Neutron Scattering - JUWEL - Forschungszentrum Jülich

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SPHERES 11 The stationary equilibrium solution is just pm =1/3for all m. When perturbed, the system relaxes into equilibrium with a time dependence of exp(−t/˜τ). Explicit solution of the linear differential equation system (13) yields ˜τ =2τ/3. According to a fundamental theorem of statistical mechanics (the fluctation dissipation theorem), the relaxation by which a slightly perturbed system returns into equilibrium has the same time dependence as the pair correlation function in equilibrium. Therefore, we can employ the solution of (13) to write down the self-correlation function of the protons that constitute our methyl group. Fourier transform yields then the incoherent scattering function S(q, ω) =a(q)δ(ω)+b(q) Γ ω2 . (14) +Γ2 The first term describes elastic scattering. The second term, the Fourier transform of the exponential exp(−t/˜τ), is a Lorentzian with linewidth Γ=˜τ −1 ; such broadening of the elastic line is often called quasielastic. 4 Preparatory Exercises 1. Relate the relative wavelength spread δλ/λ to the relative energy spread δE/E. 2. In SPHERES, useable detectors are located at scattering angles 2θ ranging from 12.5 ◦ to 134 ◦ . Calculate the corresponding wavenumbers in ˚A −1 . Recommendation: use the following constants in atomic units: �c = 1973 eV ˚A and mnc 2 = 940 MeV. 3. Convert dynamic range and resolution of SPHERES into GHz. To make contact with optical spectroscopy, you might also wish to convert into cm −1 . 4. Empirically, it is found that the centre of the resolution function can be fitted by a Gaussian a exp(−E 2 /2/σ 2 ). Derive an expression that relates the Gaussian standard deviation σ to the FWHM. 5. Note that the above mentioned fit applies only to the very centre of the resolution function. How does a Gaussian look like on the lin-log representation of Fig. 6? And a Lorentzian? 6. In SPHERES, the distance sample-analyzer is 2 m. Calculate the time neutrons need for a round trip sample-analyzer-sample, and deduce the rotation frequency of the chopper. 7. Assume that the monochromator motion is perfectly sinusoidal. Sketch how the measuring time per energy channel varies with �ω. 8. Draw a sketch of the expected backscattering spectrum S(q, ω) of a ferromagnetic material with I �= 0. 9. Assume a hyperfine splitting of ΔE =2μeV. To which temperature do you have to cool the sample to observe a 10% difference between the probabilities of energy gain and energy loss scattering? 10. How do you expect ΔE to evolve when the sample is heated towards the Curie or Néel temperature?
12 J. Wuttke 11. Calculate the moment of inertia, I = � mr2 ⊥ , of a methyl group. Verify that the rotational constant B = �2 /(2I) has a value of about 670 μeV. 12. Expand V (φ) around a potential minimum, and compare the resulting Schrödinger equation with that of a harmonic oscillator. Show that the splitting of oscillator levels is of the order of meV. 13. Draw a coordinate system energy-versus-angle. Sketch V (φ), the harmonic approximation, the ground state’s ψ(φ), and the lowest oscillator energy levels. What does that imply for the validity of the oscillator approximation? 14. Sketch the expected spectra for different temperatures. 5 Experiment Procedure 5.1 The experiment itself After an initial discussion, the group chooses which experiment to perform: hyperfine splitting, methyl group tunneling, or participation in an ongoing research project. For a given chemical composition, the group computes the sample thickness that yields 90% transmission. Depending on the group’s interest, a sample is prepared, or a standard sample is used. The tutor shows how to insert the sample in the instrument’s cryostat. Using the instrument’s graphical user interface, starting a measurement is rather trivial. Log entries are written to the instrument log wiki. 5.2 Raw data reduction The program SLAW is used to convert raw neutron counts into S(Q, ω). It is parametrized by a script, called Slawfile. The tutor provides a sample script, which is then modified to convert the results of the current experiment. SLAW can save S(Q, ω) in a variety of output formats. Most relevant are plain tabular formats recttab and spectab, and a self-documenting format y08 required by our standard dataanalysis software FRIDA. 5.3 Data evaluation In a first approach, labcourse participants should analyse plain tabular data using whatever allpurpose data-analysis software they are used to. 1. Plot a representative selection (choose a few Q) of measured spectra. 2. Determine the FWHM of the elastic line, and of the inelastic lines if there are any. 3. Try to fit these lines with a Gaussian, with a Lorentzian, with a squared Lorentzian.
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Member of the Helmholtz Association
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Forschungszentrum Jülich GmbH Jül
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RESEDA 3 1 Introduction Neutrons ar
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RESEDA 5 various length values l ac
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RESEDA 7 In this expression, a norm
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