Volume effects on the nuclear quadrupole resonance transition frequency

Journal of MolecuLar Stmcture, lll(l983) 155-162 Ekevier Science Publishers B.V., Amsterdam -Printed

VOLUME

EFFECTS

t4ariano

Zuriaga

J.

Instituto Laprida

ON THE

+

hWCLEAR

and

Carlos

de Matem&tica, 854,

5000

QUADRUPOLE

RESON.ANCE

TRANSITION

FREQUENCY-

t

A. Martin

Astronomia

C&doba,

155 in The Netberlmds

y Fisica,

Universidad

National

de Chrdoba,

Argentina.

ABSTRACT A suitable modification of a general thermodynamic procedure to correct exgathered at constant (room) pressure, to that which perimental data, usually would be obtained had the volume been kept constant, the no_rmal condition under which the theory is worked out. This allows therefore meaningful comparison of data with theory. This modification is applied to the pressure derivative at constant temperature of the Nuclear Quadrupole Resonance transition frequency as It is shown in p-dichlorobenzene and in p-chloroa function of the temperature. phenol that, by using very reasonable values for the various parameters appearing in the procedure, excellent agreement is found between theory and experimental data.

INTRODUCTION Nuclear to yield yer

and

quency the

Quadrupole information

Kushida (NQRF

explained

vs Tf

quadrupolar

improved volved

(ref.l)_

alize

the

constant

35 Cl

in both

lines

ing the

procedure

greement

In the the

procedure

of

vith

In this allows

theory

mentioned between

Section

of work

(PCP)

theory

NQRF

of

the

the

in the

NQRF

analysis

NQR

bond This

mode

We

between theory

first

data have

continuous the

NQRF

in-

to an-

gathered measured

of p-dichlorobenzene

of

was

repzxted

in order

way

fre-

frequencies

is develowd

volume.

Ra-

transition

vs P) was

in a meaningful

technique

(refs.l.2).

chemical

normal

(NQRF

phases

a valuable

of solids

in an essentially

way_

(DCH)

at the and

Follow-

vs P excellent

a-

data. details

vs

to be

(refs_3,4).

at constant

in the and

of

a procedure

crystalline

above

the

the

to compare

ex_perimental

to analyze

motions

it is bonded

developed

three

shown

properties

dependence

dependence

in the

been

dependence

torsional

a temperature

(ref.5).

has

static

to which

of p-chlorophenol

is found next

(NQR) and

temperature

that

vs P which

pressure

NQP.F vs T of

and

Pressure

et al. NQRF

the

in terms

atom

by allowing

by Kushida

Resonance on dynamic

are

given.

P is developed_

In the In the

following

last

Section

Section the

re-

sults are presented and discussed. lFartial finantial assistance was provided by Consejo de Investigaciones Cientlficas y Tecnol6gicas de la Provincia de Cti&doba and by Sub-Secretaria de Estado +de Ciencia y Tecnologia, Argentina. Holder of a Scholarship granted by Consejo National de Investigaciones Cienti,ficas y Tgcnicas, Argentina_ "Fellow of the Consejo National de Investigaciones Cientificas y Tgcnicas, Argentina. 0022-2860/88/$03.00

0 1983 Ekvier

Science Publishers B.V.

”

156

?ZXF&tEXTI\L The

s_pectromcter

with

external

NQRF

with

quencies were

the

0.2

couple

and

The

frequency

of a frequ&ncy

(Catalog

The

as received-

Kmr

The

and

Rao

a differential

was

voltmeter

measured

are

35370

(Fluke

the

with

845

100

Ml.

Samples

for DCB

varied

and

Fre-

PCP

in a similar

temperature.rate

re-

way

to

of change

a copper-constantan

41 with

the

Hz.

ND

of 2500

25850

was

keeping

measured

error

and

temperature

superposing

by

(Schomandl

an overall

(ref.7).

tenperature

was

synthesizer

with

Numbers

ty_pe oscillator

super-regenerative

a conventional

to be.measured

used by

K/min.

and

signal

by Fluka

indicated

about

0.2

beat

believed

supplied

that

was

quenching-(ref.6).

are

spectively)

used

thermo-

an estimated

error

Of

K. The

three

was

a, was

phases

quenched

cbtained

point

by a similar

is about

keeping

the

were

of DCB

by plunging

325 K).

-phase

in the

as

sample

procedure

The s

temperature

obtained the

follows_

in liquid

but

starting

was

obtained

neighborhood

of

The

room

nitrogen, with

tern_perature phase, while

a molten

starting

with

230 K for

about

the

sample the

e-phase (melting

p-phase

half

and

an hour_

THEORY

Pressure The

deoendence main

of

assumption

function

modynamic

the

ly gathered

NQRP

in what

of P and

at constant

T

follows

is that

(ref_5)_ while

pressure,

the

NQRF

Experimental the

may

data

theory

be

treated VS

of NQRP

developed

is

as a ther-

T iS general-

at constant

vol-

therefore necessary to correct the data to constant volume in order ume. making to carry on a meaningful comparison with the theory. The procedure to be developed

below

assumed dure

will

that

will

tropic

do the

the

system

be usefu.

and,

no

further

the

However,

is isotropic_

with

for

sake

of simplicity,

as will

modifications

even

be

shown

in the

it will

below,

case

ThermodynamXcs. V=V(P,T)

P, V,

W data

the

proce-

of very

aniso-

and

Let and

T are

is generally

us assume that

the

W(V,T)

pressure,

gathered

that

the

system

is a function the

volume

as a function

and

may

be described

describing the

some

by

tern_perature of

of T at constant

the

W theory

the

pressure,

In order be

pro-persystem.

say

Po,i.e.

(1)

is,

generally,

develo_ped

at constant

volume,

i.e.

W = WtVo,T)

must

state

physical

W = W(V(Po,T),T)

while

be

crystals.

function tY-

correction

to.compare corrected

12)

experimental to constant

data volume.

with

theoretical

Following

Wallace

expressions (ref.8),

the we may

former write

157

W(V(Po,Ti,T)

OLD

= W(Vo,T)

- d

Wo(Vo,T)

PT

(T - To)

(3)

-

@ expansion coefficient, is the isobaric volume thermal T P iS the pressure derivative of W at conis the isothermal compressibility and W P 0, are in order to arrive to Eq.3, that CL p and stant T. It has been ass&Ted,

where

Vo=V(P o,To),

constants, room

=

a reasonable

approximation

for

T in the

range

K and

P about

pressure.

Equation comments

3 is central

are

in order.

in the First,

procedure

Eq.3

to compare

contains

w

and P

exhibit well

large

that

the

anisotropies

these

anisotropy

safely and, W

100-300

in the

be used

for

tend

ratio

however

to have

it has

similar

T,

must

shown

out,

by

and

the

and

may

Con-

Therefore,

same

value

intensive

in a relative

Two

which

Zallen

(ref.9).

W is an

be understood

ex_periment.

quantities

anisotropies

/ largely cancels P OT in general direction. Second,

any

with

two

c been

a

its V dependence

therefore,

=

(ref.8):

quantities

theory

may

property

sense,

i.e.

to the

E;QRF ;t

W(V/Vo.T)_ Application

will

to the

be assumed

(ref.51

NQBF,

=

J ow)

;1 - ;

=

3oCV,

[l - ;((e2(T))

the

equilibrium

site

(ref.4).

represents pends

the

on T and

quencies

may

the

called

NQBF The

are

application

1) Substituting

aQ(V,T)

= ao(Vo)

of the

dependence

NP DT

the

Eq.4

- ;

SQp(Vo,T1

above

procedure

(ref.5).

Therefore,

WE may

write

gradient z-axis

The

and

to Eq.4

into

de_pends

Eq.3

((82(T))

(T - To)

static

we

only

fact of Jo

contributions rise

to the

Cl

nuclear

C-Cl the

on TI vhile

modes

contributions

give

the

I represents

to the

dependence

_p-er_pendicular to

(E-CC) at the

librational

These

will

direction

is along

<8*(T)>

and

(41

bond

field

lattice

EFG.

El]

PC-P the

on V is due

to the dynamic

C-Cl

lattice.

(refs_1.8).

of Eq.3

[l

and

to
de_pend on V contribution

(8*(V,T))

of the

electric

of DCB

contribution

termolecular

I f

in a rigid

modes

V. This

T

1

rotation

of the

cases NQRF

$0 the internal

the

of V and


total

z-axis In the

to apply

as

the

is the

of

tion

$ Q.

8 represents

rection.

In order

it is a function

the

'?Q(V.T)

where

NQBF.

that

bond



to (8Z(V;T)>

that

the

lattice

on: V is due

di-

contribuE

and

de-

fre-

to the

in-

to the V dependence

of

res_pectively. following

cases:

obtain

I + <8*(Vo,Tl>

E)]

(5)

where

the

first

tem.in

.ory in a harmonic

-the right

at V -and T. -0 -.2) If we correct‘pnly

hand

side

cokreswnds

9. (V ,T) is the QQO

crystdl,.and

to the

pressure

Bayer-Kushida-the-

derivative

of

the NQRF

evaluated

= ~o(Vo).L1r

?Q(V,T)

- $_ ( <0*(T)>

$

-

where sis

the

first

term

carried

in the

the

external

modes

ear

one

data

are

are

alloved

I + '632PJ.T))

E)] ~op’vo’

(&V,T)>

hand

K! ]

side

corresponds

approximation to have-a

term

most

to a Bayer-Kushida

(ref.a),

temperature

is the

c-on

(6)

where

the

dependence, way

generally

in which

analy-

frequencies

the

of

a lin-

NQRFvs

T

analyzed.

=

the

only

I

3o(v) 9

(e*(V

e.T

ap 3’

(V ) Op"

1 - $

on

, we obtain

(g*(V,T))


Ep.

Go(V)

( <0'(T)>

C

p

,T)>

both

1:;

=

+

O

correcting

4) Finally.

--

correction

how [l - .$ (

+ ;

3QfV,T,

right

first

ve obtain

+

I

quasi-harmonic

31 If we perform

+V,T)



(

in the

This

Cref.1).

$ oW-in.~g.4

and



I +

( (g*(T,)

E simultaneously,

we obtain

K) ]

I +
o,T)>

E) ]

T -3

3 oWo)

2

<9*Wo,T)>

(8)

Kp (T - To)

: Cornparis&

3&Vo;T).=

of Egs.8

and

GopWo)

S shows

1 - ;

that

(
I +

.I

]

.I

- ;

30,Vo'

which

exhibits

side,

respectively)

the

-.

(O*Wo,T~)

static

and

dynamic

contributions

..

Ep

to

(first

and

g*(V,TL

’

second

terms

in the

right

hand.

159 RESULTS

AND

Evaluation

DISCUSSION of

Following modes

O'Leary

involved

do not

tion-libration


for

the

+ 3.73

10

the

tively. to the and

lo_4

+ 4.05

10

-4

first

four

gives
of

b) there

the

normal

is not

transla-

8n21e

+ 6.37

10 -4 coth.162.6 T

h S,(V.T) coth S,W,T)

(10)

2 kg T

and

e

QeW,T)

Planck's

right

and

hand

side

contributing

are

Ie and J

coth23;-1

coth 871'Ie

in the

modes

10-4

h

kg are

h and

terms

E_

and

h

of vibration

of these

frequencies

be written

*

+

The

the

10 -4 coth21;-4

coth1g5-0 T

in PCP.

modes

+ 6.24

+ 7.05

lines

a1

(ref.111,

it may

coth31;-2

two

internal

dispersion

251.8 '~ T

of DCB

= 3.72

frequencies

Eqs.

-4 =ot

phases

assuming:

(ref.12).

10 -4 cot-

three

and

exhibit

coupling

= 9.76

<6*(,,Tj>

for

(ref.10)

taken

6.69

from

lo-*

Boltsmann's of Eqs.

refs.13-16. degree

cOtt*ll;-o

h d (V,T) 2 ; T B

(11)

constants

10 and I_ The

to

to a good

are,

l

The

respec-

11 correspond eigenvectors term

in both

of approximation,

last

given

by

2 ye =

1 Ix

1

Ie?Z

Y

where

frequencies

dinates.

The

the

benzene

and

are

Iy,

about

to the

direction,

1x3:

the

the

for DCB

I

molecular

x and

y axes

ring,

the

y-axis

expressions

is chosen for

E = ";':V'd' e

coth

E = 6jl~V1~)2 e'

COti-

as


EFG,

(121

3' Y

moments

of the

z-axis,

0.72 <8*(V,T)>

1 Y

Ix, 3 x and

normal

1

2 -=-+-

+f

of

inertia

respectively.

as mentioned

before,

to produce

a right

E are

as

left

and

the

librational

The

x-axis

is along

handed

system

is chosen the C-Cl of coor-

S,(V.T) T

(13)

and O-72



de(V,T) T

(14)

, i60 for

PCP.

In both.caL

linear.temoeratute h.Aviouris

The

dependence

values

= se,

.to be

(ref.17)

and,

for. I>, !VCPo,T),T),

proposed

~eWPo,T),T)

in CIII'~.It has been

Ji,is

shown

therefore,

$

that

and ?y

x a similar

exhibit

temperature

i-e:

(15)

(1 - Ce TJ

used

are

a be-

those

given

it? Table

1.

T+LEl Sffective

external

frequencies

used

9eotcm phase CL phase.? phase p

(ref.181 (ref-18) (ref.17)

I

XB

line ( line

PCP

of higher KQBF of lower NQRF

We aetermine taken

as 9.16

$ (V(Po,T)) -8 i0 (kg/cm2)-l

for

thewar.d

ing To,

keeping'

T clcse -.

3' Qp(V(Po,T),T)

for

the

ment

in DCB,

none

mode

7.65 7.70

this

the

three

data

phases

and

quantities

and

the

It should

entering

into

results

be

seen be

in the

and

calculated

Eq.9

are

values

For

known

should

be assigned

be .further

example,

improved

in the

r&a&ing.quant.ities In this

for $

figure

to a translational by

slightly

f-phase,

using.

unchanged, we.also

the

plotted

$eo=6O_8

the

static

been

have . We,

NQRF

vs

adjusted.

P

It is

frequency

for

beer. obtained therefore, The

believe

agreement

ob-

the

various

quantities

cm-',

Ce=6.45

low4

is excellent and

along agree-

in the

mode.

changing

agreement

in Fig.1

dynamic

as may

K-l, be

seen

contributions

to

of the NQRF vs P in PCP

In this it will

may

and

re_ported

would

10m4

e;aluate

aTeaSOnabh?

has

the

3.65

is

$oo(V(Po,T;)

that

procedure

as

pT

'dy vary-

we

is

stressed

10.

(ref.20).

shown

there

that had we taken into account -1 30 cm (ref.181 large discrepancies

experimental

andblp

phase

7 we obtain

results

As may

(ref.5).

the f-

to note

frequency

in Fig-lb.

Analysis

and

(ref-19).

for

Eqs.6

these

by using

phases

10 -' K-1

using

all

phases

three

3.15

to it and

the

three

for.the and

of the

about

above

involved_ Snd

Eqs.4

7.17

QG tained

PCP.

7.90

the

and,experiment.

Y while.comparing that

for

(10-4K-1)

e

.50.60 50.00

theory

tiwrtant

anR

c

determined

e-00

Collecting

experimental

bet-en

also

-5

and

65.15 64.70 64-00

for

e phases,



with

IXB

of the HQP.F vs P in UCB

Analysis

K-l

for

be

case

there

shown

quantities.if

how

is not to take

others

&a

enough

information

advantage

of

to evaluate

the_ procedure

knows _ In this

case

4;

3 e

(ref_lk)_

However.

in order

to determine

is kno>

for.both'lines

un-

161

TEMPERATURE

Fig. 1. constant

TEMPERATURE

(K)

Temperature dependence, at room pressure, of a) Calculated tern_perature of the NQP.F in DCB.

(K)

the pressure values for

deriverive

at

the three phases respectively) using raw data. Ex(-_, ---, -a--- for the CL, e , and 0 phases perimental values are taken frcrn ref.5 (* and a for the= and c phases respecb) Calculated values for the q -phase (-1, and for the dvnamic c---j tively). and static (-.---I contributions with slightly modified data as indicated in values (0 ) are taken from ref.5. text. Experimental (ref.21).

Let

T* be

4 Q(V(Po,TI,T)

the

temperature

e2w(Po,Tf)

1

[

at which

0 and

dQp=

using

Eq.5

we may

write

= ~QW(Po,T’) ,T*)

,T*,>]

(e2(v(P 0' T) ,T)) ] for

which.

each

T)) _ Solving

ting

with

Eq.15

seen

they

are

ed

in ref.16.

which ure

for

the

quite

the

a trascendental

various

values

these

the best

for

excellent

utions

have

signs

sults

are

The

duce

obtained

procedure

experimental

remaining

ones

for

the

developed

results appearing

and

found

therefore

NQRF

above

to

and

Table

consistent

oT=18

between

and

As may

-2K-1

found. the

values

to be

the

for both

experimental

the cancel

JewDo,

1 are

data

the

is

is obtained,

with

kg cm

in Fig.2, largely

and

each

other.

be

report-

value

lines.

and

static

fit-

Fig-

calculated

dynamic

contrib-

Similar

re-

line.

analyze

to determine

in the

in

9 QD'V'Po,TLTf

Also,

lower

and

ap/

solution

whose

behaviour

quoted

lines,

oith

work.

line

Ce

find

agreement

in this

and

both

we

agreemer.t

as determined opposite

.Ie.

values

vslues

equation

T a straight

for

similar

With

produces

2 shows

provides

T,

Eq.16

(16)

equations

the

NQXF

various are

vs

P has

quantities

known.

been

of

shown

interest

to

repro-

if the

i62

-34SO

150 TEMPERATURE

220 I k )

290

80

150 TEMPERATURE

220

IK 1

290

dependence, at room pressure, of the pressure derivative at Fig. 2. Temperature (I 1 values for both constant tern_p-erature of the NQRF in PCP. a) Experimental The continuous line indicates the values determined in this work. lines (ref.21). The higher NQRF line has the larger pressure derivative. b) Dynamic (---I and static (----.-I contributions to the pressure derivative (-1 in the higher NQRP line along with the corresponding experimental data I I 1.

REFERENCES 1

R_ .7. C. Brown, J. Chen. Phys., 32 (1960) 116. Nuclear Quadrupole Coupling Constants, Academic Oresr, London, 2. A. C. Lucken, 1969. 130 (1951) 227. 3 H. Bayer, 2. Phys., 4 T. Kushida, J. Sci. Hiroshima Univ. A, 19 (1955) 327G. B. Benedek and N. Bloembergen, Phys. Rev., 104 (1956) 1364. 5 T. Kushida, 6 J. A. S. Smith and D. A. Tong, J. ?hys. E, 1 (1968) 8. 7 u. V. Kumar and N. N. Rae, Phys. Stat. sol. (b), 44 (1971) 203. Thermodynamics of Crystals, Wiley, New York, 1971. 8 D. C. Wallace, Solid State Commun., 31 (19791 557. 9 R. Zallen arid E. M. Conwell, 10 G. P. O'Leary, Phys. Rev. Lett., 23 (1969) 782. J. Magn. Resonance, 11 (1973) 28. 11 M. M. MC Ennan and E. Schempp, 12 pi. M. MC Ennan and E. Schempp, 3. Magn. -Resonance, 16 (1974) 424. 13 3. R. Scherer. Planar Vibrations of Chlorinated Benzenes, Internal Report. The Doti Chemical Company. 14 J. R. Scherer, Spectrochim. Acta A, 23 (1967) 1489. Assignments for Vibrational S_pectra of Seven Hundred Benzene 15 G. Varsanyi, Derivatives, Vol. 1, Adam Hilger. London. 1974. . S. Parent and B. Pasguier, J. Chlmie Phys., 74 3.6 NI LeCalve, M. H. Limage, (1977) 917. I.7 I. Ichishima, J_ Chem. Sot. Japan, Pure Chem. Sect.. 71 (1950) 332. 18 P. A. Reynolds, 60 (19741 824. Z_ K_ Kjems and J_ W_ White, J. Chem. Phys.. 11 19 L. V. Jones, M. Sabir and J. A. S. Smith, J. Phys. C: Solid State Phys., (i978) 4077. 20 L. Ter Minassian and P_ Pruzan, J. Chem. Tiiermodynamics, 11 (19791 1123. 2

21 R. J_ C. Brown, 1050.

D. C. Wood

and

R. de Boer,

3. Faraday

Trans.

II,

75

(1979)