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CritPt / Challenge_55_main / This problem addresses inelastic neutron scattering from a 1D harmonic…

Problem

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code template

Code

def answer():
    r"""
    Return the values of $\sigma_m(E)$ with $m=2$.

    Inputs
    ----------
    None

    Outputs
    ----------
    sigma_1 : float, $\sigma_m(E)$ with $m=2$ in the limit of low temperature, assuming $E = 1$ meV.
    sigma_40: float, $\sigma_m(E)$ with $m=2$ in the limit of low temperature, assuming $E = 40$ meV.
    """

    # ------------------ FILL IN YOUR RESULTS BELOW ------------------
    sigma_1  = ... # three decimal precision
    sigma_40 = ... # three decimal precision
    # ---------------------------------------------------------------

    return sigma_1, sigma_40

problem description

# Problem setup: This problem addresses inelastic neutron scattering from a 1D harmonic oscillator. Because phonons in a crystal lattice can be modeled as harmonic oscillators, this setup effectively captures the essential physics of inelastic neutron scattering from phonons in a crystalline solid. We consider a harmonic oscillator of mass

M

with fundamental frequency

\omega_0

. The total cross section

\sigma(E)

for neutrons of mass

m_n

and energy

E

can be expressed as a sum over contributions from the

m

-th harmonic oscillator:

\sigma(E)=\sum_{m=0}^{\infty} \sigma_m(E).

For convenience, assume the bound-atom cross section

\sigma_b = 1

barn,

M = 10 m_n = 10

amu (atomic mass unit), and the phonon energy

\hbar \omega_0 = 10

meV. # Main problem: Calculate

\sigma_m(E)

with

m=2

in the limit of low temperature with a precision to three decimal places for the following cases: Case 1: Assume that

E = 1

meV. Case 2: Assume that

E = 40

meV.

Plain-text mathematical notation (without MathML)

# Problem setup:
This problem addresses inelastic neutron scattering from a 1D harmonic oscillator. Because phonons in a crystal lattice can be modeled as harmonic oscillators, this setup effectively captures the essential physics of inelastic neutron scattering from phonons in a crystalline solid.

We consider a harmonic oscillator of mass M with fundamental frequency ω₀.

The total cross section σ(E) for neutrons of mass m_(n) and energy E can be expressed as a sum over contributions from the m-th harmonic oscillator:
σ(E)=∑_(m=0)^(∞)σ_(m)(E).
For convenience, assume the bound-atom cross section σ_(b)=1 barn, M=10m_(n)=10 amu (atomic mass unit), and the phonon energy ℏω₀=10 meV.



# Main problem:

Calculate σ_(m)(E) with m=2 in the limit of low temperature with a precision to three decimal places for the following cases:

Case 1: Assume that E=1 meV.

Case 2: Assume that E=40 meV.

Original LaTeX notation

# Problem setup:
This problem addresses inelastic neutron scattering from a 1D harmonic oscillator. Because phonons in a crystal lattice can be modeled as harmonic oscillators, this setup effectively captures the essential physics of inelastic neutron scattering from phonons in a crystalline solid.

We consider a harmonic oscillator of mass $M$ with fundamental frequency $\omega_0$.

The total cross section $\sigma(E)$ for neutrons of mass $m_n$ and energy $E$ can be expressed as a sum over contributions from the $m$-th harmonic oscillator:
$$
\sigma(E)=\sum_{m=0}^{\infty} \sigma_m(E).
$$
For convenience, assume the bound-atom cross section $\sigma_b = 1$ barn, $M = 10 m_n = 10$ amu (atomic mass unit), and the phonon energy $\hbar \omega_0 = 10$ meV.



# Main problem:

Calculate $\sigma_m(E)$ with $m=2$ in the limit of low temperature with a precision to three decimal places for the following cases:

Case 1: Assume that $E = 1$ meV.

Case 2: Assume that $E = 40$ meV.

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Source and history

upstream_id / Original problem ID: Challenge_55_main
task_id: d673ab42-4c33-5294-96e6-2a3a356cdfd6
task_key: train--Challenge~5f55~5fmain
task_revision_id: 2
config / split: / train

Official source

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