Study of the saturation of a multilevel spin system : Mn++ in Zns

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Study of the saturation of a multilevel spin system : Mn++ in Zns

C. Blanchard, A. Deville, B. Gaillard

To cite this version:

C. Blanchard, A. Deville, B. Gaillard. Study of the saturation of a multilevel spin system : Mn++

in Zns. Journal de Physique, 1976, 37 (12), pp.1475-1481. �10.1051/jphys:0197600370120147500�.

�jpa-00208550�

Centre de

Saint-Jérôme,

¹³³⁹⁷

Marseille,

^Cedex

4,

^France

(Reçu

^le

29 juin 1976, accepté

^{le 23 aofit}

1976

Résumé. ²⁰¹⁴ Aux basses températures, les raies de structure fine et hyperfine ^{de Mn2+ dans ZnS}

sont

couplées

par diffusion. On a étudié la saturation continue d’une raie à deux températures

2014

4,2 ^{et 39 K -} encadrant la température caractéristique pour laquelle ^les temps de relaxation et de diffusion sont égaux (20 K). ^{On a d’autre} part ^{observé à} 1,34 ^{K une} anomalie dans la loi de saturation, qui ^ne peut être due à une accumulation des phonons résonnants. On a mis en évidence un raccour-

cissement de T1 ^{avec la} puissance ^incidente.

Abstract. ²⁰¹⁴ At low temperature ^{the fine and} hyperfine ^{structure lines of Mn2+ in ZnS are}

coupled

by diffusion. Continuous saturation of one line was studied at two temperatures ²⁰¹⁴ 4.2 and 39 K 2014 above and below the characteristic temperature for which relaxation and diffusion times are

equal (20 K). ^{Moreover at 1.34} K, ^an anomalous behaviour of the saturation law was observed which cannot be due to an accumulation of resonant phonons. ^{A reduction of} T1 with incident power has been observed.

1. Introduction. ^- In

previous

paper

[1, 2],

^we

described

pulse

saturation

experiments

^{on the multi-}

level system

Mn"

in cubic ZnS

(I

⁼

5/2, S

⁼

5/2).

We showed the

existence,

^{at low}

temperatures (T

²⁰

K),

^of

spectral

diffusion with characteristic time

TD (TD

^{= 0.5}

ms).

The width of the saturated line which has to be considered for continuous satu- ration measurements should be

quite

^different

depend- ing

^on whether the bath temperature ^is greater ^or lower than that for which the relaxation time

T,

is

equal

^to

TD.

In section

2,

^we

present

^and

interpret

^{the measu-}

rements made at 4.2 K and 39

K,

^where

T1

^{= 0.28 s}

and 8 x

10-’

s

respectively (pulse experiments [1]).

We do not make the usual but unrealistic treatment where all relaxation

probabilities

^are

equal.

^{We show}

how to relate the time deduced from continuous saturation

experiments

^{to that} measured in

pulse experiments

^when

Tl

>

TD.

In section

3,

^we discuss the anomalous behaviour of the

saturating

process under strong r.f. field for low temperature

( T

^{= 1.34}

K).

This behaviour cannot be attributed to an accumulation of resonant

phonons.

In order to know whether

T1

^{is r.f.} power

dependent,

we have used a modified version of Look and Locker’s method

[3],

which is well suited for these measure-

ments. The results are discussed in section 4. We

finally give

^some

concluding

^remarks.

2. Continuous saturation

experiments.

^{- 2 .1 SATU-}

RATION OF THE ABSORPTION LINE AT 4.2 K. ^- The concentration of the

crystals

^was

^10-4

^atom g Zn per ZnS mole. We used a

superheterodyne

^{X band}

spectrometer

( f F

^{= 30}

MHz)

^{which was}

frequency-

locked on an

auxiliary temperature-stabilized cavity,

and allowed us to detect the

absorption

^line

directly.

Field

sweeping

^effects observed for

long T,

^were

thus avoided.

We measured

X"(roo, Hl),

^the

peak amplitude

^of

the

absorption signal (resonant frequency mo)

^for

different values of the r.f. field 2

H,

^cos roo t present

on the

sample.

^{In our}

experiments Hi

^{= 0.6}

P(G’, W),

P is the incident _power on the

cavity. Figures

^{1 and 2}

plot

^{the ratio a}

X"(o-)O, O)IX’(coo, Hl)

^{versus P in the}

range 30 nW P ¹⁰⁰

JlW. They

^{show a linear}

variation of a versus P. These results can be understood

by considering

^{that a diffusion} process establishes a

spin

temperature ^{for the}

spin

^{levels. In a}

population model,

^we have the

following expression

^{for a : :}

(*) ^{E.R.A. no 375.}

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0197600370120147500

1476

FIG. 1. ^- Variation of a ⁼ x"(roo, 0)/x"(.wo, Hl) ^{versus incident} powerP(P ¹² pW - H // [III]W - 11/2, - 1/2 > -+ I - 1/2, - 1/2 )

transition).

10

FIG. 2. ^- Variation of a

versus P (P

³⁰⁰ IlW - ^{H //} [III]W - 1/2,

- 1/2 > -+ / - 1/2, - 1/2 > transition).

Tl

^{is the}

spin-lattice

relaxation

time, p(wo)

^{is the}

shape

factor of the unsaturated

absorption

^{line :}

where

f(w)

^{is the}

signal intensity,

^{and the}

integral

is

performed

^over the whole spectrum. ^{For the}

! 1/2, - 1/2 > -+ I - 1/2, - 1/2 > transition,

Experimentally,

^{we find a = 2 for P = 6}

Jl W.

^This

leads to

Tl

^{= 0.33 s. which} compares

favourably

to the 0.28 s obtained

by

^the

pulse

^method.

2.2 SATURATION OF THE DERIVATIVE OF THE ABSORP- TION LINE. ^- We studied the same transition as

previously

^{with the same} orientation of the static field.

We used lock-in detection

(20 Hz) giving

the deriva- tive of the

absorption

^line

dX"IdH.

^In

figure ^3,

^we

plot

the ratio

FIG. 3. ^- Variation of

versus the rotating ^{r.f. field} H1

(T

^{= 39 K - H //} [III]W 1/2,

- 1/2 ) --- I - 1/2, - 1/2 ) transition).

versus

H1; [dX"(w, Hl)/d1lJM

^{is the}

peak amplitude

of the derivative of the

absorption

line. We still have

Hi

^{= 0.6}

P(G2, W).

^We determined

p

^{for r.f.} powers in the 1.8

ptW-1.8

mW range. The

experimental

conditions are such that even at the

highest

^r.f. power,

we do not fulfil the conditions of Redfield’s model

[4].

The characteristic results are shown below :

The

spin-lattice

relaxation times differ

by

^{a factor}

3

000,

whereas the

corresponding

factor for the r.f.

powers is 15

only. Although

^{we cannot} compare

directly

the values of the two incident powers, since

one involves the

absorption

line and the other its

derivative,

^{this cannot}

explain

^{such a difference}

in the ratios. We must consider that the satura- tion does not occur

by

^{means of the same} process in the two

experiments.

^We will show that the results for T ⁼ 39 K are consistent with the saturation of the

11/2, 1/2 > --+ 1 - 1/2, - 1/2 )

transition

only,

whereas the whole spectrum ^{was saturated at 4.2 K.}

To

interpret

the results of

figure 3,

^{we have}

to consider a multilevel

system (MI

^{= -}

1/2, Ms

^{= -}

5/2, ...,

⁺

5/2).

The different levels ^are numbered from 1 to 6

by increasing

^{value of}

Ms.

We will call _Pi the

population

of level i. Levels 3 and 4 are

coupled by

^{the r.f. field}

with transition

probability

^{U. The}

spin-lattice

^transi-

tion

probability

^{from i}

toward j

^is

Wij.

^{We have the}

following

^relations

(see

^ref.

[1], [2])

between the

Wij :

^v

obtained

following

the method used

by Llyod

^and

Pake

[5].

The calculation

gives :

Ak4

is the cofactor of the 4th column element in the kth row in the determinant of

the pi

coefficients.

More

explicity :

where z ⁼

a/b.

We thus get ^{for a an}

expression

^{similar to Bloem-}

bergen’s [6],

^with

k/2

^{to be}

compared

^to

TI.

^{When the}

transition line

shape

^is lorentzian

(width Aco), P

^is

given by

^the

following expression :

The coefficient 9 in

(6)

^comes from the ratio :

The theoretical

expression (5)

^cannot describe the

experimental

results. This suggests that the observed line is not

homogeneous.

^{We will use the}

independent homogeneous spin-packets

^model

[7]

^{and we will}

consider that the distribution law

h(w - mo)

^{of the}

spin-packets

is lorentzian with a width Am*. The

absorption

^{line of a}

spin-packet

will be obtained from

expression (3) using

^{for the}

shape

factor of the tran- sition

probability

^{U a} lorentzian line with width A(o.

The

expression

^for

#

^{is thus :}

try

experimental by

the

pulse

^{method are} consistent with our

interpretation

of continuous saturation

experiments.

The evolution of

A 34(t)

^after

saturating

the 3 --+ 4 transition is :

The reduced

eigenvalues Ailb

^of

equation (2) (where

U ⁼

0),

^were calculated for z

varying

^between

1/16

and

16,

^and classified

by increasing

^values

(A,

⁼

0).

To

study

^the

amplitudes Ai,

^{we must} first know the values of the

populations

^at the end of the satu-

rating pulse,

^{which in our}

experiments

^was

by

^far

longer

^than any

IlAi.

These values are obtained from

(2),

^{with U - oo and}

using

^{the new variable}

The

amplitudes A2, A3, A5

^{versus z are}

plotted

in

figure 4; A4

^and

A6

^are

¹⁰⁸

^{times weaker than}

As.

For a

given

^{value of z,}

the Ai

^are obtained from the reduced

quantities Ailb,

^b

being

determined from continuous saturation

through (4).

^{The time constants}

of interest are

plotted

ⁱⁿ

figure

^5.

Experimentally,

^we have observed an

exponential

return to

equilibrium,

^{with a} 80 ps time constant.

This is consistent with the

interpretation

of continuous saturation

experiments

^{if z has a} value between 0.5 and 2. We cannot

hope

^{for a} better determination of z, because of the strong variation of the relaxation

probabilities

^with

temperature.

^We

experimentally

observe -

I/Â2

^{if z =}

0.5, - 1/Â3

^{if z = 2 and}

-

I/A2 - - 1/ Â3

^{if z = 1.}

3.

Study

^of the saturation law at 1.34 K. ^- We

see

(Fig.

^{1 and}

2)

^that

a(P)

^{does not follow} expres- sion

(1)

^for

high

^r.f. power P. We have taken into account various

hypothesis

which failed to

explain

the

experimental

^{results. We} therefore made _expe-

1478

FIG. 4. ^- Variation of the amplitudes A2, A3 ^and A5 ^versus

Z = W65/W64-

FIG. 5. ^- Variation of the time constants - I/A2, - IIA3 ^and

- IIA5 ^{versus z =} W65/ W64. 80 ps is the time constant of the observed exponential signal (saturating pulse » relaxation time

constants, ^{T =} 39 K - H % [III]w).

riments

using

^a modified version of Look and Locker’s method

[3],

which allowed _easy determination of

Tl

versus the r.f. _{power P.} We observed a

shortening

^of

Tl

with

increasing

P. We will discuss these different

points successively.

3.1 DISCUSSION OF THE CONTINUOUS SATURATION EXPERIMENTS. ^- a was measured

using

^{the same}

experimental

conditions as at 4.2

K,

^{and two}

samples

of different sizes

(3 x 5 x 8

^{mm and 1 x 2 x 3}

mm)

obtained from the same

crystal.

^{For low r.f.}

fields,

a varies

linearly

^{with P.} In the linear

region

^{a = 2}

is obtained for P ⁼ 2

JlW.

This leads to

T1

^{= 1.0 s,}

in

agreement

with the results of the

pulse

^method

(7B

^{T = 1.18 s.}

K,

T ¹⁰

K).

This shows that the model considered at 4.2 K is still valid for low r.f.

fields. For

increasing

powers, the saturation of the

absorption

^{line is more} difficult than

expected

^from

(1).

This effect is more

important

^{for the}

large crystal :

for P ⁼ 100

Jl W,

^{the a value is 32} for the small

crystal

and 13 for the

large

^one, instead of 51.

The observation of a size effect suggests ^{that the}

mean number of resonant

phonons

may be increased from _no

(thermal equilibrium value)

^{to n, either}

by spins (phonon-bottleneck)

^or

by piezoelectric

^effect.

We will discuss these two

hypothesis successively,

for a

spin 1/2.

Phonon-bottleneck has been treated

by Faughnan

and

Strandberg [8]

^for

pulse

saturation

experiments,

and

by

Scott and Jeffries

[9]

^for continuous saturation.

They supposed

^an

exponential

^{return of n to no,}

with time constant iph, and introduced the bottleneck factor 7. Under severe bottleneck

(0-

>

1)

^{and after}

a

saturating pulse,

^the

spins

^and

phonons

^come

first to

equilibrium

with time constant i" -

TphlU,

then the

spin-phonon

system reaches the bath tempe-

rature with time constant r’ ⁼

u Tl.

^If

Tl

^oc

T

then ’t" oc T - 2. For continuous saturation

where _p is the

population

difference between the two

spin levels,

^{and S’ = 2}

UTi,

^{where lJ is the}

probability

of the transition induced

by

the r.f. field. If a -

0,

u - ao = 1 + S’.

Expression (9)

shows that

a/oco

> 1 for _any r.f. field. This means that saturation is easier with

phonon-bottleneck.

^Our

experimental results,

inconsistent with the

preceding ^features,

^{cannot be}

explained by

^a

phonon-bottleneck.

Because of the strong

piezoelectricity

^of

^ZnS,

one may consider that the

crystal,

^{submitted to a}

weak electric

field,

absorbs acoustical _energy,

leading

to an increase of the mean number of resonant

phonons.

^{We can}

study

this situation

using Faughnan

and

Stranberg’s

^model

[8]. n and

p ^are

given by

^{a set}

of

coupled equations :

In

equation ( 10b),

^the

quantity

^{l U} describes the

Locker’s method

[3]

^{in the}

following

way. Constant

magnetic

^{field was set at the resonance of}

the 11/2,

-

1/2 ) -t - 1/2, - 1/2 )>

transition. The

crystal

was submitted to a known r.f. _power

by switching

on the fast diode

(Fig. 6) ;

^{at the same} time the 30 E.S.R.

FIG. 6. ^- Block diagram ^{of the} experimental apparatus ^for T,

measurements by successive field-sweeps. ^{We use two Helmoltz coils} (0 ²⁵ mm, 120 turns) ^{situated on} both sides of a slotted TEo 12

cavity.

lines were swept

by

^a

triangular magnetic

^field

Hpp

of 450 G

peak-to-peak amplitude

^and

period Tf Tl.

During

the successive _sweeps the

spin

system ^came

progressively

^to saturation. The r.f. field was then switched off for a time

long enough

^to

bring

^the system ^{back to}

equilibrium.

^The theoretical

analysis

of this

experiment [10]

^{shows that the two}

following

conditions must be fullfilled :

1)

^The

sweeping

^{rate of the}

magnetic

^{field must be}

such that the variation of the resonant

frequency

in a time

T2

⁼

(yHL) - 1 be

much less than the time width

yHL.

This leads to

2)

^The

spin

temperature ^{must not vary}

greatly during

^a passage

through

^{the resonant} line. This leads to

Under the

preceding conditions, during

^{the field}

A

knowledge

^{of r and}

v(aJ)/v(0)

then allows the determination of

T1

^{in the} presence of the r.f. field.

In these

experiments Wral = nyHf/Hpp. ^Comparison

of the results obtained

by

^{this method and}

by

^conti-

nuous saturation

requires

some care. For

example,

for a

given

^incident power ^P

since the absorbed r. f. _energy is not the same in the two

experiments ( WSat

is weaker than W defined

in (1)).

In order to

satisfy

^the

previous conditions,

^we chose :

For the

large sample

^{we obtained :}

(This

^value compares

favourably

^{with the 880 ms time}

constant obtained

by

^the

pulse

saturation

method)

Since the observed spectrum ^{was somewhat}

complex,

it is difficult to make a

precise

determination of the

envelope.

^This

gives

^{us a 20}

%

^{error in the}

T1

^measu-

rement. We could not obtain similar information from the small

crystal

because of the _poor

signal-to-

noise ratio.

A decrease of

Tl

^with

increasing

^{P has}

already

been observed

by ^Davis, Strandberg

^and

Kyhl [11]

ⁱⁿ

pulse experiments

^on

Lal-xGdx(C2HsS04)39 H20 (x

^{= 5 x}

10-3)

^{at 4.2 K.}

They suggested

^{that this}

decrease

came from an increase

(in

the conventional

sense)

of the lattice

temperature.

^{Marr and}

Swarup [12], making

continuous saturation measurements on a

sample

^{with x =}

10 - 2,

observed that the

spin

system

was difficult to saturate. Their

suggestion

^{of a}

phonon-

bottleneck cannot be retained because this would lead to

a/ao

>

1,

^whereas

they

^observed

cx/cxo.

^1.

Finally

^the

hypothesis

of Davis et al. is at the present

1480

U)

FIG. 7. ^- Saturation of the absorption signal by successive field-sweeps ^which gives values of T1(P). (T ^{= 1.34 K - H //} [III]w).

a) ^{P = 0.1} mW ; T1 ^{= 600 + 120 ms.} b) ^{P = 0.9} mW ; T1 ^{= 125 + 25 ms.}

time the

only

^valid

interpretation

^{of the}

experimental

results on

gadolinium ethylsulfate.

Our continuous saturation

experiments

^{have been}

made at 1.3 K where the relaxation takes

place through

a direct process. It is then difficult to

explain

^our

results

by simple

^lattice

heating.

We still consider

a

spin 1/2.

^Let

T( U)

be the lattice temperature ^when the transition

probability

^induced

by

the r.f. field is U.

In continuous saturation

experiments

^{we obtain the}

quantity G/p,

where G is a coefficient

independent

of

U,

and of the variation of the lattice temperature from its value

To

^{for U = 0. For a}

paramagnetic

center

following

the Curie law and

relaxing by

^a

direct _process,

expression ( l0a)

^{with n =} no

gives :

where A and B are two coefficients

independent

^{of T}

and U. It is easy ^to show from

expression (11) that,

for

any U,

^{the value a will be greater than that obtained}

without lattice

heating.

This is the contrary ^{to our} observation. If one retains the

assumption

^{of lattice}

heating,

it is therefore _necessary to reach the Raman

region.

^{For ZnS :}

Mn2 +,

^{this means that the lattice}

temperature ^{must be} greater ^{than 10}

K,

^{which one can}

hardly accept.

4. Conclusion. ^- A

difficulty

may appear in the

interpretation

of continuous saturation

experiments

made on a multilevel system when there is a tempe-

rature-independent

^diffusion process. It may then be _necessary to use different models

according

^{to the}

temperature

range. This

difficulty

may be removed

by making simultaneously pulse

measurements.

For ZnS :

Mn 2+

it has been confirmed that :

- at 4.2 K the whole spectrum is saturated and behaves like a

homogeneous

line because of

spin diffusion;

- at 39 K one saturates

only

the considered transition. The

experimental

saturation law was

justified by considering

^a model where the six fine- structure levels are

coupled by spin-lattice relaxation,

and

using

^the

independent spin packet

^formalism.

Comparing

the results obtained

by

^both

methods,

it was shown that the ratio ^{z =}

W5/2-3/2/W5/2-1/2

References

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J. Physique ³⁶ (1975) ^1151.

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[3] LOOK, ^D. C., LOCKER, ^D. R., Phys. ^{Rev. Lett. 20} (1968) ^987.

[4] REDFIELD, ^A. G., Phys. ^{Rev. 98} (1955) ^1787.

[5] LLOYD, J. P., PAKE, ^G. E., Phys. ^{Rev. 94} (1954) ^579.

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