# Atomic Models Throughout History

By

Ervin Chia

,

Chua Wei Hong

,

Hillson Hung

,

Nemo Chen

●

Published 2022-08-15

Towards the end of the 1800s, most problems in classical physics were considered solved; they were either solved ingeniously or were decided that they had too much computational cost. The problems left, in principle, were solvable, given enough time to calculate (back then, by hand). People widely thought that physics was nearing its limit. However, it was eventually to be found that classical theories failed to account for various phenomena in atomic physics.

The atomic model was a huge point of debate for scientists back then; only the most creative of methods would allow them to investigate something smaller than any microscope could hope to see. First, a quick summary of what an atom has:

• Protons: particles with a small unit positive electrical charge, known as an elementary charge +e.
• Neutrons: particles with a very similar mass to a proton, carries no electrical charge.
• Electrons: particles with a negative electric charge, -e. Also significantly lighter than protons and neutrons.

The plum pudding model was proposed by J.J. Thompson in 1904, in which the discovery of electrons, which are very small particles that carry a singular unit of negative charge, led him to envisage atoms as negatively charged electron particles held together by a cloud of positive charge. Protons and neutrons were not discovered yet.

Ernest Rutherford investigated the Plum pudding model and quickly realised that it was wrong; the famous Rutherford Scattering experiment showed that the atom had to be composed of an extremely dense nucleus surrounded by a sea of electrons. Scientists were largely satisfied with the Rutherford model based on whatever experiments they could do with an atom, albeit for a gap in understanding of how the electrons behave in the atom.

Consider just one of the electron. Since the nucleus is positively charged with +Ze, and the electron is negatively charged with -e, there is an attractive Coulumb force \mathbf{F}_{E} between them given by

\begin{equation}
\mathbf{F}_{E}=-\frac{1}{4\pi\epsilon_{0}}\frac{Ze^{2}}{r^{2}}\hat{\mathbf{r}}
\end{equation}

where \epsilon_{0}=8.854\times10^{-12}\text{m}^{-3}\text{kg}^{-1}\text{s}^{4}\text{A}^{2} is a constant, r is the distance between the nucleus and the electron, and \hat{\mathbf{r}} is the radial unit vector. With this attractive force, shouldn't the electron fall into the nucleus? Not neccessarily! Recall how planets orbit around a star under the influence of an attractive gravitational force. Motivated by the planetary model, Rutherford suggested that the electron is to orbit around the nucleus under the influence of the Coulomb force.

There is however something peculiar about charges (that do not apply for uncharged masses such as planets): When charges accelerate, they emit radiation (light) and lose energy. This means that the electron will loss energy (continuously) during the orbit and will spiral into the nucleus. A planetary model atom should not even last for a second!

Enter Niels Bohr. He knew of Planck's and Einstein's new idea of a quantised light particle (didn't quite like the idea though), and also Rutherford's new planetary atom (this idea he loved!). He did not regard the abovementioned shortcoming of Rutherford's model as a no-go. His intuition led him to believe that the workings of the subatomic world should not be ruled by classical theories. Through a series of educated guesses, Bohr then assumed that when electrons orbited around the nucleus, not any kind of orbits would do; only specific orbits would be allowed in the atom. After working for a year on this problem, he had a breakthrough when he came to realise that Rutherford's planetary model with discrete allowed orbits can be used to derive the emission/absorption spectrum of hydrogen. With that he laid down his atomic model with three postulates (fancy word for assumptions).

The 3 postulate of Bohr's model of an atom are:

1. Electrons move in a circular orbit around the nucleus, much like how the earth orbits around the sun.
2. The angular momentum of the electron is quantized.
3. Any change in the electron’s energy is caused by the emission or absorption of a quantized packet of light known as a photon.

Let us look into each of the postulate in detail.

The electron orbits the nucleus in circular motion described by the following equation,

\begin{equation}
\mathbf{F}=-\frac{m_{e}v^{2}}{r}\hat{\mathbf{r}}
\end{equation}

where F is the (resultant) centripetal force exerted on the object in motion, m_{e} is the mass of the electron, r is the radius of the circular motion, and v the speed of the electron in circular motion. We also know that the electric force exerted on the electron to be given by Eq.(1).

In the absence of all other forces, the electric force is the only force providing for the centripetal force. This gives us

\begin{equation}
m_{e}v^{2}=\frac{1}{4\pi\epsilon_{0}}\frac{Ze^{2}}{r}
\end{equation}

Much like how momentum is a measure of a body’s tendency to remain in its linear motion, angular momentum is a measure of a body’s tendency to remain in its rotary motion. For the electron in circular motion, the angular momentum L is given by

\begin{equation}
L=m_{e}vr
\end{equation}

At this point, Bohr took a bold step and and believed that angular momentum is quantized:

\begin{equation}
L=n\frac{h}{2\pi}=n\hbar
\end{equation}

where n is a positive integer (i.e. n=1,2,3,... ), and h is Planck's constant. \hbar=h/2\pi which is read as h bar, is a convenient shorthand notation. Putting Eq.(5) into Eq.(4), we have

m_{e}v=\frac{n\hbar}{r}

Putting this into Eq.(3), we have

\begin{align}
\frac{n^{2}\hbar^{2}}{r^{2}} & =m_{e}\frac{1}{4\pi\epsilon_{0}}\frac{Ze^{2}}{r}\nonumber \\
r & =\frac{4\pi\epsilon_{0}n^{2}\hbar^{2}}{m_{e}Ze^{2}}\,,\,n=1,2,3,\dots
\end{align}

Note that now the radius of orbit is quantised, meaning that we can only find electrons at fixed radii from the proton; no closer or further. The speed of the electron is also fixed; it is neither too fast nor too slow!

You may have heard of "energy levels" of an atom. These refer to the possible discrete energies of an electron can have based on it's quantised radius of orbit. To work out the energy levels, we need to digress a little.

The potential energy of an electron experiencing Coulomb force Eq.(1) is given by¹

V=-\frac{1}{4\pi\epsilon_{0}}\frac{Ze^{2}}{r}

We write the potential energy with a negative sign because the electrons are bound to the nucleus: the electron would require extra energy to escape from the electric attraction from the positively charged nucleus.

Total energy of the system is given by

\begin{align*}
E & =\text{Kinetic energy}+\text{Potential Energy}\\
 & =\frac{1}{2}m_{e}v^{2}-k\frac{Ze^{2}}{r}\\
 & =\frac{1}{2}\frac{1}{4\pi\epsilon_{0}}\frac{Ze^{2}}{r}-\frac{1}{4\pi\epsilon_{0}}\frac{Ze^{2}}{r}\quad\text{(see Eq.(3))}\\
 & =-\frac{1}{8\pi\epsilon_{0}}\frac{Ze^{2}}{r}
\end{align*}

Putting the quantised radius Eq.(5) into the above, we have

\begin{equation}
E=-\frac{Z^{2}e^{4}m_{e}}{32\pi^{2}\epsilon_{0}^{2}\hbar^{2}n^{2}}\,,\,n=1,2,3,\dots
\end{equation}

Eq.(6) gives discrete or quantised energies of an electron. The value of the energy depends on the integer value of n. The higher the n, the higher (less negative) the energy. The electron can transit from one energy level to another. But to do so, it must gain or lose the exact amount of energy difference by absorbing or emiting a photon with the specific energy.

\begin{equation}
hf=\left|E_{\text{final}}-E_{\text{initial}}\right|
\end{equation}

For example, when an electron to fall from n=3\rightarrow1, it will emit a photon of energy

E=hf=\frac{Z^{2}e^{4}m_{e}}{32\pi^{2}\epsilon_{0}^{2}\hbar^{2}}\left(\frac{1}{1^{2}}-\frac{1}{3^{2}}\right)=\frac{Z^{2}e^{4}m_{e}}{36\pi^{2}\epsilon_{0}^{2}\hbar^{2}}

The curious case of the Rydberg formula was finally solved: the hydrogen spectra had puzzled scientists for many years until the day Bohr published his model. Bohr's model accurately predicted every single transition the Rydberg formula gave while giving meaning to the seemingly unknown coefficients present in it. The true success of the Bohr model comes from the results of the quantized energies and its ability to derive the Rydberg formula, which was first found empirically from experimental data. To this day, despite some shortcomings, the Bohr model remains an extremely important concept in modern physics.

Through Bohr's model, we found that energy transitions in the hydrogen atom between its different energy levels absorb or emit light. Furthermore, they should be extremely sharp and distinct: the energy levels are significantly far from each other. This means that every atom has a specific set of wavelengths that it would emit. The whole group of emissions could be further classified into the specific energy level that is involved in the transition; every transition between distinct energy levels can be uniquely identified by the light emitted or absorbed.

The energies of each level can be computed using equation \ref, varying n for the different energy levels. Since energy cannot be created or destroyed, then we know that when a transition occurs from one state to another, the difference in energy levels must exactly correspond to the energy of the photon emitted or absorbed. Transitions that involve the innermost shell, n=1, is eponymously known as the Lyman series and is predominantly ultraviolet. As such, we are unable to observe it with our eyes (we can most definitely detect it with sensors).

On the other hand, transitions to the second energy level n = 2, known as the Balmer series, lie predominantly in the visible light spectrum. In fact, the Balmer series is exactly the specific wavelengths of light observed in a Hydrogen lamp. The hydrogen emission spectrum is observed when hydrogen de-excites to the second energy level. The signature purple glow seen in a hydrogen lamp is a mixture of these colors, and can be observed clearly with a proper setup in the darkness. This is known as the emission spectra of the element. Since it is the transition that is quantized, we can just as equally try to measure the effects when hydrogen absorbs the energies needed to excite to higher energy levels. Since the other colors are not accepted by the hydrogen atom, we can shine a continuous spectrum of light onto Hydrogen and collect the light that passes through. The hydrogen would have absorbed the light that it can accept, and allow the rest through. The collected light is then analyzed, and we do indeed find the dark fringes where the emission spectra would have been. This equivalent method of finding the spectra is known as the absorption spectrum of the element.

At this point, we are able to mathematically predict the absorption and emission spectra of hydrogen and hydrogen-like atoms using Bohr's model. Any other system involving more than one electron is simply too difficult to calculate by hand (three body problem), and are usually approximated by other methods in quantum mechanics or computationally modelled with Schrodinger's model.

Despite its successes in recovering the empirical observation of the Hydrogen emission by a solely theoretical approach, Bohr's model suffered from several shortcomings. Most notably, it was only able recover the emission spectrum of Hydrogen and other one-electron systems. It is unable to reproduce the results of atomic systems with two or more electrons.

In 1926, Erwin Schrödinger proposed a revolutionary new proposal of using wave mechanics as a way to approach the atomic model. His model builds upon Bohr's model in an attempt to further refine and overcome its shortcomings. It is also alongside this paper where he published the now eponymous Schrödinger equation which will be discussed in further detail later. The basis behind Schrödinger's modification hinges from work contributed by Louis de Broglie. In 1924, de Broglie proposed that our previous distinction between waves and particles, one that is well established at the height of classical mechanics, was to be blurred. He hypothesised that particles, when sufficiently small or fast, begin to exhibit wave-like properties; and when waves begin to carry enough momentum, they begin to obtain particle-like properties. The distinction that is broken is summarised in the following equation, known as the de Broglie relation:

\begin{equation}
\lambda=\frac{h}{p}
\end{equation}

where p represents momentum, a fundamental property of particles, and \lambda , the wavelength, a fundamental property of waves. This was just like how light was originally thought of as waves, but eventually wave-particle duality became accepted. According to de Broglie, everything exhibits wave-particle duality. This property was eventually experimentally verified when electrons were found to behave like waves under specific conditions. This held enormous consequences; the de Broglie relation could be used to justify the quantization that Bohr proposed.

Go to Activity 7.

Applying the concept of matter waves into the Bohr model allows us to understand quantisation more intuitively. Waves exhibits interference. An electron-in-orbit behaving like a wave would loop back to itself and if a standing wave^{2 3} is not formed, the "electron wave" would destructively interfere with itself. As a result, only specific setups that form standing waves were allowed to exist, creating these quantized orbits. These specific setups would eventually come to be known as atomic or electron orbitals.