# Presentation: 831229

## Scaling and Formulary of Cross Sections for Ion-atom Impact

***Igor D. Kaganovich***
***Plasma Physics Laboratory ***
***Princeton University***

## In Collaboration with

**Theory ****(PPPL)****:**** **
**E. A. Startsev, R. C. Davidson, **
**S. R. Kecskemeti, A. Bin-Nun, T. Bender**
**Experiment**
**D. Mueller and L. Grisham ****(PPPL)**** **
**R. L. Watson, V. Horvat, K. E. Zaharakis, and Y. Peng ****(Cyclotron Institute, TX A&M University)**
**Atomic electron functions**
**T. Koga ****(Muroran Institute of Technology, Japan)**

## Ion-atom Stripping Cross Sections

*e*

**Ion – atom collisions occur**
**Injector**
**Accelerator**
**Fusion chamber**

## Outline

**Motivation**
**Why do we still need to study cross sections?**
**What do we have to do?**
**Experimental Results**
**	****	****Multi Electron Stripping **
**	****	****Extensive data base for theory validation**
**Theory**
**Theoretical approaches** **to cross section evaluation**
**New fit formula**

## Motivation: Why study cross sections

**Heavy Ion Fusion needs data for ion-atom collisions **
**low charged state  **
**Speed up to 0.2c**
**Various pairs of ion and atoms **
**No experimental data until driver is build**
**Need to rely on theory**

## Motivation: What do we have to do? 

**Collect as much experimental data for ion-atom stripping cross sections**
**Validate theory**
**Build atomic codes**

## Theoretical Approaches to Cross Sections Evaluation

**Classical description of electron motion**
**Quantum mechanical**** **
**Born approximation**
** **

**N. Bohr**
- K. Dan. Vidensk. Selsk. Mat.- Fys. Medd., (1948)
**H. Bethe**
- Ann. Phys. (Leipz.) (1930)

**Solving for 3D Schr****ö****dinger equation for electron wave-functions  in many electron system is impossible => approximations **

## Detailed Comparison Between Classical and Quantum Mechanical Calculations 

- where P(q) is the probability of electron stripping from the projectile when the electron acquires the momentum q, and d/dq is the differential cross section for scattering with momentum q.

- See, I. Kaganovich, et al., Phys. Rev. A **68**, 022707 (2003)

## Born Approximation

**Classical Mechanics**

- Classical mechanics prescribes the electron velocity distribution function (EVDF) for hydrogen-like orbitals as a microcanonical ensemble

- The Probability of Electron Stripping From the Projectile

## Classical Calculation Works Well If Projectile Velocity is Comparable to the  Electron Orbital Velocity

## Classical Calculation Works Well If Projectile Velocity is Comparable to the  Electron Orbital Velocity

## Detailed Comparison Between Classical and Quantum Mechanical Calculations of Electron Striping by Atomic Nitrogen

| , 
10-16cm-2 | Quantum | Classical |
| --- | --- | --- |
| H-, 0.75eV | 0.1 | 1.34 |
| I-, 3eV | 0.08 | 0.47 |
| Cs+, 25eV | 0.045 | 0.1 |

- V=32a.u.

- Classical Calculation Does not Works If Projectile Velocity is NOT Comparable to the  Electron Orbital Velocity

## Historical Overview of Fit Formulas for Ionization by Fully Stripped Ions

**Thompson 1908**

- Classical mechanics under assumption V>>vnl=0

**Gerjuoy after 1968**

- Classical mechanics under assumption V>>vnl account for finite vnl

## Historical Overview of Fit Formulas for Ionization by Fully Stripped Ions

**Bethe 1930**

- Quantum mechanics in Born approximation under assumption
- V>>vnl ,2Zpv0

## Developing Simple Fit Formulas making use of limiting cases and expanding beyond their limits

- G is a function of only v/vnl

**Gerjuoy 1966 **
**Ruztherford scattering averaging over all angles leading to ionization**
**Gryzinskli 1965**
**Similar but used artificial f(v****nl****) to match Bethe formula at large v**
**Bates and Griffing ****1952** **used Born Approximation **** **

## Ionization Cross Sections of Hydrogen by Fully Stripped Ions Experiment vs Theory 

- Ionization Cross Sections of Hydrogen by Fully Stripped Ions Experiment vs Theory

**Classical mechanical calculation               Quantum-mechanical calculation **
**GGV - the due to Gerjuoy                                     Bethe – Bethe formula        Gryz - the Gryzinski fit                            BA- in the Born approximation. **
**. **

**All values are in atomic units: the ionization potential *****I******H***** =1/2****, *****v*****0****=2.2 10****8*****cm*****/*****s*****, *****a*****0**** =0.529 10****8*****cm*****.**

## Failure of the Simple Formulas

**Did not account for the electron circulation around nucleus** **! **

**Collision time ~ circulation time **

## Advanced fits

**Gillespie fit 1983 **

**Rost and Pattard fit 1996 **

**PPPL 2004**

## We Developed New Fit 

**    ****Ionization cross sections of *****hydrogen***** by fully stripped ions showing the scaled experimental data and the theoretical fits**

- BA - the Born approximation
- Gillespie - Gillespie’s fit
- "New" denotes the new fit

**Velocity is scaled on**
**Cross section - on**

- See, I. Kaganovich, et al., _[arxiv](http://arxiv.org/abs/physics/0407140)__[. 0407140](http://arxiv.org/abs/physics/0407140)_, Phys. Plasmas **11** 1229 (2004).

## Ionization Cross Sections of Helium by Fully Stripped Ions Experimental Data

**Exp – ****experimental data**
**CDW-EIS - ****theoretical calculations based on modified Born approximation**

## Applying New Fit to He Cross Sections

**Exp – ****Scaled experimental data**
**CDW-EIS – ****Scaled theoretical calculations**
**New- ****New fit**

## Applying New Fit to Li Cross Sections

- Ionization cross sections of lithium by *H*+ and He+2 . The symbols show experimental data. Shown in the figures are: (a) the raw data and the Bethe formula; (b) the scaled data together with the fit function.

## Applying New Fit to Zp studies for ionization Cross Sections

- Scaled ionization cross sections (*σ/ σ**Bohr*) of helium by fast fully striped ions as a function of charge for ion energies (a) 2.31 MeV/amu, experimental data from Ref. [72] and (b) 6 Mev experimental data from Ref. [32]. Shown in the figures are the raw data, the calculation, see review paper for references.

## Adiabatic scaling v<<vnl

## Olson’s scaling for total electron loss

- Electron capture and total electron loss cross sections of hydrogen by protons
- showing both experimental data presented in Ref. [42] and theoretical fits. Classical trajectory
- Monte Carlo (CTMC) calculations are shown together with fit formula for CTMC
- calculations Eq.(28) and quantum mechanical calculation of Ref. [79].

## Conclusions

**We have recently investigated theoretically and experimentally the stripping of more than 18 different pairs of projectile and target particles in the range of 3-38 MeV/amu to study the range of validity of both the Born approximation and the classical trajectory calculation. **
**The new scaling in for the ionization and stripping cross sections of atoms and ions by fully stripped projectiles has been proposed.**
**We have developed hybrid method, which combine both approximations and helps to identify which method is more trustworthy and produces more reliable results than either of approximations separately.**

## Experiments on TX A&M  Cyclotron

**Ion beams 3.4-38 MeV/amu **
**Limited range of Z/M**

## Clear Evidence of Multiple Electron Stripping

**Charge distributions Kr****+7**** and Xe****+11 **
**3.4MeV/amu**
**N****2**** 4mTorr gas**
**Peak ~ gas pressure =>**
**Single event of multiple electron loss**
**not many collisions**

## Multiple Electron Stripping in Ions with Many Electrons

**Multiple electron stripping occurs if **
**number of electron   (N****nl****) **
**one-electron cross section **nl** **
**                                  **
**                                 ****N**nl** **nl>a2

## Multiple Electron Losses Cross Sections 3.4MeV/amu Kr+7 in N2 

**     ****Kr****+7**** has 29 electrons**
**      ****Average number of electron lost per collision 1.9 **

## Multiple Electron Losses Cross Sections  3.4 MeV/amu Xe+11 in N2 

**    ****Xe****+11 **** has 43 electrons**
**      ****Average number of electron lost per collision 2.0 **

## Extensive Experimental Data Base for Theory Validation

*e*

- Energy 3-38 MeV/amu
- Kr 7+ and Xe +11                         in N2;
- Ar +6
- Ar +8                                           all in He, N2, Ar and Xe.
- He+
- N +6

- See, D. Mueller, et al., Laser and Particle Beams **20**, 551 (2002); Physics of Plasmas, **8**, 1753 (2001).
-  Similar Olson et al, GSI (2003)

## Electron Loss Cross Sections for 10.2 MeV/amu Ar+6 in Various Gases

**   ****Average number of electron lost per collision:**
**He 1.45 **
**N  1.57  **
**Ar  1.77**
**Xe  1.96**

- Xe

- Ar

- N

- He

## Theoretical Approaches to Cross Sections Evaluation

**Classical description of electron motion**
**Quantum mechanical**** **
**Born approximation**
** **

**N. Bohr**
- K. Dan. Vidensk. Selsk. Mat.- Fys. Medd., (1948)
**H. Bethe**
- Ann. Phys. (Leipz.) (1930)

**Solving for 3D Schr****ö****dinger equation for electron wave-functions  in many electron system is impossible => approximations **

## Comparison of Experiment with Calculations 

**Table.  The total electron-loss weighted cross-sections of He****+****  on 30MeV/amu compared with the calculated cross sections in units of 10****–22**** m****2****.**

| Target  | Experiment  | Theory
Born  | Theory
Classic  |
| --- | --- | --- | --- |
| He | 0.4±0.1 | 0.30 | 0.69 |
| N | 1.9±0.1 | 2.4 | 4.1 |
| Ar | 7.3±0.4 | 9.0 | 11.5 |
| Xe  | 23.±1. | 47 | 36 |

## Comparison of Experiment with Calculations 

**Table.  The total electron-loss weighted cross-sections of N****+6****  on 38MeV/amu compared with the calculated cross sections in units of 10****–22**** m****2****.**

| Target  | Experiment  | Theory
Born  | Theory
Classic  |
| --- | --- | --- | --- |
| He | 0.06±0.01 | 0.044 | 0.046 |
| N | 0.34±0.04 | 0.34 | 0.36 |
| Ar | 1.64±0.03 | 1.58 | 1.58 |
| Xe  | 6.29±0.04 | 10.30 | 6.50 |

## Hybrid Method of Cross Sections Calculations

**Classical mechanics**
**Born approximation of quantum mechanics**

- Schematic of
- atom potential

- See, I. Kaganovich, et al., Nuclear Instruments and Methods in Physics Research, **544**, 91 (2005); ** **Physics of Plasmas **11** 1229 (2004).** ****  **

## Results of Hybrid Method: He+

**Table.  The total electron-loss weighted cross-sections of He****+****  on 30MeV/amu compared with the calculated cross sections in units of 10****–22**** m****2****.**

| Target  | Experiment  | Theory
Born  | Theory
Classic | Theory
Hybrid |
| --- | --- | --- | --- | --- |
| He | 0.4±0.1 | 0.30 | 0.69 | 0.30 |
| Ar | 7.3±0.4 | 9.0 | 11.5 | 9.1 |
| Xe  | 23.±1. | 47 | 36
 | 33 |

**Indicates the valid approximation **

## Results of Hybrid Method: N+6 

| Target  | Experiment  | Theory
Born  | Theory
Classic 
 | Theory
Hybrid |
| --- | --- | --- | --- | --- |
| He | 0.06±0.01 | 0.044 | 0.046 | 0.44 |
| Ar | 1.64±0.03 | 1.58 | 1.58 | 1.68 |
| Xe  | 6.29±0.04 | 10.30 | 6.50
 | 6.9 |

**Table.  The total electron-loss weighted cross-sections of N****+6****  on 38MeV/amu compared with the calculated cross sections in units of 10****–22**** m****2****.**

**Indicates the valid approximation **

## Detailed Comparison Between Classical and Quantum Mechanical Calculations 

- where P(q) is the probability of electron stripping from the projectile when the electron acquires the momentum q, and d/dq is the differential cross section for scattering with momentum q.

- See, I. Kaganovich, et al., Phys. Rev. A **68**, 022707 (2003)

## Born Approximation

**Classical Mechanics**

- Classical mechanics prescribes the electron velocity distribution function (EVDF) for hydrogen-like orbitals as a microcanonical ensemble

- The Probability of Electron Stripping From the Projectile

## Born Approximation

**Classical Mechanics**

- The Differential Cross Section for Scattering with Momentum q

- For Coulomb Potential Zqu=Zcl!!!!

## The Differential Cross Section for Screened Coulomb Potential e-r/r

- The Differential Cross Section for Screened Coulomb Potential e-r/r

- Plot of the differential cross section for shielded Coulomb potential for v=32a.u.

- Quantum tends to the Rutherford cross section for q>>1

- Classical tends to the Rutherford cross section <<1, q>>2/v

- 

- q>1 uncertainty principal

## The Differential Cross Section for Scattering with Momentum q

- Shown in the figure is a comparison of the ionization probabilities and the effective charges  in quantum and classical mechanics for  3.2GeV I- ions colliding with a nitrogen atom. Ionization of only the outer electron shell (3eV) is considered.

## Detailed Comparison Between Classical and Quantum Mechanical Calculations of Electron Striping by Atomic Nitrogen

| , 
10-16cm-2 | Quantum | Classical |
| --- | --- | --- |
| H-, 0.75eV | 0.1 | 1.34 |
| I-, 3eV | 0.08 | 0.47 |
| Cs+, 25eV | 0.045 | 0.1 |

- V=32a.u.

- Note, for N+7 ******qu****/ ********cl****=0.57ln(v/v****nl****) +1.26**