Introduction to Atomic Physics

Introduction to Atomic Physics

• Atomic physics scientifically explores the structure of atoms, their energy states, and interactions with particles, electric, and magnetic fields.
• Quantum mechanics has proven to be applicable in atomic physics, establishing itself as a cornerstone of modern physics.
• An atom's configuration is a small nucleus of protons and neutrons surrounded by rapidly moving electrons.
• The atomic diameter is typically around 10^-10 meters, while the nucleus is about 10^-15 meters.
• Protons and neutrons possess approximately the same mass, around 1.00728 and 1.00867 amu, respectively, and are about 1800 times heavier than electrons.
• Neutrons are electrically neutral, while protons have a positive charge of +1.6 x 10^-19 Coulombs.
• The charge of a proton is exactly opposite the negative charge of an electron.
• In a neutral atom, electrons orbiting the nucleus equals to the number of protons in the nucleus.

Atomic Number and Isotopes

• An important equation for atomic components is A = Z + N, where A is the atomic mass number, Z is the atomic number, and N is the neutron number.
• The atomic number (Z) signifies the number of protons in the nucleus, characterizing a chemical element.
• A denotes the atomic mass number; N denotes the number of neutrons within the nucleus.
• Elements are characterized by Z (number of protons); elements sharing the same Z but differing in N and A are termed isotopes.
• A hydrogen atom has A=1 indicating one proton, one electron, and zero neutrons.

Early Atomic Models

• Scientists throughout history have attempted to understand the structure of the atom by proposing various models.
• In 1808, John Dalton proposed that each element consists of identical atoms.
• In 1811, Amedeo Avogadro suggested elements consist of particles with two or more atoms combined.
• Avogadro termed these molecules, estimating molecules in hydrogen or oxygen gas to be pairs of atoms.

Dalton's Atomic Model

• Dalton's atomic model was based on experimental results.
• The model claimed that all matter consists of small particles called atoms.
• Atoms are indestructible and resist changes.
• Atoms cannot be created, destroyed, divided, or transformed into other types.
• The conclusions were based on the law of conservation of mass in the late 1700s.
• Every element is characterized by a unique atomic mass.
• All atoms of a single element have identical shape, size and mass.
• Atoms combine in chemical reactions in small whole-number ratios to form what is known as molecules.
• Atoms of different types can form molecules with simple whole-number ratios like carbon monoxide and carbon dioxide which have differing quantities of carbon and oxygen atoms.

Thomson's Atomic Model

• J.J. Thomson discovered the negatively charged electron in 1897.
• Atoms are large positively charged bodies with negatively charged electrons scattered throughout.
• The model is known as the "plum-pudding" model from 1904.
• Electrons sit in a uniform sphere of positive charge, like blueberries in a muffin.

Rutherford's Experiment and Conclusions

• In 1911, Ernest Rutherford, found the majority of alpha particles passed through a gold foil.
• Some alpha particles deflected at angles, with a fraction bouncing back.
• Rutherford concluded the atom contains a dense, positively charged nucleus at its center.
• The positive charge and most of the mass are concentrated in the nucleus.
• The atom mostly contains empty space with relatively smaller, negatively charged electrons.

Bohr's Atomic Model

• In 1913, Niels Bohr (a student of Rutherford) proposed his shell model of the atom to explain stable electron orbits around the nucleus.
• Classical mechanics and electromagnetic theory made Rutherford's model unstable, because charged particles moving along a curve emit electromagnetic radiation.
• Bohr stated each orbit has a unique energy level, with distance from nucleus determining the force on electrons.
• Energy is absorbed when electrons move to a higher orbit, and released when electrons fall to a lower energy orbit.

Modern Quantum Atomic Model

• In 1926, Erwin Schrödinger employed mathematical equations to describe the probability of locating electrons in specific locations via the Schrödinger equation.
• It defines electron position as a region where location is highly probable.
• In modern models, atoms contain a nucleus of protons and neutrons surrounded by an ill-defined cloud of electrons.
• Representing electrons as a cloud accounts for their probabilistic behavior.
• Quantum mechanics continues to advance atomic theory and subatomic particles called Quarks were found in the 1960s.

Hydrogen Atom Spectroscopy

• Atomic spectra arise from transitions between the energy states (levels) of individual atoms.
• These are analytical tools which determine the composition of matter.
• Studying the sun's radiation spectrum aided in understanding its atomic composition.
• Scientists observed specific emission spectra lines when hydrogen atoms were excited.
• Spectra were observed to be not continuous.
• Understanding quantum mechanics describes how spectral lines arise.
• Lines in the hydrogen spectrum can be classified into series named after their discoverers, which correlate to the energies from transitions between energy levels.
• The wavelengths can be calculated using a simple enough relation involving Rydberg's constant.

Rydberg Constant and Spectral Series

• The mentioned relation accurately predicts the spectral lines.

• The equation for wavelength (and energy via E=hv) involving transition from n1 to n2 state was presented:

• _R_H is Rydberg's constant equating to 10973731.6 m^-1 or 1.097 x 10^7 m^-1.

• Note that n1 < n2 holds true.

• Lyman series (n1=1), Balmer series (n1=2), Paschen series (n1=3) appear.

• E = hv relates to the energy of electromagnetic waves.

• C = vλ relates to the speed of electromagnetic wave which travels at 3 * 10^8 m/s.

Spectrum Types and Angular Momentum

• Continuous spectrum happens when light wavelengths vary continuously, while line spectrum happens when light wavelengths come in separate values

• The angular momentum of an electron with mass m and velocity v going along a orbit with radius r satisfies the following relation:

• Where n is a positive integer and known as the principal quantum number.

Time Dilation

• Special relativity, proposed by Albert Einstein in 1905, introduced a new understanding of space and time.
• Laws of physics are the same in all inertial frames of reference.
• The speed of light (c = 300,000 km/s) is constant for all observers.
• The theory challenges classical Newtonian physics.
• Time dilation is a relativistic effect where time passes slowly for a moving observer.
• The clock on the spaceship runs slow or about 6 years while the Earth observer measures 10 years.
• This becomes significant at speeds approaching that of light.
• GPS satellites use time dilation to correct their clocks, ensuring accurate positioning because atomic clock rates are offset by about 38 microseconds per day.
• Particle accelerators use time dilation to achieve high particle energies.
• v2 Equation: Δt′ = Δt √1 − 𝑐2

Length Contraction

• Length contraction happens as object velocity approaches light speed.
• Length reduces along direction of motion for stationary observer.
• Hendrik Lorentz independently contributed to this phenomenon, called Lorentz contraction. 𝑣2
• L′ = 𝐿 √1 − 𝑐2

Relativistic Mass Increase

• As objects moves closer to light speed, its relativistic mass increases.
• Acceleration becomes more difficult and requires infinite energy to reach the speed of light. m
• m′ = 2 √1− 𝑣2 𝑐

Simultaneity

• Simultaneity is challenged in special relativity; simultaneous events for one observer may not be simultaneous for another in relative motion.
• Ultimately, Einstein concluded that simultaneity is relative.

Equations for Relativistic Energy and Momentum

• New equations for energy and momentum reduce to classical equations at low speeds.
• These equations are essential for understanding particles at relativistic speeds, confirmed by particle physics. m 𝑐2
• Energy: E = 2 √1− 𝑣2 𝑐 m𝑣
• Momentum: p = 2 √1− 𝑣2 𝑐

Mass-Energy Equivalence

• Mass and energy are equivalent is explained by the equation E = mc².
• The equation explains the energy that is released during nuclear reactions and the foundation for nuclear weapons and energy.
• Mass converts into sunlight during nuclear fusion of hydrogen atoms in the sun, which creates solar power.

Spin Orbit Coupling and Fine Structure

• Conventional atomic theory can not account for spectral lines consisting of closely spaced sets.
• Transitions between energy levels which is characterized by the principle quantum number n was not sufficient.
• Besides Bohr and Sommerfield's model, emission spectrum of hydrogen had fine structure, additional lines which could not be explained above.
• Additional relativistic correction in Bohr and Sommerfield's model was also able to explain certain fine structure.
• The theory was unable to account for alkali atoms however, which has a sole electron orbiting outside a spherically symmetric core.
• Uhlenbeck, Goudsmit and Dirac introduced electron spin, originating the idea of spin-orbit coupling, accounting for fine structure.

The Quantum Physics of Spin–Orbit Coupling

• A magnetic dipole can be explained as an example of spin orbit coupling in a magnetic field.
• The energy of said magnetic dipole is represented as E = ½ I w2.
• An electron acts like a magnet due to spin.
• An electron orbiting the nucleus behaves like a magnet.
• The interaction of the spin and orbital motion of the electron will cause it to couple and this is called spin-orbit coupling.
• The orientation between the two magnetic moments causes the orbital energy to be slightly altered.
• The orange color of the sodium lamp, due to Na D line, can be used as a paradigm line with the emission at 17000 cm^-1.
• Spin-orbit interactions satisfy relativistic and quantum mechanics, using the more complex Dirac equation.
• Looking closely at the transitioned lines reveals a separation into two bands.

Fine-Structure Doubling and Angular Momentum

• Fine-structure doubling of spectral lines stems from magnetic interaction between spin and orbital angular momenta, termed spin-orbit coupling.
• Spin-orbit's classical model includes an electron circling nucleus that find itself in a magnetic field.
• The magnetic field applies to and interacts with the electron’s spin causing a kind of internal Zeeman Effect.
• Each electron has orbital angular momentum L and spin angular momentum S that contribute to total angular momentum J.
• Orbital angular momenta Li of electron are coupled into L and this is equivalent to S being coupled into Si J = L + S (1). Quantized angular momenta L, S, and J have magnitude:
• l=√(l+1)ħ, s=√(s+1)ħ, and J=√(j+1)ħ (2). j = l + s= l ± ½ (3), where j is l coupled with s.
• If l = 0, then, j has the single value j=1/2 ((j = – ½ is not allowed as j is non-negative
• Jz, the J component in the z direction, is Jz = mj ћ (4), where mj = (-j), (-j + 1),... , (j – 1), (j) (5).

Examples of Jz Orientation

• When l= 1 the orbital and spin angular-momentum vectors cannot be exactly parallel/antiparallel (showed in Figures)
• Possible Jz orientations for l = 1 (equation 3) j = 3/2 and j=1/2 that correspond to l = 1, with equation (4), mj = (-j), (-j + 1),... , (j – 1), (j).
• For the j=3/2 state, equation (4) and (5) gives mj = −3/2 , −1/2, 1/2 , 3/2

Quantum Numbers and Orbitals

• Four quantum numbers describe electrons (n, l, ml, ms).
• The first three define certain orbital, while the last defines the electron occupation number.
• 'n' the principal quantum number defines electron level and orbital size
• Orbitals sharing 'n' are in the equal shell
• Hydrogen atoms with n=1 electrons are in the base state while having n=2 electrons would indicate excitation. Number of equal value 'n' orbitals is n².
• 'l' (0 to n-1) the secondary quantum number defines shell shape and distinguishes shell. Letters prevent "n and l" confusion
• The energy (n) and the shape (l) specifies direction, and this 'ml equal to -l to +l specifies orbitals.
• Every subs hell with 2l + 1 orbitals is divisible by the number, where s has one, p has three etc as shown.

Spin and Diamagnetism Exclusion

• The spin is defined by ms, as 'spin axis' orientation'. It may measure as -0.5 to 0.5
• A spinning electron forms magnetic field in one out of two ways
• Two electron share an orbital which leads to spin pairing, hence diamagnetism
• Paramagnetic atoms with an unbalanced are weakly magnetic.
• "No two electron can share all 4 key numbers" Pauli states.
• Hence every orbitals has at most 2 electrons spinning and opposite direction.

Hund's Rule and Atomic Orbitals Definition

• In quantum mechanics, atomic orbitals are equations representing location and wave properties of atom electrons. Orbitals are for calculating probabilities of electrons near atom nucleus.
• Atomic orbitals solve the Schrodinger Equation and describe likely locations.
• The Schrödinger Equation defines atomic orbitals.
• The description of time is separate in the Schrodinger Equation.
• In reference to hydrogen-like atoms these equations are: (equations shown)

Essential Points of Atomic Orbitals

• A space has a high chance of containing a pair of electrons.
• Orbitals reveal electron placement in an atom.
• Energy levels/subshells define different shapes where electrons are found.
• Solutions determine Schrodinger's equations.
• Orbitals show boxes linked by levels using small arrows.

Electronic Configuration Rules

• 3 rules organize the e configuration.
• The Aufbau rule says the electrons fill the lowest energy
• Pauli Exclusion says the orbitals contain at most two electrons and that they are anti-aligned
• Hund's rule says that sublevels contain a single electron in each orbital till occupied, anti-aligned to increase spin.
• You first full the second then first main number
• Filling follows "s" or the sublevel marking
• Li or Lithium and Be or Beryllium are examples