PHY 554 - Fundamentals of Accelerator Physics
Homework: RF Acceleration with ELEGANT
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OBJECTIVE
Simulate a 100 MeV electron passing through an RF cavity and study
the dependence of its energy gain on RF phase.

PARAMETERS
Particle: electron             Initial kinetic energy: 100 MeV
RF frequency: 500 MHz          RF voltage: 10 MV
RF cavity length: 0.3 m

FILES AND RUN COMMAND
  lattice.lte    ELEGANT lattice
  run.ele        ELEGANT command file

Run:
  elegant run.ele

RF cavity definition:
  RFC: RFCA, L=0.3, VOLT=10e6, FREQ=500e6, PHASE=0, CHANGE_P0=1

TOTAL: 20 POINTS


QUESTION 1 - RF CAVITY (3 POINTS)
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(a) Explain the meanings of L, VOLT, FREQ, and PHASE.
(b) Calculate the RF period for 500 MHz.
(c) Estimate the distance traveled by a 100 MeV electron during one
    RF period. Compare half this distance with the 0.3 m cavity length.


QUESTION 2 - RF PHASE SCAN (11 POINTS)
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(a) ELEGANT defines pCentral = p/(m_e c). Explain why

      K = m_e c^2 [sqrt(1 + pCentral^2) - 1]

    gives the kinetic energy, and use it to verify that the initial
    beam kinetic energy is 100 MeV. (1 point)

(b) Scan PHASE from -180 deg to +180 deg in 10 deg steps. For each
    phase, find the final kinetic energy and calculate

  Delta E = E_final - 100 MeV.

Plot Delta E [MeV] versus PHASE [deg] and fit the data with a
sinusoidal function. The scan may be performed manually or automated.
(10 points)


QUESTION 3 - COMPARISON WITH RF THEORY (2 POINTS)
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Compare the simulation with

  Delta E(phi) = A cos(phi - phi_0).

(a) Determine A and phi_0 from the phase scan.
(b) Compare the maximum energy gain with that expected from 10 MV.


QUESTION 4 - PHYSICAL INTERPRETATION (4 POINTS)
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(a) Find the phases where the energy gain is approximately zero.
(b) Find the phase and energy loss for maximum deceleration.
(c) Explain why the energy gain depends on RF phase.
(d) Explain why maximum acceleration should not be assumed at PHASE=0.


SUBMISSION
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Submit a report with answers to Questions 1-4, including the phase-scan
data and plot with sinusoidal fit for Question 2.

Submit to: jma1@bnl.gov
Due: October 14, 2026
