QII: IV, Bohr’s Model of the Atom and Splits
In this post, we will explore:
- Single Split Experiment
- Double Split Experiment
- Bragg’s Law
- Bohr’s Postulates
- Atomic Energy States
- A Critique of Old Quantum Theory
As a result, we will find evidence supporting the quantum understanding of our world.
Double Split Experiment
Thomas Young, in 1801, showed the wave nature of light. He fired monochromatic light from a slit, to pass through two parallel slits (S1 and S2). On a screen, an interference pattern forms. The bright bands correspond to interference maxima. The dark bands show the interference minima.
In addition to maxima and minima, there is constructive and destructive interference. Constructive interference occurs when sigma is of an integer value and is when there is a maxima.
When sigma is equal to an odd integer multiple of λ / 2, the waves will be destructive interference with a dark fringe.
The intensity for a double split is:
For
Single Split Diffraction
The intensity distribution for a single-split diffraction is as follows:
Bragg’s Law
In Bragg’s law, as two waves hit a lattice, the schematic set up is as such:
And constructive interference occurs when:
Bohr’s Postulates
Bohr outlined four postulates to make a model of atomic structures.
- The circular orbit of an electron has Coulombic attractions between it and a nucleus
- The electron’s orbit is limited by a finite quantity of angular momentum
- The electron does not give off electromagnetic energy while it constantly accelerates; it’s total energy E remains constant
- If an electron changes its motion, it emits Electromagnetic radiation
4.6 Bohr’s Model
To Bohr, he found that the quantization of the orbital angular momentum leads to a quantization of its total energy.
The total energy is of an electron in Bohr’s model is:
For a neutral hydrogen atom, Z = 1, for a singly ionized helium atom, Z = 2, and for a doubly ionized lithium atom Z = 3; this continues along the periodic table.
This equation can be used to create an energy-level diagram for a hydrogen atom.
The binding energy is numerically equal to the energy of the lowest state where n =1.
The energy level diagram corresponds to the different spectral lines.
4.7 Finite Nuclear Mass
In 4.6, we assumed that the mass of the atomic nucleus is much larger than that of the mass of the electron; this makes the nucleus remain fixed in space. It’s a good approximation for Hydrogen but not so for other elements.
Mathematically, we can derive an approximation μ to be included in the Energy equation in the denominator.
4.7 Atomic Energy States
The Bohr model states that the total energy of an electron is quantized. Direct evidence of this came from the Frank and Hertz experiment. In this experiment, emitted electrons are accelerated and collide with a gas in a vacuum.
Electrons are emitted from C, accelerated to reach the anode at A because of the potential V that is applied. Some move past the gaps at A to give off a current at I.
Experimental results demonstrated that as the voltage of the applied potential increased, the current increases; however when V = 4.9 eV, the current drops. This must be because of some interaction between the gas in the chamber and the electrons. At this point of the local maxima, Frank and Hertz found a spectral line, providing evidence for the quantization of energy of atoms.
