Phase and distance effects in rf modulation of Mössbauer radiation under stationary conditions

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Phase and distance effects in rf modulation of Mössbauer radiation under stationary conditions

N.I. Ognjanov, L.T. Tsankov

To cite this version:

N.I. Ognjanov, L.T. Tsankov. Phase and distance effects in rf modulation of Mössbauer radiation under stationary conditions. Journal de Physique, 1983, 44 (7), pp.865-870.

�10.1051/jphys:01983004407086500�. �jpa-00209670�

Phase and distance effects in rf modulation

of Mössbauer radiation under stationary ^conditions

N. I. Ognjanov ^{and L. T. Tsankov}

Department ^{of Atomic} Physics, University ^of Sofia, ^{1126 -} Sofia, Bulgaria (Reçu ^le 30 juillet 1982, accepté ^{le 8}

1983)

Résumé.

Les effets de phase apparaissant ^dans

système constitué d’une source Mössbauer et d’un absorbant Mössbauer, ^{vibrant avec une} fréquence ultra-sonore, ^{sont étudiés} théoriquement. ^{Il est} démontré que l’intensité de la radiation sans recul dépend ^{de la} différence absolue de phase ^entre les vibrations de la source et de l’absor- bant, ainsi que du déphasage effectif, ^dû

temps ^{fini de} propagation ^{de l’onde} électromagnétique ^{entre eux} (effet

de distance). ^{La relation entre} l’effet de distance et la théorie générale de la relativité est discutée de manière critique.

Abstract.

The phase effects, appearing ^{in a} system consisting ^{of a Mössbauer source and}

^Mössbauer absorber, both vibrating ^at ultrasonic frequency,

theoretically investigated. It is shown that the intensity ^{of the trans-}

mitted recoilless radiation depends ^{on both the absolute} phase shift between the vibrations of the source and those of the absorber, ^{and on the effective} dephasing ^{due to} the finite propagation time of the electromagnetic

between them (the ^so-called

^distance effect »). The relation between the distance effect and general relativity theory ^is critically ^discussed.

Classification Physics ^Abstracts

03.40K - 76.80

04.40

1. Introduction.

In the present paper ^we would like to pay attention to the possibilities ^of investigating certain effects

depending ^{on the} phases of ultrasonic vibrations

applied ^{to a Mossbauer source or} absorber, and/or

on the distance between the components ^{of a more} complex system of vibrators under stationary (i.e.

time-independent) conditions of observation (some non-stationary ^have been studied in [2, 3]). ^This work,

is based on the formalism proposed ⁱⁿ [1], ^which

will be .referred ^{to as} part ^I.

2. Phase and distance effects in a two-specimen system.

Obviously, phase effects under stationary ^conditions

could only ^{be observed in} systems containing ^more

than one vibrating ^resonant specimen. ^The simplest

system of that kind consists of a Mossbauer source

and an absorber, ^both performing coherent harmonic

rf vibrations. For the sake of simplicity ^{we will assume}

that the conditions (19) ^and (29) ^of part ^{I are} fulfilled,

i.e.

It is assumed also that the source and the absorber possess single ^resonant lines unshifted with respect ^to each other, ^{centred at} angular frequency coo. Then the functions fo(m) ^and f1(ro) ^{have the} following ^form [4, 5] :

where D stands for the effective resonant thickness of the absorber. The spectral distribution of the trans- mitted radiation I(w) ^can be calculated by ^{means of} equations (24) ^and (28’) ^of part ^{I :}

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01983004407086500

866

where

Since f1(ro - pQ) ,;-, ^{1 and} fo(ro - pQ) ;-- ⁰ if w - roo - pQ I ^» F and 0 >> r, ^then fl(ro - kQ)

and fl(ro - lQ) may be replaced by ¹ - bsk ⁺ 6.,k fl (Co - sQ) ^{and 1} - ðsl ⁺ ðs1fl(ro - sQ) respectively.

Then the dependence of Ski ^{on k} and 1 is reduced to some multiple factors of the form ðsk ei(k-l)t/I etc. Sums over

k and 1 can be carried out using the summation theorem for Bessel functions [6]. The final result is :

where

x is determined by

It is seen that the spectrum consists of a sum of satellite lines, ^{centred at} coo + sQ (s

1, 0, 1,...).

The shape of the individual lines is, ⁱⁿ general, non-Lorentzian. The intensity ^at the maximum of the s-th satel- lite is :

The full intensity ^S may be obtained using equations (14), (18’) ^and (28’) ^from part ^{I :}

Since Q » r, ^{it is} easily ^{seen that the} factor 1 fl(W) 12 ^{appearing in the integrand could be neglected for}

k # ^{0. The} remaining integrals ^{are well known and we} give directly the result :

here Io ^{is a modified} Bessel function. Taking ^{into account that}

we obtain :

(the inessential factor nr j2 ^is omitted).

1(m) ^{and S} depend ^{on the} phases of the rf vibrations and on the distance source-absorber through Aol, A02’ t/1 ^{and x. We} shall consider below the expression (9) only, since it is more convenient for theoretical investiga-

tion and more accessible for experimenting.

The dependence ^{of interest} appears in a purest ^{form when} A0 = A1 = ^{A. Then} (9) ^{becomes :}

where 0 denotes the ^« effective phase ^{shift » : :}

It is seen from (10) ^{that S is a} periodical function of 0 with a period ^{2 n and is} symmetric ^with respect ^{to the} line 8 = . If 2 A 2.4 (the ^{first zero} of Jo), ^{S is} monotonically increasing ^{in the interval} [0, n] ^{and has absolute}

maximum at 0 = 7r. If 2 A > xk (Xk being ^{the k-th zero of} J 0)’ ^{S has k local} (and absolute) maxima of the same

magnitude ^{at the} points ^{2 A} sin -

Xk. The local minima of S ^{are at} points ^{2 A} sin 0 ^{= yk, Yk being the k-th}

local extremum of Jo. ^A plot ^{of S vs.} 0 for several amplitudes ^{A and D}

^{10 is} presented ^on figure ^1.

Fig. ^1.

^Full intensity ^{of the} transmitted radiation

dephasing 0 between the source and the absorber for D

10 and for several values of the reduced amplitude ^A. So denotes the intensity ^{at 0}

^0.

Concluding ^this part ^{of the} work, ^we would like to cite here, for the sake of completeness, ^an expression

for the intensity of the transmitted radiation for a system consisting ^{of that} just considered plus ^{a second resonant}

absorber having ^identical properties ^{as the first one :}

868

Further we shall discuss briefly ^the possibilities ^for observing the considered effects in the case of incoherent modulation. According ^to equation (30) ^of part I, ^{we have :}

where Scoh is defined by (9), Sincoh is the total intensity in the incoherent case, and Pi(Ai, gi) is the mutual distribu- tion function of Ai, (Pi-

In accordance with the discussion in part I, ^we may consider a system ^{with a casual} (or closely ^to casual) phase distribution, ^{and a} Rayleigh distribution of the amplitudes Ao, A, (see Eq.(32), part I). ^{If we take} equal

width parameters ^{in both} distribution functions we obtain :

Integral (13) ^{does not} result in a convenient analytical expression. Numerical calculations show that, although ^the averaging ⁱⁿ (13) decreases the effect, it remains large enough ^for experimental investigation. ^This

is demonstrated in figure 2, ^where Sincoh is plotted ^vs. dephasing ^{0 for A}

^{1, D}

^{10. For} comparison, ^the depen-

dence of Soh ^{VS. 0 is} plotted ^{in the same} figure ^{for the same} parameter ^values.

Fig. ^2.

^{The same}

ⁱⁿ figure ^{1 in the case of} incoherent vibrations with Rayleigh distribution of the amplitude (the Sincoh curve). ^The S.h ^{curve relates to} the coherent case (casual distribution of the amplitude).

3. The distance effect and general relativity theory.

The intensity of the transmitted radiation (10) depends ^{on both the «} absolute » phase shift between the vibra- tions of the ^source and the absorber and on the « effective dephasing » ^caused by the finite propagation ^{time of}

the electromagnetic ^{wave between} them. This last effect, usually ^{known as} «the distance effect » [7-9] ^{will be}

discussed here in more detail because of its hypothetical connection with the general relativity theory (GRT).

For the first time the dependence ^{of S on the} distance source-absorber was discussed by ^B6mmel [7] ⁱⁿ

connection with GRT. B6mmel considers the ^case go

9, and proposes ^a linear behaviour of S when the distance source-absorber and/or ^the acceleration of the system ^are changed.

In reference [8] ^{a new} formula has been obtained analogous ^to equation (24), part ^{I for n}

1, ^{2. It} yields ^a quadratic dependence ^{of S on} the distance when go

({Jl and the distance is small. Probably, this fact has been taken as a basis by the authors of the later work [9] ^to conclude that the effects considered in [7] ^and [8] ^are

different and that additional experiments ^should provide ^a clarification of the phenomenon.

In our opinion, ^the problem of the theoretical description of the distance effect could be solved correctly by ^{means of a more careful account} for the connection of this effect and GRT. First of all, ^{it is} necessary ^to point

out that the existence of an effect, depending ^on the acceleration of the system, would contradict the principle

of local inertiality

^{one of the} fundamental principles of GRT. A GRT approach ^is required if the processes

are considered in the self frame of the source, ^or that of the absorber (or ⁱⁿ any other accelerated frame). ^{We are}

going ^to show how this might ^{be done. In} this section we use the standard coordinate annotations of GRT ; ^the

Lorentz metric tensor has the form flllv

diag ( - 1, 1, 1, 1). ^{We restrict to} po

cP 1 in order ^to obtain the distance effect in its purest ^form. Furthermore we set xrO)’ ^the coordinates of the equilibrium ^{state of the source.}

Then, taking ^{into account} the sinusoidal law of motion (1), part I, ^{it is} easy ^to derive the components gllv ^{of the} metric tensor in the self frame of the ^source. The non-vanishing ^{terms are :}

where x/l are coordinates in the rest frame of the source.

The time it takes a light signal ^to pass from point (x1i, x2, x3) ^{to the} point (x’ ⁺ dxl, ^{X2 +} dx2, ^{XI +} dx3)

in a gravitational field, ^{if a} synchronization ^is made with the _proper time at each point, ^is [10] :

Substituting g ⁱⁿ (14) ^and integrating ^{over an} isotropic geodesical connecting ^{the source and the absorber,}

we obtain the time of propagation of the radiation from the source to the absorber :

where x° is the moment of irradiation of the signal.

The consequence of (15) is that if a monochromatic wave from the source has at its rightside boundary ^the

form f°(w) eiro(t-to), ^{then the same wave would have at the left} boundary of the absorber the form (carrying ^out

the usual transforms [1]) :

Fig. ^3.

Relative transmitted intensity SIS,

distance Ax between the source and the absorber (the

distance effect »)

for D

10, CPo

CPt, and for several values of the reduced amplitude ^A.

870

with respect ^{to the} proper time of the absorber. Since the symbol ^m appearing ⁱⁿ f(w) denotes the frequency ⁱⁿ

the _proper time, ^then (16) ^is just ^the necessary expression for calculation of the wave penetration through ^the

absorber. The field of the radiation on the rightside boundary of the absorber is :

Further deliberations are fully analogous ^to those used in [1] ^and yield again equation (10) (remembering

that _go

( Pi, x(0)

0). Equation (10) ^{can be} easily obtained also for go :A ^(p1, but in this case the effect is not

purely gravitational ^{even in the rest} frame of the source (absorber).

So, the distance effect _may be obtained in two ways

in the laboratory frame it is caused by ^{the finite} velocity ^of light signals, which results in an effective dephasing of the vibrations of the source and the absorber,

and in the rest frame of the system ^{it is a} result of the retarding ^of electromagnetic signals ^{in a} gravitational ^field.

As in both cases the considerations are made in planar space, the distance effect could hardly ^be regarded ^as

a test for the GRT. A rather more interesting ^case represents ^the possibility (at ^{least in} principle) ^of measuring

the velocity ^of light by ^means of the distance effect, ^{when the} signal ^is propagating ^one way instead of in a closed chain as in the classical spectroscopic experiments. ^Such measurements are desirable for some general ^theoretic

reasons (see [11]). ^{It is} possible that the distance effect could be regarded ^{also as a test} for the local inertiality principle (in ^{the case of} quasimacroscopic motion), but this would involve some special considerations, ^which

are not the subject of this work.

Numerical calculations show that the discussed distance effect is accessible experimentally. ^A plot of s/S0 ^vs.

distance source absorber Ax for A

1 to A

5, ^D

10, Dj2 n

³⁰ MHz, go

91 is drawn ^on figure ³ (So denotes the intensity of transmitted radiation at Ax

0).

References

[1] OGNJANOV, ^N. I., TSANKOV, ^L. T., part I in this issue.

[2] MONAHAN, ^J. E., PERLOW, ^G. J., Phys. ^{Rev. A 20} (1979) ^1499.

[3] PERLOW, ^G. J., Phys. Rev. Lett. 40 (1978) ^896.

[4] LYNCH, ^{E. J.,} HOLLAND, ^R. E., HAMMERMESH, M., Phys. ^{Rev. 120} (1960) ^513.

[5] HARRIS, ^S. M., Phys. ^{Rev. 124} (1961) ^1178.

[6] GRADSTEIN, ^I. S., RIJIK, ^I. M., ^Tablici integralov,

summ, rjadov ⁱ proizvedenij, M., Fizmatgiz (1963)

p. 502.

[7] BÖMMEL, ^H. E., in Proc. 2nd Int. Conf.

^Mössbauer Effect, Saclay, France, ¹⁹⁶¹ (N.Y., ^J. Willey ^&

Sons, 1962), p. 229.

[8] MITIN, ^A. V., Kvantovaja Elektron. 3 (1976) ^840.

[9] MAKAROV, ^E. F., MITIN, ^A. V., Usp. ^{Fiz. Nauk 120} (1976) ^55.

[10] IVANICKAJA, ^O. S., Lorentzev bazis i gravitacionnie effekti

Einsteinovskoj ^teorii tjagotenija, ^Minsk (1979), ^{Nauka i} Tehnika, p. 54.

[11] VASIL’EV, ^B. V., preprint JINR Dubna P13-9411 (1975).