The Neutron Star Surface Crystal · AI Generator #GEN-A535CAE6

Physics · modern-physics

Problem

A neutron star has formed a thin crystalline crust on its surface composed entirely of iron-56 nuclei arranged in a face-centered cubic (FCC) lattice. The neutron star has a mass

M = 2.8 \times 10^{30}\;\text{kg}

and radius

R = 12\;\text{km}

. The density of the crystalline crust is

\rho = 4.0 \times 10^{14}\;\text{kg/m}^3

. Due to the extreme gravitational field, the lattice constant of this crystal differs significantly from terrestrial iron crystals. An X-ray photon with energy

E_0 = 25\;\text{keV}

is emitted from the neutron star surface and observed by a distant detector. (a) Calculate the lattice constant

a

of the iron-56 FCC crystal on the neutron star surface. (b) If this X-ray photon was produced by a characteristic K

_\alpha

transition in iron-56, determine the observed photon energy

E_{obs}

as measured by the distant detector, accounting for gravitational redshift. Given: Mass of iron-56 nucleus =

9.29 \times 10^{-26}\;\text{kg}

G = 6.67 \times 10^{-11}\;\text{N⋅m}^2/\text{kg}^2

c = 3.00 \times 10^8\;\text{m/s}

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