Accelerating Molecular H$_2^+$ Beam in AGS and RHIC

Following the successful acceleration of molecular H$_2^+$ beams in the Booster, this paper proposes testing their acceleration to higher energies in the AGS and RHIC while analyzing potential challenges and the associated benefits for BNL.

The Gist

Imagine the Brookhaven National Laboratory (BNL) as a massive, high-speed racetrack for subatomic particles. Usually, these racers are single protons (hydrogen atoms stripped of their electron). But this report proposes a daring new strategy: racing molecular hydrogen ions ($H_2^+$).

Think of a standard proton as a solo runner. A molecular hydrogen ion ($H_2^+$) is like a tandem bicycle: it has two protons (the riders) tied together by a single electron (the chain holding them). The goal is to see if we can get this "tandem bicycle" to race all the way up to the highest speeds in the lab's biggest ring (RHIC), reaching energies of 100 GeV.

Here is the breakdown of the paper's claims, using simple analogies:

1. The Big Question: Can the Tandem Bike Hold Together?

The scientists have already successfully raced these tandem bikes in the smaller "Booster" track (reaching 1 GeV). Now, they want to test them on the bigger tracks: the AGS (up to 12 GeV) and the massive RHIC ring (up to 100 GeV).

The main worry is that the bike might fall apart due to two specific forces:

• The "Magnetic Wind" (Lorentz Effect):
  Imagine the tandem bike riding through a strong magnetic field. In the bike's own frame of reference, this magnetic field transforms into a powerful electric wind blowing sideways.

  • The Risk: If the bike goes too fast, this "wind" gets so strong it might rip the chain (the electron) off, causing the two riders (protons) to fly apart.
  • The Finding: The math suggests the bike will be fine on the medium-sized track (AGS). However, on the massive RHIC track at very high speeds (50–100 GeV), the wind might get strong enough to break the chain. The paper says we must test this immediately before the track shuts down to see if there is a "speed limit" where the bike breaks.

• The "Crowded Room" (Beam Gas Collisions):
  Even in a vacuum, there are a few stray air molecules floating around.

  • The Risk: If the tandem bike crashes into a stray air molecule, the impact could knock the chain loose.
  • The Finding: The vacuum in the lab is incredibly empty. The paper calculates that even in the worst-case scenario, the bike would travel for over 3 minutes before hitting a stray molecule. This is much longer than the time it takes to complete a lap, so this isn't a major problem.

2. Why Bother? The Benefits of the Tandem Bike

If the scientists prove they can race these tandem bikes at high speeds, it offers several unique advantages for the lab:

• A Cheaper, Smarter Fuel Source:
  Currently, getting high-speed protons is expensive and requires complex machinery. Using these tandem bikes allows the lab to strip off the electron with a simple thin foil, turning the tandem bike into two solo runners (protons) right on the track. This is a lower-cost, flexible way to get the proton beams needed for other experiments (like medical research or neutron sources).

• A Built-in Calibration Tool for the Future (EIC):
  The lab is planning a future "Electron-Ion Collider" (EIC). If they use these tandem bikes, every single "rider" (proton) comes with a built-in "passenger" (an electron) moving at the exact same speed.

  • The Analogy: Imagine every car on a highway has a passenger in the back seat. When the cars collide with an incoming electron beam, the passenger (electron) can crash into the incoming electron.
  • The Benefit: This creates a predictable, known type of crash (called Møller scattering). Scientists can use these crashes as a "ruler" or "calibration tool" to check if their detectors are working perfectly, ensuring their measurements of other collisions are accurate.

3. The Bottom Line

The paper is a call to action. Before the lab shuts down for maintenance, they need to run tests to see if the "tandem bicycle" can survive the extreme speeds of the RHIC ring without the "magnetic wind" tearing it apart.

If it works, it opens the door to cheaper proton beams and provides a perfect, built-in calibration system for the future Electron-Ion Collider. If it doesn't, they need to know that limit now so they can plan for the next decade. The paper also suggests testing other "vehicles" (like $H_3^+$ or $D_2^+$) in the future, but the immediate focus is strictly on the $H_2^+$ tandem bike.

Technical Summary

Technical Summary: Accelerating Molecular H+2 Beam in AGS and RHIC

Problem Statement
Prior to the scheduled shutdown of the Relativistic Heavy Ion Collider (RHIC), there is an urgent need to determine the feasibility of accelerating molecular hydrogen ions (H+2) to high energies (up to 100 GeV/u) within the RHIC and Alternating Gradient Synchrotron (AGS) rings. While the successful acceleration of H+2 beams from the Electron Beam Ion Source (EBIS) to 1.0 GeV/u in the Booster has been demonstrated, scaling this to higher energies introduces significant physical constraints. The primary challenge is identifying whether fundamental limits exist that would prevent H+2 acceleration at RHIC energies. Two specific mechanisms are identified as potential barriers to beam survival:

1. The Lorentz Effect: In the rest-frame of the ion, the strong transverse magnetic field of the bending magnets transforms into a strong electric field. If this field exerts a force sufficient to overcome the molecular binding energy, the H+2 molecule may dissociate, leading to beam loss.
2. Dissociation via Beam-Gas Collisions: Collisions between the H+2 beam and residual gas molecules in the vacuum chamber can strip the molecule apart.

Methodology
The report employs a semi-classical estimation and conservative cross-section assumptions to evaluate these limiting effects across different accelerator stages (Booster, AGS, and RHIC).

• Lorentz Effect Analysis: The study calculates the electric field ($E'_x$) experienced in the ion's rest frame as a function of the laboratory magnetic field ($B_y$) and the relativistic boost factors ($\gamma, \beta$). The resulting "Lorentz-separation-potential" ($E'_x \cdot a_0$, where $a_0$ is the Bohr radius) is compared against the known dissociation energy of H+2 (2.65 eV). Calculations are performed for kinetic energies ranging from 1 GeV/u (Booster) to 137.5 GeV/u (EIC maximum).
• Dissociation Effect Analysis: The study assumes a conservative dissociation cross-section ($\sigma$) of $5 \times 10^{-19} \text{ cm}^2$ for H+2, based on H- stripping data and Bethe-Born scaling. Using typical vacuum levels (ranging from $10^{-10}$ to $2 \times 10^{-9}$ torr) and beam gas number densities, the collision mean-free-path ($\lambda$) is calculated. Beam lifetime ($\tau$) is estimated using the relation $\tau \approx \lambda / \beta c$.
• Proposed Experimental Plan: The authors propose conducting tests to accelerate H+2 beams in the AGS (up to 12 GeV/u) and the RHIC Blue ring (up to 100 GeV/u). The objective is to empirically determine the maximum achievable energy and quantify beam loss at each stage.

Key Results and Findings

• Lorentz Effect Limits:
  • At Booster (1 GeV/u) and AGS (12 GeV/u), the calculated Lorentz-separation-potential (0.01 eV and 0.20 eV, respectively) is well below the 2.65 eV dissociation threshold. The report concludes that acceleration in AGS is unlikely to be prohibited by this effect.
  • In RHIC, the potential rises significantly with energy. At 50 GeV/u, the potential is 1.17 eV, and at 100 GeV/u, it reaches 4.67 eV, exceeding the dissociation energy. At the EIC maximum of 137.5 GeV/u, the potential reaches 8.82 eV. The report indicates that an upper energy limit "might be encountered" in the RHIC regime due to the "strong field atomic regime," necessitating urgent testing to confirm if a hard limit exists.
• Dissociation via Vacuum:
  • Even under worst-case vacuum assumptions for AGS ($2 \times 10^{-9}$ torr), the collision mean-free-path is orders of magnitude larger than the ring circumference.
  • For RHIC, assuming a conservative cross-section, the expected beam lifetime is estimated to be greater than 3 minutes. This suggests that beam-gas collisions are not the primary limiting factor for the proposed tests.
• Kinematic Feasibility for EIC:
  • The report details the kinematics of H+2 beams in the context of the future Electron-Ion Collider (EIC). Because H+2 contains two protons and one electron at a 2:1 ratio moving at the same velocity, collisions would inherently produce a precise 2:1 ratio of electron-proton ($e^- + p$) to electron-electron ($e^- + e^-$) interactions.

Significance and Proposed Benefits
The successful acceleration of H+2 beams in AGS and RHIC is presented as a significant advancement in accelerator physics with several specific benefits for Brookhaven National Laboratory (BNL):

1. Cost-Effective Proton Beam Option: H+2 beams from EBIS offer a lower-cost alternative to Tandem or Linac sources for generating high-flux proton beams. This is particularly relevant for facilities like the NSRL, HEET, and the U-line test beam.
2. Enhanced RHIC/EIC Capabilities:
   • Dual-Beam Operation: Accelerating H+2 would allow RHIC to provide a unique combination of proton and electron beams simultaneously within the same bunch structure.
   • Luminosity Calibration: The fixed 2:1 ratio of protons to electrons in the H+2 molecule allows for precise determination of electron-proton collision luminosity by tagging Møller scattering ($e^- + e^-$) events. This anchors cross-section measurements to well-known QED standards.
   • Detector Calibration: The kinematics of Møller scattering events (where one electron's trajectory determines the other's) provide an over-constrained system ideal for calibrating tracking detectors and particle identification (PID) systems.
3. Future Applications: The technology could enable new options for high-intensity proton applications, including proton therapy, spallation neutron sources, isotope production, and neutrino generation, particularly in linear accelerators where Lorentz dissociation effects are absent.

Conclusion
The report concludes that while AGS acceleration appears safe from Lorentz dissociation, the upper energy limit for RHIC remains uncertain and requires immediate experimental verification before the facility shutdown. Confirming the viability of H+2 acceleration is critical for planning future detector designs and commissioning strategies for the EIC and other BNL facilities. Future studies should also explore other molecular beams (H+3, D+2) and partially stripped ions (3He+).