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PHASE COHERENCE OF FREQUENCY MODULATED MÖSSBAUER GAMMA RAYS
J. Cashion, P. Clark
To cite this version:
J. Cashion, P. Clark. PHASE COHERENCE OF FREQUENCY MODULATED MÖSS- BAUER GAMMA RAYS. Journal de Physique Colloques, 1979, 40 (C2), pp.C2-44-C2-46.
�10.1051/jphyscol:1979214�. �jpa-00218512�
JOURNAL DE PHYSIQUE
Collogue C2, supplement au n° 3, Tome 40, mars 1979, page C2-44
PHASE COHERENCE OF FREQUENCY MODULATED MOSSBAUER GAMMA RAYS
J.D. Cashion and P.E. Clark
Department of Physics, Monash University, Clayton, Victoria 3168, Australia
Résumé.-Une source Mossbauer montée sur un cristal piézoélectrique qui module la fréquence, donne des "sidebands" symétriques. On montre que l'inclusion dans le système d'un filtre résonant modulé en fréquence, doit donner des intensités inégales pour les "sidebands" quand la source et le fil- tre sont modulés en phases opposées. L'asymétrie a été observée auparavant sans commentaire; elle est plus importante quand l'indice de modulation est petit.
Abstract.- Frequency modulated sidebands in the spectrum of a Mossbauer source mounted on a piezo- electric crystal are known to be symmetrical. It is shown here that inclusion of a frequency modu- lated resonant filter into the system should result in unequal intensities of corresponding posi- tive and negative sidebands when the source and filter are modulated in antiphase. This asymmetry has been obtained previously but not commented upon and is more significant at low modulation
indices.
1. Introduction.- When a Mossbauer source or absor- ber is attached to a piezoelectric crystal driven
at a radio frequency U) (where Ka) > V, the observed linewidth), frequency modulated sidebands may be observed at energies ±nMio (n integer) on either side of the Mossbauer line. The intensity of the n sideband is proportional to exp(-m2)I (m2) IM where m is the modulation index and I is the modi-
n
fied Bessel function. This differs from classical FM theory, where the intensities are proportional to the Bessel function J (m), because of weighting by the Rayleigh distribution function on the assump- tion that the phonons are incoherent /l/. Substan- tial agreement with the modified intensities has been found /2,3/ and the incoherence explained /2/
in terms of the short lifetimes of the phonons of energy KOJ, Substantially poorer agreement has been found /4/ when the sidebands are produced by magne- to-elastic coupling between the sample and a radia- ted r.f. field.
2. Theory.- Consider the situation in which the emitted spectrum from a frequency modulated Moss- bauer source is analyzed by a single line absorber.
The classical theory of FM gives the amplitude as
CO
A = J (m) sinW t + £ j (m) [sin(td +nu)t+(-l)nsin (u) -nw) tj
where co is the gamma ray frequency. We shall igno- re for the moment the modification caused by the insertion of the Rayleigh distribution function or other similar appropriate function since this does not alter the conclusions. It can be seen that the amplitude of the sidebands at frequencies w + nto
and w - no), corresponding to the annihilation and creation of n phonons respectively, are unequal when n is odd. However since the Mossbauer spectrum obtained will correspond to the intensity rather than the amplitude, this phase information is lost.
The phase information may be retained by inserting a resonant filter which is also frequency modulated in such a way that there is a definite phase rela- tionship between the two. The transmitted flux from this coupled system now depends directly on the phase difference between the two components and may be analyzed in a conventional fashion by a second absorber.
An experiment of this type was carried out by Asher et al. /5/ who used two identical PdFe absorbers mounted on opposite sides of a piezoelectric crys- tal and illuminated by a single line source mounted on the same driving arm so that there was no net motion between them. The transmitted flux was ana-
lyzed in energy by a stationary PdFe foil. Examina- tion of the line intensities obtained by them show that for all four driving voltages the first side- band at positive velocity is larger than the cor- responding sideband at negative velocity, although it is only for the lowest voltage results that the ratio of unity is not included in their error bars.
This asymmetry was ignored by Asher et al. who stated that the transmitted sidebands should appear to be symmetrical.
3. Experimental arrangement.- In order to investi- gate further the coherence properties of phonons excited by a piezoelectric crystal, a slightly
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1979214
simpler experiment was s e t up a s i n f i g u r e 1 .
Piezoelectric
5 k 0 ~
SS KFC N DetectorSource Filter Analyzer
F i g . 1 : Schematic diagram of experimental arrange- ment.
The source and s t a i n l e s s s t e e l f i l t e r a b s o r b e r (na- t u r a l abundance of 5 7 ~ e ) were both mounted on p i e - z o e l e c t r i c q u a r t z c r y s t a l s a t r e s t i n t h e l a b o r a t o - r y frame and t h e e n r i c h e d potassium f e r r o c y a n i d e (KFCN) a n a l y z i n g absorber mounted on a conventional c o g s t a n t - a c c e l e r a t i o n d r i v e system. The source- f i l t e r combination used i s n o t t h e optimum one f o r t h i s experiment because of t h e non-zero isomer s h i f t between t h e c o n s t i t u e n t s . The c r y s t a l s were nominally i d e n t i c a l X-cut, t h i r d harmonic q u a r t z c r y s t a l s tuned t o 8 MHz and t h e metal f o i l s were a t t a c h e d w i t h A r a l d i t e f a s t - s e t t i n g epoxy.
Radio frequency power was o b t a i n e d from a Hewlett- Packard Model 8690B v a r i a b l e sweep o s c i l l a t o r w i t h an A.R.C. Model ARC-30L a m p l i f i e r which gave a maximum o u t p u t of approximately 130 V corresponding t o a modulation index of approximately 25.
4. R e s u l t s . - Applying t h e r . f . modulation a t 10.5 MHz t o b o t h t h e p i e z o e l e c t r i c c r y s t a l s i n p a r a l l e l and a n a l y z i n g t h e t r a n s m i t t e d f l u x with t h e KFCN a b s o r b e r gave a symmetrical spectrum s i m i l a r t o t h a t o b t a i n e d by many p r e v i o u s workers with modula- t e d systems. The r e l a t i v e phase of t h e two c r y s t a l s was t h e n r e v e r s e d by r o t a t i n g t h e f i l t e r assembly through 180' and t h e r e s u l t i n g spectrum showed t h a t t h e f i r s t p o s i t i v e sideband had i n c r e a s e d a t t h e expense of t h e corresponding n e g a t i v e sideband and was now of g r e a t e r i n t e n s i t y than t h e c e n t r a l peak, a c o n f i g u r a t i o n not p r e v i o u s l y observed.
The p r e l i m i n a r y r e s u l t s were f i t t e d u s i n g a sum of e q u a l l y spaced L o r e n t z i a n s and i n c l u d i n g t h e e f f e c t s of t h e next t h r e e sidebands o u t s i d e o u r v e l o c i t y range assuming t h a t t h e i r i n t e n s i t i e s were given by In. The r a t i o s of t h e i n t e n s i t i e s of t h e corresponding p o s i t i v e and n e g a t i v e sidebands were c a l c u l a t e d f o r each spectrum and Table I shows t h e r a t i o s f o r t h e a n t i p h a s e spectrum d i v i d e d by t h a t f o r t h e in-phase spectrum.
Table I : R a t i o of t h e i n t e n s i t i e s of corresponding sidebands f o r t h e a n t i p h a s e experiment d i v i d e d by t h e corresponding q u a n t i t i e s f o r t h e in-phase expe- riment. The number i n b r a c k e t s i n d i c a t e s t h e e r r o r i n t h e l a s t s i g n i f i c a n t f i g u r e .
n : 1 : 2 : 3
I I 1
I I I
R a t i o of ~ n t e n s i t i e s : 1.12(10): 1.00(10): 1.04(10)
,
17 1 I
5. Discussion.- It i s e a s i e s t t o c o n s i d e r t h e i n t e r - p r e t a t i o n i n terms of t h e r e s o n a n t s c a t t e r i n g of gamma r a y s from t h e f i l t e r which may t h e n be r e l a - t e d t o t h e i n t e n s i t y of t h e t r a n s m i t t e d f l u x v i a t h e o p t i c a l theorem. A s c a t t e r e d gamma r a y of f r e - quency w + w can have p a r t i c i p a t e d i n one of an
Y
i n f i n i t e s e r i e s of p o s s i b l e e v e n t s , e.g. emission i n t h e z e r o phonon l i n e from t h e source and t h e n s c a t t e r i n g with t h e simultaneous a n n i h i l a t i o n of one phonon i n t h e f i l t e r , o r emission w i t h a phonon a n n i h i l a t i o n and s c a t t e r i n g r e s o n a n t l y , o r any s e t of m u l t i p l e phonon e v e n t s whose a l g e b r a i c sum of phonons l o s t by t h e o s c i l l a t i n g system e q u a l s one.
The dominant terms w i l l be p r o p o r t i o n a l t o J (m) J l ( m ) . I n t h e case of t h e f i r s t n e g a t i v e sideband t h e dominant terms w i l l be p r o p o r t i o n a l t o J (m) x J-l(m),, where t h e B e s s e l f u n c t i o n s have t h e property J-n = ( - I ) ~ J ~ . The i n f i n i t e s e r i e s each comprise terms of a l t e r n a t i n g s i g n which become c l o s e r i n magnitude a s t h e modulation index i n c r e a s e s . Thus t h e e f f e c t should be more marked a t low modulation i n d i c e s a s e v i d e n t from t h e r e s u l t s of Asher e t a l . The d e s c r i p t i o n given i s based on c l a s s i c a l theo- r i e s b u t t h e e s s e n t i a l d e t a i l s a r e n o t a l t e r e d i n a quantum mechanical t r e a t m e n t . A complete d e r i v a - t i o n of t h e t h e o r y and f u r t h e r experimental r e s u l t s w i l l be published elsewhere.
The r e t e n t i o n of phase i n f o r m a t i o n i s an important p r e - r e q u i s i t e f o r many u s e f u l t e c h n i q u e s and t h i s o b s e r v a t i o n could l e a d t o s e v e r a l new a p p l i c a t i o n s . One l o g i c a l a p p l i c a t i o n could be t h e measurement of t h e decay of phonon coherence by e l a s t i c s c a t t e r i n g i n s t e a d of t h e more usual measurement of phonon l i f e t i m e s .
Acknowledgements
This work i s supported by t h e A u s t r a l i a n Research Grants Committee and t h e A u s t r a l i a n I n s t i t u t e f o r Nuclear Science and Engineering. We a r e g r a t e f u l f o r h e l p f u l d i s c u s s i o n s w i t h P. Lloyd.
JOURNAL DE PHYSIQUE
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FREQUENCY MODULATION O F THE 6,2 k e V MOSSBAUER STATES OF l * ' ~ a
P.J.
