Relativistic Doppler Shift

The experimentally observed isomer shift, (5exp, includes a relativistic contribution, which is called second-order Doppler shift, sod> and which adds to the genuine isomer shift d. 

Now let us study Fig. 12 again and seek the direction (l) in which the traveler should experience zero Doppler shift. She should not look backward because in that direction there is a redshift. Nor should she look directly sideways because even then there is a redshift, the relativistic transverse redshift. As forward is the blueshift, she should look slightly forward. The way to find the zero Doppler shift for incoming light (a) is as follows ... 

Thus the Doppler shift is the difference between the Doppler-shifted wavelength (k) and the original wavelength (A0) divided by Aq- The numerator is the classical Doppler redshift from a moving light source, while the denominator represents the red-shift caused by the relativistic time dilation resulting from the total velocity, which is independent of the direction of motion. 

The existence of a relativistic temperature-dependent contribution to the chemical isomer shift was pointed out independently by Pound and Rebka [6] and by Josephson [7]. The emitting or absorbing atom is vibrating on its lattice site in the crystal. The frequency of oscillation about the mean position is of the order of 10 per second, so that the average displacement during the Mossbauer event is zero. However, there is a term in the Doppler shift which depends on v, so that the mean value is non-zero. 

The expression for the Doppler shift given in ihe chapter and in Problem 8-8 is an approximation that works al relatively low speeds. The relativistic expression for the Doppler shifi is... 

Any astronomically measured frequency shift consists of several components, including the chemical shift, described here. Other contributions include relativistic gravitational redshifting, a distance-dependent redshift caused by the topological curvature of space-time, and a Doppler shift where the source is in relative motion. 

Example 9.1 A parallel beam of Ne ions accelerated by 10 keV moves with the velocity = 3 x 10 m/s. When the beam is crossed perpendicularly by a single-mode laser beam tuned to a transition with X = 500 nm, even ions with Vx =Vy=0 show a quadratic relativistic Doppler shift of Av/v = 5 X 10 , which yields an absolute shift of Av = 250 MHz. This should be compared with the linear Doppler shift of 600 GHz, which appears when the laser beam is parallel to the ion beam (Example 4.6). 

The rate of radiation from dipolar emitters moving with high velocities is appreciably reduced compared to the rate of ordinary electric-dipole radiation at Doppler-shifted frequencies. This reduction is due to the magnetization arising in a dipolar emitter moving relative to the detector, which is revealed by a nonrelativistic treatment. Additional corrections to the emission rate are encountered at relativistic velocities. The nature of all these corrections is discussed below. They are then included in the analysis of spontaneous and stimulated emission from relativistic channeled particles (electrons and positrons) in a crystal, as a model example. In this system, the particle propagates relativistically in the... 

Most experiments in this field are perfonned with highly relativistic particles (10 . Y/ 10 ). Then channeling radiation is strongly concentrated about the forward direction z and consists of spectral peaks whose frequencies (usually in the x- or y-rsnge) are the dipole transition frequencies cOgg multiplied by the huge Doppler shift (l-q /c) 2Y. ... 

The second-order Doppler shift, (5sod> which is often called temperature shift, is a peak shift related to the relativistic Doppler effect originating from the thermal motion of the nuclei. If the Mossbauer atom has a speed u, and moves in a direction making angle a with the direction of the y ray it emits, then the v frequency of the emitted y ray will differ from the Vo frequency it had if the atom had been at rest. The v frequency is related to Vo in the following way ... 

The considerable Doppler shift in all collinear-beam experiments has opened up a few general applications beyond the spectroscopy of particular atomic or molecular systems. The scope of such applications ranges from simple beam velocity analyses to precision experiments related to metrology or problems of fundamental physics. These latter include the calibration of high voltages and measurements of the relativistic Doppler effect, for which the atomic transition frequency provides an intrinsic clock. 

The measurement of absolute beam velocities, or the calibration of voltages, is already quite sensitive to the relativistic quadratic term in the Doppler shift formula (7). In fact, this transverse Doppler shift, caused by the time dilatation factor y = (1 - j8 )" , was first observed in the spectral lines of fast-moving hydrogen atoms from a 30-keV beam of H2 ions, viewed along and opposite the direction of propagation. Comparable accuracy in the percent range was also achieved in Mossbauer experi-ments, and more recently the time dilatation factor on the muon lifetime was determined to 1 x 10". ... 

Beam-Laser Techniques. By exchanging the exciting foil in the beam-foil teclmique for a focused laser beam, selective excitation can be obtained and the problem of cascades is eliminated [9.142]. By directing the laser beam at a certain angle Q with respect to the ion beam considerable Doppler shifts in the interaction with the fast beam are obtained. The relativistically correct formula for the interaction wavelength A, experienced by the ions that are illuminated by a laser of wavelength g is given by... 

The usual experimental set-up works in transmission geometry as shown in fig. 5 (top) The strength of resonance absorption is determined by the reduction of y-ray intensity reaching the detector. Not taking into account the hyperfine interactions, to be discussed later, and neglecting a small relativistic effect (the second-order Doppler shift) one has an exact energy match for the zero-phonon lines of N(E) and Resonance absorption is now strongest. 

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