Spektrallinjers breddning

Radiative Processes

• Spectral lines possess a finite width even for atoms at rest due to the uncertainty principle

Line Broadening Mechanisms

• These mechanisms describe why spectral lines are not infinitely narrow

Natural Broadening

• It is a consequence of the finite lifetimes of excited states.
• The uncertainty principle dictates that energy levels have an inherent width, expressed as ΔE Δt ~ ħ.
• ΔE represents the energy uncertainty, and Δt represents the time an electron spends in an excited state.
• The uncertainty in emitted/absorbed light's frequency is Δν ~ 1/Δt.
• The spectral line adopts a Lorentzian profile: φ(ν) = (Γ / 4π²) / ((ν - ν₀)² + (Γ / 4π)²).
• ν₀ denotes the line's central frequency, and Γ the damping constant.
• The damping constant Γ sums transition probabilities from a state to all lower levels.
• The profile is normalized: ∫₋∞ to ∞ φ(ν) dν = 1
• The full width at half maximum (FWHM) equals Γ / 2π.
• Natural width is typically very small, around 10⁻⁸ eV.

Doppler Broadening

• Atoms in gases move randomly at various velocities, causing Doppler shifts in observed radiation frequency.
• The frequency observed ν relates to the rest frame frequency ν₀ by ν = ν₀ (1 + v/c) where v is the atom's velocity component along the line of sight.
• Atomic velocities are Maxwell-Boltzmann distributed: P(v) dv = (m / (2πkT))^(1/2) * e^(-mv²/2kT) dv.
• m signifies atom mass, T the gas temperature, and k Boltzmann's constant.
• The spectral line shows a Gaussian profile: φ(ν) = (1 / (√(π) Δν_D)) * e^(-(ν - ν₀)² / Δν_D²).
• Δν_D stands for the Doppler width, given by Δν_D = ν₀ √(2kT / mc²).
• The FWHM of the line is 2√(ln 2) Δν_D.
• Doppler broadening predominates in astrophysical plasmas.

Pressure Broadening

• Atomic collisions interrupt the emission or absorption process, effectively shortening exited state.
• Collision broadening, is the other name for this mechanism.
• Pressure broadening depends on gas density and temperature
• The types are:
  • Van der Waals broadening results from neutral atom interaction.
  • Stark broadening arises from charged particle interaction.

The Curve of Growth

• It plots line's equivalent width versus absorbing gas's column density.
• The equivalent width W is the width of a rectangle with the same area as a spectral line: W = ∫₋∞ to ∞ (I₀ - Iᵥ) / I₀ dv = ∫₋∞ to ∞ (1 - e^(-τᵥ)) dv.
• Here, I₀ = continuum radiation intensity and Iᵥ = intensity at frequency ν
• τᵥ = optical depth at frequency ν.
• Three sections:
  • Linear section: At small column densities, the optical depth is small, and equivalent width is proportional to the column density: W ∝ N.
  • Square root section: At larger column densities, the optical depth at line center grows and the equivalent width increases with column density: W ∝ √N.
  • Logarithmic section: At high column densities, the line saturates and the equivalent width shows logarithmic growth with column density: W ∝ ln N.
• By measuring equivalent width and comparing it to a theoretical curve of growth the curve of growth can determine the column density of an absorbing gas.

Guía breve para la calidad del agua de pozo privado

• This guide provides information on testing and protecting private well water

Protegiendo su salud

• In private well water bacteria, viruses, pesticides and other pollutants may cause contamination. Testing the water helps ensure it is safe to drink.

¿Qué contaminantes debo analizar?

• These are the contaminants to tests for, at least yearly:
  • Total coliform bacteria
  • E. coli
  • Nitrates
  • Total dissolved solids (TDS)
  • pH
• And every 3-5 years:
  • Arsenic
  • Barium
  • Fluoride
  • Lead
  • Manganese
  • Radio
  • Uranium
• If it is near certain activities you may also wants to test for pesticides, for example if the well is near a farm

¿Dónde puedo analizar mi agua?

• The water can be analysed at a state-certified laboratory. A list is provided by the state's health offic or by the environmental resources department

¿Cómo interpreto los resultados de mis pruebas?

• The EPA has standards for some contaminants in piblic drinking water. Actions should be taken to remedy situations when test results are beyond these standards.

¿Qué puedo hacer para proteger mi agua de pozo?

• To protect well water, here are a list of helpful practices:
  • Inspect the well regularly
  • keep your well properly sealed
  • Avoid pouring chemicals and waste near the well
  • Maintain the septic tank correctly
  • Stay careful when using fertilizers and pesticides

¿Dónde puedo obtener más información?

• Further information can be obained from:
  • The local health office
  • The state environmental resources department
  • The U.S. Environmental Protection Agency (EPA)
  • The U.S. Geological Survey

Pasos a seguir

• What is needed to be done:
  • Sample water
  • Send sampel to state-certified lab for analysi
  • Interpret the test results
  • Remedy any problems
  • Take action to protect your water

Preocupaciones estacionales

• Useful season related information:
  • Spring: Test total coliform bacteria and E. coli because of snow and rain runoff
  • Autumn: Test nitrates after harvesting

Teorema de Bayes

• Describes the probability of an event, based on prior knowledge of conditions that may be related to the event.

• Expressed by:

  P(A|B) = (P(B|A)P(A)) / P(B)

• Where:

  • P(A|B) = conditional probability of A, given that B occurred
  • P(B|A) = conditional probability of B, given A occurred
  • P(A) and P(B) = the probabilities of A and B occurring independently

História

• Named after Thomas Bayes, who provided an equation that updates beliefs with evidence.
• Richard Price edited and published the work in 1763.
• Pierre-Simon Laplace rediscovered and developed the theorem in 1774, not knowing of Bayes.

Derivação

• From the definition of conditional probability:

  P(A|B) = P(A ∩ B) / P(B) P(B|A) = P(B ∩ A) / P(A)

• Solving for P(A ∩ B):

  P(A ∩ B) = P(A|B)P(B) = P(B|A)P(A)

• Dividing by P(B):

  P(A|B) = (P(B|A)P(A)) / P(B)

Aplicações

• Applications are:
  • Medicina: calculating disease probability given a test result.
  • Finanças: assessing investment risk given economic conditions.
  • Aprendizado de Máquina: updating beliefs about parameters given data.
  • Filtragem de Spam: identifying spam emails based on contained words.

Exemplo

• If a test has 99% accuracy and 1% of the population has the disease, the probability of a person having the disease given a positive test result is only about 50% due to the low prevalence.

Chemical Principles

The Properties of Gases

• Gas pressure:
  • Pressure = Force / Area
  • Measured in N/m^2 = 1 pascal (Pa)
• Atmospheric pressure:
  • Measured using a barometer
  • 1 atm = 760 mm Hg
  • 1 atm = 760 torr
  • 1 atm = 101.325 kPa

The Gas Laws

• These are the gas laws and relationships:

Boyle's Law

• Volume is inversely proportional to pressure at constant temperature
  • P ∝ 1/V (constant n, T)
  • P ∙ V = constant
  • P1V1 = P2V2

Charles's Law

• Volume is proportional to temperature at constant pressure.
  • V ∝ T (constant n, P)
  • V/T = constant
  • V1/T1 = V2/T2

Tension

• Pulling force exerted by a string, cable, chain, or similar object on another object.
• Denoted as T, measured in Newtons (N).

Understanding Tension

• Tension is a pulling force.
• It acts along the direction of the string.
• Tension is equal throughout the string if the string is massless and there are no pulleys.
• At any point, the tension force pulls equally in both directions.

Key Scenario Examples:

• Block Hanging Vertically:
  • Tension T = mg (where m is mass, g is acceleration due to gravity).
• Block Pulled Horizontally:
  • Tension T = ma (where a is acceleration) if the block is accelerating
  • T - f = ma if there is friction (f is the friction force).
• Inclined Plane Tension:
  • Resolving forces into components needed
  • Tension balances force from gravitational component+ frictional forces.

Tension in Systems with Multiple Objects

• Tension can vary in each string.
• Analyze each object separately with free-body diagrams, then apply Newton's Second Law.
• Ideal scenarios involve massless string approximation and Ideal Pulley.
• Real-world applications: structures, elevators, mechanical systems.