.025-0.024-0.023-0.022-0.021-0.02

-0.019-0.018-0.017-0.016-0.015-0.014-0.013-0.012-0.011-0.01

-0.009-0.008-0.007-0.006-0.005-0.004-0.003

-20 -15 -10 -5 0 5 10 15 20

Bx[

Gs]

y[mm]

Precession Coil with current plate

x=5mmx=0mm

x=10mm

(d) The x-component of the field along the y-axis

for different values for x for the coil with current

sheet.

Figure 4.15: Simulation of the current sheet:(for precession coil from Fig. 4.13) The plots in (a)and (b) show that for both cases - with and without current sheet - the main component of the field inthe plane z = 0 is along the x-axis inside the coil. Therefore the x-components of the field were plottedalong the y-direction for the region where the beam passes to compare the two setups. The beam has across-section of 25mm × 30mm, so the interesting region is y = −15...15mm. x = 0 corresponds to thecenter of the coil, whereas x = 10mm is the surface of the coil. Only graphs for positive x values areshown because the ones for negative values are similar. The comparison between (c) and (d) shows thatthe current sheet improves the situation a factor 10-20. Apart from that the fields also get a little morehomogenous. These simulations were done without taking the mu-metal-yoke into account, the resultsrepresent the pure combination of guided coil windings and the current sheet.

On the other side it is simply necessary to insulate the windings of thecoils from each other, because the wires themselves are not insulated.So grooves were milled in a way that there is a distance of 0, 1mmbetween two wires (s. Fig4.13).

• A solenoid of 100 windings over 110, 1mm with a current of 2A makes

4.2 DESIGN 67

a field of

B = 12, 5[Gsmm

A]

n

L[mm]I[A] = 22, 7Gs. (4.23)

Multiplied by the depth of the coil of 21mm (from center of wire tocenter of wire) passing neutrons will exactly see an inner field integralof 47, 7Gscm which is close to the wanted 44, 4Gscm.

• In order to eliminate the currents in the x-y-plane the grooves includean additional design feature. The grooves are made in a way that thesloped part of one winding is only at one of the smaller faces of therectangular coils (s. Figs. 4.13 and 4.16(a) ). Therefore the currentelements in z-direction are only at that side. To compensate them acurrent sheet is put on that side through which the back current fromthe coil windings flows in inverted direction relatively to that in thewindings (s. Fig. 4.16(e)). This also solves the problems due to theconnection cables (s. Fig. 4.16(d)). Because now both connectionsare mounted at almost the same points, the fields of the connectioncables are removed by twisting the cables. The improvement due tothis design was simulated again using the software FARADAY. Theresults, which are shown in Fig. 4.15, yield that the principle works.Probably it can be even more optimized by changing the cross-sectionof the current sheet. Until now the cross-section through which thecurrent goes is 3mm× 10mm because of the included connections. Byreducing it to 1mm×10mm the current would be forced to flow nearerthe coil and compensation would be even better.

• The mu-metal-yoke was especially fabricated 6 that the coil body wouldexactly fit in it with a tolerance of +0−0, 2mm to improve the shieldingof the outer field to the maximum possible. The grooves in the coil bodyalso allowed to guide the wires of the first and last windings respectivelyat the very edge of the body. The distance between wire and mu-metalis only a tenth of a mm (s. Figs. 4.13 and 4.14). The mu-metal-yoke is2mm thick and has outer dimensions 104, 1mm× 99mm× 30mm. Themu-metal is 10mm deeper than the coil body to improve the screeningof the outer field.

• The whole coil was inserted into an outer aluminum frame of outerdimensions 134, 5mm× 134, 5mm× 42mm. This frame is closed on allsides besides of the beam window in the front and back plate. Aroundthat frame there is an outer field-screen out of 0, 5mm mu-metal whichwas also closed all around the frame beside of the beam windows. The

6http://www.sekels.de/

68 CHAPTER 4: MUPAD

squared dimensions of the front and back side of the outer frame actu-ally allow to use the same coil design for δ and ϕ coils (s. Fig. 4.16(c)).

• The wires in the beam window have no direct thermal contact to thecoil body. To prevent deformation by heat expansion some possibilityfor air cooling was included. The air enters one of the holes from topor bottom of the coil into the beam window and leaves between thewires (s. Fig. 4.13). This feature also enables MuPAD to be appliedfor bigger wave vectors.

All aluminum parts and the current sheets where manufactured in the centralworkshop of the the department of physics of the Technische UniversitatMunchen7.

Transmission and Small Angle Scattering

As already mentioned the wires of the PCs are made out of pure Al with-out insulation layer, e.g. aluminum-oxide. This choice has been made afterSmall Angle Neutrons Scattering (SANS) experiments were performed withwires with different insulation layers. The experiments were originally donefor NRSE-coils of the NRSE-TAS instrument at the FRM-II in Munich, Ger-many ([Kel04]). Nevertheless the results are also very interesting for MuPAD.The choice of aluminum for the conducting material of the wire for the coilswas evident because Cu is fairly much absorbing and other alternative ma-terials like Mg have poor mechanical properties.Measurements were carried out on the NG-7 SANS instrument ([NG7]) by J.Cook at NCNR-NIST ([NCN]). As samples two different types of aluminiumwires with diameter 1mm and anodization layer8 were used. To reduce smallangle scattering (SAS) by hydrogen inside the oxide, the wires were boiled inD2O (deuterated) with different final pressures. One sample consists out ofone layer of 20 wires placed neatly parallel together with vertical orientation.The measurements were done with λ = 5A and the detector at 12m. Fig.4.17 shows the results. They show that all anodized wires have a peak at ap-proximately Q = 0, 017A−1 which corresponds to a density correlation lengthof 370A in real space (s. Fig. 4.17(a)). The 2d plots show that the scat-tered peaks are oriented along axis of the wires. When the oxide is removed,the peaks vanish (Figs.4.17(c) and 4.17(a)). Table 4.1 shows the amount ofsmall angle scattering by the different samples. The transmission losses are

7http://www1.physik.tu-muenchen.de/einrichtungen/zw/zw2.htm8Both are from ’Wesselmann Umwelttechnik’(http://www.anoxal.de/), but the type 2

is not longer available

4.2 DESIGN 69

(a) the coil body

(b) the coil inside the mu-metal yoke

(c) the complete precession coil

(d) the connections

(e) the current sheet

Figure 4.16: A MuPAD precession coil: (a) The coil body out of anodized aluminum with groovesto guide the wires and a beam window. Grooves only have a slope on the small face on the right side ofthe coil. (b) Precession coil winded with pure Al-wire inserted in the mu-metal-yoke. The additional coilon the yoke is for demagnetization of the yoke. (c) Complete precession coil inside Al-frame and outermu-metal screen. The 0.1mm distance between the wires to insulate them from each other can be seen.On top the connection for air cooling is shown. (d) The connections to the coil. They are twisted tominimize magnetic fields from them. (e) The current sheet to minimize magnetic fields in x-y-plane.

70 CHAPTER 4: MUPAD

likely due to isotropic scattering from the hydrogen in the anodization layer.It scatters just a certain percentage out of the detector region. The valuesin the column ’scattering’ are the corresponding count rates on the peak atQ = 0, 017A−1.first of all, the data show that the transmission for wires without an insu-

Sample final Transmission Scatteringpressure[bar] (%) (Count. Rate [sec−1])

type 1 not boiled 0.97565 2630.58type 1 3 0.978867 1300.72type 1 6 0.976266 1113.29type 1 7 0.976617 1109.07type 1 9 0.979101 1011.51type 1 13 0.979242 1177.15type 1 oxide removed 0.990181 88.2586type 2 not boiled 0.969383 383.468type 2 5 0.972904 287.12type 2 15 0.974288 326.224type 2 oxide removed 0.993184 23.6744

Table 4.1: SANS data for different anodized al-wires

lating anodization layer is approximately 1% higher. Certainly this doesn’tseem to be a lot, but this is only for one layer. For the MuPAD PCs eightlayers have to be taken into account. Further on, the SAS is significantlyhigher for the anodized wire. This can be improved by boiling the wires inD2O. But still for the best value achieved the small angle scattering is ap-proximately by a factor 12 lower for both wire types if the anodization layeris removed. Certainly the SAS will be lower by a factor 5-10 for the wavelengthes used for MuPAD (2, 36A and 1, 53A) as it scales with λ2. But stillthe measurements can be negatively influenced by SAS. The SAS peak foundat Q = 0.017A−1 for λ = 5A in the measurements can be easily expressed ina scattering angle, as

Q =4π

λsin(θ), (4.24)

thus θ = arcsin(Qλ/4π) (4.25)

= 0, 4. (4.26)

In several experimental situations this will already be more than the usualbeam divergence (in the order of 10−60′) in TAS experiment and the mosaicspread of the monochromator and analyzer crystals. This will lead to

4.2 DESIGN 71

(a) Compilation of data for all samples. The two black lines represent the

samples type 1 and 2 with removed anodization wire. For them the small

angle scattering disappears. The left peak is the tail of the direct beam not

cut off by the beam stop.

(b) Untreated 2d data for anodized wire. Images look similar for all samples.

(c) Untreated 2d data for wire with removed anodization layer. The peak does not appear

for this sample

Figure 4.17: SANS-measurements on Aluminium wires. Measurements done with λ = 5A andDetector at 12m

72 CHAPTER 4: MUPAD

• loss of intensity, because the neutrons scattered at the coil wires willnot be reflected at the analyzer any more because they can not fulfillthe Bragg condition (s. Eq.(3.3)).

• more background signal.

• slight depolarization of the beam because the neutrons scattered at thecoil wires will see different field integrals than the others.

Some other measurements done on D11 ([D11]) at the ILL by P. Lindnerare in agreement with the one above. Al wire with 1mm diameter and withdifferent insulating coatings was tested: varnished Al wire, anodized wirefrom ’Wesselmann Umwelttechnik’, wire coated with ’Polyurethan’9 (alsoused for some coils of instrument EVA at ILL [Maj03]) and finally pure Alwire without any coating. The data is shown in table 4.2. Also here the purewire has the best transmission and the lowest amount of SAS. That the SASis higher in this experiment is due to the samples. This time the wire wasalready winded on an real aluminum body, so that two layers of wire are inthe beam.

Sample Transmission Scattering(%) (Count. Rate [sec−1])

without anodization 0.9863 730varnished 0.9158 2100urethan 0.9440 1150anodized 0.9680 3350

Table 4.2: SANS data for al-wires with different insulations([Lin03])

Therefore both measurements show that the best solution was to take thealuminum wire without anodization for the MuPAD PCs.

Fields of Precession Coils

Inner and outer field of the coils were measured to verify that the coil designworks properly. All the results are shown in Fig. 4.18.The inner field was measured with a Lakeshore Gaussmeter Model 410 10 anda longitudinal probe which was entered through a hole in the top of the coilwhich is also used for the air cooling (s. Fig. 4.13). The Gaussmeter wasused in its range from 0 to 200Gs where it has an accuracy of reading of

9Spray ordered from http://www.cramolin.de/10http://www.lakeshore.com/mag/ga/gm410po.html

4.2 DESIGN 73

2% and a resolution of 0.1Gs. Within the accuracy of this probe the fieldis homogenous over the beam window. Further on a curve ’current versusinner field’ was recorded to test the linearity of the coil. The result is shownin Fig. 4.18(a). The data was fitted with

B(I) = aI + b, (4.27)

where a = 10.54± 0.04068[GsA

] and b = −0.297pm0.00657[Gs]. It shows thatwithin the measurement errors the precession coil is absolutely linear up to5A.The outer fields were measured with the Bartington Single Axis FluxgateMagnetometer Mag-01H 11 with the longitudinal probe Mag B and the transver-sal probe Mag C. The probes were operated in the range 0 − 20mGs and20 − 200mGs where they have accuracies of 1µGs and 10µGs respectively.The magnetometer switches automatically between those ranges and has anzero offset of ±50µGs. To measure the outer stray fields the coil was cen-tered in the double upper MuPAD sample shielding. To measure only thefield produced by the coil a differentiation measurement was performed: thefield without any applied current was measured first and subtracted fromthe field measured with applied currents 1A and 2A. All field componentswere measured along the beam axis (x-direction, coil center at x=0) centeredon the beam window with respect to the coordinate frame defined in Fig.4.18(b). The fields over the cross section of the beam are similar to a highextend so that we only measured along that single axis. The results areshown in Figs.4.18(c) to 4.18(h).To estimate the outer field integrals the data is fitted on both sides of thecoil with exponential functions of the form

f(x) = ai exp(−x

ti). (4.28)

The field integral was calculated by integrating over these functions from11mm to ∞ and −∞ to −11mm for positive and negative side of the x-axisrespectively. The limit of ±11 corresponds to the outer faces of the coil.The different field integrals are shown in Table 4.3. For the y-component thepositive side was difficult to fit. Due to that the field integral of the negativeside was multiplied by 2 to estimate the sum of both.The data show that the field integrals in y and z direction are a factor 1130and 940 smaller than the inner one. Only the x component has a field integralwhich is quite big. It is only 130 times smaller than the inner one. Alreadyin the examination of the fields in the x-y-plane due to the slope of the

11http://www.bartington.com/mag01.htm

74 CHAPTER 4: MUPAD

coil windings it was recognized, that the biggest component is that in thex-direction. This problem might be solved by further improvement of thecurrent sheet, which should be made thinner (s. section ’Final Coil design’).Due to the small current correction (s. Appendix B, p.117 and Fig. B.1),there is only one case in the present setup which needs angles of rotationbigger than 90. That is in case the polarization should be turned fromz to -z direction. Therefore the rotation errors for 1A give a very goodimpression of the accuracy of the coils. They prove that only the inner fieldintegral has to be considered to calculate the turning angle of a precessioncoil. Apart from that the small deviations even can be taken into account inthe calibration of MuPAD.Therefore the turning angle of one precession coil can be adjusted just by

setting the right current to generate an appropriate inner field integral. FromEq.(3.18) the angle can be calculated

ϕ = 2π · 2916[rad

sGs]B[Gs]

l[m]

v[ msec

], (4.29)

where the length of the inner field is l = 0, 021m and the inner field iscalculated with Eq.(4.27). The relationship between the wavenumber andthe velocity of a neutron is

v = 629, 6[mA

s]k[A−1]. (4.30)

Inserting all in (4.29) we get the precession angle of the polarization vectoraround the inner field axis of one PC dependent on the wavenumber of theneutrons and the current in the coil

ϕ = 6, 44[rad

AA]

I[A]

k[A−1]. (4.31)

Table 4.3: Outer field integrals of one precession coil for applied current of 1A and 2A respectively.Turn errors of the coil due to these outer fields were calculated for neutrons with wavenumber 4, 1A−1.These currents correspond to a turning angle of 89, 9 and 179, 7 by the inner field of the coil respectively(calculated with Eq.(4.27)).

field 1A 2Acomp. field int.[mGscm] turn error[] field int.[mGscm] turn error[]

x 151,97 0,62 319,29 1,28y 18,53 0,08 35,78 0,15z 22,49 0,09 45,96 0,18

4.2 DESIGN 75

-1 1.5

4 6.5

9 11.5

14 16.5

19 21.5

24 26.5

29 31.5

34 36.5

39 41.5

44 46.5

49 51.5

54

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5

Bz[

Gs]

I[A]

a = 10.536+/- 0.04068b = -0.297+/- 0.00657

f(x)=a*x+b

(a) Inner field of the coil center versus the current

through the coil.

(b) Coordinate frame used to measure the outer

fields.

-1 0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

-140 -120 -100 -80 -60 -40 -20 0 20 40 60 80 100 120 140

Bx[

mG

s]

x[mm]

a1 = 88.179+/- 4.96871t1 = 16.126+/- 0.49202a2 = 115.436+/- 3.79383t2 = -14.678+/- 0.22188

f(x)=a1*exp(-x/t1)g(x)=a2*exp(-x/t2)

(c) Field Bx along the axis of the beam for 1A.

-1 0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

-140 -120 -100 -80 -60 -40 -20 0 20 40 60 80 100 120 140

Bx[

mG

s]

x[mm]

a1 = 242.361+/- 13.40190t1 = 14.269+/- 0.27157a2 = 233.530+/- 15.54531t2 = -14.583+/- 0.34010


(d) Field Bx along the axis of the beam for 2A.

0

1

2

3

4

-140 -120 -100 -80 -60 -40 -20 0 20 40 60 80 100 120 140

By[

mG

s]

x[mm]

a2 = 19.123+/- 0.66631t2 = -12.062+/- 0.16878

g(x)=a2*exp(-x/t2)

(e) Field By along the axis of the beam for 1A.

0

1

2

3

4

-140 -120 -100 -80 -60 -40 -20 0 20 40 60 80 100 120 140

By[

mG

s]

x[mm]

a2 = 34.482+/- 2.37161t2 = -12.504+/- 0.30750

g(x)=a2*exp(-x/t2)

(f) Field By along the axis of the beam for 2A.

-11

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

-140 -120 -100 -80 -60 -40 -20 0 20 40 60 80 100 120 140

Bz[

mG

s]

x[mm]

a1 = -21.904+/- 1.24209t1 = 14.620+/- 0.44894a2 = -100.055+/- 19.56282t2 = -5.497+/- 0.26703


(g) Field Bz along the axis of the beam for 1A.

-11

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

-140 -120 -100 -80 -60 -40 -20 0 20 40 60 80 100 120 140

Bz[

mG

s]

x[mm]

a1 = -44.791+/- 2.53598t1 = 14.479+/- 0.44098a2 = -202.651+/- 42.66045t2 = -5.565+/- 0.29433


(h) Field Bz along the axis of the beam for 2A.

Figure 4.18: Fields of a PC: (a) shows the inner field of the coil. (c)-(h) show the x-, y- andz-component of the outer field along the beam axis. They were fitted with a function f(x) = ai exp(− x

ti)

to calculate the outer field integral. The field in y-direction on the positive x-axis could not be fittednicely. (b) is the coordinate frame to measure the fields. The coil itself is in the x-region from x=-11...11(with wire).

76 CHAPTER 4: MUPAD

Because in inelastic measurements on a TAS the wavenumber changes afterthe neutrons have been scattered at the sample, it has to be taken intoaccount separately for the precession coils in the entrance and exit arm ofMuPAD.In all the precession coils developed within this work seem to be perfectlysuitable to turn the polarization vector with high accuracy.

Figure 4.19: One Coupling Coil of MuPAD. The small pictures show (f.l.t.r) the adiabatic entranceside, the coil inserted in the entrance (exit) of the zero field chamber and the non-adiabatic side with themu-metal yoke.

4.2 DESIGN 77

4.2.3 Coupling Coils

Thermal neutrons can not pass mu-metal shields of mm-thickness withouta serious loss in polarization, as the field strength in this material is in theTesla-range. Therefore a solution is needed to guide the polarization into(out of) the MuPAD zero field chamber. For a wide range from thermal toultracold neutrons, numerous devices have been built, to guide the neutronpolarization into an area, which is screened by mu-metal against outer fields.Experiments like NRSE-spectrometry ([Kli03],[Sta02]) rely on such devices.Neutrons penetrate through holes in the shield with their spins aligned withwell-defined compact guide fields. The transition from the outer field to theguide field (from the guide field to the outer field) is chosen as adiabatic,whereas the transition from the guide field into the screened area (from thescreened area to the guide field) is strictly non-adiabatic, in order to conservethe polarization direction into (out of) the zero field region.A picture of the guide field designed for ’MUPAD’ is shown in Fig. 4.19: Neu-trons from a high field region - normally an external guide field (10−100Gs)- enter adiabatically into a ’coupling coil’ (exit the ’coupling coil and enteradiabatically in the outer guide field), made of rectangular windings. On theentrance side (exit side), the coil wires may are bent out of the beam to as-sure a adiabatic transition into (out of) the coil. Towards the exit (entrance)of this coil the guide field gets more and more homogenous, and the influenceof the outer field disappears. With its exit (entrance) side the coupling coilpenetrates into the screened area. Here the return field of the coil is guidedby a mu-metal-yoke - similar to the one of the PCs - which is put around thatside of the coil. In this way the return field of the coil does not interact withthe neutron beam because it will be guided through the mu-metal. On thisside a non-adiabatic transition takes place which conserves the polarizationdirection into (out of) the zero field chamber.The MuPAD coupling coils have outer dimension of 50mm×50mm×200mm(H × W × D). The coil body is out of anodized aluminum to provide goodheat conduction in order to allow relatively high fields without active cool-ing. Its body is also made with grooves in order to guide the wires similarto the PCs. Here slope is on the side of the coil where the wires are bentout of the beam, which allows that for the rest of the coil the wires areall exactly parallel to the outer edges of the body. This should generatean even more homogenous field. The wire has a diameter of 0.8mm and isanodized. Wire without anodization can not be used for the coupling coilbecause of the bent part. The coil has 60 windings. The dimension for thebeam window are 38mm × 30mm (H × W ). All inner surfaces are covered

78 CHAPTER 4: MUPAD

with cadmium to avoid scattering from the coil surfaces. In order to improvethe non-adiabatic transition the mu-metal-yoke for the coupling coil was pro-duced with the same efforts as for the PCs. The yoke is out of 2mm thickmu-metal. The yoke is 100mm long and is manufactured in a way that thecoil fits exactly in it in order reduce the gap between coil and mu-metal (s.section 4.2.2).There are two different setups for the coupling coils. For the normal mode ofoperation during a measurement, the inner field of the coupling coil is alignedparallel with outer guide field which usually points in the direction perpen-dicular to the scattering plane. In this case there is no special care needed tofulfill the adiabatic condition; the coupling coil is only a prolongation of theouter guide field into the zero field chamber with a non-adiabatic transitionat the end to conserve the direction of the polarization vector. But in thesecond mode which is useful for the calibration of MuPAD (s. section 4.3) thecoil is turned by 90 around the beam axis so that its inner field is parallel tothe scattering plane in order to have the polarization vector entering the zerofield chamber in the scattering plane. In that case the adiabatic transitioninto (out of) the coil should turn the polarization vector exactly by this 90.The field strength Bcc in the coupling coil with length l needed for that casecan be determined from the adiabaticity condition γlBcc >> ωBcc (s. section3.2.2). ωBcc = π

2vl

is the angular frequency with which the field turns withinthe coupling coil for a passing neutron of velocity v. If an adiabaticity pa-rameter of 4 (s. Eq.3.20) corresponding to ≥ 96% conserved polarization (s.Table 3.1) is assumed the field strength has to be

Bcc =4π · v2γ · l . (4.32)

For a neutron with wavelength of λ = 1, 53A (k = 4.1A−1), respectivelywith velocity of v = 2586m

s, γ = 2, 913kHz

Gsand l = 0, 2m we obtain as field

for the coupling coils Bcc = 18, 5Gs, which can be achieved without addi-tional cooling. The coupling coil can be turn around the beam axis insidethe entrance (exit) of the zero field chamber.

Analyzer Coil

To measure the polarization with the help of a single crystal polarizer orpolarizing supermirror usually a flipper is installed in front of it to flip thepolarization vector by 180 (s. section 3.2.1). This kind of task can be done

4.3 CALIBRATION OF MUPAD 79

with the help of the coupling coil on the exit side of the zero field cham-ber. To do so, the polarity of the current through the coil is inverted. Butas the Coupling Coil is only a more fancy guide field this will not flip thespin. Guide fields conserve the polarization vector for parallel or antiparallelalignment with respect to the field direction (s. Eq.(3.17)). Therefore thissingle feature won’t flip the polarization vector. But in combination with anadditional circular coil, making a field of 10 − 20Gs parallel to the beam,the polarization vector can be inverted with respect to the analyzer axis ofthe polarizing analyzer by inverting polarity in the coupling coil. This sup-plementary coil is called the ’Analyzer Coil’. Due to the different polaritiesdifferent adiabatic transitions take place because the fields of analyzer coiland coupling coil superpose in different ways. Therefore one time the polar-ization vector arrives at the analyzer with parallel and the other time withantiparallel orientation with respect to the analyzer axis respectively. This isequivalent to making a π-flip. This is explained in more detail in Fig. 4.20.

4.3 Calibration of MuPAD

In order to perform measurements with MuPAD a well defined calibrationprocedure is necessary. On one hand measurements will be more reproducibleand on the other their accuracy will be enhanced by careful adjustment.

4.3.1 Mechanical Adjustment

Before tuning the parameters of MuPAD’s precession coils, it is adjustedmechanically. Mechanical calibration can be divided into two parts. Thefirst part copes with positioning of all six coils with respect to the beam.The second part is to vary the magnetic shielding of MuPAD in order toassure best guidance of polarization through the zero field chamber.

Positioning of MuPAD Coils

1. The precession coils have to be adjusted so that the inner fields of coil1 and 4 are parallel to the scattering plane and perpendicular to thebeam and the inner field of coil 2 and 3 are perpendicular to scatteringplane and to the beam. The scattering plane is assumed to be exactlyparallel to the tanzboden. It can be safely assumed that the direction

80 CHAPTER 4: MUPAD

(a) The analyzer coil is the circular coil on the left side. On the right side there is the

coupling coil inserted in the shielding. The device between them is a graphite filter. The

outer guide field is inside the analyzer housing behind the analyzer coil.

(b) In black the field lines of all coils are shown. The polarization vector is shown in red. Neutrons with

polarization vector parallel to the field of the coupling coil enter from the right side. Between coupling coil

and analyzer coil the fields of the two superpose. As on the upper part of this drawing the fields superpose

to zero because they are oriented antiparallel whereas on the lower part they superpose to a finite field the

field turns adiabatically anticlockwise from pointing downwards to pointing to the right. Between the outer

guide field and analyzer coil their two fields to zero on the lower part of the drawing whereas on the upper

part they superpose to a finite field. Therefore here the field turns adiabatically clockwise from pointing to

the right to pointing down again.

(c) As the field direction of the coupling coil is inverted the fields of coupling coil and analyzer coil

superpose to zero on the upper part whereas they superpose to a finite field magnitude on the lower part.

Therefore the adiabatic field transition from coupling coil to analyzer coil turns in the opposite direction as

in (b). But as for the guide field side the situation is the same as in (b) the polarization vector arrives with

inverted direction to that in (b). This is equivalent to making a π-flip.

Figure 4.20: Functionality of coupling coil and analyzer coil as a flipper device.


of the inner fields is exactly defined by the outer dimensions of the alu-minium coil frame, as the coil windings are well defined by the groovesin the coil body which are parallel to the outer faces of the coil withinthe standard CNC-precision of 1/100mm corresponding to ≈ 1/100.Therefore we only need to align the lower surface of each coil parallelto the tanzboden. This should lead to perfect alignment with respectto the beam within the validity of our assumptions.There are some difficulties in this procedure. The outer mu-metalscreen of the precession coil is not exactly parallel to the coil frames.Apart from that already the support of the coils should be exactlyaligned. Therefore we first align the arms with the help of levels. Thecoils are mounted inside the arms with the help of X9512 rails and car-riers. In a second step we adjust the coils on their X95 carrier. Afterthis we put the coils on their carriers inside the arm and assume thatthey are now well adjusted.

2. All polarization measurements performed with MuPAD are done withrespect to the incident polarization. As the coupling coil (CC) in thefirst arm of MuPAD defines the incident polarization, its adjustment isextremely crucial. For this adjustment we make the same assumptionas for the precession coils. Therefore we only need to align the surfacewhich is perpendicular to the assumed inner field direction absolutelyparallel to the tanzboden.This alignment is done with one precision level (accuracy 0, 0286)13

glued to one of the surfaces which should be parallel to the tanzbodenand with a second level which is glued to one of the surfaces whichshould be perpendicular to the tanzboden and parallel to the beam.The coil can be fully turned around the axis parallel to the beam andcan be slightly turned around the other two axes perpendicular to thecoils surfaces by adjustment screws. The coil should be firstly adjustedfor the position where its inner field is perpendicular to the tanzbodenso that the first level is centered. Then it should be turned by 90

and the position should be corrected until the second level is also cen-tered. This should be done iteratively do achieve the best results. Theresulting positions are marked for both cases on the instrument.

3. Since the same assumptions as above are valid for the CC in the sec-ond MuPAD arm which is used to analyze the final polarization, the

12http://www.newport.com/store/product.asp?lang=1&id=3008 andhttp://www.linos-katalog.de/

13http://www.feinewerkzeuge.de/level.htm

82 CHAPTER 4: MUPAD

adjustment for this coil is done in exactly the same manner.

4. For the alignment of the zero field chamber both coupling coils and theanalyzer coil should already guide the polarization perfectly into andout of the chamber. Therefore the optimal current should be foundfor all of them. Both CCs should be initially set to a start value ofapproximately 1A which generates a field inside of them of 12Gs. Forthe analyzer coil no general estimation can be made, as its currentdepends strongly on its distance from the coupling coil. This distanceis different for every TAS. Then a scan of final polarization over thecurrent of CC1 should be performed in direct beam geometry14 of theTAS (~ki = ~kf , scattering angle θ = 0). The current in CC1 should beset to the current value corresponding to the maximum in polarization.Afterwards the same scan should be done for CC2. As a third stepthe current through the analyzer coil should be scanned. This shouldbe repeated iteratively. Experience from NRSE technique show that inmost cases two iterations are enough ([Kli03]).

Alignment of the magnetic shielding

1. The magnetic shielding is made out of two main components: the armswhich host all coils and which move together with the arms of the TASand the shielding of the sample region. The second arm and the sam-ple shielding will move relatively to each other causing friction wherethe two parts touch each other. They should be as close as possibleto each other to provide a good magnetic connection without damag-ing themselves or handicapping the movement of the TAS. Optimumadjustment is done by roughly centering the upper and lower samplecylinder with respect to the first arm. Then the second arm, whicheasily moves on rails (s. section 4.2.1), should be softly pressed againstthe cylinder by a rubber band. The arm should be moved around thesample cylinders by moving the TAS’ second arm. Places with a lot offriction should be located and the cylinders adjusted in a way to reducethese spots to a minimum by repeating the procedure.

2. In a last step the slit between the upper and lower sample shieldingcylinders has to be adjusted. If the slit is to narrow, the neutrons maytravel through passages with too big fields disturbing the polarization.

14Note that in this geometry all three coordinate frames used before are equivalent:xi = xf = x, yi = yf = y. For the z-direction this is generally true.


To prevent this, the final polarization for each slit size should be mea-sured and the slit size with maximum polarization should be chosen forthe experiment.

4.3.2 Calibration of MuPAD precession coils

The accuracy of MuPAD is highly dependent on the precision of the angularencoding of the four precession coils. Hence, the correlation between pre-cession angle and coil current is of big importance. The inner and even thenegligible outer field of the precession coils (s. section ’Fields of PrecessionCoils’) scale linearly with the current and the precession angle is directlyproportional to the magnitude of the field. The relation between precessionangle and coil current can be generally expressed as

ϕ(I) = ΩI, (4.33)

where Ω is a proportionality factor. The experimentally measured polariza-tion is the projection of the polarization vector onto the analyzer axis (s.section 3.2.1). Scanning the final polarization via the current of a preces-sion coil with inner field perpendicular to the initial polarization vector thenresults in

Pf = P0 cos(ϕ) = P0 cos(ΩI), (4.34)

where P0 is the magnitude of the initial polarization vector15. By accuratelyrecording the final polarization over the coil current for several periods ofthe cosine we determine the proportionality factor Ω. The accuracy of Ω islimited by counting statistics and the setting accuracy of the coil current.The Kepco power supplies used for MuPAD have an accuracy of ±0, 5mA.Note that the precession angle is also dependent on the magnitude of thewavevector of passing neutrons k (s.Eq.(4.31)). We define a proportionalityfactor ω which is independent of k:

Ω(k) =ω

k. (4.35)

The imperfectness of the magnetic shielding allows parasitic magnetic fieldsto enter in the zero field chamber and to turn the polarization vector addi-tionally. Assuming that the precession angles due to the parasitic fields arevery small (because the field are also very small) the action of the different

15In realitiy P0 is smaller than the magnitude of the initial polarization due to imper-fectness of polarizing analyzer.

84 CHAPTER 4: MUPAD

components of these fields can be regarded as decoupled16. Then the effectof the parasitic fields can be taken into account as an additional phase shiftin Eq.(4.34):

Pf = P0 cos(ϕ) = P0 cos(ω

k(I + Ip)), (4.36)

where Ip17 is the current corresponding to that phase shift. With that the

searched relation between angle and current is

ϕ(I) =ω

k(I + Ip) (4.37)

With that knowledge all four precession coils can be calibrated in the directbeam geometry of the TAS.

1. The δi- and δf precession coils, which turn the polarization vectoraround the yi- and yf - axes respectively (which are the same for thedirect beam geometry), can be calibrated if the fields of both couplingcoils are oriented along the z-axis. The polarization vector is thenguided into and out of the zero field chamber parallel to the z-axis.By scanning the final polarization over the current of one of both coilsEq.(4.36) is recorded and ω and Ip can be determined for each δ-coil.

2. If the coupling coils are turned by 90 the polarization vector is con-served into and out of the chamber aligned along the yi = yf -axis.The ϕ-coils which turn the polarization around the z-axis can thereforebe calibrated by measuring Eq.(4.36) for each coil respectively in thissetup.

This calibration procedure is only as simple as this if the parasitic fieldsinside the chamber are very small (parasitic field integrals should correspondto precession angles αpara ≪ 2) and if they do not change with the scatteringangle θ. If they are dependent on θ, MuPAD has to be calibrated on eachpoint of (~Q, ω)-space corresponding to a choice θ, ki and kf . There is noeasy calibration procedure for such a situation. It is strongly dependent onthe specific situation and will therefore not be described in detail. This is ageneral experience in three dimensional polarization analysis ([Reg]).

16Generally rotations of vectors around different axes do not commutate. However, inthe case of angles α ≪ 1 commutation is well fulfilled. S. Appendix A

17Note that also Ip is dependent on k

Chapter 5

Measurements

To verify the principle of MuPAD two measurements were performed. InApril 2004 the four precession coils were tested inside the magnetic shieldingof the Neutron Resonance Spin Echo Option ZETA (Zero Field Spin Echo forThree-Axis-Spectrometer)([Kli03]). The shielding of ZETA was mounted onthe thermal TAS IN3 at the ILL ([IN3]) and the precession coils were installedinside the shielding. Within this measurement the full functionality of thecoils was demonstrated. The polarization vector could already be turned inthree dimensions. Because the zero field chamber of ZETA is not designedto have the for MuPAD required high screening factors, no measurements ona real sample were done.In June 2004 the complete setup of MuPAD, including zero field chamberand coils, was tested six days on the TAS IN22. As sample the already wellknown MnSi was chosen. The experiment was performed in cooperationwith Louis-Pierre Regnault.

5.1 Setup on TAS IN22

This first version of MuPAD was designed to fit on the Three-Axis-SpectrometerIN22 ([IN2]) at the Institut Laue Langevin in Grenoble, France. IN22 is in-stalled at the end position of the thermal supermirror guide H25. It can beequipped with a large area (140mm × 120mm) vertically focusing Heuslermonochromator and a large size (150mm × 100mm) horizontally focusingHeusler analyzer crystal for polarization analysis. Both of them allow a largeflux of polarized neutrons with a polarization of about 0.882 for 2, 662A (flux:0, 6 · 10−7cm−2s−1) and 0.866 for 4, 1A (flux: 0, 8 · 10−7cm−2s−1) neutrons.

85

86 CHAPTER 5: MEASUREMENTS

(a) The setup of IN22 (picture taken from [IN2]).

(b) MuPAD installed on IN22 Three-Axis-Spectrometer at the Institute Laue Langevin in Grenoble, France.

On the left side the monochromator housing can be seen. In the center MuPAD is mounted on the sample

table hosting the orange cryostat with the sample. On the right side analyzer housing and the detector are

shown.

Figure 5.1: IN22 and MuPAD

5.2 CALIBRATION 87

The whole instrument is fully non-magnetic including the ’Tanzboden’ onwhich the instrument is moved by air pads.The spectrometer is controlled by a DIGITAL ALPHA-STATION 200 run-ning under OPEN-VMS via a VME/OS9 electronics. The currents neededfor the two coupling coils and the four precession coils of MuPAD were sup-plied by one unipolar power supply of type Sorenson DCS 40-25 (couplingcoil on entrance side) and five bipolar power supplies of type KEPCO BOP20-10M1 (the four precession coils and the exit coupling coil). They are con-trolled via the workstation by a IEEE interface card. The KEPCO BOPs arecalibrated to set the current through the coils with an accuracy of ±0, 5mA.The currents in the precession coils for each setting were calculated by theadditional program pMuPAD (s. Appendix B). The calculated currents forMuPAD were then entered into the IN22 control.As the beam is polarized and analyzed along the negative z-axis of the anal-ysis frame on IN22 and pMuPAD was programmed to calculate the coil cur-rents for the case of production and analysis along the positive z-axis theanalysis frame was inverted:

x 7→ −x

y 7→ −y (5.1)

z 7→ −z

5.2 Calibration

After MuPAD was mounted onto IN22 it was mechanically adjusted andcalibrated using the procedure described in section 4.3. The calibration wascarried out with ki = kf = 2.662A−1. A PG-filter was installed betweensample and Heusler analyzer.The currents found for the coupling coils were 0, 8A for the entrance couplingcoil and 1A for the exit one. The optimal current through the analyzer coilwas 3A. With this setup flipping ratios between 16 and 21.5 correspondingto a polarization between 0.875 and 0.907 were achieved, which matches wellwith the maximum polarization for the Heusler crystals (s. section 5.1).The four precession coils were firstly calibrated in the direct beam. We foundthe following proportionality factors for them

1http://www.kepcopower.com/bop.htm


ωδi= 6, 713[ rad

AA]

ωϕi= 6, 744[ rad

AA]

ωϕf= 6, 827[ rad

AA]

ωδf= 6, 912[ rad

AA]

A calibration curve for the first precession coil is shown in Fig. 5.2.Further on we tried to estimate the phase shifts Ip for the coils producedby parasitic fields in the zero field chamber. In the direct beam geometry,where the situation is more clear, we found big field integrals along the x-and y-axes2: In both directions the corresponding errors of rotation were. 14. Along the z-direction the field integral corresponded to a precessionangle error of below 0, 5.Additionally these field integrals had a dependency on the scattering angle θ.This was verified by mounting a graphite sample on the sample table of IN22inside the MuPAD shielding. Field integrals where then remeasured for thetwo graphite Bragg peaks (100) and (200) corresponding to scattering anglesof 41, 20 and 89, 10 respectively. The field integral in z-direction seemed tobe independent whereas the big components in the x and y-direction3 were

-1

-0.5

0

0.5

1

-3 -2 -1 0 1 2 3

P(5

sec)

I[A]

P0 = 0.8973+/- 0.000720w = 6.7131+/- 0.002896ip = 0.0112+/- 0.000783

f(x)=P0*cos(w/2.662*(x+ip)) PC calibf(x)

Figure 5.2: The graph is a calibration curve for the δi-precession coil with ki = kf = 2.662A−1.Shown is the dependency of final polarization at the analyzer of the coil current. The red dots are themeasured values. The green doted line is a fit curve to distinguish the proportionality factors ω and thephase shift Ip (s. section 4.3).

2In direct beam geometry all coordinate frames used in section 4.1 are equivalent.

5.2 CALIBRATION 89

also dependent on the position of the TAS. But no systematic dependencywas found.The strong field integrals along the x- and y-direction can be explained by acoupling of the guide fields from the Heusler monochromator to the first Mu-PAD arm and from the second MuPAD arm to the Heusler analyzer. Thesefields point in the negative z-direction and have magnitudes of approximately20Gs at the exit of the monochromator and analyzer housing. They declineover a distance of approximately 250mm to almost zero. The MuPAD armshad only a distance of approximately 100mm to the exits of the housings.Therefore the field lines of the guide field were captured by the shielding ofthe arms. The strongest fields inside the chamber were measured at the innerside of the arms where field lines seemed to jump over the hole in the armswhich opened to the sample shielding in the x-y-plane. Over a distance ofapproximately 50 − 100mm they were in the order of 150mGs. Such a fieldintegral would give a turning error of ϕ ≈ 9, 4. This already explains themeasured values. Assuming that at the position of the arms the magnitudeof the guide fields is still about 10Gs, the field inside the arms would be10Gs/100 = 100mGs (the shielding factor of the arms is around S=100). Allthese calculations are only a rough estimation, but explain the relatively bigfields inside the zero field chamber of MuPAD.The fields inside the sample shielding were still in the region of beyond 1mGsas described in the last chapter. Therefore the parasitic fields seem to be aneffect due to a design of the arms. The situation can be improved by

• also designing a double shielding for the arms at least by a factor 10.

• designing a special guide field between the MuPAD arms and monochro-mator and analyzer respectively which avoids to magnetically saturatethe mu-metal shielding additionally.

• replacing the Heusler polarizer and analyzer with benders.

To be able to perform the measurement on the MnSi sample despite thisdifficulties, supplementary compensation coils were wound on the shieldingof the arms to compensate the influence of the guide field partially. Afterthis setup was optimized, we were able to turn the polarization vector re-producible on the two Bragg peaks of the graphite with an accuracy of lessthan 1, 5. Because of the dependency of the parasitic fields on the scatteringangle a general set of phase shift parameters Ip for the four precession coilscould not be distinguished. Therefore we set these parameters to zero be-

3Note that for a scattering angle θ 6= 0 the components x and y turn with respect tothe scattering vector ~Q.


cause correcting them on each single point was not possible within the shorttime available for the tests.

5.3 Final Measurement

5.3.1 The sample: MnSi

The Experiment was carried out on a single crystal of MnSi of large size(5cm3). MnSi has the cubic space group P213 with the lattice parametera = 4, 558A (s. Fig. 5.3(b)). The four Mn atoms are situated at (u, u, u),(1

2+ u, 1

2− u,−u), (1

2− u,−u, 1

2+ u) and (−u, 1

2+ u, 1

2− u) with u = 0.138.

This cubic structure lacks inversion symmetry.MnSi is an itinerant-electron ferromagnet with the Curie temperature Tc of29K and an ordered magnetic moment of 0.4µB on each Mn atom. For fieldslarger than 0, 6T MnSi is ferromagnetic. Its magnetic structure in absenceof any outer field is a long-period ferromagnetic spiral with the propagationvector (2π/a)(ζ, ζ ζ) with ζ = 0.017 resulting in a period of approximately180A ([RBFE02]).The phase diagram is shown in Fig. 5.3(a).

(a) Magnetic phase diagram of MnSi. Inset

shows the projection of Mn atomic positions

on a (100) plane. This atomic arrangement

lacks a center of symmetry. Graph taken from

[SCM+83].

(b) The crystal structure of MnSi. For

clarity only the Mn atoms are shown .

Picture taken from [RBFE02]

Figure 5.3: The properties of MnSi

Long-period magnetic structures like the magnetic spiral in MnSi can be

5.3 FINAL MEASUREMENT 91

explained in terms of a Dzyaloshinski-Moriya (DM) interaction (s. [Dzy58]and [Mor60]), which arises due to the noncentral arrangement of the mag-netic Mn atoms in the unit cell (s. also Fig. 5.3(b)). The superstruc-ture is caused by an instability of the ferromagnetic structure with respectto small ’relativistic’ spin-lattice or spin-spin interactions. Bak and Jensen[BJ80] demonstrated in 1980 theoretically that the noncentral arrangementof the Mn atoms leads to a free energy of the spin system containing a termiD~k( ~Sk × ~S∗

k) representing the DM interaction. D describes the strength of

the interaction and ~Sk the magnetic moments in reciprocal space. This freeenergy can be only minimized if the spin structure changes to

~S(~r) =1√2[~S~k exp(i~k~r) + ~S∗

~kexp(−i~k~r)]

=1√2[(~α~k + i~β~k) exp(i~k~r) + (~α~k − i~β~k) exp(−i~k~r)] (5.2)

~α~k ⊥ ~β~k (5.3)

|~α~k| = |~β~k| (5.4)

which describes a right (D > 0, −~k ‖ ~α~k × ~β~k) and respectively a left handed

(D < 0, ~k ‖ ~α~k × ~β~k) spiral (s. Appendix C.1). Experimental results clearlyshow that the spiral in MnSi is righthanded ([SCM+83]). The paramagnetic

space group P223 of MnSi allows four different ordering wavevectors ~k1 =k(1, 1, 1), ~k2 = k(−1,−1, 1), ~k3 = k(−1, 1,−1), and ~k4 = k(1,−1,−1) withk = 2π

aζ. Each propagation vector corresponds to a possible magnetic domain

with a magnetic spiral propagating along its direction. In an experiment witha multiple domain crystal all these four domains can be observed.

5.3.2 The Satellite Peaks

Because the MuPAD setup on IN22 does not allow to tilt the sample cryostat,the MnSi crystal was aligned before on TAS IN20 on a micro goniometerwhich could be mounted within the cryostat. All measurements were per-formed with fixed kf = 2.662A−1 at T = 1.5K and therefore in the phase wereMnSi has long range ferromagnetic spiral ordering. To prove that MuPADworks correctly we firstly remeasured the right-handedness of the magneticspiral in MnSi, already known from results of G. Shirane et al ([SCM+83]).The measurements were performed in the plane defined by the reciprocalspace directions [011] and [100]. Two of the four possible propagation vec-

tors are parallel to this plane and could be observed, namely ~k1 and ~k4. The


magnetic chiral structure of MnSi described through (5.2) leads to two mag-

netic satellites at the positions ~Q = ~τ ± ~k in reciprocal space around eachnuclear Bragg reflection at ~Q = ~τ for each propagation vector ~k:

(dσ

dΩ

)

el.

= r20

(2π)3

2v0

∑

~τ

δ3( ~Q + ~k − ~τ)

|~S⊥~k|2 − iP x0

(

Sy

⊥~kSz∗⊥~k

− Sz⊥~k

Sy∗⊥~k

)

+ r20

(2π)3

2v0

∑

~τ

δ3( ~Q − ~k − ~τ)

|~S⊥~k|2 + iP x0

(

Sy

⊥~kSz∗⊥~k

− Sz⊥~k

Sy∗⊥~k

)

,

(5.5)

where v0 is the volume of the reciprocal lattice cell, P x0 is the x-component of

the initial polarization vector with respect to the analysis frame (s. (2.18)),

and ~S⊥~k = ~Q × (~S~k × ~Q) 4. For an unpolarized neutron beam (~P0 = 0 ⇒P x

0 = 0) the intensities on both satellites are the same but for P x0 6= 0 they

are different. Left- and righthanded spirals have characteristic patterns: Forthe initial polarization vector fixed parallel to the x-direction (along ~Q),one satellite has a bigger intensity with respect to the other depending onthe handedness of the spiral. For righthanded spirals one satellite peak isbig whereas for lefthanded spirals the other one is big. By inverting thepolarization state to be antiparallel to ~Q the intensities on the satellites areexchanged for each handedness state. Polarization analysis therefore enablesus to distinguish righthanded from lefthanded spirals.We did the analysis on the satellites around the Bragg peaks [111] and [011].

The initial polarization vector was put antiparallel to the scattering vector ~Q(x-direction in analysis frame) in each case by using the precession coils in thefirst MuPAD arm. For each Bragg reflex we scanned over the four satellites.Because the resolution ellipsoid5 was big with respect to the distance of thesatellites to the Bragg reflex we were not able to do constant energy scans(ω = 0) along the propagation vector, as one usually would do, becausethen the satellites could not have been separated from the Bragg reflex6.Therefore the scans were performed parallel to the [100]-direction betweentwo satellites. The orientation of the resolution ellipsoid and the performedscans are shown in Fig. 5.4 for the [111] and in Fig. 5.5 for the [011] Bragg

4S. also (2.14). The result is derived in appendix C; see most of all section C.35Due to the beam divergence and the mosaicity of the monochromator and analyzer

crystals the intensity is not only recorded for a point in the 4D (~Q, ω)-space but for a 4Dvolume, called the resolution ellipsoid, by a TAS instrument. (s. [Dor82] and [CN67])

6In principle the situation could have been improved by using collimators. But thenthe whole TAS instrument should have been realigned. Due to lack of time we desisted todo so.


(a)The situation at the [111] Bragg peak. The blue arrows show the positions of the four magnetic

satellites due to the propagation vectors ~k1 and ~k4; red doted lines show the two constant energy scansperformed to find the position and intensity of satellites. The resolution ellipsoid is shaded in green. P isthe direction of the initial polarization.

0

5000

10000

15000

20000

25000

30000

35000

40000

45000

0.9 0.925 0.95 0.975 1 1.025 1.05 1.075 1.1

Inte

nsity

(Cnt

s pe

r 30

s)

(ζ,0.983, 0.983)(r.l.u.)

T=1.5Kkf=2.662Å−1

NSFSF

(b)Constant energy scan along the red doted line denoted by 1 (parallel to [100]-axis around the point[1 0.983 0.983])in (a). The spin flip contribution (SF) to the scattering cross-section shows the satellites

−~k1 (left) and ~k4 (right). The non-spin flip contribution shows the Bragg peak which is striked due tothe orientation of the resolution ellipsoid.

0

5000

10000

15000

20000

25000

30000

35000

40000

45000

0.9 0.925 0.95 0.975 1 1.025 1.05 1.075 1.1

Inte

nsity

(Cnt

s pe

r 30

s)

(ζ, 1.017, 1.017)(r.l.u.)

T=1.5Kkf=2.662Å−1

NSFSF

(c)Constant energy scan along the red doted line denoted by 2 (parallel to [100]-axis around the point[1 1.017 1.017])in (a). The spin flip contribution (SF) to the scattering cross-section shows the satellites~k1 (right) and −~k4 (left). The non-spin flip contribution shows the Bragg peak.

Figure 5.4: Magnetic satellites of MnSi around [111]


peak respectively. Due to the resolution ellipsoid the Bragg peak can alwaysbe seen in the recorded scans. Its position is slightly shifted because of theslope of the resolution ellipsoid. It can be separated from the satellites byseparating the cross-section into spin flip (SF) and non-spin flip (NSF) con-tributions with respect to the axis of initial polarization. Because the nuclearcontribution does not turn the polarization vector (s. Eq.(2.50=), the nu-clear contribution is only on the diagonal terms of the polarization tensor, itconserves the spin direction) its intensity is only in the NSF-channel of thecross-section. The precession coils in the second arm were used to analyze theneutrons with polarization vector parallel (SF) and antiparallel to ~Q (NSF)respectively.To prove that the the spiral structure of the measured MnSi-crystal is righthanded, the intensities on the four satellites were calculated using the mag-netic structure (5.2) and the formalism derived in appendix C. In the cal-

culations a initial polarization vector of ~P0 = (−0, 9 0 0) was used. Thecalculated and measured intensities for the [111] Bragg peak are shown inTable 5.1. The comparison of calculated and measured intensities on thesatellites shows that the satellite peaks were suppressed for a right-handedspiral. Therefore we remeasured that the spiral is righthanded. That theyare not quantitative the same is due to the non-ideal situation with the res-olution ellipsoid. This proves MuPAD works well in classical polarizationanalysis where only the projection of the final polarization vector on the in-titial one is measured. The same was done for the [011] Bragg peak (s. Table5.1). Around the [111] Bragg reflection the intensities for the two differentpropagation vectors are different whereas at the [011] they are identical. Thisis due to the geometric selection rule (s. section 2.2.4). The magnetic mo-ments of the spiral structure are perpendicular to the propagation vector.Only moments which are perpendicular to the scattering vector ~Q contributeto the scattering process. As the mutual orientation of the moments with ~Qis different for the both spirals along the two different propagation vectorsaround the [111] peak the intensities are different for each of them. But forthe two propagation vectors around [011] the scattering vector is a symmetryaxis and therefore their intensities are identical.

5.3.3 Chirality

The chiral term ir20(〈y

⊥ ~Q†z⊥ ~Q

(t)〉−〈z⊥ ~Q

†y⊥ ~Q

(t)〉) in the scattering cross-section

and final polarization vector is only non-zero if the sample possesses spiralordering of its magnetic moments.


(a)The situation at the [011] Bragg peak. The blue arrows show the positions of the four magnetic

satellites due to the propagation vectors ~k1 and ~k4; red doted lines show the two constant energy scansperformed to find the position and intensity of satellites. The resolution ellipsoid is shaded in green. P isthe direction of the initial polarization.

0

20000

40000

60000

80000

100000

−0.05 −0.025 0 0.025 0.05

Inte

nsity

(Cnt

s pe

r 70

s)

(ζ,0.983,0.983) (r.l.u.)

kf=2.662Å−1T=1.5K

↑ 550.000 Cnts NSF

NSFSF

(b)Constant energy scan along the red doted line denoted by 1 (parallel to [100]-axis around the point[0 0.983 0.983])in (a). The spin flip contribution (SF) to the scattering cross-section shows the satellites

−~k1 (left) and ~k4 (right). The non-spin flip contribution shows the Bragg peak which is striked due tothe orientation of the resolution ellipsoid.

0

20000

40000

60000

80000

100000

−0.1 −0.075 −0.05 −0.025 0 0.025 0.05 0.075 0.1

Inte

nsity

(Cnt

s pe

r 70

s)

(ζ,1.017,1.017) (r.l.u.)

kf=2.662Å−1T=1.5K

↑ 1.040.000 Cnts NSF

NSFSF

(c)Constant energy scan along the red doted line denoted by 2 (parallel to [100]-axis around the point [0

1.017 1.017])in (a). The spin flip contribution (SF) to the scattering cross-section shows the satellites ~k1

(right) and −~k4 (left). They are suppressed due to the choice of the initial polarization vector along thenegative x-axis. The non-spin flip contribution shows the Bragg peak. In the SF contribution a part ofthe Bragg peak appears. This is due to inaccurate alignment of the polarization vector antiparallel to ~Q.

Figure 5.5: Magnetic satellites of MnSi around [011]


Table 5.1: The table shows calculated and measured intensities for the four magnetic satellites aroundthe [111] and the [011] Bragg peak. The calculated intensities are in arbitrary units. They are calculated forthe righthanded structure (5.2) of MnSi. The calculated and measured intensities are in good agreement.This proves the right-handedness of the magnetic structure in MnSi.

[111]Satellite calc. Int.[A.U.] meas. Int.[Cnts/30secs] Position[r.l.u]

~k1 0,05 1.906 [1,027 1,017 1,017]

−~k1 0,95 16.659 [0,982 0,983 0,983]~k4 0,43 10.656 [1,010 0,983 0,983]

−~k4 0,13 2.673 [0,990 1,017 1,017][011]

Satellite calc. Int.[A.U.] meas. Int.[Cnts/70secs] Position[r.l.u]~k1 0,05 23.166 [0,006 1,017 1,017]

−~k1 0,78 66.956 [0,012 0,983 0,983]~k4 0,78 82.000 [-0,016 0,983 0,983]

−~k4 0,05 21.335 [-0,008 1,017 1,017]

There are two terms of the polarization tensor in Eq.(2.50), which are es-pecially suitable to distinguish the chiral term on the magnetic satellites ofMnSi. These are the two off-diagonal terms

P yx =iIz + Tchiral

σy, (5.6)

and P zx =−iIy + Tchiral

σz(5.7)

of the polarization tensor, where Iy and Iz are the y- and x component ofthe imaginary part of the nuclear-magnetic interference term and Tchiral isthe chiral term given in Table 2.1. σy and σz are the cross-sections for theinitial polarization vector along the y- and z-direction respectively. P yx andP zx correspond to analyzing the x-component of the final polarization vectorfor the initial polarization vector aligned along the y- and z-direction respec-tively. The nuclear-magnetic interference term is zero in MnSi as the nuclearand magnetic contributions appear at different positions in reciprocal space:the magnetic satellites at ~Q = ~τ ± ~k and the Bragg peak at ~Q = ~τ . Hence,measuring these two terms of the polarization tensor results in directly mea-suring the chiral term. We did so using the four MuPAD precession coils toturn the polarization vector in the proper directions.Around each of the [111] and [011] Bragg peaks a constant energy scan withzero energy transfer with these polarization analysis settings was performed.We scanned along the direction of the ~k4 propagation vector because for this


choice the mutual orientation of the resolution ellipsoid and the satellites al-lowed to separate the satellites and the Bragg reflex. The situation is shownin Fig. 5.6(a) for only the [111] reflex. It was similar for the [011] reflex. Thescan around the [111] reflex was made with initial polarization vector alongthe y-direction whereas the one around the [011] with initial polarization vec-tor along the z-direction. In both the x-component of the final polarizationvector was analyzed. The scans are shown in Fig. 5.6.In the graphs the quantity I+ − I− is plotted against the position in recip-rocal space. From section 3.2.3 we know that the polarization is measuredexperimentally like

P =I+ − I−

I+ + I− . (5.8)

With the settings polarization analysis settings above we are measuring

P =Tchiral

σy/z. (5.9)

Therefore the graphs show the quantity

I+ − I− = P (I+ + I−)

=Tchiral

σy/z(I+ + I−), (5.10)

which is directly proportional to the chiral term. This quantity was chosenfor plotting because this avoided big error bars.The measured terms show the characteristic pattern for a righthanded or-dering of the magnetic spiral. This was proved by calculating the chiralterm with the formalism in appendices C.3 and C.47. Therefore MuPADalso works when off-diagonal terms of the polarization tensor are measured.In both pictures the intensity of the Bragg peak can be seen partially in thecenter. This is due to not absolute perfect turning of the polarization vector.We will treat this in the section 5.3.6.

5.3.4 Full Polarization Analysis

We measured the full polarization tensor on the four satellites of each of thetwo Bragg reflexes [111] and [011] in order to separate the different contri-bution to the scattering cross-section.

7For the calculation one has to consider the inversion of the analysis frame on IN22; s.(5.1)


(a)The scans to measure directly the chirality were performed along the red doted line. The situation issimilar for the scan around the Bragg reflex at [011]. Scanning along this direction allowed to separatethe signal of the satellites from the Bragg signal due to the mutual orientation of resolution ellipsoid andthe satellites.

-15000

-10000

-5000

0

5000

10000

15000

0.95 0.96 0.97 0.98 0.99 1 1.01 1.02 1.03 1.04

I+ -

I- (Cnt

s pe

r 70

s)

(-ζ,ζ,ζ)(r.l.u)

MnSi Chirality in yx at [1 1 1]

T=1.5Kkf=2.662¯-1

I+ - I-

(b)Constant energy scan with zero energy transfer performed around the [111] Bragg peak with initialpolarization vector set into the y-direction and x-component of the final polarization vector analyzed.I+ − I− is directly proportional to the chiral term. The two peaks correspond to the intensity of thechiral term on the magnetic satellites. In the center the part of Bragg intensity which appears due to notperfect turning of the polarization vector through MuPAD was removed.

-100000

-50000

0

50000

100000

0.97 0.98 0.99 1 1.01 1.02 1.03

I+ -

I- (Cnt

s pe

r 70

s)

(1-ζ,ζ,ζ) (r.l.u.)

MnSi Chirality in zx at [0 1 1]

T=1.5Kkf=2.662¯-1

I+-I-

(c)Constant energy scan with zero energy transfer performed around the [011] Bragg peak with initialpolarization vector set into the y-direction and x-component of the final polarization vector analyzed.I+ − I− is directly proportional to the chiral term. The two peaks correspond to the intensity of thechiral term on the magnetic satellites. In the center the part of Bragg intensity which appears due to notperfect turning of the polarization vector through MuPAD was removed.

Figure 5.6: Chirality on magnetic satellites of MnSi around [111] and [011]


The real and imaginary part of the nuclear-magnetic interference term andthe magnetic-magnetic interference term are zero in MnSi. By assuming sowe get a system of equations out of (2.50) and (2.51) which serves to sep-arate the nuclear contribution, the magnetic contributions along the y- andz- directions and the chiral term (s. Table 2.1 for exact definitions). Theequations are

σyP yx = Tchiral, (5.11)

σzP zx = Tchiral, (5.12)

σxP xx = P0σN − P0My − P0M

z + Tchiral, (5.13)

σyP yy = P0σN + P0My − P0M

z, (5.14)

σzP zz = P0σN − P0My + P0M

z, (5.15)

σ∗ = σy = σz = σN + My + M z, (5.16)

σx = σN + My + M z − P0Tchiral, (5.17)

where P0 is the magnitude of the initial polarization vector. The solutionfor this system of linear equations of the variables σN , My, M z and Tchiral isgiven through:

σN =[

P xx(1−P0P yx)−P yx+P yy+P zz

P0+ 1

]σ∗

4= Nσ∗ (5.18)

My = [P yy − P xx(1 − P0Pyx) + P yx] σ∗

2P0= Y σ∗ (5.19)

M z = [P zz − P xx(1 − P0Pyx) + P yx] σ∗

2P0= Zσ∗ (5.20)

Tchiral = P yxσ∗ = P zxσ∗ = Tσ∗ (5.21)

The different polarization terms were extracted out of the data files of IN22and the different contributions were calculated with the help of the programSNPSoft8. The results are summarized in Table 5.2.The data show that for the two satellites, produced by the magnetic spiralalong the ~k1-propagation vector, the situation cannot be clearly interpreted.The reason for that is the unfavorable orientation and size of the resolutionellipsoid on these satellites, which we already know from the scans for thelocalization of them (s. e.g Fig. 5.5(a)). Hence, while measuring the inten-sities on these satellites, we always count additional intensities due to thenuclear contribution on the Bragg peak in the center which is also touchedby the resolution ellipsoid. This can be seen by the big counting rates for thenuclear contribution, which were distinguished. Also it seems that for each+~k1 satellite we did not measure on the right point because on them almostno magnetic contribution is found. This is because this satellite was almost

8Author: Cyrille Boullier


Table 5.2: The four different contributions of the scattering cross-section σN , My , Mz and Tchiral

were separated by full polarization analysis on the four magnetic satellites of each of the Bragg peaks [111]and [011].

S.P. [111] (~k1=[1.03 1.017 1.017], −~k1 =[0.98 0.983 0.983],~k4=[1.01 0.983 0.983], −~k4 =[0.99 1.017 1.017])

~k1 N Y Z T0, 88 ± 0, 029 0, 01 ± 0, 029 0, 01 ± 0, 029 0, 06 ± 0, 003σN [cnts/s] My[cnts/s] M z[cnts/s] Tchiral[cnts/s]

5.547 ± 161, 42 55, 5 ± 161, 42 55, 5 ± 161, 42 332, 8 ± 18, 86

−~k1 N Y Z T0, 36 ± 0, 085 0, 27 ± 0, 085 0, 27 ± 0, 085 −0, 59 ± 0, 003σN [cnts/s] My[cnts/s] M z[cnts/s] Tchiral[cnts/s]

438, 3 ± 103, 28 323, 8 ± 103, 28 326, 8 ± 103, 28 −712, 5 ± 3, 65

~k4 N Y Z T0, 01 ± 0, 025 0, 05 ± 0, 025 0, 90 ± 0, 025 −0, 50 ± 0, 005σN [cnts/s] My[cnts/s] M z[cnts/s] Tchiral[cnts/s]6, 86 ± 34.3 34, 30 ± 34.3 617, 40 ± 34.3 −343, 0 ± 3, 3

~−k4 N Y Z T0, 003 ± 0, 068 0, 17 ± 0, 068 0, 91 ± 0, 068 0, 62 ± 0, 004

σN [cnts/s] My[cnts/s] M z[cnts/s] Tchiral[cnts/s]2, 1 ± 48, 55 121, 4 ± 48, 55 649, 7 ± 48, 55 357, 0 ± 2, 86

S.P. [011] (~k1=[0.01 1.017 1.017], −~k1 =[-0.01 0.983 0.983],~k4=[0.0125 0.983 0.983], −~k4 =[-0.01 1.017 1.017])

~k1 N Y Z T1, 01 ± 0, 021 −0, 06 ± 0, 021 −0, 06 ± 0, 021 −0, 04 ± 0, 001σN [cnts/s] My[cnts/s] M z[cnts/s] Tchiral[cnts/s]

31.510, 0 ± 655, 15−1.871, 9 ± 655, 15−1.871, 9 ± 655, 15−1.247, 9 ± 31, 19

−~k1 N Y Z T0, 27 ± 0, 057 0, 40 ± 0, 057 0, 55 ± 0, 057 −0, 58 ± 0, 003σN [cnts/s] My[cnts/s] M z[cnts/s] Tchiral[cnts/s]

740, 9 ± 156, 41 1097, 6 ± 156, 41 1372, 8 ± 156, 41 −1591, 5 ± 8, 23

~k4 N Y Z T0, 006 ± 0, 037 0, 35 ± 0, 037 0, 57 ± 0, 037 −0, 89 ± 0, 002

σN [cnts/s] My[cnts/s] M z[cnts/s] Tchiral[cnts/s]9, 1 ± 56, 35 533, 0 ± 56, 35 868, 1 ± 56, 35 −1.355, 0 ± 3, 05

~−k4 N Y Z T0, 007 ± 0, 048 0, 55 ± 0, 048 0, 47 ± 0, 048 0, 48 ± 0, 003

σN [cnts/s] My[cnts/s] M z[cnts/s] Tchiral[cnts/s]10, 7 ± 73, 87 846, 5 ± 73, 87 723, 3 ± 73, 87 738, 3 ± 4, 62


completely suppressed (s. section 5.3.2) in the setup we used to locate itsposition.The situation on the ~k4-satellites is more convenient. Here the resolutionellipsoid of the TAS only enclosed the satellite. Especially the ~k4-satellitesaround the [111] Bragg peak gives good results. On this position of the TASthe situation concerning the parasitic fields in the zero field chamber seemedto be more relaxed. If the different contributions on these two satellites arecalculated with the formalism of appendix C the results given in Table 5.3are received9. They are calculated assuming that the magnitude of the initialpolarization vector is 0,9.Looking at the calculated intensities we see that the chiral contribution hasapproximately the same intensity on the both peaks but opposite sign. Thisis the case for the measured intensities on the satellites. Further on the in-tensities of the magnetic contribution along the z-axis should be equal onboth satellites. This effect can be understood by looking at the geometricselection rule (s. section 2.2.4). Due to the selection rule only componentsof the magnetic moments in the sample which are perpendicular to the scat-tering vector ~Q contribute to the scattering process. The components alongthe z-axis are perpendicular to ~Q independent of its position as the z-axis isdefined to be perpendicular to ~Q. Therefore the intensities on both satellitesare the same. For the components in the x-y-plane this is different as hereonly the projection of them on the y-axis contributes to the scattering inten-sity. As ~Q is different for both satellites despite their identical orientationthe magnetic contributions along the y-axis are different on each satellite.The measured data is in good agreement with that. The intensities of themagnetic contributions along the z-axis are identical on the two satelliteswithin their error bars. Also the ratios between the two contributions arein good agreement with the calculated data. We conclude that in situationswhere the influence of parasitic magnetic fields in the zero field chamber islow, MuPAD already shows reasonable performance.

Table 5.3: Calculated values for the four different contributions of the scattering cross-section σN ,My , Mz and Tchiral on the two magnetic satellites ~k4 and −~k4 around the Bragg peaks [111]. Theintensities are in arbitrary units. σN = 0 is assumed on the satellites.

~k4 σN [A.U.] My[A.U.] M z[A.U.] Tchiral[A.U.]0 0, 025 0, 250 −0, 159

−~k4 σN [A.U.] My[A.U.] M z[A.U.] Tchiral[A.U.]0 0, 030 0, 250 0, 174

9Also here the inversion of the analysis frame on IN22 has to be considered; s. (5.1)


5.3.5 Inelastic measurements

Figure 5.7: The magnon dispersion relation of MnSi measured at the [011] Bragg peak by Ishikawaet al in 1977 (picture taken from [ISTK77]).

MuPAD is designed to do especially inelastic measurements on a TAS instru-ment. Therefore also inelastic measurements were performed with the sameMnSi crystal. Fig. 5.7 shows the magnon dispersion relation of MnSi mea-sured at the [011] Bragg peak by Ishikawa et al in 1977 ([ISTK77]). Becauseof the short time available for the test measurement only very few inelasticmeasurements could be processed due to the low inelastic counting rates onthe crystal (1-2cnts/sec in the SF channel).We performed a constant scan energy with energy transfer ~ω = 3meV inthe surroundings of the [011] Bragg peak along [ζ 0.983 0.983] to locate theexcitation to perform a full polarization analysis on a reasonable point of itafterwards. From Fig. 5.7 we expected the magnon at [0.2 0.983 0.983] Thescan was done in classical polarization setup; the initial polarization vectorwas turned in the x-direction by the precession coils, while the NSF andSF channel were also analyzed along the x-direction. The scan is shown inFig. 5.8. As expected the SF channel shows that the maximum intensityof the magnon is at [0.19 0.983 0.983]. Surprisingly the NSF channel showsalso a peak on this point. To reveal more information about this point we


0

20

40

60

80

100

120

140

160

180

200

0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.2 0.21 0.22 0.23 0.24 0.25 0.26

Inte

nsity

(Cnt

s pe

r 17

0s)

(ζ,0.983,0.983) (r.l.u.)

kf=2.662Å−1T=1.5K

h/ω=3meV

NSFSF

Figure 5.8: Constant energy scan with energy transfer performed with ~ω = 3meV . The scanwas performed with initial polarization vector along the x-direction and the x-component of the finalpolarization vector analyzed. The SF-channel shows the magnon at q = [0.200](r.lu.). No backgroundwas subtracted. In the NSF channel a ’ghoston’ can be seen (s. Text for detail).

performed a full inelastic polarization analysis on the peak with the energytransfer ~ω = 3meV . The result is shown in Table 5.4(a) and is correctedfor background. The background was estimated by additionally performinga full polarization beside the excitation at [1.3 0.983 0.983] with the sameenergy transfer.The measured polarization matrix shows clearly that the peak at this pointcannot be of any magnetic origin. The magnitudes of the diagonal terms ofthe matrix are all identical within the errorbars. This is impossible in thecase of a non-zero magnetic contribution because then all three terms shouldbe different (s. Eq.(2.50)). Therefore the peak in the NSF channel clearly isof nuclear origin. Due to the large size of the crystal used, it is very likelya ’ghoston’, which is produced by multiple scattering of neutrons on an ex-citation and a Bragg peak afterwards or vice versa ([RRL04]). Probably itarises from the incoherent intensity of a Bragg peak due to the nuclear spinswhich is scattered a second time on the measured excitation. This wouldalso explain the decrease in polarization on the ’ghoston’. The proof of thissupposition cannot be given in this work as it goes far beyond it. Neverthe-less the polarization analysis performed on this point is useful to estimatethe accuracy of MuPAD in inelastic measurements. This will be treated insection 5.3.6.To be able to separate the different contributions on the magnon we did a


second full polarization analysis on the point [0.15 0.983 0.983] beside the’ghoston’ with the same energy transfer. The same background was used asfor the ’ghoston’ point. The results are given in Table 5.4(b). The data showthat due to low count rates and therefore bad statistics on this point theerrorbars are big. Hence, trying to separate the different terms is not reason-able with these data. Because no time was left we were unable to performmore measurements and count longer on this point10.

Table 5.4: The polarization matrix for the full inelastic polarization analysis at [0.19 0.983 0.983]and [0.15 0.983 0.983] with energy transfer of ~ω = 3meV . The data are background corrected. Thebackground was estimated on the point [1.3 0.983 0.983] for both. For the background correction weaveraged over the diagonal terms and the off-diagonal terms of the full polarization matrix measuredthere. For each term of the matrices was counted approximately 0,5h for NSF and SF channel. ~P ′ is thefinal polarization vector and ~P the initial polarization vector.

(a)[0.19 0.983 0.983]

~Px y z

x 0.54 ± 0.01975 −0.05 ± 0.02237 −0.03 ± 0.02239~P ′ y −0.03 ± 0.02308 0.56 ± 0.01886 0.04 ± 0.02219

z −0.08 ± 0.02274 −0.12 ± 0.02226 0.57 ± 0.01905

(b)[0.15 0.983 0.983]

~Px y z

x −1.1 ± 0.10686 0.05 ± 0.09601 0.23 ± 0.08388~P ′ y 0.01 ± 0.08514 −0.05 ± 0.10237 0.13 ± 0.08610

z 0.05 ± 0.09034 0.14 ± 0.09177 −0.19 ± 0.10373

5.3.6 Accuracy of MuPAD

For any examination concerning the accuracy in turning the polarizationvector, all terms of the polarization tensor should be exactly known for theconsidered point in (~Q, ω)-space. The most convenient way to test the ac-curacy of a three dimensional polarization analysis device like MuPAD isto measure the whole polarization tensor on a Bragg reflex. All contribu-tions besides the nuclear σN = 〈N ~QN †

~Q(t)〉 contribution are zero there. The

polarization tensor (2.50) then looks like:

10Already to obtain the data set given in Table 5.4 we counted approximately one hourfor each point of the matrix. As one matrix hat 9 terms and three were measured thistook already 27 hours.


Figure 5.9: As the polarization vector is not changed, when the neutron beam is scattered on a Bragg

peak, it should be still aligned along its initial direction along ~P‖. Therefore any deviation in a direction

perpendicular to the initial direction, here denoted with ~P⊥, enables us to estimate the accuracy of MuPAD

in turning the polarization vector. The error expressed as an angle of rotation is then α = arctan( P⊥P‖

).

~P ′ =

σN

σx 0 00 σN

σy 00 0 σN

σz

P x0

P y0

P z0

, (5.22)

where the cross-sections for the different initial polarization vectors are allidentical for this case:

σx = σy = σz = σN . (5.23)

Therefore the relation between final and initial polarization vector is

P ′x0

P ′y0

P ′z0

=

1 0 00 1 00 0 1

P x0

P y0

P z0

. (5.24)

Hence, the direction of the polarization vector is not changed in the scatter-ing process.We performed tests on 3 different Bragg peaks, namely the [200], [111] and[011] reflexes. The values are presented in Table 5.511. From the off-diagonalterms which are non-zero the accuracy of MuPAD can be estimated. Thisis explained in Fig. 5.9. The errors of rotation are given in Table 5.6. Thedata reveals that the accuracy of MuPAD during the experiments on MnSiwas better than / 8 in any direction. But there are also components whichshow much better accuracy up to / 1. The errors are nearly identical onthe three different Bragg peaks. The small deviations can be explained bydifferent magnitudes of the parasitic magnetic fields in the chamber for thedifferent TAS positions (s. section 5.2). From that we can conclude that theMuPAD precession coils turn the polarization vector very reliable and repro-ducible. We have to consider that due to the unfavorable mutual orientationof the resolution ellipsoid and the magnetic satellites the resolution ellipsoidprobably also intersected with the satellites belonging to the ~k1 propagationvector, when counting on the Bragg peak. Therefore some of the inaccuracyon the Bragg peak can possibly be from some additional intensity due to the

11The values are calculated from the measured values with SNPSOft written by CyrilleBoullier.


magnetic contributions on the satellites. This would explain why the termsyx and zx of the polarization tensor are non-zero (s. Table.5.5). The onlycomponent which would then still be big would be the xy-term which couldnot be explained by intensity from the satellites. Hence, only this compo-nent would lead to a real error in turning the polarization vector and theperformance of MuPAD can be even considered to be better as the figures ofthe Tables 5.5 and 5.6 demonstrate.A second measurement was performed to estimate the accuracy of MuPAD. In section 5.3.3 we measured directly the chiral terms on the two ~k4 satel-lites. The scans show that in the center between the satellites a part of theintensity of the Bragg peak appears in the terms P yx and P yz. This signi-fies that on the position of the Bragg peak a component of the polarizationvector aligned along the y- or z-direction respectively before the scatteringprocess points in x-direction after the scattering at the sample. We see fromEq.(5.24) that on a Bragg peak the y- and z-components of the initial polar-

ization vector ~P0 are never turned into the x-direction during the scattering

Table 5.5: Measured polarization matrices on the three Bragg reflexes [200],[111], [011]. ~P ′ is the final

polarization vector and ~P the initial polarization vector.

(a)[200]

~Px y z

x 0.92108 ± 0.00162 −0.05766 ± 0.00415 −0.11281 ± 0.00410~P ′ y 0.12336 ± 0.00413 0.91630 ± 0.00167 0.06285 ± 0.00413

z 0.07300 ± 0.00414 0.02352 ± 0.00417 0.90871 ± 0.00173

(b)[111]

~Px y z

x 0.90812 ± 0.00083 −0.07140 ± 0.00196 −0.10513 ± 0.00194~P ′ y 0.12211 ± 0.00194 0.90868 ± 0.00083 0.03375 ± 0.00196

z 0.06768 ± 0.00195 0.01222 ± 0.00194 0.90509 ± 0.00084

(c)[011]

~Px y z

x 0.91263 ± 0.00085 −0.07871 ± 0.00203 −0.10192 ± 0.00201~P ′ y 0.12428 ± 0.00203 0.90912 ± 0.00085 0.03349 ± 0.00204

z 0.05783 ± 0.00203 0.00486 ± 0.00202 0.90949 ± 0.00085


process. This can only be due to uncorrect turning of the polarization vectorin MuPAD. To estimate how big the error of rotation in this process is, thex-component of the final polarization vector for the initial polarization vectoralso parallel to x was measured while performing a constant energy scan withω = 0 over the [111] Bragg peak along the direction of the ~k4 propagationvector. The scan is shown in Fig. 5.10(a). Fig. 5.10(b) shows the scan ofFig. 5.6(b), but without the Bragg intensity being removed at the center.From the two pictures wee see that the intensity on the Bragg reflex wasapproximately I0 = 2.200.000

70[ cnts

sec] and that the part of the Bragg intensity

which was turned by error in the y-direction was Ierr = 150.00070

[ cntssec

]. Fromthis we can calculate a error of rotation like this:

α = arctan(Ierr

PI0

) ≈ 3, 9. (5.25)

Thus, also this test proves the good performance of the MuPAD precessioncoils.The ’ghoston’ found in the inelastic measurements at [0.19 0.983 0.983] inthe section 5.3.5 enables us to estimate the accuracy of MuPAD in inelasticmeasurements. As the three diagonal terms of the polarization matrix

Table 5.6: Angular accuracy of MuPAD in turning the polarization vector estimated on differentBragg peaks. The angular errors are calculated from the data in Table 5.5. α1 and α2 are the deviationsin the two direction perpendicular to the initial polarization vector ~P .

(a)[200]

~Px y z

α1[] 7, 7 ± 0, 246 −3, 6 ± 0, 267 −7, 1 ± 0, 254

α2[] 4, 5 ± 0, 259 1, 5 ± 0, 277 4, 0 ± 0, 268

(a)[111]

~Px y z

α1[] 7, 7 ± 0, 119 −4, 5 ± 0, 127 −6, 6 ± 0, 122

α2[] 4, 3 ± 0, 126 0, 8 ± 0, 134 2, 1 ± 0, 132

(a)[011]

~Px y z

α1[] 7, 8 ± 0, 123 −4, 9 ± 0, 129 −6, 4 ± 0, 156

α2[] 3, 6 ± 132 0, 3 ± 0, 139 2, 1 ± 0, 136


0

500000

1e+006

1.5e+006

2e+006

0.96 0.98 1 1.02 1.04

Inte

nsity

(C

nts

per

70s)

(−ζ ζ ζ)


kf=2.662Å−1T=1.5K

↑ 2.200.000 Cnts NSF

NSFSF

(b)Constant energy scan with zero energy transfer performed around the [111] Bragg peak with initialpolarization vector set into the x-direction and x-component of the final polarization vector analyzed.The NSF contribution shows the intensity of the Bragg peak of approximately 2.200.000cnts/70s. Thetwo small peaks in the SF channel are the magnetic satellites.

−160000

−140000

−120000

−100000

−80000

−60000

−40000

−20000

0

0.95 0.96 0.97 0.98 0.99 1 1.01 1.02 1.03 1.04

I+ −

I−(C

nts

per

70s)

(−ζ,ζ,ζ)(r.l.u)


T=1.5Kkf=2.662Å−1

I+ − I−

(c)The same scan as in Fig. 5.6(a) but without the Bragg intensity being removed.

Figure 5.10: Error estimation on the [111] Bragg peak.

Table 5.7: Angular accuracy of MuPAD in turning the polarization vector estimated on the spurionat [0.19 0.983 0.983]. The angular errors are calculated from the data in Table 5.4(a). α1 and α2 are the

deviations in the two direction perpendicular to the initial polarization vector ~P .

~Px y z

α1[] −3, 2 ± 4, 35 −5, 1 ± 4, 54 −3, 0 ± 4, 27

α2[] −8, 4 ± 5, 36 −12, 1 ± 5, 26 4, 0 ± 3, 77

5.4 OUTLOOK 109

measured on the ’ghoston’ are all equal whereas the off-diagonal terms arealmost zero it has clearly nuclear origin. We can estimate the accuracy ofMuPAD in inelastic measurements equivalently to the Bragg peaks in thisspecial case. The inaccuracy calculated from the polarization matrix mea-sured on the spurion (s. Table 5.4(a)) is given in Table 5.7. The data showthat even in inelastic measurements, which are more challenging than theelastic ones, MuPAD performs quite well. On two terms the turning er-rors are big(8 and 12). But for the rest of the terms the accuracy is veryreasonable (. 5), if one considers the unfavorable situation concerning theparasitic fields in the MuPAD arms.To judge these results right, one has to take into account that the calibrationfor this measurement was made on a small graphite crystal (17×25×2mm3)whereas the measurements where performed on a big MnSi crystal (5cm3).This is a quite different situation as for a bigger crystal a larger cross-sectionof the beam is used. Therefore the beam travels through more inhomo-geneous magnetic field regions which decreases the overall accuracy of theinstrument. Apart from that we have to consider that in principle MuPADcan be calibrated for a single point (~Q0, ω0) especially to reach much higheraccuracy. Therefore only a purely nuclear Bragg peak like the above used[200] in the environment of the point (~Q0, ω0) has to be used to correct theerrors of rotation. Afterwards MuPAD can be operated in the surroundingsof that point with a gain in accuracy.

5.4 Outlook

Despite the difficulties concerning the parasitic fields in the arms, Mu-PAD successfully demonstrated its performance in the test measurements onMnSi. Results already known from measurements of Shirane ([SCM+83])et al on MnSi were well reproduced. Additionally the chiral terms due tothe spiral ordering of the magnetic moments in MnSi were measured for thefirst time (to our knowledge) on off-diagonal terms of the polarization tensor.MuPAD demonstrated an accuracy of turning the polarization vector betterthan 8 in elastic measurements. On most terms the accuracy was betterthan 5 and on a few terms even better than 1. Even in the performedinelastic measurements, which are always more challenging, MuPAD provedto work reasonable. The accuracy in inelastic experiments is better than 5

degrees for most of the terms. Apart from that MuPAD could be operatedvery reliable and reproducible for all performed measurements, which enablesus to systematically calibrate it for any given point in reciprocal space. The


experimental situation for MuPAD was far from the optimum because manypossible corrections could not be used due to the very short time for the firsttests (6 days). Taking this into account the performance of MuPAD is quiteimpressive.The sources for the parasitic fields in the chamber, and therefore for theobserved inaccuracy, were located during the test (s. section 5.2) and can beremoved for a next generation MuPAD. Apart from that even some improve-ments can be done on the precession coils which worked very reliable andstable during the test measurements. Hence, a next generation MuPAD willcertainly improve to an accuracy in turning the polarization vector of lessthan 1 which will result in a very competitive option for three dimensionalpolarization analysis on Three-Axis-Spectrometers.

Chapter 6

Conclusion

The magnetic moment of the neutron make it an unique probe to reveal in-formation about microscopic magnetic properties of condensed matter. Sincethe advent of neutron scattering techniques the study of magnetic structuresand excitations has been one of its prominent domains. The use of polarizedneutrons since the late 1960’s opened new possibilities to explore magnetismand to separate the different magnetic contributions from the pure nuclearscattering. Finally, with the development of techniques for three dimensionalpolarization analysis the whole available information about the scatteringprocess could be extracted. To encode the vectorial information about themagnetic scattering potential in the polarization of the beam, the polariza-tion vector has to be manipulated in 3D upstream and downstream of thesample. This is realized with high precision Larmor precession devices. Azero field region around the sample prevents the depolarization of the neu-tron beam and therefore the loss of information. This method proved to bea very powerful tool in neutron scattering over the last years.The present work is dedicated to the construction and test of a new type ofsuch instruments, called MuPAD. It is especially adapted for inelastic threedimensional polarization analysis on a Three-Axis-Spectrometer (TAS) andwas tested on TAS IN22 at the Institut Laue Langevin, Grenoble. Design,construction, first experiments and their evaluation were the major tasks ofthis work.In the first test measurements in June 2004 MuPAD demonstrated its perfor-mance and stability. Several elastic and inelastic measurements on a MnSicrystal proved the feasibility of MuPAD for three dimensional polarizationanalysis. To our knowledge the chiral term on the magnetic satellites ofMnSi was measured for the first time on off-diagonal terms of the polar-

111

112 CHAPTER 6: CONCLUSION

ization tensor. The overall accuracy of MuPAD in turning the polarizationvector was found to be better than 5. On some terms the accuracy was evenbetter than 1. The precession coils showed impressive reproducibility andreliability.The rather moderate accuracy of 5 is due to some difficulties with the mag-netic shielding, which houses the four precession coils up- and downstreamof the sample. The magnetic shielding was affected there by the strong fieldsof the Heusler polarizer and analyzer. This led to parasitic magnetic fieldsinside the chamber which explain well the precession error of 5. A futureimproved version of MuPAD will avoid Heusler crystals but will rely on lowfield polarizers. Such a new version of MuPAD will easily reach an overallaccuracy in turning the polarization vector of less than 1.Due to this first successful demonstration, an improved version of MuPADwill be rebuilt for the FRM-II in Munich, Germany, and SINQ at the PSI inVilligen, Swiss ([Bon]). In the near future MuPAD will be an important toolto:

• study complex non-collinear magnetic structures (in diffraction).

• precisely determine form factors in antiferromagnets with several Bra-vais sublattices whose structure is described by a propagation vectork=0 (in diffraction).

• study magnetic fluctuations in superconducting materials below Tc

which requires a zero-field chamber (in diffraction and inelastics).

• to investigate ’hybrid’ correlation functions (i.e. those mixing differentmagnetic degrees of freedom or the nuclear-magnetic interference term).This is probably the most challenging task.

• be applied even to fundamental physics (e.g., electro-weak force).

Additionally MuPAD is well suited to perform classical polarization analysis,where only the diagonal terms of the polarization tensor are measured ininelastic measurements. Due to its compact design it will allow higher fluxupon the sample compared to the commonly used setup with guide fields andHelmholtz coils.

Appendix A

Commutation of rotarymatrices

In general two rotations of a vector around different axes do note commutate.This can be easily verified in terms of rotary matrices. The matrix

Rx(α) =

1 0 00 cos α sin α0 − sin α cos α

(A.1)

describes a rotation around the x-axis by α whereas the matrix

Rz(β) =

cos β sin β 0− sin β cos β 0

0 0 1

(A.2)

denotes a rotation around the z-axis by β. The two possible multiplicationsof them result in

T1 = Rx(α)Rz(β) =

cos α sin α 0− sin α cos β cos α cos β sinβsin α sin β − cos α sin β cos β

(A.3)

and

T2 = Rz(β)Rx(α) =

cos α sin α cos β sin α sin β− sin α cos α cos β sinα cos β

0 − sin β cos β

. (A.4)

113

114 CHAPTER A: COMMUTATION OF ROTARY MATRICES

Cosine and sine are defined as the real and imaginary part of the expansionof the exponential function:

cos x = 1 − x2

2!+

x4

4!− ... =

∞∑

i=0

(−1)i x2i

(2i)!(A.5)

sin x = x − x3

3!+

x5

5!− ... =

∞∑

i=0

(−1)i x2i+1

(2i + 1)!. (A.6)

For x ≪ 1 we can neglect all terms higher than the linear term. Assumingα ≪ 1 and β ≪ 1 the following approximations can be made:

T1 ≈

1 α 0−α 1 β0 −β 1

(A.7)

and

T2 ≈

1 α 0−α 1 α0 −β 1

, (A.8)

where we used that sin α sin β ≈ αβ ≈ 0 because this is a quadratic term.Within this approximation T1 and T2 are the same. Thus for α ≪ 1 andβ ≪ 1 the two rotations commutate.To get a feeling for the interval were this conclusion is valid we insert somevalues: So we assume that for angles smaller than 6 rotary matrices can be

x[] x[rad] x − sin(x) 1 − cos(x)1 0,017 8, 9 · 10−7 0,000153 0,052 2, 4 · 10−5 0.00146 0,01 0,00019 0,0055

regarded be as commutating matrices.

Appendix B

pMuPAD

pMuPAD is a small but powerful program written in Python 2.3.3 ([Pyt])which calculates the currents to be set in the four different precession coilsof MuPAD for any given incident and final polarization vector1. Python waschosen for this task due to several reasons:

• It is freely available.

• It is platform independent and runs under most operating systems.pMuPAD is tested under Microsoft Windows XP and Debian Linux.

• The control program of IN22 is being remade in Python. Thereforeit will be very easy to implement pMuPAD code for calculating thecurrents in the control program.

• MuPAD should possibly be rebuilt for the FRM-II in Munich, Germany.There Python is the standard programming language. Also there, animplementation of the source code will be more easy.

pMuPAD is a command line client which operates in the same way as theprogram MAD ([Far]) for controlling TAS instruments at the ILL. It under-stands the following basic commands:

set or se: Sets an instrument parameter. The syntax is

se par val

where par is any instrument parameter and val is the value it shouldbe set to. Different parameters can be set within one command like

1Author: Marc Janoschek. The source code is freely available. Contact: [email protected]

115

116 CHAPTER B: PMUPAD

se par1 val1 par2 val2 ... parn valn

or

se par1 val11 val12 par2 val21 val22 val32...

In the first form (parameter-value) pairs are given whereas in the sec-ond only the first parameter within one parameter group is given withseveral values. The program automatically sets the next parameter in-side that group to the next given value. All parameters and parametergroups are given in TableB.1.

print or pr: Prints the value of a parameter. The syntax is

pr par

list or ls: Lists all parameters and values ordered in parameter groups.

help or ?: Prints a simple introduction to each command similar to thisone. By typing

cmd ?

more detailed instructions to the command cmd is displayed

quit or exit: Exits the program and saves all values to a file which can bereloaded at the next start up of the program by answering the firstquestion by typing yes.

If the typed command has a syntax error the program will mark the placeof the error and give a short error message. In most cases it is extremelyhelpful to correct a command.Basically pMuPAD calculates the currents of the four precession coils (ip1...ip4)from the inverse function of Eq.(4.37)

I(ϕ) = ϕk

ω− Ip, (B.1)

where the proportionality factor ω is given for each coil by the program pa-rameter wi (i=1...4) and the phase shift Ip by pi. They should be enteredafter the calibration of MuPAD (s. section 4.3). For the wavenumber k thevalue of TASParameters ki(initial beam) or kf(scattered beam) is inserteddepending on the position of the respective coil. The angle is calculated foreach coil from ki, kf, a4 (the name of the scattering angle in MAD) andthe polarization vectors by applying the algebra presented in the section 4.1.Additionally pMuPAD is capable of calculating the angles and the corre-sponding initial and final polarization vectors from given currents. If one

117

parameter is changed all parameters depending on it will be automaticallyrecalculated. They can then be displayed with ls or pr. The command lineclient is shown in Fig. B.2.pMuPAD has one more feature called the small current correction. Inseveral cases there are two ways corresponding to two sets of angles (δ, ϕ) orcurrents (ip1, ip2) respectively, to turn the polarization vector to a certaindirection. This situation is explained in Fig.B.1. pMuPAD is programmedto automatically choose the one which needs the smaller currents, becausethis also reduces the outer return fields.During the measurement on IN22 several more powerful commands wereadded to the program to generate jobfiles which could be interpreted by thejobfile-interpreter of the control program of IN22. These commands are notyet documented in the help function. These are

makejob or mjob: Generates a complete jobfile for measuring all nineterms of the polarization tensor on the point selected by the parametersa4, ki and kf. The syntax is

mjob scan commands [jobfilename]

where scan commands is a string containing the scan commands forIN22 and jobfilename is the name of the file generated job data shouldbe written to. The last argument is optional.

makescan or mscan: Generates a jobfile for a complete scan by reading ina file with simulated scan data generated by MADSIM. This file has tobe named ’scan.dat’ and to be saved in the same folder as pMuPAD.From this file pMuPAD takes the values of ki, kf and a4 for each point tobe scanned and calculates all four currents for this configuration. Theseare then saved in a jobfile as a series of single point scans. That wasthe only way to perform scans with MuPAD because the IN22 programis not yet capable of calculating the currents for the precession coil.

mscan scan commands [jobfilename]

where scan commands is a string containing the scan commands forIN22 and jobfilename is the name of the file generated job data shouldbe written to. The last argument is optional.


Figure B.1: Two sets of angles (δ, ϕ) correspond to each position the polarization vector should beturned to. They can be transformed into each other like δ2 = −δ1 and ϕ2 = −sgn(ϕ1)π − |ϕ1|. pMuPADautomatically chooses the set with the smaller magnitudes of the angle ϕ.

119

Table B.1: Parameters of pMuPAD.

Parameter Group Paramter Unit Descripion

CoilAdjust w1 [

A] the proportionality factor of PC1

w2 [


w3 [


w4 [


p1 [

A] the phase shift of PC1

p2 [


p3 [


p4 [


CoilAngles thi [] angle δi of PC1phi [] angle ϕi of PC2phf [] angle ϕf of PC3thf [] angle δf of PC4

CoilCurrents ip1 [A] current in PC1ip2 [A] current in PC2ip3 [A] current in PC3ip4 [A] current in PC4

Polarization pix component of initial polarizationvector along x

piy component of initial polarizationvector along y

piz component of initial polarizationvector along z

pfx component of final polarizationvector along x

pfy component of final polarizationvector along y

pfz component of final polarizationvector along z

TASParameters a4 [] angle between incident and finalwavevectors

ki [A−1] incident wavenumberkf [A−1] final wavenumber


Figure B.2: The command line of pMuPAD.

Appendix C

Chiral Magnetic structures

MnSi, which was the sample for the test measurements of MuPAD, has achiral ordering of its magnetic moments. This appendix looks deeper in thedescription of such a magnetic system. Apart from that, a guide to calculatethe elastic cross-section at the satellite-peaks produced by such a structureis given.

C.1 Chiral structure in real space

It will be shown that the magnetic structure

~S~l =1√2[~S~k exp(i~k~l) + ~S∗

~kexp(−i~k~l)]

=1√2[(~α~k + i~β~k) exp(i~k~l) + (~α~k − i~β~k) exp(−i~k~l)] (C.1)

~α~k ⊥ ~β~k (C.2)

|~α~k| = |~β~k| (C.3)

~α~k × ~β~k ‖ ±~k (C.4)

found by Bak and Jensen [BJ80] in 1980 for MnSi describes a chiral ar-rangements of the magnetic moments. MnSi exhibits this structure for tem-peratures below TC = 29.5K and outer magnetic fields below B = 0.6T . ~lis a lattice vector denoting the discrete positions of the atoms carrying themagnetic moments. By looking at the more general expression

121

122 CHAPTER C: CHIRAL MAGNETIC STRUCTURES

Figure C.1: Chirality of a spiral structure; to define a chirality the thumb of the considered hand

points into direction of the propagation vector ~k and all the other fingers point in the direction of theturning sense. a) lefthanded b) righthanded c) righthanded d) lefthanded. Note that a) and d) describethe same lefthanded spiral and b) and c) the same righthanded spiral respectively.

(a) The screw is screwed into the the

nut. The screw thread describes a right

handed spiral.

(b) The same screw is screwed out of

the nut. The propagation vector and

the direction of rotation are therefore

inverted. But still this is a right handed

spiral. The spiral of the screw thread

can be described by both situations.

Figure C.2:

C.2 CHIRAL STRUCTURE IN RECIPROCAL SPACE 123

~S±~l

=1√2[(~α~k + i~β~k) exp(±i~k~l) + (~α~k − i~β~k) exp(∓i~k~l)]

=1√2[(~α~k + i~β~k)cos(~k~l) ± i sin(~k~l) + (~α~k − i~β~k)cos(~k~l) ∓ i sin(~k~l)]

=√

2(

~α~k cos(~k~l) ∓ ~β~k sin(~k~l))

(C.5)

the chirality can be identified. ~k is the propagation vector of this structureand ~α~k and ~β~k are defining the direction of a magnetic moment at the position~l of an atom. As ~α~k and ~β~k are being modulated by the cosine and sine factorsthis clearly describes a magnetic spiral. Four different cases can be identified:

a) ~S+~l

: ~k ‖ ~α~k × ~β~k lefthanded

b) ~S+~l

: −~k ‖ ~α~k × ~β~k righthanded

c) ~S−~l

: ~k ‖ ~α~k × ~β~k righthanded

d) ~S−~l

: −~k ‖ ~α~k × ~β~k lefthanded

These four cases are illustrated in Fig. C.1. Actually two of them describethe same spiral equivalently at a time. This can be understood best bylooking at a screw. A standard screw is always righthanded. But there aretwo ways to describe that right-handedness (dexterity). As someone screws

the screw into a thread (propagation vector ~k0) it turns clockwise; screwing

it out (propagation vector −~k0) it turns counterclockwise. This signifies that

inverting both the propagation direction ~k and the rotational direction atthe same time leads again to the same spiral representing exactly the samechiral structure. Fig. C.2 explains this situation. Therefore it is alreadyenough to choose one if the equations ~S±

~lto describe right- and lefthanded

spirals. The handedness can be chosen by selecting the right direction of ~kin correspondence with ~α~k × ~β~k.

C.2 Chiral structure in reciprocal space

The quantity which really matters in a magnetic neutron scattering exper-iment, is the Fourier transformation of the magnetic structure. It will be


derived here quickly. ~Q is the scattering vector in the scattering experiment.

~S±~Q

=∑

~l

~S±~l

exp(i ~Q~l) (C.6)

=~S~k

2

∑

~l

exp(i( ~Q ± ~k)~l) +~S∗

~k

2

∑

~l

exp(i( ~Q ∓ ~k)~l) (C.7)

= ~S~k

(2π)3

√2v0

∑

~τ

δ3( ~Q ± ~k − ~τ) + ~S∗~k

(2π)3

√2v0

∑

~τ

δ3( ~Q ∓ ~k − ~τ) (C.8)

The last step was made by using the identity∑

~l exp(i ~Q~l) = (2π)3

v0

∑

~τ δ3( ~Q−~τ), where ~τ is a reciprocal lattice vector and (2π)3

v0is the volume of the re-

ciprocal lattice unit cell (s. [Squ78]). Analyzing this equation it can alreadybe seen that for each of the two cases (±) there will be exactly two peaks

produced by such magnetic structure at the positions ~τ ± ~k. As every pointin reciprocal space denoted by ~τ gives a Bragg peak in the scattering cross-section those two peaks are called chiral satellites to a bragg-peak. These twosatellites exactly correspond to the two representations of one spiral struc-ture. This can be seen by choosing for example only ~S−

~Q. Then the first

peak will appear at ~Q = ~τ + ~k with a coefficient ~S~k = (~α~k + i~β~k) whereas

the second one will be at ~Q = ~τ − ~k with a coefficient ~S∗~k

= (~α~k − i~β~k). We

see that both the propagation vector (~k → −~k) and the rotational direction

((~α~k + i~β~k) → (~α~k − i~β~k)) are inverted at the second peak with respect to thefirst. In polarization analysis we can chose between the two representationsby selecting the right initial polarization.

C.3 Neutron scattering cross-section of a chi-

ral structure

This section is a guide to calculating the magnetic cross-section at the satel-lite positions in reciprocal space supposing a chiral ordering of the momentslike specified above. The quantities to be known for this are the propagationvector ~k of the structure,the handedness of the structure and the scatteringvector ~Q and the initial polarization vector.The scattering cross-section is given by (2.40). The nuclear contribution andthe nuclear-magnetic interference term are zero at the position of the satel-lites because they appear at the positions ~Q = ~τ ± ~k in reciprocal space,

C.3 NEUTRON SCATTERING CROSS-SECTION OF A CHIRAL

STRUCTURE 125

which is different from the position of the Bragg peak at ~Q = ~τ . Thus, thepartial differential cross-section on a magnetic satellite is

d2σ

dΩdE ′ =k′

k

1

2π~

∫

dt exp(−iωt)

r20〈~⊥ ~Q~†⊥ ~Q

(t)〉 − ir20~P0〈~⊥ ~Q × ~†⊥ ~Q

(t)〉

.

(C.9)

This expression is only valid in the analysis frame defined in (2.18). Becausewe look at a static structure only, we replace the correlation functions bytheir limiting values as t → ∞1:

limt→∞

〈~⊥ ~Q~†⊥ ~Q(t)〉 = 〈~⊥ ~Q〉〈~

†⊥ ~Q

〉 (C.10)

limt→∞

〈~⊥ ~Q × ~†⊥ ~Q(t)〉 = 〈~⊥ ~Q〉 × 〈~†⊥ ~Q

〉. (C.11)

The elastic partial cross-section is then:(

dσ

dΩ

)

el.

= r20

〈~⊥ ~Q〉〈~†⊥ ~Q

〉 − i ~P0〈~⊥ ~Q〉 × 〈~†⊥ ~Q〉

. (C.12)

The thermal average of the magnetic interaction vector (2.14) is then giventhrough

〈~⊥ ~Q〉 = ~Q × (~S+~Q× ~Q), (C.13)

where ~S+~Q

is the chiral structure in (C.8) and ~Q is a unit vector in direction

of ~Q. We choose the ’+’ representation for the following calculation whichmeans that ~k ‖ ~α~k×~β~k for lefthanded spirals and −~k ‖ ~α~k×~β~k for righthanded

spirals. Eq.(C.13) shows that only components of ~S+~Q

perpendicular to ~Q

contribute to the scattering process. These are the y- and z-componentswith respect to the analysis frame as its x-direction is defined to be parallel

to ~Q. Therefore if we construct ~S+~Q

in the analysis frame only its y and z

components have to be taken into account. We will define the axes of theanalysis frame like this2:

~x =~Q

| ~Q|, (C.14)

~z =~Q × ~p

| ~Q × ~p|, (C.15)

~y =~z × ~x

|~z × ~x| , (C.16)

1A~Q(t) = A~Q

(∞) + A′~Q(t), each operator is composed out of a time-dependent and a

time-independent part; s. chapter 4 of [Squ78].2this defines a righthanded coordinate frame conform with the definition in (2.18)


where ~Q is the scattering vector and ~p is a second vector parallel to thescattering plane, e.g. ~ki or ~kf . With this definition we can transform anyvector ~vsample defined in the coordinate frame of the sample into the analysisframe like

~va =

xx xy xz

yx yy yz

zx zy zz

· ~vsample (C.17)

= M ~Q · ~vsample, (C.18)

where xi and so on is the component i of ~x in the frame of the sample. Thisis basically projecting the vector ~vsample along the axes of the analysis frame.

Note that this matrix is dependent on ~Q. Therefore it is different at anypoint in reciprocal space. The geometrical situation is shown in Fig. C.3.As the x-components of the magnetic moments will not participate in thescattering process, we define a new transformation matrix which will takethis into account:

M⊥ ~Q =

0 0 0yx yy yz

zx zy zz

(C.19)

The magnetic interaction vector can then be expressed like

〈~⊥ ~Q〉 = M⊥ · ~S+~Q

= M⊥

~S~k

2

∑

~l

exp(i( ~Q + ~k)~l) +~S∗

~k

2

∑

~l

exp(i( ~Q − ~k)~l)

=~S⊥~k

2

∑

~l

exp(i( ~Q + ~k)~l) +~S∗⊥~k

2

∑

~l

exp(i( ~Q − ~k)~l), (C.20)

with ~S⊥~k = M⊥ · ~S~k.

In a last step we construct the components ~α~k and ~β~k of the magnetic mo-ments, which are defined through Eqs.(C.2)-(C.4). As both should be per-

pendicular to the propagation vector ~k, which is parallel to the scatteringplane, we define them as

~α~k = ± m√2(

~k × ~z

|~k × ~z|), (C.21)

~β~k =m√2~z, (C.22)

C.3 NEUTRON SCATTERING CROSS-SECTION OF A CHIRAL

STRUCTURE 127

where m describes the magnitude of the moments and 1√2

is a normalization

factor3. The case (+) corresponds to a antiparallel orientation (righthanded

spiral) of ~α~k × ~β~k and ~k whereas (−) corresponds to a parallel orientation(lefthanded spiral) of them. Then we have

~S~k =m√2(±(

~k × ~z

|~k × ~z|) + i~z), (C.23)

and ~S†~k

=m√2(±(

~k × ~z

|~k × ~z|) − i~z). (C.24)

Inserting ~S~k and ~S†~k

in Eq.(C.20) and the magnetic interaction vector in the

cross-section (C.12) gives the final result4

(dσdΩ

)

el.= r2

0

(M⊥ ~Q · ~S+~Q)(M⊥ ~Q · ~S+

~Q)† − i ~P0

(

(M⊥ ~Q · ~S+~Q) × (M⊥ ~Q · ~S+

~Q)†

)

(C.25)

=r20

2

∑

~l~l′

exp[

i( ~Q + ~k)(~l −~l′)]

~S⊥~k~S∗⊥~k

− i ~P0

(

~S⊥~k × ~S∗⊥~k

)

+r20

2

∑

~l~l′

exp[

i( ~Q − ~k)(~l −~l′)]

~S∗⊥~k

~S⊥~k − i ~P0

(

~S∗⊥~k

× ~S⊥~k

)

(C.26)

= r20

(2π)3

2v0

∑

~τ

δ3( ~Q + ~k − ~τ)

|~S⊥~k|2 − iP x0

(

Sy

⊥~kSz∗⊥~k

− Sz⊥~k

Sy∗⊥~k

)

+ r20

(2π)3

2v0

∑

~τ

δ3( ~Q − ~k − ~τ)

|~S⊥~k|2 + iP x0

(

Sy

⊥~kSz∗⊥~k

− Sz⊥~k

Sy∗⊥~k

)

,

(C.27)

where ~S⊥~k = M⊥ ~Q · ~S~k. From that we see that a magnetic chiral structure

gives two satellites at the positions ~Q = ~τ +~k around any Bragg peak at thereciprocal lattice position ~τ . For unpolarized neutrons (~P0 = 0) both satel-lites will have the same intensity whereas for polarized neutrons they will havedifferent intensities. By changing the direction of polarization (~P x

0 → −~P x0 )

the intensities of the both satellites can be exchanged, which is equivalent to a

3Like this ~α~k, ~β~k

and ~k are expressed in a righthanded basis.4In the first step the identities ~a × (~b + ~c) = ~a × ~b + ~a × ~c and

∑

~lexp

[

±i ~Q~l]

=

(2π)3

v0

∑

~τ δ3( ~Q − ~τ) are used. The terms including ~Sk~Sk and ~S∗

k~S∗

k cancel out because ofthis. In the last step we used that Sx

⊥~k= 0.


change between the two representations of one spiral. Due to this mechanismit can be distinguished between right- and lefthanded spirals.

C.4 The different magnetic contributions

The magnetic scattering cross-section can be separated in three differentcontributions: the magnetic contributions along the y- and z-direction re-spectively (there is no component along the the x-direction because of thegeometric selection rule) and the chiral term which are defined in Table 2.1.They are given by:

My/z = ir20

(2π)3

2v0

(

Sy/z

⊥~kS

y/z∗⊥~k

)[∑

~τ

δ3( ~Q + ~k − ~τ) +∑

~τ

δ3( ~Q − ~k − ~τ)

]

(C.28)

Tchiral = ir20

(2π)3

2v0

∑

~τ

δ3( ~Q + ~k − ~τ)(

Sy

⊥~kSz∗⊥~k

− Sz⊥~k

Sy∗⊥~k

)

−

− ir20

(2π)3

2v0

∑

~τ

δ3( ~Q − ~k − ~τ)(

Sy

⊥~kSz∗⊥~k

− Sz⊥~k

Sy∗⊥~k

)

(C.29)

Tchiral is only non-zero for a spiral structure like (C.1).

Figure C.3: The coordinate frames of the sample (qx, qy and qz) and of the analysis frame (x,y andz). The picture shows the scattering vector Q and a Bragg reflex at the lattice vector τ together with thetwo satellites denoted by the propagation vectors k and −k.

Bibliography

[BJ80] P. Bak and M. Høgh Jensen. Theory of helical magnetic structuresand phase transitions in mnsi and fege. J.Phys C: Solid St.Phys., 12:L881–5, 1980.

[Blu63] M. Blume. Polarization effects in the magnetic elastic scatteringof slow neutrons. Phys. Rev., 130:1670–1676, 1963.

[Bon] P. Boni. private communication.

[CN67] M. Cooper and R. Nathans. Acta Cryst., 23:357–367, 1967.

[CR96] N. F. CAMPAGNA and C. REBIZANT. Using caeto design magnetic shielding solutions. Technical report,http://www.integratedsoft.com/, 1996.

[Cry] http://www.ill.fr/pages/science/DPT/SampleEnvironment/cryo equip.html/.

[D11] http://www.ill.fr/YellowBook/D11/.

[Dor82] B. Dorner. Coherent Inelastic Neutron Scattering in Lattice Dy-namics. Springer Verlag, 1982.

[Dzy58] L. Dzyaloshinskii. A thermodynamic theory of weak ferromag-netism of antiferromagnets. J. Phys. Chem Solids, 4:241, 1958.

[Fan57] U. Fano. Description of states in quantum mechanics by densitymatrix and operator techniques. Rev. Mod. Phys., 29(7493), 1957.

[Far] E. Farhi. http://www.ill.fr/tas/practical/MadHelp/index.html.

[Far00] E. Farhi. Tas help pages : Graphite filter.http://www.ill.fr/tas/practical/PGFilter.html, 2000.

[GG87] R. Golub and R. Gahler. A neutron resonance spin echo spec-trometer for quasi-elastic and inelastic scattering. Phys. Lett. A,267-268(43), 1987.

129

130 BIBLIOGRAPHY

[Gut32] P. Guttinger. Das Verhalten von Atomen im magnetischenDrehfeld. Z.Phys., 73:169–184, 1932.

[Han] J. Bork Vacuumschmelze Hanau. private communication.

[Her98] Hercules course, 1998.

[IES] http://www.integratedsoft.com/.

[IN2] http://www.ill.fr/YellowBook/IN22/.

[IN3] http://www.ill.fr/YellowBook/IN3/.

[ISTK77] Y. Ishikawa, G. Shirane, J. A. Tarvin, and M. Koghi. Magneticexcitations in the weak intinerant ferromagnet mnsi. Phys. Rev.B, 16:4956–4870, 1977.

[Jan] O. Janoschek. http://www.multivitamin-graphics.de/.

[JSB+02] M. Janoschek, M. Schulz, P. Boni, R. Gahler, and S. Prokudaylo.’mupad - proposal for 3-dimensional polarisation analysis in neu-tron scattering based on mu-metal shields and conventional coils.Poster at PNCMI, Julich, Germany, 2002.

[Kel04] T. Keller. private communication, 2004.

[Kli03] S. Klimko. ZETA, A zero field spin echo method for very highresolution study of elementary excitations and first applications.PhD thesis, Technische Universtt Berlin, 2003.

[Lin03] P. Lindner. private communication, 2003.

[Lov84] S. W. Lovesey. Theory of Neutron Scattering from CondensedMatter Vol I/II. Oxford Science Publications, 1984.

[Maj03] J. Major. private communication, 2003.

[Mal99] S. V. Malayev. Nuclear-magnetic interference in the inelastic scat-tering of the polarized neutrons. Physica B, 267-268:236–242,1999.

[Mor60] T. Moriya. Anisotropic superexchange interaction and weak fer-romagnetism. Phys. Rev., 120:91, 1960.

[MRK69] R. M. Moon, T. Riste, and W.C. Koehler. Polarization analysisof thermal neutron scattering. Phys. Rev., 181(920), 1969.

BIBLIOGRAPHY 131

[NC02] M. A. Nielsen and I. L. Chuang. Quantum Computation andQuantum Information. CAMBRIDGE UNIVERSITY PRESS,2002.

[NCN] http://www.ncnr.nist.gov/index.html.

[NG7] http://www.ncnr.nist.gov/instruments/prtsans.html.

[ORVG75] A.I. Okorokov, V. V. Runov, V. I. Volkov, and A. G. Gukasov.Sov. Ohys. JETP, 42:300, 1975.

[Pro04] S. Prokudaylo. Calculations for Neutron Resonance Spin Echo.PhD thesis, Technische Universitat Munchen, 2004.

[Pyt] http://www.python.org/.

[Rai87] W. Raith. Bergmann-Schaefer: Elektro-Magnetismus, volume 2.de Gruyter, 1987.

[RBFE02] B. Roessli, P. Boni, W.E. Fischer, and Y. Endoh. Chiral fluc-tuations in mnsi above the curie temperature. Phys. Rev. Lett.,88(23), 2002.

[Reg] L. P. Regnault. private communication.

[RGF+03] L. P. Regnault, B. Geffray, P. Fouilloux, B. Longuet, F. Man-tegezza, F. Tasset, E. Lelivre-Berna, E. Bourgeat-Lami,M. Thomas, and Y. Gibert. Spherical polarimetry on the three-axis spectrometer in22. Physica B, 335:255–258, 2003.

[RGF+04] L. P. Regnault, B. Geffray, P. Fouilloux, B. Longuet, F. Man-tegazza, F. Tasset, E. Lelivre-Berna, S. Pujol, E. Bourgeat-Lami,and N. Kernavanois et al. Spherical neutron polarization analysison the three-axis spectrometer in22. Physica B, 350:E811–E814,2004.

[RRL04] H. M. Rønnow, L. P. Regnault, and J. E. Lorenzo. Chasing ghostsin reciprocal space - a novel inelastic neutron multiple scatteringprocess. Physica B, 350:11–16, 2004.

[RS79] M. Th. Rekveldt and F. J. Van Schaik. Three-dimensional neu-tron polarimeter calibration procedure. Nuclear Instruments andMethods, 166:277–288, 1979.

[Sch97] F. Schwabl. Quantenmechanik fur Fortgeschrittene. Springer Ver-lag, 1997.

132 BIBLIOGRAPHY

[SCM+83] G. Shirane, R. Cowley, C. Majkrzak, J.B. Sokoloff, B. Pagonis,C. H. Perry, and Y. Ishikawa. Spiral magnetic correlation in cubicmnsi. Physical Review B, 28(11), 1983.

[Squ78] G. L. Squires. Introduction to the theory of thermal neutron scat-tering. Dover Publications, 1978.

[Sta02] C. Stadler. Linienbreiten von Phononen - erste Messungen mit’ZETA’ am ILL, Grenoble. Master’s thesis, Technische Univer-sitat Munchen, 2002.

[STRG79] F. Van Schaik, J. De Blois T. Rekveldt, and F. De Groot. Exper-imental set-up and process program for three-dimensional time-dependent neutron depolarization measurements. Nuclear Instru-ments and Methods, 166:289–298, 1979.

[Tas89] F. Tasset. Physica B, 156-157:627, 1989.

[Tas98] F. Tasset. Calibration of the zero-field neutron polarimeter cry-opad ii. Physica B, 241-243:177–179, 1998.

[Tas99] F. Tasset. Spherical neutron polarimetry with cryopad-ii. PhysicaB, 267-268:69–74, 1999.

[TR95] F. Tasset and E. Ressouche. Optimum transmission for a 3heneutron polarizer. Nuclear Instruments and Methods in PhysicsResearch, A(359):537–541, 1995.

[VAC88] VAC. Magnetische Abschirmungen, 1988.

[Vla61] V. V. Vladimirskii. Magnetic mirrors, channels and bottels forcold neutrons. Sov. Phys. JETP, 12:740, 1961.

[YKP91] Y. B. Yildir, B. W. Klimpke, and K. M. Prasad. Three dimen-sional analysis of magnetic fields using the boundary elementmethod. Technical report, Joint EEIC/ICWA, 1991.

Acknowledgements

Many people contributed to the success of this diploma work. I would liketo thank:

Prof. Peter Boni for welcoming me at his chair E21 of the Technical Uni-versity of Munich and for giving me the opportunity to work on thisbeautiful project. Thank you for answering all my questions despitethe big spacial distance between us.

Dr. Roland Gahler for supervising me during the whole diploma work.He supported me where he was able to do so, but still trusted in mycapabilities and left me a lot of freedom during the realization of Mu-PAD. Thank you for welcoming me so warmly in Grenoble and forso many conversations, not only about physics, but about the wholespectrum of life.

Dr. Serguei Klimko for accompanying me throughout the whole projectand providing a lot of help and advice during design, construction andtesting of MuPAD. He gave me many possibilities for discussions andhelped me to get over many difficulties. Thank you for being a colleagueand friend.

Andreas Emmert for the nice working atmosphere he helped to createwithin this project. I specially want to point out how much he helpedme during the construction and later on during the test measurementsof MuPAD. He put a lot of work in helping with the design of the zerofield chamber. Thank you for the effort you put in this project. Weworked hard together but we still had a lot to laugh about together.Thank you for your superb introduction into snowboarding!

Dr. Louis-Pierre Regnault for giving me the possibility to install Mu-PAD on the CRG TAS IN22. His support and advice during the testmeasurements and later on during data treatment where very helpful.

133

134 BIBLIOGRAPHY

I am especially grateful to him for taking the time to discuss my nu-merous questions about three dimensional polarization analysis. Yourwonderful humor helped me to understand many problems more easy.

Dr. Mechthild Enderele for spending a lot of time to give me an intro-duction into magnetic scattering and polarization analysis. She helpeda lot during both - preliminary and final - measurements. Additionallyshe saved our final test measurement by prealigning our MnSi-crystalon IN20 during her lunch break. Thank you for giving me so manypossibilities for discussions.

Klaus Thillhosen for his help with many small but still important detailsof my project. Thank you very much for moral encouragement during’bad times’ and for so many advice for the practical side of life inGrenoble.

Cyrille Boullier for helping me to fight through the basics of three dimen-sional polarization analysis data treatment. His introduction to pythoninspired me a lot. I really enjoyed our basket ball matches which helpedto keep my head clear. Thank you for introducing me into the ’frenchway’ of Grenoble’s nightlife.

Dr. Bjorn Fak for spending quite some time to discuss details about MnSiwith me.

Dr. Eddy Lelievre-Berna for giving me the possibility to discuss a lotabout CryoPAD with him. Thank you for presenting MuPAD at thePNCMI 2004 in Washington D.C.

Dr. Franz Demmel for his help during the preliminary experiments onIN3.

Christiane Thillhosen for helping me with all the administrative stuff Ihad to cope with. Your friendly way really made it more than conve-nient to work with you.

Central workshop of the physics department of the TUM for produc-ing endless numbers of special parts for me. Every single part was readywithin time and all were perfectly the way I wanted them. Thanks es-pecially to Manfred Pfaller and Armin Braunschadel.

Pierre Thomas and his team for helping me very often when I needed somespecial parts which should have been ready for yesterday.

All the technicians of IN22 for there expertise help and advice before

BIBLIOGRAPHY 135

the experiments on IN22: Frederic Mantegazza, Pascal Fouilloux, Ben-jamin Longuet and Bernard Geffray.

Ian Gartshore, the technician of IN3, for making some special parts to fixmy coils inside the ZETA shielding.

My brother Oliver for designing the really awesome MuPAD Logo! Withthat dancing cow MuPAD is really the coolest option at the ILL. Thankyou for so many things that I cannot tell them all here. Thank you forproofreading this work.

Lucija for proofreading this work. Thank you for the wonderful ’coffee’breaks filled with interesting conversations.

My Family not only for supporting me during my whole life but most ofall for giving me the freedom and encouragement to realize my owndreams.

Kati for giving me a wonderful and very interesting home during the firstthree month in Grenoble. I really enjoyed staying in your house. Thankyou for so many interesting discussions about life, religion and moral.

Ruben for keeping the groove alive. You are the friend who accompaniedme through all different phases of my life. Thank you for calling alwaysat the right moments of my life.

Helena for giving me a lot of moral back up during the several difficultphases of this work. Thank you for bringing so much sunshine into mylife.

Doms for the numerous chats about physics, life and philosophy we hadthroughout our whole studies. Thanks for many sad and happy mo-ments we shared together.

All my friends in Grenoble for partying with me during the rare timesI spent outside of the institute. I’m sure without you folks makingme keep in mind another side of life this year would have been veryblue. So thanks a lot: Andi, Box, Caro, Carmen, Christian, Cecile andCyrille, Denise, Elena, Grazi, Jochen, Lucija, Marc and Marc, Maurits,Pims, Rudi, Sophie, Stefan, Thorsten, Vincent. Thank you for manyinteresting conversations and for listening to my worries.

MuPAD 3D Polarization Analysis in magnetic Neutron Scattering

Documents

Transcript of MuPAD 3D Polarization Analysis in magnetic Neutron Scattering