A Note About Beam Depolarization

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1 A Note About Beam Depolarization Jaideep Singh University of Virginia Deember 2, 2008 Basi Mehanism of Beam Depolarization Ionizing radiation inreases the nulear spin relaxation in the target hamber. Also known as beam depolarization, it is essentially a two step proess. First, the beam ionizes an 3 He atom whih results in an free eletron and an atomi ion 3 He +. There is also the possibility that the atomi ion bonds with an neutral 3 He atom to form an moleular ion 3 He + 2. Seond, interations with 3 He ions indue 3 He nulear spin flips. Therefore, the total relaxation rate due to ionization by the beam is given by: [ ] [ ] ionization rate mean number of nulear spin flips Γ beam = per target hamber atom per atomi ion [ ] eletrons atomi ions reated = atoms in t n per unit time per eletron a + n m 2 [ ] I total energy lost = n a + n m 3 e mean energy per ion V t [He] t [ ] de = I ρ dx L t [He] t n a + n m 4 e E i V t [He] t [ ] I de = n a + n m 5 e E i ρ dx = Γ ion n a + n m 6 where I is the eletron beam urrent, E i is the mean energy for ion-eletron pair reation, is the mean ross setional area of the target hamber, Γ ion is the ionization rate per 3 He atom in the target hamber, and n a & n m are the average number of spins lost per atomi ion reated due to interations with atomi & moleular ions respetively. 2 Beam Energy Lost to Ionizing Interations The energy lost to ollisions per unit density per unit length is given by the elebrated Bethe-Bloh formula and, for an eletron beam, it is []: [ ] de ρ dx = 2πre 2 m e 2 Z [log β 2 [γ ] 2 [γ + ] δ + 2 log me 2 I BB F γ 2 C ] s Z 2πre 2 m e 2 = 6.85 ev/amagat/m 8 F γ = [+ 2γ γ ] 2 log2 [ ] 2 8 γ γ 2 & γ = = E beam β 2 m e 2 9 where Z is the target atomi number, β= v/ is the eletron veloity relative to the speed of light, I BB is the mean exitation potential of the target material, δ is the density orretion, and C s is the shell orretion. 7

2 parameter value omments Z 2 atomi number I BB 4.8 ev mean exitation potential C s 0 for shell orretion δ 0 0 Y a Y Y m [ 3 He] amg for density orretion atm & 20 o C Table : Bethe-Bloh Formula Parameters for Eletron-Helium Interations. All values taken from [2]. The shell orretion is signifiant only when the inident eletron veloity is roughly equal to or slower than the bound eletron orbital veloity. For JLab beam energies, this is not the ase; therefore the shell orretion will be negleted C s = 0. The density orretion δ is given by [, 2]: δy = δ 0 exp [2 Y Y 0] [ ] Y Y 0 2Y Y a + [δ 0 2Y 0 Y a ] Y m Y Y Y Y 0 2Y Y a Y <Y 0 <Y Y Y = log βγ & Y a,0, = Y a,0, log [ 3 He]/[ 3 He] 0 where Y a, Y 0, Y, m, and [N] 0 depend on the target material at atm & 20 o C and for 3 He are listed in Tab.. For a 3 He density of 8.3 amg or higher, the equivalent beam energy for Y = Y is 700 MeV or less. Therefore for typial 3 He experiments at JLab, we get: [ ] ] de = 4πre 2 m e [log 2 Ebeam [N] log ρ dx GeV 0 amg 4πr 2 em e 2 = 50 kev barn = 3.70 ev/amagat/m Mean Energy for Helium Ion-Eletron Pair Creation The mean energy per ion-eletron reation has been measured in helium a number of times, see Tab. 2. The early measurements found about 32 ev per pair. As later authors noted on more than one oasion [3, 4, 5, 6], these early measurements were performed on insuffiiently pure helium samples. Later measurements, whih took great are to purify the helium sample, obtained results about 0 ev per pair higher. We need to know the value for pure He beause we are interested in knowing how many He ions are reated. Consequently, we use a weighted average of five modern measurements that went to great lengths to purify their He sample. As a side note, the mean energy per ion-eletron reation E i is entirely different than the mean exitation potential I BB. It is merely a oinidene that they have nearly the same value for He. We are finally in a position to alulate the mean ionization rate per atom: Γ ion = ee i [ ρ de dx ] I = β I 4 where e is the elementary harge, I is the beam urrent, is the mean ross setional area of the target hamber, and β is tabulated in Tab. 3 for various beam energies. The dependene of β on the beam energy is soft; onsequently the mean ionization rate per atom within 5 perent over all JLab energies is: Γ ion = m 2 µa hr I = 2 hrs I 0 µa 2.0 m 2 5 2

3 E i ev year omments ref purified in haroal at liquid air temps, possible double ionization of He? [7] purified in haroal at liquid air temperatures [8] value listed in [9] and [0] [] tank He at 99.95% purity with traes amounts of N 2 and O 2 [2] ited in [4, 3] [4] 32.5 ± He/Ar/CH 4 mixture [5] 29.7 He with 0.3% Ar purified with haroal at liquid air temperatures [3] 26.0 ± was purified, but not pure enough? [6] 42.7 ± purified with haroal at liquid air temperatures [4] tank He with less than 0.02% N 2 [7] 44.2 ± purified with Ca-Mg hips at 470 o C [3] 46.0 ± two sets of He samples with different purifiation methods [5] 42.3 ± purified with haroal at liquid air temperatures [8] 40.3 ± purified with haroal at liquid air temperatures [9] 55,60 ±5% 957 used He-ethylene mix, but applied an impurity orretion [20] 29.9/35.2 theoretial alulation for impure He sample theoretial alulation for pure He [6] 42.7, [2] sensitivity to impurities disussed, but no original soures listed [] E i weighted mean = 43.2 ± 0. ev Table 2: Mean Energy per Ion-e Pair Creation in He Gas. Only measurements performed on arefully purified samples * are used in the alulation of the weighted mean. The different measurement tehniques and their respetive sensitivities to impurities are disussed in the 958 review artile by Valentine and Curran [22]. [ ρ ] de dx β hr µa/m 2 E beam GeV η % Table 3: Variation of Ionizing Energy Loss Parameters with Eletron Beam Energy. The seond olumn is the energy lost to ollisions relative to the value at 2 GeV. The maximum relative ionization ontribution from radiation, η, is estimated assuming a 3 He density of 0 amg and a target hamber length of 40 m. 3

4 4 Spin Relaxation Due to Atomi and Moleular Helium Ions Atomi ions ontribute to polarization loss due to a hyperfine interation between the 3 He nuleus and the unpaired eletron in the atomi ion. Beause harge exhange ours readily, eletrons from highly polarized neutral atoms jump to lowly polarized atomi ions. The newly formed atomi ion partially depolarizes until it undergoes harge exhange and so on. The umulative effet is at most one nulear spin flip [23]. In addition to this proess, moleular ions also lose polarization to the rotational degrees of freedom via a spin-rotation interation [24]. Before estimating the number of spin flips indued by both proesses, it is useful to first estimate the fration of ions of both types and their typial lifetimes. First we write down the rate equations for the frationof atomi ions h a and moleular ions h m in the target hamber, where we have assumed h a,h m : dh a /dt = +Γ ion h a /τ a & τ a dh m /dt =+k m h a [He] 2 t h m/τ m & τ m = k n [N 2 ] t + k m [He] 2 = [N t 2] t k n + k n [He] t where k m, k n, k n,&k n are the rate onstants for moleular formation, atomi ion harge exhange, binary moleular harge exhange, and three body moleular harge exhange, see Tab. 4. The mean atomi and moleular ion lifetimes are τ a and τ m. The equilibrium frations are obtained from setting the rates to zero and give: Γ ion h a = Γ ion τ a = k n [N 2 ] t + k m [He] 2 t h m = k m[he] 2 t τ mh a = Γ ion [N 2 ] t k n + k n [He] t k n[n 2 ] t k m [He] 2 8 t Under our onditions, we find τ a,τ m 00 ps and h a,h m 0 5, whih justifies our previous assumption that there are very few ions. The presene of a foreign gas suh as N 2 greatly limits the lifetime of moleular ions. Whereas moleular ions have the potential to depolarize many nulei, their effet is greatly redued beause they are so short lived. Relaxation due to moleular ions is disussed in [24] and they derive an expression for n m of the following form: γm N Γ ion n m = h h m Q m n m = γm N h m h Γ ion Q m 9 where γ m N/h = 29 MHz is the moleular spin-rotation oupling onstant and Q m is the unitless relative relaxation rate that depends on the magnitude of the magneti field and the density of 3 He. Sine Q m an be at most, the maximum value for n m is given as: n m γmn h [N 2 ] t k n + k n[he] t + k n[n 2 ] t k m [He] t Aording to [23] the mean number of spin flips due to an atomi ion n a when the atomi harge exhange rate τex is muh slower than the hyperfine preessional frequeny A a /h, n a τ ex + τ a 2τ ex + τ a & τ ex = k ex[he] t 2 where k ex is the binary atomi harge exhange rate onstant. Under our onditions, τex 50 GHz is muh faster than A a/h =8.66 GHz. Physially, this means that eletron and nuleus have very little time to interat before the eletion is exhanged to another atom. This has the effet of suppressing the the hanes of a nulear spin flip and therefore redues the overall relaxation rate. Over a 3 He density range of 9 amg to 2 amg and a N 2 to 3 He density ratio range of 0.5% to 2%, the following parameteriation reprodues the more general alulation from [23] to better than 3%: where h and ρ are given by: n a = [ ρ ] h ρ 22 h =[ 3 He] t /0 amg & ρ = 00 [N 2 ] t /[ 3 He] t 23 4

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