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On the theory of coherent ultrasonic modulation of Mössbauer gamma-radiation
N.I. Ognjanov, L.T. Tsankov
To cite this version:
N.I. Ognjanov, L.T. Tsankov. On the theory of coherent ultrasonic modulation of Mössbauer gamma- radiation. Journal de Physique, 1983, 44 (7), pp.859-864. �10.1051/jphys:01983004407085900�. �jpa- 00209669�
On the theory of coherent ultrasonic modulation of Mössbauer gamma-radiation
N. I. Ognjanov and L. T. Tsankov
Department of Atomic Physics, University of Sofia, 1126 - Sofia, Bulgaria (Reçu le 30 juillet 1982, révisé le 14 février 1983, accepte le 8 mars 1983)
Résumé. 2014 Une méthode unifiée et relativement simple est proposée pour la description théorique du passage des quanta gamma, émis par une source vibrante avec une fréquence ultra-sonore, à travers un système constitué
de n (0 ~ n ~) absorbants vibrants avec des fréquences, amplitudes et phases différentes. Des expressions générales reproduisant les résultats particuliers antérieurs, sont obtenues pour la distribution spectrale de la
radiation transmise et pour son intensité totale. Le traitement repose sur la théorie classique de dispersion.
Abstract. 2014 A unified and relatively simple method is proposed for theoretically describing the transmission of Mössbauer gamma-quanta, irradiated by a coherently rf vibrating source and emerging through a system consisting of n (0 ~ n ~) resonant absorbers, coherently vibrating with different frequencies, phases and amplitudes. General results for the spectral distribution and for the full intensity of the radiation are derived, reproducing particular results obtained up to now. These considerations are based on classical dispersion theory.
Classification
Ph ysics Abstracts
03.40K - 76.80
1. Introduction.
The modulation phenomena involving Mossbauer
radiation have been intensively investigated - both theoretically and experimentally during the last
20 years (see e.g. the review article [1]). These investi-
gations extend our knowledge of the interaction of
gamma-radiation with matter and create a basis for
some new methods for studying the elementary
excitations in solids.
An essential task of the theory is the description
of the modulation phenomena in the case of coherent ultrasonic vibrations of resonant samples - the
so-called coherent ultrasonic modulation of gamma- quanta (CUSM). The wide variety of problems
related to CUSM makes it desirable to develop a
unified method for their solution. The subject of the
present work is to propose such a method. Taking advantage of the superposition principle, we consider independently the modulation of every monochro- matic component of the photon wave packet. This
allows us to obtain general results for the dependence
of the spectral distribution and of the full intensity
of the modulated radiation on various factors
(including the often neglected distance modulation
term).
The general arrangement we shall consider is as
follows : the collimated radiation from a Mossbauer
source passes through n absorbers containing resonant
nuclei (0 n oo ). For the sake of conciseness
we shall summarize some definitions and annotations : 1. The oscillations of a resonant specimen are
called coherent if all the points of the specimen
vibrate with the same frequency, amplitude and phase.
2. The quantities x(0)’ X(1), ..., x(n) denote the coor-
dinates of the source and the absorbers (x(0)
x (1) ... x(n»). It is assumed that the linear thick-
ness of the resonant specimens is negligible with
respect to the distances between them.
3. The function fo(w) stands for the amplitude
of a monochromatic component of the photon wave packet when the source is at rest. Here w denotes
the angular frequency of the considered component.
If one deals with a single M6ssbauer line, characte- rized by a natural linewidth r and centred at fre- quency coo, then fo(m) becomes :
but the following considerations are not bound to this special case.
4. The functions fj(w) stand for the frequency
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01983004407085900
860
response of the j-th absorber ( j = 1, 2, ..., n), i.e.
if the incident monochromatic wave on the absorber is characterized by angular frequency w and complex amplitude A, then the transmitted wave will have the same frequency and amplitude j(m) A. The
correctness of this description as well as the concrete
form of Jj(w) are discussed in [2, 3].
5. kp = (k1, k2, ..., kp) is a combined subscript.
If the opposite not specified, kj varies from - oo to oo.
Further, it is assumed that the source and the absor- bers undergo coherent vibrations, following the
relation :
The principal problem of CUSM is to calculate the spectral distribution I(w) and the full intensity S
of the transmitted radiation as a function of fj(w),
X(i), aj, Qj, qJj and the observation conditions.
2. Theory.
According to the superposition principle the propagation of each monochromatic component of the photon
wave packet can be considered independently. The field of such a component is :
where to is the moment of creation of the excited nuclear state that provides the M6ssbauer photon.
If the source performs vibrations following (1), then (2) becomes a phase-modulated field. The phase modu-
lation propagates with group velocity, which is equal to c in vacuum. The field is
Here the vibration of the source is neglected in the phase of the modulation term
The correspondence factor, if encountered, would result in an additional time-dependent phase shift of the
wave (2). This shift is of the same order of magnitude as -
( c 2 . 0*
A typical value of this quantity is 10-’o andtherefore it is possible to neglect the discussed phase shift for every monochromatic wave, which contributes
significantly to the wave packet of the Mossbauer photon.
Accounting for the motion of the first absorber, the following expression can be obtained for the field on
its left boundary :
Then, using the well-known expansion :
where Jk are the Bessel functions of order k, we obtain :
It is seen from (6) that the field of the considered wave at the left boundary of the first absorber is a sum of monochromatic waves with frequencies 6) + ki Do - 11 D1. Hence, using the above definition of fj(w), we
obtain the following relation for the field at the rightside boundary of the absorber :
This field is felt by an observer vibrating with frequency D1. The radiation propagates further, so we regard
the field at x > x(1) to be produced by a vibrating source. The field of this source (if it was not vibrating) is :
where
Performing similar considerations for the remaining absorbers, we obtain the field of the considered wave
after it has passed through the entire system of absorbers :
where
Each term appearing in (8) represents a monochromatic wave with frequency w’
Waves with this frequency appear in the resultant field of a discrete set of initial waves and a summation over
their amplitudes should be done. Renaming w’ -+ w, we obtain the transmitted radiation field :
where
and f,. + 1 «(J)) = 1 is defined.
Hence, the spectral distribution is :
If the moment of creation of the excited nuclear state to is not measured, an integration over it should be
performed and (11) becomes :
where
862
Now we would like to discuss the calculation of the full intensity S of the transmitted radiation. It is neces- sary to point out that this quantity is invariant and therefore it may be calculated in the self frame of the n-th
absorber, i.e. Fknln should be considered rather than Fkn+tln+t. When S is being calculated, two general types of experiments should be different from each other : with resonant (i.e. frequency-dependent) and with non-
resonant registration of the transmitted radiation. In the first case the partial waves with different frequencies are distinguishable and do not interfere with each other; so, their intensities should be summed over a) :
or, if to is not measured :
If the detector registers non-resonantly, the amplitudes of the partial waves should be summed. In fact, this
treatment is suitable for most of the experimental arrangements, as the M6ssbauer analyser can be regarded as
the (n + 1 )-th absorber of the system when Qn+ 1 = 0 is set and a proper form of fn + 1 is chosen.
The relations for S in this case are :
In (18) we average over t in the interval (0, oo), since the integrand is in general a non-periodic function, although the amplitude of its monochromatic components declines rapidly with the growth of their period.
Formulae (10) to (18) represent a general solution of the principal problem of CUSM in the case of harmonic oscillations of the components of the system. These results can be generalized for any arbitrary pattern of motion by expanding the displacements in Fourier series or integrals.
3. Approximations and special cases.
Now we would like to consider some approximations and special cases of the formulae (10) to (18) :
1. In (10) the dependence of the Bessel functions arguments on co, kj, lj, 92j can be neglected. Actually, f0(w) is practically zero except in a small neighbourhood of some central frequency mo. The length of this interval is typically 10-12 w0 and the Jk do not change if ro is kept in the neighbourhood. On the other hand, aolc is
also of order 10-12. So, the term becomes significant only if at least one of the sub-
scripts ks, Is is of a very high order (e.g. 109). Bessel functions of such an order are practically zero for every value of their argument. Thus we can approximate (10) and (28) (see below) with :
2. The integrals over to and t in (16), (17), (18) can be approximately commutated with these over co and W"
(exact calculation of similar integrals and evaluation of some approximations can be found in [ 1, 4]). The results
are :
3. The case
is of special interest. This is the case actually investigated so far. The most straightforward way to simplify the
results in this case is to repeat the previous deliberations, making use of the trigonometric identity :
where
and ({Jj-l,j is obtained from the relations :
The final results are :
where
864
Formula (27’) corresponds to (16’) and (16), (17), (17’), (18), (18’) do not change.
4. If besides the approximations 1 and 2 we assume
that
r being the natural linewidth, then a closed form of the expressions for I and S may be obtained using the
summation theorem for Bessel functions. A brief demonstration of this technique can be found in [8].
It is not difficult to derive the well-known results from the relations considered above. Results similar to
(24) have been obtained in [5] for the case n = 1, 2 by
means of simultaneous solution of the nuclear density
matrix equations together with the Maxwell equations.
Time dependence of the transmitted radiation for the
case n = 1 is investigated in [4].
4. Incoherent modulation.
The approach, proposed above, can be extended in order to describe incoherent vibrations as well. In the last case the quantities Aj, Tj should be considered
as statistical variables and the expressions for S and I
should be averaged over their statistical distribution.
Hence,
Here Rcoh stands for a measurable quantity in the
coherent case, Rlncoh denotes the corresponding quan-
tity in the incoherent case, and pi(Ai, oi) is the mutual distribution function of Ai and qJi.
The main problem in the case of incoherent vibra- tions is the investigation of statistics p(A, 9) for a
concrete resonant sample and its dependence on the
experimental conditions. Most of the investigations published so far deal with a system consisting of one
resonant sample at stationary observation conditions.
In this case the phase of the rf vibrations has no effect
on the experimental results; so, attention has been
paid to the amplitude distribution only. The most general theoretical results are obtained in [6, 7], showing that the amplitude statistics is that of Ray- leigh-Rice :
where Io is a modified Bessel function of order 0.
Two limiting cases of (31 ) are the casual distribution
(i.e. full amplitude coherency) at y -+ 0 and the Ray- leigh distribution (full amplitude incoherency) at
x -+ 0 :
Most of the experimental results agree reasonably
with (32) (see e.g. [9, 10]), but in some cases a certain degree of coherency has been observed (see e.g. [11,
12]).
Phase distributions have been little studied up to now. It is often considered that in the case of incohe- rent vibrations the phase distribution is uniform, but
some experiments [4] show that it might be close to the
casual distribution. A detailed discussion of this
problem is not the subject of this paper. We would like
only to point out that a necessary condition for observ-
ing a phase coherency is that the thickness of the resonant sample should be considerably smaller than the wavelength of the ultrasound generated in it.
References
[1] MAKAROV, E. F., MITIN, A. V., Usp. Fiz. Nauk 120
(1976) 55.
[2] LYNCH, F. J., HOLLAND, R. E., HAMMERMESH, M., Phys. Rev. 120 (1960) 513.
[3] HARRIS, S. M., Phys. Rev. 124 (1961) 1178.
[4] MONAHAN, J. E., PERLOW, G. J., Phys. Rev. A 20 (1979) 1499.
[5] MITIN, A. V., Kvantovaja Elektron. 3 (1976) 840.
[6] SADIKOV, E. K., Fiz. Tverd. Tela 19 (1977) 1650.
[7] SADIKOV, E. K., DIDKIN, V. A., Izv. VUZ Fiz. 22
(1979) 7.
[8] OGNJANOV, N. I., TSANKOV, L. T., part II in this issue.
[9] MISHORY, J., BOLEFF, D. I., in Mössbauer Effect Methodology, ed. I. J. Gruverman, vol. 4 (New York-London, Plenum Press) 1968, p. 13.
[10] CRANSHAW, I. E., REVIARI, P., Proc. Phys. Soc. 90 (1967) 1059.
[11] PFEIFFER, L., HEIMAN, N. D., WALKER, J. C., Phys.
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[12] MKRTCHYAN, A. R. et al., Phys. Status Solidi B 92 (1979) 23.
