Neutrons 101

Properties of Neutrons Canadian Powder Diffraction Workshop June 2010 UQTR

What is a neutron?

• The neutron is a subatomic particle with no net electric charge.

Nucleus

• Neutrons are

usually

bound (via strong nuclear force) in atomic nuclei. Nuclei consist of protons and neutrons—both known as nucleons. • The number of protons determines the element & the number of neutrons determines the isotope, e.g.

15 N and 14 N have 7

p

and 8

n

and 7

n

respectively.

Instability of free neutron and mass

• • Free neutrons are unstable; they undergo b -decay, mean lifetime ~ 885.7 ± 0.8 s.

They cannot be stored for long free!

•

n 0 → p + + e − + ν e

• Mass is

slightly larger

than that of a proton

Neutrons have a

spin

• Spin,

s

, is a quantum number: neutrons are

spin-half, s=1

/2 • Angular momentum

S

 

s

(

s

 1 ) • Particles with angular momentum have a magnetic moment,   

g q

2

m S

Spin

s

Angular Momentum

S

Moment  Note: Although neutral,

q

= 0, the neutron is made up of quarks— electrically charged particles. The magnetic moment of the neutron is ultimately derived from the angular momentum of

spins

of the individual quarks and of their

orbital motions

.

Electrons have a spin too.

• Orbital and spin (

s

= 1/2) angular momentum give rise to moments and magnetism

m

L

m

s magnetic moments in solids. • Get additional magnetic diffraction peaks from the lattice of ordered spins (beyond our course).

Characterizing Neutrons By….

E k



T



v

1  meV cm -1 THz K Å ms -1 Linear

E

meV 1 0.1240 4.136 8.616e

-2 81.807 5.227e

-6 Reciprocal Square-reciprocal Root Root-reciprocal Square

Neutron Conversion Factors k

 cm -1 THz 8.006 1 33.336 0.6949 659.8 4.216e

-5 0.2418 0.02998 1 2.083e

19.78 1.265e

-2 -6

Key

1 THz  4.136 meV 1 Å  3956 ms -1 1 Å  19.78 THz 1 meV  437.4 ms -1 1 meV  9.045 Å 1 ms -1  5.227e

-6 meV

T

K 11.605 1.439 48.00 1 949.4 6.066e

-5 9.045 25.68 4.447 30.81 1  Å 3956

E

= 4.136

v v

= 3956/   = 19.78/  2

v

= 437.4



E

 = 4.447/ 

v E

= (

m

/2)

v

2

v

ms -1 128.4 3956 1 437.4 154.05 889.5

E



h

2 2

m

 2

E



h



E



k B T E

 1 2

mv

2

E

  2

k

2 2

m

Neutron Sources

Neutrons must be

liberated

from their bonds Binding energy of the nuclei ~MeV

Fission Reactor

• U 235 +

n

(thermal) • ~2 MeV neutrons produced – Fission neutrons move at ~7% of the speed of light – Moderated (thermal) neutrons move at ~8 times the speed of sound.

http://upload.wikimedia.org/wikipedia/commons/9/9a/Fission_chain_reaction.svg

• This is around 7700 times slower!

Spallation Source

• Spallation=“blowing chunks” (

p

,

n

) • hydride ion (H  targets  ) source  moderators  proton accelerators instruments http://www.isis.rl.ac.uk/

Moderation/Slowing-down

-neutrons as particles (“gas”)

Maxwellian

• Distribution of velocities of particles as f(T) – neutrons behave like a gas.

• Maxwell-Boltzmann distribution-the most probable speed distribution in a collisionally dominated system consisting of a large number of non-interacting particles.

– describes the neutron spectrum to a good approximation (ignoring  -dependent absorption).

Elastic Collisions

• Elastic* collisions between the nucleus and the neutron transfer energy. Simon Steinmann, Raul Roque: Creative Commons Attribution ShareAlike 2.5

An elastic collision is a collision in which the total kinetic energy of the colliding bodies after collision is equal to their total kinetic energy before collision.* Good moderator nucleus = Low mass + low absorption cross-section + high scattering cross-section.

E



k B T E

 1 2

mv

2

Moderators

• Moderated neutrons take on the average kinetic energy of the moderator, set by its

T

.

How many collisions are necessary to moderate a 2MeV fission neutron to a 1eV neutron?

~16 for light water, which take place in about 30 cm of travel.

Reactor simulator

0.0016

0.0014

0.0012

0.0010

0.0008

0.0006

Moderators & the Maxwellian

E



h

2 2

m

 2 lambda (Angstrom) Cold Source H2 20K NRU D2O 333K Hot Source Graphite 2303K

Note:

Hot source

increases the number of high-

E

(

v

2 ), short  neutrons, but does so by spreading out the dist’n, thereby reducing the flux at any ,( or

v

, or

E

, ….).

0.0004

0.0002

0.0000

0 2000 4000 6000 8000 velocity (m/s) 10000 12000

E

 14000 1 2

mv

2

Cold source

reduces the spread to only very long  and increases the flux at those 

•

Wave-Particle Duality

Neutrons have a wavelength

de Broglie hypothesis

: all matter has a wave-like nature • Neutrons have an associated wavelength,  , diffract and have wave-like properties

E



h

 ;  

h mv k

 2   Strictly “angular” wavenumber

E

  2

k

2 2

m

 r

h

~ Planck' s constant;

m

~ mass;

v

 ~ wavelengt h;  ~ frequency;

k

~ velocity;

mv

~ momentum; ~ wavenumber

Waves

http://upload.wikimedia.org/wikipedia/commons/5/5c/Plane_wave.gif

http://upload.wikimedia.org/wikipedia/commons/1/12/Spherical_wave2.gif

Plane Waves

• A constant-frequency wave whose wavefronts (surfaces of constant phase) are infinite parallel planes of constant amplitude normal to the wavevector,

k

.

k

Huygens-Fresnel Principle

Each point of an advancing wave front is the centre of a fresh disturbance and the source of a new train of waves. The advancing wave is the sum of all secondary waves arising from http://upload.wikimedia.org/wikipedia/commons/a/a4/Christiaan_Huygens-painting.jpeg

points in the medium Christiaan Huygens 1629-1695 already traversed.

Plane wave passing through a 4  -slit: Note secondary spherical wave sources A classical, very simple way of seeing the relationship between plane wave (beams) and spherical waves (scattering from individual particles)

Ocean plane waves passing through slits

http://www.physics.gatech.edu/gcuo/UltrafastOptics/OpticsI/lectures/OpticsI-20-Diffraction-I.ppt

Scattering lengths and cross-sections

Spherical Waves

• Wave energy is conserved as wave propagates • Energy of the wavefront spreads (radiates) out over the spherical surface area, 4 

r

2 .  Energy/unit area decreases as 1/

r

2 .

• Since energy  intensity

E

 Amplitude 2 .

Amplitude of a spherical wave  1/

r

Interaction Strength

Neutrons interact via the strong nuclear force (nuclear scattering).

What is a scattering length?

Spherical wave 10 -15 m • Nucleus is a point with respect to . 10 -10 m • Treat the incoming monochromatic neutron beam as a plane wave of neutrons with single

k

• Neutrons scatter from individual nuclei (secondary source): –

independently

of angle as

spherical waves

– scattered wave amplitude   1/

r

• Proportionality constant:

b – scattering length

 

b

exp(

ikr

)

r

Scattering Length,

b

• Can be positive or negative!

• A

positive b

can be explained simply in terms of an impenetrable nucleus which the

n

cannot enter – D ~ 180°.

• A

negative

D ~ 0°.

b

is due to “

n

+ nucleus” forming a compound nucleus – • More generally,

b

is

complex b

=

b

’+ i

b

”– the

b

” imaginary component is related to absorption and is frequency-dependent.

Scattering Length,

b

 Cross-section, s  (

r

)  (

r

) * defines a probability density of finding neutron at

r

from the nucleus The surface area of a sphere at radius,

r

 

b

exp(

ikr

)

r

4 

r

2 s  4 

r

2  *  4 

r

2  2  4 

b

2 Not forgetting our identities: exp(

ikr

)  cos(

kr

) 

i

sin(

kr

) cos 2 (

kr

)  sin 2 (

kr

)  1

Cross-section

U is “as big as a barn.” • The interaction probability is the likelihood of a point-projectile hitting the target area (the

cross section

, σ).

• • Each nucleus thought of as being surrounded by a a characteristic area.

Barn

= 10 −28 m 2 , ~ the cross sectional area of U.

• Cross-sections for different processes: scattering, absorption, fission… • They are not constant, but energy-dependent There are also units of sheds, and outhouses…but not used for neutrons….

Energy dependence of cross sections

Cold Fast

Note:

• Resonances at high-energy • Constant plateau of scattering cross-section • Strong (1/

v)

dependence of absorption – related to the time spent near the nucleus (probability of capture).

Good neutron shielding Cold

An absorber:

113

Cd

Resonances Fast Shielding materials: 1) 2) Moderators e.g. H thermalize fast neutrons Attenuators: e.g. H strong scatterers like a diffusing screen (pearl light bulb) 2) Thermal absorbers Cd, 10 B, Gd ( 6 Li) ENDF/B-VII Incident-Neutron Data – 60pp for 113 Cd!

http://t2.lanl.gov/data/neutron7.html

Coherent & Incoherent Scattering

• Scattering nucleus at a given position in a crystal may be either: (i) different isotope (ii) different nuclear spin state [(iii) different element (diffuse scattering)] • •

Mean

measure of expected value -

– coherent scattering

interference effects – average structure – Bragg diffraction

Std deviation

measure of dispersion – “spin”/“isotopic”

– incoherent scattering

single particle dynamics

b co



E

(

b

) 

b

 

x i b i b inc



Var

(

b

) 

E

(

b

2 ) 

E

(

b

) 2  

x i b i

2  ( 

x i b i

2 

..which leads to comparison to X ray scattering

Form Factors

• The form factor,

f

(

Q

) is the Fourier Transform of the scattering density r (r) – for neutrons it is in the form of a d -function – for X-rays the electron cloud distribution.

f

(

Q

)   0  r (

r

) exp 

i

Q  r 

dr

X-ray form factor

X-ray beam: Incident plane wave (in phase) Path difference: therefore, X-rays have a form factor that causes amplitude to die off as a function of 2 q

Bigger object (cation), faster drop off

Form factors

Bigger angle: greater path difference, more drop off X-rays • Nucleus is infinitesimal point wrt • • neutron wavelength No destructive interference Isotopic dependence Neutrons

Scattering lengths and cross-sections s  4 

r

2  *  4 

r

2  2  4 

b

2

Summary

• Spin, charge etc • Energy, velocity, wavelength • Moderation • Cross section, scattering length • X-rays vs. neutrons