Energy partition and distribution of excited species in direction-sensitive detectors for WIMP searches

Transcription

1 Cygnus 013 Toyama, Japan 10-1 June 013 Energy partition and distribution of excited species in direction-sensitive detectors for WIMP searches Akira Hitachi Kochi Medical School Key words: S T =S e + S n quenching factor q nc =η/e LET el = dη/dx Bragg-like curve: dη/dr PRJ Don t miss ISON in late this year PANSTARRS c/011 L4

2 Head-tail discrimination The Bragg-like curve for head-tail detection The distribution of the electronic energy η deposited in the detector gas as a function of the ion depth, i.e. projected range R PRJ. Δη/ΔR PRJ It is an averaged one dimensional presentation. For slow ions, it is not given by the electronic stopping power, S e = (de/dx) e. One needs the Lindhard factor q nc = η/e.

3 Nuclear Stopping Power Lindhard Interaction potential A screened Rutherford scattering U( r) = ( Z1Z e / r) φ( r / a) φ( r / a) : Fermi function [ exp(-r/a B ): Bohr] /3 /3 1/ a: Thomas-Fermi type a = a0( Z1 + Z ) screening radius A universal differential cross section dσ = πa t t dt 3 / f ( t 1/ θ = ε ( T / Tm ) = ε sin The nuclear stopping power ) f(t 1/ ) dε dρ n ε = dx 0 f (x) ε t 1/ = ε sin(θ/)

4 Stopping Powers The nuclear stopping power S n < T( E) > πpdp S n ( E) = = Nσ < T >= Nσ T( E, θ( p)) λ( E) σ 0 = πn <T(E)> is the mean energy transferred in an elastic collision, and λ(e) = 1/Nσ T( E, θ) = ( A 4A A θ E sin A ) S n can be expressed by the analytical expression [Birsack 1968] S n = de ds 4πaNA1 Z1Ze = A + A 1 lnε ε (1 ε 3/ ) 0 Td ( p ) for all Z 1, Z The electronic stopping power S e An atom moving though an electron gas of constant density. Using a Thomas-Fermi treatment, [Lindhard & Scharff} S e = π 1 ξe 8 e a0 / 3 / 3 3/ ( Z1 + Z ) v ξ 1/ 6 e Z 1 0 S 1/ e = kεk = 0.1 ~ 0. Z Z v for all Z 1, Z

5 Stopping Power Lindhard Low energy v < v 0 =e /ħ The generalized range and energy ρ = RNA 4πa ( A 1 A1 + A ) d ε /d ρ ε = kev 10 kev Xe/Xe (k=0.165) 1 kev A r/a r (k=0.144) N e/n e (k=0.138) 100 kev H e/h e (k=0.106) S e = k ε1/ 10 kev ε = E Z 1 Z aa e ( A + A) S n ε 1/ Nuclear S n and electronic S e stopping powers as a function of energy ε for k=0.15. The Thomas-Fermi treatment becomes a crude approximation at the extreme low energy and separation of the nuclear scattering and electronic stopping becomes uncertain. LSS theory is not very reliable at ε < 0.01, i.e., below 10 kev for Xe ions in Xe. In this sense, almost all the stopping theory available is not reliable below ε < 0.01.

6 Lindhard factor q nc = η/ε The stopping powers contain only a part of the necessary information to obtain the quenching factor, q nc = η/ε ratio. The differential cross section in nuclear collisions is needed for the integral equations. For Z 1 = Z, + = ʹ ε ν ε ν ε ε ν ε ν ρ ε ε t t t f t dt d d e ) ( ) ( ) ( 1/ 0 3 / 1/ ε ρ ε k d d e = η as a fn of ε for k = 0., 0.15, 0.1 ε = η + ν

7 Electronic Linear Energy Transfer (LET el ) LET el -dη/dr = -Δη/ΔR R: the range = -(η 1 -η 0 )/(R 1 -R 0 ) for quenching calc. etc. The true range R is given by the total stopping power R T = (de/dx) total -1 de Birks eq. LET el should be used, not S el. The Bragg-like curve for TPC The projected range, R PRJ, may be used (depth) b) E a) E1 η 1 E0 η 0 Δ E Δη q ΔΕ RPRJ R1 R0 R

8 Stopping Power and LET Xe ions in Xe TotalStopping Power de/dx (MeVcm/mg) Electronic LET Nuclear Stopping Power 1.0 Electronic Stopping Power ENERGY (kev) Fig. 1 The stopping power and the electronic LET as a function of the recoil energy for Xe in Xe. HMI & Lindhard

9 Bragg-like curve.5 Xe/Xe 00 kev.0 Δη / Δ R PRJ (kev cm / µ g) kev 100 kev Ar/Ar Ne/Ne ions kev Projected range ( µ g/cm ) Bragg-like curves, dη/dr PRJ for recoil ions in rare gases. The ions enter from the right hand side. Points are plotted at every 5, 10 or 0 kev The projected ranges are taken from SRIM. Head and tail detection of WIMPs

10 He in He (+5%C 4 H 10 ) He in He+5%C 4 H 10 Santos 008 The measured points are considerably smaller than Lindhard and SRIM. There are passes to no ionization He* + M He + M(excitation, dissociation, non radiative decay) The W-value can depend on He pressure and dopant pressure.

11 W-value Contributions of 1 S and 3 S may be added for slow ions in He. He( 1 S), He( 3 S) and He ( 1 S) and He ( 3 S) are true metastables. He( 1 S) and He( 3 S) were not produced directly by fast particles; however can be produced directly by slow ions. Resonance trapping for He( 1 P). (low pressure) 10keV e in He Platzmann 1961 W α /W β 1 for rare gases fast ions A part of energy will diffused out of the chamber. The W-value can depend on energy and type of the incident particle at low energy. Energy (ev) Contribution to ionization and excitation.

12 Lindhard factor for Z 1 Z The formation of accurate general solutions become quite complicated. The integral equation for the ion Z 1 in the matter Z is, dσ 1 : the diff. cross section for an elastic colli. between Z 1 and Z. T < T m = γe ; γ = 4A 1 A /(A 1 +A ) Characteristic energies: 3 4 / 3 1/ 3 1 E c A ( A + A ) Z Z 500 ev, A + A ) A Z 15 ev, where The power law approximation for very low energy, The straggling: { ν ( E T) ν ( E) ( )} ν ʹ 1( E) S1 e = dσ ν T η = CE Ω 3/ 1/ 1 γ η = γ 14 CE c / 1/ 1/ 1 1/ C = E1 c + γ E c E c ( 1 1 1/ 7 16 Good for heavy ions such as Pb ions in α-decay, for E < E 1c, E c, 3 / 3 Z / = Z1 + Z / 3

13 Lindhard factor for recoil ions in α-decay ================================================================================ Recoil ion 06Pb 08Tl 08Pb Ec Energy kev gas expt calc expt calc expt calc kev Ar 0.1a Xe a H He b b CH CH C3H CO C + O C O N Dry air N + O CS C + S S Al (for CS) Recoil ion 14Pb 10Pb Energy kev 11 calc 147 calc CS SRIM by P.J. C + S Al (for CS) ================================================================================= calc : the power law approximation by Lindhard which is only good at E<Ec. expt : W(alpha)/W(recoil) from: a G.L.Cano, Phys. Rev. 169, 7 (1968). b W.G. Stone & L.W. Cochran, Phys. Rev. 107, 70 (1957).

14 Binary gases 1.0 Binary gases such as CS, CF 4 S T the Bragg rule compound correction Simple approximation for q nc to consider only one element at a time. C in CS C in C, S in CS S in S N uclear quenching factor CS C/Al p.l C k = 0.17 C/S asym. S/Al p.l. S k = 0.15 Pb/C Pb/(C+S) Pb/Al 0. S/C asym. Pb/S Energy (kev) 4 Pb CS 00 ΔNi solid: C ions in C, S ions in S dashed: Asymptotic eq. C and S in Al, : Snowdon-Ifft expt. dη/dr PRJ (M ev cm /m g) kev 100 kev 50 kev S 00 kev 100 kev C 00 kev (W = 19 ev ) (ions cm /µg) 100 Power law approximation 1/3(Pb in C)+/3(Pb in S) 10 kev prelim inary Projected Range (µg/cm )

15 CF 4 N uclear quenching factor CF 4 Pb C F k = 0.17 k = 0.13 dη/drprj (M ev cm /m g) CF 4 10 kev 50 kev F 50 kev 100 kev 100 kev 00 kev C 00 kev ΔN i (io n s c m /µg) Energy (kev) Projected range (µg/cm ) [C]:[F] = 1:4

16 fast ions The maximum impact parameter b max is given by Bohr s impulse principle b v / E max= 1 v: the incident ion velocity : Planck s constant divided by π E 1 : the lowest excitation energy. δ-ray penumbra Excitation by the light ions (p, α) described theoretically in terms of perturbation by an incident point charge. core b max

17 Stopping power for slow light particles The Coulomb effect the deflection and deceleration of the projectile in the field of the nucleus. The Threshold effect the energy delivered to e - must be as large as I. D. Semrad, PRA (1986) I.S.Tilinin, PRA (1995) H-C (1) He-N () I.S.Tilinin, PRA (1995)

18 SRIM or TRIM Based on The B-K theory originally TRIM was for light particles and at high energy. It adapted the LSS theory for slow heavy ions. SRIM, NIMB 01 The measurement of stopping power for heavy ions of low velocity is quite difficult Usually, S T = S e + S n is measured and often Lindhard is assumed for S e. S el LSS, Ne 40keV A. Fukuda, The spectrometer determines the energy quite accurately; however, the measurement does not take contribution from θ > 0.

19 Heavy-particle collisions at low energy When atomic projectile goes hard (wide deflection angle small impact parameter) collision with atom, the large inelastic energy losses occur at characteristic internuclear distances. Showers of fast-electrons are thrown out. Inelastic energy losses Q for Ar + -Ar collision e - e - e - (ArAr) + Molecular Orbit (MO) Kessel & Everhart, PR146 (1965)

20 Molecular orbital (MO) collision theory v c << v e, the electronic motion of the system is expected to adjust adiabatically to the changing position of colliding nuclei. The transient formation of molecular orbitals (MO s) MO s can connect different shells. An electron occupies a MO and become excited during the collision to a higher energy at smaller separation. It contains vacancies after the collision. Coordinate system for a quasimolecule composed of an electron e - and two nuclei A and B. J. Eichler, Lectures on ion-atom collisions Energy levels of diabatic (H + -like) molecular orbitals of the Ar-Ar system.

21 W. Lichten, PR 131 (1963)

22 Nuclear quenching factor q nc = η/ε ratio Some puts: q nc ε Sel ( ε ) ( ε ) = RHS: integrated value S ( ε ) + S ( ε ) LHS: differential value nc el

23 Wave optics using ultrasound S P U 0 The wave U 0 observed without deflecting object delays π/ in phase compared to the wave U sa passing on the central axis. U sa Hitachi&Takata, Am J Phys 78, 678 (010). Hitachi, J Acoust Soc Am, 131, 463 (011). Books by Born&wolf, Jenkins&White, etc.