On the use of Liouville relaxation supermatrices in Mössbauer studies

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On the use of Liouville relaxation supermatrices in Mössbauer studies

F. Hartmann-Boutron, D. Spanjaard

To cite this version:

F. Hartmann-Boutron, D. Spanjaard. On the use of Liouville relaxation supermatrices in Mössbauer

studies. Journal de Physique, 1975, 36 (4), pp.307-314. �10.1051/jphys:01975003604030700�. �jpa-

00208255�

ON THE USE OF LIOUVILLE RELAXATION

SUPERMATRICES IN MÖSSBAUER STUDIES

F. HARTMANN-BOUTRON and D. SPANJAARD

Laboratoire de

Physique

^des

Solides,

Université

Paris-Sud,

⁹¹⁴⁰⁵

Orsay,

^France

(Reçu

^{le 22 novembre}

1974)

Résumé. ²⁰¹⁴ Dans une publication antérieure

(F.

Gonzalez-Jimenez, P. Imbert et F. Hartmann- Boutron, Phys. ^{Rev. B 9} (1974),

95),

^{nous avions} exprimé ^{la forme} de raie Mössbauer en présence

de relaxation, ^en fonction d’une certaine supermatrice de relaxation R. Dans le présent article,

nous établissons une expression rigoureuse ^{de R} qui ^{est valable} quelle ^{que soit la} température, ^et

nous relions R à la supermatrice de relaxation S qui caractérise la relaxation de la matrice densité 03C3 de l’atome radioactif. Nous examinons ensuite le problème général des moyennes ^sur le réseau dans le calcul des effets de relaxation thermique, ^{et à titre} d’exemple ^nous établissons l’expression ^de

la forme de raie Mössbauer d’un émetteur.

Abstract. ²⁰¹⁴ In a previous ^paper

(F.

Gonzalez-Jimenez, ^P. Imbert and F. Hartmann-Boutron, Phys. ^{Rev. B 9}

(1974), 95),

^we have related the Mössbauer spectrum in the presence of relaxation ^to

a certain relaxation supermatrix ^{R. In} this paper ^we derive a rigourous expression ^{of R which is}

valid at arbitrary temperatures ^{and we relate it to} the relaxation supermatrix S which characterizes the relaxation of the density matrix 03C3 of the radioactive atom. We then discuss the

general problem

of lattice

averaging

^{in the} computation of relaxation effects and as an illustration, ^{we derive the} expression for the Mössbauer lineshape ^{of an emitter.}

Classification Physics ^Abstracts

4.200 ^- 8.610 ^- 8.680

Introduction. ^- In this _paper, we want to revise

slightly

^one of the results of ref.

[1]

^{i. e : the definition}

of the relaxation

supermatrix R

associated with

M,

which comes into

play

in eq.

(16)

^and

(23)

^of

[1] and

whose elements are defined

by

^footnote

(7)

^of

[1].

We will show here that the

supermatrix R

^{must be}

the

transposed

matrix of the relaxation

supermatrix S

associated with the

density

^{matrix 6 of} the radioactive atom.

Explicit expressions

of S and R are then derived and are

compared

with the results

of previous

^theories.

Finally,

ⁱⁿ view of future

applications

^{to Môssbauer}

transmission and

scattering

^we

give

^some

general

rules for

performing

^lattice averages in relaxation

problems.

1. Liouville évolution

superoperators

^{for an isolated}

atom without relaxation. ^- Let us first assume that

we have an isolated atom described

by

^a

complete

orthonormal set of

functions 1 a ), 1 b ), 1 c), 1 d)

and whose evolution is

govemed by

^a hamiltonian

Jeo [1].

^The relevant evolution operator

Uo(t)

^{is :}

In terms of this operator ^{the time}

dependence

^{of the}

density

^{matrix a of this atom in the}

Schrôdinger

representation

^{is :}

with :

where Xxo

is the Liouville superoperator

[1]

^asso-

ciated with the hamiltonian

Ho.

On the other

hand,

^{in the}

Heisenberg representation

the time

dependence

^{of an atomic}

observable Q

^{is :}

with : ’

In

explicit

^{form :}

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01975003604030700

308

From this it follows that :

i. e. the Liouville matrix associated with

°’11o(t)

^{is the}

complex conjugate

^{of the}

transposed

^{matrix of the}

Liouville matrix

corresponding

^to

CU o(t).

2. Standard treatment of relaxation. ^- We must now consider the case where the atom is

coupled

to a quantum ^bath

by

^a relaxation hamiltonian

H1,

the hamiltonian of the bath and its

eigenstates being

denoted

respectively by XB

^and

by 1 r >, 1 r’ >.

The standard

approach

^to relaxation effects

[2, 3]

consists in

calculating

the relaxation of the

density

matrix u of the atom. For

this,

^one first derives the

equation

of motion of the total

density

matrix 0 of the atom

plus

the bath in the interaction

representa-

tion

[3].

^{Let us define :}

The

equation

^of evolution of 0* is :

This

equation

^is

integrated by

successive

approxi-

mations

([2] chap.

^{VIII eq.}

(32)

^and

(60)) :

It is then admitted that since the heat

capacity

^{of the}

lattice is _very

large,

^the

density

^{matrix 0 can be facto-}

rized as :

where _po is the Boltzmann

density

matrix of the lattice.

Further

approximations

^{which are discussed}

by Abragam ([2] ^chap.

^{VIII p.}

276) ^finally

^{lead to the}

well-known master

equation :

We must now translate these results into terms for the evolution

superoperators.

3. General

properties

^{of the évolution} superoperators in the présence of relaxation. ^- From the total hamil- tonian of the system ^{atom + bath}

we can construct the evolution

operator :

In order to transform to the interaction

representation

one may write

[4] :

-

Il ^- il ^{1 - 1}

with :

and U’

being

the solution of the

equation :

In terms of U the evolution of the total

density

matrix 0 is :

or

alternatively

in thé interaction

representation :

This

equation

^is

equivalent

^to eq.

(12) (see

^also

[2]

chap.

VIII eq.

(30)).

^{We may therefore}

^perform

^the

factorization of 0* and then take the _average over the lattice variables. We obtain :

or, after

going

^{back to} the normal

representation :

(this equation

could also be obtained

by performing

the factorization of 0

directly

ⁱⁿ eq.

(19)

^{but we find}

this method less

convincing).

We shall define

’BJ(t) by :

’BJ(t)

will be the evolution superoperator ^{of J in the} presence of relaxation.

Eq. (23) gives

^{us an}

expression

of

V(t)

in the form of a lattice _average.

On the other hand in the interaction

representation,

the average value of ^an atomic

variable Q

^{is :}

which,

^after

using

eq.

(21)

and the invariance of the trace under circular

permutation

may be cast into the form :

In the normal

representation

^we also have that :

(28)

with :

We shall define :

flL(t) being given by

eq.

(29). Using

^this

equation

and _eq.

(23)

ⁱⁿ

explicit

form it is _easy to show

(by rearrangement

of the lattice

indices)

^{that :}

which is identical to the relation

(8) already

^derived

in the absence of relaxation.

As is well known the

general

^{solution of the} eq.

(14)

of evolution of the

density

matrix has the

form,

^{in the}

normal

representation :

S

being

^a relaxation matrix which will be

specified

below. It follows that :

and,

after relation

(31) :

where R is the

transposed supermatrix

^{of the}

(real) supermatrix S :

Notice that once S or R has been

determined,

^we may

compute

^the

properties

^{of the atom as if there were}

no

lattice ; simply we

^must

everywhere replace ctto(t)

and

’U o(t) by flL(t)

^and

’U(t) (details

^{of the}

procedure

are

given

ⁱⁿ

Appendix I).

In other words the

practical

^effect of relaxation is ^to add a

dissipative

^{term to} the atomic Liouville hamil- tonian.

It is also worthwhile

mentioning

^{that if 6 =} UB, Boltzmann

density

^matrix

corresponding

^to

Jeo,

^we

have the

identity :

We shall now

apply

these results to the case consi- dered in ref.

[1].

4.

Application

^to relaxation effects in Môssbauer spectra

[1] [6].

^{- Let us consider a} Môssbauer emit- ter with a Môssbauer excited state 1 and

corresponding

electronuclear

kets 1 f >

^{and with a Môssbauer}

ground

state

Ig

^and

corresponding kets 1 g ).

In the

Appendix

^of

[5]

^{we have}

computed

^the

Môssbauer

intensity I(cv)

^emitted

by

^a

powder

source, under the

assumption

^{that state 1 is fed}

by

^{a radio-}

active transition from an upper nuclear state

I;.

^This

was achieved

by

^first

performing

^a

complete

^calcula-

tion of the radioactive cascade in the interaction

representation

^with

respect

^to the radiative

coupling

and then

going

^{back to the}

Schrôdinger

represen- tation. The final result of the calculation is _repre- sented

by

eq.

(AI)

^of

[5].

The formula relative to

a

powder

^{source is}

readily

deduced from this _equa- tion

by averaging

^over

02

CP2

(which

^introduces

a Kronecker

symbol bM2M2 ).

Let us first consider the case where there is no

relaxation. Then

[5] :

where T is the inverse nuclear

lifetime,

^M is the nuclear electric or

magnetic multipole

^moment which induces the Môssbauer

transition,

^and

o1n(t")

^{is the}

density

matrix of state I

just

^after

feeding by

radioactive

decay

of the _upper state

I;

^at time t"

(in practice

it is inde-

pendent

^of

t").

Let us set : t’ = t - i’ and t" = t’ - 7:".’ Then :

or in compact ^{form :}

where we have introduced the operators :

310

Let us now write the trace in _eq.

(37)

ⁱⁿ

explicit

^form

before

introducing

^the Liouville evolution _super- operators :

After the definition of

Vo,

eq.

(6),

the bracket is

equal

^{to :}

Let us now introduce relaxation. In the

preceding paragraph

^we have shown that in the _presence of relaxation

vo

^{must be}

replaced by v.

After substi-

tuting

eq.

(33)

^into

(43), using

eq.

(31)

^and

(34),

report-

ing

^into

(42)

^and

performing

^the double time

integral

we

finally

^{obtain :}

i. e. eq.

(9)

^{of ref.}

[6].

In the case where the

temperature

^is

high compared

to the level

spacing

in the excited Môssbauer state, its

density

matrix remains

always equal

^{to the unit}

matrix and the second

parenthesis

may be

replaced by ôfl f3

^We then obtain the

expression

^{for the}

high

temperature spectrum :

This

expression

^was

already

derived in ref.

[1]

^{but the}

supermatrix R

^as defined in footnote

(7)

of this refe-

rence was in

reality S

^{and not}

ST

^{as it}

ought

^{to be}

(notice

^{however that at}

high temperature S = ST

^{= R}

and that this condition is fulfilled in the

practical example

^{worked out in ref.}

[1]).

^{Our error arose from}

the fact that we calculated

directly

the relaxation of

pM

^{instead of}

proceeding

^{to a full}

analysis

^{of the}

problem.

Let us now consider the structure of the relaxation matrices S and R. Let 6 be the

density

matrix for the total set of states associated with both the excited and the

ground

^{nuclear states 1 and}

Ig.

^{In terms of}

projection operators Pl

^and

PIs

^{one may write :}

The evolution

operator U,

eq.

(15),

which does not contain radiative

couplings (see

^the definition

of ^Jeo

in

[1 ])

^{has no} matrix element8 between 1 and

Ig.

^There-

fore

by

^{virtue of} eq.

(23)

each of the matrices in the R.H.S. of _eq.

(46)

^is

coupled only

with itself. Conse-

quently,

the relaxation

supermatrix

^S factorizes into the submatrices :

The

physical meaning

^of

^ull

^and

(II 1 S II)

is obvious.

These are the

density

matrix of state 1 and the associated relaxation matrix. As concerns

uIlg

it is

usually

^zero but it becomes

temporarily

^excited

during

the émission of the _{y ray,} since the two nuclear states 1 and

Ig

^are

^{coupled by}

the transition operator ^{M. The} relaxation matrix which characterizes the relaxation of M is

clearly :

Consequently

^{in the first}

parenthesis

^{of eq.}

(44)

^R

reduces to the submatrix

(Ig ^{1 1 su} 1 Ig I)

^{while in the}

second

parenthesis, S

^{reduces to}

(II 1 s II).

Let us now

give

^the

explicit expression

^{for S. We}

assume that JC1 ^{has the same form and}

properties

^as

in ref.

[1] :

^:

a 1 = L Kq

^F4.

^Starting

^{from eq.}

⁽¹⁴⁾

q

one

expands

the double commutator and introduces lattice correlation functions :

where :

,

Finally

when all calculations are

completed,

^it

turns out that the

equation

of evolution of a has the form of _eq.

(32)

^{with a} relaxation

supermatrix S given

by :

Notice that if we associate with

Jet

^{a Liouville} ope- rator

(JC*)’, Sd,&-

^{can also} be obtained from the for- mula :

1

where _r,

r’,

rl, r2 represent ^the

degrees

of freedom of the lattice.

We have demonstrated above that the relaxation matrix R relative

to Q

^{is the}

transposed

^{matrix of S.}

Using

^this

result,

^{let us} calculate the relaxation

equation of Q

^{in the case where}

Q only

^has

diagonal

matrix elements :

This

equation

shows that in this case the sum of the elements of a row of R is zero.

This result is

satisfactory

in connection with a

problem

^{which we have been}

considering recently,

that of the Môssbauer

lineshape

^{of an}

^Yb170

^trivalent

ion in the _presence of relaxation between two electro- nic Kramers doublets

[7].

^{If we assume that the}

hyper-

fine structure in each doublet is

completely

^anisotro-

pic,

^{of the} type

Alz Sz

⁺

P(3 I’ - I(I

⁺

1)), ^(this being

^an

example

^{of a}

diagonal Q),

^our

approach

to the

problem

^{should be}

equivalent

^{to the old sto-}

chastic

approach [8].

^{This means} that the relaxation matrix R must reduce to a Markov matrix. But an

essential

property

^{of such a} matrix is that the sum of the elements of one row is zero. As an

example,

^when

the

system jumps

^{between two}

frequencies

^which

are not

equally probable :

This is how we discovered that one should have R =

ST..

5.

Comparison

with Hirst’s relaxation

équation

^and

Môssbauer

lineshape.

^{- It is worth}

mentioning

^that

the value of R which can bè deduced from _eq.

(51), (35)

is still different from that associated with the eq.

(4)

^{of Hirst’s}

Paper [9].

Indeed

using

eq.

(30), (34), (35), (51)

^{of the}

present

paper it is

possible

^to show that the

equation

^{of motion}

of an

operator

^M in the normal

representation

^{is :}

In our

opinion

^{the reason} for the differences between this

equation

and Hirst’s

equation (which

^are espe-

cially appearant

ⁱⁿ the correlation functions of the first and fourth relaxation

terms)

^{is the}

following.

As a function

of H*(t)

the evolution operator ^U’

in the interaction

représentation

^is

given by :

The exact

expression

for the derivative of 7* in the interaction

representation

^is

(see

^eq.

(21)) :

where in the

expression

^of

U’,

to ^{= 0.}

The master eq.

(14)

^{for Q*} amounts to

approximat- ing

^this

equation by :

where now to ^{= - oo .}

Let us use the same

approximation

^{to derive an}

equation

of motion for

Q *(t)

^from eq.

(27) :

312

Upon substituting expression (56)

^{of U’ in this} equa- tion we arrive at

(up

^to second order in

H1*) :

Notice that this

equation

differs from the relaxation eq.

(14)

for 6* both

by

^the

position

of po and

by

^the

order

of Je!(t - i) and Je!(t)

^{in the double commuta-}

tor. This is due to

the’

fact that in

passing

^{from U’}

to U’t the two time arguments t1

and t2

^{in the double}

time

integral of eq. (56)

^are inverted. It _may be checked that the _eq.

(60)

^{leads to a} relaxation matrix R which is indeed

equal

^to

ST.

A further check is the

following.

^{Let us start from}

the

equation

^{derived from} eq.

(12) by

factorisation of the

density

^matrix

0,

^and

drop

^{the term linear}

in H*1

which we know would

disappear

^at the end of

té

calculation. Then :

Let us consider

(by

definition of

Q *,

^see eq.

(27) ; Q *(0)

⁼

Q).

After

rearrangement

^{of the trace} this becomes :

or

altematively

^after

performing

^{the standard}

approximations :

in

complete

agreement ^with eq.

(60) (compare

^with

ref.

[2] chap. VIII,

eq.

(44)).

On the other hand

according

^to this author

[10],

Hirst’s _eq.

(3)

^must

be interpreted

^{as :}

For a quantum ^bath eq.

(60)

^and

(66)

^are

clearly

different

(since [Po, Je1] =1 0)

^{and this is the reason}

for the

discrepancy.

Notice that Hirst’s _eq.

(3)

^{is what}

is obtained

by using

^the method of successive

approxi-

mations in the same way ^as for the

density matrix,

in order to solve an

equation

^{of the} type :

Our results

clearly

indicate that in the case of M*

this method leads to an incorrect

handling

of the time

variables,

^{i. e. the} past and the future are inverted.

Incidentally

this conclusion is in

agreement

^{with a} remark formulated more than twenty years ago

by

Gell-Mann and

Goldberger ([12]

^eq.

(3.13)

^and

(3.14)).

As a final

remark,

^{let us} also mention that _eq.

(8)

of Hirst

[9]

for the Môssbauer

lineshape

^is

only

^valid

for emission from a

stationary

^{excited state.}

Indeed,

it is

clearly

stated in the text of

[9]

^{that the row vec-}

tor W which enters into the

expression

^for

G(t)

p. 660 is time

independent,

all the evolution in the excited state

being

included in the

quantity M(t).

^In

reality,

formulae like _eq.

(8)

^of

[9]

^{are well}

adapted

^{to the}

description

^{of slow} passage

lineshapes

^{in NMR or of}

light absorption

^{from an atomic}

ground

^{state. But as}

shown

by comparison

^{with our own results} eq.

(39)

and

(44) they

^cannot describe radiation emission from

a transient excited state in a

cascade,

^{unless Boltzmann}

equilibrium

^{in this state} is achieved before emission

(in

^{which case}

^{6(1/T )}

^{in eq.}

⁽³⁹⁾

^{can be}

replaced by

^the

Boltzmann

density

matrix of level

I).

^{As a further}

proof,

^{let us} recall that in

[5]

^we have checked that

by integration

^of eq.

(A .1 )

^over 0153, ^we did recover the

expression

for the Nuclear Orientation

signal (eq. (A. 3))

^{as we}

ought

^{to. In cascades}

involving

, transient states, ^{this N.0}

signal

^{is a direct measure-}

ment of the

quantity u(1/r),

^{whose value}

depends

^on

relaxation effects in this state. The nuclear

Korringa

relaxation time of

Ag109

^{in Fe at} very low

temperatures

has

already

been determined

by

this method several years ago

[13].

Note added ⁱⁿ

proof.

^- Notice that while Hirst’s formulas involve

only

^one

time,

^Blume has derived

an

expression

^{for the Môssbauer}

lineshape

^which

contains an

integration

^{over two times}

([14]

^eq.

(4)).

We have been able to show that this last

expression

is

exactly equivalent

^{to our}

equations (37)-(39).

^We

thank Dr. Blume for

calling

^{our attention to his}

work. ^’

6. Conclusion. ^- In conclusion we feel that we have elaborated a consistent

description

of relaxation

phenomena

within the frame of the Liouville forma- lism. The main results of this _paper are

represented by

eq.

(33), (34), (35).

APPENDIX 1

We have demonstrated in the main text that

(for t > 0) :

and

These results show us how to

compute

^lattice averages

involving

^two operators ^{U and} Ut whose

positions

relative to po ^are

Up.

^{U t}

(and

^circular

permutations).

Now we have

(for t

> to >

0) :

On the other

hand,

^after eq.

(33) :

This

gives

^{us a rule for}

taking

the lattice _average of

products

^of

operators

^{U and ut of the} type :

and also of ^more

general products :

where

A, B, C,

^{D are atomic}

operators.

As an

example :

Let us stress that this

breaking

of the lattice _average is made

possible

^both

by

the fact that _eq.

(A. 5)

contains successive ordered time intervals which ^are

long compared

^{to the} memory ^{of the}

lattice,

and

by

the

exponential

^{form of}

V(t),

eq.

(A. 3).

Let us now

apply

these results to the

computation

of atomic observables. In the absence of

relaxation,

a

physical

atomic observable is a trace over

atomic

variables of a

product :

In the _presence of

relaxation,

^{one must}

replace Uo by U, r(0) by 0(0)

^{and the} trace must involve both atomic and lattice variables. 0 is then factorized as

po Q,

leading

^{to a lattice} average ^{over a}

product

^like

eq. (A. 5).

The standard method for

computing

^the

properties

of the atom in the _presence of relaxation should therefore be the

following :

1) Ignore

relaxation and _express the atomic _quan-

tity

of interest as a trace over a

product of operators Uo, Uô

^{and u}

(plus possibly

inserted atomic

operators).

2) Rearrange

^{the trace in such a} way that all the

Uo

be on the left of 6 and all the

UÓ

^{on the}

right,

^with

positive

^time arguments

(corresponding

^to

integration variables)

^{ordered in the same} way ^as in eq.

(A. 5).

While

performing

^the

rearrangement,

^one may ^use transient

Ui

^{s with}

negative arguments.

3)

^{Write the trace in}

explicit form,

relate the matrix elements of the

UO, UJ

^to

Liouville operators ^Vo.

Finally,

make the substitution

’BJo -->

^9Y. This is how eq.

(43)

^was derived in the main text.

314

APPENDIX II

As shown

by

^this paper, ^{one must} be _very careful in

deriving expressions

for the Môssbauer

lineshape, although

^{at first}

sight

^the

position

^and

sign

of the time argument in the correlation function of M would seem

of little

importance,

^{since one looks at the real part}

of a

complex

number. A

complete analysis

^{of the}

emission _process led us to the exact

expression (eq. (39)) :

Strickly speaking

^this

expression

^is different from eq.

(5) of [1]

^{and from} eq.

(8) of [6]

and should

replace

these

equations.

^{As concems ref.}

[6],

eq.

(9)

^{of this}

paper ^as well ^as

subsequent equations

^{were derived}

from the correct

equation (7)

^{and are therefore correct.}

Eq. (1)

^of

[6]

^{is also correct.}

On the contrary, eq.

(15)

^{of ref.}

[1]

differs from eq.

(45)

^{of the} present paper in that the order

of f and g

in the Liouville kets is inverted

(this

has the effect of

changing

^the

signs

^{of the}

energies). However,

in eq.

(72)

^of

[1]

^{there was an}

interchange of f and g

in

going

from the left hand side to the

right

^{hand side.}

It follows that all

equations following

eq.

(72)

^are

entirely

^correct. Notice that _eq.

(45)

^{of the} present paper agrees with _eq.

(27)

of Clauser and Blume

[11].

References

[1] GONZALEZ-JIMENEZ, F., IMBERT, P., HARTMANN-BOUTRON, F., Phys. ^{Rev. B 9} (1974) 95.

[2] ABRAGAM, A., ^The principles of ^Nuclear Magnetism (Oxford University Press) 1961, Chap. ^VIII.

[3] WANGSNESS, ^R. K., BLOCH, F., Phys. ^{Rev. 89} (1953) ^728.

[4] MESSIAH, A., Mécanique Quantique, ^{tome I} (Dunod) 1962, Chap. VIII, § ^14.

[5] HARTMANN-BOUTRON, F., SPANJAARD, D., ^J. Physique ³³ (1972) 285.

[6] HARTMANN-BOUTRON, F., Phys. ^{Rev. B 10} (1974) ^2113.

[7] CHOPIN, C., HARTMANN-BOUTRON, F., SPANJAARD, D., Communication to the International Conference on the

Applications of the Mössbauer Effect. Bendor (France)

Sept. 1974, to be published ^{in J.} Physique Colloq. 35

(1974) ^{C 6-433.}

[8] ^See Chap. ^{X of ref.} [2], eq. (53), p. 448.

[9] HIRST, ^L. L., ^J. Phys. & Chem. Solids 31 (1970) ^655. According

to the author, in eq. (4) of the article the frequency argu- ments of the four spectral densities should be reversed.

In

addition Jq

^{should be} replaced by J-q ^{in the last two}

terms (HIRST, ^L. L., private communication).

[10] HIRST, L. L., private communication.

[11] CLAUSER, M. J., BLUME, M., Phys. ^{Rev. B 3} (1971) ^583.

[12] GELL-MANN, M., GOLDBERGER, ^M. L., Phys. ^{Rev. 91} (1953) ^398.

[13] STONE, ^N. J., FOX, R. A., HARTMANN-BOUTRON, F., SPANJAARD, D., ^J. Physique Colloq. ³² (1971) ^{C 1-897.}

[14] BLUME, M., Hyperfine ^structure and nuclear radiations, Eds.

E. Matthias, ^{D. A.} Shirley (North Holland) 1968, p. 911.