interactions of particles with matter alfredo ferrari, cern, geneva vösendorf, may 27 th 2015...

Interactions of particles with Matter

Alfredo Ferrari,CERN, Geneva

Vösendorf, May 27th 2015

Accelerators for Medical Applications (CERN Accelerator School)

Alfredo Ferrari 2

Caveat: The talk is intentionally qualitative with minimal math, and no in-

depth discussion* … it is aimed at illustrating some general and well known concepts

about particle atomic and nuclear interactions… … in particular the nuclear physics part is kept at a (sub?)minimal

level and the maximum energy considered is limited at few hundreds MeV

May 27th, 2015

It will likely be disappointing for many and maybe still obscure for non-experts, I apologize in advance

Credits, in particular but not only: A.Mairani, V.Patera, P.Sala, F.Salvat, PDG…

* Extra material with more details is available in another file on Indico

Alfredo Ferrari 3May 27th, 2015

Overview:Photon interactions Compton Photoelectric Coherent scattering Charged particle atomic interactions: (average) stopping power Landau fluctuations Multiple scattering Bremsstrahlung and Pair

production radiation lengthNuclear interactions Elastic/Non-elastic hN nuclear interactions hA nuclear interactions AA nuclear interactions Photonuclear interactions

Neutronics: Reaction types Evaluated data files Examples of evaluated cross sections CaveatsWhat matters for what: Electron machines Shielding Hadron Therapy Isotope productionMiscellaneous Deuteron stripping ElectroMagnetic DissociationHigh energy showers ElectroMagnetic showers EM component of hadronic showers Spatial development of hadronic

showers

Alfredo Ferrari 4

Cross section metrics:

May 27th, 2015

1 mb 10 mb 100 mb 1 b 10 b 100 b 1 kb 10 kb 100 kb

350 g

cm

-2

30 c

m

350 k

g c

m-2

300 m

350 m

g c

m-2

0.3

mm

3.5

mg

cm

-2

3

m

35 k

g c

m-2

30 m

3.5

kg

cm

-2

3 m

35 g

cm

-2

3 c

m

3.5

g c

m-2

3 m

m

35 m

g c

m-2

30

m

Pb

11.4

g c

m-3

20 g

cm

-2

10 c

m

20 k

g c

m-2

100 m

20 m

g c

m-2

0.1

mm

0.2

mg

cm

-2

1

m

2 k

g c

m-2

10 m

200 g

cm

-2

1 m

2 g

cm

-2

1 c

m

200 m

g c

m-2

1 m

m

2 m

g c

m-2

10

m

C

2 g

cm

-3

Barn (b) = 10-24 cm2

g/mole][]molebcm[6.0 b][g/mole][

]molebcm[])(g/cm[

b][g/mole][

]molebcm[]g/cm[]cm[

1-1-21-1-2

121

1-1-231-11

APNP

N

P

N

P

N

P

ZN

mass

electrons

P

N

mass

atoms

P

N

volume

atoms

AAvA

Av

A

Av

A

Av

A

Av

A

Av

A

Av

May 27th, 2015 Alfredo Ferrari 5

Charged particle (atomic)interactions

Alfredo Ferrari 6

Charged particles dE/dx

Two main problems: Compute the average energy loss, for a given particle, in a given material, at a

given energy Compute the distribution of actual energy losses around the average value (energy

loss fluctuations) not discussed today

The problem is to compute the moments of the energy loss distribution

It is a central problem both in dosimetry and in general in radiation physics

May 27th, 2015

All energy transfers to the target medium are in the end mediated by Coulomb interactions of charged particles following atomic or nuclear

interactions

Alfredo Ferrari 7

Coulomb collisions among charged particles

Rutherford cross section (mprojm << MtargM):

using the 4-momentum transfer the cross section becomes:

In this form the cross section is no longer dependent on the (m << M) assumption and it works in every frame!

Finally, using, T=q2/2M (T=target recoil energy):

2sin4 422

22222

p

cmrZz

d

d eeRuth

)cos1(22

sin2 22 pqpq

42

22222

2

4

q

cmrZz

dq

d ee

M

m

T

cmrZz

dT

d eee22

22222

May 27th, 2015

The dependence on the recoil energy is essentially given by the 1/T2 term.

It is therefore clear from such formulae, that low energytransfers are much more likely than large ones.

Alfredo Ferrari 8

Coulomb collisions: considerations

For a given projectile/energy combination: The cross section per atom is given by

Z times the cross section on one electron ( Z x 12) 1 time the cross section on the atomic nucleus (1 x Z2)

The q2 (4-momentum transfer) dependence is the same for light/heavy target/projectile Energy losses due to interactions on atomic electrons are Mnucleus/me times larger than

those on atomic nuclei (T=q2/2M) for the same q2

Angular deflections are the same for the same q2

Energy losses are dominated by interactions with electrons (so called electronic stopping power) by a factor Mnucleus/(Z me) = mamu/me A/Z and are computed as the sum of: Close collisions (collisions energetic enough to be ~ on free electrons) Distant collisions (lower energy transfer, interaction involving the whole atom)

Angular deflections are mostly due to interactions on atomic nuclei by a factor Z

May 27th, 2015

Alfredo Ferrari 9

The maximum energy transfer, Tmax, to an electron is dictated by kinematics and given by:

2

222

max

21

2

Mm

Mm

cmT

ee

e

Common approximation

Close collisions: secondary electrons

The cross section for producing an electron of energy Te for an incident particle of kinetic energy T0 = ( − 1)Mc2 (note now Mmproj) and charge z is given for spin 0 and spin ½ particles by:

Similar expressions hold for e-e- (Møller, Tmax=T0/2) and e+e- (Bhabha, Tmax=T0) scattering. In all cases the dependence on the energy of the secondary electron is mostly due to the 1/T2 term.

May 27th, 2015

2

20

22

2

2

2

2

11

2

McT

T

T

T

T

cmr

dT

d e

max

e

e

ee

e

For spin ½ only

ATTM

m

M

pcmT e

e /500

142 0012

22

max

For a 200 MeV/n ion Tmax~ 400 keV RCSDA H2O~2 mm, for 20 MeV e- RCSDA H2O~45 mmTmax determines the extent of the buildup region and of the electronic (dis)equilibrium!

Alfredo Ferrari 10

The (unrestricted) electronic (ionization) stopping power for charged particles heavier than the electron can be obtained summing up distant and close collisions for spin 0 or spin 1/2 particles as:

while for electrons (Tmax =T0/2) and positrons (Tmax =T0) is given by:

Unrestricted dE/dx for heavy particle:

Z

CLzzL

McT

T

I

Tcmzcmrn

dx

dE eeffeee

hv

2)(2)(24

12

)1(

2ln

22

2122

0

2max2

22max

22

2

222

ln 44 relativistic rise

May 27th, 2015

Mean excitation energy

)..( volumeunitperelecnP

ZNn

A

Ave

For spin ½

only

2

22

2

20

2

22 1

8

12ln

121

2

1ln

2

I

Tcmnr

dx

dE eee

el

32

2

2

20

2

22

1

4

1

10

1

1423

12

12ln

2

I

Tcmnr

dx

dE eee

po

Densitycorrection

Barkas (z3) term Bloch (z4)

termShell

corrections

Effective charge

Alfredo Ferrari 11

dE/dx and range examples:

May 27th, 2015

for =1.8 b Nucl. Int. dominated

dE/dx dominated

Continuous Slowing Down Approximation (CSDA) range, RCSDA, or simply R = total amount of matter traversed by a particle of energy E0 whenever the energy losses are the average ones

It is a useful concept for (heavy) charged particles up to the energy where nuclear reactions dominate.

Since dE/dx is approx. a function only of the particle velocity, , and of its charge squared, z2, the following scaling property holds:

Eg the range of an particle of momentum p is approximately equal to the range of a proton of momentum p/4, same momentum per nucleon, eg of energy Tp = T/4 in the non relativistic regime, and again TpT/4 in the relativistic case

Ex

ER

meanEcsda d

d

d1

0

0

b

abaaaa

ab

abbbbb M

MppzMR

zz

MMpzMR ,,

/

/,,

22

Alfredo Ferrari 12

dE/dx: considerations:

dE/dx in a given material depends only on the particle velocity, , and charge, z

thus particles with the same velocity and charge have roughly the same energy loss.

if one measures distances in units of dx, g/cm2, the energy loss is weakly dependent on the material, as it goes like Z/A plus the logarithmic dependence on I

Obviously dE/dx depends on the projectile charge squared In practice, due to shell corrections, dE/dx never behaves like 1/Ek at low

energies The energy loss, when plotted as a function of =p/Mc, has a broad minimum

at 3-3.5. This minimum is almost constant up to very high energies, if the restricted

energy loss (that is the energy loss due to energy transfers smaller than some suitable threshold) is considered

In practice, most relativistic particles have energy losses in active detectors close to the minimum and are called minimum ionizing particles, or mip’s

May 27th, 2015

Alfredo Ferrari 13

Energy loss: examples (from PDG)

May 27th, 2015

Alfredo Ferrari 14

Bremsstrahlung: The classical rate of energy loss for a charged particle experiencing an acceleration a is given by:

In reality things are much more complex and atomic screening plays a major role in determining the actual bremsstrahlung rate,

May 27th, 2015

The cross section differential in v = w/E0 (w photon energy, E0, incident particle energy)::

Here fc is an higher order correction (the so called Coulomb correction), 1, 2 are the (elastic) screening functions for nuclear bremsstrahlung, and 1, 2 are the (inelastic) screening functions for (incoherent) bremsstrahlung on atomic electrons.

4

1

4)1(

3

2

log3

2

4

1log

3

1

43

4

3

44

d

d

2121

112

2242

Zv

ZZ

fZvvm

m

v

Zzr

v

σc

p

eebrem

Alfredo Ferrari 15

Bremsstrahlung spectra: examples

May 27th, 20151/Z2 v dbrem/dv for e- (red)/e+ (blue) at 12 MeV on Pb (left) and 12 GeV on Al (right)

Alfredo Ferrari 16

Pair production:The matrix elements of the bremsstrahlung are related to those of pair production by the substitution k - k and p - p, where p is the four momentum or either the incident particle in the bremsstrahlung emission or the four momentum of one of the pair of particles in the pair production.

In another way, they Feynman diagram of pair production is the same as the bremsstrahlung one, rotated of 90 , where one of the electron lines, now going back in time, becomes the (outgoing) positron line

May 27th, 2015

where again fc is the Coulomb correction, and 1, 2 are the same (elastic) screening functions as for nuclear bremsstrahlung, while 1, 2 the same (inelastic) screening functions as for (incoherent) bremsstrahlung on atomic electrons.

4

1

4)1(

3

2log

3

2

4

1log

3

1

41

3

4

3

44

d

d2121112

2

242

ZuuZ

ZfZuu

m

mZzr

u

σc

p

ee

pair

The integration of the pair production cross section over all possible angles brings to the usual cross section differential in u = E+/k , as, for instance, reported in the review of Tsai:

Alfredo Ferrari 17

Pair production: examples

1/Z2dpair/du+ for different incoming photon energies (in units of mec2)

May 27th, 2015

Alfredo Ferrari 18

Radiation length X0

Integrating the bremsstrahlung cross section gives the result

where the radiation length X0 has been introduced:

X0 is the length over which the initial energy is reduced to 1/e

In a fully analogue manner:

Zf

Z

LLE

P

N

m

mZzr

vvvE

P

NXE

x

E

c

fsradfs

radA

A

p

ee

brem

A

A

brem

11

18

14

d

dd

d

d

0

2

242

1

001

00

Zf

ZL

LPN

mm

ZzrX c

fsradfs

radA

A

p

ee

11

181

41

2

2420

May 27th, 2015

0

2

2421

0

1

7

911

54

1

9

74

d

dd X

Zf

Z

LL

P

N

m

mZzr

uu

P

Npairc

fsradfs

radA

A

p

ee

pair

A

Apair

3/2

3/1

1194log

15.184log

Zm

mL

Zm

mL

e

pfsrad

e

pfsrad

Bremsstrahlung (cm2/g) scales as (same as pair production):

A

ZX

21

0

Alfredo Ferrari 19

The Molière cross section for particle-Nucleus Coulomb scattering accounting for screening:

2224

4222

411

14)(

aa

eeeMol E

cmZZzr

Coulomb collisions: Molière cross section

T(MeV) a (mrad)

Mol (kb) R (g/cm2) -1 (g/cm2)

0.1 38 900 0.0187 4.98x10-5

Al 1.0 8.4 346 0.555 1.30x10-4

10.0 1.14 308 5.86 1.46x10-4

0.1 317 517 0.0311 6.66x10-4

Pb 1.0 35 784 0.784 4.39x10-4

10.0 4.5 793 6.13 4.34x10-4

May 27th, 2015

22224

42222

2224

42222

22

2

)(

4

)21cos1()2

1cos1(

)cos1(

d

d),(

d

d

d

d

a

ee

a

ee

a

Ruthscr

RuthMol

E

cmrZz

E

cmrZzK

Alfredo Ferrari 20

The Molière distribution is an approximate result which holds for a specific single scattering cross section (the Molière one), for path-lengths not too short, and angles not too large. It is expressed as an universal function, which depends only on one parameter B.

Probability of scattering through an angle after travelling a total step length t:

where the scaled variable is given by:

and B is solution of the transcendental equation (cc and bc are material dependent constants):

n

n

Mol

uuJuu

nf

fB

fB

tF

4

uln

4ed

!

1

sin...

11e2d2d,

224/u-

0 0

2

1

221

2

2

MCS: Molière distribution

Bethe correction, not present in the original

theory

E

t

Bcc

c

c2

21

200lnlntb

bBB c

May 27th, 2015

Single scatt. tailFor >>c f11/4

Gaussian term

A Gaussian approximation for the MCS distr. (like ) can often be found

However it is not precise and, most important, it ignores the tails!

00

ln038.012.19

X

t

X

t

prms

Alfredo Ferrari 21

Cross section metrics: X0

May 27th, 2015


X0:: C 42.7 g cm-2 (21.3 cm), Cu 12.86 g cm-2 (1.44 cm), Pb 6.37 g cm-2 (0.56 cm)

20 g

cm

-2

10 c

m

20 k

g c

m-2

100 m

20 m

g c

m-2

0.1

mm

0.2

mg

cm

-2

1

m

2 k

g c

m-2

10 m

200 g

cm

-2

1 m

2 g

cm

-2

1 c

m

200 m

g c

m-2

1 m

m

2 m

g c

m-2

10

m

C

2 g

cm

-3

350 g

cm

-2

30 c

m

350 k

g c

m-2

300 m

350 m

g c

m-2

0.3

mm

3.5

mg

cm

-2

3

m

35 k

g c

m-2

30 m

3.5

kg

cm

-2

3 m

35 g

cm

-2

3 c

m

3.5

g c

m-2

3 m

m

35 m

g c

m-2

30

m

Pb

11.4

g c

m-3

>>>>>Charged particle

Coulomb interactions

Alfredo Ferrari 22

Energy loss e+/e-: examples, Water and Lead

Critical energy

May 27th, 2015

Critical energy

Critical energy the energy at which collision and radiative energy losses are equal

Given that: Bremsstrahlung scales as

(same as pair production):

Electronic stopping power (dE/dx) scales (roughly) as:

® The critical energy approximately scales as:

Z

Ec

1

A

ZX

21

0

A

Z

dx

dE

Bragg peaks: ideal proton caseThe Bragg peak is a pronounced peak on the Bragg curve (laterally integrated depth-dose curve) which plots the energy loss of ionizing radiation during its travel through matter. For protons, alpha particles and heavy ions, the peak occurs immediately before the particles come to rest. This is called Bragg peak, for William Henry Bragg who discovered it in 1903.

Bragg peaks: ideal proton case


Curves (200 MeV p on Water): Red: pure CSDA, no nucl. int. Green: MCS + CSDA Blue: CSDA + nuclear int. Purple: no MCS, no nucl. int, Landau fluct. on. Light Blue: full calculation

Alfredo Ferrari 24

Electron energy losses: complete examples, H2O and Pb

May 27th, 2015

Right: depth-dose curve for20 MeV e- on WaterLeft: depth dose curve for10 MeV e- on Lead Purple: pure CSDA

(bremss. included) Cyan: MCS + CSDA Red: full calculation Blue: secondary

electrons (E> 10 keV) contribution

Green: Bremss. photon contribution

Note that the effect of Multiple Coulomb Scattering (MCS) is

much more important than for the 200 MeV

p example


Nuclear reactions

Nuclear reactions: elastic and non-elastic

In order to understand Nucleus-Nucleus (AA) and Hadron-Nucleus (hA) nuclear reactions one has to understand first Hadron-Nucleon (hN) reactions, since nuclei are made up by protons and neutrons

In general there are two kind of nuclear reactions (for both hN and hA/AA) elastic and non-elastic:

Elastic interactions are those that do not change the internal structure of the projectile/target and do not produce new particles. They transfer part of the projectile energy to the target (lab system) Or equivalently they deflect in opposite directions target and projectile

in the centre-of-mass system (CMS) with no change in energy. There is no threshold for elastic interactions

Non-elastic reactions are those where new particles are produced and/or the internal structure of the projectile/target is changed (e.g. exciting a nucleus). Any specific non-elastic reactions has usually an energy threshold below

which the reaction cannot occur (the exception being neutron capture)

Nuclear reactions: elastic and non-elastic C1


Nuclear interactions: generalities

Alfredo Ferrari 27

From hN to hA cross sections:

Proton (Neutron) Carbon cross sections computed in self-consistent Glauber

approach accounting for inelastic screening

starting from NN ones (left)May 27th, 2015

Pion production threshold ~290

MeV

NN cross sections+

Nucleus ground state+

Glauber calculus

Alfredo Ferrari 28

nA, pA Cross sections:

May 27th, 2015

Hadronic interactions are mostly surface effects hadron nucleus cross section scale with the target atomic mass A2/3 (RnucA1/3)

Neutron onCarbon

Neutron onLead

Proton onLead

reac minimum maximum penetration

Alfredo Ferrari 29

Cross section metrics: int

May 27th, 2015


int:: C 85.8 g cm-2 (42.9 cm), Cu 137.3 g cm-2 (15.3 cm), Pb 199.6 g cm-2 (17.5 cm)

20 g

cm

-2

10 c

m

20 k

g c

m-2

100 m

20 m

g c

m-2

0.1

mm

0.2

mg

cm

-2

1

m

2 k

g c

m-2

10 m

200 g

cm

-2

1 m

2 g

cm

-2

1 c

m

200 m

g c

m-2

1 m

m

2 m

g c

m-2

10

m

C

2 g

cm

-3

350 g

cm

-2

30 c

m

350 k

g c

m-2

300 m

350 m

g c

m-2

0.3

mm

3.5

mg

cm

-2

3

m

35 k

g c

m-2

30 m

3.5

kg

cm

-2

3 m

35 g

cm

-2

3 c

m

3.5

g c

m-2

3 m

m

35 m

g c

m-2

30

m

Pb

11.4

g c

m-3

30

Simplified scheme of Nuclear interactionsTarget nucleus description (density, Fermi motion, etc)

Preequilibrium stage with current exciton configuration and excitation energy(not discussed in this lecture)

Generalized IntraNuclear cascade

Evaporation/Fragmentation/Fission

γ deexcitation

t (s)

10-23

10-22

10-20

10-16

Glauber-Gribov cascade with formation zone

High energies (above few

GeV)

May 27th, 2015 Alfredo Ferrari

Alfredo Ferrari 31

The projectile is hitting a “bag” of protons and neutrons representing the nucleus. The products of this interaction can in turn hit other neutrons and protons and so on. The most energetic particles, p,n, ’s (and a few light fragments) are emitted in this phase

… INC, a bit like snooker…

May 27th, 2015

…it is in this phase that if energy is enough extra “balls” (new particles) are produced (contrary to snooker). The target “balls” are anyway protons and neutrons, so further collisions will mostly knock out p’s and n’s

Alfredo Ferrari 32

When the excitation energy is similar or below the separation energy (energy needed to break the nuclear binding and emit a nucleon), prompt photons are emitted during gamma deexcitation

Evaporation:

May 27th, 2015

The process is terminated when all available energy is spent the leftover nucleus, possibly radioactive, is now “cold”, with typical recoil energies MeV. For heavy nuclei the initial excitation energy can be large enough to allow breaking into two major chunks (fission).

After many collisions and possibly particle emissions, the residual nucleus is left in a highly excited “equilibrated” state (compound nucleus). De-excitation can be described by statistical models which resemble the evaporation of “droplets”, actually low energy, isotropically emitted, particles (p, n, d, t, 3He, alphas…) from a “boiling” soup characterized by a “nuclear temperature”Since only neutrons have no Coulomb barrier to overcome, neutron emission is strongly favoured.

Thick target examples: neutrons


197Au(p,xn) @ 68 MeV, stopping target Data: JAERI-C-96-008, 217 (1996)

9Be(p,xn) @ 113 MeV, stopping target Data: NSE110, 299 (1992)

Evaporationneutrons

Cascade, preequilibrium neutrons

Double differential neutron yield, energy

spectra at various angles

34

1 A GeV 208Pb + p reactions Nucl. Phys. A 686 (2001) 481-524

Example of fission/evaporation

Quasi-elastic

Spallation

Deep spallationFission

Fragmentation

Evaporation

• Data• Model final result• Model after cascade• Model after preeq

May 27th, 2015 Alfredo Ferrari

The production of specific residuals is the result of the very last step of the nuclear reaction, thus it is influenced by all the previous stages

Alfredo Ferrari 35

Residual NucleiRight: color scale of isotope production as a function of neutron excess (x-axis) and atomic number (y-axis) for various proton energy/target combo’s. The black line is the stability line Particle beams tend to produce

proton rich isotopes because of the preference for evaporating neutrons rather than charged particles

Isotopes produced by fission are typically neutron rich (at least for fission on actinides)

there is an obvious complementarity between the two techniques

May 27th, 2015

Ion-Ion reaction cross sections:


Energy/nucleon (MeV)

reac

(mb)

Energy/nucleon (MeV)

reac

(mb)

fmA2.1MeVAA

4.11 3/1/3/13/1

2A/BBA

BA

BACoul

k

CoulBAreac R

ZZV

E

VRR

Carbon onCarbon

4He onLead

Evaporation+ prompt

“Heavy” ion nuclear interactions:


b

fmA2.1 3/1

/

2

A/BBA

BAreac

R

RR

Fast stageCascadePreeq.

Projectilespectators

Targetspectators

Evaporation of the projectile-like fragment: low energy particles isotropically in its rest frame highly energetic

ones, strongly forward peaked, in the lab frame (including the final residual)!

Contrary to hA collisions there could be a projectile-like residual flying forward

Examples of (thin target) neutron emission spectra:

Alfredo Ferrari 38

C on C @ 135 MeV/n(0,15,30,50,80,110 deg)

Ar on C @ 95 MeV/n(0,30,50,80,110 deg

Neutron double differential spectra at various angles

Symbols: exp. data

Projectile evaporation forward

peakHigh energy tails (>> Ebeam/n)

May 27th, 2015


Monoenergetic and SOBP for 12C:

~70% reductiondue to nucl. Int.

XC

Left: 12C @ 400 MeV/n in water

Depth-dose distribution (top)

and beam attenuation

(bottom)

Right: Spread Out Bragg Peak, global

depth-dose distribution (top) and detail of ion

fragment contributions

(bottom)

Photonuclear reactions: Pb(,x)


Vector Meson Dominance

Giant Dipole Resonance (GDR)

Delta (1)

Quasideuteron


Neutronics

Alfredo Ferrari 42

Low energy neutron interactions:

At energies below 10-20 MeV, the specific nuclear structure of individual isotopes starts to play a major role, and cross sections are no longer a smooth function of A (mass number), rather…

Evaluated nuclear data files (ENDF, JEFF, JENDL...) typically provide neutron (cross sections) and secondary particles (sometimes

only neutrons) inclusive distributions for E < 20 MeV for all channels. Recent evaluations include data up to 150/200 MeV for a few isotopes

are stored as continuum + resonance parameters

May 27th, 2015

Neutrons, being the only “stable” (T1/210 min) neutral particle are the dominating component at low energies. They undergo elastic and non-elastic nuclear interactions until in most cases they are thermalized and captured by a nucleus (n + A

ZX A+1ZX + ’s). The

slowing down is mostly accomplished via elastic interactions since non-elastic ones (apart capture) have thresholds

“Low” energy neutrons: Thermal neutrons: Maxwellian distribution (most probable energy @ 293 K ~

0.025 eV) Epithermal neutrons, resonance neutrons, slow neutrons: 0.4 eV – 0.1 MeV Fast neutrons: > 0.1 MeV


2

,

,*

)(cos12

ZAn

ZAnnkinrec Mm

MmET

Non-elastic: (n,2n) (>~8 MeV), (n,3n), (n,p), (n,), (n,d), (n,np)… Eth ~ several MeV

Inelastic: (n,n’) (’s emitted with the n), Eth~MeV’s (even-even nuclei), ~keV’s (heavy odd-odd nuclei)

Elastic: no thresh., energy transfer (*= cms scattering angle) Proton target: Trec max=Ekin n, <Trec>=1/2 Ekin n

Lead target: Trec max=0.019 Ekin n, <Trec>=0.009 Ekin n (for isot. scatt., En <0.5 MeV)

Capture: no thresh., important in the thermal (and resonance) regions, mostly n + A

ZX A+1ZX + ’s, notable exceptions:

3He(n,p)3H, Q = 764 keV 14N(n,p)14C, Q = 626 keV 10B(n,)7Li, Q = 2790 keV

Neutron data: examples (eg from

http://www.oecd-nea.org/janis/):


ENDF/B-7.1: USJENDL-4.0: JapanJEFF-3.1.2: Europe…TENDL-2013:Model (TALYS)

Some of them include data for incident charged particles as well, and/or evaluations up to 150/200 MeV for some isotopes

http://www.oecd-nea.org/janis/

http://www.oecd-nea.org/janis/

… or from http://www.nndc.bnl.gov :


http://www.nndc.bnl.gov/

Typical neutron cross section

tot

thermal

1/v

resolvedresonance

region

1eV 1keV

Resonances energy levels in compound nucleus A+1Z*


unresolved resonance region

Ekin incident neutron

resonancespacingfew eV

Resonance spacing too dense overlapping resonances

Before Doppler broadening(capt<<tot)

Alfredo Ferrari 47

Low Energy Neutron Cross sections: C

May 27th, 2015

Blue: total cross sectionGreen: elastic cross sectionRed: capture cross section

kT @ 293 K

0 capt~0.0035 b

Alfredo Ferrari 48

Low Energy Neutron Cross sections: 56Fe

May 27th, 2015

Blue: total cross sectionGreen: elastic cross sectionRed: capture cross section

0 capt~2.8 b

Alfredo Ferrari 49

Low Energy Neutron Cross sections: 113Cd

May 27th, 2015

kT @ 293 K

0 capt~21000 bBlue: total cross sectionGreen: elastic cross sectionRed: capture cross section


Photon (atomic) Interactions

Alfredo Ferrari 51

Photon interactions:

May 27th, 2015

Alfredo Ferrari 52

Photon cross sections: scaling The photoelectric macroscopic cross section (cm2/g) scales

as:

The Compton macroscopic cross section (cm2/g) scales as:

The coherent (Rayleigh) macroscopic cross section (cm2/g) scales as:

The pair production macroscopic cross section (cm2/g) scales as:

A

Zpe

5

A

Zincoh

A

Zcoh

2

A

Zpair

2

May 27th, 2015

Alfredo Ferrari 53

Compton scattering: dynamicsKlein-Nishina cross section (see for example Heitler, “The Quantum Theory of Radiation”):

2

2

22 cos42

'

'

'

4

1

k

k

k

k

k

kr

d

de

KN

Let e be the polarization vector of the incident photon, and e’ that of the scattered one: 'cos ee

Split into the two components, and || to e respectively (actually with e’ to the plane (e,k’), or contained in the plane (e,k’) ):

222 cossin1cos

22

2

22||

2

22 cossin42

'

'

'

4

12

'

'

'

4

1

k

k

k

k

k

kr

d

d

k

k

k

k

k

kr

d

dee

k

e

k’

e’

May 27th, 2015

The effect of polarization is important for polarized photon beams (eg synchrotron radiation

source)!! It breaks the scattering azimuthal symmetry!!!

Alfredo Ferrari 54

Atomic electrons are bound atomic corrections are important

Atomic corrections:

Mild effects on Compton (except at low energies):

Small angles ( small energy/momentum transfers) suppressed

Compton line broadened by bound electron motion

Dominant effect on coherent scattering

Atomic effect dominant, only small angle scattering is left

Medium-large angle scattering suppressed because of loss of coherence

Under certain approximations atomic effects can be described via the inelastic and elastic atomic form

factors

May 27th, 2015

Alfredo Ferrari 55

Compton cross section: Gold*

*From the Penelope manualGreen curve: Klein-Nishina formula.Dashed curve: Waller-Hartree approximation.Black curve: relativistic impulse approximation.

May 27th, 2015

KN 1/k

Z0=Z 8/3re2

Alfredo Ferrari 56

Photoelectric effect: just a reminder (more in the backup slides)

Z Z*

Te k - Ui

k

The incident photon is absorbed by an atomic electron which is emitted with kinetic energy roughly equal to the incident photon energy minus the (electron) binding energy. The atom is left in an excited state

The electron must be bound to fulfill energy-momentum conservationWhat is required to describe fully p.e. interactions?

Cross sections for each atomic shell Angular distribution of photoelectrons Effect of (possible) photon polarization De-excitation of atomic ions left after the interaction

Fluorescence (X-rays) (radiative, between shells) Auger emission (electrons, between shells) Coster-Kronig emission (electrons, intra-shell)

May 27th, 2015

Alfredo Ferrari 57

Photon cross sections: summary

Compton dominated

Compton dominated

Photoelectric dominated

Photoelectric dominated

Pair dominated

Pair dominated

p.e.=photoelectric cross section; incoh=Compton cross section;coherent=Rayleigh cross section; nuc =photonuclear cross section;N=pair production cross section, nuclear field;e=pair production cross section, electron fieldMay 27th, 2015

Photonuclear

Alfredo Ferrari 58

How to make good use of (unwanted) nuclear interactions:

Blue: CalculationRed: data

Green: dose profile

May 27th, 2015

Z (mm)

Photo

n y

ield

[sketch and exp. data taken from F. Le Foulher et al IEEE TNS 57 (2009), E. Testa et al, NIMB 267 (2009) 993. Exp. data have been reevaluated in 2012 with substantial corrections]

E> 2 MeV, within few ns

from spill

Z (mm)

GANIL: 90 deg prompt* photon

yields by 95 MeV/n 12C in PMMA

* Prompt photons: nuclear de-

excitation photons emitted in nuclear

interactions

Alfredo Ferrari 59

Unwanted nuclear physics turned useful: ß+ isotope prod.

May 27th, 2015

p beam

Thanks for your attention!

Dose map ß+ emitters produced in nuclear interactions

map

NIST Database:http://physics.nist.gov/PhysRefData/Xcom/Text/chap4.html


References/Literature I E.A. Uehling, Annual Review of Nuclear Science, 4, 315 (1954) E.A. Uehling, Annual Review of Nuclear Science, 4, 315 (1954) Nuclear Science Series, Report n. 39, “Studies if penetration of charged

particles in matter”, National Academy of Sciences – National Research Council, (1964)

H.A. Bethe, J. Ashkin, “Passage of Radiations Through Matter”, Experimental Nuclear Physics, Vol. I, Ed. E. Segre` (1960)

R.D. Birchoff, “The passage of fast electrons through matter”, Encyclopedia of Physics, Vol. XXXIV, Springer-Verlag (1958)

ICRU Report n. 37, “Stopping Powers for Electrons and Positrons”, (1984) ICRU Report n. 49, “Stopping Powers and Ranges for Protons and Alpha particles”,

(1993) ICRU Report n. 73, “Stopping of Ions Heavier than Helium”, (2005) !!Be careful

to get also the errata and addenda (2009)!! ICRU Report n. 63, “Nuclear Data for Neutron and Proton Radiotherapy and for

Radiation Protection”, (2000) H.W.Koch, J.W.Motz, “Bremsstrahlung Cross-Section Formulas and Related Data”,

Review of Modern Physics 31, 920, 1959 J.W.Motz, H.A.Olsen, H.W.Koch, “Pair Production by Photons”, Review of Modern Physics

41, 581, 1969May 27th, 2015 Alfredo Ferrari 62

Useful bibliography/sites:

References/Literature I Yung-Su Tsai, “Pair Production and Bremsstrahlung of charged particles”, Review of

Modern Physics 46, 815, 1974 “Monte Carlo Transport of Electrons and Photons”, eds T.R.Jenkins, W.R.Nelson and A.

Rindi, Plenum Press 1988 E.Gadioli, and P.E.Hodgson, “Pre-equilibrium Nuclear Reactions”, Clarendon Press,

Oxford, 1992 “Reference Input Parameter Library”, Nuclear Data Sheets 110, 3107, 2009, https://www-nds.iaea.org/RIPL-3/ A.Ferrari, P.R.Sala, “The physics of High Energy reactions”, Proc. of the Workshop on

Nuclear Reaction Data and Nuclear Reactors Physics, Design and Safety, World Scientific, A.Gandini, G.Reffo eds, Vol.2, p.424 (1998),

http://www.fluka.org/content/publications/1996_trieste.pdf http://www.nist.gov/pml/data/index.cfm http://www.nist.gov/physlab/data/star/index.cfm http://www.nist.gov/pml/data/xcom/index.cfm http://www.nist.gov/pml/data/radiation.cfm http://www.nndc.bnl.gov/ http://www.oecd-nea.org/janis/


Useful bibliography/sites:

interactions of particles with matter alfredo ferrari, cern, geneva vösendorf, may 27 th 2015...

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