Atoms and the Periodic Table

The idea of the atom as being the irreducible constituent of matter is central to our current understanding of nature. According to Feynman, if there was one sentence that he could communicate to a future civilization that has lost all scientific knowledge, it would be the atomic hypothesis, namely that `` all things are made of atoms - little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another''. This single sentence contains the key to regaining almost all of the current theories of science.

Brief History of Atom

The ancients have conjectured that matter is made out of atoms for many millennia. The word ``atom'' comes from the Greek word ``uncuttable'' coined by the Greek philosopher Democritus around 500 B.C. For the ancients, the atom was only an idea without any material entity that could be identified as an atom, and without any specific properties that could be empirical tested. For this reason, the Greek idea of the atom cannot be taken to be a scientific hypothesis. In fact, comparing the statement of Feynman regarding the atomic hypothesis with the Greek idea, we clearly see that the formulation given by Feynman refers to material entities, and the properties of attraction and repulsion are empirically verifiable properties. Hence, we can take the formulation of the atomic hypothesis by Feynman to be a scientific hypothesis. In contrast to Democritus, other Greek philosophers such as Anaxagorus felt that matter consist of a layered structure, and that one could never reach the final constituents of matter. Buddhist philosophers, applying dialectical reasoning, came to the conclusion that all material entities are composite, since any material entity can always be resolved into dialectical opposites. They therefore concluded that matter must be infinitely divisible, and cannot have any ultimate constituents. Before one dismisses Anaxagorus and the Buddhists, one should note an irony of history that, although the twentieth century saw the vindication of the idea of the atom, nature seems more like an onion, with one layer after another layer of substructure. An even more radical challenge to the atomic hypothesis is posed by string theory in which the ultimate constituents of nature are not point like objects that we associate with an atom, but rather extended objects such as strings, sheets, and from three to nine dimensional structures, that are thought to be the smallest constituents of matter. If matter is made out of continuous objects like a string, then we are led to the view that matter is infinitely divisible, since this is the property of any continuous body. From the string point of view, the atom is just one of the many layers in the organization of nature, and is only an effective description of matter. In the latter half of the eighteenth century, chemists returned to the ancient idea of the atom. In studying how to combine different materials, chemists found that there were two distinct forms of combination. The first case was of a mixture such that any amount of the two material could be combined, such is the case in combining salt with water. The second the case was of a chemical reaction, where fixed proportions of two materials had to be combined. For example, in dissolving copper in sulphuric acid, an exact proportion of both material are needed, and an excessive supply of, say, copper results in part of the copper being left undissolved. Chemists then went on to conclude, by the early 1800's, that there must be elementary constituents that were being combined in chemical reactions, and finally came up with the idea of irreducible elements, what we now know as atoms, and went on to conclude that compounds - the present day term for which is molecules - are made from the elements. Physicists in the nineteenth century took up the idea of the atom resulting from the study of heat and thermodynamics. They studied how a gas behaves under different pressure and temperature. The kinetic theory of gases was based on the idea of a gas being composed out of large collection of atoms, and quantities such as pressure and temperature were understood to result from the properties of the atoms which composed the gas. The field of statistical mechanics explained the phenomenon of heat and entropy based on the idea that atoms compose all material entities. By the end of the nineteenth century, the following questions were left unanswered: what is the size of an atom? how much does it weigh? and most importantly, what are the laws that describe atomic phenomenon? As we have already discussed briefly, the first great success of quantum mechanics was to provide a complete theory to explain the existence, as well as the properties, of atoms. In this chapter, we will complete the discussion started with the Bohr atom, and show how the entire Periodic Table of elements can be understood from the principles of quantum mechanics.

Atoms: Nuclei and Electrons

A complete theoretical explanation for the existence of atoms comes from quantum theory. Most of our ideas about atoms in general, including the most complex ones, are based on our understanding of the hydrogen atom (hydrogen atom), which is composed of a bound state of a single electron with a proton. To understand the ingredients needed for a complete quantum mechanical understanding of the atom, we revisit the Bohr atom to understand its correct results as well as its shortcomings. The key idea of Bohr, validated by later developments, is the existence of pure numbers, called quantum numbers, that describe a quantum state, and in particular, the atom. The Schr $\ddot{\mathrm{o}}$dinger's equation gives the following description of the hydrogen atom, and which has been verified to a very high degree of accuracy by numerous experiments. An atom is an electrically neutral object, and is composed of a heavy, positively charged nucleus, surrounded by an approximately equal number of negatively charged electrons. Every proton has a charge equal in magnitude, but of opposite sign, to that of an electron. The nucleus is made out of an equal number of protons and neutrons. The nucleus has a size of about $10^{-15}m$ and is almost a million times smaller than the size of the atom, which is about $10{-9}m$. The size of the electron is around $10{-13}m$. Hence, in studying the atom, both the nucleus and the electron can be considered to be point-like, and for example, the Coulomb potential between the nucleus and the electrons is written as if they were pointlike particles. Protons and neutrons have a mass of about $1 GeV/c^2$, and an electron has a mass about 2,000 times smaller, that is, about $0.5 MeV/c^2$. (Recall $1eV$ is the energy gained by an electron in moving across $1$ volt.) Since the nucleus of an atom is about 4,000 times heavier than the electrons which surround it, all the internal motion of an atom consists of electrons moving around a nucleus that can be taken to be stationary. For motion of the entire atom, all of its mass can be taken to come from the nucleus. To excite the electrons of an atom, it is sufficient to impart to it energy of the range of 1-100 eV. On the other hand, to excite the protons and neutrons of the nucleus needs energy of the range of 5-100 MeV. Most phenomena that we encounter in chemical,biological, thermal and macroscopically observed processes involve energies of the range of a only few eV's, the properties of an atom in this range of phenomena results solely from the behaviour of its electrons. Consequently, we conclude that, given the large mass of the nucleus, and its high excitation energy, the sole function of the nucleus in an atom is to provide the atom with a pointlike mass and positive charge, but is otherwise taken to be inert. The only entity which is active in ordinary circumstances are the electrons of the atoms, and by the state of the atom is meant the state of the electrons in the atom.

Atoms and Quantum Numbers

An atom is described by its wave function $\Psi$. For the case of the hydrogen atom, the nucleus is considered to be at rest, with the electron being bound to it by the electrostatic Coulomb potential. The wave function $\Psi$ determines the probability of the electron being found a various distances from the nucleus.

The electron in the atom has angular momentum, and this is reflected in the wave function of the electron. Although one cannot think that the electron is "revolving" around the nucleus as a classical particle, a sense of angular momentum is present in quantum mechanics. The electron in the atom has an average a velocity, and hence, loosely speaking, in a bound state is moving around the nucleus; hence it can have angular momentum. One should not take this picture too literally, since the electron can have zero angular momentum in spite of ``revolving'' around the nucleus, something that is not allowed by classical mechanics. The higher the angular momentum of the electron, the more likely that it is far from the nucleus. Furthermore, angular momentum, just like energy and linear momentum, is absolutely conserved in quantum mechanics. An electron in an atom is fully described by four quantum numbers. They are the following.

1. The principal quantum number $n$ that expresses the quantization of energy, namely $E_n$ with $n=1,2,...\infty$.
2. The orbital quantum number $l$ expresses the quantized values of the total angular momentum of the electron in an atom, and has values $L=\hbar\sqrt{l(l+1)}$, with $l=0,1,2,....n-1$. The quantum number $l$ is said to be angular momentum of the electron.
3. Given that the angular momentum of an electron is a vector quantity, quantum mechanics requires that the angular momentum vector point in only discrete directions. Choosing an arbitrary direction, say, the $z$-axis, quantum mechanics then dictates that the $z-$component of angular momentum, namely $L_z$ take only certain discrete values, namely $L_z=m\hbar$, where the integer $m$ is the magnetic quantum number and is given by $m=-l,-l+1,...., l-1,l$
4. And finally, the theory of relativity requires that all electrons have an intrinsic angular momentum, called spin, and denoted by vector ${\bf S}$ such that its $z$-component $S_z=s\hbar$, where $s$ is the spin quantum number and can take only two (2) values (along the $z$-axis), namely -1/2 and +1/2.

In sum, the wave function of the electron in an atom is described four quantum numbers, and is symbolically represented by $\Psi_{nlms}$. We briefly discuss below the physical significance of these quantum numbers.

Principal Quantum Number and the Bohr Atom

Recall in our discussion on Bohr's model for the atom in Section 11.4, there were two key formulae that underpin the Bohr atom, namely, that the magnitude of the angular momentum of the atom is quantized, and which in turn leads to the quantization of the energy of the electron in the atom. Recall that we have, from eqs.(11.13) and (11.14) , that the energy of the hydrogen atom is given by

$$E_n = -\frac{E_{\mathrm{Rydberg}}}{n^2}=-\frac{13.6}{n^2}eV~;~n=1,2,..\infty$$ (12.1)

$$E_{\mathrm{Rydberg}} = \frac{mk^2e^4}{2\hbar^2}=13.6 eV$$ (12.2)

The most important success of the Bohr atom is the correct prediction of the energy levels of the hydrogen atom. In fact, according to some authors, the most important equation in atomic physics is the Bohr formula for the energy levels of the hydrogen atoms. The principal quantum number n expresses the discrete and quantum nature of the atom's energy.

Figure 12.1: Energy Levels of the Hydrogen Atom

According to the Bohr atom, the lowest energy of the atom, called the ground state, has $n=1$. The higher the energy of the atom, the larger is the value of $n$, and the the further away, on the average, is the electron from the nucleus. In the limit that $n\rightarrow \infty$, the (binding) energy of the electron in the atoms becomes zero, and the electron becomes free from the nucleus. The energy levels of the hydrogen atom are given in Figure 12.1. The process of the electron gaining enough energy to escape from its bound state (in an atom) is termed as ionization. For example, the ionization of an electron in the ground state of the hydrogen atom is $13.6eV$. The different values of $n$ can be represented by concentric shells around the nucleus, with shells with larger values $n$ having a larger radius. This shell structure of the hydrogen atom is shown in Figure 12.2.

Figure 12.2: The Shell Structure of the Hydrogen Atom

Recall from eq.(11.1) that the energy of a photon with wavelength $\lambda$ is given by

$$E=N\frac{hc}{\lambda}   ;   N=1,2,...\infty$$ (12.3)

If an atom is subjected to radiation, it can absorb energy from photons. Suppose the hydrogen atom is in an excited state denoted by $n$ and with energy $E_n$, and that it absorbs one quantum of photon by making a quantum transition to a higher energy state with energy given by $E_k$. From energy conservation, we must have

$$\mbox{\rm {Energy of absorbed photon}}=\mbox{\rm {Increase in the energy of the atom}}$$ (12.4)

and which yields

$$\frac{hc}{\lambda} = E_k-E_n=E_{\mathrm{Rydberg}}(\frac{1}{n^2}-\frac{1}{k^2})$$ (12.5)

$$\Rightarrow \frac{1}{\lambda} = R(\frac{1}{n^2}-\frac{1}{k^2})  ;  n,k=1,2,...\infty$$ (12.6)

$$\mathrm{where}$$

$$R = \frac{E_{\mathrm{Rydberg}}}{hc}=1.097\times10^7m^{-1} :\mbox{\rm {Rydberg's constant}}$$ (12.7)

In other words, only certain wavelengths of light, given by above formula can be absorbed (and re-emitted) by the hydrogen atom. Other wavelengths of light will not be absorbed since this would entail a violation of energy conservation. The spectrum of the hydrogen atom is given by the wavelengths of light that are absorbed by it, and is illustrated in Figure 12.3, with the inset being the spectral lines.

Figure 12.3: The Absorption Spectrum of the Hydrogen Atom

To experimentally determine which frequencies an atom absorbs, we shine on it light containing a large range of wavelengths. The electron in the atom absorbs only those wavelengths of light allowed by eq.(12.8), and in doing so makes a quantum transition to a higher energy level. The higher energy state is unstable, and electron makes a transition to a lower energy state by re-radiating the absorbed photon, which is then detected. Since only a discrete set of wavelengths are absorbed, the radiation emitted by the atom has high intensity for only a few select wavelengths, and these are called the spectral lines for the atom in question. Transitions from highly excited states to the ground state $n=1$ are known as the Lyman series of spectral lines, and to $n=2$ are called the Balmer series and so on. A typical wavelength of the Lyman series has wavelength of $\lambda\simeq 10^{-7}m$, which is in UV range, whereas for the Balmer series has wavelength of $\lambda\simeq 4\times 10^{-7}m$ which is in the violet range of the electromagnetic radiation. The energy levels of the hydrogen atom shown in Figure 11.1 have been reconstructed from the study of the spectral lines of the hydrogen atom, and constitute the biggest achievement of the Bohr atom. The energy levels of the hydrogen atom responsible for the spectral lines of the hydrogen atoms are shown in Figure 12.4.

Figure 12.4: Spectral Lines of the Hydrogen Atom

Quantization of Angular Momentum

There a number of problems with the Bohr atom. Recall from eq.(11.10), that according to Bohr the angular momentum of the hydrogen atom is given by

$$L_{Bohr}=n\hbar   ;   n=1,2,....\infty    \mathrm{incorrect}$$ (12.8)

The ground state of the hydrogen atom, given by $n=1$, is experimentally determined, and has zero angular momentum, contradicting the prediction of Bohr which requires it have a non-zero value equal to $\hbar$. In fact, Bohr could not guess that angular momentum needed a new quantum number, and had incorrectly identified the quantum number for angular momentum with $n$. The correct result is obtained by solving Schr $\ddot{\mathrm{o}}$dinger's equation, and one obtains that the magnitude of total angular momentum for the hydrogen atom, for a state with energy $E_n$, is given by

$$L=\hbar\sqrt{l(l+1)}   ;  l=0,1,2,....n-1.$$ (12.9)

Quantum number $l$ denotes the orbital quantum number of the electron, and reflects the angular momentum carried by the electron. From the above formula, we immediately see that for the ground state of the hydrogen atom, given by $n=1$, has zero total angular momentum.

Magnetic Quantum Number

Recall angular momentum of the atom is a vector given by ${\bf L}=(L_x,L_y,L_z)$. The Heisenberg Uncertainty Principle implies that no experiment can fully determine the value of all three components of angular momentum L; all that experiments can determine is the exact value of only one of them, say, $L_z$. Since a component cannot exceed the total magnitude of angular momentum, we have

$$L_z=m\hbar   ;  m=-l,-l+1,.....l-1,l$$ (12.10)

and is shown in Figure 12.5 . $m$ is called the magnetic quantum number as this is the component of angular momentum that couples to an external magnetic field which can be taken to lie along the z-direction.

Figure 12.5: Magnetic Quantum Number

Spin

Figure 12.6: Spin up and down

The electron has intrinsic angular momentum called spin ${\bf S}$, which is a vector, and has a fixed value. Note ${\bf S}$ can point in only two directions with values for $S_z=\pm\frac{1}{2}\hbar=s\hbar$. Due to the fact that the projection of spin ${\bf S}$ can take only two values, we can conclude from eq.(12.12) that the spin has an angular momentum of $s=\frac{1}{2}\hbar$, where $s$ is called the spin quantum number of the electron. Hence the electron is said to be a fundamental particle with spin $1/2$. From the general expression for angular momentum given in eq.(12.11), the magnitude of spin angular momentum is given by

$$S = \hbar\sqrt{\frac{1}{2}(\frac{1}{2}+1)}$$ (12.11)

$$= \frac{\sqrt{3}}{2}\hbar$$ (12.12)

Spin originates in the theory of special relativity. The electron has intrinsic spin, and like its charge and mass, is an inherent property of the electron. All classical angular momentum is based on the idea of a particle rotating about some axis; although electron spin does have a sense of an axis as expressed by $S_z$, in no sense can one think of the electron as a classical spinning particle. The electron itself is pointlike, and hence it cannot have any classical angular momentum as this would need the electron to rotate about the point that it is occupying, which is clearly not possible. Furthermore, unlike all classical magnetic fields which are the result of the movement of charge, due to its spin the electron carries a magnetic field even when it is not in motion, and hence is an example of a quantum mechanical source of magnetic fields without the motion of any charge.

Degeneracy of States

Note that we have discussed the possible states that an electron can have in a hydrogen atom, and found that to describe its wave function $\Psi_{nlms}$ we need four quantum numbers. In discussing quantum numbers, for brevity, one dispenses with writing the full wave function $\Psi_{nlms}$(which contains the electron's probability distribution in space), and instead one denotes the state simply by its quantum numbers, namely $n,l,m,s$. Note that, once we fix $n$, the energy of all the states with the allowed values of $,m$ and $s$ is determined only by $n$, and in fact is given by the Bohr formula

$$E_{nlms}=-\frac{E_{\mathrm{Rydberg}}}{n^2}$$ (12.13)

In other words, the energy does not depend on the the quantum numbers $l,m$ and $s$. A situation such that many different quantum states, specified by the various values of the quantum numbers $l,m,s$, have the same energy, is termed to be degenerate, and the degeneracy is given by the number of states that have the same energy. We count the number of states of the hydrogen atom that have the same energy by fixing $n$, and then calculating how many different quantum numbers correspond to a given $n$.

1. Quantum number $n$=1,2,3.....$\infty$.
2. The quantum number $l$ takes values $0,1,2...n-1$.
3. The quantum number $m$ takes $2l+1$ value for a given $l$.
4. The spin quantum number $s$ has two values, regardless of the value of $n$.

Hence, the degeneracy of energy $E_n$ of the hydrogen atom is given by the total number of states that have energy $E_n$. Since, for a fixed $n$, all the states with different values of $l,m$ and $s$ have the same energy, by a simple counting we have the degeneracy given by by

$$\sum_{l=0}^n 2\times(2l+1)=2n^2$$ (12.14)

In other words, for every allowed energy $E_n$, or what is the same thing, for every $n$, there are $2n^2$ states that have the same energy. In the shell picture of the hydrogen atom, we now see that every shell has a degeneracy of $2n^2$. This fact, in essence, contains the logical kernel for explaining the entire periodic table of elements, and has enormous importance in the study of chemistry and biology. We summarize the quantum numbers of an electron in the hydrogen atom in Table 12.1.

Table 12.1: Quantum Numbers of the Electron in the Hydrogen Atom

Quantum Numbers	Range
Principal Quantum Number	$n=1,2,3...\infty$
Orbital Quantum Number	l=0,1,2,...n-1
Magnetic Quantum Number	m=-l,-l+1,.-1,0,1,..l-1,l
Spin Quantum Number	s=-1/2,+1/2
Total Degeneracy of Energy	2$n^2$

Exclusion Principle

The last ingredient we need for understanding the structure and stability of atoms was formulated by Wolfgang Pauli, in 1925, and is called the exclusion principle.

The exclusion principle states that, in general, no two electrons can occupy exactly the same quantum state.

For the atom, the exclusion principle implies that if an atom has more than two electrons, then no two electron can have the wave functions $\Psi_{nlms}$ with exactly the same set of quantum numbers $n,l,m,s$.

Example. Helium atom consists of a nucleus with two protons and two neutrons, and two electrons bound to the nucleus. Suppose the state of one of the electrons is in the lowest possible energy state given by $0,0,0,+1/2$; then the second electron, due to the exclusion principle, cannot be in the same state, and the best it can do to occupy the lowest energy state available to be in the state $0,0,0,-1/2$. This configuration of Helium's electrons is shown pictorially in Figure 12.7; the nucleus is shown much larger than its actual size for indicating the number of protons and neutrons.

Figure 12.7: Helium Atom and Exclusion Principle

Note to the first box ($n=1$) in Figure 12.10 corresponds to Helium atom's electrons, with the electron's spin pointing in opposite directions as required by the exclusion principle.

The Pauli exclusion principle was made on an ad hoc basis, and it was only the later development of quantum field theory that led to a more complete understanding. In essence, the structure of spacetime, as dictated by the special theory of relativity, allows only two kinds of fundamental particles, called fermions and bosons.

1. Fermions, which can have either spin $1/2$, or at most spin $3/2$, and obey the exclusion principle, in that no two fermions can occupy the same state. The most familiar fermion is the electron, followed by the proton, neutron and so on.
2. Bosons have an integer spin of either $0,1$ or at most $2$, and do not obey the exclusion principle. In contrast, any number of bosons can all occupy the same state. In particular, the equilibrium state for a collection of bosons is for all them to occupy the lowest energy state, and leads to a phenomenon known as boson condensation. The most familiar boson is the photon, which has spin $1$.

Clearly, if the electron was a boson there would be no atoms as we know since all the electrons would be in the lowest energy state corresponding to $n=1$. The exclusion principle is the reason for the elaborate structure of atoms, is the basis of all of chemistry, and provides us with a principle for organizing the multielectron system bound to the nucleus of an atom .

The Periodic Table

Recall an atom, in general, consists of $Z$ number of protons with net positive charge of $+Z$ and an approximately equal number of electrically neutral neutrons. (The number of neutrons are larger than the number of protons for many nuclei, and isotopes denote atoms whose nuclei does not have an equal number of protons and neutrons.) The nucleus is surrounded by $Z$ number of electrons with a total electrical charge of $-Z$, resulting in an electrically neutral atoms. An atom is determined by the number of protons in the nucleus. The number of stable atoms found in nature is about 90, and it is estimated that the maximum number of atoms that can exist is 126. If one studies the studies the physical and chemical properties of the various atoms, a pattern is seen to emerge, wherein the physical and chemical properties of the atoms seem to approximately repeat themselves. The word ``periodic'' expresses this repetitive (periodic) structure present in atoms.

Figure 12.8: The Atomic Number Z Plotted Against the Melting Point of Atomic Material

All macroscopic collection of atoms at room temperature are either solid or gaseous, with the exception of mercury and bromine that exist in the liquid state. All bulk material composed out of a particular atom become solids at low enough temperature; in Figure 12.8 the atomic number $Z$ of the atom is plotted against its melting temperature. We can clearly see that the temperature approximately repeats itself, and heavy vertical lines have been drawn to mark this periodic behaviour. There are numerous other properties, such as ionization energy, density of atoms of a particular kind, and so, also show a periodic pattern. This pattern was recognized, in 1869, by Russian chemist Dmitri Medeleev, who proposed the Periodic Table. At that time, many of the elements of the Periodic Table had not yet been discovered, and Mendeleev made many (successful) predictions of what the missing elements should be, including properties such as atomic numbers and so on. The predictive power of the Periodic Table is a canonical exemplar of the scientific method, and illustrates the unique feature of the leading theories of science. The periodic table of elements is given in Figure12.9. The periodicity of the table is in the horizontal direction. The first problem that needs to be addressed is why do atoms appear to have a periodic structure? If one goes down vertically along any column, the atoms have similar properties. This is the other outstanding feature of the periodic table of elements that needs to be explained.

Figure 12.9: The Periodic Table

The explanation offered by quantum mechanics for the structure of atoms is the following. A crude first approximation to the multielectron system of an atom is to ignore the interactions amongst the electrons. We assume(incorreclty) that the multielectron system is such that the electrons interact only with the nucleus, and not with each other. With this approximation we have a hydrogen-like situation, with the electrons being described by wave functions $\Psi_{nlms}$ of an electron in a hydrogen atom. Note the important fact that, even for complex atoms with a high value of $Z$, the electrons are completely described by the four quantum numbers $n,l,m,s$; however, the simple degeneracy of the hydrogen atom no longer holds for more complex atoms. In the approximation that we are considering, the energies of the electrons are given by Bohr's expression for energy, with the only difference being that the charge of the proton in the hydrogen atom is replaced by $Z$, the charge of the nucleus in question. Hence, from eqs.(12.3) and (12.4) , the energy of the atom with nucleus having charge $Z$ is given by replacing the charge of the proton in the hydrogen atom by charge $Z$, and yields

$$E_n = -\frac{E_{Z}}{n^2}=-13.6\frac{Z^2}{n^2}eV ; n=1,2,....\infty$$ (12.15)

$$E_Z = Z^2E_{\mathrm{Rydberg}}=\frac{mk^2Z^2e^4}{2\hbar^2}=13.6Z^2 eV$$ (12.16)

Since electrons are fermions, the Pauli exclusion principle dictates that they cannot all occupy the lowest energy state of the atom. Hence, the electrons will arrange themselves into successive shells, labeled by the principal quantum number $n$. Each shell can accommodate $2n^2$ number of electrons, as these are the number of states available in each shell. One can think of the electronic configuration of an atom as being similar to a circular amphitheater. The concentric energy shells correspond to successive circles that have increasing number of seats, and with the nucleus being the stage. As one goes to atoms with larger and larger number of electrons, the electrons arrange themselves in a manner so as to occupy the seats in the shells closest to the stage.

Noble Gases

When one shell is completed for some $n$, all the electronic states for that shell are occupied. This in particular means that we have electrons with all values of quantum number $m$ with values $-l,-l+1,...,l-1,l$; hence the net projection of the value of $L_z$ should be zero. A state with $L_z=0$ is spherically symmetric; this means that other electrons will not be attracted to this atom, and furthermore, this atom will not easily lose its electrons. Atoms with a completely filled shell are the noble gases (He, Ne, Ar, Kr, Xe and Rn) and signal the end of one of the periods of the periodic table. An example of the arrangement of electrons for the noble gas Neon is given in Figure 12.10, with its 10 electrons distributed in two energy shells given by $n=1$ and $n=2$. The $n=2$ shell is completely filled up, as is expected for a noble gas.

Figure 12.10: Arrangement of 10 Electrons in Two Shells for Neon

To sum up, the periods in the periodic table are the result of the complete filling up of one of the shells of energy. Since there are $2n^2$ number of states available in the $n$-th shell, we should expect the periodic table to have the following periodicity as shown in Table 12.2.

Table 12.2: Expected and Observed Periodicity in the Periodic Table

n th Energy Shell	1	2	3	4	5	...
Number of atoms with period $=2n^2$	2	8	18	32	50	72...
Observed number of atoms with period	2	8	8	18	18	36..

We see from Table 12.2 that there are significant deviations of the observed structure of the Periodic Table from the prediction of the simple minded approximation that we have considered. To get the actual structure of the periodic table by solving the Schr $\ddot{\mathrm{o}}$dinger's equation is even today an unsolved problem, since it involves solving an intractable multielectron system. However, without any detailed computation, we can qualitatively see where the deviations from the simple picture based on the hydrogen atom starts to take place. In the simple approximations that we considered above, we completely neglected the interactions amongst the electrons. The electrons repel each other due to the Coulomb potential. This leads to a removal of the degeneracy of states that holds for the hydrogen atom, and, in general, the electrons in the atom depend not only on the value of the quantum number $n$, but start to depend on quantum numbers $l,m,s$ as well. Hence, instead of completely filling up one shell before going on to the next, it may so happen that the high angular momentum states for some value of $n$, have a higher in energy than a low angular momentum state belonging to the next $(n+1)$-th shell. This is in fact precisely what happens by the time one has reached the $n=3$ shell. Neon $(Z=10)$ is the noble gas that is the last element of the $n=2$ shell, and, as expected, is inert and unreactive. After Neon, we begin to populate the $n=3$ shell with the first two atoms, namely, sodium and magnesium, and then followed by six more elements (aluminum through argon). This accounts for the 8 atoms in the $n=3$ shell. Following argon, we ``should'' be able to put in 10 more elements according to the simple hydrogen-like counting that we are doing. However, at this point, due to the interactions of the electrons, and the fact that the electrons in the filled shells start to ``shield'' - and effectively reduce - the charge of the nucleus felt by the outer electrons. For this reason, both potassium $(Z=19)$ and calcium $(Z=20)$ choose to occupy the $n=4,l=0$ state as it has a lower energy than the $n=3,l=2$ states of the previous shell. One can imagine how intricate the actual structure of the atoms become as one goes to larger and larger $Z$ atoms with more and more electrons. The chemical and physical properties of atoms are primarily determined by the number of electrons in their outer most shell, since these are the electrons that are free to interact with forces external to the atom. Hence, we expect atoms with the same number of electrons in the outer shell to have similar physical and chemical properties. This is the explanation of why the elements in the vertical column of the periodic table have similar properties.

Alkali Metals and Halogens

Atoms with different atomic number $Z$, but with only one electron in the outer shell, comprise the first column of the periodic table, and are called the alkali metals (H, Li, Na, K, Rb, Cs, Fr). All the alkali metals have similar chemical properties, are highly reactive, and are easily ionized. Figure 12.11 shows the electronic configuration of Sodium (Na), with a single electron in the outer most shell, called the valence electron.

Figure 12.11: Sodium as an Alkali Metal

Similarly, if two atoms are say one short of completely filling up their outer most shell, they should be chemically similar. The halogens (F, Cl, Br, I and At), comprising the seventh column of the periodic table, are one electron short in filling up their outer most shell, and all have similar chemical properties. An example of a halogen is Chlorine (Cl) and its electronic configuration is shown in Figure 12.12. As already mentioned, if an atom has a completely filled outer shell, it will be inert belong to the noble gases.

Figure 12.12: Chlorine as a Halogen

In this way, all the elements of the periodic table can be accommodated in the shell structure of atoms. The periodic structure of the elements, as well as the similarity in their chemical and physical properties, can be understood, both qualitatively and quantitatively, from the principles of quantum mechanics.

Spectral Lines of an Atom

To empirically verify the energy levels and angular momentum states occupied by the electrons of an arbitrary atom, the spectral lines for the atom are determined in a manner similar to the one discussed in the case of the hydrogen atom. The outer electrons of the atom experience a shielded nucleus, since the electrons in the inner shells partly cancel the positive charge of the nucleus. Hence, the ionization and absorption spectrum of the outer electrons is in the range of 1-20 eV and wavelength of $10^{-7}$ respectively, similar to the hydrogen atom. The noble gases have the maximum ionization potential, as is expected. Note that the electrons in the inner shells experience the full unshielded charge of the nucleus. Since the attractive force is much larger in atoms with large $Z$, the binding energy goes as $Z^2$ times the binding energy of the hydrogen atom, as given in eq.(12.17). Hence, for example, the spectral lines of a high $Z$ atom such as molybdenum, will have extra spectral lines due to absorption of radiation by the inner shell electrons ,in addition to the hydrogen like spectral lines due to the outer shell electrons. For example, the Lyman lines for a high $Z$ atom will have wavelength shorter by a factor of $Z^2$. For $Z=50$, the wavelength is approximately $$\frac{10^{-7}}{2500}\simeq10^{-9}m$$, which is the wavelength in the X-ray range.

Figure 12.13: X-ray scattering results for Molybdenum

Molecules and Quarks and Gluons

There are a number of different directions that one can go from the atom. Atoms exert an attractive force on each other, and can form bonds resulting in the formation of bound states of atoms,known as molecules. There are two basic forms of bonds, namely the covalent and ionic bonds. The study of molecules is the subject matter of chemistry. Nature exhibits a bewildering variety of molecules, which can have anything from a few to hundreds of atoms. Macromolecules can have anything from thousands to millions of atoms. Large macromolecules constitute the microscopic basis of life, and are the subject matter of Molecular Biology. A typical protein is a molecule resulting from the bonding of about 10,000 atoms; a DNA molecule, the basis of all of life, can have anywhere from 50,000 to a few million atoms. One can go in another direction, and take large aggregates of atoms, and form bulk matter out of them. They can form exotic materials such as semiconductors, superconductors, plastics, alloys and so on. The study of materials, and of bulk matter in general, is the subject matter of condensed matter physics as well as that of materials science.

The study of what composes the nucleus is the vast subject of nuclear physics, and focuses on understanding the properties of nuclei starting from protons and neutrons. The strong nuclear force binds the protons and neutrons to form the nucleus. The proton and neutron in turn are seen to be composed out of more fundamental particles called quarks. The nuclear force binds quarks to form protons and neutrons. The nuclear force field is called the gluon field. The gluon is force field similar to the electromagnetic field, but far more complex and counterintuitive. For example, the gluon field's attraction is so strong that the quarks are permanently confined inside protons and neutrons, and can never become a free particles such as, for example, the electron. High energy physics studies the structure of quarks and gluons, and has finally led to the idea of strings as the underlying quantum of all of reality. Strings constitutes all of physical reality, including all forms of matter, all force fields as well as the geometry of spacetime.