Channeling in direct dark matter detection III: channeling fraction in CsI crystals
aa r X i v : . [ a s t r o - ph . C O ] N ov Preprint typeset in JHEP style - HYPER VERSION
Channeling in direct dark matter detection III:channeling fraction in CsI crystals
Nassim Bozorgnia
Department of Physics and Astronomy, UCLA, 475 Portola Plaza, Los Angeles, CA90095, USAE-mail: [email protected]
Graciela B. Gelmini
Department of Physics and Astronomy, UCLA, 475 Portola Plaza, Los Angeles, CA90095, USAEmail: [email protected]
Paolo Gondolo
Department of Physics and Astronomy, University of Utah, 115 South 1400 East [email protected]
Abstract:
The channeling of the ion recoiling after a collision with a WIMP changes theionization signal in direct detection experiments, producing a larger signal scintillation orionization than otherwise expected. We give estimates of the fraction of channeled recoilingions in CsI crystals using analytic models produced since the 1960’s and 70’s to describechanneling and blocking effects. ontents
1. Introduction 12. Channeling fractions 6Appendices 11A. Crystal structure and other data for CsI 11B. Temperature dependence of lattice constant 12
1. Introduction
Understanding the possibility of having channeled ion recoils in crystals is important in directdark matter experiments measuring ionization or scintillation signals [1]-[3]. These experi-ments search for Weakly Interacting Massive Particles (WIMPs) composing the dark matterhalo of our galaxy through the energy they deposit in collisions with nuclei within a crystal.Channeling occurs when the nucleus that recoils after being hit by a dark matter particlemoves off in a direction close to a symmetry axis or symmetry plane of the crystal. Chan-neled ions suffer a series of small-angle scatterings with lattice nuclei that maintain them inthe open “channels” between the rows or planes of lattice atoms. Thus they penetrate muchfurther into the crystal than in other directions and give 100% of their energy to electrons(their quenching factor is Q = 1), producing more scintillation and ionization than they wouldproduce otherwise. In scintillators like NaI (Tl) or CsI (Tl), channeling increases the observedscintillation light output corresponding to a particular recoil energy. In dark matter searches,CsI (Tl) crystals are used by the KIMS collaboration [4]. In this paper we give upper boundsto the geometric channeling fraction of recoiling ions in CsI crystals.We proceed here in a similar manner as we did for NaI, a very similar crystal, in a previouspaper [5] (we have also already considered channeling of recoiling ions in Si and Ge crystals ina different paper [6]). We use a continuum classical analytic model of channeling developedin the 1960’s and 70’s, in particular by Lindhard [7]-[16]. In this model the discrete series ofbinary collisions of the propagating ion with atoms is replaced by a continuous interactionbetween the ion and uniformly charged strings or planes. The screened atomic Thomas-Fermipotential is averaged over a direction parallel to a row or a plane. This averaged potential, U or U p , is considered to be uniformly smeared along the row or plane of atoms, respectively,which is a good approximation if the propagating ion interacts with many lattice atoms in– 1 –he row or plane by a correlated series of many consecutive glancing collisions with latticeatoms. We are going to consider just one row or one plane, which simplifies the calculationsand is correct except at the lowest energies we consider.In order for the scattering to happen at small enough angles so that channeling is main-tained, the propagating ion must not approach a string or plane closer than a critical distance r c or x c respectively ( r is the transverse distance to the string and x is the distance perpen-dicular to the plane). The critical distance depends on the energy E of the ion and on thetemperature of the crystal. Ions which start their motion close to the center of a channel (forexample ions incident upon the crystal from outside), far from a row or plane, are channeledif the angle their trajectory makes with the row or plane is smaller than a critical angle, ψ c that depends on the critical distance of approach r c or x c , and are not channeled otherwise.Nuclei ejected from their lattice site by WIMP collisions are initially part of a row or plane,so they start their motion from lattice sites or very close to them. This means that “blockingeffects”, namely large-angle interactions with the nuclei in the lattice sites directly in frontof the recoiling nucleus site, are important. In fact, as argued originally by Lindhard [7], ina perfect lattice and in the absence of energy-loss processes, the probability that a particlestarting from a lattice site is channeled would be zero. This is what Lindhard called the “Ruleof Reversibility.” However, any departure of the actual lattice from a perfect lattice due tovibrations of the atoms, which are always present, violate the conditions of this argumentand allow for some of the recoiling lattice nuclei to be channeled.There are several good analytic approximations of the screened Thomas-Fermi poten-tial and each leads to a different expression for the transverse continuum string and planepotentials, U ( r ) and U p ( x ) respectively. As in Ref [5] here we use Lindhard’s expression,because it is the simplest and allows to find analytical expressions for the quantities we need.The transverse averaged continuum potential of a string as a function of r , relevant for axialchanneling, was approximated by Lindhard [7] as U ( r ) = Eψ
12 ln (cid:18) C a r + 1 (cid:19) , (1.1)where C is a constant found experimentally to be C ≃ √ ψ = 2 Z Z e / ( Ed ).Here Z and Z are the atomic numbers of the recoiling and lattice nuclei respectively, d is the spacing between atoms in the row, a is the Thomas-Fermi screening distance, a =0 . Z / + Z / ) − / [17, 8] and E = M v / E is the recoil energy imparted to the ion after a collision with a WIMP, E = | ~ q | M , (1.2)where ~ q is the recoil momentum. The string of crystal atoms is at r = 0. The transverseaveraged continuum potential of a plane of atoms, relevant for planar channeling, given by– 2 –indhard [7] as a function of x is U p ( x ) = Eψ a "(cid:18) x a + C (cid:19) − xa , (1.3)where ψ a = 2 πnZ Z e a/E , n = N d pch is the average number of atoms per unit area, N isthe atomic density and d pch is the width of the planar channel. Also, the axial channel width d ach is defined in terms of the interatomic distance d as d ach = 1 / √ N d , with N the atomicdensity. The plane is at x = 0. Examples of axial and planar continuum potentials for Csions propagating in the < > axial and { } planar channels of a CsI crystal are shownin Fig. 1. The potentials for Cs and I ions are practically identical, because Z Cs ≃ Z I (seeAppendix A). H nm L U H k e V L
100 Channel, Cs ions a Cs PlanarAxial
Figure 1:
Continuum axial (black) and planar (green/gray) potentials for Cs ions, propagating inthe < > axial and { } planar channels of a CsI crystal. The screening radius shown as a verticalline is ¯ a Cs = 0 . For a “static lattice,” that here means a perfect lattice in which all vibrations are ne-glected, the critical distances of approach r c and x c are given in the Eqs. 1.4 and 1.5, expres-sions that were derived in Ref [5] . The critical distance for axial channeling is r c ( E ) = Ca s (cid:20) √ z cos (cid:18)
13 arccos (1 − z/ z ) / (cid:19) − (cid:21) , (1.4)where z = 9 E d / (8 EC a ). For planar channeling we will follow the procedure of defining a“fictitious string” introduced by Morgan and Van Vliet [10, 16]. They reduced the problem ofscattering from a plane of atoms to the scattering of an equivalent row of atoms contained in– 3 – strip of a certain width centered on the projection of the ion path onto the plane of atoms.Thus, x c ( E ) ≡ ¯ r c ( E ) , (1.5)where ¯ r c ( E ) is the critical distance obtained from Eq. 1.4 for the fictitious string. Along thefictitious row, the characteristic distance ¯ d between atoms needs to be estimated using dataor simulations which are not available for a CsI crystal. As explained in Ref. [5], the choiceof ¯ d equal to the average interdistance of atoms in the plane d p , i.e. ¯ d = d p , yields a lowerbound on x c , which translates into an upper bound on the fraction of channeled recoils intoplanar channels.So far we have been considering static strings and planes, but the atoms in a crystalare actually vibrating with a characteristic (one dimensional rms) amplitude of vibration u ( T ) which increases with the temperature T . In principle there are modifications to thecontinuum potentials due to thermal effects, but we take into account thermal effects in thecrystal through a modification of the critical distances found originally by Morgan and VanVliet [10] and later by Hobler [16] to provide good agreement with simulations and data. Foraxial channels it consists of taking the temperature corrected critical distance r c ( T ) to be, r c ( T ) = p r c ( E ) + [ c u ( T )] , (1.6)where the dimensionless factor c in different references is a number between 1 and 2 (seee.g. Eq. 2.32 of Ref. [11] and Eq. 4.13 of the 1971 Ref. [10]). For planar channels, followingHobler [16] we use a similar equation x c ( T ) = p x c ( E ) + [ c u ( T )] , (1.7)where again c is a number between 1 and 2 (for example Barret [17] finds c = 1 . c = 2). We will mostly use c = c = 1 in the following, to tryto produce upper bounds on the channeling fractions.In Appendix B it is shown that the variation of the lattice size with temperature, char-acterized by the variation of the lattice constant a lat with temperature, has a negligible effecton the channeling fractions. This is why we ignore this effect (not only in this paper but alsoin our previous papers [5, 6]).We use the Debye model to account for the vibrations of the atoms in a crystal. The onedimensional rms vibration amplitude u of the atoms in a crystal in this model is [8, 15] u ( T ) = 12 . (cid:20)(cid:18) Φ(Θ /T )Θ /T + 14 (cid:19) ( M Θ) − (cid:21) / , (1.8)where M for a compound is the average atomic mass (in amu), i.e. for CsI, M = ( M Cs + M I ) / T are the Debye temperature and the temperature of the crystal (in K), respectively,and Φ( x ) = x R x tdt/ ( e t −
1) is the Debye function. With M Cs = 132 . M I = 126 . M = 129 . u in CsI is plotted in Fig. 2 as a function ofthe temperature T . The crystals in the KIMS experiment were kept at 0 ◦ C in 2007 [19].Currently the operating temperature of the crystals is 20 ◦ C [20]. The vibration amplitudeis u = 0 . ◦ C, and u = 0 . ◦ C. H K L H n m L a Cs > a I u Figure 2:
Plot of u ( T ) for CsI (Eq. 1.8 with M = ( M Cs + M I ) / Using the temperature corrected critical distances of approach r c ( T ) and x c ( T ) (Eqs. 1.6and 1.7) or the static lattice critical distances r c and x c (Eqs. 1.4 and 1.5), we obtain the cor-responding critical axial and planar channeling angles ψ c (see Ref. [5] for details). Examplesare shown in Figs. 3 to 5, for c = c = c and c = 1 or c = 2 as indicated.Fig. 3 clearly shows how the critical distances and angles change with temperature fora Cs or I ion propagating in the < > axial and { } planar channels of a CsI crystal,with temperature effects computed with c = c = c = 1. At small energies the static criticaldistance of approach is much larger than the vibration amplitude, so temperature correctionsare not important. As the energy increases, the static critical distance of approach decreases,and when it becomes negligible with respect to the vibration amplitude u , the temperaturecorrected critical distance r c becomes equal to c u . In this case, since u ( T ) increases with T , the critical distance r c ≃ c u becomes larger with T , and therefore the critical channelingangle becomes smaller. Notice that for the 100 channels, the widths of axial and planarchannels are the same, d ach = d pch and r c = x c .Fig. 4 shows the same effects for the < > axial channel. In this channel, the criticaldistance (the minimum distance to a string to maintain channeling) becomes larger than theradius of the channel at energies below a few keV, shown in the figures. This means thatnowhere in the channel an ion can be far enough from the string of lattice atoms for channelingto take place (thus the critical channeling angle is zero). The exact calculation of the energy– 5 – ch (cid:144) r c = x c
600 °C 77 K293 KStatic lattice H keV L H n m L < > and < channels, c = static lattice77.2 K293 K600 °C AxialPlanar H keV L Ψ c H d e g L < > and < channels, c = Figure 3:
Static (green) and temperature corrected with c = c = c = 1 (black) (a) critical distancesof approach (and u ( T ) in red) and (b) the corresponding critical channeling angles, as a function ofthe energy of propagating Cs or I ions (they are practically the same for both) in the < > axial(black) and { } planar (green) channels. Here d ach = d pch . d ach (cid:144)
600 °C 293 K 77 KStatic lattice H keV L H n m L < > axial channel, c = static lattice77.2 K293 K600 °C H keV L Ψ c H d e g L < > axial channel, c = Figure 4:
Same as Fig. 3 but for the < > axial channel. at which this happens would require considering the effect of more than a single row of atoms(which we do not do here) thus our results at these low energies are only approximate. Noticethat for the 111 channels, the < > axial and { } planar channels do not have the samewidths, d ach = d pch , and we only show the critical distance and angles for the axial channel.Figs. 5(a) and 5(b) show the static and T -corrected critical distances and angles repec-tively at several temperatures for traveling Cs or I ions in the 100 axial and planar channelswith c = c = c = 2.
2. Channeling fractions
In our model, a recoiling ion is channeled if the collision with a WIMP happened at a distance– 6 – ch (cid:144)
22 u
600 °C 293 K 77 KStatic lattice H keV L H n m L < > and < channels, c = AxialPlanarstatic lattice77.2 K293 K600 °C H keV L Ψ c H d e g L < > and < channels, c = Figure 5:
Same as Fig. 3 but with c = c = c = 2. large enough from the string or plane from which the ion was expelled. Namely, channelinghappens if the initial position of the recoiling motion is r i > r i, min or x i > x i, min for an axialor planar channel respectively. Here r i, min and x i, min depend on the critical distance, r c or x c (and through it on the energy E and the temperature T ) and on the angle φ the initial ion’smomentum makes with the string or plane. In Ref. [5] we obtained the following expressionsfor both distances: r i, min ( E, φ ) = Ca r(cid:16) C a r c (cid:17) exp (cid:0) − φ/ψ (cid:1) − − d tan φ, (2.1)with r c given in Eq. 1.6 and x i, min ( E, φ ) = a C − (cid:20)q x c a + C − x c a − sin φ/ψ a (cid:21) (cid:20)q x c a + C − x c a − sin φ/ψ a (cid:21) − d p tan φ, (2.2)where x c is given in Eq. 1.7.We take the initial distance distribution of the colliding atom to be a Gaussian with aone dimensional dispersion u , and to obtain the probability of channeling for each individualchannel we integrate the Gaussian between the minimum initial distance and infinity (a goodapproximation to the radius of the channel; see Ref [5] for details). The dependence ofthese probabilities on the critical distances enter in the argument of an exponential or an erfcfunction. Thus any uncertainty in our modeling of the critical distances becomes exponentiallyenhanced in the channeling fraction. This is the major difficulty of the analytical approachwe are following.In order to obtain the total geometric channeling fraction we need to sum over all the in-dividual channels we consider. Taking only the channels with lowest crystallographic indices,100, 110 and 111, we have a total of 26 axial and planar channels, as explained in Appendix– 7 – of Ref. [5] (CsI and NaI have the same crystal structure). Here “geometric channelingfraction” refers to assuming that the distribution of recoil directions is isotropic. In reality,in a dark matter direct detection experiment, the distribution of recoil directions is expectedto be peaked in the direction of the average WIMP flow. The integral over initial directionsis computed using HEALPix [21] (see Appendix B of Ref. [5]).Fig. 6 shows upper bounds to the channeling probability computed for each initial recoildirection direction ˆ q and plotted on a sphere using the HEALPix pixelization for (a) a E = 200keV and (b) a 1 MeV Cs ion at 20 ◦ C with c = c = 1 assumed for the temperature effects.The red, pink, dark blue and light blue colors indicate a channeling probability of 1, 0.625,0.25 and zero, respectively. The results are practically identical for an I ion.Fig. 7 shows upper bounds to the channeling fractions of Cs recoils for individual channels,for T = 20 ◦ C and assuming c = c = 1. The black and green (or gray) lines correspond tosingle axial and planar channels respectively. The upper bounds of the channeling fractionsof planar channels are more generous than those of axial channels because of our choice of x c in Eq. 1.5. This does not mean that planar channels are dominant in the actual channelingfractions. Figure 6:
Upper bounds on the channeling probability of a Cs ion (for an I ion the figure would bepractically identical ) as function of the initial recoil direction for a (a) 200 keV and (b) 1 MeV recoilenergy at 20 ◦ C (with c = c = 1). The probability is computed for each direction and plotted ona sphere using the HEALPix pixelization. The red, pink, dark blue and light blue colors indicate achanneling probability of 1, 0.625, 0.25 and zero, respectively. Upper bounds to the geometric channeling fractions of Cs and I ions as function ofthe recoil energy are shown in Figs. 8 and 9 with thermal effects taken into account with c = c = 1 and c = c = 2, respectively.Notice that we have not included here any dechanneling due to the presence of impuritiesin the crystal (such as Tl atoms), which would decrease the channeling fractions presented.We intend to return to this issue in a later paper. As we see in Fig. 8 and 9, when nodechanneling is taken into account, the channeling fraction increases with energy, reaches amaximum at a certain energy and then decreases as the energy increases. This maximum– 8 – H keV L F r ac ti on c = c =
1, T =
293 K
Figure 7:
Upper bounds on the channeling fractions of Cs recoils as a function of the recoil energy E when only one channel is open, for T = 293 K with temperature corrections included in the criticaldistances with the coefficients c = c = 1. Black and green/gray lines correspond to axial and planarchannels respectively. Solid, dashed, and dotted lines are for 100, 110, and 111 channels respectively.The corresponding figure for an I ion would be practically identical.
600 °C 293 K 77.2 K273 K H keV L F r ac ti on c = c = Figure 8:
Upper bounds on the channeling fraction of Cs (solid lines) and I (dashed lines) recoils asa function of the recoil energy E for T = 600 ◦ C (orange/medium gray), 293 K (green/light gray), 273K (black), and 77.2 K (blue/dark gray) in the approximation of c = c = 1 without dechanneling. occurs because the critical distances decrease with the ion energy E , which makes channelingmore probable, while the critical angles also decrease with E , which makes channeling less– 9 –
00 °C 293 K273 K 77.2 K H keV L F r ac ti on c = c = Figure 9:
Same as Fig. 10 but for c = c = 2.
600 °C 293 K 77.2 K273 K H keV L F r ac ti on Static lattice
Figure 10:
Same as Fig. 10 but for c = c = 0 (static lattice), which provides an extreme upperbound (any larger values of c and c , which can reasonably be as large as 2, yield smaller factions). probable. At low E the critical distance effect dominates, and at large E the critical angleeffect dominates.As shown in Fig. 8, the channeling fraction for CsI (Tl) is never larger than 5% at 293 K(with c c c c = c = 2instead. However, since we do not know which are the correct values of the crucial parameters c and c for CsI, we could ask ourselves how the upper bounds on channeling fractions wouldchange if the values of these parameters would be smaller than 1 (please recall that for othermaterials and propagating ions the values of these parameters were found to be between 1and 2). The values of c and c cannot be smaller than zero, thus Fig. 10 shows our mostgenerous upper bounds on the geometric channeling fraction, obtained by setting c = c = 0,namely by neglecting thermal vibrations of the lattice (which make the channeling fractionssmaller as T increases) but including the thermal vibrations of the nucleus that is going torecoil (which make the channeling fraction larger as T increases). Although it is physicallyinconsistent to take only the temperature effects on the initial position but not on the lattice,we do it here because using c = c = 0, namely a static lattice, provides an upper bound onthe channeling probability with respect to that obtained using any other non-zero value of c or c . Even in this case, the channeling fractions at 293 K cannot be larger than 10%.To conclude, let us remark that the analytical approach used here can successfully de-scribe qualitative features of the channeling and blocking effects, but should be complementedby data fitting of parameters and by simulations to obtain a good quantitative descriptiontoo. Thus our results should in the last instance be checked by using some of the manysophisticated Monte Carlo simulation programs implementing the binary collision approachor mixed approaches. Acknowledgments
N.B. and G.G. were supported in part by the US Department of Energy Grant DE-FG03-91ER40662, Task C. P.G. was supported in part by the NFS grant PHY-0756962 at theUniversity of Utah. We would like to thank Prof. Sun-Kee Kim for providing us with impor-tant information about the KIMS experiment.
A. Crystal structure and other data for CsI
CsI is a diatomic compound that has two interpenetrating face-centered cubic (f.c.c.) latticestructures displaced by half of a lattice constant with 8 atoms per unit cell. The latticeconstant of CsI crystal is a lat = 0 . a lat is explained in Appendix B.The atomic mass and atomic number of Cs and I are M Cs = 132 . M I = 126 . Z Cs = 55 and Z I = 53.With respect to the Thomas-Fermi screening distance, for Cs recoils from a mixed rowor plane we use the average¯ a Cs = ( a CsCs + a CsI ) / . , (A.1)– 11 –here a CsCs = 0 . Z / + Z / ) − / = 0 . a CsI = 0 . Z / + Z / ) − / =0 . a CsCs because the row or planefrom which the recoiling Cs ion was emitted contains only Cs atoms. Similarly, for I recoilsfrom a mixed row or plane, we use¯ a I = ( a II + a CsI ) / . , (A.2)where a II = 0 . Z / + Z / ) − / = 0 . a CsI correspond to an I ion scatteringfrom an I and a Cs lattice atom, respectively. For I recoils from a pure row or plane we use a II , since the row or plane the recoiling ion is emitted from is made of I ions only.To compute the interatomic spacing d in axial directions and the interplanar spacing d p in planar directions, we have to multiply the lattice constant by the following coefficients [8]: • Axis: < > : 1 / < > : 1 / √ < > : √ / • Plane: { } : 1 / { } : 1 / √ { } : 1 / √ B. Temperature dependence of lattice constant
In general the lattice constant, a lat is temperature dependent. The change in a lat withtemperature depends on the thermal expansion coefficient, β of a crystal. For CsI (Tl), β = 54 × − ◦ C − . To find the change in the lattice constant at a temperature T and thelattice constant at 20 ◦ C, we have[ a lat ( T ) − a lat (20 ◦ C)] /a lat (20 ◦ C) = β ( T − ◦ C) , (B.1)where a lat (20 ◦ C) = 0 . ◦ C for CsI (Tl). When T changesfrom 20 ◦ C to 600 ◦ C, the change in the lattice constant (using Eq. B.1) is only 3.1%. Thischange in a lat between 20 ◦ C and 600 ◦ C results in a negligible change in the channelingfractions. As an example the Cs channeling fractions with c = c = 1 and c = c = 2 forthe two choices of a lat (20 ◦ C) and a lat (600 ◦ C) are shown in Fig. 11(a). As the two curvesare very similar, we use a lat (20 ◦ C) for all three crystal temperatures in this paper.We can use Eq. B.1 to find the temperature dependent lattice constant for NaI (Tl), Siand Ge crystals. For these three crystals, the coefficient of thermal expansion is β NaI(Tl) =47 . × − ◦ C − , β Si = 2 . × − ◦ C − , and β Ge = 5 . × − ◦ C − . When T changes from20 ◦ C to 600 ◦ C, the change in the lattice constant of NaI (Tl) is 2.75%. In Si and Ge, we cango to higher temperatures, and the maximum temperature that we considered in our previouspaper on Si and Ge [6] was 900 ◦ C. When T changes from 20 ◦ C to 900 ◦ C, the change in thelattice constant of Si and Ge is 0.23% and 0.52% respectively.– 12 – lat H
600 °C L a lat H
20 °C L c = 1c = 2 H keV L F r ac ti on C s CsI Crystal, T =
600 °C a lat H
600 °C L a lat H
20 °C L c = 1c = 2 H keV L F r ac ti on N a NaI Crystal, T =
600 °C
Figure 11:
Channeling fraction of (a) Cs ions propagating in a CsI crystal and (b) Na ions propagatingin an NaI crystal as a function of the recoil energy E for T = 600 ◦ C with a lat (600 ◦ C) (black) and a lat (20 ◦ C) (green/gray) for the two choices of c = c = 1 or 2. c = 1 c = 2 a lat H
20 °C L a lat H
900 °C L ´ - ´ - ´ - H keV L F r ac ti on Si Crystal, T =
900 °C c = 1 a lat H
900 °C L a lat H
20 °C L c = 2 ´ - ´ - ´ - H keV L F r ac ti on Ge Crystal, T =
900 °C
Figure 12:
Channeling fraction of (a) Si ions propagating in a Si crystal and (b) Ge ions propagatingin a Ge crystal as a function of the recoil energy E for T = 900 ◦ C with a lat (900 ◦ C) (black) and a lat (20 ◦ C) (green/gray) for the two choices of c = c = 1 or 2. The Na channeling fractions with c = c = 1 and c = c = 2 for the two choices of a lat (20 ◦ C) and a lat (600 ◦ C) are shown in Fig. 11(b) for an NaI crystal. Fig. 12 shows thechanneling fractions for Si and Ge for the two choices of a lat (20 ◦ C) and a lat (900 ◦ C).Clearly, the change in the curves is negligible. Thus we always used the value of a lat measured at 20 ◦ C in this paper as well as in our previous papers [5, 6].
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