Neutrons as probe particles V pseudo Fermi

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Muriel Walters
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~ 510 13 cm 510 thermal neutrons 5 A E ~ 5

1 Neutrons as probe particles V pseudo Fermi Nuclear interaction range (~10-13 cm) b r R b ~ cm 510 thermal neutrons 5 A E ~ 5 mev for thermal neutrons ~ Å for thermal neutrons b

2 Penetration depth of neutrons, X rays and electrons Beam intensities:

3 Cross section and scattering length The interaction of neutrons with matter takes place principally via nuclear forces characterized by a radius of action comparable with the one of the atomic nucleus (10-14 m). The deepness of the potential, V 0 is of the order of few (10-60) MeV. Such value does not vary systematically with atomic number or with atomic weight as in the case of X-Ray scattering where electron density is all that matters. The total electronic function describing the scattering event of the neutron is: ( r) e ik r with f scattering amplitude (a complex number, scattering implies delay and dephasing) independent of the scattering angle θ. The diffusion probability is thus isotropic. f r e The reason for this behavior is connected to the value of the wavelength of the de Broglie wavelength of the thermal neutron which is much larger than the size of the nucleus. The quantity of motion p and the scattering parameter d, yield an angular momentum L L d p L l( l 1) The distance corresponding to l=1 is: d ikr L p l( l 1) l( l 1) 1 many order of magnitude larger than the size of the nucleus for thermal neutrons. scattering in s wave, Born approximation holds Neutrons interact weakly also through their magnetic moments with the electron density. Since the latter extends over all the atom, the resulting scattering intensity will decrease with the scattering angle k 1 5

4 The scattering amplitude f has the dimension of length. scattering length : value of f in the limit of large wavelengths b lim f Once the neutron penetrates into the nucleus its wavefunction can be described by a stationary wave of amplitude /r. Its amplitude and derivative have to match the plane wavefunction outside of the nucleus at the radius of the atomic nucleus, R k. Three possible cases: 1) Neutron does not penetrate into the nucleus cannot b=r k ) Weak interaction: the neutron is scattered b>0 3) Strong interaction: the neutron is initially absorbed and then reemitted b<0 H has a negative value of b, while b is positive for D. An appropriate isotopic mixture can thus give any average value of b. This is very useful when embedding an object in water, since a proper choice of the isotopic mixture can eliminate the contrast in parts of the object, making them invisible.

5 Total and differential cross sections 4 4 f u t r e f r ut ikr outgoing incident outgoing r ik incident e u 4 4 b f t u neutron velocity R position in space (in barn 10-8 m ) Differential cross section ) ( 4 b f d d t t f does not depend on angle scattering potential approximated ) ( ) ( r b m r V by the Fermi pseudopotential Total cross section ) ( ) ( R j r b m r V which for a lattice reads:

depends on the nature of the nucleus and on the particluar isotope involved in the collision d d 0 costan te d d d d d I,, dd N I N Total cross section: t = a + s s scattering cross section s d d d a

6 depends on the nature of the nucleus and on the particluar isotope involved in the collision d d 0 costan te d d d d d I,, dd N I N Total cross section: t = a + s s scattering cross section s d d d a absorption cross section d dd dd Bi The different terms are described by the scattering length, connected to the radius of the potential well and thus of the order of fm. For a composite material the scattering length is the average of the lengths weighted on the density of the components The scattering length depends on energy

7 Coherent and incoherent scattering Since materials correspond to isotopic mixtures with different scattering lengths, the scattering intensity consists of diffraction peaks superposed on an unstructured incoherent background. d d N iqr bje j1 j i iqr ij i b b i j e isotope A with concentration p isotope B with concentration 1-p bb i j pb A ( 1 p) b B p(1 p)( ba b ) B coherent incoherent ij d q ( r r ) d d d coh inc b Np ij e i j diffraction peaks p(1 p) b b ( 1 p)( ba bb ) N A B unstructured background Incoherence results not only the isotopes mixture but also from magnetic scattering (Ni: only 3.4 barn out 18 are coherent, for H only barn out of 80)

9 Elastic vs inelastic scattering k 1 k 0 Inelastic scattering corresponds to energy momentum transfer to phonon or magnon baths of the sample. Elastic scattering Inelastic scattering k 1 k 0 q el k 1 q el k 0 k 0 > k 1 E 0 > E 1 E > 0 the neutron looses energy to the sample k 0 sin q k 1 k 0 q k 0 < k 1 E 0 < E 1 E < 0 the neutron gains energy Accessibile kinematic region: q = k 0 k 1 conservation of momentum E s k 1 k0 1 conservation of energy E 0 Scan curve q k 0 k 1 k1 k0 q/k 0 cos Notice: for light and X rays E = ħ c k, = c k: The set of accessible Q and is different than for massive particles E loss /E 0

10 Applications: Neutron radiography:

11 Crystallography Determination of lattice spacing Investigation of Phase Transitions Odd Miller index channels are open for the ordered phase only

12 Lattice dynamics: Phonon dispersion Vertical and horizontal cuts in the -q plane are obtained by changing two parameters at a time as possible using the three axis spectrometer, discussed below

13 Magnetic Scattering the neutron interacts via its magnetic dipole moment n =g n s B expressed in Bohr magnetons, with g n = -1,91 anomalous giromagnetic factor s neutron spin. Atomic scattering length a m = g n r e f(q)m r e =e /mc classical electron radius (g n r e =1, cm) f(q) magnetic form factor M projection of the magnetization in the direction to q. a m is small since it is deterrnined by the weak magnetic interaction with the electronic orbitals of the sample; f(q) is peaked at small q since the orbitals extend over the dimension of the atom. magnon cross section has a strong q dependence.

Magnetism Determination of magnetic superlattices Ferro-para magnetic phase transitions and magnetization curves The intensity due to magnetic

14 Magnetism Determination of magnetic superlattices Ferro-para magnetic phase transitions and magnetization curves The intensity due to magnetic scattering goes to zero at the Neel temperature, T N. The beams in the example do not vanish above T N since the structure of MnV O 4 is not bcc

15 Magnetic topography Topografia Spin up Spin down Visualization of magnetic domains

16 Magnetic dynamics Magnons, magnetic excitations in which the spin flip is delocalized Magnon dispersion is quadratic in exchanged momentum

$Neutron Small Angle Scattering Small Angle Scattering relies on diffraction on structures larger than just the atomic size, such as big molecules or clusters that can be hundreds times larger than an$

17 Neutron Small Angle Scattering Small Angle Scattering relies on diffraction on structures larger than just the atomic size, such as big molecules or clusters that can be hundreds times larger than an atom. Neutron Small Angle Scattering is a non-destructive method for the determination of microstructures with lengths between 0.5 nm and 500 nm. It is based on the principle that an initially parallel neutron beam is scattered by inhomogeneities of a sample. These can be due to fluctuations in density, concentration or magnetisation. The method is widely used in biological structure research (viruses, proteins, enzymes), polymer research (conformation analysis), energy and environmental engineering (porosity and surfaces of filter materials and catalysts) and the development of new materials. Similar and complementar information can be retrieved by Small Angle X-Ray

Production of neutron beams Neutrons are generated at high energy by nuclear reactions. The cost of a source depnds on the amount of energy ending up in each neutrone (MeV/n).

Experiments are approved by international scientific committees SPALLATION to spall = scheggiare Bombarding a heavy nucleus (Uranium/Tantalum) with high energy (~ GeV) protons FISSION Typically of

18 Production of neutron beams Neutrons are generated at high energy by nuclear reactions. The cost of a source depnds on the amount of energy ending up in each neutrone (MeV/n). Neutron beams are expensive. Experiments are approved by international scientific committees SPALLATION to spall = scheggiare Bombarding a heavy nucleus (Uranium/Tantalum) with high energy (~ GeV) protons FISSION Typically of Uranium A thermal neutron hits U. the nucleus splits into two lighter nuclei (e..g Br e La) and releases energy (00 MeV) + ~.5 swift neutrons Chain reaction when more than one neutron per fission is retained in the system. Various particles (protons, neutrinos ) + ~ 0 neutroni(/proton with energies of some MeV ~1 neutron (/fission) with energy of MeV Neutrons must be moderated, i.e. brought to thermal energies (5 mev), by inelastic scattering with light atoms (hydrogen, graphite, methane)

19 Neutron Sources: an important flux of neutrons is available only at large scale facilities, like the Laue-Langevin laboratory in Grenoble. Fission Reactors: The most commonly used reaction to produce neutrons involves fission of 35 U: n+ 35 U= 36 Ba+ 56 Kr+Nn+ +b The cross section is largest for low energy neutrons, i.e. for energies < 1 ev and involves the production on average of N.5 neutrons per fission event. 00 MeV are released, distributed for 10 MeV in rays, MeV in the neutrons and the rest in the other fission products. The emitted neutrons are peaked at 1 MeV, energy at which they are easily captured by the much more abundant U 38 isotope. To maintain the chain reaction the neutrons must be slowed down by immerging the fissile material in water or better heavy water and trying to minimize the lost neutrons. The reactor is screened by a several meter thick concrete, Fe and B wall which blocks rays and neutrons.

20 The beam collimators are of stainless steel and B 4 C, materials with a high stopping power for rays and transparent to neutrons. The experiments are placed at the end of beamlines distributed around the nuclear reactor. The moderator is cooled to room temperature in order to obtain a thermal distribution centered at 5 mev, corresponding to translational velocities of 00 m/sec.

21 Spallation Sources: p +A= Nn+ + pp + spallation products pp stays for small mass particle-antiparticle pairs A is a heavy nucleus. The incident proton must have an energy E>0.5 GeV. It is produced either with a protosincrhotron (diameter 100 m) or by a linear accelerator (length 1 km) or by a combination of both techniques. Typically 10 to 0 high energy neutrons are produced in each collision and fluxes of n/sec concentrated in a small solid angle are generated (high brilliance). The power needed to operate a spallation source is NE 100 kw much smaller than the one of a reactor generating neutron beams with comparable brightness. The spallation source can be operated in continuous or pulsed modes (useful for TOF).

$Lenses and waveguides are possible since neutrons are slowed down inside materials (e.g. quartz) and one can define an effective refraction index.$

22 Neutron Optics: An efficient use of neutron beams implies waveguides to lead them to the experiment and lenses to focus them on the sample. Lenses and waveguides are possible since neutrons are slowed down inside materials (e.g. quartz) and one can define an effective refraction index. n 1 - (n i b i ) / n i = i th -atomic species density b i = scattering length = neutron wavelewngth Typically : 1-n At =60, the deflection =(n-1) [tg +tg(- )] =3 10-6, Corresponds to just a few arc seconds. prisms are dispersive elements Neutron Lense Since n<1 convergent neutron lenses are concave.

23 Neutron Waveguides are based on total reflection of thermal neutrons for grazing angles c <0.1 ( 56 Ni) n = sen( i )/sen( r ) = sen(/ - c ) = cos c = 1- ½ c = 1 - (n i b i ) / yielding: c = <b>/ Thanks to waveguides the experiments may be placed far away from the nuclear reactor. They are built of quartz covered by a thin film of 58 Ni whose critical angle is even larger than for 56 Ni. Larger critical angles can be reached using non periodic multilayers consisting of materials with different refraction index. This method allows to build neutron mirrors and neutron polarizers, too.

24 Typical set up for a neutron beamline. For = nm the distance between lens and exit slit S 4 is 10 m. The effect of gravitation must be considered in computing the trajectories.

$Polarization Definition: P=(N + -N - )/(N + +N - ) Polarized neutron beams are obtained by: 1) Transmission through polarizing filters, based a) on the spin dependence of the diffraction cross$

25 Polarization Definition: P=(N + -N - )/(N + +N - ) Polarized neutron beams are obtained by: 1) Transmission through polarizing filters, based a) on the spin dependence of the diffraction cross section (polycristalline iron, P0.6); b) on the difference in the cross section for collisions in singlet and triplet state (crystals doped with Nd 3+, P=100% at E= 5 mev); ) Reflection off a magnetized mirror; critical angle: c = (N(b a m )/ ) with b and a m nuclear and magnetic scattering lengths the highest polarization is obtained for a Ti and Gd mirror. P98% for Ge-Fe multilayers. 3) Bragg reflection from a magnetized single crystal.

26 Neutron detectors: neutrons have no electric charge. Detectors are based on conversion of the energy released in nuclear reactions: Gas phase detectors: 3 He doped with 10 B Reactions: 3 He+n=p+ 3 H 10 B+n 7 Li * + 3 He+n releases 0.8 MeV distributed in a ratio of 3:1 on proton and Tritium. The electrically charged proton ionizes the gas. B +n has a higher cross section. The energetic particle ionizes the gas. The so generated electrons cause further ionizations. The current is collected on a filament. It is proportional to the energy released in the nuclear reaction so that neutron events can be distinguished from those generated by other ionizing particles. Since the ionization potential of 3 He is 0 ev, each events produces 1.6 fc. Assuming that such charge is collected within 1 sec and discharged on a 10 M resistor, one obtains 160 mv impulses. Typically a.5 cm diameter detector with 0 bar pressure has a detection efficiency of 90% for neutrons with =1 Å. 3 He detectors can be made position sensitive by collecting the current with more than one filament.

$Neutron Monocromators: Neutron monochromators are based on Bragg diffraction on monocrystals typically Cu or pyrolytic graphite.$

27 Neutron Monocromators: Neutron monochromators are based on Bragg diffraction on monocrystals typically Cu or pyrolytic graphite. Such crystals have the advantage of being ideally imperfect - (mosaicity 0.1 to 1 ) so that diffraction can occur over a sufficiently large energy window to allow for high beam fluxes (even the most efficient neutron sources are weak!). At =1 Å Cu has a reflectivity peak in the (331) diffraction channel of 30%. Pyrolytic graphite performs even better (70%). Efficiency improves at low temperature, T, because of the Debye Waller factor. I(T)=I o e -WT. Monochromatization can be achieved also with velocity selectors

28 Inelastic diffusion: Time of Flight Chopper

3 axis spectrometer It consists of Bragg monochromators used both to monochromatize and to analyze the beam. It can achieve a typical resolution E/E~1% i.e. 0.

29 3 axis spectrometer It consists of Bragg monochromators used both to monochromatize and to analyze the beam. It can achieve a typical resolution E/E~1% i.e. 0. mev Direct geometry I( S, A ) at fixed M Inverse geometry I( S, M ) at fixed A For both geometries each tern ( S, M, A ) defines a value of Q. ( M, A ) define. It is possible to perform measurements As a function of exchanged energy keeping Q costant.

30 sample Bragg monochromator chopper L 0 L 1 detector M Sample Bragg analyzer A mono M Crystal monochromator ToF spectrometer Detector direct 3-axis geometry inverse 3-axis geometry

31 A 3 1 A A 3 A A A A A cos sin cos sin k d d d E E m m m m M 0 1 M, ~, Q S k k Q I M 3 0 M M cos k d m Q k 0 k 1, ~ cos ) (, A A Q S k k Q I Q k 0 k 1 A A, ~ ) (, Q S k k k Q I Fixed initial energy and scattering angle (direct geometry) Fixed scattering energy (inverse geometry) Incident energy and angle are varied, ~ cos, M 0 M Q S k Q I Three-Axis Spectrometer (constant-q mode)

During the flight time in the magnetic field the spin precess.

32 Spin Echo Technique The direction of the neutron spin is preserved in the inelastic interaction with the sample. During the flight time in the magnetic field the spin precess. Inelastic scattered neutrons are slower and precede over a larger angle. Energy Resolution: E/E=10-6 =4 nm E=10-8 mev Typical diffusion energies Apparatus at ILL Grenoble

33 -Spin flipper (campi magnetici incrociati)

34 Larmor precession of the neutron spins

Neutron Instruments I & II. Ken Andersen ESS Instruments Division

Neutron Instruments I & II. Ken Andersen ESS Instruments Division Neutron Instruments I & II ESS Instruments Division Neutron Instruments I & II Overview of source characteristics Bragg s Law Elastic scattering: diffractometers Continuous sources Pulsed sources Inelastic