What Frequency Does an Atom in a Cubic Lattice Absorb Radiation?

• catkin
Hz.In summary, we are given an atom of mass 5 \times 10^{-26} kg in a cubic lattice with spring constant 100 N/m. Using the equations F = ma, F = -kx, and a = -\omega^{2}x, we can estimate the natural frequency of oscillation of the atom as 4 \times 10^{13} Hz. However, the book gives the answer as 7.12 \times 10^{12} Hz, which can be obtained by calculating f = ω / 2π.
catkin
[SOLVED] SHM: atom in a lattice

Homework Statement

An atom of mass $$5 \times 10^{-26}$$ kg is in a cubic lattice with all bonds between adjacent pairs of atoms having a spring constant about 100 N/m. Estimate the natural frequency of oscillation of the atom. If it is able to absorb radiation at this frequency, what part of the electromagnetic spectrum does it absorb?

Homework Equations

$$F = ma$$ (Newton 2)
$$F = -kx$$ (Hooke)
$$a = -\omega^{2}x$$ (SHM)

The Attempt at a Solution

From Newton 2 and Hooke
$$a = -kx / m$$
Substituting in SHM
$$-kx / m = -\omega^{2}x$$
$$\omega = \sqrt{k/m}$$
$$= \sqrt{\frac {100} {5 \times 10^{-26}}}$$
$$= 4 \times 10^{13}$$ Hz ct1sf

My difficulty is that the answer given in the book is $$7.12 \times 10^{12}$$ Hz. What have I done wrong? Using $$\omega = \sqrt{k/m}$$ and the given value of m, k would have to be ~2.55 N/m. ?

The ans given is correct. Check your arithmetic.

Shooting star said:
The ans given is correct. Check your arithmetic.
Arithmetic check:
$$\sqrt{\frac {100} {5 \times 10^{-26}}}$$
$$= \sqrt{\frac {100} {5}} \times 10^{13}}$$
$$= \sqrt{20}\times 10^{13}}$$
$$= 4.472\times 10^{13}}$$

I still can't see my mistake

Ah! f = ω / 2π

1. What is SHM in relation to atoms in a lattice?

SHM stands for Simple Harmonic Motion, which is the repetitive back and forth motion of a system around a stable equilibrium point. In the case of atoms in a lattice, SHM refers to the oscillation of atoms around their equilibrium positions within a crystal lattice structure.

2. Why do atoms in a lattice exhibit SHM?

This is because the atoms in a lattice are connected by bonds, which act as springs. When the atoms are displaced from their equilibrium positions, these bonds exert restoring forces that cause the atoms to oscillate back and forth, resulting in SHM.

3. How does the mass of the atoms affect SHM in a lattice?

The mass of the atoms does not affect the frequency of SHM, but it does affect the amplitude (or size) of the oscillation. Heavier atoms will have smaller amplitudes compared to lighter atoms, but they will still exhibit SHM.

4. Can the temperature of a lattice affect SHM?

Yes, the temperature of a lattice can affect SHM. At higher temperatures, the atoms in a lattice have more energy and can vibrate with larger amplitudes, resulting in a higher frequency of SHM. At lower temperatures, the amplitude and frequency of SHM will decrease.

5. How is SHM in a lattice related to the properties of materials?

SHM in a lattice is related to the mechanical properties of materials such as elasticity and stiffness. The frequency and amplitude of SHM can affect the strength and flexibility of a material, making it an important factor to consider in material design and engineering.

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