2. The Interaction of Matter and Light#
2.1. Spectral Lines#
Some philosophers during the Renaissance considered strict limits to human knowledge, while other natural philosophers used experimentation to probe reality. In 1800, William Wollaston showed that a number of dark spectral lines were present within the a rainbow-like spectrum using sunlight. By 1814, Joseph von Fraunhofer cataloged 475 of the dark lines (i.e., Fraunhofer lines) in the solar spectrum. Fraunhofer determined that the wavelength of one prominent dark line in the Sun’s spectrum corresponds to the yellow light emitted by salt in a flame. The new science of spectroscopy was born with the discovery of the sodium line.
Wavelength (nm) |
Name |
Atom |
Equivalent Width (nm) |
---|---|---|---|
385.992 |
0.155 |
||
388.905 |
0.235 |
||
393.905 |
2.025 |
||
396.849 |
1.547 |
||
404.582 |
0.117 |
||
410.175 |
0.313 |
||
422.674 |
0.148 |
||
434.048 |
0.286 |
||
438.356 |
0.101 |
||
486.134 |
0.286 |
||
516.733 |
0.065 |
||
517.270 |
0.126 |
||
518.362 |
0.158 |
||
588.997 |
0.075 |
||
589.594 |
0.056 |
||
656.281 |
0.402 |
2.1.1. Kirchoff’s Laws#
The foundations of spectroscopy (and modern chemistry) were established in the 1800s with Robert Bunsen and Gustav Kirchhoff. Bunsen’s burner produced a colorless flame that was ideal for studying the colors produced by burning a range of substances. Bunsen and Kirchoff designed a spectroscope that passed the light of a flame spectrum through a prism. They determined that the wavelengths of light absorbed and emitted by an element fit together like a lock and key. Kirchoff determined that 70 of the Fraunhofer lines were due to iron vapor. In 1860, Kirchoff and Bunsen developed the idea that every element produces its own pattern of spectral lines and thus the elements had “fingerprints”. Kirchoff summarized the production of spectral lines in three laws (i.e., Kirchoff’s Laws):
A hot, dense gas or hot solid object produces a continuous spectrum with no dark spectral lines.
A hot diffuse gas produces bright spectral lines (emission lines).
A cool, diffuse gas in front of a source of a continuous spectrum produces dark spectral lines (absorption lines) in the continuous spectrum.
2.1.2. Applications of Stellar Spectral Data#
Using the development of spectral fingerprints, a new element helium was discovered spectroscopically in the Sun (in 1868) and found on Earth in 1895. Another line of investigation was measuring the Doppler shifts of the spectral lines. For most cases at the time, the low-speed approximation (
By 1887, the radial velocities of Sirius, Procyon, Rigel, and Arcturus were measured with and accuracy of a few km/s. The rest wavelength
What is the radial velocity of the star Vega, if its
Applying the low-speed approximation for the radial velocity, we get:
import numpy as np
from scipy.constants import c
def calc_radial_vel(l_obs,l_rest):
#l_obs = the observered wavelength
#l_rest = the rest wavelength measured in a lab
#the units of l_obs must equal l_rest
rad_vel = (l_obs-l_rest)/l_rest
return rad_vel*c
Ha_obs = 656.251 #Vega H-alpha in nm
Ha_rest = 656.281 #rest H-alpha in nm
Vega_RV = calc_radial_vel(Ha_obs,Ha_rest)/1000 #convert from m/s to km/s
print("The radial velocity of Vega is %2.3f km/s." % np.round(Vega_RV,3))
The radial velocity of Vega is -13.704 km/s.
The negative sign for the radial velocity indicates that Vega is approaching the Sun (i.e., moving towards us). Some stars also have a measured proper motion
Vega’s proper motion is
The velocity through space is determined by combining the radial velocity with the proper motion. From the previous problem, we know that Vega’s radial velocity
The tranverse velocity can be calculated as
Finally, the velocity through space can be determined through the hypotenuse of two vector components,
#get constants from scipy; https://docs.scipy.org/doc/scipy/reference/constants.html
from scipy.constants import astronomical_unit, year
def mu2SI(mu):
#mu = proper motion (in arcseconds/year)
#mu * (arcseconds in a radian)/(seconds in a year)
mu *= (as2rad/year)
return mu
def PC2km(r):
#r = distance in pc
r2AU = r/as2rad #pc converted to AU
r2km = r2AU*AU #convert to km
return r2km
AU = astronomical_unit/1000. #(in km) Astronomical Unit
as2rad = (1./3600.)*(np.pi/180.)
mu = mu2SI(0.34972)
r = PC2km(7.68)
v_theta = r*mu
print("The transverse velocity of Vega is %2.1f km/s." % np.round(v_theta,1))
v_Vega = np.sqrt(v_theta**2+Vega_RV**2)
print("Vega's velocity through space is %2.1f km/s" % (v_Vega))
The transverse velocity of Vega is 12.7 km/s.
Vega's velocity through space is 18.7 km/s
The average speed of stars in the solar neighborhood is about 25 km/s, where the measurement of a star’s radial velocity is also complicated by the motion of the Earth (29.8 km/s) around the Sun. Astronomers correct for this motion by subtracting the component of Earth’s orbital velocity along the line-of-sight from the star’s measured radial velocity.
2.1.3. Spectrographs#
Modern methods can measure radial velocities with an accuracy of
where
where
2.2. Photons#
A complementary description of light (in contrast to waves) uses bundles of energy called photons. The energy is described in terms of Planck’s constant h and is recognized as a fundamental constant of nature like the speed of light c. This description of matter and energy is known as quantum mechanics, which uses Planck’s discovery of energy quantization. The first step forward was made by Einstein, who demonstrated the implications of Planck’s quantum bundles of energy.
2.2.1. The Photoelectric Effect#
The photoelectric effect describes the light energy necessary to eject electrons from a metal surface. The electrons with the highest kinetic energy
Einstein’s solution describes the light striking the metal surface as a stream of massless particles (i.e., photons). The energy of a single photon of frequency
What is the energy of a single blue photon of wavelength
We can find the energy of a single photon in units of
from scipy.constants import h
def energy_photon(l,units):
#l = lambda; wavelength in nm
if units=='eV':
hc = 1240 #eV*nm
else:
#h = 6.62607004e-34 #(m^2*kg/s) Planck's constant
hc = h*c*1000/1e-9
return (hc/l)
E_blue_eV = energy_photon(450,'eV')
E_blue_J = energy_photon(450,'J')
print("The energy of a single blue photon (450 nm) is %1.2f eV or %1.2e J" % (np.round(E_blue_eV,2),E_blue_J))
The energy of a single blue photon (450 nm) is 2.76 eV or 4.41e-16 J
Einstein deduced that when a photon strikes the metal surface, its energy may be absorbed by a single electron. The photon supplies the energy required to free the electron from the metal (i.e., overcome the binding energy). The work function
where the cutoff frequency and wavelength are
2.2.2. The Compton Effect#
Another test of light’s particle-like nature was performed by Arthur Compton. He measured the change in the wavelength of X-ray photons as they were scattered by free electrons. Through special relativity, the energy of a photon is related to its momentum
Compton considered the interaction (or “collision”) between a photon and a free electron (initially at rest). The electron is scattered by an angle
where
2.3. The Bohr Model of the Atom#
2.3.1. The Structure of the Atom#
With the discovery of the electron (by J.J. Thomson), it showed that the atom is composed of smaller parts usually consisting of electrically neutral bulk matter (equal parts of negative and positive charges). Ernest Rutherford discovered in 1911 that an atom’s positive charge was concentrated in a tiny, massive nucleus using
2.3.2. The Wavelength of Hydrogen#
Many natural philosophers (now known as chemists) produced a wealth of experimental data. Fourteen spectral lines of hydrogen had been precisely measured before the 20th century. The spectral lines were described by their wavelengths and categorized using greek letters (
where
with
Series Name |
Symbol |
Transition |
Wavelength (nm) |
---|---|---|---|
Lyman |
121.567 |
||
102.572 |
|||
97.254 |
|||
91.18 |
|||
Balmer |
656.281 |
||
486.134 |
|||
434.048 |
|||
410.175 |
|||
397.007 |
|||
388.905 |
|||
364.6 |
|||
Paschen |
1875.10 |
||
1281.81 |
|||
1093.81 |
|||
820.4 |
What are the first 3 wavelengths for the Balmer, Lyman, and Paschen lines? (Compare to Table 2 in Ch 5 of the textbook)
The first 3 wavelengths of each series can be estimated using the Rydberg equation using integers
Then, the first term of each series is given as
from scipy.constants import Rydberg
def calculate_lambda_mn(m,n):
#m,n = integers
if m < n:
return (1./R_H)*(m**2*n**2)/(n**2-m**2)
else:
print("Please choose appropriate values for m,n (m<n)")
return 0
R_H = Rydberg*1e-9 #Rydberg constant in nm
Balmer = np.zeros(3)
Lyman = np.zeros(3)
Paschen = np.zeros(3)
wavelength = [Balmer,Lyman,Paschen]
for i in range(1,4):
Balmer[i-1] = calculate_lambda_mn(2,2+i)
Lyman[i-1] = calculate_lambda_mn(1,1+i)
Paschen[i-1] = calculate_lambda_mn(3,3+i)
Series = ["H","Ly","Pa"]
letter = ["$\\alpha$","\$beta$","$\gamma$"]
for i in range(0,3):
for j in range(0,3):
print("%s%s is %3.3f nm." % (Series[i],letter[j],wavelength[i][j]))
H$\alpha$ is 656.112 nm.
H\$beta$ is 486.009 nm.
H$\gamma$ is 433.937 nm.
Ly$\alpha$ is 121.502 nm.
Ly\$beta$ is 102.518 nm.
Ly$\gamma$ is 97.202 nm.
Pa$\alpha$ is 1874.607 nm.
Pa\$beta$ is 1281.469 nm.
Pa$\gamma$ is 1093.520 nm.
Although this formula worked for hydrogen, physicists of the day wanted a physical model that could be applied more broadly. The simplest “planet-like” model of an electron orbiting a proton was the most attractive, but Maxwell’s equations predicted a basic instability. A particle in a circular orbit experiences an acceleration, where Maxwell showed that accelerating charges emit electromagnetic radiation. Under this model, the electron would lose energy by emitting light as it spirals down into the nucleus in only
2.3.3. Bohr’s Semiclassical Atom#
In 1913, Niels Bohr noted that the dimensions of Planck’s constant are equivalent to angular momentum and perhaps the angular momentum of the orbiting electron was quantized. Bohr hypothesized that orbits with integral multiples of Planck’s constant (i.e., a specific angular momentum), the electron would be stable and would not radiate in spite of its centripetal acceleration.
To analyze the interactions between a proton and electron, we start with the mathematical description given by Coulomb’s law. The electric force between two charges (
where
where
which can be solved for the kinetic energy
The electrical potential energy
The total energy
The kinetic energy is a positive quantity, which implies that the total energy is negative and indicates that the electron and proton are bound. To ionize or free the electron, it must receive at least an amount of energy equal to
Getting back to Bohr’s contribution, he suggested that the angular momentum
which can be rewritten using the kinetic energy as
Solving the second line of the equation above for the radius
where
The principal quantum number
The spectral lines are produced by electron transitions within the atom (i.e., moving between energy levels). Balmer tried to produce a formula that would predict the wavelength that resulted from a transition from
where
What are the first 3 wavelengths in the Balmer lines using this method?
In this method, we can evaluate the wavelengths using
where
from scipy.constants import physical_constants
R_H = physical_constants['Rydberg constant times hc in eV'][0]
def calculate_lambda_hl(n_h,n_l):
#n_h = higher state
#n_l = lower state
if n_h > n_l:
return (1240/R_H)*(n_h**2*n_l**2)/(n_h**2-n_l**2)
else:
print("Please choose appropriate values for n_h,n_l (n_l<n_h)")
return 0
for i in range(0,1):
for j in range(0,3):
B_lam = calculate_lambda_hl(j+3,2)
print("%s%s is %3.3f nm." % (Series[i],letter[j],B_lam))
H$\alpha$ is 656.196 nm.
H\$beta$ is 486.071 nm.
H$\gamma$ is 433.992 nm.
The above results and those produced from Balmer’s formula are for light traveling in a vacuum. The empirical measurements are made in air, where the speed of light is slightly slower by a factor of 0.999703 and
This discussion has focused on electrons that transition from a higher (large
A hot, dense gas or hot solid object produces a continuous spectrum with no dark spectral lines. The continuous spectrum is described by the Planck functions
and . The wavelength where reaches its peak intensity is given by Wien’s displacement law.A hot, diffuse gas produces bright emission lines, when an electron makes a downward transition. The energy lost is carried away by a single photon.
A cool, diffuse gas in front of a source of a continuous spectrum produces dark absorption lines in the continuous spectrum. The lines are produced when an electron absorbs a photon of the required energy to jump from a lower to higher energy orbit.
Despite the success of Bohr’s model, it describes only the simplest arrangement of particles. In reality, the electron is not really in a circular orbit nor is its position well-defined. There is more structure to the atom than Bohr realized, where the fine structure can come into play during and after a star’s main-sequence.
2.4. Quantum Mechanics and Wave-Particle Duality#
2.4.1. de Broglie’s Wavelength and Frequency#
Louis de Broglie (a French prince) posed the following question: If light could exhibit the characteristics of particles, might not particles sometimes manifest the properties of waves? In his PhD thesis, de Broglie extended the wave-particle duality of light to all other particles. Einstein’s theory of special relativity showed that photons carry both energy
The de Broglie wavelength and frequency describe massless photons, as well as, massive electrons, protons, neutrons, etc. This seemed outrageous at the time, but the hypothesis was confirmed by many experiments including the interference pattern produced by electrons in a double slit experiment. The wave-particle duality lies at the heart of the physical world, where wave properties describe the propagation and the particle nature manifests in interactions.
Compare the wavelengths of a free electron moving at
The de Broglie wavelength can be calculated for any object, where
from scipy.constants import h, c, m_e
def deBroglie_lambda(m,v):
#m = mass in kg
#v = speed in m/s
return h/(m*v) #de Broglie wavelength in meters
m_UB = 94 #kg; mass of Usain Bolt; https://en.wikipedia.org/wiki/Usain_Bolt
#m_e = 9.10938356e-31 #kg; mass of electron; https://en.wikipedia.org/wiki/Electron
#h = 6.62607015e-34 #J/s Planck constant
#c = 2.99792458e8 #speed of light in m/s
v_e = 0.01*c
v_UB = 12.42 #fastest speed in m/s
lambda_e = deBroglie_lambda(m_e,v_e)
lambda_UB = deBroglie_lambda(m_UB,v_UB)
print("The wavelength of a free electron moving at 0.01c is %1.3e m or %1.3f nm." % (lambda_e,lambda_e/1e-9))
print("The wavelength of Usain Bolt moving at 12.42 m/s is %1.3e m." % lambda_UB)
print("The wavelength of a free electron is %1.0e times larger than for the fastest sprinter." % (lambda_e/lambda_UB))
The wavelength of a free electron moving at 0.01c is 2.426e-10 m or 0.243 nm.
The wavelength of Usain Bolt moving at 12.42 m/s is 5.676e-37 m.
The wavelength of a free electron is 4e+26 times larger than for the fastest sprinter.
In a double-slit experiment, a photon or electron passes through both slits to produce the interference pattern on the other side. The wave doesn’t convey information about where the particle is, rather only where it may be. As a result, the wave is described by a probability with and amplitude
2.4.2. Heisenberg’s Uncertainty Principle#
A probability wave
Show code cell content
import numpy as np
import matplotlib.pyplot as plt
lw = 2
fs = 'x-large'
col = (218./256,26./256,50./256)
theta1 = np.arange(0,4*2*np.pi,0.01)
y1 = np.sin(3*theta1)
theta2 = np.arange(-np.pi,2*np.pi,0.01)
y2 = np.zeros(len(theta2))
th_cut = np.where(np.logical_and(0<= theta2, theta2 <= np.pi))[0]
y2[th_cut] = 2*np.sin(theta2[th_cut])*np.sin(10*theta2[th_cut])
fig = plt.figure(figsize=(10,3),dpi=150)
ax1 = fig.add_subplot(121)
ax2 = fig.add_subplot(122)
ax1.plot(theta1,y1,'-',color=col,lw=lw)
ax1.plot(theta1,np.zeros(len(y1)),'k--',lw=lw)
ax2.plot(theta2,np.zeros(len(y2)),'k--',lw=lw)
ax2.plot(theta2,y2,'-',color=col,lw=lw)
ax1.axis('off'); ax2.axis('off')
ax1.text(0.47,-0.1,'(a)',transform=ax1.transAxes)
ax2.text(0.47,-0.1,'(b)',transform=ax2.transAxes)
fig.savefig("Fig07.png",bbox_inches='tight',dpi=150);
Werner Heisenberg developed a theoretical framework for this inherent “fuzziness” of the physical world. The product of the uncertainty in the position and momentum must be equal to or larger than
What is the minimum speed and kinetic energy of an electron in a hydrogen atom?
The minimum speed and kinetic energy is determined using uncertainty principle for the electron’s momentum
where the momentum is determined using the de Broglie wavelength (Eq. (2.16)).
The minimum kinetic energy is calculated using the equation for kinetic energy using the minimum speed, or
from scipy.constants import hbar, eV, physical_constants
a_o = physical_constants['Bohr radius'][0] #Bohr radius of hydrogen atom; Delta x
p_e = hbar/a_o #uncertainty in the electron's momentum p
v_min = p_e/m_e #minimum speed of the electron
K_min = p_e**2/(2*m_e) #minimum KE of the electron
#eV = 1.602176634e-19 #J in 1 eV
print("The minimum speed of the electron is %1.2e m/s or %1.3fc." % (v_min,v_min/c))
print("The minimum KE of the electron is %1.2e J or %2.1f eV" % (K_min,K_min/eV))
The minimum speed of the electron is 2.19e+06 m/s or 0.007c.
The minimum KE of the electron is 2.18e-18 J or 13.6 eV
2.4.3. Quantum Mechanical Tunneling#
Light changes its behavior at boundary transitions, where this is most easily seen for a light beam traveling from a prism (glass) into air. The light may undergo total internal reflection if its incident angle is greater than a critical angle
The electromagnetic wave does enter the air, but it dies away exponentially (i.e., becomes evanescent). When another prism is placed next to the first, then the evanescent wave can begin propagating into the second prism without passing through the air gap between them. Photons have tunneled from one prism to another. The limitation to this effect is through Heisenberg’s uncertainty principle, where the location of a particle (i.e., photon) cannot be determined with an uncertainty less than its wavelength. For barriers that are only a few wavelengths wide, a particle can suddenly appear on the other side. Barrier penetration is important for radioactive decay and inside stars, where nuclear fusion rates depend upon tunneling.
2.4.4. Schrödinger’s Equation and the Quantum Mechanical Atom#
Heisenberg’s uncertainty principle prevents the use of classical models to treat the electron-proton like a planetary system. Instead, the electron orbits within a cloud of probability (i.e., orbital) where the more “dense” regions corresponding to where the electron is most likely to be found. In 1926, Erwin Schrödinger developed a wave equation that can be solved for the probability waves that describe the allowed values of a particle state (e.g., energy, momentum, etc.) and its propagation through space.
The Schrödinger equation can be solved analytically for the hydrogen atom and it produces the same set of allowed energies obtained by Bohr. However, Schrödinger found that two additional quantum numbers
where
The
However, hydrogen gas in a star is not isolated and the electrons will feel the effect of the stellar magnetic field. Due to the external magnetic field the energy levels are no longer degenerate and the spectral lines are observed to split. The simplest case is described by the Zeeman effect, where the
Today, the Zeeman effect is used to monitor the magnetic field of the Sun by taking spectra near sunspots. Interstellar clouds may contain very weak magnetic fields (
Note
Another commonly used unit of magnetic field strength is gauss, where
Calculate the change in frequency under the magnetic field of an interstellar cloud using the mass of the electron for the reduced mass (
The change in frequency
from scipy.constants import elementary_charge, m_e
def calc_Zeeman_freq(B,mu):
#B = magnetic field strength in Tesla (T)
#mu = reduced mass
return q_e*B/(4*np.pi*mu) #returns frequency in Hz
q_e = elementary_charge #1.60217662e-19 #Coulombs; electron charge
B_IC = 2e-10 #magnetic field of interstellar cloud
delta_nu = calc_Zeeman_freq(B_IC,m_e)
print("The change in frequency from the Zeeman effect is %1.1f Hz." % delta_nu)
The change in frequency from the Zeeman effect is 2.8 Hz.
2.4.5. Spin and the Pauli Exclusion Principle#
More complicated patterns of magnetic field splitting led physicists to discover another quantum number related to the spin angular momentum S of an electron. This is not a classical top-like rotation but purely a quantum effect. The spin vector has a constant magnitude
with a
Quantum mechanics was at odds with Schrödinger’s (non-relativistic) wave equation and Einstein’s theory of special relativity. Paul Dirac worked to combine the two theories and succeeded in writing a relativistic wave equation for the electron that naturally included the spin of the electron. His relativistic wave equation extended the Pauli exclusion principle by dividing the world of particles into two groups: fermions and bosons.
Fermions are odd-integer half-spin particles (e.g.,
, , ) such as electrons, protons, and neutrons. Fermions obey the Pauli exclusion principle, which explains the structure of white dwarfs and neutron stars.Bosons are integer spin particles (e.g., 0,
, , ) such as photons. Bosons do not obey the Pauli exclusion principle.
The Dirac equation also predicted the existence of antiparticles. A particle differs from its antiparticle in its charge and magnetic moment (e.g., electron
2.4.6. The Complex Spectra of Atoms#
Four quantum numbers (
The Zeeman effect shows only three transitions occur between
2.5. Homework#
Problem 1
Barnard’s star is an orange star in the constellation Ophiuchus. It has the largest known proper motion
(a) Determine the radial velocity of Barnard’s star.
(b) Determine the transverse velocity of Barnard’s star.
(c) Calculate the speed of Barnard’s star through space.
Problem 2
The Sun’s spectrum contains two spectral lines (Sodium D lines) 588.997 nm and 589.594 nm.
(a) If a diffraction grating with 300 lines per millimeter is used to measure the Sodium D lines, what is the angle between the second-order spectra for each of the wavelengths?
(b) How many lines must the grating have to resolve the sodium D lines?
Problem 3
The ejection of an electron leaves a grain of dust with a positive charge, which leads to heating within an interstellar cloud. If the average energy of the ejected electron is about 5 eV and the process is particularly effective for UV photons (
Problem 4
Consider the case of a “collision” between a photon and a free proton (initially at rest) given a scattering angle
(a) What is the characteristic change in the wavelength of the scattered photon (in nm)?
(b) How does this compare with the Compton wavelength,
Problem 5
To demonstrate the relative strengths of the electrical and gravitational forces between the electron and proton in the Bohr atom, suppose the hydrogen atom were held together solely by gravity. Determine the radius of the ground-state orbit (in nm and AU) and the energy of the ground state (in eV).
Problem 6
Calculate the energies and vacuum wavelengths of all possible photons that are emitted when the electron cascades from the
Problem 7
An electron in an old TV set reaches a speed of about
Problem 8
A White dwarf is a very dense object, with its ions and electrons packed extremely close together. Each electron may be considered to be located within a region of size
Problem 9
An electron spends roughly
(a) Determine the uncertainty
(b) Calculate the uncertainty