Nuclear magnetic ordening in fluorene

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Nuclear magnetic ordening in fluorene

P. Verheij, W. Wenckebach

To cite this version:

P. Verheij, W. Wenckebach. Nuclear magnetic ordening in fluorene. Journal de Physique I, EDP

Sciences, 1993, 3 (11), pp.2259-2267. �10.1051/jp1:1993243�. �jpa-00246866�

Classification

Physics

Abstracts

75.25 76.60 76.70E

Nuclear magnetic ordening in fluorene

P.F.A.

Verheij

and W.Th. Wenckebach

Faculty

of

Applied Physics,

Delft

University

of

Technology,

P.O.B. 5046, 2600 GA

Delft,

The Netherlands

(Received

24 March 1993, revised 28 June 1993,

accepted

28

July

1993

)

R6sumd. Ce travail discute la possibilit4 d'obtenir de l'ordre

magn4tique

nuc14aire dans

des cristaux mo14culaires de fluorene _par

polarisation dynamique

nucl6aire

(DNP)

suivi _par

desaimantation

adiabatique

nuc16aire dans le r6f6rentiel tournant

(ADRF).

On trouve que cet ordre est obtenu de la

fawn

la

plus

facile orientant Bo ^I I l'axe

c et choisissant T _< 0- Alors

une

polarisation

nuc16aire de 26% suffit pour obtenir

un ordre

ferromagn6tique. L'importance

du fluorene est donn6e _par la

possibilit6

d'utiliser des

triplets photoexcit6s

_pour la DNP-

Alors,

pendant

et

aprbs

la ADRF l'6chantillon _ne contient pas des centres

paramagn6tiques

et l'6tude de l'ordre

magn6tique

nucl6aire est

plus

_pure.

Abstract The present _paper considers the

possibility

to obtain

a

magnetic ordering

of the proton

spins

in molecular

crystals

of fluorene

using dynamic

nuclear

polarization (DNP)

followed

by

adibatic

demagnetization

in the

rotating

frame

(ADRF).

It is found that such _an

ordering

is most easily reached for T < 0 and 80

II c-axis. Then _a polarization of

only

26% needs to be

reached

by

DNP in order to obtain _a

ferromagnetic ordering.

The

importance

of fluorene lies

in the

possibility

to _use

photoexcited

triplet states for DNP. Then,

during

and after the ADRF the

sample

is free of

paramagnetic

centres and _a cleaner

study

of the

ordering

_can be made.

1 Introduction.

Nuclear

spins

may order

magnetically

due to nuclear

exchange

interactions _as

occurring

_e.g.

in

~He iii,

_or to indirect

spin-spin

interactions _as observed _e.g. in

TmP04 (2],

or

simply

to

long

_range

dipolar

interactions. The first

example

of this latter

dipolar magnetic ordering

of nuclear

spins

was

produced by Abragam

and coworkers.

They

created this

ordering

in two

steps. First,

the nuclear

spins

were

polarized by Dynamic

Nuclear Polarization

(DNP).

Then

an adiabatic

demagnetization

in the

rotating

frame

(ADRF)

_was

performed

_on the Nuclear

Magnetic

Resonance

(NMR)

transition to cool the

spin system

^down to sub-micro Kelvin

temperatures.

Such

experiments

have been

performed

_on

simple

cubic

systems

formed

by

the nuclear

spins IF

_" of

~~F

in

CaF2 _(3],

the nuclear

spin ILi

_" ^~ of

~Li

and

IF

_"

IH

_" of

2 2 2

19F

or protons ^{in LiF and LiH}

[4,

5] ^{and the}

hexagonal quasi

two-dimensional

system

^of

proton

2260 JOURNAL DE

PHYSIQUE

I N°11

spins

in

Ca(OH)2 (6, 7].

^{A second method} to

produce dipolar magnetic ordering

^{of nuclear}

spins

_was introduced

by

Huiki et al. [8].

Using

brute force adiabatic

demagnetization they

produced

_an

antiferromagnetic ordering

of the

spins

of the Cu nuclei in metalic _copper.

The

present

_paper discusses the

feasibility

of _an

improvement

of the method introduced

by Abragarn

and coworkers. In their first

cooling step (conventional DNP)

_{one uses} microwave

irradiation to transfer the electron

spin polarization

of

permanent paramagnetic

centres to the nuclear

spins. However,

these

paramagnetic

centres lead _to nuclear

spin-lattice

relaxation and hence to fast

heating

of the nuclear

spin system

^{after the}

ADRF,

_so the ordered _state is

rapidly destroyed.

The _use of Microwave Induced

Optical

Nuclear Polarization

(MIONP) _[9,

_10]

instead of DNP would solve this

problem.

Thus

far,

the best

example

is

protonated

fluorene

doped

with

phenanthrene. Using UV-light,

the

phenanthrene

molecules _are

photo-excited

into

their lowest

triplet state, creating

electron

spins

S

= I.

Operating

at T ₌ 1A

K,

in _a

magnetic

field

(80(

_" 2.7

T,

and

using

a microwave

frequency

of 75 GHz to transfer the electron

spin polarization

to the protons, a proton

spin polarization

of 42% has been achieved

ill].

After

shutting

off the excitation

light,

the

phenanthrene

molecules

decay

to their

diamagnetic ground

state and the nuclear

spin-lattice

relaxation becomes _very slow. At IA K and 2.7 T

dipolar

order relaxes with _a time _constant of 2.3 x

10~

s, which is _more than three orders of

magnitude longer

than in the _presence of

paramagnetic

centres

ill].

The aim of this article is to

investigate

how much nuclear

polarization

is

required

to create

magnetic dipolar

ordered _states in fluorene. Thus _{we can} evaluate whether MIONP is devel-

oped

far

enough

to start studies of

dipolar magnetic ordening

of nuclei. As is well

known,

after

cooling

the nuclear

spins

with DNP and _a

subsequent ADRF,

these

spins

still

experi-

ence a

strong externally applied magnetic

field

Bo,

_so

they

order due to the trucated

dipolar

interaction

7i

=

~j _{A,j (31]1]}

-1'.

IJ) (I)

>,J

In this Hamiltonian

~~~

~~

~~~ j~~~~~~

~°~ _(~

~ j) (2)

u

A

"

~' (3)

where r~ ^{is the vector}

connecting spins

I~ and

IJ,

_9~j is the

angle

between _r~ and

Bo

and r,j =

(r,j (. ^Finally,

~iI is the

gyromagnetic

ratio of the nuclear

spins.

By choosing

the

sign

of the

frequency change during

the ADRF _one has the

possibility

to

create at will

negative

_or

positive spin temperatures.

^In the former _case the _energy tends to _a

maximum,

while in the latter _case it tends to _a minimum. The _case of the truncated

dipolar

interaction allow for two

types

^of

orderings, longitudinal orderings

where the nuclear

spins

are directed

parallel

_or

anti-parallel

to the

externally applied magnetic

field and transverse

orderings

where

they

_are oriented in the

plane perpendicular

to

80.

In the effective field

approximation,

each

longitudinal magnetic

structure with _{an energy}

+ED

has _a

counterpart

in _a transverse

magnetic

structure with _{an energy}

ED (12]. Therefore,

to

predict

which

2

magnetic

structure

actually

_{occurs, one}

only

needs to determine which

longitudinal

structures

correspond

to maximum and minimum _energy.

Subsequently,

_{one can} deduce

directly

whether transverse structures with

higher

_or lower

energies

^exist.

The

crystaline

structure of fluorene is very

complicated.

As shown in

figure I,

the _or- thorhombic unit cell contains 4 molecules and 40 proton

spins [13].

^Therefore we did not

venture to calculate the space Fourier transform of the

coupling

constants

A,j _[14]. Instead,

to

determine the structure of the ordered

phases,

_{we use a} Monte Carlo simulation of the nuclear

spin system.

^{This method has} proven ^to

give

reliable results for truncated

dipolar

interactions between nuclear

spins

in _a

simple

cubic

arrangement

[15] ^and the full

dipolar

interaction be-

tween nuclear

spins

in _a face centered cubic lattice

[16].

^For

negative spin

temperatures, we

always

find _a

longitudinal ferromagnetic ordering,

while for

positive temperature

^the

ordering

is

always

transverse

ferromagnetic

_or helicoidal.

Next, using

the molecular field

approximation [12, 14],

we calculate the critical

polarization,

which must be

minimally produced by

MIONP in order _to reach _an ordered state after the ADRF. For

negative spin temperature

^and

Bo

^I

c-axis,

we find that

only 26%

nuclear

spin polarization

is needed for this _purpose. In exper-

iments the observed critical

polarization

is

usually

about 1.3 times

higher [12],

^which is still below

42%.

So _a

ferromagnetic ordering

of the proton

spins

is within reach.

b-axis

I

~

a-axis

Fig.

I. The

crystallographic

unit cell of

fluorene,

with the definitions of the

crystal

_axes.

2. Monte Carlo simulations.

In the

proposed experiment,

nuclear

dipolar magnetic ordening

in fluorene is created

by

first

polarizing

the nuclei

by

MIONP. After MIONP the strong ^external

magnetic

field

Bo

is oriented

along

one of the

crystaline

_a, b _{or c-axes}

(see Fig. I)

and _an ADRF is

performed by applying

a

strong

rf-field with _a

frequency

uJ and

sweeping

this

frequency

towards the nuclear Larmor

frequency

_UJI

= ~iI

(Bo( Finally,

the

amplitude

of the rf-field is reduced to zero. Note that the strong

magnetic

field is still

present.

To describe the nuclear

spin system

we use a frame of reference

rotating

with the nuclear Larmor

frequency

about the direction of the external

magnetic

field. Then the nuclear

spin system

is described

by

the truncated

dipolar

interaction

(I)

between the nuclear

spins.

For the

prediction

of the

properties

of the ordered states _we calculate the

dipolar

_energy,

using

the molecular field

approximation.

As discussed in the introduction _we

only

need _to calculate

the _energy

ED

of so-called

longitudinally

ordered states, where the

spins

P _are oriented either

parallel

_or

anti-parallel

to

Bo.

The _energy of

transversely

ordered _{states can} be deduced from

2262 JOURNAL DE

PHYSIQUE

I N°11

these values. Then this _energy is

given by

~D j ~ ~~jPiPj> (4)

'>J

where _{p, =<}

Ij

_>

II

is the

polarization

of the i~~

spin.

In order to determine the structure of the

dipolar ordening, I.e.,

to determine for each

spin

^whether _p, is

positive

_or

negative,

_we first

consider the _case that the

dipolar temperature

is very close to _zero

[21],

so pi m ~l.

Then,

~D ~ _{~ljflJ' (~)}

l,j

where

f~

is the structure

parameter: f~

₌ +I if the

spins

P and IJ _are

parallel, fij

₌ -I in the

opposite

_case. We note that the

dipolar

interaction is

long

_range so the _sum

(5)

is

generally dependent

_on the

shape

of the

sample.

We _{use a} Monte Carlo simulation

[17, 15]

^{to determine the}

spin configuration I.e.,

the factors

f~, yielding

the

highest

and lowest value of

ED

Our model consists of _a small

rectangular crystal

with six

crystallographic

unit cells

[13], containing

in total 240

proton spins.

The Monte Carlo calculation is started with random values for _pi. Then the

dipolar

_energy

ED

is calculated

using equation (5).

We restrict the _sum to

spins

^where _r~ < 8

1. Thus,

_on the _average 48 to 96

neighbouring spins

_are taken into each sum,

depending

_on the distance

between the

spins

and the surface of the

crystal.

Then _we allow _one

spin

to

flip

and calculate the

dipolar

_energy

again.

^This new

dipolar

_energy,

E[,

is

compared

with the

previous

_one,

ED

We

accept

^the

spin flip

if

e~i

,

> R

(6)

where R is _a

pseudorandom

rational number between 0 and I determined in the _way described in

[18], kB

is Boltzmann's constant and

TD

is chosen _to be ~l nK which is far below the

expected

transition

temperature.

^If the

spin flip

is not

accepted,

the

spin

is set back in its

original

orientation. We continue this

procedure by sequentially flipping

all 240

proton spins.

After

passing

about 20 times

through

these 240

spins,

_a stable

spin configuration

is reached where the

dipolar

_energy is maximal for TD " -I nK and minimal for

TD

₌ +I nK.

In order to characterize the thus obtained

spin configuration,

_we

perform

_{a space} Fourier transform

yielding

240

p(k)

#

~ _{pze~~'~', (7)}

1=1

and

investigate

whether

p(k)

has _a

profound

maximum. If such _a maximum is found for _a value of k such that

=

27r(k(~l

is of the order of the distance between the nuclear

spins,

the order is

antiferromagnetic.

If however

=

27r(k(~l

is

comparable

to the dimensions of the

model,

the nuclei order in _a

ferromagnetic

domain _structure.

We

performed

these Monte Carlo simulations for the external

magnetic

field

Bo

^I to the _a, b

an c-axis. As _an

example figure

2 shows the

development

of the total

magnetization

and the total energy of the

spin system during

the Monte Carlo

procedure

for the _case that

magnetic

field is oriented

along

the

crystal

b-axis.

Figure

3 shows the

b-component

of the _space Fourier

transform

(7)

of the obtained structure as a function of

kb.

A

profound peak

is observed for

kb

_" 0.42

1

_~.

Table I sumniarizes the results for the

dipolar

_energy for each direction of

Bo

and for both

signs

^of

TD.

^The first column

gives

^the

sign

^of

TD.

^The next column

gives

the direction

M~

~~

^0.4

lx lo

(a.u.> ^~~

0

oi

o

lxlo~

°0 _{0.5x10~ lx10~}

Fig.

2.

Example

of

a Monte Carlo simulation of

ED

and

magnetization

of the model. Orientation

of the

magnetic

field is

along

the

b-axis,

TD _" ~l nK.

0 0.2 DA

k~

IA

^'~

Fig.

3.

Space

Fourier transform of the obtained ordered

phase

with the

magnetic

field

along

the b-axis, TD

" -I nK.

of

Bo.

The

following

3 columns show the dimensions _{ea, eb} and _ec of the model used for the Monte Carlo simulation in the _a, b and c-direction

(a

=

8.3651,

_b

=

5.6541,

c =

18.745

1,

see

Fig.

I

).

The sixth column

presents

the

dipolar

_energy of the

longitudinal orderings

simulated in our Monte Carlo calculations. Note that this value still

depends

_on the

shape

of the

crystal.

Therefore the results _are

given

in

arbitrary

units.

Using

the

argument presented

in the introduction _we have calculated the maximum and minimum

dipolar energies

of the transverse

orderings.

These are shown in the last column of table I. As for

TD

<

0,

the energy should be

maximal,

the

ordering

is

always longitudinal,

while for

TD

>

0,

where the energy should be

minimal,

the

ordering

is

always

transverse.

Furthermore,

for _a

given

direction of the

magnetic

field the structure of the

longitudinal ordering

at

negative temperature

is the _same

2264 JOURNAL DE

PHYSIQUE

I N°11

Table I. Monte Carlo results for

ED

The units _are

arbitrary

Note that the results

depend

on

sample shape.

model size

ED ED

to _{ea eb ec}

< 0 _a 3a 2b _c

+0.889 +0.154

< 0 2a 3b _c

+0.777 +0.189

< 0 _c 2a 3c

+0.671

> a c -0.445

> 0 b 2a 3b _c -0.378 -0.388

> 0 _c 2a 3c -0.266 -0.336

as the structure of the transverse

ordering

at

positive temperature.

We _now consider these ordered structures in _more detail. Table II

presents

the values of

ka,

_kb and

kc

where

p(k)

has _a

profound

maximum for _the

longitudinal orderings expected

at

TD

_< o. Note that the _same values of

ka,

kb and

kc

are

expected

for the transverse

orderings

expected

at

TD

> 0. The last colunin shows the

wavelength

=

27r(k(~l

^of the ordered struc-

ture. From table

II,

it is evident that k is

always

found to be

parallel

to

Bo. Furthermore,

all values ₌

27r(k(~l

are much

larger

than the

typical

distance between nuclear

spins

and of the order of the size of the model used for the Monte Carlo simulation. The values of

are also incommensurate with the size of the unit cell

excluding

_some kind of

long wavelength

"antiferromagnetic" ordering.

Therefore _we conclude that at

negative spin temperatures

fer-

romagnetic longitidinal orderings

_are obtained.

Again using

the

argument

^of the

introduction,

this _means that at

positive temperature ferromagnetic

transverse

orderings

_are

predicted.

We

note however that in this latter _case, the

ordering might

also be helicoidal

[12].

Table II. Monte Carlo results for the structure of the

longitudinal ordenings

at

TD

< 0.

TD Bo

model size

ka ka ka

=

I to _{ea eb e~}

)~~ )~l )~l 27r/(k(

< 0 _a 3a 2b _c 0.46 0.00 0.00 13.6

< 0 b _2a 3b _c 0.00 0.42 0.00 14.8

< 0 _c 2a b 3c 0.00 0.00 0.33 18.8

3.

Properties

of the

ferromagnetic phases.

We _now

proceed

to calculate the

dipolar

_{energy per}

spin,

the critical temperature ^{and the} critical

polarization

needed to reach the three

ferromagnetic phases

found above. One could

use the Monte Carlo method for this _purpose

[19]. However,

to _save numerical time _we used the well _proven molecular field method introduced

by

Goldman

[14].

^{We first consider the}

orderings

at

negative temperature.

Front the Monte Carlo

siniulations,

_we conclude that the nuclear

spin

system

^orders as a so-called ferro-sandwich for all three orientations of the

magnetic

field. This

ordering

is characterized

by

_a domain

structure,

where the domains have the

shape

of

large

flat

pancakes,

with their surface

perpendicular

to the

externally applied magnetic

field.

We first calculate the

dipolar

_energy of the nuclear

spin _system

at

TD

_" 0. Goldman

[14]

calculated the

dipolar

_energy of the i~~

spin

in the _case of _a ferro-sandwich. The result is

given by

~~ ~~

~~~ ~ ^~)~~~~ ~

~~~

j #

I r~j ^{< R}

Here the contribution of all

spins

IJ within _a

sphere

with radius R is calculated

by

discrete

summation,

^{while the} contribution of

spins

IJ that _are further _away is

approximated by

_an

integral.

To obtain the total

dipolar

_{energy per}

spin,

we need to calculate the _sum of the

dipolar energies

of the individual

spins

ED

_"

f

El. (9)

2n

Note that _a factor is introduced because the interaction between _two

given spins

is _encoun- tered twice in the _sums in

equations (8)

and

(9).

In fluorene there _are 40 different

proton

sites in the

crystallographic

unit cell.

However,

due to the

symmetry

^{of the}

crystal (Pnam),

there

are

only

_n

= 12

magnetically inequivalent proton

sites _so the summation in

equation (9)

needs to be taken _over these 12 sites

only.

The calculation of

ED

is

performed numerically.

To

study

the convergence of the _sum in

equation (8),

_we increase the radius R

stepwise

to 200

1

_{where the}

sphere

contains about 1.2 x

10~ spins.

Clean _curves with

increasing

R _are obtained. From these _courves the limit for R _{- cc} is obtained. The results _are

presented

in the first three _rows of table III. We estimate the relative accuracy of these limits to be 2 _x

10~~

Table III.

Properties

for the

ferromagnetic ordening

of the

proton spins

of fluorene at

TD

< o for diiTerent orientations of the

magnetic field,

the

dipolar energies

_are

given

_per

spin.

TD Bo

T~

P~

(

(I~K)

< 0 _a

+0.940

-0.469 0.40

< 0

+0.543

-0.271 0.85

< c

> 0 _a -0.470

+0.235

0.80

> 0 b -0.271

+0.136

1.70

> 0 _c -0.540

+0.269

0.52

2266 JOURNAL DE

PHYSIQUE

I N°11

The critical

spin temperature Tc

where the

phase

transition from the

paramagnetic

to the

ferromagnetic phase

_occurs is calculated from

ED using _[20]

Tc

=

~

ED- (10)

2kB

The values of

Tc

_are also

given

^{in the first} three _rows in table III. We note however that _in

our case these results _can

only

be

approximate

because the Weiss field is different _at different

proton sites,

_so the so-called

strong

constraint [21] ^is not fulfilled.

Again using

the

argument presented

in the

introduction,

also the values of

ED

and

Tc

for the transverse

orderings

at

positive spin temperature

can be found. These _are shown _in the last three _rows of table III.

Finally

_we determine the critical

polarization Pc,

which must be reached

by

MIONP in order to obtain the

ferromagnetic orderings by

a

subsequent

ADRF.

According

to Goldman

[14]

p _~e I

@lLo 'fjh~

I

f _~

l 3

cos29~

^~

~ 8 47r

kBT~

_n

r3

^II

i=i _j iJ

The

dipole

_{sum converges}

rapidly

and is calculated

numerically

with the _{same accuracy as}

ED

For both

positive

and

negative temperature

^{and for each} of the three directions of

Bo

the

resulting

critical

polarizations

_are also

given

in table III. Note that the result for

TD

> 0 and

Bo

^I b-axes is

larger

than I.

Evidently,

our method fails for this _case.

4 Conclusion.

Using

Monte Carlo methods _we find that the

magnetic dipolar

ordered

phase

of the proton

spins

of fluorene is

expected

to be

longitudinally ferromagnetic

for

negative

nuclear

spin temperature

and transverse

ferromagnetic

or helicoidal for

positive temperature. Using

the molecular field

method of Goldman

[14]

we calculated the critical

polarizations

for each of these

orderings.

It is found that the _case that

TD

< 0 and B

ii c,axis is the most favourable for

creating

_a

dipolar magnetic ordering

in fluorene.

Then, according

to the molecular field

approximation

_a proton

spin polarization

of

only 26%

is needed. It is noted that in actual

experiments

the critical

polarization

is

typically

a factor 1.3

higher _[14]. Still,

_as at present 42Yo _can be reached with

MIONP,

such _an

ordering

is well within reach.

Acknowledgements.

The authors thank J. Schmidt for many

helpful

and

stimulating

discussions. This work is

part

^{of the} research programme of

"Stichting

_voor Fundamenteel Onderzoek der Materie

IF-O-M-)" financially supported by

"Nederlandse

Organisatie

_voor

Wetenschappelijk

Under- zoek

(N.W.O.)".

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