MuPAD 3D Polarization Analysis in magnetic Neutron Scattering

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Physik-Department MuPAD 3D Polarization Analysis in magnetic Neutron Scattering Diploma thesis submitted by Marc Janoschek September 2004 Technische Universit ¨ at M ¨ unchen

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Transcript of MuPAD 3D Polarization Analysis in magnetic Neutron Scattering

magnetic Neutron Scattering
1 Introduction 5
2 Theory of Polarization Analysis 9 2.1 Neutron scattering cross section . . . . . . . . . . . . . . . . . 9 2.2 Magnetic scattering . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2.1 The magnetic interaction potential . . . . . . . . . . . 11 2.2.2 The magnetic interaction vector . . . . . . . . . . . . . 12 2.2.3 Magnetisation of the sample . . . . . . . . . . . . . . . 13 2.2.4 The geometric selection rule . . . . . . . . . . . . . . . 13
2.3 Polarization of a neutron beam . . . . . . . . . . . . . . . . . 14 2.4 Polarization in neutron scattering experiments . . . . . . . . . 18
2.4.1 Polarized neutron scattering cross-section . . . . . . . . 18 2.4.2 Polarization of the scattered beam . . . . . . . . . . . 21 2.4.3 The polarization tensor . . . . . . . . . . . . . . . . . . 23
3 Polarization Analysis on a Three-Axis-Spectrometer 25 3.1 Three-Axis-Spectrometer (TAS) . . . . . . . . . . . . . . . . . 25 3.2 Equipping a TAS for Polarization Analysis . . . . . . . . . . . 28
3.2.1 Polarizer . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.2.2 Turning the Polarization Vector . . . . . . . . . . . . . 30 3.2.3 Polarizing analyzer . . . . . . . . . . . . . . . . . . . . 35
3.3 Setups for Polarization Analysis . . . . . . . . . . . . . . . . . 37 3.3.1 Classical polarization analysis . . . . . . . . . . . . . . 38 3.3.2 Three Dimensional Polarization Analysis . . . . . . . . 40 3.3.3 CryoPAD . . . . . . . . . . . . . . . . . . . . . . . . . 43
4 MuPAD 45 4.1 The Principle of MuPAD . . . . . . . . . . . . . . . . . . . . . 45 4.2 Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.2.1 Zero Field Chamber . . . . . . . . . . . . . . . . . . . 53
3
4.2.2 Precession coils . . . . . . . . . . . . . . . . . . . . . . 61 4.2.3 Coupling Coils . . . . . . . . . . . . . . . . . . . . . . 77
4.3 Calibration of MuPAD . . . . . . . . . . . . . . . . . . . . . . 79 4.3.1 Mechanical Adjustment . . . . . . . . . . . . . . . . . 79 4.3.2 Calibration of MuPAD precession coils . . . . . . . . . 83
5 Measurements 85 5.1 Setup on TAS IN22 . . . . . . . . . . . . . . . . . . . . . . . . 85 5.2 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 5.3 Final Measurement . . . . . . . . . . . . . . . . . . . . . . . . 90
5.3.1 The sample: MnSi . . . . . . . . . . . . . . . . . . . . 90 5.3.2 The Satellite Peaks . . . . . . . . . . . . . . . . . . . . 91 5.3.3 Chirality . . . . . . . . . . . . . . . . . . . . . . . . . . 94 5.3.4 Full Polarization Analysis . . . . . . . . . . . . . . . . 97 5.3.5 Inelastic measurements . . . . . . . . . . . . . . . . . . 102 5.3.6 Accuracy of MuPAD . . . . . . . . . . . . . . . . . . . 104
5.4 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6 Conclusion 111
B pMuPAD 115
C Chiral Magnetic structures 121 C.1 Chiral structure in real space . . . . . . . . . . . . . . . . . . 121 C.2 Chiral structure in reciprocal space . . . . . . . . . . . . . . . 123 C.3 Neutron scattering cross-section of a chiral structure . . . . . . 124 C.4 The different magnetic contributions . . . . . . . . . . . . . . 128
Bibliography 129
Acknowledgements 133
Chapter 1
Introduction
The basic properties of thermal neutrons make them a highly suitable probe to 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 scattering process which yield information on the structure of the scattering sys- tem.
• Being uncharged particles, neutrons only interact with the scattering system via nuclear forces. Therefore there is no Coulomb barrier to overcome. Neutrons can deeply penetrate in the sample. In difference to X-ray scattering the scattering lengths are no monotonic function of the atomic number. They vary strongly for neighbouring cores in the table of elements.
• Energies of thermal neutrons and of elementary excitations in con- densed matter are of the same order. By analyzing neutron energies in the scattering process, sample dynamics can be studied easily.
• Due to its magnetic moment the neutron interacts with the magnetic fields generated by the electrons of a magnetic sample. On account of this the neutron scattering cross-section does not only include nuclear contributions but also magnetic ones.
By also analyzing the spins of the scattered neutrons, usually regarded in terms of neutron polarization, it is possible to gain additional information about the scattering sample. In 1963, Blume [Blu63] derived the change of the polarization vector upon elastic scattering. This clearly showed that nuclear and magnetic contributions to the scattering cross-section, which are
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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 used in polarization dependent measurements. Central in this experimental ar- rangement is the existence of a magnetic guide-field all the way through the instrument, from the polarizing monochromator to the analyzer. This guide-field is used to adiabatically conserve the projection of the initial neu- tron polarization to the guide field in smooth rotation towards a particular direction at the sample position. The neutron spin is conserved or flipped there with regard to the quantization axis, given by the guide field due to the microscopic scattering process. The guide field conserves only the projection of of final polarization to the guide field towards the analyzer. Thus only the projections of the final polarization on the incident polarization, corre- sponding to the diagonal terms of the polarization tensor, can be measured in 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 the scattering process this means a loss of information, namely all off-diagonal terms 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 diffraction experiments. 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 and reflectometry techniques. It is based on a zero field chamber, realized through a double superconducting Meissner-shield, in which neutrons enter through a non-adiabatic field-transition which conserves the polarization. Inside the zero-field region the neutron spin is not precessing. The incident and final polarization-vector can be turned in any arbitrary direction using a total of four coils outside and between the Meissner-shields. This method is known as ’spherical neutron polarimetry’ (SNP) because the polarization vector is turned in terms of spherical coordinates. The technique of spherical neutron polarimetry (SNP) developed since 1989 at the Institut Laue Langevin (ILL), Grenoble, has proven to be a powerful tool to solve magnetic structures in elastic scattering, which where intractable before. For inelastic scattering one big advantage over conventional analysis consists in the zero-field sample environment which is important when type-II superconductors 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-
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sis. The main task of this diploma work was to built and test the new non- cryogenic polarization analysis device ’MuPAD’ (Mu-Metal Polarization Analysis Device) - an alternative setup to the existing ’CRYOPAD’ - for a Three-Axis-Spectrometer (TAS). ’MuPAD’ relies on two compact preces- sion coils with well defined field geometry and magnetic screens up and down stream 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 through the instrument. All is based on existing components from NRSE-instruments (Neutron Resonance Spin Echo) where the scattering sample is in zero field as well ([GG87],[Kli03]). The low cost of ’MUPAD’ and its ease of handling are attractive. The theory of polarization analysis is introduced and experimental tech- niques for polarization analysis on a TAS are illustrated. The principle of the experimental approach ’MuPAD’ is discussed and its construction dur- ing this work is described in detail. Field measurements and calculations of the central MuPAD precession coils are explained and analyzed. Finally the successful test measurements performed with MuPAD on a MnSi sam- ple on the thermal TAS IN22 at the ILL are presented. For the first time the chiral term of a magnetic sample was measured on off-diagonal terms of the polarization tensor. The performance of MuPAD will be analyzed and propositions 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
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 unpolarized neutrons 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 in the parallel or antiparallel state with respect to any chosen quantization axis. The angular distribution and the energy of neutrons scattered by the sample are analyzed to reveal information about the sample. In Fig.2.1 the geometry of such an experiment is shown. The measured quantity in such an experi-
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ment is the partial differential cross-section denoted by ([Lov84],[Squ78])
d2σ
ddE ′ =
number of neutrons per second with a certain final
energy between E′ and E′ + dE′ that are scattered by the

λ′,σ′
λ, σ|V † ~Q |λ′, σ′λ′, σ′|V ~Q|λ, σδ(~ω + Eλ − Eλ′),
(2.1)
where Φ[area−1time−1] is the flux of the incident neutrons, therefore the number 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 of the initial and scattered neutron wave vector respectively. ~ ~Q = ~~k − ~~k′
and ~ω are the momentum transfer and the energy transfer in the scattering process. ~Q is called the scattering vector. The states of the scattering sys- tem and neutron spin before the scattering are given through the quantum numbers λ and σ; the neutron energy is given by Eλ. The primed quantities describe the same quantities after scattering. pλ and pσ are the probabilities for the scattering system and the neutron spin for being in a specific state λ and σ. Because the beam is unpolarized the probabilities for the up and down states of the neutrons spins are equal. Therefore pσ = 1
2 for all σ. Av-
eraging over λ and σ takes into account all possible initial states before the scattering process, while summing over λ′ and σ′ considers all possible final states after the scattering process. The δ-function assures the conservation of energy during the scattering process. V ~Q is the Fourier transformation of the 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 a neutron with wave vector ~k will be scattered into the direction (θ, ) due to the interaction with the sample, represented by V ~Q, undergoing a momentum
transfer ~ ~Q and a energy transfer ~ω) and then being detected by an detector that covers a solid angle d 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 magnetic interaction 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 the scattering length and ~ri the position of the nucleus i in the sample. It contains information about the structure of the scattering sample. The magnetic part is composed out of the magnetic scattering length r0 and ~⊥ ~Q, generally called the magnetic interaction vector. The magnetic interaction vector holds information about magnetic properties of the sample. The Pauli-matrices denoted by
σx =
(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 and dynamics of samples, the magnetic interaction which is dependent on these properties of the scattering system will be examined. The magnetic interac- tion of the neutron with the sample is the interplay between the magnetic dipole moment of the neutron carried by its spin and the magnetic fields generated by the electrons of the sample. The operator corresponding to the magnetic dipole moment of the neutron is
~µn = −γµn~σ, (2.4)
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 of Eq.(2.3). The interaction of the magnetic dipole moment of the neutron with a 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)
12 CHAPTER 2: THEORY OF POLARIZATION ANALYSIS
the magnetic field produced at a point ~R from the electron is
~Bs = curl ~A, ~A = µ0

Here µB = e~
2me is 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
~p × ~R
R3 . (2.9)
at the point ~R. Because ~L = ~R × ~p is the angular momentum of the electron this field is denoted by ~BL.
The geometric relationship between ~R and the positions of the neutron ~r and of the electron ~ri is explained in Fig.2.2. In total the magnetic interaction potential is then
Vm = −µ0
where ~Ws = curl( ~s × ~R
R3 ), (2.11)
2.2.2 The magnetic interaction vector
The quantity really included in the partial differential cross-section (2.1) is the Fourier transformation of the interaction potential multiplied by (m/2π~
2). For magnetic scattering it is given by2
(m/2π~ 2)
where ~⊥ ~Q = ∑
(2.14)
is the magnetic interaction vector. r0 = 5.391 · 10−15m is a collection of all the 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 the sample 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 the magnetic interaction vector can be derived ([Squ78]):
~⊥ ~Q = ~Q × (~ ~Q × ~Q) (2.15)
~ ~Q = − 1
~M(~r) exp(i ~Q~r)d3r (2.17)
~M(~r) describes the local magnetisation of the sample at the point ~r. The magnetisation operator and therefore also the magnetic interaction vector contain vectorial information about the three dimensional distribution of the magnetic 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. The evaluation of this expression proofs that only components of ~ ~Q perpendicular
14 CHAPTER 2: THEORY OF POLARIZATION ANALYSIS
(a) The geometric selection rule: Only com-
ponents of ~~Q perpendicular to ~Q contribute
the scattering process. Further on these com-
ponents are denoted as ~ ⊥~Q
.
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). Because of that the frame of reference in polarization analysis experiments is usually chosen 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 it will 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 due to the interaction of the neutron spin and the magnetic fields of the in the sample. The neutron spins in a neutron beam are generally described by the concept of neutron polarization. Therefore we will develop a descriptive picture of neutron polarization before the theory of neutron polarization analysis 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
(2.19)
which describes the superposition of eigenstates being parallel or antiparallel to the chosen quantization axis in a two-dimensional Hilbert space H2. From now on they will be call the up and down states respectively. |a|2 and |b|2 are the probabilities of the system being in the up or the down state. Because the probability that the system is in any of these state is 1 the normalization condition
χ†χ = |a|2 + |b|2 = 1 (2.20)
has to be fulfilled. This can be generally achieved by the following choice for a and b:
a = cos θ
2 ei
(2.23)
Then Eq.(2.20) is true for any choice of θ and . With that choice we will be able to understand Eq.(2.19) in a more descriptive way.
χ = cos θ
2 ei
≡ |θ, (2.24)
|θ, 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 =
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 . By solving the eigenvalue equation for this operator, namely
(~n~σ)χ = λχ (2.26)
sin θei −(cos θ + λ)
)( u v
λ = +1 : χ+ =
2
2
(2.29)
are gained. ~n~σ projects the components of the spin of the considered system onto the unit vector ~n. Thus, the inserted eigenvalues show that for the two calculated 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, ), which originally describes the superposition of up and down states to a given general state in the made choice of eigenfunctions, also denotes the three dimensional orientation of the spin of that state in the corresponding frame of reference. Hence, the spin 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. The vector ~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 quantum computation for qubits represented by a spin-1
2 -system.([NC02])
We have to take into account that the spin is a vectorial quantity in the concept of polarization. Thus, the polarization is defined as a unit vector pointing in the direction of the neutron spin, given by the expectation value of 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 spin state ([Fan57]). The polarization vector ~P of the spin state defined as a superposition of up and down states in Eq.(2.19) is then
~P =

=


cos θ
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
~Pi =<< ~σ >>beam, (2.34)
where N is the number of neutrons in the beam and ~Pi is their corresponding polarization. Therefore the polarization is defined as a property of a neutron beam. 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 completely polarized beam | ~P0| = 1; and for a partially polarized beam 0 < | ~P0| < 1.
Figure 2.4: Bloch-sphere S2 with Bloch-vector ~n
18 CHAPTER 2: THEORY OF POLARIZATION ANALYSIS
2.4 Polarization in neutron scattering exper-
iments
In conventional neutron scattering experiments only direction and energy of the scattered neutrons are analyzed. More information about the sample is gained by additionally analyzing the spins of the neutrons. The additional information serves to separate the nuclear and the magnetic contributions to the scattering process, which superpose in unpolarized neutron scatter- ing experiments (s. Eqns. (2.1) and (2.2)). The polarization vector of the neutron beam incident on the sample is aligned along a known direction be- fore the scattering process. The change of polarization during the scattering process is analyzed afterwards. Hence, this technique is generally known as polarization analysis.
2.4.1 Polarized neutron scattering cross-section

pσσ|V † ~Q |σ′σ′|V ~Q|σ. (2.35)
The density matrix ˆ in Eq.(2.32) can be represented through
ˆ = ∑
σ
pσ|σσ|, (2.36)


~Q V ~Q ˆ) = Tr(ˆV †
~Q V ~Q)
d2σ
λ,λ′
pλTr ˆλ|V † ~Q |λ′λ′|V ~Q|λδ(~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-section for polarized neutrons is gained:
d2σ
0λ|~⊥ ~Q~†⊥ ~Q |λ +
+ r0 ~P0
N ~Q|λ ]
|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 out of four different terms. These are
• the nuclear contribution N ~QN † ~Q
which contains the pure nuclear scat-
tering,
• the magnetic contribution ~⊥ ~Q~†⊥ ~Q which contains the pure magnetic
scattering,
~Q which 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 (
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-section depends 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 problem can be solved by also analyzing the final polarization after the scattering process which reveals additional information about the sample. In a last step we formulate Eq.(2.39) in terms of correlation-functions:
20 CHAPTER 2: THEORY OF POLARIZATION ANALYSIS
d2σ
0~⊥ ~Q~†⊥ ~Q (t) +
(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 are defined by
A ~QA† ~Q (t)
= ∑
exp(−iHt/~)|λ, (2.41)
where H is the Hamiltonian of the scattering system and A†
~Q (t) = exp(iHt/~)A†
~Q in 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~
= 1
2π~
dt exp(−iωt)λ|A ~Q|λ′λ′| exp(iHt/~)A† ~Q
exp(−iHt/~)|λ

= ∑
λ,λ′
pλλ|A ~Q|λ′λ′|A† ~Q |λδ(~ω + Eλ − Eλ′). (2.43)
2.4 POLARIZATION IN NEUTRON SCATTERING EXPERIMENTS 21
In the last step the integral representation for the δ-function for conservation of energy was used:
1
2π~

dt exp{i(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 all possible initial spin states of the beam and the summing over all possible final states. The constant of proportionality is determined by normalization:
~P ′ = Tr(ˆV † ~Q ~σV ~Q)/Tr(ˆV †
~Q V ~Q). (2.46)
~P ′ d2σ
ddE ′ = k′
λ,λ′
pλTr ˆλ|V † ~Q |λ′~σλ′|V ~Q|λδ(~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 scattering process:
~P ′ d2σ
ddE ′ = k′
0 ~P0λ|~⊥ ~Q~†⊥ ~Q
|λ +
( ~P0~⊥ ~Q)|λ +
N ~Q|λ )
~Q ~⊥ ~Q|λ
|λ }
22 CHAPTER 2: THEORY OF POLARIZATION ANALYSIS
Again the several parts are expressed as correlation functions for the case of inelastic scattering:
~P ′ d2σ
ddE ′ = k′
0 ~P0~⊥ ~Q~†⊥ ~Q
(t)( ~P0~⊥ ~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 r2 0
the magnetic contribution.
lection rule!
magnetic interference term.
nuclear-magnetic interference
ical meaning
2.4.3 The polarization tensor
Before looking at the different terms of the final polarization vector it is more convenient to formulate Eq.(2.48) as a tensor equation
~P ′ = P ~P0 + ~P
(2.50)
where σi is the partial differential cross-section in Eq.(2.39) with the initial polarization pointing in direction i, where i can be x, y and z. The tensor P is called the polarization tensor. The coordinate frame in which this expression is valid, is defined in (2.18). The final polarization vector can be separated into two parts:
• P ~P0 is dependent upon the initial polarization vector ~P0. The terms contained in this part rotate the initial polarization.
• ~P is completely independent of ~P0. The terms represented by this part produce polarization even from an unpolarized beams. Samples which have properties that result in such terms are actually used to polarize an unpolarized neutron beam.
All terms in Eq.(2.50) are of the form k′
k 1
~Q (t). The
corresponding correlation-function for each term is given in Table (2.1). An important thing to recognize is that in all terms containing magnetic contri- bution there’s no x-component. This is because of the geometric selection rule. The cross-section (2.39) can also be expressed in terms of the given notation:
d2σ
ddE ′ = σN + My + M z − P x 0 Tchiral + P y
0 Ry + P z 0 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 different contributions that usually superpose in the cross-section. This can be done over 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 the microscopic magnetic properties of a sample.
24 CHAPTER 2: THEORY OF POLARIZATION ANALYSIS
Chapter 3
Polarization Analysis on a Three-Axis-Spectrometer
Various types of neutron scattering instruments use polarization analysis to obtain information upon the magnetic properties of matter. As MuPAD was especially designed to be installed on a Three-Axis-Spectrometer (TAS) for inelastic measurements, we only deal with polarization analysis on this type of instrument.
3.1 Three-Axis-Spectrometer (TAS)
The principle of Three-Axis-Spectroscopy consists in directing neutrons with a certain incident wavevector ~ki on the sample and analyzing the scattered neutrons in dependency of their wavevector ~kf . Here we measure the partial differential 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 the relationships of momentum and energy conservation during the scattering process:
~ ~Q = ~~ki − ~~kf (3.1)
f ), (3.2)
where mN is the mass of the neutron. The neutrons coming from the source have a broad wavevector distribution. Thus, the incident wave vector ~ki is
25
THREE-AXIS-SPECTROMETER
selected by first order Bragg reflection from a known single crystal monochro- mator:
~ki sin θM = 1
,
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 source is divergent only the momentum transfer ~~τ is defined exactly. Though it is possible to satisfy Eq.(3.3) for different neighboring ~ki and θM . The use of crystals with mosaicity1 as monochromator even relaxes the exact definition of ~τ and allows much higher neutron fluxes at the sample position. This is demonstrated in Fig. 3.1. The direction and therefore also the magnitude of ~ki is often defined exact enough just by the alignment of monochromator and sample. By use of collimators or diaphragms the divergence of the beam can 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 volume in reciprocal (denoted by blue shaded surface) is selected around a mean wavevector ~kI which allows much higher neutron flux upon the sample.
1A crystal consisting of several small single crystals whose crystal planes are slightly tilted against each other.
3.1 THREE-AXIS-SPECTROMETER (TAS) 27
~kf sin θA = 1
,
where, analog to the monochromator crystal, θA is the Bragg angle at the analyzer, dA and τA are the lattice spacing and the reciprocal lattice vector of the analyzer crystal, and λf is the wavelength of the scattered neutron. To suppress contamination of the beam through higher order Bragg reflection a filter (e.g. graphite) is used which scatters out a certain bandwidth of wave vectors ([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 recorded by varying ~ki and ~kf reveals the interaction potential V ~Q between the sample and the neutrons. As with a TAS instrument momentum and energy transfer can 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 analyzer crystals the intensity is not only recorded for a point in the 4D (~Q, ω)-space but for a 4D volume, called the resolution ellipsoid, by a TAS instrument. (s. [Dor82] and [CN67])
28 CHAPTER 3: POLARIZATION ANALYSIS ON A
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sis
For polarization analysis the following modifications of a TAS-instrument are required.
3.2.1 Polarizer
Because the neutrons produced in a fission reaction do not have a preferred direction for their spins, a general neutron beam is unpolarized. Hence, the neutron beam has to be polarized first before any polarization measurements can be performed. There are basically two methods which are commonly used 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 ferromagnetic thin 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 are the nuclear and magnetic scattering length, respectively. The + and − case describe the reflection angle for neutrons having spin antiparallel or parallel to the direction of magnetization of the film. For the ideal case b = p all reflected 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 for λ ≈ 4A and Fe50Co48V2 as magnetized film). This problem is usually solved by producing films out of multiple magnetic and non-magnetic layers, called supermirrors. Typical combinations of materials are
• Fe/Si
• Co/Si
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- or C-form, in order to polarize the beam completely by multiple reflection. As wafers high transparent substrate material (e.g. Si) is used, on which the supermirror is deposited by sputtering techniques. Such a device is placed downstream 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 which do have the magnetic property
~⊥ ~Q = ~ † ⊥ ~Q
. (3.6)
Such a crystal is then employed as a monochromator crystal. A magnetic field is applied in such a way that all its magnetic moments are saturated and aligned perpendicular to the scattering vector ~Q (remember the geometric selection 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-section for such a crystal is obtained:
d2σ
0λ|~2 ⊥ ~Q
as for the nuclear contribution N ~Q = N † ~Q
is always true3. The final polariza-
tion in Eq.(2.48) is then:
~P ′ d2σ
ddE ′ = k′
λ
pλ2λ|N ~Q~⊥ ~Q|λδ(~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 is absorbed and not scattered.
30 CHAPTER 3: POLARIZATION ANALYSIS ON A
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~P ′ =
λ pλ
0λ|~2 ⊥ ~Q
|λ } . (3.9)
Hence, the diffracted beam from such a single crystal is completely polarized parallel to the magnetic moments of the crystal, if there is a Bragg reflection with the property r0|~⊥ ~Q| = |N ~Q|. For example if we apply the magnetic field on the crystal in direction z (in correspondence with the analysis frame we defined for magnetic scattering in (2.18)) the magnetic interaction vector will look like
~⊥ ~Q =
This leads to the final polarization
~P ′ = 2r2
0 2
r2 0
(3.12)
The geometric situation is sketched in Fig. 3.3. There are several crystals showing 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 is necessary to turn the polarization vector in several different initial directions
3.2 EQUIPPING A TAS FOR POLARIZATION ANALYSIS 31
Figure 3.3: Polarizing the beam at a single crystal.
to get the full information about the magnetic properties of a sample from Polarization Analysis. Because its spin endows the neutron also with a magnetic moment, it will interact with an applied magnetic field. The spin will precess around the axis of the magnetic field. As the polarization vector of a neutron beam is a property of the whole ensemble of all neutron spins in the beam, it also changes under the influence of a magnetic field. There are two methods to turn the polarization with the help of a magnetic field.
Larmor precession
The change of the polarization vector under the influence of a magnetic field is quantum mechanically calculated in the Heisenberg representation where the time-dependency is expressed via operators.
OH = exp[ i
~ H(t − t0)], (3.13)
and d~σH(t)
THREE-AXIS-SPECTROMETER
where H is the Hamiltonian of the considered system and O any valid opera- tor in the Schrodinger representation. The Hamiltonian H = −~µn
~B describes the interaction between the magnetic moment of the neutron ~µn (s. Eq.(2.4))
and a time independent homogenous magnetic field ~B. The change of the polarization vector ~P = ~σ(s. Eq.(2.31)) with time is then
d~σH(t)
0 0 B ) 4
~P (t) = ~σ(t) =


=



, (3.17)

σx(0)2 + σy(0)2 and P = σz(0) are the components of the polarization vector perpendicular and parallel to the magnetic field axis 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 Larmor precession of the polarization around the axis of the magnetic field. The component 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 polarization vector 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 time the neutrons need to pass the magnetic field:
= γlB[Gs]t[sec]
= 2π · 2916[ rad
sGs ]B[Gs]
] , (3.18)
4This already solves the general case, because the frame of reference can be always turned to have the magnetic field along the z-axis.
3.2 EQUIPPING A TAS FOR POLARIZATION ANALYSIS 33
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 turn polarization. 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 beam is perfectly aligned with a magnetic field, its direction will be conserved. Therefore such a magnetic field is also called a guide field. A field whose direction turns slowly with respect to the Larmor frequency ωl can be used to guide the polarization vector into another direction. The polarization vector will just precess around the slowly changing field direction. As the field direction changes scarcely during one full precession of the polarization the polarization vector will adapt to the new direction of the field. This can be 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
dt ) ≈ 0 (3.19)
The first term can be neglected because of the very slow turning of the field whereas the second is equal to zero in view of Eq.(3.17) as ~P is parallel to ~b.
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The principle is shown in Fig.3.5. In this context two different types of field transitions are distinguished:
• ωl = γlB >> ωB: (ωB = d( ~B/| ~B|) dt
is the frequency of the rotation of the magnetic field) The change of the field is so small with respect to its magnitude (which of course affects the precession frequency ωl) that the polarization vector is able to follow the field rotation. This is called an adiabatic transition.
• ωl = γlB << ωB: The change of the field with respect to its magnitude is so big that the polarization vector can’t follow the field rotation. The polarization vector is conserved along its original direction and begins to precess around the new field direction after the transition. This is called 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 turning field (s. also [Gut32],[Vla61],[Her98]). Some values are given in Table 3.1.
E 3 4 8 15 P [%] 95 96 98 99.5
Table 3.1: Different sets of the adiabaticity parameter E and the corresponding conserved polarization along 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 turning magnetic field.
3.2 EQUIPPING A TAS FOR POLARIZATION ANALYSIS 35
3.2.3 Polarizing analyzer
After the scattering process at the sample, the final polarization can be measured 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 in the state parallel to the magnetization axis of a magnetized thin film will be reflected. Now imagine a beam of neutrons already having a polarization ~P ′ 6= 0 before the reflection at the film. Because the different components of the 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 can be measured. The magnetization of the film represents such a quantization axis 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 the spins 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|2 and |b|2 are the probabilities for the spins being in the up or down state respectively (s. also Eq.(2.19)). A detector behind the supermirror counts the reflected neutrons. In a dedicated time t = tcount it will detect only those reflected neutrons, which are in the up state. If the direction of magnetization in the film is reversed, only the down states will be reflected. Now counting for the same time tcount only the neutrons which are in the down state are detected. Therefore out of the intensities I+ and I− for the up and down channel the probabilities for the neutrons being in up and down state are quickly calculated:
|a|2 = I+
I+ + I− (3.23)
|b|2 = I−
I+ + I− (3.24)
THREE-AXIS-SPECTROMETER
Eq.(3.22) then already gives the polarization of the beam along the axis of magnetization 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 the measurement device. For practical reasons the magnetization direction of the supermirrors won’t be 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 using a magnetic field (s. section 3.2.2).This has obviously the same effect. Such a device is called a ’π-flipper’.
Single Crystal Analyzer
When using this type of polarizer as a polarization analyzer in principle everything that is true for the supermirror type is also true. But here the determination of the polarization can be seen even more easily. Again the polarization vector before the reflection of the beam on the Bragg peak of the single crystal is denoted as ~P ′. Also a π-flipper is installed in front of the crystal. Remembering the properties for a single crystal polarizer which is magnetized along the z-direction (~⊥ ~Q = ~ †
⊥ ~Q , and |N ~Q| = r0 and
~⊥ ~Q =
that Bragg peak looks like (s. Eq.(2.39))

d
± = 2r2
− are the cross-sections for flipper on and off respectively.
Measuring those two quantities also gives us the projection of polarization vector along the quantization axis:
3.3 SETUPS FOR POLARIZATION ANALYSIS 37
I+ − I−
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 direction by Larmor precession or adiabatic field transitions (s. section 3.2.2).
3. The neutrons are scattered at the sample, therefore the polarization vector 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 changed between those four steps. But in fact the situation is different. The harsh environment 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 fields in the range of several Gs,
• electromagnetic fields produced in different frequency ranges and with different amplitudes produced by power electronics.
• magnetic fields produced by sample environments e.g. like supercon- ducting magnets, which are used to magnetize magnetic samples on other instruments in the surroundings with inner fields of several Tesla.
Any of these magnetic fields (s. section 3.2.2) will turn the polarization vector. None of these field sources is known in such a way that their influence
38 CHAPTER 3: POLARIZATION ANALYSIS ON A
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could be considered in the experiments. The magnetic moment of a neutron is quite sensitive to these magnetic fields. Even small fields in the mGs-range may be enough to disturb the experiment. For example if a neutron beam with wavelength of 1A corresponding to a velocity of 3956m
s is traveling 0, 5m
(in the range of typical distances between polarizer and sample) through a homogenous field of 300mGs (magnitude of earth field!) the components of the polarization vector perpendicular to the direction of the field will be already turned by (s. Eq.(3.18))
= 0, 69rad ≈ 40
around the axis of the field. A setup for polarization analysis has to prevent the 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 have the same direction as the initial polarization of the beam generated at the polarizer. 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 of the installed field is well above the disturbing field, the polarization will be conserved in such a field. Typical magnitudes for the fields are between 10 and 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 the scattering process. As the guide fields point in the direction of the polariza- tion vector incident on the sample, all components of the final polarization vector perpendicular to that direction start precessing around the axis of the guide field and are therefore lost for the measurement. This is called par- tial depolarization as the magnitude of the polarization vector remains well above zero in this process. Fig. 3.6(a) shows this situation, which is in fact equal to that of a non-adiabatic field transition, where instead of the polar- ization vector the field is turned instantly. Altogether this signifies that the method is only suitable to analyze the component of the final polarization vector parallel to the incident polarization vector. By using adiabatic field transitions (s. section 3.2.2) at least the incident polarization vector can be turned upstream of the sample into any wanted
3.3 SETUPS FOR POLARIZATION ANALYSIS 39
(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
40 CHAPTER 3: POLARIZATION ANALYSIS ON A
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direction. But downstream of the sample still only the component of the final polarization vector parallel to the guide field direction can be turned back by adiabatic transitions into the direction of the analyzer axis (remember section 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 the one of the initial polarization. The use of adiabatic field transitions is ex- plained in Fig. 3.6(b). The adiabatic field transitions are usually realized with a combination of permanent guide fields between polarizer and sample and sample, and analyzer, and a special setup of coils around the sample axis. 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 the polarization tensor in Eq.(2.50), where the direction of the initial and final polarization vector are the same, thus the diagonal terms only:
P =

.
Hence, clearly some of the additional information which can be revealed by doing polarization analysis is not measured by this kind of setup. It was first used by Moon et al.([MRK69]). Because it was the first setup for polarization analysis it is called ’Classical Polarization Analysis’.
3.3.2 Three Dimensional Polarization Analysis
Only the diagonal terms of the polarization tensor can be measured with classical polarization analysis. This is due to the presence of guide fields in the sample region where the initial polarization vector is transformed in the scattering process. The guide field depolarizes any component of the polar- ization vector perpendicular to the direction of the guide field. But if there were no disturbing fields along the path of the polarized neutron beam no guide fields would be needed to conserve the polarization and therefore this problem would be bypassed. The disturbing fields can be removed from the beam area just by building a magnetic screen around. Such a ’zero field chamber’ is supposed to shield the 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’t turn the polarization significantly in the view of the experimental precision. For example if a neutron beam with wavelength of 1A corresponding to a
3.3 SETUPS FOR POLARIZATION ANALYSIS 41
(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
42 CHAPTER 3: POLARIZATION ANALYSIS ON A
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velocity of 3956m s
travels 2m (in the range of typical TAS instrument length) through a homogenous field of 1mGs, the components of the polarization vector perpendicular to the direction of the field will be turned by ≈ 0, 5
around the axis of the field. Therefore for such neutrons zero field would be something in the order of slightly beyond 1mGs over the whole length of the instrument. There are basically two methods to screen volumes of magnetic fields:
• The Meissner-Ochsenfeld effect of a superconductor can be used: a superconductor which is cooled beyond its critical temperature Tc - where it enters the superconducting phase - expells any magnetic field out 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 enclosed by it holds only a field which is several orders of magnitude lower than the 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 devices are generally a combination of several coils which produce well-defined ho- mogenous fields in order to turn the polarization vector with high precision by Larmor precession into any wanted direction. They have to be designed in a way that the zero field condition for the rest of the neutron flight path is not 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 loosing any component of the final polarization vector, as they are all conserved in the zero field chamber until one is selected by the Larmor precession device. Hence, the combination of a zero field chamber and high precision Larmor precession techniques enables to measure all components of the polarization tensor 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 the first Larmor precession device and measure the appropriate component of the 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 of information in magnetic neutron scattering. Because the incident and final polarization vector can be turned in any direction this technique is called ’Three Dimensional Polarization Analysis’ or ’Spherical Neutron Polarimetry’.
3.3 SETUPS FOR POLARIZATION ANALYSIS 43
3.3.3 CryoPAD
CryoPAD (Cryogenic Polarization Analysis Device) was the first setup for three dimensional polarization analysis at the ILL. It was presented by Tas- set in 1989 (s. [Tas89], [Tas98], [Tas99]). First it was only used in diffraction experiments to disentangle complicated magnetic structures. In 2003 a mod- ified CryoPAD setup (called CryoPADUM) for inelastic polarization analysis measurements 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 out of niobium. The polarization vector is manipulated in ’3D’ upstream and downstream of the sample by the combination of a nutation field (motorized rotation 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 outer Meissner screen. The setup is shown Fig.3.8. For more details we refer to the 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 about that and the topic is still open for discussion. These drawbacks are:
• The use of a Meissner screen as zero field chamber is not perfect. Any field, present before the superconducting transition takes place, is ac- tually not removed out of the enclosed volume but trapped inside of it. The only way to use the superconductor as a shielding anyway is to cool 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 magnetic field can enter in the superconductor, it will keep any magnetic field out of the enclosed volume as long as it is still in the superconduct- ing phase. During cooling down, temperature differences between the different 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 shielding beyond the critical temperature, which makes it inconvenient to operate the device.
• The device always has to be cooled down in the second shielding half a day before it is ready to be operated. Only after being cooled down it can be mounted on the TAS. Often during the transportation the field inside increases again.
44 CHAPTER 3: POLARIZATION ANALYSIS ON A
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• If the device runs out of liquid helium or nitrogen the superconducting phase breaks down and magnetic fields enter in the chamber. After- wards it needs at least half a day to cool down the chamber inside the mu-metal shielding. Therefore it has to be dismounted again and to be put 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 zero field chamber out of a mu-metal screen together with highly accurate Lar- mor precession devices also based on the use of mu-metal. Similar devices have already been built in the seventies for use on SANS and reflectometry instruments ([ORVG75], [STRG79], [RS79]). Therefore the idea was born to built such an option also for a Three-Axis-Spectrometer. This concept is named MuPAD.
(a) Scheme of CryoPAD (picture taken from [Tas99]).
(b) CRYOPADUM
Figure 3.8: CryoPAD: (a) Shows a scheme of the CryoPAD setup. (b) The nutation devices can be seen on the left and right of the niobium Meissner screen in the center.
Chapter 4
4.1 The Principle of MuPAD
The option MuPAD uses a mu-metal screen to realize a zero field chamber as discussed in section 3.3.2. Its design is described in the section 4.2. As a polarized beam can not pass a mu-metal screen with a thickness in the range of several mms without being depolarized, it has to be guided through special devices - called the coupling coils - into and out of the zero field chamber. These devices are presented later in section 4.2.3. At present we assume a perfect zero field chamber capable of maintaining the polarization. For three dimensional polarization analysis Larmor precession devices are needed to turn the polarization vector upstream and downstream of the sample arbitrarily in three dimensions. Assuming the polarized beam to en- ter in the zero field chamber with the polarization vector perpendicular to the scattering plane (this direction will be further on denoted as zi-axis), it will pass two regions of perfectly homogenous magnetic fields of well defined length along the incident beam direction (denoted as xi axis, xi ~ki). The first field is perpendicular to the beam in the scattering plane (denoted as yi-axis) whereas the second points along the zi-axis. Direction and length of the two fields are fixed, but their magnitude can be varied. These two fields will be realized by two identical coils, called the precession coils. Their special 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 are presented in section 4.2.2. The arrangement of those two coils with respect to the beam is shown in Fig.4.1.
Figure 4.1: Two precession coils with inner homogenous fields perpendicular to each other are shown. The return fields of the coils do not intersect with the beam area. This arrangement of two coils is sufficient to turn the polarization vector in three dimensions.
From Eq.(3.17) we know that the polarization vector of a neutron beam passing through such a field will be turned by a certain angle around the axis of the field depending on the magnitude of the field and the velocity of the neutrons. These angles will be called δi and i for the first and second coil 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 the angle δi around the yi-axis
Tδi =
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 i
Ti =
cos i − sin i 0 sin i cos i 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:
cos i cos δi − sin i cos i sin δi
sin i cos δi cos i sin i sin δi
− sin δi 0 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 the initial polarization vector into any arbitrary direction before the neutron beam is scattered at the sample (s. also Fig. 4.1). The same setup of precession fields is used for the scattered beam, only their sequence is exchanged. Apart from that, they have to be considered in a coordinate frame with respect to the direction of the scattered neutron beam which 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 the coordinate frame used before the scattering process will be denoted as the initial frame of reference. Thus, after being scattered at the sample the beam will pass a homogenous field pointing in the z-direction of the same length as the fields before the scattering process. The action of this field on the polarization vector is then described by the following rotary matrix:
Tf =
cos f − sin f 0 sin f cos f 0
0 0 1
. (4.4)
Afterwards it passes the second field pointing in the yf direction whose action on the polarization vector is then:
Tδf =
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 neutron beam polarized along the z-direction enters into the zero field chamber denoted by the blue volume through the incident coupling coil. The polarization vector is turned by two precession coils with homogenous fields 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). Therefore the polarization vector is not changed in the scattering process. Now the second pair of precession coils downstream of the sample is used to turn the x-component of the final polarization in the direction of the analyzer axis (z-axis). This component is guided out of the zero field chamber to the analyzer by the exit coupling 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 principle of MuPAD are shown. The one in red is the analysis frame in which the change of the polarization vector is described. The two others are denoted by indices i and f and are the ones of the beam incident on and scattered by the sample respectively. They are called initial and final frames.
4.1 THE PRINCIPLE OF MUPAD 49
sample is denoted by the matrix
Tf (δf , f ) = Tδf · Tf
=


cos f cos δf − sin f cos δf sin δf

(4.6)
This second pair of fields enables MuPAD to manipulate the polarization vector after the scattering process. The setup is sketched in Fig. 4.2 and will be discussed in more detail now. The coordinate frame for polarization analysis is defined in (2.18). It will be denoted as analysis frame in the following. To measure a single term P ij of the polarization tensor the polarization vector incident on the sample has to be turned in direction i whereas the component j of the polarization vector after the scattering process has to be analyzed and therefore to be turned in the 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 initial and final frame of reference are transformed into the analysis frame by being turned 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 considered in the process of turning the polarization vector. The whole process from the polarization vector entering into the chamber to the analysis will be examined in more detail now. The initial polarization vector aligned along the z-axis will be put along each of the axis of the analysis frame. This is done in the initial frame because the rotation matrices Tδi
and Ti
are only valid there.
x-axis: The initial polarization vector has to be turned first by δi = π/2 around the yi-axis to orient it in the scattering plane. Then by turning it by i = βi around the z-axis, (s. Fig. 4.3) it is aligned along the x-axis (s. Eq.(4.3)):
~Pi = P
cos π/2
. (4.7)
To prove that the polarization is really aligned along the x direction in the analysis frame, ~Pi will be transformed into the analysis frame by
50 CHAPTER 4: MUPAD
~P a i = P
cos(−βi) − sin(−βi) 0 sin(−βi) cos(−βi) 0
0 0 1
. (4.8)
y-axis: The initial polarization vector first has to be turned by δi = π/2 around the yi-axis to orient it in the scattering plane. Then by turning it by i = βi + π/2 around the z-axis, it is aligned along the y-axis:
~Pi = P
cos π/2
~P a i = P
cos(−βi) − sin(−βi) 0 sin(−βi) cos(−βi) 0
0 0 1
sin2 βi + cos2 βi
. (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 chamber and 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 scattering process into a final polarization vector depending on the sample properties and the direction of the incident polarization vector. We assume the initial polarization vector in the analysis frame will be transformed into the final polarization
~P a f =
4.1 THE PRINCIPLE OF MUPAD 51
where a, b and c are the components of the final polarization in x, y and z direction of the analysis frame respectively. Each component of (4.11) has to be analyzed. Because the rotation matrix Tf (δf , f ) is only valid in the final
frame we express ~P a f in terms of it by turning the coordinate frame by −βf
around the z-axis:
cos(−βf ) − sin(−βf ) 0 sin(−βf ) cos(−βf ) 0
0 0 1
c
Now each component of the final polarization vector is analyzed:
x-component: To analyze the x-component we have to turn ~P a f inside the
scattering plane first by f = βf and then turn the component which is now along the xf axis by δf = −π/2 into the direction of the z-axis which 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



=


. (4.13)
As only the component aligned along the z-axis will be guided out of the zero field chamber by the exit coupling coil to the polarization analyzer, that choice of f 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 a f inside
the scattering plane by f = βf − π/2 and then turn the component which 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



=


. (4.14)
Only the z-component of this is analyzed, thus this choice of f and δf
corresponds to analyzing the y-component of ~P a f .
z-component: Nothing at all has to be done , since the z-component of ~P a f
is already aligned along the analyzer axis, namely the z-axis.
Hence, we showed that with the combination of two precession coils upstream and downstream of the sample respectively, all terms P ij of the polarization tensor (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 be calculated. Then for each of the three components of the polarization vector incident on the sample (corresponding to the choices derived above for the angles δi and i) the three components of the final polarization vector are analyzed by adjusting the precession fields to the corresponding angles δf
and f . This results in nine measurements for each point in ( ~Q, ω)-space to be made. As the precession fields are produced by coils the adjustment of the field is achieved by simply setting the right current in the coils. Therefore the whole principle 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
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
new curce Hysteresis
Figure 4.4: The hysteresis and new curve of mu-metal measured with a sample in form of stamped rings of dimensions 28, 5mm × 20mm − 0, 8mm ([Han]).
The zero field chamber of MuPAD is made out of mu-metal. Mu-metal is an alloy out of 77% Ni, 16% Fe, 5% Cu, 2% Co. It is a material with a very high relative permeability µr. As it is ferromagnetic µr, depends on the magnetic flux through the material. The hysteresis curve is shown in Fig. 4.4. For screening magnetic fields from several mGs up to several hundred Gs the mu-metal can be regarded as linear with µr ≈ 30.000. After perfect heat treatment, µr can reach values near 70.000. Minor mechanical stress however, will reduce µr and the value of 30.000 is more realistic for well treated 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 the magnetic conductor and A is its cross-section. Thus it is energetically of advantage for a magnetic field to float through a material with a high relative permeability. Therefore a volume which is enclosed by a highly permeable material, e.g. mu-metal, will be screened from outer fields because the field lines are deformed by the presence of the material in a way that the majority of 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 significantly compensate the effect of the higher µr. Fig. 4.5 demonstrates the shielding of 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 for screening factors are . 100 for a single layer of mu-metal depending on the size and geometry of the volume to be screened. Apart from that multiple layers 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 should be screened. In the right picture the volume to be screened is enclosed by a highly permeable material with µr >> 103. The magnetic field lines are deformed so that they travel through the screen with the much lower reluctance (s. Eq.(4.15)). Therefore the magnitude of the field Bi inside the volume is now much 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. The sample shielding consists of two separate parts: an upper and a lower double cylinder out of mu-metal with a slit in between them which can be varied from 40 to 100mm centered on the beam height. The slit allows the neutron beam to enter the shielding without loss of polarization. A single double cylinder with only one entrance and one exit hole is not possible as MuPAD is operated on a TAS where the scattering angle varies with the selected point in ( ~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 fixed between them for both - inner and outer - cylinders at a ’blind spot’ where the beam is never analyzed. Supplementary ten 15-sections serve as movable lamellas to establish a magnetic connection at all other places apart from the one where the beam exits. They are planned to be moved automatically in and 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 rubber bands (s. Fig. 4.7). The analyzer arm of the TAS can be moved around the sample axis by −30 ≤ θS ≤ 110 when MuPAD is mounted. The two lower cylinders have a closed bottom whereas the upper ones are open in order to be able to insert the sample cryostat. Good magnetic screening is provided in 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]):
+ 1, (4.18)
where St and Sl are the screening factors transverse and longitudinal to the cylinder axis respectively. d is the thickness of the cylinder wall, D its diameter and L its length. N is the ’demagnetization factor’ which depends on the fraction L/D (s. also [VAC88]). For a double cylinder there’s a multiplying effect for the transverse screening factor. It is then calculated like this:
Si = µr 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 the double 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 of the inner cylinders is 360mm whereas the one of outer is 404mm. As a first approximation we will treat upper and lower cylinders as one for inner and outer cylinders respectively. This should be possible as we provided a good magnetic connection between them with the lamellas and the 90-connection segment. In that case they are approximately 850mm long. For that L/D is 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 shielding is 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) The slit between upper and lower double cylinders can be seen. On the left side the slit between the outer cylinders is still open. on the right the movable magnetic connections lamellas of the inner cylinders can be 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 approximately 1.000. The difference to the calculated values is due to several reasons: on one hand the connected upper and lower cylinders are certainly much worse than one continuous cylinder and on the other hand the sample and the beam are not at the center of the cylinders. Apart from that it is difficult to say how big µr - the most important factor in the equations - really is. Mu-metal is 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 field of magnitude 0, 3Gs to be screened, we find an inner field of approximately 0, 3mGs in the sample shielding. The polarization vector of 4.1A neutrons therefore will be turned inside the shielding (s. Eq.(3.18)):
= 2π · 2916[ Hz
Gs ]0, 3[mGs]
The Arms
The arms of MuPAD are a compact prolongation of the shielding on the monochromator and analyzer side of the TAS. Each of them hosts a pair of precession 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 are just as big as needed to host the coils. As we assume the precession coils do not have any outer return field in the beam leading to crosstalk between two coils (s. Fig. 4.10), the coils were put together as near as 20mm. Thus the main part of the arm is a nearly cubic shielding (a ≈ 180mm) which hosts the precession coils. The cube has a lit which is fixed by screws in order to be able to insert the precession coils. On the entrance or exit side respectively 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 of 100mm. Therefore it is nearly 1.5 times as long as its diameter is wide. This ratio was chosen because field lines only enter approximately as deep as one diameter through a hole in a shielding3. With that design almost no field lines will penetrate as deep as the length of the tube in the shielding which is 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 a non-adiabatic field transition. See section 4.2.3 for more detail). The other side of the cube towards the sample cylinders is closed with a 60-section of a 404mm diameter cylinder of length 320mm, which is welded to it. Hence good 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. All shielding parts of the arms are also out of 2mm thick mu-metal. For details see 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. The arm 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 the sample shielding cylinders can not be manufactured perfectly circular, good contact between arms and sample shielding requires a special construction for optimal contact. The arm is mounted on a track system4 which allows radial movement of the arm shielding towards the cylindrical sample shield- ing. The arm is pulled to the nearest possible position towards the cylinder by 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 hole inside the mu-metal. But for bigger path lengthes the bigger µr of mu-metal makes it more favorable to go through the mu-metal for the lines. Thus entering more deeply is energetically worse for the