Orientational disorder in Ni(NH3)6I2. Evidence for rotation-translation coupling

(1)

HAL Id: jpa-00246778

https://hal.archives-ouvertes.fr/jpa-00246778

Submitted on 1 Jan 1993

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Orientational disorder in Ni(NH3)6I2. Evidence for rotation-translation coupling

P. Schiebel, A. Hoser, W. Prandl, G. Heger, P. Schweiss

To cite this version:

P. Schiebel, A. Hoser, W. Prandl, G. Heger, P. Schweiss. Orientational disorder in Ni(NH3)6I2.

Evidence for rotation-translation coupling. Journal de Physique I, EDP Sciences, 1993, 3 (4), pp.987-

1006. �10.1051/jp1:1993179�. �jpa-00246778�

(2)

J.

Phys.

I France 3 (1993) 987-1006 _APRIL 1993, _PAGE 987

Classification Physics Abstracts

61.12 61.50

Orientational disorder in Ni(NH~)J~. Evidence for rotation- translation coupling

P. Schiebel

(I),

A. Hoser

(I, *),

^W. Prandl

(I),

G.

Heger (2)

and P. Schweiss _(~,

**)

(')

Institut fur

Kristallographie,

Universit&t

Tiibingen,

D-7400

Tiibingen, Germany

(2) Laboratoire L£on Brillouin, CEA-CNRS

Saclay,

F-91191 Gif-sur-Yvette Cedex, France

(Received 30

September

1992,

accepted

in

final form

16 November 1992)

Abstract Patterson and Fourier

analysis

of _neutron diffraction data of cubic

Ni(HN~)512

taken at T

= 295 K and T

=

35 K show _a

planar

_proton

density

distribution with four maxima at the

comers of _a square. We show that this observation is the consequence of _a

coupled

rotational-

translational motion of the

NH~

_group in _a two-dimensional anharrnonic

single particle potential

with

tetragonal

_symmetry. Our

potential

arsatz

gives

_a better fit to the data than conventional

crystal

structure

analysis.

1. Introduction.

Metal hexammine

compounds Me(NH~)jX~

in their

high temperature phase generally

have the cubic space group Fm3m. The central metal atom of the hexammine cluster is surrounded

octahedrally by

six

NH~

groups. With the

NH~ pyramids

on a site with

symmetry

⁴ mm it is obvious that

they

must be

orientationally

disordered.

Usually

the structures of the

high

temperature

phases

_are described

by

models with the ammine group

occupying

_a number of

equivalent sites,

I.e. its three H-atoms _are disordered between 24

positions [I].

An

early

multi-site model _was

proposed by

Bates and Stevens

[2].

They

used

point charges

to calculate minimum _energy orientations of the

NH~

_groups ⁱⁿ the nickel hexammine cluster and found 8 minimum

configurations.

It _was

suggested

that the

freezing

of _one of these

configurations

at T~ ^should

trigger

the

phase

transitions observed in all metal hexammine

compounds [3].

These

phase

transitions have been

investigated by

a

variety

of different methods. Jenkins et al.

[4, 5]

and Bates _et al.

[6]

have shown

by

Raman spectroscopy ^that ^the

NH~

orientations

are indeed involved in the

phase

transitions.

Though

it is

generally accepted

that in the

high

(*) Present address _: Institut fur Chemie und Elektrochemie, Universitht Harnover, D-3000 Hanno- ver, Germany.

(**)

On leave from

Kemforschungszentrum

Karlsruhe, INFP, D-7500 Karlsruhe,

Germany.

(3)

temperature

phases

a fast reorientation of the

NH~

_groups occurs, details of the

NH~

motion _are

largely

unknown. Models used

recently

include almost free

rotation,

rotational diffusion and discrete 120°

jumps [7, 8].

Because of the

long

term

instability

of the

samples

and the difficulties in

obtaining good single crystals

the structural information

available, particularly

in the low temperature

phases,

is very limited. In the

high temperature phase

of

Ni(NH~)~I~

Eckert and Press described the motion of the

NH~

_group

by

rotational diffusion around Ni-N bond directions and below T~

by

one-dimensional

tunnelling

in _a

potential

with a

period

of 120°.

They

observed _a

slowing

down of the

NH~

reorientation _as the

crystal approaches

_T~

= 19.7 K

[1, 9].

In our

single crystals

of

Ni(NH~)~(NO~)~ [10]

and

Ni(ND~)~Br~ [I1, 12]

we found _a

density

distribution of the

hydrogen

atoms which shows four maxima at the _comers of _a squam. This observation could be

explained

as the consequence of _a

coupled

rotational-translational motion of the

NH~

molecule in _a

single particle potential

with

tetragonal

_symmetry

[13].

In this paper we

report

a neutron diffraction

analysis

_on

single crystals

of

Ni(NH~)~I~

in the

high temperature phase

at room temperature

(T

₌ 295

K)

and at T

= 35 K. The paper is

organized

_as follows. In

chapters

2 and 3 the

experimental

details and the conventional

structure

analysis

in terms of discrete site

(Frenkel)

models _are

given, respectively.

Crystallographic phases

obtained from the Frenkel model _are used _to derive _a continuous

proton

distribution

(Chap. (3.2)).

In

chapter

4 _we describe and _use _a 2-d

potential

model in order to describe this

density

distribution. The consequences of this

potential

for the rotational-

translational motion _are discussed in the final

chapter

5

together

with the

possible

_quantum

statistical

origin

for the temperature

dependence

of the

potential

parameters.

2.

Experimental

details.

The title

compound

_was

synthesized

in _a standard way

[14].

The diffraction measurements on

single crystals

_at _room

temperature

^and ³⁵ ^K _were

performed

_on the 4-circle diffractometer Pl IO at the

ORPHEE-reactor/CEN Saclay. Experimental

details _are

given

ⁱⁿ Table I. We

Table I.

Experimental

data and

crystallographic specifications.

chemical formula

Ni(NH~)~I~

space group

Fm3m,

no. 225

lattice

sites, coordinates,

site

symmetry

Ni 4 _a

(0, 0, 0)

_m 3 _m

1 8c

(1 1)$3m

4 ^' 4 ^' 4

N 24e

(x, 0, 0)

4 _mm

H disordered

(see text)

temperature ²⁹⁵ ^K ^{35 K}

neutron

wavelength

0.831

h

sample

volume 43 mm3 ~~ ~~3

lattice _constant lo.

875(6) h

_lo

₇₈₀₍₅₎ h

No. of measured reflections 481 2 408

No. of

unique

reflections 190 216

from which I _m 2 «~ 126 206

Qmm

₌ ⁴ w sin

@~~/A

^10.933

h~1

10.857

A~1

(4)

N° 4 ORIENTATIONAL DISORDER IN

Ni(NH3)612

989

applied

a numerical

absorption

correction based _on the

geometrical description

of the

sample crystals (SHELX) [15].

An effective linear

absorption

coefficient p~ff = 1.84 _cm _was

used,

calculated from the incoherent

scattering

of the protons. ^The ^intemal ^R-factors ^calculated ^from

averaged _symmetry equivalent

reflections

Rf~

=

0.040 and

R($=

_0.026 indicate

good consistency

^of ^the data at both

temperatures.

^The calculated structure factors _were corrected for

secondary

extinction

according

to

F~

=

F( (1

0.0001 _x~~~

F(~/sin _).

x~~~ ^was included in the structure refinement and _was found to be _very small for the two data sets.

3. Structural models and refinement.

3.I CONVENTIONAL _STRUCTURE REFINEMENT. In the conventional structure

analysis

molecular disorder is described

by

Frenkel models

assuming

_a

superposition

of several

rigid NH~

_groups. The three-fold ammine molecule

(molecular

symmetry ³

m)

located _on _a site with

symmetry

4 _mm thus leads to at least 12

hydrogen positions

with

occupational weights 0.25,

if the mirror

planes

of the molecule and the

crystal

coincide. We denote the _two

possible

Frenkel

models _as model OY and model

XX,

if the molecular mirror

plane

coincides with the

0 yz or the _xxz

plane, respectively (Fig.

la,

b).

With the ammine molecule in

general position

Model III

generates

²⁴

hydrogens

with

occupational weights

0.125 for each

NH~

_group, I.e.

this model

corresponds

to 8 molecules

superimposed

with the

appropriate weights.

Model I is

simply

a

phenomenological split-atom

model with

only

_one

hydrogen

site

(xyz),

which

gives

the 8

hydrogen positions

_on a circle shown in

figure

ld with

weights

0.375. We would like to

point

out that constraints

referring

to

rigid

molecules _were not

applied during

^the calculations shown in table II. The _reason for this is that _our purpose was a

density interpolation aiming

at

stable,

I.e.

unique signs

for the Fourier

density

calculation.

a) b)

~ O O

o o

o

o o

o

~

c) d)

ad da

~ ~

~ o .

2

a a

~ ~

~~ ^o ^o

Fig. I. Frenkel models used in the SHELX refinement. a) model OY, b) model XX, c) model III, d) model1.

(5)

Table II. Results

of

the conventional

refinement

at T

= 295 K.

split-atom

models model

oy i

z = " z = o-o

u 0.0301(4) 0.0290(5) 0.0287(5) 0.0283(5)

z = y = z = 0.25

u 0.0391(7) 0.0380(8) 0.0378(7) 0.0375(7)

N _z, _y

= z = 0 0.19fi7(1) 0.19fi3(1) 0.19fi3(1) 0.1962(1)

vu 0.0349(7) 0.0332(7) 0.0335(7) 0.0332(fi)

vu = u~ 0.0447(fi)

HI z 0.23fi(t)

0.081(1) -0.0fi2(5) -0.077(2) 0.07fi9(5)

z 0.040(2) -0.05fi(5) -0.028(2) 0.0392(3)

vii 0.084(5) 0.053(4) 0.063(4) 0.0594(13)

vu 0.0t9(3) 0.103(22) 0.053(3) 0.0517(19)

u~ 0.073(5) 0.10fi(17) 0.125(15) 0.0974(fi3)

ui~ -0.013(4) -0.035(7) -0.001(9) -00047(33)

ui~ -0.007(4) 0.033(7) -0.002(7) -0.0133(1?)

vu

z 0.223(2)

-0014(3) 0.081(1) -0.0fi5(2)

z -0.080(2) 0.000 -0.0fi5

vii 0.049(4) 0.0fi4(11) 0.040(5)

vu 0.084(fi) 0.022(4) 0.072(10)

u~ 0.091(5) 0.248(31) 0.072

u13 0.018(5) 0.000 -0.030(9)

ui~ 0.012(4) 0.000 0.011(3)

«ii o-fill

z

-0.075(1)

z 0.050(1)

Vii 0.03fij3)

u~ 0.054(5)

U33 o.039j5)

ui~ -0.037(5)

u13 ~0.001(3)

Uii

Number of parameters 34 22 22 lfi

R 0.0334 0.0349 0.0370 0.03fil

Rw 0.0198 0.0182 0.0185 0.0186

RG ^0.0207 0.017fi 0.0180 0.0184

esd 2.27 1.82 2.37 2.ll

Results of the _structure refinement _at T

= 295 K and T

= 35 K _are

given

in tables II and III.

Considering

the parameters ^for ^the Ni-, ^N- ^and

I-atoms,

there is _no

significant

difference between the four models at each

temperature.

^As

expected,

the

temperature

parameters ^have

diminished at 35 K.

Though

_we did not constrain the geometry ^of ^the

NH~

^molecule ⁱⁿ the

refinement,

it is

obeyed

within

experimental

_errors in all Frenkel models at both temperatures.

The

height

of the

NH~ pyramid

is

0.380(2) A

_at _T

= 295 K and

0.378(4) h

_at _T

= 35 K. The distance between the protons ^and ^the four-fold axis at 295 K ranges ^from

0,88(2)

to

1.01(2) h

and _at 35 K from

0.85(1)

to

1.04(3)A.

For Model I these distances _are

unique, namely

r~~5~ =

0.940(4) h

_and

r~5K =

0.924(2) h.

_These _values

are in

good

_agreement with the

(6)

Ni(NH~)51~

991

Table III. Results

of

the conventional

refinement

at T

= 35 K.

Frenkel

models model

I

z = = z = 0.0

u 0.0054(1) 0.0054(1) 0.0054(1) 0.0052(1)

z = = z = 0.25

u 0.0063(2) 0.0063(2) 0.0063(2) 0.0062(2)

N _z,V

= z = 0 0.1983(0) 0.1983(0) 0.1983(0) 0.1983(0)

«11 0.0079(2) 0.0080(2) 0.0080(2) 0.0079(2)

HI _x 0.230(1)

0.080(1) ~0.049(1) .0.028(4) 0.0390(2)

z 0.231(2) 0.074(1) 0.077(1) 0.0762(2)

vii 0.023(6) 0.028(1) 0.027(1) 0.029fi(fi)

uu 0.023(4) 0.070(9) 0.054(fi) 0.083fi(2fi)

u~ 0.040(4) 0.024(3) 0.018(1) 0.0192(fi)

u13 -0.001(3) 0.023(5) 0.014(2) -0.0168(10)

u13 0.028(4) -0.006(2) ,0.007(1) -0.0054(5)

H2 _z 0.000

y -0.078(1) 0.079(1) 0.068

z 0.238(1) 0.239(2) 0.228(1)

«11 0.047(8) 0.128(23) 0.037(4)

vu 0.017(3) 0.016(3) 0.037

u~ 0.028(3) 0.026(3) 0.028(3)

u13 0.014(2) ~0.008(2) -0.002(1)

u13 0.021(S) 0.000 -0.002

vu 0.000

z

0.065(2)

z 0.232(2)

vii 0.033(7)

vu 0.008(5)

u~ 0.020(3)

u~3 -0.009(3)

u13 0.003(4)

Number of parameters 34 22 22 16

R 0.0254 0.0265 0.0259 0.0282

Rw 0.0122 0.0129 0.0125 0.0143

Ra 0.0110 '0.011S 0.0113 0.0127

esd 3.56 3.50 3.Sl 3.67

NH~ geometry

ⁱⁿ ^the gas

phase [16] (h

= 0.37

A,

r ₌ 0.94

A)

and

corresponding

results

given by

Eckert and Press

[Ii (h

=

0.38(5) h,

r =

0.895(20) h).

At both temperatures ^the ^best ^R-factor ^is ^achieved with Model

III,

but with twice _as many parameters as used in Model I. A Hamilton _test

[17]

_on the R-factor ratio indeed indicates that the

improvement

of the R-factors of the Frenkel

models, compared

to the

phenomenological

split-atom model,

is _not

significant.

(7)

3.2 PATTERSON-, FOURIER _DENSITIES. Patterson densities show the autocorrelation function of the

crystal

structure

P

(U)

₌ p * p_

(1)

where _p

(r)

^is the

scattering length density,

and _p

(r)_

m p

(- r)

is the inverted

density.

The great

advantage

ⁱⁿ ^the calculation of P

(u

is that it _can be determined

directly,

I.e. without _any

phase problems

from the

squared,

observed structure factors F

Ill

^~

~

I~~i.

A closer

inspection

of

(I)

shows

that,

for _a disordered

crystal,

P

(u)

has three contributions _:

peaks

_p~ _* _p~ between

sharp,

I.e. well located nuclei

diffuse

peaks

_pd _* _p~ due _to localised nuclei

mapping

_a diffuse distribution very broad and shallow distributions p~ * p_~.

It is the second

type

^of

peaks

which

gives

the diffuse distribution p~

only moderately

widened

by

thermal and series termination effects.

Figure

2 shows the

hydrogen

disordered

a)

Y

Z~,

'

'~"$',

)~.i I""(

~

If""

g/I

,

~pl,

~~/

-0.25

.0.25 ^-~ ^X 0.25

b)

_0.25

Go it oQ

Q ^~ Q

Q O

jj ⁱⁱ

' ~

o o

~ QQ °

~O $3 O©

-0.25 ~ ~ 0.25

Fig.

2. a) Patterson map at T

= 295 K. b) Fourier _map at T

= 295 K taken from the F~~~. Phases are from _a model without any

hydrogen

( 0.25 _« _x, y « 0.25, _z

= 0.23).

(8)

Ni(NH3)612

⁹⁹³

density

_p~

mapped by

^Ni nucleus.

Hydrogen

contributions _are

negative,

because the

scattering length

of the

proton

^is

negative.

Fourier

density

_maps show the _true _p

(r ).

^One

needs, however,

correct

phases.

It is

assuring

for _our purpose that the _sets of

phases

calculated from the four models described in section 3. I for the individual

F~~i

all coincide. It is this

uniqueness

which makes the Frenkel model

interpolation

_a reliable tool in the

analysis

of disordered densities.

In

figure

3a-d _we show sections

through

this Fourier

density

with the data at 295 K and 35 K.

For _our further

analysis

_we _are

only

interested in the

hydrogen density.

Therefore _we calculate

a) j.o.35;o.3s,o.35)

(o.is,o.15, o.3s)

, , ~ ~

~j

I

MA

~ ⁱ

Ni O

j j

Q cl

(.0.35;0.35;0.15)

(0.15,0.15,-0.15)

b)

_0.14

o czzJ o

o CZ

o

,~;;j~, ^II

~

,

£'

~';§j

fl

((f @~/

^~

^if) J

"' Q

O ~ ^O

n ;Z÷O o

-0.14

.0.14 ^~ ^~ ^0.14

Fig.

3. Nuclear Fourier densities. a) T

= 295 K, [110] section, b) T

= 295 K, [0011 section, c) T

= 35 K, [110] section, d) T

= 35 K, [001] section.

(9)

C) j.0.35;0.35,0.35) (0.15,0.15,0.35)

"Z~=~i )~ ^'~)~

j _i~l,

~

~'>~

~ l

")

~

",'. _.,:

~~~

N

~

."~ ~

O~'~ ~S

~j

b Ni ₄

~

j ^O .>o

.>

j

L_ ,~"~~~~~,,~

$j _{._,-~}

(-0.35,-0.35,-0.15)

(0.15,0.15,-0.15)

d)

_0.14

f

^~

'(Q°Q~

p (~

,T

',j

^~ ^f'§

y

( ")

~~ d~h

~ ' #j 0 §

j ,'i

j~~,J ^(,/ li

',

_'- /

q~ f

^(,

',

$

(~ _~'

^'

^f)

~

~~

-0.14

-0.14 ~ 0.14

Fig.

3

(continued~.

the Fourier

density

of the ordered part of the structure, which is very well described

by

^the SHELX

analysis

and subtract it from the Fourier

density

derived from the

Fill

and the

phases

from Model I.

The fine details of the

proton

distribution the four maxima at the _comers of _a square visible in

figure 4a,

b cannot be detected

properly

from the Patterson densities in

figure

2.

They

are likewise

hardly

visible in the _« _raw _» Fourier maps of

figure

3. The _reason for this is the

broadening

of the

positive nitrogen density

both

by

thermal motion and series termination

effects _: the

density

of the

nitrogen

atom at

(0, 0, 0.1963)

is well visible in the

plane

(10)

Ni(NH~)J2

995

z ₌ 0.23 where the

hydrogen density

is located.

Only

after the N-contributions have been removed

(details

of the method _are

given

ⁱⁿ

[13])

does the correct

H-density

show up. Thus _we obtain the

hydrogen density p$~.

Sections at _z

=

0.23 _are shown in

figure 4a,

b.

a)

_o.14

-0.14 -0.14

- x 0.14

'~)

0.14

;

:1[..;.." ~...[l

:" ~. i

...,_ ,...

"....~.i..

"...(I? "1.::.."

-0.14

-0.14 _- _x 0.14

Fig.

4. Observed nuclear

hydrogen density

at _z ₌ 0.23 ( 0.14 _« _x, _y _« 0.14). a) T

= 295 K,

b) T

=

35 K.

(11)

4. Potenfiial refinement and anharmonic motion of the

NH3.groups.

4,I MODEL CONSIDERATIONS. In

figure

4 the

quasi

2d _nature of the

proton density

becomes evident. With these results in

mind,

_we _may reduce the three-dimensional

problem

to two dimensions, I.e. the observed proton

density

is assumed _to be

generated by

a

rigid

and

equilateral proton triangle

which

undergoes

_an anhamonic movement in _a

plane perpendicular

to the four-fold axis _at _z= 0.23. Each

single

proton _moves in a

crystal potential

Vc~

which must have the symmetry ^of ^the two-dimensional

point

group 4 _mm _:

VCR(x, y)

₌

VCR(r, ~ )

=

Ar~

₊

Br~

cos 4

~

+

Cr~ (2)

Our basic

assumption

is that

Vc~(x, y)

^is an effective

potential

which contains all time

independent

contributions due _to

neighbours acting

_upon the reference

NH~

_group.

The

geometric quantities

_are defined in

figure

5. This _ansatz is _exact up to terms

~

r~.

_The confinement of the molecules is

guaranteed

with C

»

[B [.

^For ^Am ⁰

Vc~(r, ~ )

shows

only

_one minimum at

ro=0,

^but for A<0 four absolute minima at

r~ =

~/[A[/(C _[B[ ₎

_and _saddle

_points

_at _r~

=

~/[A[/(C

₊

_[B[ ₎

_occur. _For B »0 the minima _are found at

~~

= ±

45°,

± 135° and the saddle

points

at

~~

=

0,

±

90°,

while for B _< 0 the minima _are at

~~

=

0,

± 90° and the saddle

points

at

~~

= ±

45°,

± 135°.

An additional

r~

_term which _was used earlier

[13]

^for

stability

reasons has tumed out to be unnecessary in the whole series of

compounds NiY~X~

with Y

=

NH~, ND~

and X

=

Br, I,

NO~, PF~.

After _we

dropped

^the

^r~

contribution the

strong

correlations among the

potential

parameters ^decreased

appreciably.

The

stability

condition C _»

[B

^is

always obeyed by

the

computed

B's and C's

[18].

One orientation of the proton

Wangle

^is described

by

the distance from the _center of _mass

(c.o.m.)

to the four-fold axis

R~, by

the

polar angle ~~

of R~ ^and ^the rotation

angle fl

of the whole group. The effective

potential

for _one orientation of the

proton Wangle

is

given by

the

sum of the

single particle potentials

of each proton

v(xs,

_Ys,

p )

~

=

v(Rs, ~s, p )

=

z vc~(rm, ~m). (3)

The

explicit

calculation leads

finally

to

v(Rs, ~s, p )

₌

v°

₊

v~(Rs)

₊

VW (Rs, ~s, p ) (4)

with

V°

= 3

Ad~

+

Cd~j ⁽⁵⁾

V~(R~)

= 3

)A

+

Cd~) ^R(

⁺

CR(j ⁽⁶⁾

V~(R~, ~

~,

fl

= 3

Bd~

R~ cos

(4 ~~

₊ 3

fl )

+

BR(

_cos 4

~$

j (7)

4

V°

_is

a zero

point

shift of the

potential

which is irrelevant for any

dynamical

and

thermodynamic

considerations. The second term

V~(R~),

_can be

imagined

_as the

angular

average of the effective

potential (4).

From

(6)

^it becomes evident that this

angular

_average of

(12)

Ni(NH~)~I2

997

x

Fig.

5. Geometric

quantities defining

the model.

V(R~, ~~, fl)

contains _a hamonic contribution _even if the

crystal potential

is

purely anharmonic,

I-e- A

=

0. The third _term

V'~(R~, ~~, fl)

_represents

explicitly

the rotation translation

coupling,

and it shows

clearly

the

periods

of 90° in terms of

~

and 120° in _terms of

fl

_: reference to

crystal

and molecular

symmetries, respectively.

One may do the

angular averaging

of

V~(R~, _{~~, fl )}

in _two steps : over

fl

and/or _over

~~.

^In ^the ^first case we are left with _an

oscillatory

term of the center of _mass motion

~R(,

in the latter _case the

angular dependence dissapears.

The Boltzmann

probability density

for _one

setting

is

given by

P s, ~, ⁼

i ~- ^v^{(R~, ~} ^{p y~~}

~~ ~ ~

~ ~

and _as shown in

[13]

the two-dimensional proton distribution function _p

(x, y)

is derived _as

~

~~'

_Y

" 3 Z- ^I e

~iR(x,yykT ~_ ~~j~ _~ _~~~~~~~

~_Y

with

V[~

=

Vc~(x

+ s cos y, y + s sin _y

) ( lo)

viR

=

vcR(x

+ s cos

(y

+

«/3),

_y ₊ _s sin

(y

₊

«/3)) (11)

where _s

=

,fi

_d _and

_V[R, _V[~

_and

_V(~

_refer _to _the _first, _second _and _third _proton.

The

partition

function is defined _as usual _:

Z

=

j exp(-

V

(R~, ~~, fl )/kT)

dr

(12)

The proton ^number

density

_p

(x, y)

in

crystal

_space

given

in

(9)

_may be

interpreted

_as

being given by

the Boltzmann

probability

due _to _a

proton

^located at

(x,

_y

multiplied by

the average

Boltzmann

probability

of the _two other protons on a circle with radius _s around

(x, y).

4.2 POTENTIAL REFINEMENT. We took the two-dimensional section

through

the

hydrogen

density p$~(x, _y)~

~o~~ as

input

data for _our

potential analysis.

Refinement of the

potential

JOURNAL DE PHYSIQUEI T 3, N' 4 APRIL 1993 ~~

(13)

parameters A, B,

C and the distance d from the

protons

to their center of _mass

finauy

leads to

p~~~(x, y)

shown in

figure 6a,

b. To make _a

comparison

between the observed and the calculated

hydrogen

densities easier one-dimensional sections _are

given

in

figure 7a,

b.

'~~

_0.14

-0.14

-0.14 _-

x 0.14

b)

_0.14

'°.~jo,14

- x °'~~

Fig.

6. Calculated nuclear

hydrogen density

at z = 0.23 ( 0.14 _« _x, y «

0.14).

a) T

=

295 K, b) T

= 35 K.

(14)

Ni(NH3)612

999

a)

,-,X

,-- ~,__

'

"

,

, ' '

-, '_'

g

~ x x x

cafe.

Z diw.

-0.2 -0.I O-O O-I _- xla

b)

x

x~''~x

f

~ x x

I calc.

dill.

-0.2 -0.I O-O O-I _- Vfi xla

Fig.

7. Sections

through

the observed and calculated nuclear densities. a) T

= 295 K, along the _x- axis, b) T

= 295 K, along the

diagonal,

c) T

=

35 K, along the x-axis, d) T

= 35 K

along

the

diagonal.

We

point

out the different scales

applied

for the densities in (a, b) and (c, d)

respectively.

(15)

C)

-, ~, ,-

' -'

'~~x'

^" '

W ' '

t ''

C

~

fl x x x

j

_caJc.

Z diT.

-O.2 -O.I O-O O-I _- xla

d)

'X

' ,

~,, '

,' ^-'~

,' ^' ,X ^X' _, _, ',

'" ,~X~'

,,

~

g

caJc.

~ -

O.2 ---*-V5

Fig.

7 (continued~.

(16)

Ni(NH~)61~

joey

The final

parameters

and _some related

quantities

of the

potential

refinement _are

given

in table IV. The constant c takes into account differences between the actual

averaged scattering density

and the

averaged scattering density referring

to the

non-hydrogen

atoms.

For

comparison

_we calculated R-factors

specifiqally

for the two-dimensional nuclear

density

derived from the SHELX refinement of model

I(R~~~

=

0.206,

_R$~~

=

0.193,

^R~~ ₌

0.152, R(~

₌

0.128).

It is clear that _even

though

the discrete site Model I _uses _a

larger

number of

parameters

^than ^the

potential

model the latter achieves better

agreement

with the observations.

One should

keep

in mind that the R-values

quoted

here refer _to the pure

hydrogen density.

They

must not be

compared

^with the

corresponding

numbers

given

in tables II and III which describe the agreement ^between ^the ^models ^for ^the ^total ^structure ^and ^the ^data. ^The reason for

separating

the

hydrogen density

from the total

density p~°~~'(x,

_y,

z)

lies in the

fact,

that

p~(x,

_y,

_z)

is

only

a minor part ^of

p~°~~(x,

_y,

z),

and _so any deficiencies of the models

describing

_p

~(x,

_y,

z

)

will contribute

only

ⁱⁿ a minor fashion _to the total R-values of tables II and III.

Table IV. Results

of

the

potential refinement.

N N

£ £ (p$s(Xi'Yj)~pilc(Xi'Yj)(

~~~

^~~

N N

£ £ (Pis(xj, _Yj)(

i=lj=I

N N

£ £

_Wj

j

(p is _{(Xi, yj )}

_p

Sic (Xj, yj )

)~

R~

₌ ^'" ^~

~ ~

£ £

_Wi

j

(p is

_(X~,

_yj

_)~

i I j I

T K 295 35

scale

1.304(1) 5.45(1)

c

0.043(1) 0.065(1)

d

h _0.9594(4) _0.9593(4)

A

K/h2 _232.3(2) 86.2(1)

B

K/h4 _166.0(1) _32.9(2)

C

K/h4 _776.I(1.8) 234.6(6)

R 0.070 0.123

R~

^0.059 ^0.106

r~

h

_0.617 _0.654

VCR

(r~,

45°

)

K 22. II 9.21

rs

h

_0.497 _0.568

Vc~(r~, _o)

K 14.32 6.94

R$ h

_0.123 _0.084

V°(0, _4~, _{fl )}

K 172.4 30.0

V~~(R), 45°, 0)

^K ^145.6 ^26.4

V~;~ (R), 45°,

60°

)

^K ^253.8 ^41.0

(17)

At both

temperatures

^the

single

article otential

(Fig. 8)

shows the four

expected

minima at

#

= +

45°,

+ 135° with _r

=

~~

= 0.617

h _{(295 K)}

_and _r~

= 0.654

h _{(35 K).}

_For

~

the motion of the

NH~-group

the total

potential

V

(R~, #~, fl )

as defined in

(4)

^is needed. In

a)

loco K

K

~

-0.14 -0.14

_ x 0.14

~~

0.14

~

-0.14

-0.14 _- x 0.14

Fig.

8.

Single pwticle potential,

a) T

= 295 K, b) T

= 35 K. Level lines _are spaced

logarithmically.

The insert shows the gear wheel motion

explained

in the text.

(18)

Ni(NH~)~I2

1003

figure 9a,

b _we show V

(R~, #~, fl )

for the

particular

value R~ ^at ^which ^the absolute minimum of V

(R~, 4~, fl )

=

V~~

occurs. For both

temperatures

an absolute minimum _was found at the

same

#~

and

fl values,

I-e-

#~

= 45° and

fl

= 0°. The

corresponding

distances of the center of

a)

l146

K

b)

K

Fig.

9. The total

potential

V(R~, #~, p for _one

NH~

molecule. a) T

= 295 K, R~ = 0.123

A,

b) T ₌ 35 K, _R~

= 0.084

A. Actually

the support for

V(R~,

_~b~, p ) is _a torus.

(19)

mass from the four-fold

axis,

however, are different with

R)

= 0.123

h

at room

temperature

and

R$

=

0.090

h

_at ₃₅ _K,

as could be

expected

^from ^the temperature

dependence

of the

anisotropic

_u~~

= u~~ parameter ^of

nitrogen.

As shown in

figure 9a, b,

this absolute niinimum is

only weakly pronounced compared

to the

valleys

which _are

given by

#~

₌

(2

n

+1)

45°

3/4 p

with _n

integer.

A

simple

rotation of the

H3 triangle

^around ^the

four-fold axis (R~ = 0

)

would

imply

_a constant

potential ^V°.

From the clear difference between the calculated

V(gs

~ ⁼ l 72.4

K, V(s

~ ⁼ 30.0 K and

V[$]

~ ⁼ 145.6

K, Vfl[

=

26.4 K follows

a

coupling

of rotation-translation motion.

5. Discussion.

Inspection

of

figure

9 shows broad

valleys

for

(#~, p

for _c-o-m- distances R~ ^close to the

lowest values of

V(R~, #~, ^p).

^The ^molecule may move

along

these

valleys

without

appreciable

_energy barriers if the molecular and the _c-o-m-

angles

_are

coupled

in the ratio

p/4~

= 4/3 _: this motion

corresponds

to a toothwheel with 3 teeth

(the molecule) rotating

without

slippage

in _an intemal

geared

wheel

having

4 teeth

(the

^local

environment).

This is indicated in _a

symbolic

fashion in the inset of

figure

8.

From _a _recent molecular

dynamics study [18]

there is evidence for _a motion at constant

energy close to, but not

exactly

at the bottom of the

valley.

This type ^of ^motion ^needs no

thermal

activation,

and therefore is very

likely

to be the dominant _process of rotation- translation

coupling

of the model _as well _as of the realisation in

Ni(NH~)~I~.

The pure 120°

jumps

of the

NHjND~

_groups ^referred to in the literature _are hindered

only slightly by

barriers of

= 90 K at _room

temperature,

if

they

_occur at R~ = 0.12

h.

An open

question

is the temperature

dependence

of the

potential

found for the first time

(Tab. IV, Fig. 8)

_: Is it real ? There _are indeed at least two

possible explanations.

The first would be _; the effect is due to the combined action of the thermal contraction of the lattice and the reduced thermal

amplitudes

of the

NH~

_groups as well _as the

neighbouring

I ions. The

observed reduction of the

potential

constants

by

_a factor of

~

0.3 for

A,

C and _a factor of 0.2 for C _on

cooling

from 295 K _to 35 K make this

explanation

rather

unlikely.

There is,

however,

_as an altemative

explanation,

_a strong argument

pointing

towards quantum ^statistics as the

origin

of _our observation. We shall

develop

this argument ⁱⁿ

analogy

with _a harmonic oscillator.

Assume _a ld harmonic oscillator

having

the

potential

V

=

Ax~.

In the Boltzmann

(high

4

temperature) approximation

the _mean square

amplitude

is

(x~)(

₌ ^kT/A

,

(13)

so that _we may determine the force constant A from the

observations,

_e.g. the

Debye

Waller factor from

A

=

kT/

(x~)( (14)

For low

temperatures,

^however,

(13)

has to be

replaced by

j~2) ) (15)

=

x(

Coth

(~~ ~

with

xl

=

h/Mw and _w ²

=

AIM.

If _now

(13)

where used

erroneously

in the low

temperature

^domain ^with

(x~)

~ ^m

(x~)

(~P. the

(20)

Ni(NH~)~I~

1005

resulting

force constant would become temperature

dependent

A~~

=

~

~l

tgh (hw/2 kT) (16)

xo

A~~~ is shown in

figure

10 _: at

high temperatures

_A~~~ ₌ ^A. In the quantum ^statistics

domain, however, (x~)(~P.

^shows ^the ^width ^of ^the

^ground

^state wave function

which,

if

interpreted

in

terms of

(13)

simulates _a

softening

of the

potential.

We have observed and

prooved

this kind of

quantum

^effect ^earlier ^{in the} case of _an anharmonic _rotator

(NH4

in

(NH4)~SnCl~ [19]).

In the latter _case the

NH4

units _are well

separated

in the lattice. The

importance

of

using

the

quantum

statistical

expression (15)

for the calculation of harmonic force _constants from low

temperature

data has been

pointed

out earlier

[20].

kT

O, 1. 2. ^~ ^kT

Fig,

lo. The effective

spring

constant A~w ^calculated ^from (16).

A calculation of the 35 K

density

ⁱⁿ

Ni(NH~)~I~ using

^the Boltzmann formalism

(8)-(12)

and the _room temperature

potential

shows 12

peaks [18]

instead of the four observed in

figure

4b _: the calculated 12

peaks

_are well

localised,

in contradiction with what _one may expect ^from

quantum

statistics. For _our present

example

_a

complete quantum

statistical

analysis

is _more

complicated

because translation and rotation _enter into the

play,

and less

straightforward

because the 2d model of 6

independent NH~

units for each octahedron is _an

approximation

which is

certainly

_more valid _at

high

than _at low temperatures. ^The calculations _are under way and will be

reported

later.

This

investigations

was

supported by

^the ^BMFT under contract _no. 02-PR2-TUE.

(21)

References

ill ECKERT J,, PRESS W., J. Chem.

Phys.

73 (1)

(1980)

451-460.

j2] BATES A. R., STEVENS W. H., J.

Phys.

C 2 (1969) 1573-1585.

j3] ^BATES A. R., HUGHES S. H., SOMMERFORD D. J., J.

Phys.

C 16 (1983) 2847-2859.

j4] JENKINS T. E., FERRIS L. T. H., BATES A. R., GILLARD R. D., J.

Phys.

C _: Solid State

Phys.

lo (1977) 521-526.

j5] JENKINS T. E., FERRIS L. T. H., BATES A. R., GILLARD R. D., J.

Phys.

C _: Solid State

Phys.

ll (1978) 77-81.

j6] BATES A. R,, FERRIS L. T. H,, JENKINS T. E., J.

Phys.

C _: Solid State

Phys.

12

(1979)

2945-2955.

j71 KEARLEY G. J., BLANK H., COCKCROFTK., J. Chem.

Phys.

86 (1987) 5989-5996.

j8] JANIK J. A., JANIK J. M., MIGDAL-MIKULI A., MWULI E., Acta

physica

polonica A 74

(1988)

423- 43l.

j9] ECKERT J., YOUNGBLOCD R., Solid State Commun. 44(9)

(1982)

1393-1396.

j10] HOSER A., JOSWIG W., PRANDL W., VOGT K., Molecular

phys.

56(4) (1985) 853-869.

j1II HOSER A., PRANDL W., SCHIEBEL P., HEGER G.,

Physica

B 156 &157

(1989)

85-87.

ii 2] HOSER A., PRANDL W., SCHIEBEL P., HEGER G., Z.

Phys.

B-Condensed Matter 81 (1990) 259-263.

ii 3] SCHIEBEL P., HOSER A., PRANDL W., HEGER G., Z.

Phys.

B-Condensed Matter 81(1990) 253-258.

j14] GMELINS, Handbuch der

Anorganischen

Chemie

(Verlag

Chemie).

j15] SHELDRICK G., SHELX-76, A

Program

for

Crystal

Structure Determination (1976).

j16] Handbook of

Chemistry

and Physics, R. C. Weast Ed. (Cleveland/Ohio : CRC-Press, 1986).

j17] HAMILTON W. C., Acta Cryst. 18 (1965) 502-510.

j18] SCHIEBEL P., Dissertation,

Tiibingen (1992).

j19] BRUCKEL Th., PRANDL W., VOGT K., ZEYEN C. M. E., J.

Phys.

C Solid State

Phys,

17 (1984) 4071-4088.

[20] TRUEBLOOD K. N., DuNm J. D., Acta

Cryst.

B 39 (1983) 120-133.