Radioactive sources and radiation beam production - Part 1

First part of radiation sources course

1. F. CleriSources des radiations1
Radioactive sources and
radiation beam production

2. F. CleriSources des radiations2
radioactivity radiation exposure absorbed
source dose
absorber
material
radiation beam
Screening
(Pb, concrete,...)
source
Idealized configuration of an irradiation plant

3. F. CleriSources des radiations3
What energy? What radiation?
a particles?
Line spectrum ? Ei, Ej, Ek, … β± electrons?
γ rays?
Continouus spectrum ? Emin, Emax,… neutrons ?
protons, charged fragments…?
(old units,
Roentgen= 1/3786 C/kg)
Exposure Coulomb/kg
C kg-1
sourceCuries (Ci)
Bequerels (Bq)N. of emissions/sec
Gray (Gy = 1 J kg-1)
Rad (non-SI, 1 Rad=0.01 Gy)
Absorption
Dose = energy deposited
in the tissue

4. F. CleriSources des radiations4
What energy? What radiation?
a particles?
Line spectrum ? Ei, Ej, Ek, … β± electrons?
γ rays?
Continouus spectrum ? Emin, Emax,… neutrons ?
protons, charged fragments…?
(old units,
Roentgen= 1/3786 C/kg)
Exposure Coulomb/kg
C kg-1
sourceCuries (Ci)
Bequerels (Bq)N. of emissions/sec
Gray (Gy = 1 J kg-1)
Rad (non-SI, 1 Rad=0.01 Gy)
Absorption
Dose = energy deposited
in the tissue

5. F. CleriSources des radiations5
A final quantity useful in the manipulation of radiation beams is the
SPECIFIC GAMMA CONSTANT.
It is a measurable quantity that combines the previous concepts, aimed
at establishing a relationship between exposure to a source used in the
laboratory, and the relative level of safety, i.e.:
1 s
1 m
X Bq
How much doses
do you catch
during the day?

6. F. CleriSources des radiations6
What energy? What radiation?
a particles?
Line spectrum ? Ei, Ej, Ek, … β± electrons?
γ rays?
Continouus spectrum ? Emin, Emax,… neutrons ?
protons, charged fragments…?
(old units,
Roentgen= 1/3786 C/kg)
Exposure Coulomb/kg
C kg-1
sourceCuries (Ci)
Bequerels (Bq)N. of emissions/sec
Gray (Gy = 1 J kg-1)
Rad (non-SI, 1 Rad=0.01 Gy)
Absorption
Dose = energy deposited
in the tissue
Specific gamma constant
C kg-1 s-1 Bq-1 à 1 m
c

7. F. CleriSources des radiations7
sourceCuries (Ci)
Bequerels (Bq)N. of emissions/sec
What energy? What radiation?
a particles?
Line spectrum ? Ei, Ej, Ek, … β± electrons?
γ rays?
Continouus spectrum ? Emin, Emax,… neutrons ?
protons, charged fragments…?
(old units,
Roentgen= 1/3786 C/kg)
Exposure Coulomb/kg
C kg-1
Gray (Gy = 1 J kg-1)
Rad (non-SI, 1 Rad=0.01 Gy)
Absorption
Dose = energy deposited
in the tissue

8. F. CleriSources des radiations8
Radioactive sources: a bit of history...
Enrico Fermi in his
laboratory in Rome
Frédéric and Irene
Joliot-Curie in their
laboratory in Paris

9. F. CleriSources des radiations9
The discreteenergiesmeasuredfor beta electrons(whichcame onlyfrompoorlydone
experiments), in a few cases wereclose to somegenergies. The mostlogicalexplanationfor
the γ and β emissionof radioactive elementsat thattime wasa formof resonantconversion of
energy, of electromagneticorigin.
Most scientistsstillin 1916-17 tookthe atomicweightA, and not the charge numberZ, as the
orderparameter-not verysatisfactory-in Mendeleev'stable.
The commonlyobservedemissionof particlesα as single objects, supportedthe ideaof the
nucleus as composedof a set of α particles, and playedin favorof the possibilityof having
electronsinsidethe nucleus.
–
++
–

10. F. CleriSources des radiations10
The ideaof a "nuclearelectron" wasverypopular(Rutherford, Millikan, Sommerfeld, Aston...):
the electronsinsidethe nucleus neutralizethe (almost) double charge Q.
The onlywayto have 2 unitsof positive mass for eachunit of increasein Z, wasthento
assume thata second proton and a negativecharge wereadded(i.e. twoprotons plus an
electronmakea unit charge and a double mass, the mass of the electronbeingnegligible).
This sameidealedRutherford to propose the existence of a "neutron" in the nucleus, which
hestillimaginedin 1920 (BakerianLecture in Oxford) as a pair of particles, composedof a
positive and a negativecharge.
–
+
Neutron …?

11. F. CleriSources des radiations11
But already in 1913 Wilson had demonstrated that the spectrum of b's is
continuous and not discrete.
In 1914 Moseley showedexperimentallyusingX-ray diffraction techniques thatthe
mostintense line in the X-ray spectrumof a particularelement(Ka), isrelatedto
the position of the elementin the periodictable, i.e. to itsatomicnumber, Z.
In 1928 Ornstein showed that the spin of the nitrogen nucleus is S=1. If the 14N
nucleus were composed of 14 protons and 7 electrons, or an odd number (21) of
particles with spin 1/2, then the resulting nuclear spin would have to be half an
integer.
1932 - J. Chadwick identifies the neutron with mass = proton, by using beryllium to
absorb alpha particles emitted by Radium:
9Be +4He (α) →12C +1n

12. F. CleriSources des radiations12
1932 – In their quest to produce neutrons, I. and J. Curie used α
particles from Polonium to bombard 27Al targets, and obtained - by
chance - radioactive isotopes:

24He+1327Al→1530P+n
The 30P thus produced is radioactive:
30P → 30Si + e++ v
The crucial observation is that the
emission of positrons does not stop after
the bombardment has stopped, but it
continues for several minutes
…."as for a natural radioelement".

13. F. CleriSources des radiations13
1934 - E. Fermi produces neutrons in large quantities via an Rn/Be source:
The particles α emitted by the 222Rn
react with the 9Be, and produce
neutrons in large quantities.
With this new neutron source,
the Rome group begins to bomb
pure elements, in order of increasing Z.
And in fact, for Z ≥ 9 they start to
observe a quantity of new isotopes
emitters α, β, n, γ …
86222Rn→24He+84218Po↓24He+49Be→613C*→612C+n
RnBe
"cake"

14. F. CleriSources des radiations14
Once he got to Z=92, Fermi believed
have discovered still more isotopes
trans-uranic elements. He had almost
observe on 22 October 1934, the first
artificial fission in 235U.
In 1935 O. Hahn and L. Meitner in Berlin
confirm Fermi's experiments, and
identify the 239Pu in the products.
In 1938, the Joliot-Curies in Paris also confirmed the experiences of Rome
and Berlin. But they found among the products a radioisotope with a fairly
short half-life (about 3.5 hours), with chemical properties strangely "close to
those of Lanthanum".
They «almost» identified 141La, produced by fission, and so they passed - just
like Fermi, Hahn and Meitner - next to fission!!!

15. F. CleriSources des radiations15
In fact, thistype of resultindicatedthatthe new "reactions" weremuchmore
complexthantheyimagined.
As earlyas 1934, Ida Noddack, a German chemist, hadwarnedFermi to
compare the (chemical) propertiesof the new isotopes withthoseof all
knownelements, not justwithelementsclose to uranium.
Sheproposedthat"whenheavynucleiare
bombardedby neutrons, itisplausible that
the nucleus breaks intoseveralfragments, which
willof course beisotopes of knownelements,
but far fromthe mass of Uranium…”
The poorwomanwascompletelyignored…
It wasnot untilDecember18, 1938, thatHahn and Strassmannin Berlin
identifiedthe fission of U-235, bombardedby neutrons.

16. F. CleriSources des radiations16
Againin 1934, Fermi'sgroup in Rome made anotherdiscoverydestinedto
revolutionize(and disrupt) the livesof millions of people.
Bruno Pontecorvo and EdoardoAmaldi, the youngestof the group, observed
thatthe use of a screen made of light material, suchas the plan of a wooden
table, storedby chance betweenthe neutron source and the uranium target,
greatlyincreasedthe rate of radioactive countingof the products... Theythen
triedwithPb or Fe screens, but the woodwasyettooeffective!
Fermi thereforeproposedto use somethingevenlighter,
suchas paraffin(a wax composedof a mixture of
hydrocarbonsCnH2n+x).
The Geiger count rate explodedat the end of the scale!

17. F. CleriSources des radiations17
The legendehas isthatitwastime for a meal
at noon, and theyall wentto eat. During
of the meal, Fermi foundthe explanation...
It seemsthathesaidto Pontecorvo and Amaldi:
“If youfinda good agreement withthe theory,
youhave onlydonea good measure. But
If you'relucky, you'llfinda disagreementwith
the currenttheory: thisiswhereyouhave done
a real experiment!”
Fermi proposedthatneutrons weresloweddown by collisions withhydrogenatoms
(in woodand paraffin). He thenproposedto repeatthe experimentwithwater: they
plungedthe springand detector set intothe famous"goldfishbasin" of the Institute
of Physics... The counteralmostexploded!!!
Hence... "slowed-down" neutrons more efficient thanthe veryenergeticones??

18. F. CleriSources des radiations18
The historical concepts presented here all point in the direction of
identifying events at the scale of nuclei, which could be described as
"nuclear reactions", again in analogy with chemical reactions
between atoms... But with very important differences!
Typically, we consider reactions between two nuclei,
a "projectile" and a "target", .
A nuclear reaction leads to a rearrangement of the nucleons during
the collision, the products can range from a single particle to several
particles, also accompanied by X rays and γ rays :

€
Z
A
X+
Z'
A'
Y→
Z
i
A
i
W
i
+ω
j
j
∑
i
∑ZAXZ'A'Y

19. F. CleriSources des radiations19
Each nuclear reaction must conserve several quantities:
A+A'=Aii∑Z+Z'=Qii∑E+E'=Ei+i∑ ωjj∑p+p'=pi+i∑ kjj∑J+J'=Jii∑+±1()jj∑(−1)J(−1)J'=(−1)Jii∏ ⋅(−1)j∏ (±1)j
Conservation of mass
Conservation of charge
Conservation of energy
Conservation of momentum
Conservation of angular
momentum
Conservation of parity
€
(k
j
=ω
j
/c)
(and a few otherthings... baryonicnumber, leptonicnumber, "strangeness"...)

20. F. CleriSources des radiations20
For example, nitrogen-14 isbombardedwithα particles:
whichisalsowrittenas:
If the initial energiesare not toohigh (~ not givingriseto the emission
of subnuclearparticles) the reactioncan typicallydecomposeinto
threestages:
€
7
14
N+
2
4
He→
8
17
O+p
+
€
7
14
N(α,p)
8
17
O
€
7
14
N+
2
4
He→
9
18
F
[]
*
→
8
17
O+p
+
compound or
intermediate
nucleus
Initial state Final state

21. F. CleriSources des radiations21
In fact, for a reactionproducinga compound nucleus in a veryhigh
energystate, therecan bemanydifferentmodalitiesof radioactive
decay(= differentfinal states), called"reactionchannels" (or simply
"channels").”) :
€
€
13
27
Al+p
+
→
14
28
Si
[]
*
→
12
24
Mg+
2
4
He
14
27
Si+n
14
28
Si+γ
11
24
Na+3p
+
+n
$
%
&
&
'
&
&
€
(p,α)
(p,n)
(p,γ)
(p,3p+n)

22. F. CleriSources des radiations22
Formation of a compound nucleus via
the reaction. m + M => MC*
with m = mass of the projectile, with
kinetic energy T .
The binding energy of the
projectile+target set in the compound
nucleus is BE = (M+m–MC*) .
The energies of the states
(resonances) of the compound
nucleus excited (e.g. by the excess of
kinetic energy) correspond to the
levels k=1,…4.
MC*c2
MC*
The reactionchannels correspond to the
excitedlevelsof the compound nucleus

23. F. CleriSources des radiations23
Warning: the compound nucleus is a virtual state of the
reaction, which exists only for a very short time~10-19 s
it is not easy to observe it directly!
Indirect evidence of the existence of the compound nucleus (as [28Si]* in
this case): the forward and reverse reactions have the same resonance
energies ER (= transitions between excited levels of the compound nucleus).
ER
27Al(p,a)24Mg
24Mg(a,p)27Al

24. F. CleriSources des radiations24
Compound nucleus: ”prompt" and "delayed" g

25. F. CleriSources des radiations25
€
49
115
In+n→
49
116
In
[]
*
→
49
115
In+n
49
115
In+n+γ
49
116
In+γ
48
115
Cd+p
+
$
%
&
&
'
&
&
€
→
50
116
Sn+γ
In the desintegration process
of In-116* we observe two
familles of γ:
- the γ prompt, which are
emitted by the compound
nucleus 116In*
- the γ delayed, which are
emitted by the following
deexcitation of the 116Sn.
(same as for delayed
neutrons)
Reactionchannels and
"delayed" emissions
prêts
retardés

26. F. CleriSources des radiations26
6Li
2H 4He
4He
Reaction kinematics
If the incomingand outgoingprojectile are the
same(e.g., a neutron wecall thisa scattering,
elastic(onlyvariation of momentumvector) or
inelastic(variation of momentumAND energy).C*
€
m
a
+M
A
→M
*
→m
b
+M
B
Q=ma+MA( )−mb+MB( )⎡⎣ ⎤⎦c2MeV/c2Ta≥−Qmb+MBMA⎛⎝⎜ ⎞⎠⎟
If Q < 0 the projectile a must have a threshold kinetic energy :
A genericreactionwitha compound nucleus,

27. F. CleriSources des radiations27
Let’sconsiderthe reactionTarget ( projectile , x ) Residue=> C ( p , x ) R
Neglectingthe binding energiesof atomicelectrons, wehave:
mpc2+ Tp+ mCc2= mxc2+ Tx+ mRc2+ TR
The Q of the reaction:
Q = (mp+ mC)c2–(mx+ mR)c2= Tx+ TR–Tp
(So, alsoequalto the differencebetweenen. initial and final kineticenergy)
Necessary(but not sufficient) condition :
Q + Tp> 0(i.e. Tx+ TR > 0 )

28. F. CleriSources des radiations28
WealreadydefinedQsimplyas the differenceof masses:
Q= (mp+ mC)c2–(mx+ mR)c2
In fact, if ourisotope table givesus onlythe mass defects, D, wecan also
calculatethe quantity:
Q= (mpc2–Ap) + (mCc2–AC) –(mxc2–Ax) –(mRc2–AR)
Becausein termsof A.M.U., wemust have Ap+AC=Ax+AR(conservation of A).
So, itisalso:
Q= (Dp+ DC) –(Dx+ DR)
The possibilityof a spontaneousreactionrequiresQ > 0…or evena negative
Qand a projectile withkineticenergyTp≠0,
Q+Tp> 0
(payattention to the correct definitionof
TP: laboratoryor center of mass
referenceframes?)

29. F. CleriSources des radiations29
In the laboratory frame of reference,
we can write both equations from
momentum conservation:
mpvp= mxvxcosq+ mRvRcosf
0 = –mxvxsinq+ mRvRsinf
Given the momentum as p=mv=(2mT)1/2, we transform:
(2mpTp)1/2–(2mxTx)1/2cosq= (2mRTR)1/2cosf
(2mxTx)1/2sinq= (2mRTR)1/2sinf
And again, we take the squares, and we sum the right and left sides:
mpTp–2(mpTpmxTx)1/2cosq+ mxTx= mRTR
p C
x
R
q
f
Parallel Component
Perpendicular component

30. F. CleriSources des radiations30
By the precedingdefinition, Q = Tx+ TR–Tp, wefind:
This tells us that, by measuringTxand cosq(kineticenergyand exit angle of the
ejectedparticle),all the restbeingknown, wecan determinethe Qof any
reaction.
If, on the otherhand, weknow the Q(for ex. fromthe isotope tables) wecan
invert, and findthe kineticenergyof the emittedparticleas:
whichallowsto identify, e.g. the type of particleemittedin the reaction.
Q=Tx1+mxmR⎛⎝⎜ ⎞⎠⎟−Tp1−mpmR⎛⎝⎜ ⎞⎠⎟−2mRmpTpmxTx( )1/2cosθ
Tx1/2=mpmxTp( )1/2cosθ±mpmxTpcos2θ+mR+mX( )mRQ+mR−mp( )Tp"# $%{ }1/2mR+mX

31. F. CleriSources des radiations31
In the reference frame of the center of mass (CM), kinetic energy and velocity of
the center of mass are :
After substitution :
The real energy available for the reaction is in fact:
(TCMis just the translation of the CM, therefore it is not useful for the reaction) :
According to the condition Q+ T0 ≥ 0
threshold energy
TCM=12mp+mC( )vCM2vCM=vpmpmp+mCTCM=12mp+mC( )vpmpmp+mC!"## $%&&2=12mpvp2mpmp+mC!"## $%&&=Tlabmpmp+mC!"## $%&&T0=Tlab−TCM=Tlab1−mpmp+mC"#$$ %&''=TlabmCmp+mC"#$$ %&''
Tp=Tlab≥−Qmp+mCmC#$% &'(
(see
slide26)
(vC=0 initially)
T0 = projectile energyin
the CM

32. F. CleriSources des radiations32
Definition of the effective cross-section s
Fraction of surface area transverse to the direction of the
beam to have a reaction on a nucleus/atom of the target
Naparticles of type a/time dthit the target A
Nbparticles of type b /time dtexit, being
scattered by 1 nucleus of the target A
nA= nuclei A/ unit volume
number / unit surface =nAdx
Beam a
dx
NanA
S
Nb
flux :
€
Φ
a
=
N
a
S
=
N
a
dx
Sdx
=n
a
dx
dt
=n
a
v
incident

33. F. CleriSources des radiations33
The effective cross section σfor the reaction (a,b) is by definition :
In fact, nASΔxis the total number of target nuclei, hence Nb/(nASΔx)gives
the flux of particles bscattered, by 1 targernucleus, per unit of time.
€
σ(a,b)=
fluxdiffusé
fluxincident⋅n.at.cible
=
N
b
(n
a
v)(n
A
SΔx)
=
surface
atomecible
€
dn
a
=−n
a
σn
A
()
dx
€
n
a
(x)=n
a
(0)exp(−n
A
σx)
Penetration profile along x
number of a that
interact (na • prob)
Interaction probability =σnAdxNon-Dimensional Quantity
Definition of the effective cross-section s

34. F. CleriSources des radiations34
Reaction Rate
Flux Φa= navn. of particles aper unit of surface, per unit time
Collision rate (s-1)on target nuclei A:
Using the effective cross-section σwe can introduce other coefficients defining
the ability of a material to absorb a certain type of projectile:
μ = nA σ (cm-1) Volume absorption coefficient
μρ= μ / ρ (cm2 g-1) Mass absorption coefficient
λ=1 / μ (cm) Penetration length
€
R=Φ
a
σ=
N
b
n
A
Sdx
(Reaction rate per
target nucleus)

35. F. CleriSources des radiations35
b = impact
parameter
The x-section differential in angle is so defined:
€
dσ
dΩ
=
N
b
Φ
a
=
b
sinθ
db
dθ
depends on the type of
interaction potential
(harmonic oscillator,
Coulomb, Saxon-
Woods, etc.)
N. of particles scattered atΩ=(θ,φ)/ sec,
NbdΩ= Nb2πsinθdθ≡Φadσ=
= Φa2πb db= incident flux in 2πb db
Classical diffusion theory
The number of particles scattered in Ω,Ω+dΩis equal to the n. of particles
incident in the annular ring b,b+db.
(seeRutherford ! TD 88)
€
dΩ=2πsinθdθ
= solid angle around the direction θ

36. F. CleriSources des radiations36
In the quantum theory of potential (MQ) we consider the
scattering of a particle by a potential V(r) :
The general solution of the Schrodinger eq is the sum of the homogeneous
solution (for V=0), + a particular solution, for ex. :

€
(∇
2
+k
2
)ψ(r)=
2µ

2
V(r)ψ(r)

€
k=
2µE

€
ψ
hom
(r)=exp(ik
0
⋅r)=φ
0
(r)

€
ψ(r)=φ
0
(r)+
2µ

2
dr'
e
ik⋅(r−r')
r−r'
∫ V(r')ψ(r')=

€
≈φ
0
(r)+
2µ

2
e
ik⋅r
r
dr'∫e
−ik⋅r'
V(r')φ
0
(r')=φ
0
(r)+
e
ik⋅r
r
f(θ,ϕ)

€
=φ
0
(r)+
2µ

2
dr'
e
ik⋅(r−r')
r−r'
∫ V(r')φ
0
(r')+O(2)≈
1st order Born
approximation
k0
k
dr’ = Integration on volume
of the target
approx r >> r’r’r
cible
detector
Scatteringamplitude

37. F. CleriSources des radiations37
€
f(θ,ϕ)=dr'∫e
−ik⋅r'
V(r')φ
0
(r')=e
−ik⋅r'
V(r')e
ik
0
⋅r'
is the scattering amplitude that, in the 1st-order Born approximation is just
the Fourier transform of the diffusing potential :
, q = k0–k

€
f(θ,ϕ)=dr∫e
−ik⋅r
V(r)e
ik
0
⋅r
=dr∫e
iq⋅r
V(r)
r=r', variable muette
=4πqrdr∫V(r)eiqr−e−iqr2⎛⎝⎜ ⎞⎠⎟=4πiqr2dr∫V(r)sin(qr)r
€
f(θ,ϕ)=2πr
2
dr∫V(r)sinθdθe
iq⋅rcosθ
∫ =2πr
2
dr∫V(r)e
iqrx
dx
−1
1
∫ =
x=cosq
Moreover, if the potential V(r) is simply spherically-symmetric:

38. F. CleriSources des radiations38
In QM the flux is associated with the operator current of particles :
j=−i2mψ*∇ψ−ψ∇ψ*( )≈kmˆk0+kmf(θ,ϕ)2r2ˆr
Incoming flux
Diffused flux

€
N
b
dΩ=(j⋅ˆ r )dA=
k
m
f(θ,ϕ)
2
r
2
r
2
dΩ

€
Φ
a=j⋅
ˆ
k
0=
k
m
dσdΩ=NbΦa⎛⎝⎜ ⎞⎠⎟

€
dσ=
k
m
f(θ,ϕ)
2
dΩ
k
m
€
dσ
dΩ
=f(θ,ϕ)
2
=ψVφ
0
2
=V
fi
2
Scattering
factor
Matrix element of V between
the initial and final state
(Slide 30 of Cours 1)
Hence the definition of effective cross-section
(diffused flux / incoming flux / atom) :
^
^ ^

39. F. CleriSources des radiations39
If there are multiple reaction channels available, b, b’, b”, etc., the total
probability of projectile a interaction is the sum of the independent
probabilities, and the total effective x-section is the sum of the partial
cross-sections for each channel:
The effective macroscopic cross-sections efficaces (partials and
total) for a target A are defined as:
"Macscopic" cross-section
€
σ
Tot
(a)=σ(a,b)+σ(a,b')+σ(a,b")+...
€
Σ
ab
=n
A
σ(a,b)
Σ
a
TOT
=n
A
σ
Tot
(a)
$
%
&
'
&
in cm-1, it is the same as the volume
attenuation coefficient μ
(Rather used for photons γ and X )

40. F. CleriSources des radiations40
Half-lives and line widths
The excited states i (with resonant energies ERi) have a limited half-
life, defined by characteristic time τi :

€
τ
i
=

Γ
i
Γi is the spectral linewidth i.
Γ = Γnn’ + Γnγ+ Γnn’γ+ Γnp + … is the total width (MeV).
The probability of each reaction process (= channel) is :
Γnn= neutron scattering, Γnγ= radiative capture, Γnγ= emission γ, etc.
€
probi
()
=
Γ
i
Γ

41. F. CleriSources des radiations41
For levels ERi well separated in energy, we can derive the Breit-Wigner
formula, for the effective cross-section of production of a compound
nucleus of the reaction :

a + A C* b + B
as:
The effective production x-section of the particle b in the réaction being :

€
σ
C*
=π
2
(2l+1)
Γ
a
Γ
(E−E
R
)
2
+(Γ2)
2
€
σ
ab
=σ
C*
Γ
b
Γ
product between the probability of forming
C* following the interaction of a with the
target, and the probability of exit in the
channel b = Γb/Γ
Breit -Wigner
!=E−1
In units of ℏc

42. F. CleriSources des radiations42
So we have, for example, for the elastic diffusion of a :
Also for example, the radiative captureof a neutron followed by g emission:
It should be noted that:
- for E=ER the x-section has a maximum (resonance peak)
- for E=ER ± Γ/2 the x-section is at half of its maximum, which defines the
total Γ also as the FWHM of the spectral line.

€
σ
aa
=π
2 Γ
a
2
(E−E
R
)
2
+(Γ2)
2

€
σ
nγ
=π
2
Γ
n
Γ
γ
(E−E
R
)
2
+(Γ2)
2
σnγ=4π2ΓaΓbΓ2

43. F. CleriSources des radiations43
Effective cross section for neutron
capture in n + Cd
The effective x-section varies as
s~ 1/E1/2form small E
But the De Broglie wavelength of the
particle varies as
!=ℏ
2%&∝1
&!/#
hence the total )$%$∝!#∝!
&

44. F. CleriSources des radiations44
n + 16Oè17O
The cross-section of the
compound nucleus (and of
the excited final state)
contains a series of B-W
lines corresponding to the
transitions between
discrete levels, with
successive γ photon
emission.

€
σ
C*
TOT
=
i
∑π
2
(2l
i
+1)
Γ
a
Γ
i
(E−E
i
)
2
+(Γ
i
2)
2
Γi
Γ total for the
i-th line
17O

45. F. CleriSources des radiations45
Thus, the Breit-Wigner half-heightwidth(FWHM) isa "naturalwidth" of
a line, whichcontainsinformation on the lifetimeof the intermediate
(resonant) state of the reaction:
A furthercomponent of the naturalline-widthisgivenby the Doppler
broadening, producedby temperature:
whosecontribution to the FWHM is(typically>> B-W)

€
Γ
i
=/τ
i
€
Δω=ω
R
2k
B
T
Mc
2
€
Δω
FWHM
=ω
R
8ln2k
B
T
()
Mc
2

46. F. CleriSources des radiations46

47. F. CleriSources des radiations47
This isthe explanationof Fermi's"slowingdown" experiments:
whythermal neutrons are muchmore effective at inducingfission
in 235U …
… and why, on the contrary, weneedneutrons withE > 1.5 MeV
to inducethe fission in 238U.