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**Chapter 2 – Background**

In the 1940s, physicists studied the properties of the oxygen molecule to determine how and why it absorbs electromagnetic waves in the presence of an atmosphere. They also wanted to understand how individual resonance lines are distinguishable at low pressure but disappear as atmospheric pressure increases. In this chapter, the background for the physical properties of oxygen, the equations for the shape of the attenuation curve, and the behavior of oxygen in the presence of changing air pressure are described

** Physical Properties of Oxygen**

H. Van Vleck detailed the physics of atmospheric oxygen absorption at = 5 mm in a paper published in 1947 [30], where he presented the background for why atmospheric oxygen absorbs microwaves and resonates in the vicinity of 60 GHz. Atmospheric oxygen is electrically non-polar. Most electrically non-polar molecules absorb electromagnetic waves when electrons transition between two electronic states, causing resonance to occur in the ultraviolet region. Electrically polar molecules absorb electromagnetic waves when transitions between two electrons occur in a single electronic state. Although oxygen is electrically non-polar, it absorbs electromagnetic waves close to = 5 mm when transitions occur between closely spaced electrons within a single electronic state.

Oxygen is paramagnetic, which means it has a permanent magnetic dipole moment. Generally, only molecules with electric dipoles were thought to absorb electromagnetic waves. Since Maxwell’s equations are symmetric in both the E and H fields, molecules with magnetic dipoles can also absorb waves. Molecules with electric dipoles tend to have a stronger effect on the absorption of electromagnetic waves than their magnetic counterparts. However, with the oxygen molecule, the magnetic absorption has a stronger effect on electromagnetic waves propagating through the atmosphere if the transmitting frequency is near the resonating frequency of 60 GHz.

When an electromagnetic wave collides with an oxygen molecule, electrons transition within a single electron state, which results in the molecule resonating. As the electron jumps around with a single state, the energy of the electron varies. This change in energy results in frequency of absorption and is represented by where *v** _{i,j}* is the frequency of absorption,

*E*

*and*

_{i}*E*

*is the energy of the electron before and after it moved, respectively, and*

_{j}*h*is the Planck constant. These transitions combined with the magnetic dipole moment of oxygen cause the oxygen molecule to absorb electromagnetic waves in the presence of the atmosphere

** Atmospheric Oxygen Absorption**

In the following sections, the background for the equations developed in Liebe’s model is presented. Although these equations are archaic (not in SI units), they provide the reader with an introduction to the equations discussed in Chapter 3

*Atmospheric Oxygen Absorption Formula*

*Atmospheric Oxygen Absorption Formula*

Van Vleck proposed equations for calculating the attenuation due to atmospheric oxygen around 60 GHz based on a standard attenuation equation. He first presented the general absorption formula for the specific attenuation, , in dB/km [30] where *N* is the number of molecules per cc; * _{i,j}* is the matrix element of the dipole moment connecting two stationary states

*i*,

*j*, with energy

*E*

*,*

_{i}*E*

*;*

_{j}*is the frequency corresponding to the resonant line or absorption; is the frequency of the incident radiation;*

_{ij}*k*is Boltzmann constant;

*h*is Planck’s constant; and

*T*is the temperature. The

*10*

^{6}*log*

_{10}*e*term allows to be expressed in decibels per kilometer, where

*e*= 2.718. Van Vleck’s term

*f(*

_{ij}*, )*, the structure factor, governs the shape of the absorption line and is expressed by the following unitless function:

where is the line-breadth constant or the width of the resonating frequency line. The function

*f(*

_{ij}*, )*is the early form of the line shape function discussed in 3.2.2.1 and determines the shape of resonating lines in the attenuation curve.

Equation ( 2 ) is the foundation for the atmospheric oxygen absorption formula ( 5 ). In this absorption formula ( 5 ), the indices

*i*,

*j*in equation ( 2 ) are now represented by a trio of quantum numbers

*J*,

*K*,

*S*, which describe the rotational orientation of the oxygen molecule.

*K*is orbital momentum. For oxygen, the Pauli exclusion principle allows only odd values of

*K*. Pauli’s principle says that no state exists in any quantized system in which two electrons are in the same quantum state [31].

*J*is total angular momentum and is defined by

*J*=

*K*+

*S*

where

*S*is the molecular spin and is also known as the spin quantum number. For oxygen,

*S*= 1, then the three quantum states of

*J*are

*K*-1,

*K*, and

*K*+1, which are separated by intervals of

*K*. The structure of oxygen is classified as a rho-type triplet because the quantum number

*J*has three defined states and the individual states,

*K*-1,

*K*, and

*K*+1, are so small they cause resonance to occur at microwave frequencies.

The attenuation formula for atmospheric oxygen involves the rotational behavior of atmospheric oxygen, which is characterized by the quantum numbers

*J*and

*K*, and is represented by where

*and*

_{K+}

_{K}_{–}represent the center frequency of the spectral line that corresponds to a particular

*K*state

*. The first two terms of the summation in the numerator of represent the resonant absorption of atmospheric oxygen, while the third term represents the non-resonant absorption, which is explained in Section 3.2.2.2. The shape function*

^{1}*f(*

_{ij}*, )*now includes

*and*

_{K+}

_{K}_{–}, which produces the following formulas

The values for

*and*

_{K+}*are defined in a table listed in [30].*

_{K-}The parameter

*F( )*characterizes the line shape function for the nonresonant part of the atmospheric oxygen attenuation. The parameters

_{K+}*,*

^{2}

_{K-}*, and*

^{2}

_{K0}*represent the strength or the intensity of the resonant line. The oxygen molecule carries a magnetic dipole moment with two Bohr magnetons, which are accounted for in the parameters*

^{2}

_{K+}*,*

^{2}

_{K-}*, and*

^{2}

_{K0}*. The Bohr magneton is defined as where*

^{2}*m*is the mass of an electron.

Van Vleck plotted the atmospheric oxygen absorption, which is seen in Figure 2.1. Figure 2.1 displays the theoretical calculation of the atmospheric oxygen attenuation,

*,*equation ( 5 ), where the absorption occurs in dry air with temperature

*T*= 300 K and atmospheric pressure

*P*= 1013 mbar. The atmospheric oxygen absorption is a function of frequency in GHz and the attenuation is plotted in dB/km. Rather than plot Figure 2.1 at various pressures, Van Vleck plots the attenuation for various resonance line widths. The short-dashed curve (curve with the lowest peak) represents the attenuation for = 3 GHz, and the long-dashed curve (the curve with the middle peak) represents = 1.5 GHz, where is the width of each resonance line. The solid curve represents the width of = 0.6 GHz. In this solid curve, the individual resonance lines are visible, whereas in the dashed curves, the width of the resonant line is so broad that the lines overlap each other and merge together

Chapter 1 – Introduction

1.1 Problem Overview

1.2 Problem Significance

1.3 Wireless Applications for Operation in the 60 GHz Band

1.4 Thesis Outline

Chapter 2 – Background

2.1 Physical Properties of Oxygen

2.2 Atmospheric Oxygen Absorption

2.3 Spectral Line Broadening

Chapter 3 – Atmospheric Model

3.1 Characteristics of a Propagating Radio Wave

3.2 Microwave Propagation Model (MPM)

Chapter 4 – Simulation

4.1 Communication System

4.2 ISI in Digital Links

4.3 Effects of the Atmosphere

4.4 Simulation Process

Chapter 5 – Results

5.1 Results with Initial Conditions

5.2 Variable Pressure

5.3 Variable Data Rate

5.4 Bit Error Rate (BER) Plots

Chapter 6 – Conclusion

6.1 Thesis Summary

6.2 Results Summary

6.3 Future Work

Appendix

References

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