Interaction of particles with matter (lecture 1)
Johann Collot [email protected] http://lpsc.in2p3.fr/collot UdG 1/66 A brief review of a few typical situations is going to greatly simplify the subject. Mean free path of a particle, i.e. average distance travelled between two consecutive collisions in matter : 1 λ= σ n where : total interaction cross-section of the particle n number of scattering centers per unit volume
N A for a monoatomic element of molar mass M and example : n= M specific mass ρ .
N A Avogadro number
Electromagnetic interaction : 1 m (charged particles)
Strong interaction : 1cm (neutrons ....)
Weak interaction : 1015 m≃0,1 light year (neutrinos) A practical signal ( >100 interactions or hits ) can only come from electromagnetic interaction Particle detection proceeds in two steps : 1) primary interaction 2) charged particle interaction producing the signals Johann Collot [email protected] http://lpsc.in2p3.fr/collot UdG 2/66 typical examples : photon detection
Compton scattering
2 1 Signal is induced by electrons e-
e- 3
e+
Pair production
Johann Collot [email protected] http://lpsc.in2p3.fr/collot UdG 3/66 neutral pion detection :
A π0 decays into two photons with a mean lifetime of 8.5 10-17 s.
e+
1 - e 0 e-
2 e+
Johann Collot [email protected] http://lpsc.in2p3.fr/collot UdG 4/66 neutrino detection : A 2800 MW nuclear power station produces 130 MW of neutrinos ! A detector of 1 m3 located 20 m away from the reactor core can detect 100 neutrinos/h.
e liquid scintillator ( CH + 6L ) e nuclear reactor x i e
e e p 6 e L t i n
e e
~0 p n e e Charged particles produce light 6 nthLi t4,8 MeV in the target scintillator
Johann Collot [email protected] http://lpsc.in2p3.fr/collot UdG 5/66 Interaction of charged particles with matter For heavy particles ionization and excitation are the dominant processes producing energy loss.
Particle P of Z charge state
* Excitation : PZ atom atom*PZ followed by : atom atom
Ionization : P(Z )+atom→atom + +e-+P(Z )
Ionization + excitation : P(Z )+atom→atom *++e-+P( Z )
Johann Collot [email protected] http://lpsc.in2p3.fr/collot UdG 6/66 max Te , Maximal kinetic energy transferred to an ionized electron :
m m0 e v v = c
p= m0 v atom γ=(1−β2)−1/ 2 E=( p2+m2)−1/2 Z 0 β=(1−γ−2)1/2
Hypothesis : V >
2 2 max max 2me One may show (exercise) that : T e =E e −me= 2 (In natural units , c=ħ=1) E CM /m0
2 2 1/2 where : ECM = m0me2me E total energy in center-of-mass frame Johann Collot [email protected] http://lpsc.in2p3.fr/collot UdG 7/66 Two cases :
m0 >> me , i.e. the incoming particle is not an electron and if its energy is not too big
m2 m2 2m E 2 0 e e E=γ m (ECM /m0) = ( 2 + 2 + 2 )≃1 with 0 m0 m0 m0
2 γ me ≪1 proton Ep < 50 GeV , muon Eμ < 500 MeV (medium energy range) m0
max max 2 2 then : T e =Ee −me=2me β γ
m0 = me the incoming particle is an electron
T max=(E−m ) due to undistinguishibility of electrons, max transferable e e energy = Tmax / 2
If the incoming particle is not an electron then in practice m0 >> me .
Johann Collot [email protected] http://lpsc.in2p3.fr/collot UdG 8/66 Stopping power of heavy particles by excitation and ionization in matter. Average energy loss by a charged particle (other than an electron) in matter. Bethe and Bloch formula (see Nuclei and particles, Émilio Segré, W.A. Benjamin ; Principles of Radiation Interaction in Matter and Detection, C. Leroy and P.G. Rancoita, World Scientific ; Introduction to experimental particle physics, R. Fernow ) Stopping power or mean specific energy loss charge of incoming particle density effect correction Z of medium at high energy
2 2 2 max dE MeV 0.3071 z Z 1 2me T e 2 C e − [ ] = ln − − − dx g/cm2 A g mol−1 2 2 I 2 2 Z
Atomic shell correction at low mean excitation energy energy Atomic mass of medium (not covered in this lecture, see Leroy & Rancoita) Surface mass density of medium dx = ρ dl (or mass thickness of medium) Johann Collot [email protected] http://lpsc.in2p3.fr/collot UdG 9/66 show that : (β γ)2 β2= 1+ (β γ)2
Johann Collot [email protected] http://lpsc.in2p3.fr/collot UdG 10/66 Particle identification in Alice TPC
Johann Collot [email protected] http://lpsc.in2p3.fr/collot UdG 11/66 Johann Collot [email protected] http://lpsc.in2p3.fr/collot UdG 12/66 few remarks :
dE 5 3 − ~− / non relativistic particles - for βγ < 1 : dx dE − - for βγ ~ 3-4 : dx is minimal over a large energy plateau . A particle in this state is called a minimum ionizing particle (MIP)
dE MIP MeV In media composed of light elements : − ≃2 - 2 dx g cm dE - for βγ > 4 : relativistic increase of − as ln dx wich is tempered by -δ/2 correction.
- I : mean excitation and ionization energy , I = 15 eV for atomic H and 19.2 eV for H2 I = 41.8 for He I = 15 Z0.9 eV for Z > 2
2me max 2 2 At medium energy : ≪1 T e =2me m0
2 2 2 dE MeV 0.3071 z Z 2me 2 C e − [ ]= ⋅ [ln − − − ] dx g/cm2 A g 2 I 2 Z
Johann Collot [email protected] http://lpsc.in2p3.fr/collot UdG 13/66 Density effect correction
When energy increases, stopping power decreases to a minimum (1/β2 dependance) and then starts rising again due to logarithmic term. In fact, the max. transverse electric field increases as γ but its influence is screened by nearby atoms beyond a distance of 70 (A(g)/ρ(g)Z)1/2 Å (shown by Bohr). This density effect tempers the relativistic rise. Studies have be carried-out by Sernheimer, Peierls, Berger & Seltzer (see Leroy & Rancoita). The density correction effect term, δ is given by : md 2 S1 S β γ S0 S1 1 10 for 0 for 10 < β γ< 10 : δ=2ln(β γ)+ C+ a ln( ) β γ< 10 : δ=δ0( S ) β γ 10 0 [ ln (10) ]
S I for β γ> 10 1 : δ=2ln(β γ)+ C where : C=−2ln( )−1 h νp nr c² with ν = e in which n is the density of electrons and r the classical radius of e- : r = 2.82 fm p √ π e e S show that at very high energy : β γ> 10 1 ρ(g/cm³) h ν ≃28.7 Z eV max p A(g) dE MeV z2 Z 2m T √ −( )[ ]=0.3071 [ln( e e )−1] dx 2 2 g/cm 2.A(g) (h νp) Johann Collot [email protected] http://lpsc.in2p3.fr/collot UdG 14/66 Johann Collot [email protected] http://lpsc.in2p3.fr/collot UdG 15/66 Restricted energy loss
knock-on electron (delta ray) generated by a 180 GeV muon as observed by the experiment GridPix at CERN SPS. High energy transfers generate delta rays that may espace the detector if it is too thin. So average energy deposits are very often much smaler thant predicted by B&B.
If T0 is the average maximal delta ray energy that can be absorbed in the detecting medium, a better estimate of the average deposited energy is given by :
2 2 dE MeV 0.3071 z2 Z 1 2m β γ T β2 T δ(γβ) C −( )[ ] = ( ln( e 0 )− (1+ 0 )− − e ) dx 2 −1 2 2 2 2 2 2 2 Z g/cm A (g mol ) β I 2me β γ
S At extremely high energies, when β γ> 10 1 , stopping power reaches a constant called Fermi plateau.
2 dE MeV z Z 2me T 0 −( )[ 2 ]=0.3071 ln( 2 ) dx g/cm 2.A(g) (h νp)
Johann Collot [email protected] http://lpsc.in2p3.fr/collot UdG 16/66 Fermi plateau measured in silicon
Johann Collot [email protected] http://lpsc.in2p3.fr/collot UdG 17/66 Delta rays (secondary electrons)
The differential probability to generate a delta ray of kinetic energy T is given by :
2 dw(T , E) z Z F (T ) − − =0.3071 MeV 1 cm2 g 1 dTdx 2.A(g)β2 T 2
F(T) is a spin-dependent factor.
2 T F (T )=F 0(T )=(1−β ) for spin-0 particles T max
1 T 2 F (T )=F (T )=F (T )+ ( ) for spin-1/2 particles 1/2 0 2 E
2 1 T me 1 T 1 T me F (T )=F 1(T )=F 0(T )(1+ 2 )+ ( ) (1+ 2 ) for spin-1 particles 3 m0 3 E 2 m0
Johann Collot [email protected] http://lpsc.in2p3.fr/collot UdG 18/66 Delta rays (secondary electrons)
2 For T << Tmax and T << m0 / me ,
2 dw(T , E) z Z 1 −1 2 −1 =0.3071 MeV cm g dTdx 2.A(g)β2 T 2
This allows to compute an approximate probability to generate a delta ray of kinetic energy
greater than Ts in a thin absorber of mass thickness x :
z2 Z 1 w(Ts , E , x)≃0.3071 x 2 show this expression 2.A(g)β T s
Johann Collot [email protected] http://lpsc.in2p3.fr/collot UdG 19/66 Energy straggling distribution
So far, only the average energy loss has been considered. But energy loss is subjected to large fluctuations that in thin absorbers results in asymmetric distributions. The subject is quite complex and has no general exact solutions, but a few approximate formulas help to estimate it. z2 Z For thin absorbers in which Є / T << 1 , where : ϵ=0.3071 x MeV max 2. A (g)β2 where x is mass thickness of the absorber in g cm-2.
The problem was first studied by Landau and then Vavilov. Their distribution functions are not analitic. A useful approximation of the Landau distribution is : and ∆E is the energy loss 1 1 −λ Δ E−Δ E L(λ)= exp(− (λ+ e )) where : λ= MP ∆E is the most probable √2π 2 ϵ MP energy loss.
2 ϵ Δ EMP=Δ EBethe+ ϵ (β + ln( )+ 0.194) MeV T max Johann Collot [email protected] http://lpsc.in2p3.fr/collot UdG 20/66 Energy straggling Still for thin absorbers, an improved generalized energy loss distribution that takes into account the distant collisions that are neglected in Landau's approach, can be obtained by convoluting a Landau distribution with a Gaussian distribution : 1 + ∞ −Δ E ' f (Δ E , x)I = ∫ L(Δ E−Δ E ' , x)exp( 2 )d(Δ E ') 2 −∞ 2σ √2πσ I I
736 MeV/c protons 115 GeV/c protons
Energy deposited by protons in silicon
Johann Collot [email protected] http://lpsc.in2p3.fr/collot UdG 21/66 Energy straggling
In thick absorbers in which Є / Tmax >> 1, both the Landau and the Vavilov distributions tend to a Gaussian :
2 1 (Δ E−Δ EBethe) f (Δ E , x)≃ exp(− 2 ) β2 β 2πT ϵ(1− ) 2T max ϵ(1− ) √ max 2 2
β2 with : σ≃ ϵT (1− ) √ max 2
Johann Collot [email protected] http://lpsc.in2p3.fr/collot UdG 22/66 Stopping power of electrons by ionization and excitation in matter. Incoming and outgoing particles are identical. Energy transfer is bigger .
2 2 2 dE MeV 0.3071 Z 1 T me β γ 1 1 γ−1 −( )[ 2 ]= ⋅ 2 [ ln( 2 )+ 2 ( 1−(2 γ−1)ln(2) )+ ( γ ) ] dx g/cm A(g) β 2 2 I 2 γ 16
kinetic energy of incoming electron : T=(γ−1)me=E−me
Stopping power of positrons by ionization and excitation in matter.
2 2 dE MeV 0.3071 Z 1 T m β γ β2 14 10 4 −( )[ ]= ⋅ [ ln( e )− ( 23+ + + )] dx g/cm2 A(g) β2 2 2 I 2 24 γ+ 1 (γ+ 1)2 (γ+ 1)3
When a positron comes to a rest it annihilates : e+ + e- → γ γ of 511 keV each A positron may also undergo an annihilation in flight according to the following cross section :
2 2 Z πr γ + 4 γ+ 1 2 γ+ 3 σ(Z , E)= e [ ln( γ+ √γ −1 )− ] γ+ 1 γ2−1 √ γ2−1 Johann Collot [email protected] http://lpsc.in2p3.fr/collot UdG 23/66 Stopping power of a compound medium
dE dE ≈∑ f i ∣i dx i dx
m f i , m m i = ∑ i = where fi is the massic ratio of element i m i
dE ∣ is the stopping power of element i dx i
Johann Collot [email protected] http://lpsc.in2p3.fr/collot UdG 24/66 Interaction of particles with matter (lecture 2)
Johann Collot [email protected] http://lpsc.in2p3.fr/collot UdG 25/66 Bremsstrahlung : electromagnetic radiative energy loss
A decelerated or accelerated charged particle radiates photons. The mean radiative energy loss is given by :
fine structure medium atomic number constant = 1/137 incoming particle energy
rad 2 dE MeV 0.3071 α 2 2 me E 183 − ( 2 )= π Z z ( ) ln( 1/ 3 ) dx g/cm A (g) m me Z
incoming particle mass The mean radiative energy loss of a particle of charge z and mass m is a function of the incoming particle charge state mean radiative energy loss of an electron :
rad 2 rad Electrons are much more sensitive to dE m 2 dE − (z ,m)=( e ) z (e ) dx m dx this effect.
Johann Collot [email protected] http://lpsc.in2p3.fr/collot UdG 26/66 For an electron and taking into account the Bremsstrahlung radiation induced by atomic electrons : classical radius of electron : r e=/me rad dE − Z Z 1 2 183 − e =4 N r E ln dx A A e Z 1/ 3 which can be rewritten as :
rad dE − E − e = where X0 is the medium radiation length dx X 0 then over a path x in the medium, the mean radiated energy of an electron reads :
rad x X 2 E e−=E 1−e− / 0 where x is expressed in cm or g/cm
2 716.4 A g X g/cm = and : 0 287 ZZ1ln Z1/2 In a compound medium : f −1 where f and X i are the mass ratio and the radiation length X i i 0 o=[ ∑ i ] i X 0 of element i respectively. Johann Collot [email protected] http://lpsc.in2p3.fr/collot UdG 27/66 Critical energy : energy at which the ionization stopping power is equal to the mean radiative energy loss of electrons
dE rad dE ionization E = E dx c dx c
610 MeV 710 MeV E = for liquids and solids Ec= for gas c Z+ 1.24 Z0.92
Z 1.11 A better formula that can be used for liquids and solids is : E =2.66( X (g/cm²)) MeV c A (g) 0
In literature, both the radiation length and the critical energy are tabulated for electrons. For other particles they would scale according to the square of their masses with respect to the electron mass.
Johann Collot [email protected] http://lpsc.in2p3.fr/collot UdG 28/66 2 medium Z A X0 (g/cm ) X0 (cm) EC (MeV)
hydrogen 1 1.01 63 700000 350
helium 2 4 94 530000 250
lithium 3 6.94 83 156 180
carbon 6 12.01 43 18.8 90
nitrogen 7 14.01 38 30500 85
oxygen 8 16 34 24000 75
aluminium 13 26.98 24 8.9 40
silicon 14 28.09 22 9.4 39
iron 26 55.85 13.9 1.76 20.7
copper 29 63.55 12.9 1.43 18.8
silver 47 109.9 9.3 0.89 11.9
tungsten 74 183.9 6.8 0.35 8
lead 82 207.2 6.4 0.56 7.4
air 7.3 14.4 37 30000 84
silica ( SiO2) 11.2 21.7 27 12 57
water 7.5 14.2 36 36 83
Johann Collot [email protected] http://lpsc.in2p3.fr/collot UdG 29/66 Electron-positron pair production At very high energy, direct electron-positron pair production may play an important role. dE pair − =b Z , A , E E dx pair Energy loss by photo-nuclear interaction :
example : electro-dissociation of deuteron e−d n pe− dE nucl. − =b Z , A , E E dx nucl. Total stopping power : dE tot dE ionization dE rad dE pair dE nucl. = dx dx dx dx dx which could also be dE tot − =aZ , A , E bZ , A , E E written as : dx where a(Z, A, E) is the ionization term and b(Z, A, E) the sum of the Bremsstrahlung, the pair production and the photo-nuclear terms . Johann Collot [email protected] http://lpsc.in2p3.fr/collot UdG 30/66 Multiple scattering through small angles A charged particle traversing a medium is deflected many times by small-angles essentially due to Coulomb scattering in the electromagnetic field of nuclei. This effect is well reproduced by the Molière theory. On both x and y, the angular deflection θproj of a particle almost follows a Gaussian which is centered around 0 : 1 proj 2 − ( θ ) proj proj 1 2 θ0 proj (θspace)2=(θ proj)2+ (θ proj)2 P(θ )d θ = e d θ x y 2 π θ proj 1 space √ 0 θ = θ space 2 rms rms 1 √2 − space 1 2 0 P d = 2 e d 13.6 MeV x x 20 θ = z (1+ 0.038 ln( )) 0 β p X X √ 0 0 p particle momentum X0 radiation length x medium thickness z charge state of incoming particle Large deflection angles are more probable than what the Gaussian predicts. This results from Rutheford scattering of heavy particles off nuclei. Particles emerging from the rms 1 medium are also laterally shifted : X = θ x √(3) 0 Johann Collot [email protected] http://lpsc.in2p3.fr/collot UdG 31/66 Particle Range in matter If the medium is thick enough, a particle will progressively decelerate while increasing its stopping power ( β-5/3) until it reaches a maximum (called the Bragg peak).
This stopping power profile is used in protontherapy for treating cancerous tumors
Johann Collot [email protected] http://lpsc.in2p3.fr/collot UdG 32/66 Wanjie Proton Therapy Center in Zibo ( Shandong Province)
cyclotron delivering 230 MeV protons to treat cancerous tumors 30 centers of that sort around the world.
Johann Collot [email protected] http://lpsc.in2p3.fr/collot UdG 33/66 As the result of the stochastic behavior of particles interacting in matter, it is not possible to enunciate a perfect definition of a reproducible particle range. Continuous slowing down approximation range :
T o dT RT = 0 ∫ dE 0 T dx
where T0 is the incident kinetic energy of the particle. In practice, the integration is carried out down to 10 eV . We also use the mean range
Johann Collot [email protected] http://lpsc.in2p3.fr/collot UdG 34/66 If only ionization and excitation are used to calculate R(T) (valid for heavy particles with energies < 1 GeV), the following relationship can be used :
2 M b za M a Rb M b , z b ,T b = 2 Ra M a , za ,T b M a zb M b
Particle a with za , Ma
Particle b with zb , Mb and kinetic energy Tb
One may also write for a particle of mass M and charge state z carrying a kinetic
energy T0 : M R M , z ,T = hT / M 0 z2 0
where h is a universal function of the medium ( Z, A and I fixed).
Johann Collot [email protected] http://lpsc.in2p3.fr/collot UdG 35/66 universal h function = R / M
Johann Collot [email protected] http://lpsc.in2p3.fr/collot UdG 36/66 Alpha particle range in Si
http://physics.nist.gov/PhysRefData/Star/Text
Johann Collot [email protected] http://lpsc.in2p3.fr/collot UdG 37/66 Alpha particle range in Air
Johann Collot [email protected] http://lpsc.in2p3.fr/collot UdG 38/66 Interactions of photons with matter : Photons are indirectly detected : they first create electrons (and in some cases positrons when interacting by pair production) which subsequently interact with matter. In their interactions with matter, photons may be absorbed (photoelectric effect or e+e- pair creation) or scattered (Compton scattering) through large deflection angles.
As photon trajectories are particularly chaotic, it is impossible to define a mean range. We then proceed with an attenuation law : where :
-I0 is the initial photon beam flux I 0 I(x) − x I x = I 0 e -I(x) is the photon beam flux exiting the layer of thickness x N 2 = A tot -x surface mass density of the layer (g/cm ) A -μ is the mass attenuation coefficient (cm2/g)
-σtot is the total photon cross-section per atom Johann Collot [email protected] http://lpsc.in2p3.fr/collot UdG 39/66 1 Photon mean free paths in different media : λ = μ
Johann Collot [email protected] http://lpsc.in2p3.fr/collot UdG 40/66 Photoelectric effect : e-
atome
Because of their proximity to the nucleus, electrons of the deepest shells (K,L,M...) are favored. Following the emission of a photoelectron, the atom reorganises leading to the production of X rays or Auger electrons.
- Ee = EK - 2 EL e -EL
E = E - E if E > E -EK x K L x L Production scheme of an Auger electron
Johann Collot [email protected] http://lpsc.in2p3.fr/collot UdG 41/66 Photoelectron energy :
E = E − E E = E or E or E ... e binding where : binding K L M
At low energy (Eγ /me << 1), but if Eγ >> EK : 1 K 32 2 4 5 e photo= Z Th (per atom) = 1/137 Fine structure constant 7
Thomson scattering cross section
e 8 2 =E /me Th = re with re = classical electron radius 3 me 5 3.5 photoelectric cross section strongly increases as Z and decreases as 1/Eγ
At high energy (Eγ /me >> 1) : 4 K 2 5 photo = 4 r e Z
At low energy (Eγ <100 keV) , the photoelectric effect dominates the total photon cross section
Johann Collot [email protected] http://lpsc.in2p3.fr/collot UdG 42/66 Compton scattering : Elastic scattering of a photon off an atomic electron considered
as being free (if Eγ > Ebinding )
Ee
E E ' E ' 1 E = with = E m 11−cos e ' min E 1 cotg 1 tg = e = E 2 12 (exercise , show these equations) Klein-Nishina cross section :
e 2 1 21 1 1 13 c = 2re { − ln 12} ln 12 − ( per electron ) 2 12 2 122 e 2 2 2 2 d σc re 1+ cos θγ ϵ (1−cosθγ) = 2 (1+ 2 ) ( per electron ) dΩ 2 (1+ ϵ(1−cosθγ)) (1+ cos θγ)(1+ ϵ(1−cosθγ))
atom. e cross-section per atom : c = Z c
Johann Collot [email protected] http://lpsc.in2p3.fr/collot UdG 43/66 e+e- pair production : e- e- e+ e+ atom atom
E 2m E 4 me e
1 atom. 2 2 7 109 1≪ ⇒ pair = 4 re Z ln2− Z 1/ 3 9 54
1 atom. 2 2 7 183 1 ≫ ⇒ = 4 r Z ln − 1/ 3 pair e 9 1/ 3 54 2 Z Z where X0 (g/cm ) is the atom. 7 A 1 In this high energy regime : pair ≃ radiation length 9 N A X 0 Pair production is the leading effect at high energy
Johann Collot [email protected] http://lpsc.in2p3.fr/collot UdG 44/66 Johann Collot [email protected] http://lpsc.in2p3.fr/collot UdG 45/66 Total absorption cross section : In Compton scattering, photons are not totally absorbed Let us define a Compton energy scattering cross section :
E ' atom. atom. cs = c E And a Compton absorption cross section :
atom. atom. atom. σca = σc −σcs
Massic coefficients in cm2/g :
N N = atom. ; = atom. ; = cs A cs ca A ca c cs ca N atom. N = ; = p A pair ph A photo
a = ph pca total massic absorption coefficient
= ph pc total massic attenuation coefficient
Johann Collot [email protected] http://lpsc.in2p3.fr/collot UdG 46/66 Johann Collot [email protected] http://lpsc.in2p3.fr/collot UdG 47/66 Johann Collot [email protected] http://lpsc.in2p3.fr/collot UdG 48/66 Cerenkov light emission When a particle moves faster than the phase velocity of light in the medium, an asymetric polarization of the medium builds up along the longitudinal axis in the vicinity of the particle, that leads to the production of light, which in turn creates a coherent wave front as shown on the picture. It may be understood as the photonic shock wave of a particle that moves faster than c/n where n is the refractive index of the medium.
Cerenkov light emitted by a nuclear reactor (Advanced Test Reactor in ANL) c /nt 1 cosθ= = βc t βn 1 β> Cerenkov emission has a velocity threshold n
Johann Collot [email protected] http://lpsc.in2p3.fr/collot UdG 49/66 Cerenkov light emission
The number of photons emitted per unit path length and unit wave length reads :
dN 1 1 =2π α (1− ) It is strongly peaked at short wave lengths. dx d λ λ2 β2 n2 The total number of photons per unit path length is then :
dN 1 d λ =2π α ∫ (1− 2 2 ) 2 dx βn> 1 β n λ If the variation of n is small over the wavelength region detected then :
dN 2 1 1 =2π αsin θ ( − ) dx λ1 λ2
dN 2 e.g in a wavelength interval 350-500 nm (photomultiplier tube), =390sin θ photons/cm dx dE/dx due to cerenkov light is small compared to ionization loss and much weaker than scintillating output. It can be neglected in energy loss balance of a particle. Johann Collot [email protected] http://lpsc.in2p3.fr/collot UdG 50/66 Transition radiation
This radiation is emitted mostly in the X- ray domain when a particle crosses a boundary between media of different dielectric properties.
1 The radiation is emitted in a cone at an angle : cosθ= γ
The probability of radiation per transition surface is low ~ 1/2 α (fine structure constant) The energy of radiated photons increases as a function of γ.
Johann Collot [email protected] http://lpsc.in2p3.fr/collot UdG 51/66 Deposited energy :
Generally speaking, the energy loss is never equal to the deposited energy as the radiated photons or the secondary particles may escape the medium. Deposited energy is what generates the signal in a particle detector. Deposited energy is subjected to large stochastic fluctuations. Remember : Stopping power is the mean energy loss. If the medium is thin and the number of interactions is small, the deposited energy distribution is asymmetric : it is sometimes called a Landau distribution. If the medium is thick or the number of interactions is large, the deposited energy distribution tends to a Gaussian.
There are no simple and exact analytical formulae to compute deposited energy.
Nowadays, to estimate the energy deposited in a detector or more generally in a medium we use a Monte-Carlo program which simulates the propagation of the particle through matter : e.g. Geant4 Johann Collot [email protected] http://lpsc.in2p3.fr/collot UdG 52/66 creation of electron-ion pairs : When the measured signal is a current or a charge liberated through ionizing interactions, it is useful to compute the mean number of created electron-ion pairs :
Δ E n e - ion = deposited W
where : W is the required mean energy to produce an e-ion pair W > I (mean excitation and ionization potential)
In most gazes, W ~ 30 eV.
In semiconductor detectors (Ge, Si), W is much lower : e.g. W=3.6 eV for Si and W=2.85 eV for Ge
Johann Collot [email protected] http://lpsc.in2p3.fr/collot UdG 53/66 Hadron collision and interaction lengths : When dealing with very high energy hadrons, it is somewhat useful to express the total cross-section as :
σT = σelastic+ σinelastic Only the inelastic part of the total cross-section is susceptible to induce a hadron shower. It is then useful to introduce two mean lengths :
A −2 λ T = g cm called the nuclear collision length N A σT
A −2 λ I= g cm called the interaction length N A σinelastic 95% containment of a hadronic shower can be obtained for a thickness of :
L95% (in units of λI )≃1+ 1.35 ln(E(GeV)) Then approximately 10 interaction lengths are needed to contain a 1 TeV hadronic shower.
In high A materials : λ > X which explains why hadron calorimeters are deeper than I 0 electromagnetic calorimeters. Johann Collot [email protected] http://lpsc.in2p3.fr/collot UdG 54/66 Johann Collot [email protected] http://lpsc.in2p3.fr/collot UdG 55/66 Johann Collot [email protected] http://lpsc.in2p3.fr/collot UdG 56/66 To learn more :
- Principles of Radiation Interaction in Matter and Detection, C. Leroy and P.G. Rancoita World Scientific
- Introduction to experimental particle physics, Richard Fernow, Cambridge University Press
- Particle penetration and Radiation effects, P. Sigmund, Springer
- Nuclei and particles, Émilio Segré, W.A. Benjamin
- Stopping powers and ranges for protons and alpha particles (ICRU Report 49,1993) Library of congress US-Cataloging-in-Publication Data
- Particle detectors, Claus Grupen, Cambridge monographs on particle physics
- Detectors for Particle radiation, Konrad Kleinknecht, Cambridge University Press
- Radiation detection and measurement, G.F. Knoll, J. Wiley & Sons
- Single Particle Detection and Measurement, R. Gilmore, Taylor & Francis
- Radiation detectors, C.F.G. Delaney and E.C. Finch , Oxford Science Publications
- High-Energy Particles, Bruno Rossi, Prentice-Hall
Johann- CollotIntroduction to [email protected] engineering, John R. Lamarsh, http://lpsc.in2p3.fr/collot Addison-Wesley Publishing UdG 57/66 Interactions of neutrons with matter :
Neutrons are neutral particles which only interact with nuclei.
Neutrons can be absorbed or scattered by nuclei .
With respect to their energy, neutrons can be categorized as follow :
- thermal neutrons : in thermal equilibrium with matter ,
E0 (most probable energy) = k T = 0.025 eV for T = 300 K - ultra-cold neutrons E < 2 10-7 eV - very cold neutrons 2 10-7 eV < E < 50 10-6 eV -6 - cold neutrons 50 10 eV < E < E0 =0.025 eV - slow neutrons 0.025 eV < E < 0.5 eV - epithermal neutrons : 0.5 eV < E < 1 keV - intermediate energy neutrons : 1 keV < E < 0.5 MeV - fast neutrons : 0.5 MeV < E < 50 MeV - relativistic neutrons : 50 MeV < E Johann Collot [email protected] http://lpsc.in2p3.fr/collot UdG 58/66 All neutrons may undergo elastic scattering and radiative capture (emission of photons).
Elastic scattering : It is mostly used to slow down neutrons , e.g. in a nuclear reactor.
1H , 2H and 12C are the best and preferred moderator nuclei .
Up to 10 MeV and for some target nuclei (like H), the energy spectrum of elastically scattered neutrons is approximately flat :
The probability of a neutron of mass mn and incident energy E0 to be found - after elastic scattering off a nucleus of mass m - in an energy interval dE is : dE 2 P E dE A−1 0 = where = 2 1 1− E 0 A1 The scattered neutron A is the mass E E E energy follows : 0 0 number of the target nucleus
Johann Collot [email protected] http://lpsc.in2p3.fr/collot UdG 59/66 P(E ) elastic scattering off protons is also used to 0 detect fast neutrons : detection of recoiling protons
1/(1-α)
Energy
E0 α E0 Energy spectrum of elastically-scattered neutrons (if the incident neutron energy is less than 10 MeV and the scattering process is approximately isotropic in the center-of-mass-coordinate system ) Johann Collot [email protected] http://lpsc.in2p3.fr/collot UdG 60/66 ( n , γ ) radiative capture : As neutrons are neutral particles, radiative capture may happen at very low energies (no Coulomb interaction effects)
example : Production of 239Pu in a reactor
238 239 * 239 n 92 U 92 U 92 U
239 239 - 239 - 92 U 93 Np e e 94 Pu e e
( n , γ ) radiative capture is used to produce artificial radioisotopes in nuclear reactors
It may also be used to detect neutrons and measure neutron fluences (time integrated fluxes).
In general, the capture cross section increases as the inverse of the neutron velocity : the slower the neutrons, the bigger the cross section (Gamow Law). This general behavior may be affected by capture resonances.
Johann Collot [email protected] http://lpsc.in2p3.fr/collot UdG 61/66 ( n , γ n' ) neutron inelastic scattering : neutron energy less than a few tens of MeV
gamma decay of nucleus
A A1 * A * A nZ X Z X Z X n' Z X n'
decay into neutron + excited nucleus neutron capture formation of compound nucleus
Johann Collot [email protected] http://lpsc.in2p3.fr/collot UdG 62/66 ( n , p ) and ( n , α ) reactions : A few of these reactions are exoenergetic (produce energy) The neutron is first captured to produce a compound nucleus which then decays into several products.
3 3 n He p H 760 keV σ = 5400 barns
n14 N p14 C630 keV σ = 1.75 barns n35 Cl p35 S620 keV σ = 0.8 barn
n6 Li 3 H 4782 keV σ = 940 barns
n10 B 7 Li2791 keV σ = 4000 barns
These reactions (in particular on 3He , 6Li, 10B ) are used to detect low energy neutrons.
Johann Collot [email protected] http://lpsc.in2p3.fr/collot UdG 63/66 In general, the capture cross section increases as the inverse of the neutron velocity : the slower the neutrons, the bigger the cross section (Gamow Law). This general behavior may be affected by capture resonances.
Cross section of 10B(n,alpha)7Li
Johann Collot [email protected] http://lpsc.in2p3.fr/collot UdG 64/66 neutron-induced fission : fission fragments
A A 1 * n Z X Z X N M a few neutrons 200 MeV
For some odd neutron number nuclei ( odd N), fission may occur at all neutron energies and in particular at very low energy . This is the case of : 233U , 235U , 239Pu , 241Pu
For other nuclei, fission only takes place above a neutron energy threshold, e.g. 238U : 0.9 MeV , 232Th : 1.3 MeV
Apart from its well-known application for massive energy production, fission may be used to detect neutrons.
Johann Collot [email protected] http://lpsc.in2p3.fr/collot UdG 65/66 Neutron cross sections : microscopic cross section
Macroscopic cross section is defined as : =n Σ [cm-1] number of nuclei per unit volume
tot = elastic scatteringabsorptioninelastic scattering As in the case of gammas, we can use an attenuation law : initial neutron flux
−tot x I x = I 0 e distance traversed by neutrons 1 = being the neutron mean free path in the considered medium tot
Johann Collot [email protected] http://lpsc.in2p3.fr/collot UdG 66/66