SSP Lecture 1: The Elements

**By: Tao Steven Zheng (郑涛)**

## **Atoms**

As we all know, all matter is made of **atoms**. The atoms themselves consist of a **nucleus** surrounded by a **cloud of electrons**. The nucleus, composed of **protons** and **neutrons**, is very small, on the scale of 10-15 meters. Different elements are characterized by their **atomic number**, which indicates the number of protons in the nucleus. A neutral atom has the same number of electrons as there are protons. The electrons exist around the nucleus in an electron cloud whose size is typically of the order of 10-10 meters, or 1 **angstrom** (1 Å). An atom with fewer or more electrons than its atomic number is called an **ion**, and is therefore **electrically charged**.

As of 2022, there are 118 known elements. On the **periodic table**, the elements are grouped according to their chemical properties. For each element, the top left number is the atomic number. These increase monotonically from left to right, and from top to bottom. The symbol of the element is listed under the atomic number with the name usually written underneath. The number below the name of the elements is the atomic mass number, which is measured in **atomic mass units** (amu). The atomic mass unit is defined as the mass of a proton; thus, the atomic mass number is roughly equal to the total number of protons plus neutrons in the nucleus. Electrons are so much lighter than the nucleons, that their contribution to the atom's mass is negligible. The number of nucleons is an integer, so you may be surprised to see a non-integer value listed in most cases. The explanation is that many elements come in several isotopes, each with a different number of neutrons. Some of these are stable, some are unstable to nuclear reactions.

For example, for potassium

K

19

{\displaystyle {\ce {^{19}K}}}

, the mass number is 39.0938. Potassium has 25 known isotopes, of which 3 occur naturally. The most abundant potassium isotope is potassium-39

K

39
19

{\displaystyle {\ce {^{19}\_{39}K}}}

. It is stable and its abundance is roughly 93.3% of the total amount of potassium found in nature. For comparison, the unstable isotope potassium-40

K

40
19

{\displaystyle {\ce {^{19}\_{40}K}}}

 has an abundance of 0.012% and the stable isotope potassium-41

K

41
19

{\displaystyle {\ce {^{19}\_{41}K}}}

is 6.7%.

## **Quantum Mechanics of the Atom**

To understand what are the electrons doing, we need to solve the Schrödinger equation and find what energy levels are available to them. Consider a nucleus with

Z

{\displaystyle Z}

 protons and a single electron. The interaction between the protons and electron is governed by the Coulomb potential

V
(
r
# )

−
k
Z

e

2

r

{\displaystyle V(r)={\frac {-kZe^{2}}{r}}}

. Due to the spherical symmetry of the problem, it is convenient to work with spherical coordinates. Thus, the time-independent Schrödinger equation becomes:

[

−

ℏ

2

2
m

∇

2

−

k
Z

e

2

r

]

Ψ
(
r
,
θ
,
ϕ
# )
E
Ψ
(
r
,
θ
,
ϕ
)

{\displaystyle \left[-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}-{\frac {kZe^{2}}{r}}\right]\Psi (r,\theta ,\phi )=E\Psi (r,\theta ,\phi )}

where

# k
8.988
×

10

9

N
⋅

m

2

⋅

C

−
2

{\displaystyle k=8.988\times 10^{9}\;N\cdot m^{2}\cdot C^{-2}}

.