limitations and applicability of the lindhard model for few ...peter sorensen, lbl 19 feb 2016...
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![Page 1: Limitations and applicability of the Lindhard model for few ...Peter Sorensen, LBL 19 Feb 2016 Limitations and applicability of the Lindhard model for few keV nuclear recoils 1 see](https://reader034.vdocuments.us/reader034/viewer/2022051805/5ff50910e79aca4e8f37581d/html5/thumbnails/1.jpg)
Peter Sorensen, LBL 19 Feb 2016
Limitations and applicability of the Lindhard model for few keV nuclear recoils
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see also: PhysRevD 91 083509 (2015)
Peter Sorensen (LBL)
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Peter Sorensen, LBL 19 Feb 2016
Caveats
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• This is mostly a theory talk
• No theorist has exactly solved this problem (collective many-body scattering)
• I’m no theorist
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Peter Sorensen, LBL 19 Feb 2016
Motivation
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• Measuring low-energy nuclear recoils signals is challenging
• Models can be helpful, if only as guidance
• Literature is littered with statements about how Lindhard model is not applicable at low energy, or below epsilon ~ 0.01
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Peter Sorensen, LBL 19 Feb 2016
An experimentalist descends from an ivory tower, with the Lindhard model inscribed on two tablets
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http://www.thebricktestament.com/exodus/replacement_stone_tablets/ex34_29.html
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Peter Sorensen, LBL 19 Feb 2016
Talk outline
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• Description of the Lindhard Model
• Uncertainties
• in nuclear scattering treatment
• in electron scattering treatment
• Modification of the model parameterization and solution to account for atomic binding
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Peter Sorensen, LBL 19 Feb 2016
The big picture tends to gloss over the atomic physics
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LUX, 1211.3788
pictures tend to influence our thinking
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Peter Sorensen, LBL 19 Feb 2016 7
+Z
-Z
+Z
-Z
• two body screened Coulomb nuclear scattering• average electronic scattering (stopping, really: projectile atom perturbs free electron gas)
The small picture tends to oversimplify the atomic physicsi.e. this is model, not perfect physical reality
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Peter Sorensen, LBL 19 Feb 2016
The scattering problem is simplified to effective two-body kinematics
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The origin of electronic signal: • nucleus gets a kick (from a neutron, a neutrino, dark matter)
• atom recoils• creates secondary recoils• cascade continues until atoms are thermalized
• each collision might excite or ionize a target or projectile atom
• but, individual electron collisions?? too complicated. average over electronic energy losses
etc, etc
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Peter Sorensen, LBL 19 Feb 2016
The Lindhard model, single slide version
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electronic energy
losstarget
atom after collision
projectile atom before
collision
projectile atom after collision
nuclear energy loss
•Integrate over the cascade, obtain a solution for (the energy given to atomic motion)•A parameterization of the solution is
which leads directly to
fn is what we usually call the quenching factor
reduced energy
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Peter Sorensen, LBL 19 Feb 2016
Approximations in nuclear scattering treatment
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Ziegler, Biersack, Littmark, “The stopping and range of ions in solids” (1985)
interatomic screening lengthaI = 0.8853 a0 /(Z-1/3√2) ~ 0.1
differs from single atom screening length by factor 1/√2
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Peter Sorensen, LBL 19 Feb 2016 11
What is k?
Markin et al, Phys. Rev. Lett 103 113201 (2009)
•
• all calculations (there are many) predict this basic behavior
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Peter Sorensen, LBL 19 Feb 2016 12
Approximations in electron scattering treatment
Markin et al, Phys. Rev. Lett 103 113201 (2009)
• Calculations supported by data, but
1. not a lot of data
2. non-zero x intercept is often observed
3. generic expectation for semiconductors to deviate from (drop below) velocity-proportional stopping at low energies, due to band gap (as observed by DAMIC, see Tiffenberg talk)
• Should think of liquid nobles as large band gap insulators in this context
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Peter Sorensen, LBL 19 Feb 2016
Variations in electron scattering (“electronic stopping”) calculations
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Si Ar Ge Xe
• Large uncertainty in k is possible
• Ge happens to be at a sweet spot (all calculations converge)
• Si appears to be approximately sweet
• Liquid nobles may differ (drastically) from naive Lindhard k
Land et al, Phys. Rev. A 16 492 (1977)
Ne
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Peter Sorensen, LBL 19 Feb 2016
Recap
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• The Lindhard model makes numerous approximations in order to distill solid state atomic scattering into a tractable problem• this results in quantitative predictions that appear to agree fairly well for a number of 4 frequently
used homogenous targets• it is difficult to accurately quantify the uncertainties, but a range can be inferred
• The low velocity behavior of electronic stopping is expected to decrease in materials with a band gap (i.e. materials from which one might make a detector !!)• this is difficult to quantify and data are sparse
• The model as widely disseminated does not account for atomic binding• intuitively this must make a difference at low energy• it can be re-instated in model…
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Peter Sorensen, LBL 19 Feb 2016
First simple tweak to the model: improve the parameterization
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•NB: the % error of the standard solution to the Lindhard model equation increases dramatically below ε ~ 0.01 (arrow)
•hypothesis: at the smallest energies, some irreducible amount of energy must always be wasted in atomic motion
•add a constant energy term q and re-solve the integral equation (cf. slide 9)
•result is dashed orange curve
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Peter Sorensen, LBL 19 Feb 2016
Second simple tweak to the model: account for electron binding energy
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•replace the term with and re-solve the integral equation (cf. slide 9)
•u is the average energy required to ionize an electron (the w-value)
•result is solid blue curve
•prediction of a kinematic cutoff
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Peter Sorensen, LBL 19 Feb 2016 17
Result for Si
•NB: new data from DAMIC
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Peter Sorensen, LBL 19 Feb 2016
Result for Xe
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•NB: new data from LUX
•NB: quenching applies to sum(electrons + photons)
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Peter Sorensen, LBL 19 Feb 2016 19
Result for Ar
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Peter Sorensen, LBL 19 Feb 2016
This matters if you are…
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•Searching for O(1) GeV dark matter via nuclear recoil scattering•Searching for CENNS from low-energy (e.g. reactor) neutrinos
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Peter Sorensen, LBL 19 Feb 2016
Summary
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• Lindhard model has plenty of uncertainties, but epsilon<0.01 is not particularly pathological
• Calculations of k vary immensely, may be best to treat it as a free parameter constrained by higher energy data
• Kinematic cutoff is a generic prediction of Lindhard model
• Experimental data in Ge and Xe do not appear to support this prediction… more data are essential
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Peter Sorensen, LBL 19 Feb 2016
Extra slides follow
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Peter Sorensen, LBL 19 Feb 2016
The model works pretty well!
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k=0.166
k=0.110
nuclear recoil energy [keV]
fract
ion
ofen
ergy
give
nto
elec
tron
sf n
1 10 1000.1
0.2
0.3
0.40.50.60.70.80.9
Xe
PS, C.E. Dahl, Phys. Rev. D 83 063501 (2011)
Lindha
rd
predic
tion
(k=0.157)
Barbeau et al.
Benoit et al. Nucl. Instr. Meth 577 558 (2007)
NB: new measurements from LUX extend down to ~1 keV.
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Peter Sorensen, LBL 19 Feb 2016
Is there a kinematic cutoff?
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quoting from 1005.0838
Vcutoff
the formulae, applied to nucleus-electron scattering, result in calculated cutoff recoil energies of ~39 keV in Xe and ~0.1 keV in Ge. This is not the right thing to do.
NB: as ER→0, atoms are basically standing still, but electrons have v ~ α
[10] Phys. Rev. D 36 311 (1987)
right idea, wrong physical picture: atomic scattering is not two-body kinematics