Erda Fragment Calculator – Ion Beam Analysis Precision Tool

Erda Fragment Calculator

Professional Kinematic and Energy Analysis for Ion Beam Depth Profiling

Incident Ion Mass (M1) [amu]

Mass of the projectile ion (e.g., Helium-4: 4.0026).

Please enter a positive mass.

Incident Energy (E0) [MeV]

Kinetic energy of the incoming projectile.

Energy must be greater than 0.

Target Recoil Mass (M2) [amu]

Mass of the target atom to be detected (e.g., Hydrogen-1: 1.0078).

Please enter a positive mass.

Recoil Angle (φ) [Degrees]

Angle of detection relative to the incident beam (usually 10° – 40°).

Angle must be between 0 and 90 degrees.

Primary Recoil Energy (Er)
0.960 MeV

Kinematic Factor (k)
0.4801

Mass Ratio (M1/M2)
3.9716

Recoil Velocity Ratio (v/v0)
0.6929

Formula: Er = E0 × k, where k = [4 × M1 × M2 / (M1 + M2)²] × cos²(φ)

Recoil Energy vs. Detection Angle

Visualization of energy distribution across different recoil angles for current masses.

Typical Kinematic Factors for Common ERDA Setups
Projectile (M1)	Target (M2)	Angle (φ)	Kinematic Factor (k)
4He	1H	30°	0.4801
12C	1H	20°	0.2505
35Cl	1H	15°	0.1009
197Au	12C	35°	0.1478

What is an Erda Fragment Calculator?

The erda fragment calculator is a specialized analytical tool used in Ion Beam Analysis (IBA), specifically for Elastic Recoil Detection Analysis (ERDA). This technique is essential for materials scientists and physicists who need to determine the concentration and depth profile of light elements—most notably Hydrogen—within thin films and solid samples.

By using the erda fragment calculator, researchers can predict the kinetic energy of atoms that are “knocked out” of a target sample when struck by a high-energy heavy ion beam. Unlike Rutherford Backscattering Spectrometry (RBS), which looks at the projectile’s bounce, ERDA focuses on the target atom’s forward recoil, making the erda fragment calculator indispensable for quantifying light isotopes that are otherwise difficult to detect.

Common misconceptions include the idea that ERDA can only detect hydrogen. While highly optimized for H-profiling, the erda fragment calculator can be used for any target atom lighter than the incident ion, provided the geometry and energy are correctly configured.

Erda Fragment Calculator Formula and Mathematical Explanation

The physics behind the erda fragment calculator relies on the conservation of momentum and energy in an elastic collision. The primary value calculated is the Kinematic Factor ($k$), which represents the ratio of the energy of the recoiling atom to the energy of the incident ion.

The Core Formula

The energy of the recoiled fragment ($E_r$) is determined by:

Er = E0 × [ (4 × M1 × M2) / (M1 + M2)² ] × cos²(φ)

Variable	Meaning	Unit	Typical Range
M1	Incident Ion Mass	amu	1.0 – 197.0
M2	Target Atom Mass	amu	1.0 – 30.0
E0	Incident Energy	MeV	0.5 – 100.0
φ	Recoil Angle	Degrees	10° – 45°
k	Kinematic Factor	Dimensionless	0.0 – 1.0

Practical Examples (Real-World Use Cases)

Example 1: Helium on Hydrogen (Standard Setup)

In a typical thin-film analysis, a 2.0 MeV Alpha particle (4He, mass ~4.00) strikes a polymer containing Hydrogen (1H, mass ~1.01). If the detector is placed at 30°, the erda fragment calculator reveals a kinematic factor of approximately 0.48. This means the detected protons will have a surface energy of 0.96 MeV.

Example 2: Carbon-12 on Lithium-7

Using a 5.0 MeV Carbon beam (12C) to profile Lithium (7Li) in battery electrodes at a 20° recoil angle. The erda fragment calculator computes a mass ratio of 1.71 and a kinematic factor that allows for clear separation of Li isotopes from the background noise, enabling precise stoichiometry measurements.

How to Use This Erda Fragment Calculator

Enter Projectile Mass: Input the atomic mass of your incident beam (e.g., 4.0026 for Helium).
Set Incident Energy: Input the energy from your particle accelerator in MeV.
Define Target Fragment: Enter the mass of the specific element you are profiling.
Input Geometry: Enter the recoil angle φ (the angle between the beam and the detector).
Analyze Results: The erda fragment calculator will instantly update the Kinematic Factor and the maximum recoil energy.

Key Factors That Affect Erda Fragment Calculator Results

Mass Ratio (M1/M2): The efficiency of energy transfer is highest when M1 and M2 are similar, but for ERDA, M1 must usually be heavier than M2 to allow forward recoil detection.
Recoil Angle Sensitivity: Small changes in the angle φ result in significant changes in $E_r$ due to the $\cos^2$ relationship. Precision in alignment is critical.
Stopping Power (dE/dx): As fragments travel through the material, they lose energy. The erda fragment calculator provides the “surface” energy, but actual detected energy will be lower if the fragment originates from a deeper layer.
Beam Straggling: Multiple small collisions can broaden the energy peaks, affecting the resolution of the erda fragment calculator predictions.
Detector Resolution: The ability of the Silicon Surface Barrier (SSB) detector to distinguish energies determines the smallest depth interval detectable.
Surface Roughness: Variations in the sample surface can distort the path length of recoiling fragments, causing energy shifts.

Frequently Asked Questions (FAQ)

Why is the recoil angle limited to forward angles?

Conservation of momentum dictates that in elastic collisions where M1 > M2, the lighter target atom is always pushed in the forward direction. Most erda fragment calculator configurations use angles between 10° and 40°.

Can I use the erda fragment calculator for heavy elements?

Technically yes, but detection requires M1 to be significantly heavier than M2 for effective separation, which often requires massive accelerators for elements like Gold or Iodine.

How does depth affect the calculation?

The basic erda fragment calculator gives the surface energy. For fragments at depth $x$, you must subtract the energy loss of the projectile going in and the fragment coming out.

What is the typical sensitivity of ERDA?

ERDA is highly sensitive to Hydrogen, often detecting concentrations as low as 0.1 atomic percent.

Does the erda fragment calculator account for the foil?

Standard calculators do not. In many ERDA setups, a range foil is placed before the detector to block scattered projectiles, which further reduces the fragment energy.

Why use MeV units?

Ion beam analysis occurs at the nuclear level where energy scales are millions of electron volts (MeV) to overcome Coulomb barriers or reach relevant cross-sections.

What is the kinematic factor limit?

The maximum value is 1.0 (if M1=M2 and φ=0), but practical limits are lower due to geometry and mass differences.

How does energy affect depth resolution?

Higher incident energies generally allow for deeper profiling but can decrease the near-surface resolution.

Related Tools and Internal Resources

RBS Energy Calculator: For analyzing backscattered projectile ions.
Stopping Power Reference: Essential for calculating depth from energy loss.
Thin Film Density Tool: Convert areal density (atoms/cm²) to physical thickness (nm).
SRIM/TRIM Interface: Advanced simulation for ion interactions in matter.
Cross-Section Database: Access SigmaCalc data for non-Rutherford ERDA.
Angle Converter: Switch between laboratory and center-of-mass frames easily.