The Outline Drift of electrons and ions in gases, with and without the presence of electric and magnetic fields - why this is important ? - drift of electrons and ions in electric fields - diffusion of ions in a field free gas - electron capture -drift of electrons of electric and magnetic fields, diffusion of electrons. Anna Lipniacka , Detector techniques L5 DELPHI construction, insertion of the Time Projection Chamber Muon chambers Hadronic calorimeter Electro magnetic cal. Tracking detector, Time Projection Chamber Anna Lipniacka , Detector techniques L5 Drift and diffusion- why this is important? Important in gaseous detectors working in one of two modes : - ionization detectors. Primary ionization from passing particle has to be drifted somehow to “sense device” where it has to be collected and measured. We would like to know what will be the spatial size of the ionization cloud, when will it arrive, will it arrive at all ? Will it not be “recaptured” by the surrounding gas? These detectors work often in magnetic fields (and electric fields to collect the charge) we would like to know how magnetic field modifies the movement of the charge. We would like to reconstruct the energy loss of the particle from ionization measurement. If ionization is lost somehow on our way to detection we need to correct for that. - detectors working in proportional mode. These are proportional and drift chambers. With larger field present electrons drift until they become ionizing. Primary and secondary ionization has to be collected. The pulse shape is of interest, also the drift time and diffusion of the pulse shape with the drift. From the drift time we would like to reconstruct- for drift detectors – the position of and passing particle. The pulse shape diffusion is important for the construction of the electronics. Anna Lipniacka , Detector techniques L5 TPC pads Electrons still drift to wires But charge is induced on pads next to the wires Provide extra dimension Both are readout. Might be 150 wires crossed, less pads. 4 Proportional and drift detectors There are typically wire chambers filled with gas. ionize gas and electrons are attracted by anode wires. The electric field around the wire (1/R) is high and electrons gain enough velocity to ionize more gas. This gives signal on anode wires- still proportional to the number of initial ionization electrons- "proportional counters " or MWPC. The space resolution, of the order of a fraction of a mm typically depends on the density of anode wires. Use drift chambers to make it cheaper - - --8 + -2 0 Measure drift time of electrons to anode wire to reconstruct particle passage The price to pay is left-right ambiguity, and longer drift time of electrons-> slower detectors Field shaping wires Tracking detectors contain as little material as possible, to not to "disturb" particles Anna Lipniacka , Detector techniques L5 What happens with electrons and ions in gases? electrons and ions will drift in the electric field: drift velocity and mobility, mean free path electrons and ions will diffuse due to thermal energy in a field-free gas: diffusion coefficient and mean free path ions and electrons can be neutralized before detected: ion recombination and electron capture. magnetic field will modify the drift of electrons in electric field: Lorentz angle, modification of the drift velocity diffusion of electrons in electric and magnetic fields: Electrons thermal velocities will spiral around B field, resulting in smaller diffusion. Exact pattern of what happens might depend on a gas in question. Will be discussed for most popular gas mixtures used in gaseous detectors. How do you think this gas mixtures were chosen ? Anna Lipniacka , Detector techniques L5 Drift of ions and electrons in electric field + e vD ,vD Electric field will exercise a force on an ion/electron. It will accelerate until it reaches “equilibrium velocity” at which interactions with other particles of a gas will stop it as much as E-field accelerates it. This velocity is called drift velocity . It will depend on properE ties of the drifting particle, of the gas, on the electric field. For ions we use experimental parametrization: v + D = + p0 E p + where p-pressure, p_0 = 760 Torr and is called ion mobility and depends on the gas in question. Measured in cm**2/(V s) Example : He ions in He mobility 10.2 cm**2/(V s) . For E=1KV/cm and p=p_0 we get drift velocity 10.2 *10**3 cm/s = 10.2 cm/msec ~0.1 mm/microsecond For mixture of different n gases the mobility will be combined . Mobility of ions i in the mixture of 1,,,n gases: c_k are relative volume concentrations of gases. 1 + i = n k =1 ck + ik If several types of ions are present the ones with bigger ionisation potentials will steal electrons from atoms with lower ionization potentials after 10**2-10**3 collisions. Anna Lipniacka , Detector techniques L5 Drift of electrons “ Stopping power” of the medium can be expressed in terms of mean free path of a particle interacting with the medium: 1 = n where as usual n is the concentration of interaction centers and sigma is the interaction x-section. For electrons of kinetic energies around 1 eV mean free path will be much longer than for ions, because atoms are nearly transparent due to long wavelength of electrons. We will now try to calculate drift velocity in electric field E of the “swarm” of electrons which have thermal velocity “u” and are in a medium in which they have mean free path e E In times between collisions (t) electric field E will displace thermal motion of electrons leading to an average displacement of the electron cloud : < z >~ < < t 2 >~ z u 2 e 2 Anna Lipniacka , Detector techniques 1 2 qE 2 a t >~ <t > 2 m thus < z > qE e ~ vd= < t> m u L5 Drift of electrons cont. On other hand drift velocity adjusts itself in such a way that energy lost in collisions is equal to the energy gained in the electric field qE < z >~ qE v d < t >= qE v d mu 2 = 2 e u energy gained energy lost in a collision, assuming that an arbitrary fraction delta is lost in a single collision Comparing energy lost and energy gained we get another formula for drift velocity, and we can eliminate thermal velocity u and obtain an expression for drift velocity The dependence of drift velocity on the electric 1 2 field will depend on the fact if the fraction E of energy lost depends on the electron energy and v d~ q e if the mean free path depends on electron energy2 m this will depend on the fact if collision x-section depends on the electron energy. Examples: Argon -> mean free path first increases than decreases with the energy. The result is that the drift velocity first increases quickly with E field then it stabilizes In CH_4 for energies above excitation energy the fractional energy loss decreases like 1/energy thus delta*energy = const and v_d~1/E / Anna Lipniacka , Detector techniques L5 Some numbers Typical units mm/microsecond or cm/microsecond, popular “drift” gases like argon/methane mixtures. Orders of magnitude : = 10 10 17 15 cm 2 for noble gases, might be much bigger for other gases for STP we have n= 2.69*10**9 molecules/cm**3 thus mean free path of the electron is e = 1 = 10 7 10 3 cm n for noble gases, while might be orders of magnitude shorter for other gases. For ions mean free path is of the order of 10**(-5) cm, and around 6 times larger for electrons in non-noble gases mean thermal energy at room temperature 3/2kT is around 0.035 eV. 2 mv 3 t kT = 2 2 thus thermal velocities of ions will be of the order 10**5 cm/s=1 mm/microsecond and thermal velocities of electrons will be 10**7 cm/s=100 mm/microsecond Drift velocities (experimental) for electrons are of the order 10-100 mm/microsecond. In our calculations we implicitely assumed that drift velocities are substantially smaller than thermal velocities Anna Lipniacka , Detector techniques L5 Diffusion of electrons and ions Ions and electrons will have thermal velocities with which they will disperse from the point where they were created: mu 3 < >T = kT = 2 2 the thermal energy distribution follows Boltzmann statistics ( to a large extend) 2 t P t=0 ~ exp kT In one coordinate (x ) the density dN/N distribution of electrons or ions after time t will probably be gaussian: t>0 x dN x = exp N 4 Dt 2 dx 4 D t x Anna Lipniacka , Detector techniques = 2Dt D- diffusion coefficient L5 Diffusion coefficient dN x2 = exp N 4 Dt dx 4 D t x= the diffusion coefficient is measured in cm**2/s 2Dt Since the mean diffusion will be grow with the thermal velocity : x~ ut~ 3kT m the diffusion coefficient D will decrease for heavier particles- will be much bigger for electrons than for ions for example. Example numbers : Hydrogen : 5 u t = 1.8 × 10 cm / s ; D + = 0.34 cm / s ; 2 ; cm 2 ; = 2.2 Vs 2 + cm = 13 Vs = 1.8 × 10 5 1.0× 10 5 cm ; x t = 1s 0.7 cm cm ; x t = 1s 0.35 cm Oxygen : 5 u t = 0.46 × 10 cm / s ; D + = 0.06 cm / s ; 2 Anna Lipniacka , Detector techniques + = L5 Electron capture and ion recombination Ion recombination: positive ions can recombine with electrons or negative ions. The recombination probability in a “single encounter” with an electron or negative ion will depend on the type of ion, its ionization potential for example. We describe it in terms of a recombination coefficient + dn dt = n+ n- n -> concentrations of positive and negative particles. Recombination coefficient is measured in cm**3/s . “Big” values of the coefficient are for “reactive gasses” for example oxygen and C02 the recombination coefficient is 10**{-6} cm**3/s for recombination with negative ions and 10**{-7} cm**3/s for recombination with electrons. Electron capture : electronegative gasses: 02, H20, NH3 etc can accumulate electrons. For these gases probability to capture an electron in a single collision is ~ 10**{-5}, thus on average an electron will be captured after 10**{5} collisions. How long it will take (t_a): time between collisions: Exercise: calculate it for 02. ts= s ut or n s = ut s ta = s ts = pa ut pa The result is of the order of 200 ns Anna Lipniacka , Detector techniques L5 Drift of electrons in electric and magnetic fields Coulomb force : qE, Lorentz force q v×B Simple case, magnetic field alone, or electric and magnetic field parallel : v_t 2 v t q B vt q v t B = m = = R R m For electrons = 17.6 MHz / Gauss B R B transversal velocity unchanged. The velocity parallel to magnetic field will not be changed if there is no electric field, while if there is an electric field parallel to B the velocity parallel to B will eventually reach the drift velocity characteristic for electrons in a given gas The electron will spiral around the direction of magnetic, electric field with frequency above, while moving along the electric field with drift velocity B, E Anna Lipniacka , Detector techniques L5 electric and magnetic field The general equation of motion for electron in a gas in the presence of two fields is the following: v×B m v= q E mA t stopping force due to collision with atoms This is Langevin equation. The stopping force will depend on the drift velocity and time between collisions. The drift velocity will adjust itself in such a way that the stopping force cancels the force due to EM fields- resulting acceleration will be 0. Thus we can write : 0= q E vD × B Solution: m vD E × B E B time between collisions E B B vD = 2 2 1 B2 qB ; = q m m Anna Lipniacka , Detector techniques 2 2 L5 Perpendicular E and B The solution for drift velocity is quite interesting for perpendicular electric and magnetic fields: E = E x ,0,0 , B = 0,0, B z Ex vx= 1 2 2 E x v y= 1 2 2 vz=0 Ex 2 2 v D= v x v y= 1 2 2 vy tan v D , E = = vx Lorentz angle Anna Lipniacka , Detector techniques L5 Diffusion in electric and magnetic fields Magnetic fields will decrease diffusion perpendicular to field direction by “curling down” thermal velocities : For B along z we have : DZ = D ; D X = DY = D 1 2 2 Electric fields can increase effective diffusion along E introducing the anisotropy D_L/D In practice we would like to decrease diffusion perpendicular to E-> this results in choice of B parallel to E for drift detectors, whenever possible E B Anna Lipniacka , Detector techniques L5 Detecting hadrons Charged hadrons loose energy by ionization as well and can be seen in tracking detectors. But both charged and neutral hadrons (eg neutrons) have possibility of strong interactions in matter. Typical cross-sections of strong interactions are of order of 10mbarn/nucleon. We can describe mean free path λ of hadrons in matter in terms of scattering cross-section Nn on a nucleon σ_n 24 26 2 gram ~ ln(E) 5.63cm 7cm E few λ = 0.5 10 nucleons / gram 10 mbarn = 10 cm 200 g cm 2 N Hadronic interaction lead to nuclear fragments = / and to hadronic cascade, which is usually contained in a few λ 's . Cascade particles give signal in proportional counters-the energy of the cascade is measured. Cascade should be contained for a good measurement 200 cm of iron ~ 8 λ Hadron calorimeter Anna Lipniacka , Detector techniques L5 Hadronic cascade and interaction length Hadronic interaction length is typically much longer than radiation length for electrons ( and photons). This explains why the typical size of hadronic calorimeter is much bigger than that of electromagnetic calorimeter. The development of the cascade is much harder to predict than in case of EM interactions, although Monte Carlo approaches with detailed calculations of nuclear processes make a good progress (GEANT4). For detector planing extensive beam tests and parametrized results of these are used. For example : L 95 = 9.4 ln E GeV 39 cm of Fe The energy registered in calorimeter is usually incomplete due to several effects: - escaping muons and neutrinos from pion decays - nuclear collision lead to nuclear excitations, fissions and spallations. These process result in low energy gamma rays, slow protons, nuclear fragments. Each of them with quite different types of reactions and different “visibility” in calorimeter. - as a result typically the response of the same calorimeter to pions and electrons of the same energy is ~ 30% lower for pions. Anna Lipniacka , Detector techniques L5 Calorimeter length Calorimeter depth in iron needed to contain the shower in 99% and 95% Data from two experiments and parametrisation. Anna Lipniacka , Detector techniques L5