The Bragg-like curve for the directional

Transcription

1 CYGNUS 007 July 007 The Bragg-like curve for the directional detection of dark matter Akira Hitachi Kochi Medical School Head-tail discrimination Treatment for Z 1 Z Pb in α-decay, Ar-gas mixture Binary gases CS

2 Simulating WIMP signals in a lab. WIMP Xe few 10 kev Neutron scattering experiments E rec An AXe = E (1 cos ) ( + ) θ n A A A xe = 131 E max = n Xe rec E n at θ = π

3 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. η/ R PRJ It is an averaged one dimensional i presentation. ti 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.

4 Stopping Power Lindhard Low energy v < v 0 =e /ħ The generalized range and energy ρ = RNA 4πa ε = E Z 1 Z ( A 1 A1 + A ) aa e ( A + A) dε/dρ The nuclear and electronic collisions are treated separately ε > ε = kev 10 kev 1 kev N e/n e (k =0.138) H e/h e (k =0.106) Ar/Ar (k =0.144) 100 kev Xe/Xe (k =0.165) 0165) S k e = kε 1/ S n 10 kev ε 1/ Nuclear S n and electronic S e stopping powers as a function of energy ε for k=0.15.

5 Nuclear Stopping Power Lindhard Interaction potential A screened Rutherford scattering U ( r) = ( Z 1Z 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 0 dx f (x) ε t 1/ = ε sin(θ/)

6 Stopping Powers The nuclear stopping power S n < T ( E ) > πpdp pp S n ( E ) = = Nσ < T >= Nσ T ( E, θ ( p )) λ( E) σ 0 = πn Td( p ) <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 de 4πaNA1 Z1Ze lnε = = n 3/ ds A 1 + A ε (1 ε ) for all Z, Z 1 0 The electronic stopping power S e An atom moving though an electron gas of constant density. Using a Thomas-Fermi treatment, [Lindhard & Z1Z v Scharff} Se = ξ e 8πe a0 1/ 6 / 3 / 3 3 / ξ e Z 1 1/ S e = kε ( Z + Z ) v 1 ) 0 for all Z 1, Z

7 Energy shearing in low energy For slow ions, v < v 0 = e /ħ, S e and S n are similar in magnitude. The secondaries, recoil atoms and electrons, may again go to the collision process and dtransfer the energy to new particles and so on. After this cascade process complete, the energy of the incident particle E is given to atomic motion ν and electronic excitation η. E atomic, ν heat T Bolometer electronic, η charge Q SSD, gas TPC LXe scinti. S Scintillator Electronic energy Nuclear quenching factor q nc = = η /E (Lindhard factor) Ion energy integrated for individual recoil created in the cascade integrated for individual recoil created in the cascade, the energy that went to electronic excitation.

8 Lindhard factor η/ε Lindhard factor η/ε The stopping powers contain only a part of the necessary information to obtain 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 collisions is needed for the integral equations. For Z 1 = Z, ε t t dt d + = ε ν ε ν ε ε ν ε ν ρ ε t t t f t dt d d e ) ( ) ( ) ( 1/ 0 3 / ε d 1/ ε ρ ε k d d e = η as a fn of ε for k = 0., 0.15, 0.1 ε = η + ν

9 Asymptotic equation η = ε ν Z 1 = Z, 0.1 < k < 0. large ε: ν ~ g 1 (ε) k -1 ε<1: ν ε ν~ ε k g (ε) ν ( ε ) = ε 1 + k g ( ε ) g(ε) is parameterized by Lewin & Smith (ApP 1996) g(ε) = 3ε ε ε

10 Quenching factor in rare gases Z 1 =Z Lindhard factor q nc Numerical Calc. k =0.1, 0.15, 0. Asymptotic form k = 0.1~0. ε > Xe > 10 kev, o quenching fact He/ He Ne/ Ne Ar/ Ar Xe/ Xe Xe/ Xe He & Ne satisfies ε > large W-values small N i change in the energy balance W = E i +(N ex /N i )E ex + Ε sb RN/γ ratio in gas q nc RN/γ Energy dependence in W γ and W RC. quenching facto Energy (kev) He/ He Ne/ Ne Ar/ Ar Xe/ Xe Energy Dashed curves are not reliable

11 Linear Energy Transfer (LET) LET: The energy deposited per unit length LET -de/dx for fast ions S T S e The electronic LET LET el = -dη /dx S T = S e + S n should be introduced for slow ions The ionization density The quenching calc., S/T ratio etc. is given by LET el in liquid [not by the electronic SP, (de/dx) el ] The Bragg-like curve for TPC practical & macroscopic The direction of recoil ions

12 Electronic Linear Energy Transfer (LET el ) LET el -dη/dr = - η/ R 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 E a) E1 E0 EE R T = (de/dx) total -1 de The Bragg-like curve for TPC The projected range, R PRJ, may be used (depth) η1 η0 η q Ε b) R1 R0 R RPRJ

13 Stopping Power and LET Xe ions in Xe TotalStopping Power (MeVcm/m mg) 3.0 Nuclear Stopping Power de/dx Electronic LET 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

14 electronic LET Tem porary Xe/Xe (ions cm /µg) kev Ar/Ar Ne/Ne 100 kev ions N i kev 5 kev GasPhase Range (µg/cm ) Fig. The electronic LET, dη/dr T for recoil ions in rare gases. The ions enter from the right hand side. Points are plotted at every 5 kev. Used for the electronic quenching calc in condensed media.

15 A. Hitachi, Astropt. Phys. 4, 47 (005) Gwin (196) 1. Scintil llation Eff ficienc q β e CsI(Tl) p e γ C p Si α α Ar Fe RCXe el RCXe TTL Ti(SXe) Ni(SXe) f.f. Liquid Xe U (SXe) LET (MeV g -1 cm ) Fig. Scintillation efficiency for various ions as a function of the electronic LET for liquid Xe and CsI(Tl).

16 A. Hitachi, Astropart. Phys. 4, 47 (005) 0.5 q nc ency q o gnal Effici Si Si LXe q nc (k =0.165) q TTL Reco oil/γ rati o for Energy (kev ) Fig. 4 Scintillation efficiency i for recoil ions in LXe as a function of recoil ion energy. The nuclear (q nc : Lindhard) and the total (q TTL =q nc q el ) quenching factors are shown. The results for Si are also shown for comparison. q el : exiton-exciton collision model Difference between theory and expt. is due to mainly uncertainty in γ efficiency (α/γ ratio).

17 Park, NIMA 491, 460 (00) Pécourt, Astropt. Phys. 11, 457 (1999) 0.5 recoil/ γ rati P.W. Pecourt Kudryavtsev Kudryavtsev Park 0.46% 0.18% 0.073% 0.073% 0.071% Pecourt 0.31% Tl 0.046% Tl 0.05 CsI(Tl) tem porary Energy (kev) Fig. 3 The recoil ion to γ ratio in CsI(T) as a function of recoil ion energy. The broken lines are present estimates. The solid line is fitting to the Birks-Lindhard model by Pecourt.

18 Time Projection Chamber DRIFT gas TPC (Time Projection Chamber) electronegative gas such as CS, CF 4, low diffusion The Bragg-like curve

19 The Bragg-like curve LET el -dη/dr η = - η/ Rη R: the range E a) E0 EE The Bragg-like curve for TPC The projected range (depth), R PRJ, may be used E1 η1 η0 η q Ε b) R1 R0 R RPRJ

20 Stopping Power and Bragg-like curve for He/He de/dx (MeV Vcm /mg) He in He 14 LET el Temporary TotalSP Elactronic SP Nuclear SP ENERGY (kev) Ni (arb b. un He /He E = 1-0 kev W = 46 ev 5 -de/dr PRJ Range (µg/cm ) 15 tem porary For quenching calc. LET el =-dη/dr T For TPC, the Bragg-like curve LET el =-dη/dr PRJ dη/dr T > (de/dx) el dη/dr PRJ can be lager than (de/dx) T

21 Bragg-like curve.5 Xe/Xe 00 kev 0.0 η/ R cm PRJ (kev /µg) kev 100 kev Ar/Ar Ne/Ne ions η kev Projected range (µg/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

22 η/ R PRJ for recoil ions in TPC gases Rare gases Rare gas molecule mixture Ar CO, CH 4, CF 4, CS (low concentration) S T in pure Ar or use the Bragg rule q nc for recoil ions are obtained as each ions in pure Ar the W v 0 /v plot (empirical) Binary gases CS, CF 4 S T the Bragg rule + compound correction no general methods q nc

23 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, { } ν 1( E) S1 e = dσ 1 ν 1( E T ) ν 1( E) + ν ( T ) 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 1 1 ( 1 1 The power law approximation for very low energy, η = CE 3 / The straggling: Ω 1/ 1 γ 7 7 = γ + 1/ 14 CE c 4 16 γ / η E 3 / 3 / 3 Z / = Z1 + Z E c ( 1 1 1/ 1 1/ 1/ C = E1 c + γ E c, 3 for E < E 1c, E c, Good for heavy ions such as Pb ions in α-decay

24 Recoil ions in α-decay - Power Law Approx. - Rare gases E in kev Ω /η E c He Ne Ar Xe η = 0.053E η = 0.053E η = E η = 0.01E 3 / 3 / 3/ 3 / kev kev kev MeV The asymptotic equation does not work

25 Quenching factor in Ar r Quenching Fact N Ar O Ar ions q nc,num erical Numerical Pb Nuclea 0. Ling & K nipp Power Law Appr 0.1 Ar gas Energy (kev) Experimental results in gas W(α)/W(RC) ~ q nc N, O and Ar; Phipps et al [1964] Pb; Madsen [1945], Jesse & Sadauskis [1956]

26 Lindhard factors in rare gases 1.0 q nc Q uench hing factor k = N e/n e H e/h e k = 0.14 k = Pb/He H e/h e H e/h e Pb/H e P b/h e (m easurem ent) N e/n e Pb/N e Pb/Xe Xe/Xe P b/xe (m easurem ent) 0. Xe/Xe Pb/Ne 0.0 preliminary Pb/Xe Energy (kev )

27 Recoil Pb ions in α-decay in LAr Energy Straggling WARP kev σ ~ 1% 144 kev σ ~11% Power law approximation (Lindhard) σ = Ω 1/ / η = γ + 1/ 14 CE c 4 16 γ σ ~ 14%

28 Recoil ions in α-decay - compounds - No works has been done for η/ε in compounds. A) Power low approx. q nc =η/e I) replace target CS with single element (Al): 11 kev Pb-14 Pb in Al η = 0.0E 3/ (E: kev) 0.34 II) Divide the target elements and calculate separately Pb in C and Pb in S, (Pb in C)/3 + (Pb in S) /3 Pb/C η = 0.031E 3/ (E: kev) Pb/S η = 0.00E 3/ (E: kev) 0.13 Pb/C/3 + Pb/S*/ B) Asymptotic equation Z 1 = Z 103keV Pb in He, C, N, O, S k = keV Pb q nc values are practically the same therefore not agree with expt. does not include the effect of the secondary ions

29 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).

30 Bragg-like curves for Pb ions in rare gases 10 8 Pb/He Pb/Ne Pb ions η/ R PRJ (kev cm /µ µg) 6 4 Pb/Ar Dh/DRPRJ Pb/He Dh/DRPRJ Pb/Ne Dh/DRPRJ Pb/Ar Dh/DRPRJ Pb/Xe Pb/Xe Projected range (µg/cm )

31 Bragg-like curves for Xe and Pb ions in Xe, Xe 100 kev 00 kev 00 kev, η/ R R PRJ (M ev cm /g) 1,500 1, Xe 50 kev 50 kev Pb 100 kev kev 0 kev preliminary Projected R ange (µg/cm ) 0 The LET el for Pb ions in a-decay and kev Xe ions are also quite similar therefore q el for those ions in liquid Xe are expected to be close.

32 Quenching factor in LXe qnc Xe ions in Xe qnc Pb ions in Xe M easurem ent qt Xe ions in Xe qt Pb ions in Xe scale 0.5 q nc 0.4 quench hing factor 0. Xe q T = q nc q el q nc RC/γ γ ratio 0.1 Pb q T Energy (kev ) Xe

33 Quenching factor in Ar light ions Empirical formula in Ar Z 1 < Z, v 0 /v >.5 W RN (ev) = 15.3(v 0 /v) q nc = W α / W RN = 6.4/[15.3 (v 0 /v) +6.1] empirical W Phipps et al.pr 135,A36 (1964) C N O Recoil ina-decay Ar He H fit W-value extrp W 0 = 6.4 ev Ar gas v 0 /v

34 Rare gas molecule mixture Ar (75%) + CS (5%) mixture. The recoil atom ionize mostly the host Ar, then the electron is immediately attached to CS. Ar Ar + +e - e - + CS CS - The CS - ions drift to the proportional region and give the signal. It is necessary to obtain q nc and R PRJ to obtain the Bragg-like curve for TPC. S t i l t A t i th i di t bl It i f i l f t S atom is close to Ar atom in the periodic table. It is fairly safe to consider S ion as Ar ion with the same velocity in the calculation for q nc. Then, the mixture may be regarded as Ar (9%) + C (8%). The value of q nc for C ions in Ar can be estimated by the W-v 0 /v plot.

35 Bragg-like curves for light ions in Ar.0 Armixture 100 kev 00 kev Ar/Ar ions η/ R (kev cm PRJ /µg g) kev 50 kev S/A r F/Ar C/Ar 35 kev H/Ar A r recoilion in A r H/Ar C/Ar F/Ar S/Ar kev Projected range (µg/cm ) Th l f R bt i d b SRIM i A The values for R PRJ are obtained by SRIM in pure Ar The W-values can be different in pure and mixture gases.

36 Binary gases Binary gases such as CS, CF 4 S T the Bragg rule + compound correction (e.g. SRIM) R PRJ The Lindhard factor q nc no general methods Approximations A) Replace the molecule with single element e.g., C in CS C in Al then use 1) the power law approximation (at the low energy) not appropriate for the light ions. ) the asymptotic equation (Z 1 = Z, k = 0.1 ~ 0.) B) Consider only one element at a time. e.g., C in CS C in C, S in CS S in S

37 Lindhard d factor for binary gases CS CS C k = 0.17 ar Q uenching Factor S k = 0.15 N ucle 0. Pb pw l 0.0 Preliminary Energy (kev ) solid: C ions in C, S ions in S dashed: Asymptotic eq. C and S in Al Power law approximation 1/3(Pb in C)+/3(Pb in S), : Snowdon-Ifft expt.

38 The Bragg-like curves for recoil ions in CS 4 Pb CS 00 Ni g) dη/dr PRJ (M ev cm /m kev 100 kev 50 kev S 00 kev 100 kev C 00 kev (W = 19 ev ) (ions cm /µg) kev preliminary Projected Range (µg/cm )

39 Binary gases comparisons, Th & Mes. The approximations for S ions in CS are fairly ygood. Those for C ions gives smaller q nc values for > 80 kev. Pb ions is given by the power law approximation. CF 4 contains only light ions also the structure for CF 4 is the same as CH 4. SP calc. compounds corrections? W-value particle & energy dependence? Pb ions the power law approx. does not work as in hydrocarbons? NEEDS measurements for q nc

40 Stopping power for compounds The stopping power for compounds are obtained using the Bragg- rule: the weighted sum of the elemental matter. mix S / [ev A /atom] = NiS N i i The compound correction is needed d for some compounds in low energy. compounds containing mostly H, C, N, O, and F 1) The outer shell electrons have different orbitals in the compound than in element matter. The core and bond approximation Ziegler & Manoyan, NIM B35, 15 (1988) ) The triplet states in those molecules are metastable which is not excited by fast particles but can be excited effectively by slow heavy ions.

41 Summary The Bragg-like curves, dη/dr PRJ, are introduced for head-tail detection for WIMPs in gas TPCs. The estimation of Lindhard factor, q nc = η/e, for the recoil ions is the key. a) Pure gas: He, Ne, Ar, Xe Lindhard, ε > 0.01 (Xe: E >10keV) b) Rare gas-molecule mixture Ar CS, CO, CF 4 W-v 0 /v plot for light ions (empirical) c) Binary gas Some approximations are necessary. CS C ions in CS C ions in C S ions in CS S ions in C The asymptotic equation (?) d) q nc for very heavy ions in α-decay can be obtained by the power law approx. except for the hydrocarbons. The asymptotic equation does not work because it does not include the effect of the secondary ions.

42 Summary cont d The experimental results coming in the R&D process of the TPCs for dark matter detection ti are quite important t also for the low energy stopping theory. The comparison between theory and experiments have been done almost exclusively through the range and straggling. However, the Lindhard factor becomes very important at the low energy. W-value measurements at a high-enough g field q nc W(α)/W(RC) 3D construction of the track

43 Electronic Stopping Power Aarhus expt Lindhard-Scharff Land & Brennan, Atom Data Nucl Data Tables,, 35 (1978)

44 Nuclear Stopping Power Comparisons of the nuclear stopping powers D.W. Wilson et al. P.R.B. 15, 458 (1977).

45 SP HMI & SRIM 3,000 Ar/Ar S T de/ /dx [M ev /(g/ /cm )],000 1,000 HMI S n 0 SRIM S e Energy (kev )