Post on 19-Jan-2016
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
1 mb 10 mb 100 mb 1 b 10 b 100 b 1 kb 10 kb 100 kb
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
May 27th, 2015 Alfredo Ferrari 23
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
May 27th, 2015 Alfredo Ferrari 25
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
May 27th, 2015 Alfredo Ferrari 26
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
1 mb 10 mb 100 mb 1 b 10 b 100 b 1 kb 10 kb 100 kb
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
May 27th, 2015 Alfredo Ferrari 33
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:
May 27th, 2015 Alfredo Ferrari 36
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:
May 27th, 2015 Alfredo Ferrari 37
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
May 27th, 2015 Alfredo Ferrari 39
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)
May 27th, 2015 Alfredo Ferrari 40
Vector Meson Dominance
Giant Dipole Resonance (GDR)
Delta (1)
Quasideuteron
May 27th, 2015 Alfredo Ferrari 41
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
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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/):
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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
Typical neutron cross section
tot
thermal
1/v
resolvedresonance
region
1eV 1keV
Resonances energy levels in compound nucleus A+1Z*
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unresolved resonance region
Ekin incident neutron
resonancespacingfew eV
Resonance spacing too dense overlapping resonances
Before Doppler broadening(capt<<tot)
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Low Energy Neutron Cross sections: C
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Blue: total cross sectionGreen: elastic cross sectionRed: capture cross section
kT @ 293 K
0 capt~0.0035 b
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Low Energy Neutron Cross sections: 56Fe
May 27th, 2015
Blue: total cross sectionGreen: elastic cross sectionRed: capture cross section
0 capt~2.8 b
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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
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Photon (atomic) Interactions
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Photon interactions:
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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
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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!!!
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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.
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KN 1/k
Z0=Z 8/3re2
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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
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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
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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
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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
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NIST Database:http://physics.nist.gov/PhysRefData/Xcom/Text/chap4.html
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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/
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Useful bibliography/sites: