* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download 2 The interaction of energetic particles with material
Monte Carlo methods for electron transport wikipedia , lookup
Atomic nucleus wikipedia , lookup
Photon polarization wikipedia , lookup
Quantum electrodynamics wikipedia , lookup
Bremsstrahlung wikipedia , lookup
Eigenstate thermalization hypothesis wikipedia , lookup
Standard Model wikipedia , lookup
Identical particles wikipedia , lookup
Nuclear structure wikipedia , lookup
Double-slit experiment wikipedia , lookup
Renormalization wikipedia , lookup
Introduction to quantum mechanics wikipedia , lookup
ALICE experiment wikipedia , lookup
Photoelectric effect wikipedia , lookup
Future Circular Collider wikipedia , lookup
Relativistic quantum mechanics wikipedia , lookup
ATLAS experiment wikipedia , lookup
Elementary particle wikipedia , lookup
Compact Muon Solenoid wikipedia , lookup
Electron scattering wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
2 The interaction of energetic particles with material In order to detect particles, they need to interact with (detection) material. Four important interactions are: - the Electro-Magnetic (EM) interaction of a energetic charged particle with matter; Bremstrahlung the interaction of photons (gamma’s, X-rays) with matter Cerenkov radiation The Electro-magnetic (EM) interaction In short: if a fast charged particle transverses a thin layer of material, then a force will be applied between this particle an all charged constituents in the material (fig.1). The energy transfer may result in charge separation in the form of creation of ion-electron pairs, or electron-hole pairs. These two processes lead to charge or light signals which can be amplified and detected. Charged particle Energy E Energy E’ dX Fig.1: After crossing a layer of material, a charged energetic particle looses energy dE. Calculation of force between charged particle (i.e. electron) in material and the fast particle (1) and the particle (2) in the material. Energy loss due to e.m. interactions, also referred to as "collisions", In the laboratory system S' particle 1 is moving, particle 2 is in rest. Consider a system S in which particle 1 is in rest, so particle 2 is moving in a static electric field. E⊥ = cos θ Z1 e / r2 = Z1 e b / r3 Movement particle 2 in S θ r Z1 e 1 x = -v1 t (energy particle is assumed to be high, -> b is assumed to be constant) 2 b t = 0: distance between 1 and 2 is minimal In the laboratory system S' particle 2 is in rest, so there is no effect from the (time dependent) magnetic field caused by particle 1. time in S' ! In S': E ⊥' = γE ⊥ = γZ1eb /r 3 electric field strength r 2 = b 2 + x 2 = b 2 + v12t 2 = b 2 + γ 2 v12t'2 3/2 E ⊥ = γZ1eb /(b 2 + γ 2 v12t' 2 ) Then: integration along the X-axis. Note that, due to their mass differences, the energy transfer to (light) electrons is much larger than to a nucleus. ∞ ∆p ⊥ = ∫ F dt = ∫ Z 2e γ Z1edt ⊥ −∞ (b 3 2 + γ 2 v12t 2 )2 ∞ t 2Z1Z 2e 2 ∆ p = ∆ p ⊥ = Z 2e γ Z1e = 1 v1b b 2 (b 2 + γ 2 v12t 2 )2 −∞ 2 ∆E = energy loss (∆p) 2m = 2Z12 Z 22e 4 b 2 v12m (Particle 2 is in rest -> non-relativistic calculation of energy loss possible) Interactions with electron: m = me, Z2 = 1 ∆ E(e) 2Z 2e 4 = 2 12 ∆ E(nucleus) b v1 me Interactions with nucleus with mass number A and atomic number Z: m = Amp ≈ 2Zmp, Z2 = Z 2mp 2Z12 Z 2e 4 = ≈ 4000 /Z b 2 v12 2Z mp Z me -> Energy loss due to collisions is dominated by interactions with the electrons (NB: we are comparing interactions with 1 electron to interactions with the nucleus of an atom, a non-ionized atom has Z electrons) Then: integration over b for many participating charge carriers (electrons): We have looked at the interaction between a charged particle and a single isolated electron or nucleus. In reality we are dealing with the combined effect of interactions with many electrons (and nuclei). The effect of this was estimated by N. Bohr. He considered interactions with atoms, each with Z2e electrons. Particle 1 is passed through the centre of a thin shell of atoms and the net energy loss is calculated. Then one integrates over b (the "impact parameter") to obtain dE/dx e Number of electrons in shell: ne 2πbdb dx with ne = number of atoms per cm3 b ∆E = r db Z1 e dx 4πZ12e 4 n edbdx bv12m 2Z 2e 4 b max db 4π n e Z12e 4 b dE = 2π n e 1 2 ∫ ln max = 2 m v b min m v dx b e 1 b e 1 min We first need to define the minimum and maximum values of b: 3 To find bmax we note that (b 2 + γ 2 v12t 2 )2 determines the size of the transversal field strength, from this it follows that only during a time of the order of b/(v1γ) the "collision" takes place. This time should be short relative to the time characterizing the orbital frequencies of the electrons in the atoms, otherwise the effect of the binding of the electrons cannot be neglected (Bohr considered harmonically bound charges for distant collisions, but numerically the results are about the same as for the approach discussed here). Define ω to be the characterizing orbital frequency, then: bmax = γv1/ ω. The energy transfer cannot exceed a certain maximum, as seen earlier. This maximum is for m2 >> me: 2meβ2γ2 2Z12e 4 dE 4π n e Z12e 4 m v 3γ 2 Z e2 = ln e 12 one obtains bmin ≈ 1 2 so: 2 With: ∆ E = b 2 v 2m me v1 Z1e ω dx γmv1 1 e The result shows for low energies decreasing dE/dx for increasing E and for higher energies an increase in dE/dx, due to the increase in γ and v approaching c. The properties of the material enter via ne, i.e. only the electron density matters. Or: dE/dX = C1/v12 ln C2 v13γ2 dE/dx for pions as computed with Bethe-Bloch equation dE/dx divided by density ρ (approximately material independent) slope due to 1/v2 relativistic rise due to ln γ ρ about proportional to ne, as ne = na Z = NA ρ Z / A, -> ne ≈ NA ρ / 2 high βγ: dE/dx independent of βγ due to density effect, "Fermi plateau" From PDG, Summer 2002 Fig.2 Typical curves of dE/dX for pions as a function of their energy. The value drops for increasing energy, and after reaching a minimum (‘minimum ionizing particle’ the value rises slowly (‘relativistic rise’). Calculation of distribution function of dE/dX: Φ(W )dW dx = n a dσ(W ) dW dx dW number of atoms per unit of volume Probability energy loss between W and W+dW ∞ "Primary Ionization rate": q = n a ∫ 0 dσ(W ) dW ; q dx is the probability for energy dW loss in dx Define χ(W,x)dW as the probability for energy loss between W and W+dW in a length x of absorber and then write down the "transport equation" for a a length of dx of absorber at position x in the absorber: x dx Number of particles with energy loss between W and W+dW in x+dx (I) = number of particles with energy loss between W and W+dW in x (II) number of particles with energy loss > 0 in dx (III) + number of particles with energy loss between W-e and W-e+dW in x (IV) and energy loss e in dx I : Nχ(W,x + dx)dW II : Nχ(W,x)dW III : Nχ(W,x)dW qdx ∞ IV : N ∫ χ(W − ε,x)dWΦ(ε)dxdε ε= 0 ∞ Nχ(W,x + dx)dW − Nχ(W,x)dW = N ∫ χ(W − ε,x)dWΦ(ε)dxdε − Nχ(W,x)dW qdx ε= 0 or: ∂χ(W,x) W = ∫ Φ(ε)χ(W − ε,x)dε − qχ(W,x) ∂x ε= 0 We skip the evaluation but present the results instead: From PDG, Summer 2002 Fig. 3: typical Landau distributions. The tail can be associated with ‘head-on’ collisions. Number of events Not a simple example of a Landau curve ! Part of electrons stop in detector, rest either loses part of their energy when passing through the detector or backscatter out of the detector or create escaping bremsstrahlung Fig.4. Another example of dE/dX: electrons with energy of 1 MeV traverse through a Si detector (layer thickness 0.53 mm). The curve has partly a Landau shape, but deviates towards the highest energy. Some electrons are stopped, and thus absorbed in the Si layer. This causes the clear peak in the right-hand side of the spectrum. The measured spectrum may be compared with a MonteCarlo simulation in which the dE/dX process is digitally processes. The relativistic rise can be applied for particle identification: Relativistic rise largest for gasses, can be used for measurement of β, i.e. of velocity, and therefore for particle identification, BUT many independent measurements are needed for obtaining a reasonable accuracy. This is possible with a Time Projection Chamber (TPC), to be discussed later in the course. Usually energy losses corresponding to the tail of the Landau distribution are not taken into account. For relatively low energies βγ < ~ 2 particle identification using energy loss measurement is easier. Time Projection Chamber (TPC): 2D/3D Drift Chamber The Ultimate Wire (drift) Chamber track of charged particle E-field (and B-field) Wire plane Wire Plane + Readout Pads Pad plane Fig. 5: The Time Projection Chamber(TPC). The charge signals, measured on the (cathode) strips, are proportional to dE/dX of the track passing the gaseous volume. Particle identification in the relativistic rise region with the ALEPH TPC http://alephwww.cern.ch/ALEPHGENERAL/reports/figures/detector/index.html Fig.6: Particle identifications from dE/dX as a function of the particle’s momentum. Bubble chamber photograph shows different bubble density along tracks for different particle momenta and particle type. http://physics.hallym.ac.kr/education/hep/adventure/bubble_chamber.html Fig. 7: bubble chamber picture showing clear dependence of dE/dX on the particle’s momentum. Bremstrahlung Photon (invisible) Kink in trajectory Curvature smaller after kink: due to lower momentum Hans Albrecht Bethe e Walter Heitler http://teachers.web.cern.ch/teachers/archiv/HST2002/Bubblech/mbitu/electromag-events1.htm This is another interaction between an energetic charged particle and material. The particle interacts with a bounded charged particle in material, and as a result a gamma (photon) is being created. The energy of the emitted photon and the (continuing) particle equals the original energy op the incoming energetic particle. The static particle in the material must be ‘bounded’ in order to transfer moment. Bremstrahlung can NOT occur in vacuum, although conservation of energy would allow this. [In the following: 1- note that the momentum ǁPǁ of a photon with energy E equals E/c. The speed of light c often occurs as term in equasions and cancel: see chapter Accelerators. 2- An interaction or a process between particles is often represented by a Feynman diagram, in which in- and outgoing particles are indicated.] Photons can not be radiated in vacuum Before: Ei2 - pi2 = me2 Ef,pf e E γ = pγ γ e Ei,pi Feynman diagram 2 2 2 After: (E f + E γ ) − (pf + pγ ) = me 2E f E γ = 2pf • pγ p E f = pf • γ E f < p f not possible pγ Photons can be radiated in the field of an object with charge Ef,pf e Ei = Ef + Eγ Very heavy pointlike spinless object with charge Z2 (atomic nucleus) γ e Ei,pi γ e e This diagram also contributes Classically: electric field accelerates electron. Quantum-mechanically: virtual photon exchange decreases pf, so that Ef2 - pf2 = me2 is satisfied Z2 ESRF: European Synchrotron Radiation Facility, Grenoble, France 300 m circumference booster synchrotron, 6 GeV 16 m linac, 200 MeV Fig. 8: Application of Bremstrahlung. Accelerated electrons (6 GeV) are kept in orbit by magnets with a vertical pointing B-field. While passing through this field, photons (mainly in the keV range) are emitted. The interaction of photons with material The three main interactions are: - photo-effect: the photon interacts with a electron which is in a bounded shell. The electron is emitted with an energy equal to the photon energy minus the bounding energy of the electron; - Compton effect: the photon interacts with a (possibly free) charged particle. Some of the energy of the photon is transferred to the charged particle. The photon (with reduced energy) and charged particle are outcoming particles: total energy and momentum are conserved. - Pair production: the conversion of a photon into an electron-positron pair. This can not occur in vacuum (violation of conservation laws: see Bremstrahlung: identical Feynman diagram with inverted time and charge sign!). From PDG, Summer 2002 Fig.9: Probability for photo-effect, Compton scattering and Pair Production with Lead, as a function of the photon energy. Cerenkov radiation This interaction occurs if a charged particle tranverses a medium (material) in which the local speed of light (= c/n: n is the refraction index of the medium) is SMALLER than the speed of the charged particle. The effect is very much like a jet airplane flying faster than the speed of sound: it drags a cone with ‘sonic boom’. Fig. 10: The origine of Cerenkov radiation. UV ligh is emitted from a cone which opening angle is determined by the refraction index of the medium and the speed of the particle. Fig. 11: Cerenkov light from the core of a nuclear reactor immerged in a water basin.