An alternative way of computation of amplitudes and intensities in time-dependent Mössbauer spectroscopy (TDMS)

(1)

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An alternative way of computation of amplitudes and

intensities in time-dependent Mössbauer spectroscopy

(TDMS)

I.D. Christoskov, E.I. Vapirev

To cite this version:

(2)

An

alternative

_way

of

_computation

of

_amplitudes

and

intensities

in

_{time-dependent}

Mössbauer

_spectroscopy

_(TDMS)

I. D. Christoskov and E. I.

_Vapirev

Department

of

_{Physics, University}

of Sofia,

Sofia-1126, Bulgarie

(Reçu

le 24

_juillet

1989,

accepté

sous

_forme

_définitive

le 13 novembre

1989)

Résumé. 2014

Nous exposons une autre

_façon

de

_présenter

la

_{dépendance temporelle}

des

amplitudes

et intensités de radiation obtenues dans des

expériences

de diffusion et

_{d’absorption}

résonnante. Des suites infinies de fonctions de Bessel avec des coefficients

_complexes

sont

remplacées

par des

intégrales

définies

simplement

calculables avec une seule fonction de Bessel par

point

de tabulation. Il s’ensuit une réduction du _temps_{de calcul pour}

_{l’ analyse}

des résultats

expérimentaux.

Abstract. 2014 An

alternative way of

presenting

the

_{time-dependent amplitudes}

and intensities of

radiation,

corresponding

to resonant

_absorption

and

_{scattering experiments,}

is

_developed.

Infinite series of Bessel functions with

complex

coefficients are

_{replaced by}

_simple

for calculation definite

_integrals

and the number of Bessel function calls is reduced to one _pera tabulation

point.

Thus the calculational effort for

_experimental

data

_processing

becomes smaller. Classification

Physics

Abstracts 24.30

Classical

_{electromagnetic theory}

-

damped

oscillators,

dispersive

media,

and the

correspondence

between the

_frequency

and the time domain - has been used to describe the

phenomenon

of « time filtration » both for transmission

_[1]

and resonant

_scattering

_{[2, 3].}

Since then many similar

experiments

have been

_performed,

the interest

_arising

from the fact that those

_experiments

are the

_only

ones in which

_quantum

effects in the time domain within the natural linewidth

(or

the excited state

_lifetime)

could be observed.

In the

present

work an alternative

_technique

_(again

in terms of the

_{electromagnetic}

approach)

is

_proposed

for

_presenting

the

_{time-dependent amplitude}

and

_intensity

of

radiation, corresponding

to resonant

_absorption

and

_{scattering experiments.}

1. Resonant

absorption.

Let the source

_(S)

linewidth be denoted

_by

_Fs,

and the resonant level lifetime be

,r, = 1 IF,

(h = 2 ir ).

The central

_frequency

of S is _lds. The filter

_(F)

linewidth is

T f,

its central

_{frequency - w f,}

and the effective resonant

_{thickness - B = no- 0 fa.}

(3)

444

n is the number of resonant _{absorber nuclei per}

cm2,

_{o-o - the}resonant cross-section and

f a

- the recoilless fraction of

_absorption.

The

_{frequency amplitude}

of S is

and the

_frequency

_amplitude

of F is

Hence the

_{time-dependent amplitude}

of transmitted

_gamma-quanta

will be the Fourier

_image

of the

_product

_S(,w).

_{F(w ) [1] :}

The factor

_F,lFf

is introduced in the

_exponent,

thus

_preserving

the

_absorption

line area. If we introduce the

_{following designations}

- relative shift V

=

CI) s - CI) f,

relative

broadening

rs

s

R =

r f/rs

[u

=

(R - 1 )/2],

and time in _sourcelifetime units T =

Fs t ;

the

following

expression

for

_{A,, (T,}

_{B, R,}

_{V )}

can be obtained :

J,,

are Bessel functions of order n. In TDMS transmission

_experiments

the time distribution of

counts of transmitted

_gamma-quanta

_(t

= 0

corresponds

_tothe _momentof formation of the

Môssbauer state in the

_source)

can be described

_by

the

_{squared amplitude}

Expressions (2)

and

₍₃₎

for the

_amplitude

and the

_intensity

are

_{usually applied}

for numerical simulation of

_experiments

or for

_experimental

data

_processing

_(least

_square

_fitting

for

parameter

search,

etc.).

Nevertheless, they

are not _{very convenient}because the series

₍₂₎

converges with reasonable accuracy

only

after order 5-7 and therefore for the calculation of

(2)

at a

_single

time

_point

several calls of the Bessel function are _{necessary. Here it should be}

specified

that for the sake of better _{convergency the series}

₍₂₎

must

_always

be

_absolutely

convergent.

So,

at a fixed set of

_parameters

_(B,

_R,

_{V ),}

if T becomes

_{large enough}

and the modulus of the coefficient in front of the Bessel function exceeds

_unity,

the series

₍₂₎

must be transformed

_by

means of the

_generating

function of

_{ln (z).}

At

_that,

the

_application

of the recursion formula for calculation of

_{J,,(z), n:--.}

1 leads to an

_abrupt

loss of _{accuracy in the}

vicinity

of the critical value of T.

_Therefore,

it is reasonable to make use of the reverse recursion formula

_{[6] [starting}

at a

_{sufficiently big}

order of the Bessel function n

_(e.g.

7 = 9)].

Furthermore,

the task is

_complicated

because for the purpose of

experimental

data

fitting

the transmitted

_intensity

_{Itr(T, B, R, V ),}

_together

with its derivatives with

_respect

to

all

parameters

subject

to

_{determination,}

must be calculated for a

_large

_{sequence of measured}

(4)

That is

_why

we

_suggest

a more convenient

presentation

of

expressions

(2)

and

(3).

Let us introduce the

_{quantities a}

=

(u

₊

iV )/B

and z =

ÙÔ.

If we make use of the connection

[4] :

the transmission

_amplitude

takes the form :

The

_integral

in the RHS of

₍₄₎

can be transformed :

Finally,

for

_Atr(T,

_{B, R, V )}

one obtains :

The

_intensity

takes the form :

where

In this form the effect of the resonant _{interaction appears}as an addition to _{the pure}

exponential

decay

of the excited state in the source. The

_sub-integral

function is smooth

enough

and well

_suppressed

with

time,

and for a

_reasonably

small

_step

of the

argument

(which

is the case in

_experimental

data

fitting)

even the

_simple

trapezoidal

rule for numerical

quadrature

is sufficient for

_calculating

the

_integral.

Due to the

_additivity

of the

_{integral only}

one call of _{the Bessel function per tabulation}

_point

_{is necessary, instead of several}ones in the conventional case.

Moreover,

its derivatives with

respect

of the

_parameters

_(R,

_B,

_{... )}

have a

simple

form.

Therefore,

this method

_requires

a

_{substantially}

smaller calculational effort. It has been tested in numerous

_applications

and has

_proved

to be

_fast,

accurate and

_simple

for realization. Here the

_following

must be

_{specified :}

if the values of the

_amplitude

and the

intensity

are needed for a small number of remote moments of

time,

the conventional method is more suitable since the number of calls of the Bessel function does not

_depend

on time.

2. Resonant

_scattering.

Let

/3

be the resonant thickness of a

_layer

of the

_scattering

filter.

_According

to the

_approach

(5)

446

where

The

_{following generally applicable}

relation

_[5]

between the

_amplitudes

_{Atr (T,}

_6,

_R,

_{V )}

and

Asc(T, (3,

R,

V )

exists :

Therefore,

all results demonstrated in section 1 can be

_applied

in this case. For the final calculation of

_{1 sc (T, B, R, V)}

_again

a

_simple

numerical

_quadrature

can be

_applied,

thus

eliminating

the

_necessity

to use Bessel function series and all

_problems

connected with their convergence. Modifications of this scheme have also been tested in several

applications

and have

_proved

to work

_effectively.

References

[1]

LYNCH F. _J.,HOLLAND R. E. and HAMMERMESH M. H.,

Phys.

Rev. 120

₍₁₉₆₀₎

513.

[2]

THIEBERGER P., MORAGUES J. A. and SUNYAR A. W.,

Phys.

Rev. 171

₍₁₉₆₈₎

425.

[3]

DROST H., PALOW K. and WEYER G., J.

_{Phys. Colloq.}

France 35

₍₁₉₇₄₎

C6-679.

[4]

PRUDNIKOV A. P., BRYCHKOV Yu. A., MARICHEV O. I.,

Integraly

i

_ryady.

_{Special’nye}

funkcii.

(Integrals

and Series.

_Special

_Functions),

M. : Nauka

_{(in Russian)}

1983.

[5]

VAPIREV E. I., KAMENOV P. S., DIMITROV V. and BALABANSKI D. L., Nuclear Instrum. Methods 219

(1984)

376.