Effect Of Nucleus Motion On Energy

The effect of nucleus motion on the energy of Bohr's second postulate is a significant aspect of understanding atomic structure. Bohr's second postulate describes the quantized nature of electron orbits around a nucleus, where electrons can only occupy specific energy levels. When considering the motion of the nucleus, the energy levels shift slightly, refining the accuracy of atomic models. This nuanced understanding is crucial in fields like spectroscopy and quantum mechanics, impacting technologies such as atomic clocks and MRI machines, where precise measurements of atomic behaviour are essential for accurate results. This article explores how nucleus motion influences energy levels and the broader implications in scientific and practical applications. In this article, we will discuss the concept of the Effect of Nucleus Motion on Energy and provide examples for concept clarity.

Effect of Nucleus Motion on Energy

Till now in Bohr's model, we have assumed that all the mass of the atom is situated at the center of the atom. As the mass of the electron is very small and negligible as compared to the mass of the nucleus all the mass is assumed to be concentrated at the centre of the nucleus. But actually, the centre of mass of the nucleus-electron system is close to the nucleus as it is heavy, and to keep the centre of mass at rest, both electrons and the nucleus revolve around their centre of mass like a double star system as shown in the figure. If r is the distance of the electron from the nucleus, the distances of the nucleus and electron from the centre of mass, and , can be given as

$
r_1=\frac{m_e r}{m_N+m_e}
$
and $\quad r_2=\frac{m_N r}{m_N+m_e}$

We can see that the atom, nucleus, and electron revolve around their centre of mass in concentric circles of radii $r_1$ and $r_2$ to keep the centre of mass at rest. In the above system, we can analyze the motion of electrons with respect to the nucleus by assuming the nucleus to be at rest and the mass of the electron replaced by its reduced mass $\mu_{\mathrm{e}}$, given as -

$\mu_{\mathrm{e}}=\frac{m_N m_e}{m_N+m_e}$

Now we can change our assumption and the system will look like as shown in the figure with reduced mass

Now we can derive the equation obtained by Bohr with the reduced mass also

$r_n=\frac{n^2 h^2}{4 \pi^2 k Z e^2 m_e}$

Now after replacing the electron mass with its reduced mass, the equation becomes

$\begin{aligned} r_n^{\prime} & =\frac{n^2 h^2}{4 \pi^2 k Z e^2 \mu_e} \Rightarrow r_n^{\prime}=\frac{n^2 h^2\left(m_N+m_e\right)}{4 \pi^2 k Z e^2 m_e m_N} \\ \text { or } \quad r_n^{\prime} & =r_n \times \frac{m_e}{\mu_e} \Rightarrow r=(0.529 \mathrm{~A}) \frac{m^2}{\mu Z}\end{aligned}$

But there will be no effect on the velocity because the term of mass is not present there -

$v_n=\frac{2 \pi k Z e^2}{n h}$

Similarly for energy, we can write that

$E_n=-\frac{2 \pi^2 k^2 Z^2 e^4 m_e}{n^2 h^2}$

After putting the reduced mass in the equation

$\begin{aligned} & E_n^{\prime}=-\frac{2 \pi^2 k^2 Z^2 e^4 m_N m_e}{n^2 h^2\left(m_N+m_e\right)} \\ & E_n^{\prime}=E_n \times \frac{\mu_{\varepsilon}}{m_e} \Rightarrow E_n=-(13.6 \mathrm{eV}) \frac{Z^2}{n^2}\left(\frac{\mu}{m}\right)\end{aligned}$

Thus, we can say that the energy of electrons will be slightly less compared to what we have derived earlier. But for numerical
calculations this small change can be neglected unless in a given problem it is asked to consider the effect of the motion of the nucleus.

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Solved Examples Based on the Effect of Nucleus Motion on Energy

Example 1: A positronium atom is a system that consists of a positron and an electron that orbit each other. The ratio of wavelength of the spectral line of positronium to that of ordinary hydrogen is:

1) 2:1

2) 4:1

3) 1:2

4) 1:4

Solution:

Effect of nucleus motion on the energy of the atom

$
E_n^1=E_n\left(\frac{m_N}{m_e+m_N}\right)
$
wherein
$E_n^1=$ The energy of $n^{\text {th }}$ orbital with nucleus motion
$E_n=$ The energy of $\mathrm{n}^{\text {th }}$ orbital without nucleus motion
$m_e=$ mass of electron
$m_N=$ mass of nucleus
since the two-particle has the same mass
$
\mu=\frac{m * m}{m+m}=\frac{m}{2}
$
since $\quad E_n \propto m$
$
\frac{E_n^{\prime}}{E_n}=\frac{1}{2}
$

Hence energy level of the positronium atom is half of the corresponding energy level in the H-atom.

As a result, the wavelength in the positronium atom spectral lines is twice those of corresponding lines in the hydrogen spectrum.

Hence, the answer is the option (3).

Example 2: The wavelengths involved in the spectrum of deuterium $\left({ }_1^2 D\right)$ are slightly different from that of the hydrogen spectrum, because

1) The sizes of the two nuclei are different

2) nuclear forces are different in the two cases

3) masses of the two nuclei are different

4) The attraction between the electron and the nucleus is different in the two cases.

Solution

The wavelength of spectrum is given by $\frac{1}{\lambda}=R z^2\left(\frac{1}{n_{\mathrm{f}}^2}-\frac{1}{n_2^2}\right)$ where $R=\frac{1.097 \times 10^7}{1+\frac{1}{n d}}$ where $m=$ mass of electron $\mathrm{M}=$ mass of nucleus

Masses of ${ }_1 H^1$ and $1{ }^2$ are different. Hence the corresponding wavelengths are different.

Hence, the answer is the option (3).

Example 3: A muon is an unstable elementary particle whose mass is $207 \mathrm{~m}_{\mathrm{e}}$ and whose charge is either +e or -e. A negative muon $\left(\mu^{\prime}\right)$ can be captured by a hydrogen nucleus (or proton) to form a muonic atom. Find the radius (in meters) of the first Bohr orbit of this atom -
1) $5 \times 10^{-13}$
2) $6.85 \times 10^{-13}$
3) $2.85 \times 10^{-13}$
4) $5.7 \times 10^{-13}$

Solution:

Here $m=207 m_e$ and $M=1836 m_e$; so the reduced mass is
$
\mu=\frac{m M}{m+M}=\frac{\left(207 m_e\right)\left(1836 m_e\right)}{207 m_e+1836 m_e}=186 m_e
$

According to equation $\lambda=h / p$, the orbit radius corresponding to $n=1$ is $r_1=\frac{h^2 \varepsilon_0}{\pi m_{\varepsilon} e^2}=5.29 \times 10^{-11} \mathrm{~m}$

Hence, the radius $r^{\prime}$ that corresponds to the reduced mass $\mu$ is
$
r_1^{\prime}=\left(\frac{m}{\mu}\right) r_1=\left(\frac{m_{\varepsilon}}{186 m_{\mathrm{e}}}\right) r_1=2.85 \times 10^{-13} \mathrm{~m}
$

Hence, the answer is the option (3).

Example 4: A nucleus of mass $M$ emits $\gamma_{\text {-ray }}$ photon of frequency ${ }^{\prime} \nu^{\prime}$, The loss of internal energy by the nucleus is : [Take ' $c$ ' as the speed of electromagnetic wave]

1) $h v$
2) 0
3) $h v\left[1-\frac{h v}{2 M c^2}\right]$
4) $h v\left[1+\frac{h v}{2 M c^2}\right]$

Solution:

When the nucleus emits - ray there will be recoiling of the nucleus.

$p_2 \rightarrow$ momentum of $\gamma-$ ray(photon)
$p_1 \rightarrow$ Momentum of Nucleus
By momentum conservation,
$
\begin{aligned}
& 0=p_1+p_2 \\
& 0=M v+\frac{h}{\lambda} \\
& v=\frac{-h}{\lambda M}=\frac{-h v}{M c}
\end{aligned}
$

Here negative sign implies the opposite direction motion of the nucleus (recoiling) when -ray emission takes place.

Internal energy lost by the nucleus is given to the nucleus and x-ray.

$\begin{aligned} \therefore \text { Internal energy lost } & =\mathrm{KE}_{\text {Nucleus }}+\mathrm{E}_{\gamma \text {-ray }} \\ & =\frac{1}{2} M v^2+h v \\ & =\frac{1}{2} M \frac{h^2 v^2}{M^2 c^2}+h v \\ & =h v\left[1+\left(\frac{h^2}{2 M c^2}\right)\right]\end{aligned}$

Hence, the correct option is (4).

Example 5: A diatomic molecule is made of two masses $m_1$ and $m_2$ which are separated by a distance $r$. If we calculate its rotational energy by applying Bohr's rule of angular momentum quantization, its energy will be given by ( $n$ is an integer)
$
\left(\text { use } \hbar=\frac{h}{2 \pi}\right)
$

1) $\frac{\left(m_1+m_2\right)^2 n^2 \hbar^2}{2 m_1^2 m_2^2 r^2}$
2) $\frac{n^2 \hbar^2}{2\left(m_1+m_2\right) r^2}$
3) $\frac{2 n^2 \hbar^2}{\left(m_1+m_2\right) r^2}$
4) $\frac{\left(m_1+m_2\right) n^2 \hbar^2}{2 m_1 m_2 r^2}$

Solution:

A diatomic molecule consists of two atoms of masses and at a distance apart. Let and be the distances of the atoms from the centre of mass

The moment of inertia of this molecule about an axis passing through its centre of mass and perpendicular to a line joining the atoms is

The solution is correct. So no need to change it

$\begin{aligned}
& I=m_1 r_1^2+m_2 r_2^2 \\
& \text { As } \quad m_1 r_1=m_2 r_2 \quad \text { or } r_1=\frac{m_2}{m_1} r_2 \\
& \because \quad r_1+r_2=r \\
& \therefore \quad r_1=\frac{m_2}{m_1}\left(r-r_1\right)
\end{aligned}$

On rearranging, we get

$
r_1=\frac{m_2 r}{m_1+m_2}
$

similarly $r_2=\frac{m_1 r}{m_1+m_2}$
Therefore, the moment of inertia can be written as
$
\begin{aligned}
I & =m_1\left(\frac{m_2 r}{m_1+m_2}\right)^2+m_2\left(\frac{m_1 r}{m_1+m_2}\right)^2 \\
& =\frac{m_1 m_2}{m_1+m_2} r^2 \ldots \cdots \cdots \cdots \cdot(i)
\end{aligned}
$

According to Bohr’s quantisation condition

$
L=\frac{n h}{2 \pi}
$
or $L^2=\frac{n^2 h^2}{4 \pi^2}$
Rotational energy, $E=\frac{L^2}{2 I}$
$
\begin{aligned}
& E=\frac{n^2 h^2}{8 \pi^2 I} \cdots \cdots \cdots \cdot u \operatorname{sing}(i i) \\
& E=\frac{n^2 h^2\left(m_1+m_2\right)}{8 \pi^2\left(m_1 m_2\right) r^2} \cdots \cdots \cdots \cdots \cdot u \operatorname{sing}(i) \\
& =\frac{\left(m_1+m_2\right) n^2 \hbar^2}{2 m_1 m_2 r^2} \quad\left(\because h=\frac{h}{2 \pi}\right)
\end{aligned}
$

Summary

The effect of nucleus motion on the energy of Bohr's second postulate refines our understanding of atomic structures by considering the slight shifts in energy levels due to the nucleus's movement. This adjustment is crucial for precision in fields like spectroscopy and quantum mechanics, affecting technologies such as atomic clocks and MRI machines. The solved examples illustrate how this consideration impacts the spectral lines of positronium, deuterium, muonic atoms, and the internal energy of nuclei, highlighting the practical implications of incorporating nucleus motion in atomic models.