Study of radioactive impurities in solids. Part one : radiation characteristics

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Study of radioactive impurities in solids. Part one : radiation characteristics

F. Hartmann-Boutron, D. Spanjaard

To cite this version:

F. Hartmann-Boutron, D. Spanjaard. Study of radioactive impurities in solids. Part one : radia-

tion characteristics. Journal de Physique, 1972, 33 (2-3), pp.285-297. �10.1051/jphys:01972003302-

3028500�. �jpa-00207250�

STUDY OF RADIOACTIVE IMPURITIES IN SOLIDS PART ONE : RADIATION CHARACTERISTICS

F. HARTMANN-BOUTRON and D. SPANJAARD (*)

Laboratoire de Physique des Solides (**), Faculté des Sciences, 91, Orsay (Reçu ^{le 7 octobre} 1971)

Résumé.

Le problème théorique de l’émission de rayonnement par des _noyaux radioactifs est

habituellement traité à l’aide d’un formalisme très général adapté ^{à la} physique nucléaire. Dans le présent article, ^{nous abordons cette} question

angle beaucoup plus restreint, ^mais

proposant d’établir des formules plus appropriées ^à l’interprétation ^des expériences ^de physique ^des

solides. Pour cette raison,

^donnons

traitement unifié des corrélations angulaires perturbées

et de l’orientation nucléaire,

qui permet ^de comparer facilement

deux types d’expériences.

Par ailleurs,

systématiquement

formalisme de Fano (développement ^multi- polaire de la matrice densité) qui, ^ainsi qu’on ^le

dans la deuxième partie ^de

travail,

prête particulièrement bien à l’étude des effets de relaxation et de radiofréquence.

Abstract.

The theoretical problem of the emission of radiation by radioactive nuclei is usually

treated in

very general ^form adapted ^{to nuclear} physics. ^{In this} paper

approach it within

much

scope, but with the aim of using ^the resulting formulae in solid state physics ^studies.

For this reason

give

^{unified treatment of both} perturbed angular correlations and nuclear orientation,

^to permit

easy comparison between these two types ^of experiments. ^Also

resort systematically ^{to Fano formalism} (multipole expansion ^{of the} density matrix), which, ^{in the} second part ^{of this} work, ^will appear ^as particularly suitable for the computation of relaxation and

radiofrequency ^effects.

Classification Physics Abstracts : 12.10, 18.00, ^18.20

1. Introduction.

In a number of experiments,

the characteristics of the radiations emitted by ^the nuclei of radioactive impurities ^{embedded in a} solid,

are used as a means to determine the hyperfine ^interac-

tions experienced by ^{these nuclei} (hyperfine ^{field and} quadrupolar coupling) [1]-[8]. ^This gives ^indirect

information on the electronic properties ^{of the radio-}

active atom and of the surrounding ^{matrix. Several} techniques ^{can be used, either} spectroscopic (Môss-

bauer effect) ^or non-spectroscopic (perturbed angular correlations, ^nuclear orientation). ^{In this} work, ^we

will essentially consider the latter two methods ; compared with the Môssbauer effect, they ^{have the} advantage ^of being applicable ^{to a much} greater number of nuclei and lend themselves more readily ^to

the study ^of weakly radioactive samples (especially ^of implanted nuclei). The different experimental ^situa-

tions are schematically illustrated in figure ^{1. In}

these non-spectroscopic ^{methods the} energies ^{of the}

emitted photons ^{are not measured.}

Let A be the order of magnitude ^{of the} hyperfine

interactions in the nuclear levels. It should be noted that the formulae which are used in thei nterpretation

of perturbed angular correlation experiments (Fig. la)

are only ^valid when A jkB T 1 « ^{1. On the} contrary, in 1 b and le (nuclear orientation) ^{the observed radia-}

(*) This work is based

part of the thesis of Daniel Span- jaard (réf. [1 3 ] ) .

(**) ^Associé

C. N. R. S.

LE JOURNAL DE PHYSIQUE. ^- T.

33,

2-3, FÉVRIER-MARS 1972.

FIG. 1. - (a) ^Perturbed angular correlations ; (b) ^Nuclear orientation : emission from

level I with

radiative lifetime shorter than

comparable ^to the thermal relaxation times in this level ; (c) Nuclear orientation : emission from

level I with

lifetime much longer than the relaxation times in this level.

tion will exhibit a measurable anisotropy only ^if 1 A/kB T 1 ^1.

Apart ^from this, ^the analogy between the three situations (a), (b) ^and (c) ^described by figure ^{1 is} striking ; (b) may be considered as an experiment ^of type (a) ^{in which} the first particle ^{emitted is not}

observed and (c) ^{as an} experiment ^of type (b) ^{in which}

the mean lifetime of the intermediate state tends to

infinity (1). ^{In all three cases,} the characteristics of the radiation emitted in the transition I -> Iî depend

(1) ^{So that}

^{need not}

ourselves with preceding

transitions.

19

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01972003302-3028500

strongly ^{on the} properties ^{of state I and on the} pos- sible evolution of the nucleus during the time it spends

in this state.

Previously ^{we have examined} the effects of the rela- xation in state I ^on perturbed angular correlation

experiments ((a) ^with 1 AlkB T 1 « 1) [9], [10]. Then,

in connection with some nuclear orientation measu-

rements performed ^{in Oxford, we had to consider an} experiment ^of type (b) ^{in which the} relaxation _gave rise to a partial reorientation of the nucleus in the intermediate state. This has led us to extend the clas- sical theories of relaxation (which ^{assume that}

1 AlkB T 1 « 1 ) ^{to the very low} temperature region ⁱⁿ

which this condition is not fulfilled [11 ]. ^In this way,

we have been able to interpret, very satisfactorily, ^the

nuclear orientation results concerning ’09Ag’ ^{in Fe}

and Ni [12], [13]. Finally ^some experiments [14] ^invol- ving ^{the resonant} destruction of the _{y ray} anisotropy

in situations of types (b) ^and (c) ^{have led us to consider}

the effect of radiofrequency saturation on level I.

At this point ^we have found it interesting ^{to refor-}

mulate the theory ^{of emission of cascade} radiation,

so as to obtain general ^{formulae in which} (a), (b), (c)

were particular ^{cases, and to treat the} problem ^of

relaxation and radiofrequency ^{from a more funda-}

mental view point. This is the purpose of the present work (2) which consists of two parts :

In the first part, ^{we consider} relationships ^between

the characteristics of the emitted radiation and the

properties ^{of the} intermediate state. The second part deals with effects of relaxation and radiofrequency ^on

this state.

In the first part, ^{we derive some} general ^formulae giving ^{the number and} characteristics of the photons

emitted in a 1’1 - y2 cascade (Ii -’-’+ I -’-’> _{I f)} ^assu-

ming ^{that their} energy is ^not observed. In the infi- nite temperature approximation these formulae give again ^the quantity W(kl, k2, t

t’) observed in per- turbed angular correlations experiments. ^{At any} temperature, by integrating ^over the direction k1 ^of

the first photon, ^we obtain the anisotropy W(O, ç)

of the radiation 72 ^{emitted in the second} transition ; this expression ^for W(O, ç) ^{reduces to the usual}

results when the hyperfine hamiltonian has an axial symmetry, and when the lifetime of level I is either infinite or very short compared ^{with the relaxation}

times. Finally, ^{for a finite} lifetime, ^our equations jus- tify the intuitive calculational method used in [12],

in order to take into account the partial redistribution of the populations ^{while the} nucleus is in the state I.

In order to study the evolution of the nucleus in this state we use the Fano expansion ^{of the} density

matrix [15] ^{on a basis} of irreducible tensor operators

(2) ^Practical applications and numerical values may be found in reference [13].

This formalism enables us to simplify considerably

the presentation ^and interpretation ^{of our} results. We show in particular ^{that the} perturbation ^factor

G N,N,(t@ t’) ^{used in the} theory ^of angular correlations

can be very simply ^{related to the time} variation of

In the second part ^we begin by summarizing ^how

this temporal variation is perturbed by ^relaxation

effects. We discuss the validity ^{of the} theory ^elaborated by H. Gabriel in order to treat the problem ^{of rela-}

xation when the correlation time of the fluctuating

interactions becomes longer than the lifetime of the intermediate level. Finally ^we investigate ^{the effect of} radiofrequency saturation on this level, in either the presence ^or absence of relaxation. It will be shown that if the relaxation has no time to act, the formulae obtained exhibit hard core terms when the R. F. field goes ^to infinity. ^{On the} contrary, ^{if the lifetime of} level I is much longer ^than the relaxation times (this

may be the ^case in nuclear orientation experiments),

the radiofrequency saturation makes all the u" k ^vanish

and completely destroys ^the anisotropy ^{of the emitted}

radiation. These conclusions allow a determination of the initial conditions in the experiments ^{in which}

one observes the reappearance of the y ray anisotropy,

after the R. F. field has been switched off.

II. Général characteristics of the radiations emitted in a Yl

7, cascade.

1) ^NOTATIONS AND HAMILTO- NIAN.

a) Nuclear levels.

We consider (Fig. 2)

two successive _y ray transitions between three nuclear

levels : h ^(basis functions 1 Ii, P > such that Izi

p), I(I

m) ^and If(I,,f

^y). In the absence of external

perturbations, ^the energies of these levels are given by

the eigenvalues ^{of a nuclear} hamiltonian JCN :

The external perturbations (acting inside each level),

which we will consider, ^{are of two} types : static, represented by ^a hamiltonian Jez (in practice, ^Zeeman and/or quadrupolar), ^and dynamic, represented by ^a

hamiltonian Xl(t) (in practice, radiofrequency and/or

relaxation). One should note that

In the presence of £_, the nuclear levels are split.

If Jez ^{has axial} symmetry ^with respect ^{to Oz as often}

occurs, then the basis functions defined above are

also eigenfunctions ^of (XN ⁺ JCJ ^{and one has :}

b) ^{Free radiation} field.

^{In the} plane ^wave repre-

sentation, ^the hamiltonian of the free radiation field

can be written as :

in which

ak;. ^and akl ^are photon ^{creation and} annihilation ope- rators. The index  is associated with the two possible

states of transverse circular polarization ^{of the} photon.

JeL ^{commutes with} XN, £-,, XI. ^{We shall define :}

c) Interactions between the electromagnetic field ^and

the nucleus.

We will adopt the Brink and Rose form ([3 ^{eq. (2.17’) and} (3.21)) :

with :

in which q

^{± 1 and the} TLi > ^{are, to some multi-} plicative factor, the various (electric, ^7r

^{0, and} magnetic, n

1) multipole ^{moments of} the nucleus

(Brink ^{and Rose [3] eq. (3.17) and} (3.20)).

For the TLM, ^{as well as for the rotation} operators,

we shall, ^{as in a} previous ^{work [10], use the conven-}

tions of Edmonds [16] (’). ^{Then :}

with :

where a, fi, y ^are the Euler angles ^{of the rotation which}

transforms the coordinates of the nuclear frame Oxyz

into the coordinates of a frame OXYZ which has OZ

parallel ^{to the wave vector k of the} photon (the polar angles ^{of k in the} Oxyz ^frame being ^a

ç and fl

0).

When ^one is looking only ^at pure circular polariza- tions, ^the third Euler angle y disappears ^{from the}

formulae and may then be ^set equal ^{to zero.}

(3) ^{Brink and Rose} claim that their conventions for the rota- tions

Messiah’s

But

the TLM, ^their

tions

neither those of Messiah,

^those of Edmonds.

The above conventions allow us to work out several formulae which were known previously.

d) Interaction representation.

^{We observe tran-} sitions due to spontaneous ^emission between the nuclear levels (Ii ’l 1 1 ^{, --+ ’12} _{I f).} ^The physical ^{states of}

interest are

In order to study ^the coupling between these states, ^it is convenient to turn to the interaction representation.

Let us introduce the evolution operators :

such that :

and

and define in the interaction representation :

ict) is the solution of

2) ^COUPLED EQUATIONS. EFFECTS OF SPONTANEOUS EMISSION.

As in (17) it is easy ^to establish that the

equation relating ^{the b’ s to one another are of the}

form :

We shall now assume for the sake of simplicity ^{that the}

yl and y2 transitions have a well defined multipolar

character (Ll ^for y,, L2 ^for 72). ^In this way ^we avoid

introducing mixing ratios, ^the sign ^{of which} depends critically ^{on the} phase ^{choice made for TLM with}

different L, ^{and which are} responsible ^{for the subtle}

conventions of Brink and Rose (4).

Before undertaking ^{the full} calculation of the

Let us write explicitely IXR ^{and the sum over the} photons k2, Â2 (energy, polarization, angle). ^{This becomes :}

and, ^after suming ^over e2, P2, q2 :

We must now show that, provided ^{that one} integrates (as ^{we do it} here) ^over the energy of the emitted

photon (f dk2 ), it ^{is possible, in the time integral,}

to replace t’ by t ⁱⁿ U,(t) ^and bm,(t’).

Let us set t’ = t

The quantity :

is a function of z, the time variation of which is _very slow compared with optical periods (i. ^e. Q2-’). ^{Let us}

then write schematically ^{this function} in the form :

The double integral ^{over k and t’ in eq.} (21) appears

as a sum of terms of type :

(4) ^{It should be noticed that the} signs ^{of the} mixing ^ratios disappear from ^{the formulae when}

integrates

^{the emission}

direction of the particle.

yi - Y2 cascade it is worth treating the effects of spontaneous ^{emission on} levels Ii ^{and L} Spontaneous

emission is associated with the summatidn over ki Â,

in _eq. (16) ^and the summation over k2 Î2 in eq. (17).

We shall here consider the case of level I and use the method of ref. [18] p. 544. Let us integrate [18]

and substitute the result into [17] ; there appears

a term :

p is much smaller than Q2(PIQ2 - 10-") ; ^therefore

when calculating ^the integral ^over k, p may be neglected

in the arguments ^{of the} b function and of the principal

part. ^This amounts to neglecting

ⁱⁿ (22) ^{and hence to} replace ^t’ by t ⁱⁿ (21). ^After completing ^{all the calcu-}

lations (21) ^{reduces to :}

in which :

and :

T2 and E2 characterize (5) ^the damping (« ^lifetime »)

and the self-energy ^{of level} I, ^{which are identical}

(5) The reduced matrix elements of Brink and Rose

bigger

than those of Edmonds by

^factor .J 2 1 + 1. ^{On the other hand,}

the 11 TL2 11 > depend

^the energy of the photon (B

^R

eq. (3. 29) ^and (3.30).

for all its sublevels m. The (infinite) self-energy ^is usually incorporated ^{in the} unperturbed energy of level 1.

Eq. (27), (28), (29) ^will constitute the starting point ^of

the cascade calculation.

3) ^{GAMMA-RAY CASCADE : :} GENERAL FORMULAE.

Let us assume that the sample ^is put in the container at time to. ^{Let us} integrate (28) :

in which Ha ^{is defined} by ^eq. (6).

Since we are integrating ^{over the} photon energies,

a similar argument ^{to the one used in the} preceding paragraph, ^{leads to} replacing t" by t ^{and t"’} by ^t’.

Having ^done this, ^{one can} explicitly ^{write out the}

in which we have set UA(t, t’)

UA(t) UA+ (t), ^and

uIi(t’) exp(- ^{Tl(t’ -} to)) ^{is the density matrix of the}

initial level at time t’ in the normal representation.

(More generally, ^we shall denote by J ^and p the

density ^matrices respectively ^{in the} normal and inter-

Taking ^{these results into} account, and extending

them to level Ii, ^the system of eq. (16), (17), (18) may

be replaced by :

We insert this result into (29), integrate again ^and

calculate

What we observe experimentally ^are the numbers of

photons y1 y2 emitted within certain solid angles dtf 1 dtf 2 ^without being concerned with their frequen-

cies or polarizations. Therefore, ^{we must sum over} Il,

À,1, ’Z2 ^and integrate ^{over the} lengths ^of kl ^and k2 (6).

Then we get :

Hamiltonian Ha ^{and relate} the reduced matrix elements whic happear ⁱⁿ Ha, ^to the radiative dampings Ti, T2

and self energies AE1, AE2 ^{of the two} excited levels

Ii and /. Hence :

action representation, u ^and p being ^{defined in the}

absence of radiation damping.)

(6) Some formulae pertaining ^{to the} spectroscopic ^Môssbauer

case are

given ^{in the} appendix.

In the experiments of interest to us, Ti « 1’2, ^and

t

to( 1 jTl) ^{is much} longer ^{than Larmor} periods

and relaxation times in the excited states [19]. ^{For this}

reason, it is possible ^to replace (32) by ^an expression

of type :

with

(it ^{is easy to} check that A;(k2) ^{is non-zero} only ^{for even} k). Substitution of (33) ⁱⁿ (32) gives ^the general expres- sion which will be used as a starting point ^{in the} following calculations

Upon going ^from (32) ^to (35) ^{we have} dropped ^the AE2 ^{term, since one can} demonstrate that it disap-

pears ^at the end of the calculations.

4) ^PHYSICAL INTERPRETATION.

It is possible ^to give ^a simple physical interpretation (Fig. 3) ^{of the} right ^{hand side} of eq. (35) :

up,,(t’) corresponds ^{to the} density matrix, ^{at time} t’, of the nuclei sitting ^{in the} upper level h (point (1)

in Fig. 3a).

represents (immediately ^after their arrival in the intermediate ^state I) ^the density ^{matrix of the nuclei which have} undergone, ^at time t’, ^the transition Ii

I by ^{emission of a} photon k, ^{into the solid} angle dl(Ol, qJ 1) (point ⁽²⁾

in Fig. 3a).

(7) ^The integers ki k2 which enter the summations in what follows should not be confused with the moduli of the

vectors of the photons ^{included in} Tl, F2.

the normalization of aIt being ^{chosen so as to fit the}

number of radioactive nuclei sitting ^{in state} h ^{at the} beginning time to ^{of the} experiment. ^{The trivial factor} g-ri(t-fo) then just represents ^{the decrease in} activity

of the sample during ^the experiment. ^{For the sake of} simplicity ^{we shall} drop ^{it in} what follows.

Summation over M2, M2, q2 and _Il (Edmonds 4.3.2

and 6.2.8) (’) ^{leads to the formula :}

FIG. 3.

(a) Evolution of a given ^nucleus

function of time ; (b) ^{Evolution of two} different nuclei A(» ^{and B} ( ))

^function

of time.

represents what this matrix has become at time t under the effect of the various perturbations acting ^inside

level I (point ^{(3) in Fig.} 3a). ^{Finally :}

represents the number of nuclei (per second) arriving

at time t in the fundamental level I f by ^{emission, at}

this same time t, ^{of a} photon k2 ^{into the solid} angle d1/1 2

(point ^{(4) in} Fig. ^{3a and} 3b). ^{It should be noticed}

that L( 1/1 l’ 1/12) only depends ^{on the} density ^{matrix of}

the emitting ^{nuclei at time t. But this matrix is not the}

same for all the nuclei in state I, ^{since it} depends upon t’, ^{i. e. the time at which the first} photon k, ^was

emitted (Fig. 3b).

III. Application ^to perturbed angular correlations

experiments.

The classical formulae for P. A. C.

follow from (35) by taking the « very high » tempe-

rature limit of the Boltzmann density matrix for the initial level Ii

As a matter of fact, ^{since the} hyperfine frequencies ^are

at most of the order of ^a few thousand MHz, ^this applies ^{for any} temperature higher ^{than 1 OK. In}

eq. (35) ^{it is then} possible ^to perform ^{the sum}

over p :

with Al(kl) being ^{defined in the same} way ^as A’(k,) :

Finally, upon setting :

We get ^{the result :}

with :

Gf/k2(t, t’) ^is nothing else than the perturbation factor used in perturbed angular correlation theories [20].

L(1/11’ 1/12)PAC dg/i d1/12 gives ^the expression ^{of the} integral correlation and its time derivative (setting T2 equal

to zero) gives ^the expression of the differential corrélation :

As an elementary ^{check of} eq. (43) ^{let us} calculate the total number of nuclei arriving per unit time in the fundamental level If . ^{It is equal to :}

using ^{the fact that}

et

The physical interpretation of this kind of expres- sion has already ^been given in II.4. Here one expects

an additional simplification, ^{due to} the fact that the spontaneous ^{emission of the first} photon ^has spherical symmetry. ^{In order to take} advantage ^{of this} simpli- fication, ^{it is} interesting ^to develop ^the density ^matrix

on a basis of irreducible tensor operators. ^{This will} be done in the next paragraph.

Note.

Formula (48) ^{has been} derived under the

assumption ^{that the first} particle ^{emitted in the cas-}

cade emitted in the cascade was a photon. If the first transition was an

or f3 transition, ^{or an electronic} capture, ^{the related formulae would} differ from (48) only by ^a numerical factor [21]. ^These results justify ^the procedure adopted ⁱⁿ (12), ^{in which the first transi-}

tion was an electronic capture.

V. Use of the irreducible tensor operators ^forma-

lism.

1) NOTATIONS.

Inside each nuclear level

we expand ^the density ^{matrix on a basis of norma-}

lized tensor operators ^{fitted to this level} [2] [15]. ^The formulae will be given ^{for the} intermediate level I.

By definition [10] :

It should be noted that, ^{to a constant} factor, ^the

T:’ s ^are nothing ^{else than the} operator equivalents ^of

the spherical harmonics, widely ^{used in} crystal ^field theory [22].

it is _easy to show that :

as could be expected (conservation of the total number of nuclei during ^{the radiative} processes).

IV. Nuclear orientation.

In such experiments only ^{the second} photon ^{is observed.} On the other

hand, ^one operates at very low temperatures ( 1/10 °K). ^In eq. (35) ^{we shall} consequently ^inte- grate ^over 81, çi

but no assumption ^{will be made} concerning upp, :

The expansion ^{of the} density ^{matrix can be written}

2) RELATIONSHIP BETWEEN THE TIME EVOLUTION OF THE Qk AND ^THE PERTURBATION FACTORS OF ANGULAR CORRELATIONS. - As ^a function of the evolution operator UA, ^the relationship between the density

matrix of level I at time t’ and the density ^{matrix of}

the same level at time t is :

Using eq. (44) and this last identity ^{it can be shown}

that [23].

In the presence of dissipative phenomena (rela- xation), ^{this relation allows us} the determination of the perturbation ^factors by integration ^{of the} equation

of evolution of the density matrix. In the absence of

relaxation, it is often simpler (and nevertheless safe)

to compute directly ^{the G} factors from the following relation, ^which is very easy ^to establish :

(it ^{should be noticed that this trace does not contain}

the density matrix).

3) ^{COUPLING BETWEEN} THE 6j ^{OF DIFFERENT NUCLEAR}

LEVELS BY THE PROCESS OF SPONTANEOUS EMISSION.

According ^to (36), ^the density matrix of the nuclei

arriving ^{in level} I just ^{after the} spontaneous ^emission of a photon ^with arbitrary kl, ^is (after summing

over 01 91) ^{related to their} density ^{matrix in} level h just ^{before the emission} by :

from which it follows that :

Consequently, ^the spontaneous ^emission process

Defining :

this becomes, ^{in terms of} theuk :

where lu N*(, t@ t’)]em is related to [UN t’),in ^by

a formula of type (53) ^and k t)]in ^{is related}

to [JÎ*(I;, t)] ^{by eq. (56).}

We shall now consider several particular ^{cases and}

show how these formulae are connected to the usual

expressions used in the interpretation ^{of N. 0.} expe- riments (for ^{the sake} of simplicity ^{we shall assume here}

that the nucleus does not experience ^{an R. F.} field).

a) If ^the lifetime of ^level 1 is very long.

(T2 0),

the nuclei will lose any memory of level h ^before emitting k2

Dropping explicit mention of t in the formulae, ^{as is} usually done, ^we then arrive at :

does not mix the Jl. This result which was rediscovered

recently ^{in the case of} optical transitions [24], ^had

been recognized ^a long ^{time ago} in nuclear orientation work [2]. ^{It should be noticed that :}

which _expresses the conservation of the number of nuclei during ^the spontaneous emission process.

4) RELATIONSHIP BETWEEN THE y RAY ANISOTROPY IN NUCLEAR ORIENTATION EXPERIMENTS AND THE 6k.

We have shown that the Y2 ^{radiation emitted in the}

solid angle dt/12 ^is given ^{in terms of the} density ^matrices

of the different classes of emitting ^{nuclei at time t} by (eq. (48), (36), (37), (38)) :

A formula similar to this one (but ^{with différent} notations) ^was established by ^{de Groot and Cox} [25] ]

as early ^{as 1953. For a} diagonal density matrix, ^{if we} define :

then :

in agreement ^{with the classical results} [2].

Since F2(0)

1/4 ^{7T, it is} easily ^{checked that :}

b) If ^the lifetime of ^{level 1} is very short.

The

density matrix of the nuclei will not have time to

vary during ^their stay ^{in this} level, ^{so that :}

In this case, instead of (62) ^and (64), ^{we now have} (’) :

In a more general ^{way, if one has a cascade in which}

all the intermediate ^states have a very short lifetime,

the relationship between the anisotropy of the last _y ray, and the density matrix of the initial level will have the same form as (67), (68), but the relevant Uk’ s ^will

be the products of the individual Uk’s corresponding

to all the transitions preceding ^{the last one. This}

result is also well-known [2].

c) Intermediate case : partial reorientation in the intermediate state by relaxation processes.

This situa- tion has been considered in ref. [11 ], [12]. ^{Since the} density ^{matrix involved were} diagonal, ^{the relation}

between the Boltzmann density matrix of the first state [u(Ii)], and the initial density matrix of the intermediate state [6I(t’)]in

^u,

^{was obtained by a}

transition probability calculation. [O"I(t, t’)]em ^was

then deduced from [UI(t’)] in by solving ^the equations

of evolution for the populations of level I with the

help ^{of a} computer program. Finally ^the integration

of an expression ^of type (58) furnished the _y rayani- sotropy.

It should be noticed that the same results _may also be obtained without _any integration by replacing

in (60) ^the quantities

by ^the steady ^state solutions of the detailed-balance system associated with level I. In order to obtain this system ^one simply ^{adds to the} relaxation equations ⁱⁿ

state I, ^terms describing the radiative feeding ^{of this}

level by h ^{and its radiative} decay ^to if -

VI. Remarks. 2013 1) ^The quantities Fk, eq. (59),

and Ak, eq. (34), ^{are different from zero} only ^{if k is}

even. This arises from the fact ([5] ^p. 1207) ^{that only}

the individual circular components of the radiation field are sentitive to the odd k terms of the density

matrix. Here we sum over the polarizations. ^{It is}

also clear that with a radiation of multipole ^cha-

racter L, ^{it is} only possible ^{to detect the} Jf ^with

k 2 L.

2) ^{There is a close} similarity ^{between the PAC and}

level-crossing experiments. The evolution of the nucleus or atom in a state of finite lifetime is detected : - In the first case (PAC) by ^the change ⁱⁿ intensity

of the radiation k2 emitted in a certain direction.

-

In the second case (level-crossing ^experi-

ments [26]) ^{by the} associated change ^{in its} polarization

state (under the condition of an appropriate ^{« prepa-}

ration » of the atom).

3) Finally, ^we would like to point ^{out that the} only experiments ^{which are sensitive to} relaxation pheno-

mena are :

a) The Môssbauer effect, because of the spectro- scopic ^Fourier analysis (see Appendix).

b) ^{The P. A.} C., and the unusual N. 0. experiments

involving ^{nuclear levels with} intermediate lifetimes,

because of the interference between the density

matrices of the different classes of atoms (see ^{eq. (35)}

and comments about it).

But the usual (non resonant) ^{N. 0.} experiments, involving only nuclear levels with either very short

or very long ^lifetimes compared ^with the relaxation

times, ^cannot give ^any information on the relaxation, contrary ^{to what} has sometimes been assumed. These

experiments ^can only ^{measure the} equilibrium, ^sta- tionary properties ^{of the} long-lived ^states (see ^{eq. (62)}

and (67)).

Note added in proof.

^{To be more} specific, ^{let us}

consider the case of a nuclear spin ¹ coupled ^{to an}

electronic spin ^{S. It has often been assumed} (30) ^that

N. 0. experiments ^{could be used to} discriminate bet-

ween two different situations :

--

the so-called ^« fast relaxation

situation, ⁱⁿ which case the effective nuclear hamiltonian is taken

as :

where S > is the time-averaged value of the electronic spin ^{in the external field} Ho ;

- and the so-called « slow relaxation » situation,

in which case the radiation anisotropy ^{must be obtain-}

ed from the complete hamiltonian :

our analysis shows that indeed, whatever the relaxation

rate, the radiation anisotropy ^must always ^be comput-

ed from the Boltzmann density ^{matrix associated}

with the complete Hamiltonian (70) (9)

Acknowledgements.

^This study ^{has been} greatly

stimulated by ^the experimental ^work performed by

one of us (D. S.) ⁱⁿ collaboration with the Nuclear orientation _{group of} the Clarendon Laboratory,

Oxford (N. ^J. Stone, ^{R. A.} Fox, ^{J. D.} Marsh, P. D. Johnston). ^{We also wish to thank} Dr. J. P. Des- coubes for some helpful ^remarks.

Appendix.

^{Let us} consider the case of a Môss-

bauer source. The Môssbauer level I is fed by ^an upper level Ii through the emission of a photon ki ^which

is not observed (Fig. lb). ^{On the other hand, I} decays

to the fundamental level I f ^through emission of a

« Môssbauer » photon k2 ; ^the experiment consists in

studying ^the intensity J(k2) ^{of this} emission, ^{as a} function of the energy úJ2

ck2 ^{of the} photon.

In order to deal with this situation we shall start from eq. (31) ^{in which we shall} drop ^the integration

over the modulus of k2. ^{After some} algebra, ^similar

to that of § II.3, ^and neglecting the decrease in

activity ^{of the} sample, ^{we obtain :}

with :

This formula is quite general. ^{In order to} put ^{it in a more} compact ^and physical ^{form we shall first} replace ⁱⁿ

the last line :

and then introduce two particular density matrices for level l :

and

As in § II.4, u!,(t") ^{is the} density matrix of the nuclei arriving ^{in level I} by ^radiative decay from Ii ^at time t" ;

U,,(t, t") ^{is what this} matrix has become at time t. As a function of ul,(t, t") eq. (A.l) may be written :

This general equation ^{relative to the} spectroscopic ^{case may be} compared ^{with the} corresponding equation ^for

the non-spectroscopic ^{case, deduced from} (A. 2) by summing ^over C02 (1 °) :

(9) ^The

^of

^Kondo impurity ⁱⁿ

^metal below the Kondo temperature ^{is not} considered here.

(lo) ^This equation may be transformed into _eq. (48) ^{of the text.}

It is seen that in the spectroscopic ^{case, the emission} depends (through t’), ^{on the whole} history ^{of the} emitting ^{nucleus and not} only ^{on its} instantaneous state. This is evidently ^a consequence of the ^uncer-

tainty ^relation between time and _energy.

The general eq. (A. 2) ^{takes a} simpler ^{form in two} particular ^{cases :}

a) 1jF2 ^is very long. Then, ^{the nuclei lose their}

memory of time t" and u,,(t, t") may be replaced by UI(t), density matrix of the level I at time t in the absence of radiative feeding. ^This argument applies

for t" and not for t’, because all time t" give compa-

b) ^Another simple ^{situation occurs when} the initial

level Ii is in thermal equilibrium ^at « high tempe- ratures (case of almost all Môssbauer experiments performed ^until now). ^{Then ah is} proportional ^{to the}

unit matrix

and

The corresponding expression ^for J(w,, t/J 2) ^{is identical}

to (A. 6) except ^{for the} fact that uj(t) ^is replaced by C(2 1; ⁺ 1)/(2 1 + 1). Upon summing ^{over the direc-}

tion of k2 (case ^{of a} powdered ^Môssbauer source),

it becomes :

rable contributions to the integral ^over t", ^{while the}

main contributions to the integral ^{over t’} only ^arise

from times t’ such that t - t’ is of order of the Larmor

period. ^{Then if we introduce a new evolution} ope- rator Ue, associated with the total nuclear hamil- tonian Jee :

and define

eq. (A. 2) may be ^cast into the form :

Formulae like (A. 7) ^have already ^{been used in order}

to study ^the modifications of Môssbauer spectra under the effect of random perturbation [27]. (A. 2)

and (A. 7) could also be useful in the presence of ^a coherent perturbation ^{such as a} radiofrequency ^{field ;}

it is known (Hack and Hammermesh [28]) ^{that the}

aspect ^{of the emitted} spectrum ^{is then} greatly altered,

in similarity ^{with the} Autler-Townes effect [29].

References

General references.

[1] BIEDENHARN (L. C.), ^ROSE (M. E.), ^{Rev. Mod.} Phys., 1953, 25, ^729.

[2] BLIN-STOYLE (R. J.), ^GRACE (M. A.), Handbuch der

Physik, 1957, 42, ^555.

[3] ^ROSE (H. J.), ^BRINK (D. M.), ^{Rev. Mod.} Phys., 1967, 39, 306.

[4] ^Perturbed Angular correlations Karlsson, Matthias, Siegbahn editors, ^North Holland, 1964.

[5] Alpha, Beta and Gamma Ray spectroscopy, ^Vol. 1, ² Siegbahn editor, ^North Holland, 1965.

[6] Hyperfine ^structure and nuclear radiations Matthias, Shirley editors, ^North Holland, 1968.

[7] Hyperfine interactions in excited nuclei, ^Vol. 1, 2, 3,4 G. Goldring, ^{R. Kalish} editors, Gordon and Breach, ^1971.

[8] Hyperfine interactions, ^edited by ^{A. J.} Freeman and R. B. Frankel, ^Academic Press, 1967.

0 ther references.

[9] HARTMANN-BOUTRON (F.), ^SPANJAARD (D.), Compt.

Rend., 1969, 268B, 1260.

[10] ^SPANJAARD (D.), HARTMANN-BOUTRON (F.), ^J. Phy- sique, 1969, 30, ^975.

[11] ^SPANJAARD (D.), HARTMANN-BOUTRON (F.), ^Solid

State Comm., 1970, 8, 323. Erratum : Solid State Comm., 1970, 8, ^{n° 20} p. iii.

[12] ^STONE (N. J.), ^FOX (R. A.), HARTMANN-BOUTRON (F.),

SPANJAARD (D.), 7th International Conference

Magnetism, Grenoble, 1970, ^J. Physique, 1971, 32, C1, ^897.

[13] ^SPANJAARD (D.), Thèse de Doctorat d’Etat, ^Faculté d’Orsay, 10 juin ^1971.

[14] ^FOX (R. A.), ^JOHNSTON (P. D.), STONE (N. J.), Phys.

Letters, 1971, 34A, ^211.

[15] ^FANO (U.), ^{Rev. Mod.} Phys., 1957, 29, ^74.

OMONT (A.), ^J. Physique Rad., 1965, 26, ^26.

SCHWEGLER (H.), ^Z. für Physik, 1966, 189, ^163.

[16] ^EDMONDS (A. R.), Angular Momentum in Quantum Mechanics, ^Princeton University Press, ^1957.

[17] ^BARRAT (J. P.), COHEN-TANNOUDJI (C.), ^J. Physique Rad., 1961, 22, ^329.

[18] ^BARRAT (J. P.), ^J. Physique Rad., 1959, 20, ^541.

[19] ^As

example, for the cascade

Cd109 ~ Ag109m ~ Ag109

considered in ref. [12], the lifetimes of levels Ii ^{and I}

are

respectively equal ^{to 453} days ^{and 40}

[20] ^{See for} example ^the paper by ^STEFFEN (R. M.) ^and

FRAUENFELDER (H.) ^{in ref.} [4].

[21 ] ^MORITA (M.), Lectures in Theoretical Physics, ^Vol. IV, p. 358, Boulder 1961.

[22] ^HUTCHINGS (M. T.), Solid State Physics, 1964, 16, 227.

SMITH (D.), ^THORNLEY (J. ^H. M.), ^Proc. Phys. Soc., 1966, 89, ^779.

[23] This relation has also been derived independently by

GABRIEL (H.) ^{and BOSSE} (J.), ^Proc. of the Int. Conf.

on

Angular Correlations in Nuclear Disintegra- tions, DELFT, ^1970.

[24] ^NEDELEC (O.), ^J. Physique, 1966, 27, ^660.

DUCLOY (M.) and DUMONT (M.), ^J. Physique, 1970, 31, 419.

[25] ^{De GROOT} (S. R.), ^Cox (J. A.), Physica, 1953, 19, 683.

[26] ^{See for} example ^the paper by ^STEUDEL (A.) ^{in ref.} [8], p. 210.

[27] ^BRADFORD (E.), ^MARSHALL (W.), ^Proc. Phys. Soc., 1966, 87, 731 ;

HARTMANN-BOUTRON (F.), ^J. Physique Rad., 1968, 29, 47.

GABRIEL (H.), ^BOSSE (J.), ^RANDER (K.), Physica ^Status Solidi, 1968, 27, ^{301 and} references therein.

[28] ^HACK (M. N.), ^HAMERMESH (M.), Nuovo Cimento, 1961, 19, 546 ;

GABRIEL (H.), Phys. Rev., 1969, 184, ^{359. An} explicit expression for the evolution operator ^associated with

pure R. F. field _may be found in the book

by ^RAMSEY (N. F.), « Molecular Beams », p. 429- 430. The evolution operator

also be computed

as a

function of rotation matrices by ^{the method}

used in the second part ^{of this} paper : SPAN-

JAARD

(D.), HARTMANN-BOUTRON (F.), ^{to be} published.

[29] ^AUTLER (S. H.), ^TOWNES (C. H.), Phys. Rev., 1955, 100, ^703.

HARTMANN (F.), C. R. Acad. Sci. Paris 1969, 268B, 404 and references cited therein.

[30] ^See paragraph ^V in the review paper by ^STONE (N. J.)

in ref. [7] (vol. I, p. 237).