2. Fingerprints of Matter: Spectra

Transcription

1 2. Fingerprints of Matter: Spectra 2.1 Measuring spectra: prism and diffraction grating Light from the sun: white light, broad spectrum (wide distribution) of wave lengths. 19th century: light assumed to have wave character (observation of diffraction phenomena), explained by Huygens principle (see section 3.1). decomposition of white light into colours (individual wave lengths): observation of rainbows; colours of water drops and soap films). controlled experiment: prism.

2 (Young and Freedman, 33.19) incoming beam of light is deflected angle of deflection depends on internal angle of prism and wave-length of light (index of refraction depends on wave-length) Result: white light decomposed into its components which leave prism under different angles. Dispersion into a spectrum.

3 spectral analysis using diffraction grating (further demonstration of wave character of light) (YF 36.16) incoming mono-chromatic light of wave-length λ. light scattered into all directions when passing through grid of slits (diffraction) wave property of interference cancels out certain directions determined by path difference Δ for adjacent slits: cancellation for directions where Δ equals a multiple of half a wave-length; Δ=d sin θ = m λ /2 (m: integer)

4 2.2 The puzzle of lines Fraunhofer ( ): study of sunlight using grating. Observation of very distinct dark lines in the spectrum. wavelength in m = 1 Å (

5 Laboratory analysis of - flames of salts - electric discharges of gases (e.g. neon tube) [spectrum of Helium see also figure YF, 38.8] Line spectra (each line corresponds to one wavelength/frequency): fingerprint of salt/gas Used for detection of chemical elements in stars.

6 Line spectra challenge 19 th century physics. known: light is wave (Maxwell s equations, 1861) electromagnetic waves are produced by moving (oscillating) charges (Heinrich Hertz 1887) what particles? (electrons?) why discrete frequencies? why different for different gases?

7 Progress Johann Balmer 1885: observation that Hydrogen gas spectrum is described by empirical formula. 1/λ = R (1/m 2 1/n 2 ) m=2; n integer, 3,4,5, R, parameter chosen to fit the data. Rydberg constant R = x 10 7 m -1. Physical meaning of R? explained and computed by Bohr (1913) (section 5.3) Further spectral series for hydrogen: Lyman series (m=1, n = 2,3,4, ) Paschen series (m=3, n= 4,5,6, ) Bracket series (m=4, n = 5,6,7, ) Pfund series (m=5, n = 6,7,8, )

8 paradigm change Macroscopic world: continuum Microscopic world: discreteness quantisation of energy (Franck-Hertz 1914: discrete energy levels in Mercury vapour, see section 5.4)

9 2.3 Blackbody radiation Spectra of gases of elements: individual lines More complicated molecules: emission bands Spectrum of hot solid body? Continuous, infra-red Favourite object of study: black-body, i.e. body which absorbs all incoming wavelengths

10 Black body also best possible emitter of electromagnetic radiation Spectrum well determined in 19 th century t/blackbody/black.htm Figure YF 38.31: Spectral emittance I(λ) (intensity per wave-length interval = power per unit area per wave-length interval) as a function of wave-length

11 19 th century physics could not explain the shape of this distribution and the observed relationships. Stefan-Boltzmann law for intensity I as function of temperature T: I I( ) d 0 I T 4 Stefan-Boltzmann constant Wm K Wien displacement law for frequency λm of the maximum: T m m K (YF, 38.31)

12 classical physics: Electromagnetic radiation in cavity (standing waves) exchanges energy with walls forced oscillations of charge distribution in the walls re-admission of radiation (moving charges!) at frequency of oscillation in equilibrium: energy per volume u(ω), in frequency interval [ω, ω+dω] given by u(ω)dω = k B T/(π 2 c 3 ) ω 2 dω (Rayleigh-Jeans law, 1900, 1905) (c, velocity of light) problem, total energy content of cavity, I diverges to infinity: The challenge to classical physics ultraviolet catastrophe! 0 u( d

13 1900, Planck: hypothesis: electromagnetic oscillators (in cavity wall) have vibration modes at energies which are integer multiples of hf. Planck s radiation law for energy density u(ω) (ω=2πf), fully describes the experimental data: Planck s constant h = J s is extremely small (!) and has the physical dimension of an action (J s); often used in form u( ) 2 c 3 h 2 exp 3 / k T1 B

14 average energy of an oscillator given by for high temperatures: thus <E> k B T E e hf hf / kt 1 hf / kt 1, then Taylor expansion exp( x) 1 x Planck s formula also reproduces known relationships about black body curve to show this let s rewrite u(ω) as I(λ), resulting in I( ) hc exp( hc / k T) 1 B

15 Wien displacement law di ( ( )) 0 d Stefan-Boltzmann law hc m (YF, Exercise 38.47) 4.965kT B I( ) hc exp( hc / k T) 1 B 2 hc 2 k I( T) d T T B exp( hc / kbt) 1 15c h 4 4 (YF, Exercise 38.77) for high T Rayleigh-Jeans law is limit of Planck radiation law for h / kt 1

16 Still: Planck: quantum hypothesis = mathematical trick? Einstein, Bohr et al.: quantisation is at the foundation of (sub-) atomic world (as demonstrated by interpretation of many further experiments)

17 Wien s law Comment: Blackbody spectrum of the universe T m m K hc 4.965kT What is the observed temperature of the early universe (big bang)? 1965 Penzias and Wilson (accidentally) measure the temperature of the universe to be close to 3K. More recently COBE has measured this to be K (consider also red shift) m B

18 Quantisation and molar heat capacity C C 1 n dq dt n: number of moles, Q: heat, T: temperature heat: lattice vibrations of molecules equipartition principle: energy 1/2 k B T per degree of freedom thus E tot per mol (sum of kinetic and potential energy): (YF, 18.20) E tot =3Nk B T = 3RT C= 3R, (R: gas constant, k B =R/N) law of Dulong-Petit (1819)

19 (YF, 18.21) Dulong-Petit prediction fails for low temperatures, decreasing masses and increasing stiffness of the crystal lattice, i.e. INCREASING FREQUENCY (analogous to blackbody radiation).

20 Einstein (1907), Debye (1912): oscillations are quantized! new expression for C(T) results in Agreement with experimental data C ~ T 3 at low temperatures and recovery of Dulong-Petit in limit kt