1. Dynamics of migrating ions in large LAr detectors.
WA105 meeting.
21-September-2015
Luciano Romero

2. Ion cloud
Electrons and positive ions produced within the LAr bulk are sweeped by the drift field.
The electron-ion pairs are produced within the LAr by the background.
39Ar and cosmic muons.
The ions stay a long time because of its low speed (2 1KV/cm) creating a positive charged cloud in the active volume.
Added to this cloud, the ions can be injected in the LAr by the charge detection devices placed at the anode.
This ion current is linearly dependent of the number of primary electrons reaching the anode.
Lets us define G be the gain factor defined as the number of ions injected in LAr per electron extracted in the anode.
G is proportional to the amplification of the charge detection device: G=α.A (α<1)

3. Ion current injected into the LAr liquid
The ions have a lower potential energy within the liquid than within the gas because of the displacement charge that effectively spreads the charge of the ion.
The field energy of a uniformly charged sphere of radius 𝑎 is: 𝑉= 𝑞 2 4𝜋𝜖𝑎
For liquid argon: ∆𝑉≈3𝑒𝑉; 𝑎≈1Å; 𝜖=1.5 𝜖 0
When the ion approaches the liquid surface from the gas, it is subjected to a 1/x potential due to a mirror ion.
It has been argued that the infinite barrier 1/x can preclude the ions from reaching the liquid phase.

4. Liquid vapor potential barrier
For distances of the order of the atomic radius (≈1Å) the mirror assumption is no longer valid.
The spread of charge due to displacement increases monotonically as the ion plunges into the liquid, so the potential should diminish accordingly.
All the ions travel from gas to liquid across the surface.

5. Effects of the ion cloud
Modification of the drift electric field.
With no ion cloud, the field is axial, uniform and equal to the cathode voltage divided by the detector length.
With the ion cloud, the field is still axial but not uniform.
The field increases in going from the anode to the cathode.
For a given minimal field (In the anode), the ion cloud causes an increase of the cathode voltage.
Recombination.
The ions of the cloud recombine with the primary drifting electrons causing loss of signals.
Signals produced near the cathode are the most affected ones by recombination.
Their electrons have to go across the entire ion cloud to reach the anode.

6. Recombination Recombination is a nasty problem.
It mimics electronegative impurities that capture electrons.
An ion placed in an external drift field produces an axial total electric field.
The Z axis is along the external field
The ion is at the origin.
The total field is tangent to the electron trajectories.
There is recombination when an electron trajectory meets the ion.
The recombination cross section 𝑆 𝑐𝑠 is defined as the surface holding all the recombinating trajectories.
And perpendicular to them.

7. Total electric field Ion at 0,0 immersed in an external field
SCS≈0.12 µ2
The tube defined by the dashed line includes all the recombinating trajectories.
The recombination cross section is the normal section of the tube.

8. Recombination rate The number of field lines reaching the ion is: 𝑞 𝜖
It should be equal to the number of field lines traversing the cross section:
𝐸 𝑑 𝑆𝐶𝑆= 𝑞 𝜖
The recombination rate per unit volume and unit time is:
𝑟= 𝜌 𝑖 𝑗 𝑒 𝑆𝐶𝑆= 𝜌 𝑖 𝑗 𝑒 𝑞 𝜖 𝐸 𝑑 = 𝑗 𝑖 𝑗 𝑒 𝑣 𝑖 . 𝑞 𝜖 𝐸 𝑑 = 𝑗 𝑖 𝑗 𝑒 . 𝑞 𝜇 𝑖 𝜖 𝐸 𝑑 2
ji and je are particle currents of ions and electrons.
𝑣 𝑖 = 𝜇 𝑖 . 𝐸 𝑑
vi y µi are the ion velocity and mobility.

9. Transport equations Assuming steady state and particle conservation:
0=ℎ−𝑟− 𝑑 𝑗 𝑖 𝑑𝑙
0=ℎ−𝑟+ 𝑑 𝑗 𝑒 𝑑𝑙
h is the pair production rate per unit volume and unit time.
The drift field is related to the charge density.
−𝑞 𝜌 𝑒 +𝑞 𝜌 𝑖 =𝜖 𝑑 𝐸 𝑑 𝑑𝑙
Electron density is negligible
Doing the substitution: 𝐹= 𝜖. 𝜇 𝑖 𝑞 𝐸 𝑑 2 and replacing r.
We get 3 coupled equations:
𝑑 𝑗 𝑖 𝑑𝑙 + 𝑗 𝑖 𝑗 𝑒 𝐹 =ℎ;
𝑑 𝑗 𝑒 𝑑𝑙 − 𝑗 𝑖 𝑗 𝑒 𝐹 =−ℎ;
𝑗 𝑖 = 1 2 𝑑𝐹 𝑑𝑙
Those equations can be solved numerically.

10. Approximated solution
We disregard the recombination in the ion equation.
Valid approximation for large values of G.
Or for small values of current.
We can determine analytically the ion cloud.
The drift field
And the recombination.

11. Working parameters Drift length: 6 meters Free parameters:
Like WA105
Free parameters:
Gain:
From 0 to 100.
Minimum drift field at the anode:
From 0.2 to 5 KV/cm
Pair production rate. Two scenarios:
Underground laboratory.
No cosmic muons.
The background ionization is due to 39Ar.
Surface laboratory.
The background ionization is due mainly to cosmic muons.
39Ar negligible.

12. Pair production rate underground
Activity of 39Ar: 1Bq/Kg
39Ar is a beta emitter with a Q value of 565 KeV
The average energy of the emitted electron is approximately one third of Q.
The energetic efficiency for pair production of electrons in LAr is 30 pairs/KeV.
One desintegration produce 5650 pairs.
ℎ≈ 𝑝𝑎𝑖𝑟𝑠 𝑚 3 .𝑠

13. Pair production rate in the surface
170 muons/(m2.s) at sea level.
The muons are minimally ionizing particles.
The specific energy loss of muons in LAr is:
𝑑𝐸 𝑑𝑙 =1.5 𝑀𝑒𝑉. 𝑐𝑚 2 𝑔
Multiplying by density: 𝑑𝐸 𝑑𝑙 =210 𝑀𝑒𝑉 𝑚
Let us assume an energetic efficiency for pair production of muons in LAr of 30 pairs/KeV.
ℎ≈ 𝑝𝑎𝑖𝑟𝑠 𝑚 3 .𝑠

14. Probability of recombination Underground lab
Signal produced near the cathode

15. Probability of recombination Surface lab
Signal produced near the cathode

16. Total voltage Surface lab

17. Conclusions For a 6m underground detector For a 6m surface detector.
Can be operated with fields of 1 KV/cm
Losses below 10% for G≤20.
Voltages in the order of 0.5 MV
For a 6m surface detector.
Need to be operated with fields over 5KV/cm
Losses below 20% for G≤20
Voltages over 4 MV.
Need for a very high voltage power supply.