Relativistic Doppler effect

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Relativistic_Doppler_effect 

 

Figure 1. A source of light waves moving to the right, relative to observers, with velocity 0.7c. The frequency is higher for observers on the right, and lower for observers on the left.

The relativistic Doppler effect is the change in frequency (and wavelength) of light, caused by the relative motion of the source and the observer (as in the classical Doppler effect), when taking into account effects described by the special theory of relativity.

The relativistic Doppler effect is different from the non-relativistic Doppler effect as the equations include the time dilation effect of special relativity and do not involve the medium of propagation as a reference point. They describe the total difference in observed frequencies and possess the required Lorentz symmetry.

Astronomers know of three sources of redshift/blueshift: Doppler shifts; gravitational redshifts (due to light exiting a gravitational field); and cosmological expansion (where space itself stretches). This article concerns itself only with Doppler shifts.

Summary of major results

In the following table it is assumed that for ${\displaystyle \beta >0}$ the receiver and the source are moving away from each other.

Scenario	Formula	Notes
Relativistic longitudinal Doppler effect	${\displaystyle {\frac {\lambda _{r}}{\lambda _{s}}}={\frac {f_{s}}{f_{r}}}={\sqrt {\frac {1+\beta }{1-\beta }}}}$
Transverse Doppler effect, geometric closest approach	${\displaystyle f_{r}=\gamma f_{s}}$	Blueshift
Transverse Doppler effect, visual closest approach	${\displaystyle f_{r}={\frac {f_{s}}{\gamma }}}$	Redshift
TDE, receiver in circular motion around source	${\displaystyle f_{r}=\gamma f_{s}}$	Blueshift
TDE, source in circular motion around receiver	${\displaystyle f_{r}={\frac {f_{s}}{\gamma }}}$	Redshift
TDE, source and receiver in circular motion around common center	${\displaystyle {\frac {\nu '}{\nu }}=\left({\frac {1-R^{2}\omega ^{2}}{1-R'^{2}\omega ^{2}}}\right)^{1/2}}$	No Doppler shift when ${\displaystyle R=R'}$
Motion in arbitrary direction measured in receiver frame	${\displaystyle f_{r}={\frac {f_{s}}{\gamma \left(1+\beta \cos \theta _{r}\right)}}}$
Motion in arbitrary direction measured in source frame	${\displaystyle f_{r}=\gamma \left(1-\beta \cos \theta _{s}\right)f_{s}}$

Derivation

Relativistic longitudinal Doppler effect

Relativistic Doppler shift for the longitudinal case, with source and receiver moving directly towards or away from each other, is often derived as if it were the classical phenomenon, but modified by the addition of a time dilation term. This is the approach employed in first-year physics or mechanics textbooks such as those by Feynman or Morin.

Following this approach towards deriving the relativistic longitudinal Doppler effect, assume the receiver and the source are moving away from each other with a relative speed ${\displaystyle v\,}$ as measured by an observer on the receiver or the source (The sign convention adopted here is that ${\displaystyle v\,}$ is negative if the receiver and the source are moving towards each other).

Consider the problem in the reference frame of the source.

Suppose one wavefront arrives at the receiver. The next wavefront is then at a distance ${\displaystyle \lambda _{s}=c/f_{s}\,}$ away from the receiver (where ${\displaystyle \lambda _{s}\,}$ is the wavelength, ${\displaystyle f_{s}\,}$ is the frequency of the waves that the source emits, and ${\displaystyle c\,}$ is the speed of light).

The wavefront moves with speed ${\displaystyle c\,}$, but at the same time the receiver moves away with speed ${\displaystyle v}$ during a time ${\displaystyle t_{s}=1/f_{s}=\lambda _{s}/c}$, so

$${\displaystyle \lambda _{s}+vt_{r,s}=ct_{r,s}\Longleftrightarrow \lambda _{s}=ct_{r,s}(1-v/c)\Longleftrightarrow t_{r,s}={\frac {1}{f_{s}(1-\beta )}},}$$

where ${\displaystyle \beta =v/c\,}$ is the speed of the receiver in terms of the speed of light, and where ${\displaystyle t_{r,s}}$ is the period of light waves impinging on the receiver, as observed in the frame of the source. The corresponding frequency ${\displaystyle f_{r,s}}$is:

${\displaystyle f_{r,s}=1/t_{r,s}=f_{s}(1-\beta ).}$

Thus far, the equations have been identical to those of the classical Doppler effect with a stationary source and a moving receiver.

However, due to relativistic effects, clocks on the receiver are time dilated relative to clocks at the source: ${\displaystyle t_{r}=t_{r,s}/\gamma }$, where ${\textstyle \gamma =1/{\sqrt {1-\beta ^{2}}}}$ is the Lorentz factor. In order to know which time is dilated, we recall that ${\displaystyle t_{r,s}}$ is the time in the frame in which the source is at rest. The receiver will measure the received frequency to be

Eq. 1:    ${\displaystyle f_{r}=f_{r,s}\gamma }$ ${\displaystyle ={\frac {1-\beta }{\sqrt {1-\beta ^{2}}}}f_{s}}$ ${\displaystyle ={\sqrt {\frac {1-\beta }{1+\beta }}}\,f_{s}.}$

The ratio

${\displaystyle {\frac {f_{s}}{f_{r}}}={\sqrt {\frac {1+\beta }{1-\beta }}}}$

is called the Doppler factor of the source relative to the receiver. (This terminology is particularly prevalent in the subject of astrophysics: see relativistic beaming.)

The corresponding wavelengths are related by

Eq. 2:    ${\displaystyle {\frac {\lambda _{r}}{\lambda _{s}}}={\frac {f_{s}}{f_{r}}}={\sqrt {\frac {1+\beta }{1-\beta }}},}$

Identical expressions for relativistic Doppler shift are obtained when performing the analysis in the reference frame of the receiver with a moving source. This matches up with the expectations of the principle of relativity, which dictates that the result can not depend on which object is considered to be the one at rest. In contrast, the classic nonrelativistic Doppler effect is dependent on whether it is the source or the receiver that is stationary with respect to the medium.

Transverse Doppler effect

Suppose that a source and a receiver are both approaching each other in uniform inertial motion along paths that do not collide. The transverse Doppler effect (TDE) may refer to (a) the nominal blueshift predicted by special relativity that occurs when the emitter and receiver are at their points of closest approach; or (b) the nominal redshift predicted by special relativity when the receiver sees the emitter as being at its closest approach. The transverse Doppler effect is one of the main novel predictions of the special theory of relativity.

Whether a scientific report describes TDE as being a redshift or blueshift depends on the particulars of the experimental arrangement being related. For example, Einstein's original description of the TDE in 1907 described an experimenter looking at the center (nearest point) of a beam of "canal rays" (a beam of positive ions that is created by certain types of gas-discharge tubes). According to special relativity, the moving ions' emitted frequency would be reduced by the Lorentz factor, so that the received frequency would be reduced (redshifted) by the same factor.

On the other hand, Kündig (1963) described an experiment where a Mössbauer absorber was spun in a rapid circular path around a central Mössbauer emitter. As explained below, this experimental arrangement resulted in Kündig's measurement of a blueshift.

Source and receiver are at their points of closest approach

Figure 2. Source and receiver are at their points of closest approach. (a) Analysis in the frame of the receiver. (b) Analysis in the frame of the source.

In this scenario, the point of closest approach is frame-independent and represents the moment where there is no change in distance versus time. Figure 2 demonstrates that the ease of analyzing this scenario depends on the frame in which it is analyzed.

• Fig. 2a. If we analyze the scenario in the frame of the receiver, we find that the analysis is more complicated than it should be. The apparent position of a celestial object is displaced from its true position (or geometric position) because of the object's motion during the time it takes its light to reach an observer. The source would be time-dilated relative to the receiver, but the redshift implied by this time dilation would be offset by a blueshift due to the longitudinal component of the relative motion between the receiver and the apparent position of the source.
• Fig. 2b. It is much easier if, instead, we analyze the scenario from the frame of the source. An observer situated at the source knows, from the problem statement, that the receiver is at its closest point to him. That means that the receiver has no longitudinal component of motion to complicate the analysis. (i.e. dr/dt = 0 where r is the distance between receiver and source) Since the receiver's clocks are time-dilated relative to the source, the light that the receiver receives is blue-shifted by a factor of gamma. In other words,

Eq. 3:    ${\displaystyle f_{r}=\gamma f_{s}}$

Receiver sees the source as being at its closest point

Figure 3. Transverse Doppler shift for the scenario where the receiver sees the source as being at its closest point.

This scenario is equivalent to the receiver looking at a direct right angle to the path of the source. The analysis of this scenario is best conducted from the frame of the receiver. Figure 3 shows the receiver being illuminated by light from when the source was closest to the receiver, even though the source has moved on. Because the source's clock is time dilated as measured in the frame of the receiver, and because there is no longitudinal component of its motion, the light from the source, emitted from this closest point, is redshifted with frequency

Eq. 4:    ${\displaystyle f_{r}={\frac {f_{s}}{\gamma }}}$

In the literature, most reports of transverse Doppler shift analyze the effect in terms of the receiver pointed at direct right angles to the path of the source, thus seeing the source as being at its closest point and observing a redshift.

Point of null frequency shift

Figure 4. Null frequency shift occurs for a pulse that travels the shortest distance from source to receiver.

Given that, in the case where the inertially moving source and receiver are geometrically at their nearest approach to each other, the receiver observes a blueshift, whereas in the case where the receiver sees the source as being at its closest point, the receiver observes a redshift, there obviously must exist a point where blueshift changes to a redshift. In Fig. 2, the signal travels perpendicularly to the receiver path and is blueshifted. In Fig. 3, the signal travels perpendicularly to the source path and is redshifted.

As seen in Fig. 4, null frequency shift occurs for a pulse that travels the shortest distance from source to receiver. When viewed in the frame where source and receiver have the same speed, this pulse is emitted perpendicularly to the source's path and is received perpendicularly to the receiver's path. The pulse is emitted slightly before the point of closest approach, and it is received slightly after.

One object in circular motion around the other

Figure 5. Transverse Doppler effect for two scenarios: (a) receiver moving in a circle around the source; (b) source moving in a circle around the receiver.

Fig. 5 illustrates two variants of this scenario. Both variants can be analyzed using simple time dilation arguments. Figure 5a is essentially equivalent to the scenario described in Figure 2b, and the receiver observes light from the source as being blueshifted by a factor of ${\displaystyle \gamma }$. Figure 5b is essentially equivalent to the scenario described in Figure 3, and the light is redshifted.

The only seeming complication is that the orbiting objects are in accelerated motion. An accelerated particle does not have an inertial frame in which it is always at rest. However, an inertial frame can always be found which is momentarily comoving with the particle. This frame, the momentarily comoving reference frame (MCRF), enables application of special relativity to the analysis of accelerated particles. If an inertial observer looks at an accelerating clock, only the clock's instantaneous speed is important when computing time dilation.

The converse, however, is not true. The analysis of scenarios where both objects are in accelerated motion requires a somewhat more sophisticated analysis. Not understanding this point has led to confusion and misunderstanding.

Source and receiver both in circular motion around a common center

Figure 6. Source and receiver are placed on opposite ends of a rotor, equidistant from the center.

Suppose source and receiver are located on opposite ends of a spinning rotor, as illustrated in Fig. 6. Kinematic arguments (special relativity) and arguments based on noting that there is no difference in potential between source and receiver in the pseudogravitational field of the rotor (general relativity) both lead to the conclusion that there should be no Doppler shift between source and receiver.

In 1961, Champeney and Moon conducted a Mössbauer rotor experiment testing exactly this scenario, and found that the Mössbauer absorption process was unaffected by rotation. They concluded that their findings supported special relativity.

This conclusion generated some controversy. A certain persistent critic of relativity maintained that, although the experiment was consistent with general relativity, it refuted special relativity, his point being that since the emitter and absorber were in uniform relative motion, special relativity demanded that a Doppler shift be observed. The fallacy with this critic's argument was, as demonstrated in section Point of null frequency shift, that it is simply not true that a Doppler shift must always be observed between two frames in uniform relative motion. Furthermore, as demonstrated in section Source and receiver are at their points of closest approach, the difficulty of analyzing a relativistic scenario often depends on the choice of reference frame. Attempting to analyze the scenario in the frame of the receiver involves much tedious algebra. It is much easier, almost trivial, to establish the lack of Doppler shift between emitter and absorber in the laboratory frame.

As a matter of fact, however, Champeney and Moon's experiment said nothing either pro or con about special relativity. Because of the symmetry of the setup, it turns out that virtually any conceivable theory of the Doppler shift between frames in uniform inertial motion must yield a null result in this experiment.

Rather than being equidistant from the center, suppose the emitter and absorber were at differing distances from the rotor's center. For an emitter at radius ${\displaystyle R'}$ and the absorber at radius ${\displaystyle R}$ anywhere on the rotor, the ratio of the emitter frequency, ${\displaystyle \nu ',}$ and the absorber frequency, ${\displaystyle \nu ,}$ is given by

Eq. 5:    ${\displaystyle {\frac {\nu '}{\nu }}=\left({\frac {1-R^{2}\omega ^{2}}{1-R'^{2}\omega ^{2}}}\right)^{1/2}}$

where ${\displaystyle \omega }$ is the angular velocity of the rotor. The source and emitter do not have to be 180° apart, but can be at any angle with respect to the center.

Motion in an arbitrary direction

Figure 7. Doppler shift with source moving at an arbitrary angle with respect to the line between source and receiver.

The analysis used in section Relativistic longitudinal Doppler effect can be extended in a straightforward fashion to calculate the Doppler shift for the case where the inertial motions of the source and receiver are at any specified angle. Fig. 7 presents the scenario from the frame of the receiver, with the source moving at speed ${\displaystyle v}$ at an angle ${\displaystyle \theta _{r}}$ measured in the frame of the receiver. The radial component of the source's motion along the line of sight is equal to ${\displaystyle v\cos {\theta _{r}}.}$

The equation below can be interpreted as the classical Doppler shift for a stationary and moving source modified by the Lorentz factor ${\displaystyle \gamma :}$

Eq. 6:    ${\displaystyle f_{r}={\frac {f_{s}}{\gamma \left(1+\beta \cos \theta _{r}\right)}}.}$

In the case when ${\displaystyle \theta _{r}=90^{\circ }}$, one obtains the transverse Doppler effect:

${\displaystyle f_{r}={\frac {f_{s}}{\gamma }}.\,}$

In his 1905 paper on special relativity, Einstein obtained a somewhat different looking equation for the Doppler shift equation. After changing the variable names in Einstein's equation to be consistent with those used here, his equation reads

Eq. 7:    ${\displaystyle f_{r}=\gamma \left(1-\beta \cos \theta _{s}\right)f_{s}.}$

The differences stem from the fact that Einstein evaluated the angle ${\displaystyle \theta _{s}}$ with respect to the source rest frame rather than the receiver rest frame. ${\displaystyle \theta _{s}}$ is not equal to ${\displaystyle \theta _{r}}$ because of the effect of relativistic aberration. The relativistic aberration equation is:

Eq. 8:    ${\displaystyle \cos \theta _{r}={\frac {\cos \theta _{s}-\beta }{1-\beta \cos \theta _{s}}}\,}$

Substituting the relativistic aberration equation Equation 8 into Equation 6 yields Equation 7, demonstrating the consistency of these alternate equations for the Doppler shift.

Setting ${\displaystyle \theta _{r}=0}$ in Equation 6 or ${\displaystyle \theta _{s}=0}$ in Equation 7 yields Equation 1, the expression for relativistic longitudinal Doppler shift.

A four-vector approach to deriving these results may be found in Landau and Lifshitz (2005).

Visualization

Figure 8. Comparison of the relativistic Doppler effect (top) with the non-relativistic effect (bottom).

Fig. 8 helps us understand, in a rough qualitative sense, how the relativistic Doppler effect and relativistic aberration differ from the non-relativistic Doppler effect and non-relativistic aberration of light. Assume that the observer is uniformly surrounded in all directions by yellow stars emitting monochromatic light of 570 nm. The arrows in each diagram represent the observer's velocity vector relative to its surroundings, with a magnitude of 0.89 c.

• In the non-relativistic case, the light ahead of the observer is blueshifted to a wavelength of 300 nm in the medium ultraviolet, while light behind the observer is redshifted to 5200 nm in the intermediate infrared. Because of the aberration of light, objects formerly at right angles to the observer appear shifted forwards by 42°.
• In the relativistic case, the light ahead of the observer is blueshifted to a wavelength of 137 nm in the far ultraviolet, while light behind the observer is redshifted to 2400 nm in the short wavelength infrared. Because of the relativistic aberration of light, objects formerly at right angles to the observer appear shifted forwards by 63°.
• In both cases, the monochromatic stars ahead of and behind the observer are Doppler-shifted towards invisible wavelengths. If, however, the observer had eyes that could see into the ultraviolet and infrared, he would see the stars ahead of him as brighter and more closely clustered together than the stars behind, but the stars would be far brighter and far more concentrated in the relativistic case.

Real stars are not monochromatic, but emit a range of wavelengths approximating a black body distribution. It is not necessarily true that stars ahead of the observer would show a bluer color. This is because the whole spectral energy distribution is shifted. At the same time that visible light is blueshifted into invisible ultraviolet wavelengths, infrared light is blueshifted into the visible range. Precisely what changes in the colors one sees depends on the physiology of the human eye and on the spectral characteristics of the light sources being observed.

Doppler effect on intensity

The Doppler effect (with arbitrary direction) also modifies the perceived source intensity: this can be expressed concisely by the fact that source strength divided by the cube of the frequency is a Lorentz invariant This implies that the total radiant intensity (summing over all frequencies) is multiplied by the fourth power of the Doppler factor for frequency.

As a consequence, since Planck's law describes the black-body radiation as having a spectral intensity in frequency proportional to ${\displaystyle \nu ^{3}/\left(e^{h\nu /kT}-1\right)}$ (where T is the source temperature and ν the frequency), we can draw the conclusion that a black body spectrum seen through a Doppler shift (with arbitrary direction) is still a black body spectrum with a temperature multiplied by the same Doppler factor as frequency.

This result provides one of the pieces of evidence that serves to distinguish the Big Bang theory from alternative theories proposed to explain the cosmological redshift.

Experimental verification

Since the transverse Doppler effect is one of the main novel predictions of the special theory of relativity, the detection and precise quantification of this effect has been an important goal of experiments attempting to validate special relativity.

Ives and Stilwell-type measurements

Figure 9. Why it is difficult to measure the transverse Doppler effect accurately using a transverse beam.

Einstein (1907) had initially suggested that the TDE might be measured by observing a beam of "canal rays" at right angles to the beam. Attempts to measure TDE following this scheme proved it to be impractical, since the maximum speed of particle beam available at the time was only a few thousandths of the speed of light.

Fig. 9 shows the results of attempting to measure the 4861 Angstrom line emitted by a beam of canal rays (a mixture of H1+, H2+, and H3+ ions) as they recombine with electrons stripped from the dilute hydrogen gas used to fill the Canal ray tube. Here, the predicted result of the TDE is a 4861.06 Angstrom line. On the left, longitudinal Doppler shift results in broadening the emission line to such an extent that the TDE cannot be observed. The middle figures illustrate that even if one narrows one's view to the exact center of the beam, very small deviations of the beam from an exact right angle introduce shifts comparable to the predicted effect.

Rather than attempt direct measurement of the TDE, Ives and Stilwell (1938) used a concave mirror that allowed them to simultaneously observe a nearly longitudinal direct beam (blue) and its reflected image (red). Spectroscopically, three lines would be observed: An undisplaced emission line, and blueshifted and redshifted lines. The average of the redshifted and blueshifted lines would be compared with the wavelength of the undisplaced emission line. The difference that Ives and Stilwell measured corresponded, within experimental limits, to the effect predicted by special relativity.

Various of the subsequent repetitions of the Ives and Stilwell experiment have adopted other strategies for measuring the mean of blueshifted and redshifted particle beam emissions. In some recent repetitions of the experiment, modern accelerator technology has been used to arrange for the observation of two counter-rotating particle beams. In other repetitions, the energies of gamma rays emitted by a rapidly moving particle beam have been measured at opposite angles relative to the direction of the particle beam. Since these experiments do not actually measure the wavelength of the particle beam at right angles to the beam, some authors have preferred to refer to the effect they are measuring as the "quadratic Doppler shift" rather than TDE.

Direct measurement of transverse Doppler effect

The advent of particle accelerator technology has made possible the production of particle beams of considerably higher energy than was available to Ives and Stilwell. This has enabled the design of tests of the transverse Doppler effect directly along the lines of how Einstein originally envisioned them, i.e. by directly viewing a particle beam at a 90° angle. For example, Hasselkamp et al. (1979) observed the Hα line emitted by hydrogen atoms moving at speeds ranging from 2.53×10⁸ cm/s to 9.28×10⁸ cm/s, finding the coefficient of the second order term in the relativistic approximation to be 0.52±0.03, in excellent agreement with the theoretical value of 1/2.

Other direct tests of the TDE on rotating platforms were made possible by the discovery of the Mössbauer effect, which enables the production of exceedingly narrow resonance lines for nuclear gamma ray emission and absorption. Mössbauer effect experiments have proven themselves easily capable of detecting TDE using emitter-absorber relative velocities on the order of 2×10⁴ cm/s. These experiments include ones performed by Hay et al. (1960), Champeney et al. (1965), and Kündig (1963).

Time dilation measurements

The transverse Doppler effect and the kinematic time dilation of special relativity are closely related. All validations of TDE represent validations of kinematic time dilation, and most validations of kinematic time dilation have also represented validations of TDE. An online resource, "What is the experimental basis of Special Relativity?" has documented, with brief commentary, many of the tests that, over the years, have been used to validate various aspects of special relativity. Kaivola et al. (1985) and McGowan et al. (1993) are examples of experiments classified in this resource as time dilation experiments. These two also represent tests of TDE. These experiments compared the frequency of two lasers, one locked to the frequency of a neon atom transition in a fast beam, the other locked to the same transition in thermal neon. The 1993 version of the experiment verified time dilation, and hence TDE, to an accuracy of 2.3×10⁻⁶.

Relativistic Doppler effect for sound and light

Figure 10. The relativistic Doppler shift formula is applicable to both sound and light.

First-year physics textbooks almost invariably analyze Doppler shift for sound in terms of Newtonian kinematics, while analyzing Doppler shift for light and electromagnetic phenomena in terms of relativistic kinematics. This gives the false impression that acoustic phenomena requires a different analysis than light and radio waves.

The traditional analysis of the Doppler effect for sound represents a low speed approximation to the exact, relativistic analysis. The fully relativistic analysis for sound is, in fact, equally applicable to both sound and electromagnetic phenomena.

Consider the spacetime diagram in Fig. 10. Worldlines for a tuning fork (the source) and a receiver are both illustrated on this diagram. Events O and A represent two vibrations of the tuning fork. The period of the fork is the magnitude of OA, and the inverse slope of AB represents the speed of signal propagation (i.e. the speed of sound) to event B. We can therefore write:

${\displaystyle c_{s}={\frac {x_{B}-x_{A}}{t_{B}-t_{A}}}}$ (speed of sound)

${\displaystyle v_{s}=-{\frac {x_{A}}{t_{A}}}}$        ${\displaystyle v_{r}={\frac {x_{B}}{t_{B}}}}$      (speeds of source and receiver)

${\displaystyle |OA|={\sqrt {t_{A}^{2}-(x_{A}/c)^{2}}}}$

${\displaystyle |OB|={\sqrt {t_{B}^{2}-(x_{B}/c)^{2}}}}$

${\displaystyle v_{s}}$ and ${\displaystyle v_{r}}$ are assumed to be less than ${\displaystyle c_{s},}$ since otherwise their passage through the medium will set up shock waves, invalidating the calculation. Some routine algebra gives the ratio of frequencies:

Eq. 9:    ${\displaystyle {\frac {f_{r}}{f_{s}}}={\frac {|OA|}{|OB|}}}$ ${\displaystyle ={\frac {1-v_{r}/c_{s}}{1+v_{s}/c_{s}}}{\sqrt {\frac {1-(v_{s}/c)^{2}}{1-(v_{r}/c)^{2}}}}}$

If ${\displaystyle v_{r}}$ and ${\displaystyle v_{s}}$ are small compared with ${\displaystyle c}$, the above equation reduces to the classical Doppler formula for sound.

If the speed of signal propagation ${\displaystyle c_{s}}$ approaches ${\displaystyle c}$, it can be shown that the absolute speeds ${\displaystyle v_{s}}$ and ${\displaystyle v_{r}}$ of the source and receiver merge into a single relative speed independent of any reference to a fixed medium. Indeed, we obtain Equation 1, the formula for relativistic longitudinal Doppler shift.

Analysis of the spacetime diagram in Fig. 10 gave a general formula for source and receiver moving directly along their line of sight, i.e. in collinear motion.

Figure 11. A source and receiver are moving in different directions and speeds in a frame where the speed of sound is independent of direction.

Fig. 11 illustrates a scenario in two dimensions. The source moves with velocity ${\displaystyle \mathbf {v_{s}} }$ (at the time of emission). It emits a signal which travels at velocity ${\displaystyle \mathbf {C} }$ towards the receiver, which is traveling at velocity ${\displaystyle \mathbf {v_{r}} }$ at the time of reception. The analysis is performed in a coordinate system in which the signal's speed ${\displaystyle |\mathbf {C} |}$ is independent of direction.

The ratio between the proper frequencies for the source and receiver is

Eq. 10:    ${\displaystyle {\frac {f_{r}}{f_{s}}}={\frac {1-{\frac {|\mathbf {v_{r}} |}{\mathbf {|C|} }}\cos(\theta _{\mathbf {C,v_{r}} })}{1-{\frac {|\mathbf {v_{s}} |}{\mathbf {|C|} }}\cos(\theta _{\mathbf {C,v_{s}} })}}{\sqrt {\frac {1-(v_{s}/c)^{2}}{1-(v_{r}/c)^{2}}}}}$

The leading ratio has the form of the classical Doppler effect, while the square root term represents the relativistic correction. If we consider the angles relative to the frame of the source, then ${\displaystyle v_{s}=0}$ and the equation reduces to Equation 7, Einstein's 1905 formula for the Doppler effect. If we consider the angles relative to the frame of the receiver, then ${\displaystyle v_{r}=0}$ and the equation reduces to Equation 6, the alternative form of the Doppler shift equation discussed previously.

Free-electron laser

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Free-electron_laser 

 

The free-electron laser FELIX Radboud University, Netherlands.

A free-electron laser (FEL) is a (fourth generation) synchrotron light source producing extremely brilliant and short pulses of synchrotron radiation. An FEL functions and behaves in many ways like a laser, but instead of using stimulated emission from atomic or molecular excitations, it employs relativistic electrons as a gain medium. Synchrotron radiation is generated as a bunch of electrons passes through a magnetic structure (called undulator or wiggler). In an FEL, this radiation is further amplified as the synchrotron radiation re-interacts with the electron bunch such that the electrons start to emit coherently, thus allowing an exponential increase in overall radiation intensity.

As electron kinetic energy and undulator parameters can be adapted as desired, free-electron lasers are tunable and can be built for a wider frequency range than any type of laser, currently ranging in wavelength from microwaves, through terahertz radiation and infrared, to the visible spectrum, ultraviolet, and X-ray.

 

Schematic representation of an undulator, at the core of a free-electron laser.

The first free-electron laser was developed by John Madey in 1971 at Stanford University utilizing technology developed by Hans Motz and his coworkers, who built an undulator at Stanford in 1953, using the wiggler magnetic configuration. Madey used a 43 MeV electron beam and 5 m long wiggler to amplify a signal.

Beam creation

The undulator of FELIX.

To create an FEL, a beam of electrons is accelerated to almost the speed of light. The beam passes through a periodic arrangement of magnets with alternating poles across the beam path, which creates a side to side magnetic field. The direction of the beam is called the longitudinal direction, while the direction across the beam path is called transverse. This array of magnets is called an undulator or a wiggler, because the Lorentz force of the field forces the electrons in the beam to wiggle transversely, traveling along a sinusoidal path about the axis of the undulator.

The transverse acceleration of the electrons across this path results in the release of photons (synchrotron radiation), which are monochromatic but still incoherent, because the electromagnetic waves from randomly distributed electrons interfere constructively and destructively in time. The resulting radiation power scales linearly with the number of electrons. Mirrors at each end of the undulator create an optical cavity, causing the radiation to form standing waves, or alternately an external excitation laser is provided. The synchrotron radiation becomes sufficiently strong that the transverse electric field of the radiation beam interacts with the transverse electron current created by the sinusoidal wiggling motion, causing some electrons to gain and others to lose energy to the optical field via the ponderomotive force.

This energy modulation evolves into electron density (current) modulations with a period of one optical wavelength. The electrons are thus longitudinally clumped into microbunches, separated by one optical wavelength along the axis. Whereas an undulator alone would cause the electrons to radiate independently (incoherently), the radiation emitted by the bunched electrons is in phase, and the fields add together coherently.

The radiation intensity grows, causing additional microbunching of the electrons, which continue to radiate in phase with each other. This process continues until the electrons are completely microbunched and the radiation reaches a saturated power several orders of magnitude higher than that of the undulator radiation.

The wavelength of the radiation emitted can be readily tuned by adjusting the energy of the electron beam or the magnetic-field strength of the undulators.

FELs are relativistic machines. The wavelength of the emitted radiation, ${\displaystyle \lambda _{r}}$, is given by

${\displaystyle \lambda _{r}={\frac {\lambda _{u}}{2\gamma ^{2}}}\left(1+{\frac {K^{2}}{2}}\right)}$

or when the wiggler strength parameter K, discussed below, is small

${\displaystyle \lambda _{r}\propto {\frac {\lambda _{u}}{2\gamma ^{2}}}}$

where ${\displaystyle \lambda _{u}}$ is the undulator wavelength (the spatial period of the magnetic field), ${\displaystyle \gamma }$ is the relativistic Lorentz factor and the proportionality constant depends on the undulator geometry and is of the order of 1.

This formula can be understood as a combination of two relativistic effects. Imagine you are sitting on an electron passing through the undulator. Due to Lorentz contraction the undulator is shortened by a ${\displaystyle \gamma }$ factor and the electron experiences much shorter undulator wavelength ${\displaystyle \lambda _{u}/\gamma }$. However, the radiation emitted at this wavelength is observed in the laboratory frame of reference and the relativistic Doppler effect brings the second ${\displaystyle \gamma }$ factor to the above formula. In an X-ray FEL the typical undulator wavelength of 1 cm is transformed to X-ray wavelengths on the order of 1 nm by ${\displaystyle \gamma }$ ≈ 2000, i.e. the electrons have to travel with the speed of 0.9999998c.

Wiggler strength parameter K

K, a dimensionless parameter, defines the wiggler strength as the relationship between the length of a period and the radius of bend,

${\displaystyle K={\frac {\gamma \lambda _{u}}{2\pi \rho }}={\frac {eB_{0}\lambda _{u}}{2\pi m_{e}c}}}$

where ${\displaystyle \rho }$ is the bending radius, ${\displaystyle B_{0}}$ is the applied magnetic field, ${\displaystyle m_{e}}$ is the electron mass, and ${\displaystyle e}$ is the elementary charge.

Expressed in practical units, the dimensionless undulator parameter is ${\displaystyle K=0.934\cdot B_{0}\,{\text{[T]}}\cdot \lambda _{u}\,{\text{[cm]}}}$.

Quantum effects

In most cases, the theory of classical electromagnetism adequately accounts for the behavior of free electron lasers. For sufficiently short wavelengths, quantum effects of electron recoil and shot noise may have to be considered.

FEL construction

Free-electron lasers require the use of an electron accelerator with its associated shielding, as accelerated electrons can be a radiation hazard if not properly contained. These accelerators are typically powered by klystrons, which require a high-voltage supply. The electron beam must be maintained in a vacuum, which requires the use of numerous vacuum pumps along the beam path. While this equipment is bulky and expensive, free-electron lasers can achieve very high peak powers, and the tunability of FELs makes them highly desirable in many disciplines, including chemistry, structure determination of molecules in biology, medical diagnosis, and nondestructive testing.

Infrared and terahertz FELs

The Fritz Haber Institute in Berlin completed a mid-infrared and terahertz FEL in 2013.

X-ray FELs

The lack of a material to make mirrors that can reflect extreme ultraviolet and x-rays means that FELs at these frequencies cannot use a resonant cavity like other lasers, which reflects the radiation so it makes multiple passes through the undulator. Consequently, in an X-ray FEL (XFEL) the output beam is produced by a single pass of radiation through the undulator. This requires that there be enough amplification over a single pass to produce an adequately bright beam.

Because of the lack of mirrors, XFELs use long undulators. The underlying principle of the intense pulses from the X-ray laser lies in the principle of self-amplified spontaneous emission (SASE), which leads to the microbunching. Initially all electrons are distributed evenly and emit only incoherent spontaneous radiation. Through the interaction of this radiation and the electrons' oscillations, they drift into microbunches separated by a distance equal to one radiation wavelength. Through this interaction, all electrons begin emitting coherent radiation in phase. All emitted radiation can reinforce itself perfectly whereby wave crests and wave troughs are always superimposed on one another in the best possible way. This results in an exponential increase of emitted radiation power, leading to high beam intensities and laser-like properties. Examples of facilities operating on the SASE FEL principle include the Free electron LASer in Hamburg (FLASH), the Linac Coherent Light Source (LCLS) at the SLAC National Accelerator Laboratory, the European x-ray free electron laser (EuXFEL) in Hamburg, the SPring-8 Compact SASE Source (SCSS) in Japan, the SwissFEL at the Paul Scherrer Institute (Switzerland), the SACLA at the RIKEN Harima Institute in Japan, and the PAL-XFEL (Pohang Accelerator Laboratory X-ray Free-Electron Laser) in Korea.

Self-seeding

One problem with SASE FELs is the lack of temporal coherence due to a noisy startup process. To avoid this, one can "seed" an FEL with a laser tuned to the resonance of the FEL. Such a temporally coherent seed can be produced by more conventional means, such as by high harmonic generation (HHG) using an optical laser pulse. This results in coherent amplification of the input signal; in effect, the output laser quality is characterized by the seed. While HHG seeds are available at wavelengths down to the extreme ultraviolet, seeding is not feasible at x-ray wavelengths due to the lack of conventional x-ray lasers.

In late 2010, in Italy, the seeded-FEL source FERMI@Elettra started commissioning, at the Trieste Synchrotron Laboratory. FERMI@Elettra is a single-pass FEL user-facility covering the wavelength range from 100 nm (12 eV) to 10 nm (124 eV), located next to the third-generation synchrotron radiation facility ELETTRA in Trieste, Italy.

In 2012, scientists working on the LCLS overcame the seeding limitation for x-ray wavelengths by self-seeding the laser with its own beam after being filtered through a diamond monochromator. The resulting intensity and monochromaticity of the beam were unprecedented and allowed new experiments to be conducted involving manipulating atoms and imaging molecules. Other labs around the world are incorporating the technique into their equipment.

Research

Biomedical

Basic research

Researchers have explored free-electron lasers as an alternative to synchrotron light sources that have been the workhorses of protein crystallography and cell biology.

Exceptionally bright and fast X-rays can image proteins using x-ray crystallography. This technique allows first-time imaging of proteins that do not stack in a way that allows imaging by conventional techniques, 25% of the total number of proteins. Resolutions of 0.8 nm have been achieved with pulse durations of 30 femtoseconds. To get a clear view, a resolution of 0.1–0.3 nm is required. The short pulse durations allow images of X-ray diffraction patterns to be recorded before the molecules are destroyed.  The bright, fast X-rays were produced at the Linac Coherent Light Source at SLAC. As of 2014 LCLS was the world's most powerful X-ray FEL.

Due to the increased repetition rates of the next-generation X-ray FEL sources, such as the European XFEL, the expected number of diffraction patterns is also expected to increase by a substantial amount. ^[23] The increase in the number of diffraction patterns will place a large strain on existing analysis methods. To combat this, several methods have been research in order to be able to sort the huge amount of data typical X-ray FEL experiments will generate.  While the various methods have been shown to be effective, it is clear that to pave the way towards single-particle X-ray FEL imaging at full repetition rates, several challenges have to be overcome before the next resolution revolution can be achieved. 

New biomarkers for metabolic diseases: taking advantage of the selectivity and sensitivity when combining infrared ion spectroscopy and mass spectrometry scientists can provide a structural fingerprint of small molecules in biological samples, like blood or urine. This new and unique methodology is generating exciting new possibilities to better understand metabolic diseases and develop novel diagnostic and therapeutic strategies.

Surgery

Research by Glenn Edwards and colleagues at Vanderbilt University's FEL Center in 1994 found that soft tissues including skin, cornea, and brain tissue could be cut, or ablated, using infrared FEL wavelengths around 6.45 micrometres with minimal collateral damage to adjacent tissue. This led to surgeries on humans, the first ever using a free-electron laser. Starting in 1999, Copeland and Konrad performed three surgeries in which they resected meningioma brain tumors. Beginning in 2000, Joos and Mawn performed five surgeries that cut a window in the sheath of the optic nerve, to test the efficacy for optic nerve sheath fenestration. These eight surgeries produced results consistent with the standard of care and with the added benefit of minimal collateral damage. A review of FELs for medical uses is given in the 1st edition of Tunable Laser Applications.

Fat removal

Several small, clinical lasers tunable in the 6 to 7 micrometre range with pulse structure and energy to give minimal collateral damage in soft tissue have been created. At Vanderbilt, there exists a Raman shifted system pumped by an Alexandrite laser.

Rox Anderson proposed the medical application of the free-electron laser in melting fats without harming the overlying skin. At infrared wavelengths, water in tissue was heated by the laser, but at wavelengths corresponding to 915, 1210 and 1720 nm, subsurface lipids were differentially heated more strongly than water. The possible applications of this selective photothermolysis (heating tissues using light) include the selective destruction of sebum lipids to treat acne, as well as targeting other lipids associated with cellulite and body fat as well as fatty plaques that form in arteries which can help treat atherosclerosis and heart disease.

Military

FEL technology is being evaluated by the US Navy as a candidate for an antiaircraft and anti-missile directed-energy weapon. The Thomas Jefferson National Accelerator Facility's FEL has demonstrated over 14 kW power output. Compact multi-megawatt class FEL weapons are undergoing research.

On June 9, 2009 the Office of Naval Research announced it had awarded Raytheon a contract to develop a 100 kW experimental FEL. On March 18, 2010 Boeing Directed Energy Systems announced the completion of an initial design for U.S. Naval use. A prototype FEL system was demonstrated, with a full-power prototype scheduled by 2018.

 

Degrees of freedom (physics and chemistry)

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Degrees_of_freedom_(physics_and_chemistry)

In physics and chemistry, a degree of freedom is an independent physical parameter in the formal description of the state of a physical system. The set of all states of a system is known as the system's phase space, and the degrees of freedom of the system are the dimensions of the phase space.

The location of a particle in three-dimensional space requires three position coordinates. Similarly, the direction and speed at which a particle moves can be described in terms of three velocity components, each in reference to the three dimensions of space. If the time evolution of the system is deterministic, where the state at one instant uniquely determines its past and future position and velocity as a function of time, such a system has six degrees of freedom. If the motion of the particle is constrained to a lower number of dimensions – for example, the particle must move along a wire or on a fixed surface – then the system has fewer than six degrees of freedom. On the other hand, a system with an extended object that can rotate or vibrate can have more than six degrees of freedom.

In classical mechanics, the state of a point particle at any given time is often described with position and velocity coordinates in the Lagrangian formalism, or with position and momentum coordinates in the Hamiltonian formalism.

In statistical mechanics, a degree of freedom is a single scalar number describing the microstate of a system. The specification of all microstates of a system is a point in the system's phase space.

In the 3D ideal chain model in chemistry, two angles are necessary to describe the orientation of each monomer.

It is often useful to specify quadratic degrees of freedom. These are degrees of freedom that contribute in a quadratic function to the energy of the system.

Gas molecules

Different ways of visualizing the 6 degrees of freedom of a diatomic molecule. (CM: center of mass of the system, T: translational motion, R: rotational motion, V: vibrational motion.)

In three-dimensional space, three degrees of freedom are associated with the movement of a particle. A diatomic gas molecule has 6 degrees of freedom. This set may be decomposed in terms of translations, rotations, and vibrations of the molecule. The center of mass motion of the entire molecule accounts for 3 degrees of freedom. In addition, the molecule has two rotational degrees of motion and one vibrational mode. The rotations occur around the two axes perpendicular to the line between the two atoms. The rotation around the atom–atom bond is not a physical rotation. This yields, for a diatomic molecule, a decomposition of:

${\displaystyle N=6=3+2+1.}$

For a general, non-linear molecule, all 3 rotational degrees of freedom are considered, resulting in the decomposition:

${\displaystyle 3N=3+3+(3N-6)}$

which means that an N-atom molecule has 3N − 6 vibrational degrees of freedom for N > 2. In special cases, such as adsorbed large molecules, the rotational degrees of freedom can be limited to only one.

As defined above one can also count degrees of freedom using the minimum number of coordinates required to specify a position. This is done as follows:

1. For a single particle we need 2 coordinates in a 2-D plane to specify its position and 3 coordinates in 3-D space. Thus its degree of freedom in a 3-D space is 3.
2. For a body consisting of 2 particles (ex. a diatomic molecule) in a 3-D space with constant distance between them (let's say d) we can show (below) its degrees of freedom to be 5.

Let's say one particle in this body has coordinate (x₁, y₁, z₁) and the other has coordinate (x₂, y₂, z₂) with  z₂ unknown. Application of the formula for distance between two coordinates

${\displaystyle d={\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}+(z_{2}-z_{1})^{2}}}}$

results in one equation with one unknown, in which we can solve for z₂. One of x₁, x₂, y₁, y₂, z₁, or z₂ can be unknown.

Contrary to the classical equipartition theorem, at room temperature, the vibrational motion of molecules typically makes negligible contributions to the heat capacity. This is because these degrees of freedom are frozen because the spacing between the energy eigenvalues exceeds the energy corresponding to ambient temperatures (k_BT). In the following table such degrees of freedom are disregarded because of their low effect on total energy. Then only the translational and rotational degrees of freedom contribute, in equal amounts, to the heat capacity ratio. This is why γ=5/3 for monatomic gases and γ=7/5 for diatomic gases at room temperature.

However, at very high temperatures, on the order of the vibrational temperature (Θ_vib), vibrational motion cannot be neglected.

Vibrational temperatures are between 10³ K and 10⁴ K.

	Monatomic	Linear molecules	Non-linear molecules
Translation (x, y, and z)	3	3	3
Rotation (x, y, and z)	0	2	3
Total (disregarding Vibration at room temperatures)	3	5	6
Vibration	0	3N − 5	3N − 6
Total (including Vibration)	3	3N	3N

Independent degrees of freedom

The set of degrees of freedom X₁, ... , X_N of a system is independent if the energy associated with the set can be written in the following form:

${\displaystyle E=\sum _{i=1}^{N}E_{i}(X_{i}),}$

where E_i is a function of the sole variable X_i.

example: if X₁ and X₂ are two degrees of freedom, and E is the associated energy:

• If ${\displaystyle E=X_{1}^{4}+X_{2}^{4}}$, then the two degrees of freedom are independent.
• If ${\displaystyle E=X_{1}^{4}+X_{1}X_{2}+X_{2}^{4}}$, then the two degrees of freedom are not independent. The term involving the product of X₁ and X₂ is a coupling term that describes an interaction between the two degrees of freedom.

For i from 1 to N, the value of the ith degree of freedom X_i is distributed according to the Boltzmann distribution. Its probability density function is the following:

${\displaystyle p_{i}(X_{i})={\frac {e^{-{\frac {E_{i}}{k_{B}T}}}}{\int dX_{i}\,e^{-{\frac {E_{i}}{k_{B}T}}}}}}$,

In this section, and throughout the article the brackets ${\displaystyle \langle \rangle }$ denote the mean of the quantity they enclose.

The internal energy of the system is the sum of the average energies associated with each of the degrees of freedom:

${\displaystyle \langle E\rangle =\sum _{i=1}^{N}\langle E_{i}\rangle .}$

Quadratic degrees of freedom

A degree of freedom X_i is quadratic if the energy terms associated with this degree of freedom can be written as

${\displaystyle E=\alpha _{i}\,\,X_{i}^{2}+\beta _{i}\,\,X_{i}Y}$,

where Y is a linear combination of other quadratic degrees of freedom.

example: if X₁ and X₂ are two degrees of freedom, and E is the associated energy:

• If ${\displaystyle E=X_{1}^{4}+X_{1}^{3}X_{2}+X_{2}^{4}}$, then the two degrees of freedom are not independent and non-quadratic.
• If ${\displaystyle E=X_{1}^{4}+X_{2}^{4}}$, then the two degrees of freedom are independent and non-quadratic.
• If ${\displaystyle E=X_{1}^{2}+X_{1}X_{2}+2X_{2}^{2}}$, then the two degrees of freedom are not independent but are quadratic.
• If ${\displaystyle E=X_{1}^{2}+2X_{2}^{2}}$, then the two degrees of freedom are independent and quadratic.

For example, in Newtonian mechanics, the dynamics of a system of quadratic degrees of freedom are controlled by a set of homogeneous linear differential equations with constant coefficients.

Quadratic and independent degree of freedom

X₁, ... , X_N are quadratic and independent degrees of freedom if the energy associated with a microstate of the system they represent can be written as:

${\displaystyle E=\sum _{i=1}^{N}\alpha _{i}X_{i}^{2}}$

Equipartition theorem

In the classical limit of statistical mechanics, at thermodynamic equilibrium, the internal energy of a system of N quadratic and independent degrees of freedom is:

${\displaystyle U=\langle E\rangle =N\,{\frac {k_{B}T}{2}}}$

Here, the mean energy associated with a degree of freedom is:

${\displaystyle \langle E_{i}\rangle =\int dX_{i}\,\,\alpha _{i}X_{i}^{2}\,\,p_{i}(X_{i})={\frac {\int dX_{i}\,\,\alpha _{i}X_{i}^{2}\,\,e^{-{\frac {\alpha _{i}X_{i}^{2}}{k_{B}T}}}}{\int dX_{i}\,\,e^{-{\frac {\alpha _{i}X_{i}^{2}}{k_{B}T}}}}}}$

${\displaystyle \langle E_{i}\rangle ={\frac {k_{B}T}{2}}{\frac {\int dx\,\,x^{2}\,\,e^{-{\frac {x^{2}}{2}}}}{\int dx\,\,e^{-{\frac {x^{2}}{2}}}}}={\frac {k_{B}T}{2}}}$

Since the degrees of freedom are independent, the internal energy of the system is equal to the sum of the mean energy associated with each degree of freedom, which demonstrates the result.

Generalizations

The description of a system's state as a point in its phase space, although mathematically convenient, is thought to be fundamentally inaccurate. In quantum mechanics, the motion degrees of freedom are superseded with the concept of wave function, and operators which correspond to other degrees of freedom have discrete spectra. For example, intrinsic angular momentum operator (which corresponds to the rotational freedom) for an electron or photon has only two eigenvalues. This discreteness becomes apparent when action has an order of magnitude of the Planck constant, and individual degrees of freedom can be distinguished.