Einstein Coefficients

Radiative Processes

Einstein Coefficients - Definitions

• Considers two atomic energy levels: a lower level 1 with energy $E_1$ and an upper level 2 with energy $E_2$, where $E_1 < E_2$.
• Absorption: An atom in level 1 absorbs a photon with energy $E = E_2 - E_1$, transitioning to level 2.
• The absorption rate, $R_{12}$, is given by $B_{12} n_1 \rho_{\nu}$, where $n_1$ is the number density of atoms in level 1, $\rho_{\nu}$ is the radiation field's energy density, and $B_{12}$ is the Einstein coefficient for absorption.
• Spontaneous Emission: An atom in level 2 emits a photon with energy $E = E_2 - E_1$, transitioning to level 1.
• The rate of this emission, $R_{21}$, is given by $A_{21} n_2$, where $n_2$ is the number density of atoms in level 2, and $A_{21}$ is the Einstein coefficient for spontaneous emission.
• Stimulated Emission: An atom in level 2, stimulated by a photon with energy $E = E_2 - E_1$, emits another photon and transitions to level 1.
• Rate of stimulated emission is $R_{21} = B_{21} n_2 \rho_{\nu}$, where $B_{21}$ is the Einstein coefficient for stimulated emission.

Einstein Coefficients - Relations

• In thermal equilibrium, upward and downward transition rates are equal: $n_1 B_{12} \rho_{\nu} = n_2 A_{21} + n_2 B_{21} \rho_{\nu}$.
• Solving for $\rho_{\nu}$ gives $\rho_{\nu} = \frac{A_{21}}{B_{12} \frac{n_1}{n_2} - B_{21}}$.
• In local thermodynamic equilibrium (LTE), energy levels are populated according to the Boltzmann distribution: $\frac{n_1}{n_2} = \frac{g_1}{g_2} e^{\frac{E_2 - E_1}{kT}} = \frac{g_1}{g_2} e^{\frac{h\nu}{kT}}$, where $g_1$ and $g_2$ are statistical weights, $h$ is Planck's constant, $k$ is Boltzmann's constant, and $T$ is temperature.
• Substituting into the equation for $\rho_{\nu}$: $\rho_{\nu} = \frac{A_{21}}{B_{12} \frac{g_1}{g_2} e^{\frac{h\nu}{kT}} - B_{21}}$.
• Comparing with the Planck function for blackbody radiation, $B_{\nu}(T) = \frac{2h\nu^3}{c^2} \frac{1}{e^{\frac{h\nu}{kT}} - 1}$.
• The relationships between Einstein coefficients are $B_{12} = \frac{g_2}{g_1} B_{21}$ and $A_{21} = \frac{2h\nu^3}{c^2} B_{21}$.

Economía

Definition

• Economics is the study of how societies use scarce resources to produce valuable goods/services and distribute them among individuals.

Core Concepts

• Goods are scarce
• Society must use resources efficiently
• Economics studies how people make decisions with scarcity and how these decisions affect society.

Branches of Economics

• Microeconomics: Studies behavior of individual units like markets, firms, and households.
• Macroeconomics: Analyzes the economy as a whole, including inflation, unemployment, and economic growth.
• Positive Economics: Deals with how the world actually works, based on analysis and empirical evidence (e.g., "raising the minimum wage reduces employment").
• Normative Economics: Involves value judgments about how the world should be (e.g., "the government should increase the minimum wage").

Fundamental Economic Problems

• What goods/services are produced and in what quantities?
• How are these goods/services produced?
• For whom are these goods/services produced?

Economic Systems

• Market Economy: Decisions made by individuals and firms.
• Centralized Economy: Government makes decisions.
• Mixed Economy: Combination of market and centralized systems.

Economic Instruments - Analysis

• Studies decisions on the "margin," considering additional costs/benefits of a small change.
• Marginal Cost (CM): Extra cost of producing one more unit.
• Marginal Benefit (BM): Additional benefit of consuming one more unit.
• A decision should be made when $BM \geq CM$.

Economic Instruments - Models

• Simplified representations of reality used to analyze relationships between economic variables.
• Assumptions: Simplifications to facilitate analysis.
• Variables: Factors that can change and affect results.

Economic Instruments - Data

• Numerical information used to analyze the economy.
• Sources: INE, Central Bank, etc.
• Types: Prices, quantities, interest rates.

Production Possibilities Frontier (FPP)

• Shows maximum quantities of goods/services an economy can produce using all resources efficiently.
• Costo de Oportunidad: The quantity of a good that must be given up to obtain one more unit of another good.
• Economic Growth: The FPP shifts outward, indicating the economy can produce more.
• Causes of economic growth include increased available resources and improved technology.

Circular Flow Diagram

• Illustrates the flow of money and goods between households and firms.
• Households: Provide factors of production and consume goods/services.
• Firms: Demand factors of production and produce goods/services.
• Market of Goods/Services: Households buy goods/services from firms.
• Market of Factors of Production: Firms buy factors of production from households.
• Real Flow: Flow of goods, services, factors of production.
• Monetary Flow: Payments for goods/services, salaries, rents.