Brief History

 Our understanding on what things are and what those things are made of has understandably evolved through the years. It's understandable because the closer we look at the world, the more we've had to adapt our notions to align with nature's laws. Our understanding of atoms is a superb example of our evolution of understanding.

Our understanding of atoms all started, probably on a dark and stormy night, about 2400 years ago when a cheerful, Greek philosopher named Democritus asked himself this question: how can matter be what it is?

Democritus hypothesized that if anyone were to cut something down, then cut the smaller piece again, and cut the new small piece again and again and again, eventually, they would reach a point at which the piece could not be made smaller. He called this infinitesimally small piece atomos. This is how the theory of the atom came to be, named after the word atomos. Atomos means "not to be cut." To Democritus, atoms were hard particles of all shapes and sizes made of the same material, which could bond together to form stuff.

Now, fast-forward two thousand years into the future past the dark ages to about 1800. John Dalton, some type of genius—he became an English teacher at the age of 12, go figure—conducted many experiments and concluded that the atom is like a billiard ball. He argued that:

1. Atoms make up all matter.
2. Atoms can't be divided, created, or destroyed.
3. Atoms of the same element have the same mass and properties.
4. Atoms combine to form different compounds.
5. Atoms rearrange themselves when a chemical reaction occurs.

Here's a chart of what he learned of the various kinds of atoms in existence.


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Dalton's model is a primitive Periodic Table of Elements. Although this appears incomplete to us, because it is, wouldn't physics and chemistry be much easier to learn if Dalton's model was good enough? We wouldn't have to think about charges or subatomic particles within atoms or variations of any kind.

Did someone say charges? In 1897, while experimenting with a cathode-ray tube, Joseph John Thomson discovered a negatively-charged particle. He called it the corpuscule. Eventually, it was renamed the electron.

Thompon imagined these tiny particles embedded within the atom, just like a chocolate chips in a chocolate chip cookie or raising in a plum pudding. Since the atom is neutral, he reasoned the pudding's surface was positively charged, and these little negative raisins inside made the entire pudding neutral. This is known as J. J. Thomson's plum pudding model.


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What next? Well, around 1910, some years after the discovery of negative charges within atoms, Mr. Ernest Rutherford decided to test the plum pudding model. Rutherford figured that if the atom was like a plum pudding, then its mass would be evenly distributed. To test this, Rutherford shot alpha particles, or α-particles (helium atoms stripped of their electron), through a thin gold aluminum sheet. Rutherford hypothesized that if the mass of the atom was evenly distributed, then the positively charged α-particles would breeze right through the electron-imbedded plum pudding, deflected or scattered by a few degrees at most.

Much to his surprise, although many α-particles passed straight through the gold foil, some scattered at angles larger than 90 degrees. Not only that, but some α-particles bounced straight back at an 180^o angle, a perfect example of back scattering.

To quote Rutherford, "It was quite the most incredible event that has ever happened to me in my life. It was almost as incredible as if you fired a 15-inch shell at a piece of tissue paper and it came back and hit you. On consideration, I realized that this scattering backward must be the result of a single collision, and when I made calculations I saw that it was impossible to get anything of that order of magnitude unless you took a system in which the greater part of the mass of the atom was concentrated in a minute nucleus. It was then that I had the idea of an atom with a minute massive centre, carrying a charge¹."

Go here for more on Rutherford's experiment, but suffice it to say, the backscattering led Rutherford to conclude that most of the atom's mass was focused at the massive center, which we now call the nucleus. Furthermore, that the center is positively charged. He also realized that most of the atom was empty space.



At this point the most up-to-date atomic model included vacant space around a positively charged nucleus in the center, circled by these negatively-charged "electron rings," as Rutherford called them.

Enter here a physicist called Niels Bohr. Bohr joined Rutherford's lab in 1912. Knowing about quantum mechanics as he did, Bohr decided the atom should be thought of as a mini solar system.

Instead of planets rotating around the Sun held by the gravitational force, negative electrons orbit the nucleus at different distances without losing any energy, held by the electric force² attracting them to a positive center. However, unlike classical mechanics, each atomic level, or orbital, has a fixed energy as determined by quantum mechanics³. The ones cozily living next to the nucleus are more tightly bound than the ones far, far away.

While scientists were aware that the central nuclear charge varied by element, it was Rutherford again who named the smallest positive nucleus a proton in 1920 while studying Hydrogen, the lightest element. "Proton" is Greek for "first."

The structure of the atom changed again with the arrival of James Chadwick to the scene in 1932.

Chadwick noticed that atoms of the same element often have differing properties. This meant something had to be different between these atoms, and that that something could be neither proton, nor electron, because changing protons changes the element altogether and changing electrons just changes the net charge on the atom. He deduced that another type of electrically neutral particle must reside inside the nucleus, which he dubbed the neutron. A given element can then vary depending on its number of neutrons, or on its total number of nucleons. Nucleons are all the particles in the nucleus, both protons and neutrons.



Back then, elements of the same atom that had different properties were called isotopes, and the name stuck. Isotopes of an element each have the same number of protons, the quantity of which determines which element it is, so they have different numbers of neutrons, which alter an element's properties.

On the other hand, isotones are atoms with the same number of neutrons, but a different number of protons. That makes isotones a collection of different elements with the same number of neutrons in common.

Atomic Configuration

We've unescapably encountered the periodic table⁴ before by this point in our lives. No, we're not going to study every single element and property and number on that chart. That's for chemistry class. In modern physics, we only have to know a few things: the mass number, atomic number, and the element symbol.

The standard form for writing an element is . Here X is the element, A is the mass number, and Z is the atomic number. The atomic number Z is your number of protons, and the mass number A is the number of nucleons.

Why do we call A the mass number? That's because it represents the mass of the nucleus in atomic mass units (amu), where one unit of amu, u, is defined as the mass of a neutron (m_n) or proton (m_p). The masses of a proton and neutron are nearly the same. Technically speaking, they have masses of 1.007825 u and 1.008665 u, respectively. Mathematically, 1 u = 1.66 × 10^-27 kg, so m_n ~ m_p = 1 u = 1.66 × 10^-27 kg.

Electrons have masses too, but it's a lot littler than the mass of either an electron or proton. We're talking waaaaay littler. The mass of an electron, m_e, is 5.49 × 10^-4 u. That's thousandths the mass of a proton, or 9.11 × 10^-31 kg. Yeah, electrons barely exist at all.

By using Einstein's equation⁵, E = mc², we can also write the atomic mass unit in the equivalent units of , where . This means a proton's mass is , a neutron's mass is , and an electron's mass is .

The Bohr Model

Let's return to the quantization of energy, a hallmark of quantum physics, introduced to the atomic model by Niels Bohr.

Just prior to Bohr's model, the prevailing model of the atom was Rutherford's model, in which a cloud of negative electrons orbits a small volume of positive charge at the center (the nucleus). However, there was a serious problem with this model. According to classical mechanics, when a charged particle accelerates, it emits energy in the form of electromagnetic radiation. Why, then, don't the orbiting electrons lose energy and accelerate towards the nucleus? How is it possible that atoms remain stable?
As a solution, Bohr proposed a new model, where electrons are required to orbit at a certain discrete set of distances from the center of the atom. Each of these distances, or levels, has a certain energy associated with it, and the energy of an electron can change only if it "jumps" to a different level.

One consequence of this is that the angular momentum L of an electron can only take on a certain set of discrete values, which Bohr determined to be nħ (where n is a positive integer and ħ the Planck constant).
The question we're going to work through is: what exactly are these energy levels? Can we write an expression for the energy of an electron in a given level? And what does that have to do with the price of cheese?

First, The Orbital Radii

First, let's determine what the orbital radii are.
For simplicity, let's say that the electrons orbits are circular and that they orbit at a constant speed around the nucleus. The only force on the electron is the Coulomb force due to the nucleus (we're going to pretend there's just one electron right now). Therefore, by Newton's Second Law,



where:
m is the mass of the electron,
v is the speed of the electron,
r is the distance between the electron and the center of the atom,
Z is the atomic number (number of protons) of the atom,
e is the elementary charge, and
ε₀ is the vacuum permittivity constant (approximately 8.854 × 10^-12 F/m).

Let's write this in terms of the magnitude of the angular momentum L instead of v. We want to quantize the radii and so far, the only thing we know is quantized—and how it is quantized—is angular momentum thanks to Bohr. Since the orbits are circular and the electrons are moving at a constant speed, L = |r × p| = |r × mv|= rmv, or v = L/(rm). The above equation then becomes



Solving for r gives us



Recalling how angular momentum of the electron is quantized (L = nħ), this gives us that the radius of the n^th orbital is



(Here we've added a little subscript "n" to r to emphasize that r is a sequence indexed by n, a positive integer numbering the orbitals.) So, now we know the radius of the n^th orbital in terms of nothing but a bunch of constants: Z, ħ, ε₀, e, and m.

The radius of the lowest orbital (n = 1) of the hydrogen atom (Z = 1) has a special name and symbol: the Bohr radius (insert dramatic music here):



Finally, The Energy Levels

The total energy E of the electron is the sum of the kinetic energy K and the potential energy U.

The kinetic energy K is



where we've again rewritten our expressions in terms of the magnitude of the angular momentum L instead of the speed v.

And we can't forget that according to Coulomb's Law, the potential energy U is



So, the total energy E = K + U is


Plugging in our result for r_n, we get the energy of the n^th level, E_n,



which simplifies to



This result is sometimes abbreviated as



Where is called the Rydberg energy, named after Swedish physicist Johannes Rydberg. For the hydrogen atom (Z = 1), R_E = 13.6 eV [exercise to complete when bored waiting for the bus: confirm this value by using a calculator and by being careful with units]. This is a well-known value, as it is the ground energy state (the lowest energy state) of hydrogen.

The other thing to know is that n = 1 has room for only 2 electrons, n = 2 and n = 3 have room for 8, n = 4 or 5 have room for 18, etc. See the pattern? Glance at a Periodic Table of the Elements and the pattern is clear. The level n is corresponds to the row number.



We like to think of electrons as students at a school dance. Students and electrons usually move in their own circles, doing their thing, but if the DJ plays a song with a lot of energy, then students and electrons pick up that energy and break from their usual circles to occupy an excited state. However, once the song ends and there is no more energy, students and electrons go right back to their usual circles. The only difference between students at a dance and electrons is that electrons give back that energy when they drop from their excited state to their "normal" state.

If an electron absorbs energy, it jumps to a higher orbit; if it releases energy, it drops to a lower orbital. This is a fact, and because energy is quantized, we can tell exactly how much energy.

Let's consider n = 1 and n = 2. We now know an electron orbiting the first level of a Hyrdogen atom has an energy of E₁ = -13.6 eV, we all sleep better for knowing it. We could calculate that E₂ = -3.4 eV in the same way we calculated E₁. The difference of energy of these two orbitals, ΔE, is given by ΔE = E₂ − E₁ = -3.4 − (-13.6) = 10.2 eV. That is how much energy an electron orbiting the n = 2 level would release if it dropped down to n = 1.

Why would the first level have room for that extra electron? Story time.

Let's say a Hydrogen ion floating through the universe, minding its own business, has two orbitals full, n = 1 and n = 2, when a photon crashes into it. This photon of energy E = 15.6 eV collides directly with an electron happily orbiting the nucleus at n = 1. We'd think that the electron would very happily absorb all that energy and break away a free agent, unfortunately, the hydrogen ion doesn't like that. We can't forget that atoms like having that first energy level full, but the electron must do something with that extra energy, so instead of setting the electron in n = 1 free, it forces the electron dwelling at n = 2 to drop to n = 1, which, like we calculated, releases 10.2 eV. The only thing the hydrogen atom sets free is a new photon of energy E = 10.2 eV. This extra energy is radiated away as another photon in the x-ray frequency range.

What happens with the new photon is super cool, though—not what happens in the hydrogen atom, but what happens in our eyeballs. Our eyeballs take photons like the one released by the hydrogen atom and think "color." The colors we see are all about electrons replacing others in the energy levels within atoms. Color is a relative term because the frequency could be anywhere on the electromagnetic spectrum, but we only notice the ones in the visible range.

Each color has its own specific frequency, which we can calculate with , where h is Planck's constant, f is the frequency of the photon, and λ is its wavelength. The photon of energy E = 10.2 eV released by the hydrogen atom will escape with a frequency of = 2.47 × 10¹⁵ Hz. This frequency corresponds to the "color" of light emitted when the electron drops energy levels, and this particular frequency happens to be in the far ultraviolet frequency range, so we wouldn't be able to see it.

Remember that an electron volt is a very convenient unit of measure but it's not a standard unit so the conversion to Joules isn't optional.

Take a look at the following picture. Yeah, it's like the one at the beginning of the section but with the addition of a photon.



A photon knocking an electron out of its shell is a phenomenon similar to the photoelectric effect. Even if the atom in question isn't a metal with light shining on it, a photon of the right energy or frequency may liberate an electron from orbit. Sometimes the process is called stimulated emission, other times spontaneous emission. Whatever it is, including the case of the photoelectric effect, the energy process is the same: the atom becomes an ion by losing an electron, carrying a net positive charge instead of a neutral charge.

If light shining at an oxygen atom knocked out one of its valence electrons, then we could write it with a positive charge. Like this: . Free electrons, because of their negative charge, don't stay alone very long. They too are (electrically) attractive. They join other atoms, usually positive ions because of the whole opposites-attract thing.

Ions can have extra electrons too, though, just to be clear, but those negative ions are not formed removing electrons from their orbitals, but adding more electrons to an atom's orbitals than is required for electric neutrality. The energy required for an ion to be created is called—brace yourself—ionizing energy.

In the case of bigger atoms with a lot more electrons, the electrons occupying the valence shell (the electron shell furthest from the nucleus) are easier targets than the electrons occupying the shell closest to the nucleus. The energy of the incoming photon is important too: a photon with little energy, or little momentum since E = pc,won't change much for an electron that needs more energy to move.

However, more frequently than not, an electron will change energy levels rather than leave the atom altogether. When it changes energy levels, the electron remains in the atom but at a more excited state than it was before, and it could return to its lowest possible energy level by emitting a photon corresponding to the difference in energy as exists between the energy levels.

We'd think electrons would take whatever energy's given to them while they're bound to an atom, but that's not the case. Electrons are picky—with them it's either all or nothing. If a photon with less energy than the electron's binding energy comes along, the electron says, "talk to the hand."

The photon has to offer the exact energy quantity to lift the electron to a new, permitted energy level—they're quantized, remember—or enough energy to ionize the atom. Otherwise, the two don't interact at all. Yep, they're super picky. The higher the energy of the incoming photon, the better the chance the electron has of becoming a free agent.

With how much energy would an n = 1 electron flee a Hydrogen atom with the help of the E = 15.6 eV photon? The answer is the difference between the amount of energy binding the electron to the atom—we call its binding energy, ϕ—and the incoming photon's energy E. The binding energy in the case of Hydrogen for an electron in the n = 1 level is . In this case, E_e = E − ϕ = 15.6 − 13.6 = 2 eV. This 2 eV is all kinetic energy, which as we know is just the energy from having a velocity.

The equation E_e = E − ϕ could actually be written as an inequality, since the electron will never escape unless E ≥ ϕ such that the escaped electron energy is E_e is positive. Notice with this E_e equation, we don't put another negative sign in front of the binding energy. If we did, we'd end up with 29.2 eV, which violates the conservation of energy. We can't have that.

The binding energy of larger atoms refers to the binding energies of the different valence-shell electrons, so we have to specify what n we're talking about to find the given binding energy. Sometimes a photon has the specific energy required to remove an electron from a middle orbital. In that case, the empty spot automatically gets filled by a higher-level electron. It will jump down, emit a photon of energy ΔE, which will cause an entire chain reaction of other higher-level electrons re-arranging themselves to lower levels while emitting photons. This is known as the Auger effect, and yeah, it's super cool.





Above are a few of the energy transitions possible for an electron of a Hydrogen atom. Different elements have different energy levels and therefore different colors emitted, corresponding to E = hf.

The proper term for the state of an electron with increased energy levels is excited state. This refers to any state that's not its lowest possible energy level. When electrons occupy the lowest energy orbitals possible, the atom is in its ground state. Recall that for atoms of Z > 2, both n = 1 levels and n = 2 and even more levels may be filled.

And the ionized atom? The atom captures another free electron soon enough, such is the power of the electric force. For small particles such as electrons and atoms, the electric force dominates over gravity because their masses are far too small to be much affected by the force of gravity. As it captures an electron, that electron will give off a photon of light corresponding to its change in energy level.

Atomic Spectra

We've already met the electromagnetic spectrum in optics⁸. Nice to see you again, sir.

The visible portion of the electromagnetic spectrum, as we recall, corresponds to a rainbow of colors coming from light propagating at different wavelengths⁹. With our new understanding of orbital energy levels, there are two other types of electromagnetic spectra we're now prepared to meet with: absorption and emission spectra. These two spectra refer to the colors (wavelengths) either absorbed by or emitted by atoms corresponding to the energy differences between their energy levels.

The electrons in a cool, dilute gas are lethargic, in that they're stable with electrons at their lowest possible energy levels, but if we shower it with a bunch of energy, what would happen?
If energy is added to the gas, its electrons will become excited, literally and figuratively. The electrons will absorb photons, jump up to other atomic levels, say from n = 2 to n = 3, and reflect all the other photons.

How much energy will the electrons absorb? The energy difference between the two atomic levels or ΔE = E₃ − E₂. This difference in energy determines the color of the light by E = hf, or .

Since some photons will be absorbed, we won't see a full magical rainbow in absorption spectrums. We'll see most of the rainbow, but with dark (empty) lines that correspond to the absorbed photons. The absorbed photons will later be emitted in random directions.

As we might guess and as we see from the image at the top of the page, these photons are relatively few and far between in comparison with the entire electromagnetic spectrum. The emission spectrum for a particular atom is the exact opposite of its absorption spectrum.

Each atom has a unique absorption spectrum. This means we can tell what kind of atoms or element make up a gas by shining light at it and looking at the missing dark lines.

Astronomers use this technique all the time in the study of exoplanets to figure out the composition of their atmospheres, and before that, this technique was used to study the sun. In 1814, a physicist named Joseph von Fraunhofer decided to study the solar spectrum. He noted there were a number of dark lines appearing in sunlight and measured the wavelength of each. Today these are known as the Fraunhofer lines. Each series of lines correlate to an element in the Sun.

An emission spectrum is also produced because of an atom's orbital energies, just in the reverse order. In the case of emission spectra, light passes through a hot gas (just like before) and absorption occurs as electrons jump to a higher level (just like before). Absorption, though, is quickly followed by emission as the electron transitions from a higher energy state to a lower energy state, such as going from n = 3 to n = 2, as this occurs, photons of a specific energy, ΔE, are emitted, these emitted photons are what an emission spectrum records. The emitted photons will correspond to a specific wavelength λ, which is the exact same wavelength of light absorbed in the first place. Once again, all atoms have their own unique characteristic emission spectra.

If you're an element and a physicist is looking for you, there are no chances of ever leading a private life. They can't run, they can't hide.

It's also possible for atoms to undergo transitions that "jump over" a certain orbital, i.e. from n = 3 directly to n = 1, as we saw in the last diagram of the previous subsection. The Fraunhofer lines include these transitions for hydrogen.

Determining the composition of unknown matter with the analysis of emission and absorption spectra is known as spectroscopy.