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ELEMENTS OF NEUTRON SCATTERINGELEMENTS OF NEUTRON SCATTERING

1. Basic properties of neutrons, comparison with x-ray photons2. Neutron sources3. Neutron detectors4. “Classical” neutron scattering5. Magnetic scattering6. Inelastic scattering

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1.1. Basic properties of neutrons, comparison with x-ray photonsBasic properties of neutrons, comparison with x-ray photons2. Neutron sources3. Neutron detectors4. “Classical” neutron scattering5. Magnetic scattering6. Inelastic scattering

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s 1.889 MeV, 77.0~ epn

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1. Basic properties of neutrons, comparison with x-ray photons2.2. Neutron sourcesNeutron sources3. Neutron detectors4. “Classical” neutron scattering5. Magnetic scattering6. Inelastic scattering

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Nuclear fissionNuclear fission

In reactors, fission takes place when a fissile nucleus (always 235U or 239Pu in practice, althoughthere are a few others) captures a neutron, the nucleus splits into two nearly equal-massfragments (in large variety) that together carry about 160 MeV of kinetic energy. On average,the process also promptly (~10-15 sec) produces about 2.5 neutrons for each fission.

Fission neutrons actually “evaporate” from the initially highly excited fragments and have averageenergies of about 2 MeV. Each fission event produces a total of approximately 190 MeV ofenergy: fission fragment and neutron kinetic energy, beta radiation (mostly e– because fissionfragments usually have too many neutrons), and photons ( rays), all of which appears as heat inthe reactor fuel and surroundings, and neutrino energy, which escapes. A small fraction (~0.5%) of neutrons appear after a few seconds delay time. The delayed neutrons are essential forreactor control. One neutron of the 2.5 goes on to cause another fission, usually (after a fewmicroseconds) slowing down to energies at which the fission cross-section is large. Capture incontrol rods and parasitic processes absorb about 0.5 neutron per fission. This leaves about 1neutron per fission useable for external purposes.In round numbers, fission reactors require dissipating about 200 MeV of heat energy for eachuseful neutron produced. Most (but not all) reactors used for slow-neutron scattering researchoperate in a steady mode.

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The energy distribution of prompt fission neutrons

)29.2sinh()036.1exp()( EEEf

E is in MeV

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Nuclear spallationNuclear spallation

In accelerator-driven spallation sources, high-energy particles (invariably protons of ~ 1. GeVenergy) from the accelerator impinge on a thick target of dense, high-mass-number materials,e.g., uranium, tungsten, tantalum, or mercury.Here they collide, leaving highly excited nuclei.Neutrons, protons, and pions that emerge from collisions with sufficient energy proceed tocollide again and leave further excited nuclei. The excited nuclei shed their energy by promptlyevaporating particles (by far, predominantly neutrons) until there is too little left for that process.

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Thick-target yields of neutrons as a function of incident proton energy and targetmaterial

internal cascade

spallation (approx. 40 neutrons per nucleus)

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The energy and angular distribution of neutrons emerging from a tantalum targetirradiated by 1-GeV protons. The target is 31 cm long, 7 cm wide and 20 cm high.

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ModeratorsModerators

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Paths of neutrons in a research reactor. Several fast neutrons (red) emerge from each fission (heavy dots) and proceed to collide, losing energy in each collision (progressively less red) until they cause another fission, disappear by non-fission absorption (a), or travel into the reflector. In the reflector-moderator, neutrons collide repeatedly, losing energy (less red, more green) until they come into thermodynamic equilibrium with the moderator medium (thermal neutrons, green) and live for a long time. Some return to the core to cause another fission. Others find their way into the neutron beams.

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A cavity-type cold moderator. Thermal neutrons (green) enter the 25-K liquidhydrogen from the 300-K D2O moderator. There, they collide numerous times, losing energy ateach collision (less green), and come into equilibrium with the hydrogen at 25 K (blue). Thereentrant cavity acts as a hohlraum, allowing neutrons to rattle around within, promotingthermal equilibration but permitting cold neutrons to emerge efficiently into the neutron beams.The diagram exaggerates the thickness of the L-H2 layer in relation to the diameter of the cavityand omits the necessary plumbing arrangements.

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A hot-neutron source. Thermal neutrons (green) from the 300-K D2O moderatorenter the block of vacuum-insulated graphite, heated by gamma rays and fast neutrons to about2000°C. There they collide, gaining energy in collisions (progressively yellower), and come intothermodynamic equilibrium with the hot graphite. They emerge as hot neutrons (orange) andtravel into the neutron beams.

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Flux spectra at the source surfaces of the three moderators of the FRM-IIreactor

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Spectrum of a poisoned room-temperature polyethylene moderator at IPNS

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Brightness & Fluxes for Neutron & X-Ray Sources

brightness(s-1m-2srad-1)

E/E (%) divergence(mrad2)

flux(s-1m-2)

neutrons 1015 2 10x10 1011

rotating anode

1020 0.02 0.5x10 5x1014

bending magnet

1027 0.1 0.1x5 5x1020

undulator 1033 10 0.01x0.1 1024

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Reactor ILL, Grenoble

1. The double wall of the reactor 2. The reactor's overhead crane 3. The gantry crane 4. The reactor operations hall 5. The experimental hall overhead crane 6. The experimental hall 7. Spectrometer 8. View into the reactor core 9. The reactor core (plan) 10. The main reactor pool 11. The storage pool 12. The heat exchanger

http://www.ill.fr/index_ill.html

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IN20:

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1. Basic properties of neutrons, comparison with x-ray photons2. Neutron sources3.3. Neutron detectorsNeutron detectors4. “Classical” neutron scattering5. Magnetic scattering6. Inelastic scattering

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• What does it mean to “detect” a neutron? – Need to produce some sort of measurable quantitative (countable) electrical signal– Can’t directly “detect” slow neutrons

• Need to use nuclear reactions to “convert” neutrons into charged particles• Then we can use one of the many types of charged particle detectors

– Gas proportional counters and ionization chambers– Scintillation detectors– Semiconductor detectors

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Nuclear Reactions for Neutron DetectorsNuclear Reactions for Neutron Detectors

• n + 3He 3H + 1H + 0.764 MeV• n + 6Li 4He + 3H + 4.79 MeV• n + 10B 7Li* + 4He7Li + 4He + 0.48 MeV +2.3 MeV (93%)

7Li + 4He +2.8 MeV ( 7%)• n + 155Gd Gd* -ray spectrum conversion electron spectrum • n + 157Gd Gd* -ray spectrum conversion electron spectrum • n + 235U fission fragments + ~160 MeV • n + 239Pu fission fragments + ~160 MeV

most common

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Gas detectors

MeV76.0133 HHHen

barns 8.1

5333

~25,000 ions and electrons produced per neutron (~410-15 coulomb)

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MAPS Detector Bank

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Scintillation Detectors

MeV79.4346 HHeLin

barns8.1

940

Li glass (Ce) ~7,000ZnS (Ag) - LiF ~160,000

photons per neutron

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GEM Detector Module

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Semiconductor Detectors

MeV79.4346 HHeLin

barns8.1

940

~1,500,000 holes and electrons produced per neutron (~2.410-13 coulomb)

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1. Basic properties of neutrons, comparison with x-ray photons2. Neutron sources3. Neutron detectors4. “Classical” neutron scatteringClassical” neutron scattering5. Magnetic scattering6. Inelastic scattering

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scattering length for neutrons

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consequences:

structure of a molecule seen by x-rays,clouds of electron density are denoted in red,H atoms are not detectes

structure of the same molecule seen by neutrons, clouds of nuclear density are denoted in blue, H atoms are detected (purple)

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radiography (shadow projection)

neutrons (the plastic parts are detected (H-rich))

x-rays (not sensitive to the plastic parts)

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Scattering from a set of nuclei

ijji

jibb ).(i*ed

d RRq

bi depends on the nucleus (isotope, orientation of spin relatively to the neutron spin etc.). Statistical averaging yields:

22* ||)1( bbbb ijijji

Coherent and incoherent parts of the cross-section

ij

jib ).(i2

coh

ed

d RRq Nbb22

incohd

d

Total cross-sections

2

coh 4 b 22

incoh 4 bb

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Example:scattering from a single isotope with the spin quantum number j. Two possible orientations of the neutron spin exist:• parallel spins: • antiparallel spins:

2212,21 jIjI the scattering length is b+

jIjI 212,21 the scattering length is b

probabilities of these two states are

12222

2,

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1

222

22

j

j

jj

jp

j

j

jj

jp

and the average scattering lengths

12

)1(,

12

)1(22

2

j

bjbjb

j

jbbjbpbpb

For proton with j=1/2:

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1212

p,p

cm1074.4 cm,1004.1

bb

barn 8.79 barn, 77.1 incohcoh

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Various isotopes in the sample:

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Thermal Neutrons, 8 keV X-Rays, Low Energy Electrons: Absorption by Matter

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1. Basic properties of neutrons, comparison with x-ray photons2. Neutron sources3. Neutron detectors4. “Classical” neutron scattering5.5. Magnetic scatteringMagnetic scattering6. Inelastic scattering

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Magnetic ScatteringMagnetic Scattering

• The magnetic moment of the neutron interacts with B fields caused, for example, by unpaired electron spins in a material

– Both spin and orbital angular momentum of electrons contribute to B– Expressions for cross sections are more complex than for nuclear scattering

• Magnetic interactions are long range and non-central– Nuclear and magnetic scattering have similar magnitudes– Magnetic scattering involves a form factor – FT of electron spatial distribution

• Electrons are distributed in space over distances comparable to neutron wavelength• Elastic magnetic scattering of neutrons can be used to probe electron distributions

– Magnetic scattering depends only on component of B perpendicular to Q– For neutrons spin polarized along a direction z (defined by applied H field):

• Correlations involving Bz do not cause neutron spin flip• Correlations involving Bx or By cause neutron spin flip

– Coherent & incoherent nuclear scattering affects spin polarized neutrons• Coherent nuclear scattering is non-spin-flip• Nuclear spin-incoherent nuclear scattering is 2/3 spin-flip• Isotopic incoherent scattering is non-spin-flip

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Magnetic Neutron Scattering is a Powerful ToolMagnetic Neutron Scattering is a Powerful Tool

• In early work Shull and his collaborators:– Provided the first direct evidence of antiferromagnetic ordering– Confirmed the Neel model of ferrimagnetism in magnetite (Fe3O4)– Obtained the first magnetic form factor (spatial distribution of magnetic electrons) by measuring paramagnetic scattering in Mn compounds– Produced polarized neutrons by Bragg reflection (where nuclear and magnetic scattering scattering cancelled for one neutronspin state)– Determined the distribution of magnetic moments in 3d alloys by measuring diffuse magnetic scattering– Measured the magnetic critical scattering at the Curie point in Fe

• More recent work using polarized neutrons has:– Discriminated between longitudinal & transverse magnetic fluctuations– Provided evidence of magnetic solitons in 1-d magnets– Quantified electron spin fluctuations in correlated-electron materials– Provided the basis for measuring slow dynamics using the neutron spin-echo technique…..etc

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Theory of magnetic neutron scattering for pedestrians:Theory of magnetic neutron scattering for pedestrians:

Interaction of the magnetic moment of neutron with the magnetic field produced by a moving electron

Interaction potential: Bμ .V n

Magnetic dipole moment of the neutron: σμ

Nnn

n 1.913 is the gyromagnetic factor of neutron,m

eN 2

σ

is the spin operator of neutron

The magnetic field produced by a moving electron

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rotRRc

e eeLS

RμRvBBB

magnetic dipole moment of electron Sμ Be 2

differential cross-section of magnetic neutron scattering2

2el )(.

2

1)(

d

difr

Bn qMσ

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)(rM is the component of magnetization perpendicular to the scattering vector q

rqrMrqM .i3 e)(d)( is its Fourier transformation

The magnetization has an orbital and spin component SL MMM

The spin component

jk

jkjkBS srrrM

)(2)( )3(the k-th electron in the j-th atom

We denote the average spin operator of atom j jkj sS

Then we obtain (for the spin component only)

2

.i2el e)()(

d

d

jjn

jSfr rqq

where rqrrq .i3 e)(d)( sf is the magnetic scattering factor of an atom, s is the spin density

The orbital component can be included by LS coupling:2

.i

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el e)()(d

d

jjJn

jJfgr rqq

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1. Basic properties of neutrons, comparison with x-ray photons2. Neutron sources3. Neutron detectors4. “Classical” neutron scattering5. Magnetic scattering6.6. Inelastic scatteringInelastic scattering

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This and the following pictures are taken from the lecture notes of S.K. Sinha ssinha@physics.ucsd.edu

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