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STRUCTURAL AND DYNAMICAL STUDIES
OF

LAYERED SOLIDS

BY

YIBING FAN

8.3., Peking University, 1983

DISSERTATION

Submitted to partial fulfillment of the requirements For
the degree of Doctor of philosophy in Physics

in Michigan State University, 1988

East Lansing, Michigan

A8$TRACT

STRUCTURES AND DYNAMICS STUDlES

OF LATE'HED SOLIDS

YiBihg Fan, Ph.D.

Department of Physics and Astronomy, 1988

The structural and dynamical properties of graphite intercalation com-
pounds and vermiculate clays intercalated with trimethylammonium were
studied by x-ray scattering, elastic,inelastic and quasi-elastic neutron
scattering.

He have measured the in-plane and c-axis elastic neutron scattering
from the ternary potassium-ammonia graphite intercalation compound
K(ND3)u.3C2u at room temperature. The c-axis scans establish that the
graphite layers stack in an eclipsed configuration and that there is no cor-
relation between the intercalant layers. The in-plane diffuse scattering
from the intercalant layers is well accounted for by a computer-generated
structural model in which each potassium ion is symmetrically l1-fold coor-
dinated to ammonia molecules to form closely-packed, flve-membered clusters.
This model had been previously applied to explain the in-plane diffuse X-ray
scattering results from K(NH3)11.38C211' Discrepancies between the calculated
and measured diffraction patterns are attributed to a relaxation of the ‘1-

fold symmetry of the K-ND clusters.

3

The results of an inelastic neutron scattering study of the [001] L

lattice dynamics of stage-1 potassium-ammonia graphite compounds K(ND3)KC9A

mm x = 3.1 and 1L3 are presented in chapter ill. The results snow that”. a
llbratlonal mode splits the acoustic phonon branch of these compounds at an
energy of 7 mev. The mechanisms of this coupling are discussed based on a
force constant model and the split phonon dispersion curves are described by
a linear chain model coupled with a transverse rotational oSCLllation of the
potasSlum-ammonia complex. In addition, tne interlayer force constants in
the compound are found to be appreciably softer than in the stage-2 binary
counterpart.

The quasi-elastic neutron scattering results for K(NH )XC H, which show

3 2
the coexistence of molecular rotational and translational diffusion of the
ammonia are reported in chapter IV.

For clays, neutron scattering has been used to study the basal spacing
and vibrational excitations of high oriented samples of (CH3)3NH*-
Vermiculite and its deuterated form (CD3)3ND+-vermiculite. Both forms
exhibit a basal spacing of 12.71A and a rich vibrational spectrum in the
range 20-1140 mev for Q perpendicular and parallel to the c-axis. Peaks at
28 and 37 mev are assigned as tortional modes of the methyl group, while the
peak at 51 mev is identified as a 6(C3N) bending mode. The peaks above 80
mev are assigned as the inplane optical modes of the silicate sheet. These

results are compared with infrared measurements over the same energy range

and a good agreement is obtained.

Acknowledgements

It was very Joyful for me to spend my last 5 years at erfliigan State
University. Many people have made significant contributions to the success-
flu complethxiof this thesis. Some, however, are deserving of special
mention.

First of all, my advhmny Stuart A. Solin, has spent countless hours
with me over the past five years for which I am very grateful. He introduced
me to this research field and his rich experiences guided me through every
stage of this work; ii truly appreciated his frank criticisms and other
generous help in the time of need.

I would like to thank my colaborator Dr. D.A.Neumann who has taught me
a great deal about neutron scattering. I have also appreciated his generous
hospitality and his guidance during my visits to the National Bureau of
Standards for neutron scattering experiments.

I have benefited much from working with my fellow students who have
worked in Dr. Solin's group over the years. Xiao-wen Qian has made special
contxdlnltions to my understanding of x-ray scattering and of physics when I
first joined the group. Yiyun Huang has always been helpful whenever I need
a favmn~, and was particularly helpful with sample preparation. I also thank
S. Lee, and P. Zhou for many interesting discussions.

The clay sample for chapter V were made in Professor Pinnavaia's
laboratory. I really appreciated H. Kim's patience in helping me sythesize
the clay samples.

Interactions with those people mentioned above made my stay at Michigan
Stattaihiiversity very joyful and this author will remember my MSU years for

a long time to come.

iii

Finally, research is not free. Financial support from the National
Science Foundation under grant DMR-85-17223 and from the MSU Center for

Fundamental Materials Research are also acknowledged.

1v

Table of Contents

Chapter I. Overview ..................................................... 1
1.1 Graphite Intercalation Compounds ......................... 1
1.2 Neutron Scattering ...................................... 1A
1.3 Neutron Scattering Instruments .......................... 26
References .................................................. 37

Chapter II.

Elastic Neutron Scattering Studies of the Structure of

Graphite Intercalated with Potassium and Ammonia .......... UAD
2.1 Introduction ........................................... A0
2.2 EXperimental Methods and Results ....................... A2
2.3 Calculations and Analysis .............................. SO
2.N The Structural Model ................................... 57
2.5 Calculating the Diffaction Pattern ..................... 60
2.6 Sumary and Conclusions ................................ 66
References ................................................. 68

Chapter III. Inelastic Neutron Scattering Study of Stage 1 and Stage 2

Chapter IV.

K-Anmon ia Intercalated Graphite ........................... 70
3.1 Introduction ............................................ 70
3.2 Experiment and Results .................................. 72
3.3 Discussions ............................................. 82
3.” Conclusions ............................................. 98
References .................................................. 99

Quasielastic Neutron Scattering Study of Rotations and

Diffusion in Stage 1 K-Ammonia Intercalated Graphite ...... 102
4.1 Introduction ........................................... 102
“.2 Translational Diffusive Motion ......................... 102
".3 Translational Jump Diffusion ........................... 103

Chapter V.

A.“ Rotational Jump Diffusion .............................. 106

4.5 General Case ........................................... 110
A.6 Experiment ............................................. 111
4.7 Results and Discussions ................................ 112
References ................................................. '3o

Neutron Scattering Studies oi Trimethyiammonium Vermivulite

Clay Intercalation Compounds ............................ ..129
5.1 Introduction ........................................... ’29
5.2 Experiments ............................................ 132
5.3 Results and Discussions ................... . ............ 137

Elastic Scattering ...................... ...............137
Inelastic Scattering ...................... .............147
5.“ Conclusions.................. .......... ................152
References......................... ....... .................153

v1

Chapter I. OVERVIEW

For last two decades, one of the major themes of condensed matter
physics has been the study of the low dimensional, particularly two-
dimensional, phenomena. To explore the fascination of two-dimensional
physics, many efforts have been made on layered materials, such as superlat-
tices, graphite intercalation compounds, boron nitride, transition metal
dichalcogenides, and layered silicates. As a mater of fact, advances in the
synthesis of layered materials have led to many exciting discoveries,
including the Quantum Hall Effect1 for which Klaus von Klitzing won the 1985
Nobel prize in physics. Additionally, the anisotropy of layered solids en-
riched the study of effects of dimensionality in physics. In fact, most
efforts have centered around the dimensional cross-over phenomena, i.e.,
from one to two dimensions, and two to three dimensions. Moreover, such
studies have led to some practical new devices such as the MOSFET. Thus,
this author believes that the study of low dimensional- phenomena will con-
tinue to be a major focus for. .a long time to come as an area of basic

physical phenomena.

1.1 Graphite Intercalation Compounds
The family of layered materials, can be divided into two general
categories, naturally layered materials and deliberately structured layered
materials. The former are characterized by strong intralayer interactions
and weak interlayer interactions. Examples of this type of material are
graphite, transition metal dichalcogenides, some silicates and metal
chlorides, clays, polymers and gels. The latter are typically prepared by

advanced techniques through a controlled layer by layer deposition of the

constituent species under high vacuum conditions and computer control such
as sputtering, metal organic chemical vapor deposition (MOCVD) and most inn-
portantly, molecular beam epitaxy (MBE)2. Since our interest is on the
graphite intercalation compounds and clays, I will only briefly mentitni the
studies of other layered materials with the stress on the structural and
dynamical properties. However, the connection between graphite intercalam»
tion compound (GIC) and clay intercalation compound (CIC) to the other
layered materials will be noted where applicable.

During the past century graphite has become one of the most well
studied solids known to physicists and chemists. It has a prototypical
layered structure which is shown in Fig. 1.1. The carbon atoms are
covalently bonded into a hexagonal lattice plane with a C-C distance of
luAZA, and the planes are then bonded by Van der Haals interactions with an
ABAB-H or ABCABCH stacking period of 6.70A along c-axis direction. The
anisotropy engendered by the strong covalent intraplanar bonding and the
weak van der Haals interplanar bonding is clearly reflected in the in—plane
versus the out-of-plane atomic spacings. The weak van der Haals bond can be
easily broken by charge transfer to or from the graphite carbon layer with
the resultant formation of the so called graphite intercalation compounds
(GIC)3-u. It has been reported that more than a hundred chemicals have suf-
ficient chemical activity to form GIC's3. Depending upon the charge
transfer to or from the host, the GIC is classified as a donor compound if
charge is transferred to the host while it is an acceptor compound if charge
is transferred from the host. Some of the chemical species that can form
GIC's are listed in Table 1.1 under the category of donor and acceptor.

An unusual characteristic of GIC's is the phenomenon‘5 of staging in

which a periodic sequence of host galleries is filled with guest species. A

 

 

The structure of pristine graphite.

.1

1

Fig.

 

 

 

 

Donors - Acceptors
u, K, Rb, cs, Ba, Eu, Yb, NiClZ,CoC12, MnClz, 5503'
Sr, Sm, Ca Ast, H1403, sac15

 

 

Table 1.1 Some examples of donor and acceptor intercalants of graphite.

stage n compound is one in which n carbon layers separate two adjacent in-
tercalants layers as shown in Fig. 1.2. Staged structures with n up to 10
have been reported for alkali-metal GIC's3. The physical basis for the
staging phenomenon6 is the strong interatomic intercalant-intercalant inter-
action relative to the intercalant—graphite interaction, thereby favoring a
close-packed in-plane intercalant arrangement. The introduction of each in-
tercalate layer adds a substantial strain energy as the crystal expands to
accommodate it, thereby favoring the insertion of a minimum number of inter-
calate layers, consistent with a given average intercalate concentration.
In addition, the ionized intercalant produces a static electric field giving
rise to a long range but screened repulsive interaction between intercalants
in different layers, also favoring their maximum separation. Thus for a
given intercalate concentration, the minimal energy state corresponds to a
close-packed in-plane intercalate arrangement with the largest possible
separation between a minimum number of intercalant layers. This collective
interaction results in the staging phenomenon. The fact that the<xuy
stage—1 intercalated clays have been reported is probably due to relatively
strong ionic bonding between intercalant and host. The situation for the
intercalate transition metal dichalcogenide, where only stage-1 and stage-2
have been reported, is between the above two cases. The staging phenomenon

undoubtedly enrichs the study of layered materials.

The most heavily studied GIC system is the alkali-metal intercalation

compounds. These donor compounds are easily synthesized and display a

variety of interesting physical phenomena including staging disorder7-8, two
. . . 9-11 . . 12-13 .
dimentional melting and difquion , complex in-plane
1A-16 17-19

discommensuration-domain structures , and superconductivity The

181 STAGE 2nd STAGE 3rd STAGE 41h STAGE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

A A ~ A ~ A a
e... 0000-, 0000? eeeeq

A A — A ~ A _
eeee

A B 8 ~ 8 —

eeeeJ

eeee a - * A

A eeee-la

A ‘ 000. ....d

 

 

 

' CARBON HEXAGON LAYER

e e e -, LAYER or INTERCALATE
. (ALKALI new. ATOM)

. Fig. 1.2 A schentic representation of phenomenon of staging. The symbols C
and 6 represent carbon and quest layers, respectively. The solid lines
represent the carbon layers and indicate the lateral displacement of the

carbon atoms (solid circles). The dashed lines represent the intercalated

layers.

so called two—zone method for synthesis20 of the alkali-metal compounds is
shcuui in Fig. 1.3. Fig. 1.“ shows the K/C ratio vs. temperature difference
between the two zones. The plateaus in the diagram correspond to pure
stages. As can be seen, the higher the stage, the narrower the temperature
range over which it exists. This means that the synthesis of pure high
stage samples is more difficult than that of low stages. Although stages as
high as stage-1O have been reported, these are not likely to be tnnwe stage
materials and no doubt consist of an admixture of several stages7. The
relative temperatures of the zones for different alkali metals and different
stages are listed in table 1.2.

Recently, solid state physicists have shifted their attention ihknn the
binary compounds (BGICs), in which a single chemical species is intercalated
into the host, to ternary compounds (TGICs) which contain two different
intercalants. The ‘I‘GICs dramatically extend the range of composition which
can be studied at constant stage and, therefore, enhance the variety of‘
properties and phenomena that can be explored using these layered solids.
These TGICs can be generally divided into four typesZI, namely: homogeneous,
heterogeneous, trilayer and localized 'l‘GIC's (Fig. 1.5). The structures
shown in Fig. 1.5. are limited to the stage-1 stacking sequence and repre-
sent;cNily the most highly ordered inplane structure in each type. Several
other structural forms within a given category are possible since the par-
ticular structure which a given TGIC exhibits will depend on external
parameters such as pressure and temperature. For instance, K(NH3)A.3C24 is
a type I TGIC, it exhibits a disordered in-plane structure at room tempera-
ture and shows an ordered hrmflane structure below 185 K. We now discuss

the four types of TGIC individually.

Alkali Metal HOPG Sample

 

 

 

\\\\\\\\\\\\\\\\\\\\\

 

TA W

9-!

Fig. 1.3 A schentic diagram of the two-zone method used to make alkali-

graphite intercalation compounds. TA is the alkali temperature and T is the

G
graphite temperature, and N is the heater wire.

 

kca

 

 
     
    
 

K024 _

KC“

0|
l

K/C (x104)

 

 

 

 

L L L
ICC 200 300

Fig. 1.“ Isobar for growth of graphite-K compounds showing Intercalant

Uptake versus temperature difference TG-TA between graphite and intercalant

(From ref. 20). Note the temperature ranges for staging stability, as

indicated for different stage compounds.

10

 

K Rb (.3
T9,: 250%. TA: 208°C TA ='19-i-:C'

 

Stage T,;(°C) are) arr)

 

225—320 2 15-330 200—85
350-400 375-430 475-530
450480 450480 550

color--

 

Table 1.2 T“ and T6 for preparation of alkali metal compounds with stages

1<n<3,

 

 

 

 

 

 

HOIOOINIOUO HITIIOOINIOUS
ln-Piane c-Axla
Subofltutlenal Subatitutlenal
O O O O O O
O O O O O O
O O O O O O

 

 

I].

TRILAYII
c-am
Interstitial

LOCAUZED

Island
Substitution“

l
1
ll
1
ll

 

l]

 

 

 

 

[]

ll
ll
ll

(d

Fig. 1.5 A schematic representation of the four basic types of ternary

graphite intercalation compound. The sol id lines represent the carbon layers

and indicate the lateral displacement of the carbon atoms (small solid

sircles). The large open and closed circles and rectangles represent inter-

ca la ted species .

12

The type I TGIC is the homogeneous (Fig. 1.5a) form in which the mixed
intercalant layers in each gallery are indistinguishable from one another.
The two intercalating species form a monolayer in the same gallery axui they
can form commensurate, incommensurate, discommensurate ordered structures or
disordered structures depending upon the external parameters and the lJlteP-
nal interactions in the compound. Non-crystalline layers can also exhibit
both compositional and topological disorder. In the compositional disorder
case, the hexagonal sites of the graphite layers are randomly occupied by
species A or B for which the terms "lattice-gas" or "lattice-lituiid"
apply22. In the tapological disorder case, the ZD structure of the interca-
late layers may be that of a binary liquid or even a glass both of which may
exhibit short-range correlations. The lateral positional c-axis correlation
between layers in a type I TGIC may be long range ordered or short range or-
dered or even uncorrelated as is the case for a lattice-gas/liquid or glass.

When the two intercalating species are confined to different galleries
in the host structure, the second type of TGIC is formed, i.e.,. the
heterogeneous structure shown in Fig. 1.5b. As in the case of the
homogeneous TGIC's the most highly ordered form is a 30 superlattice. There
are derivative structures of lower order in which the intercalate layers are
incommensurate, or topologically disordered. A new form of disorder, the 1
dimensional lattice-gas/liquid, may result from a c-axis random stacking
sequence. In addition, many other variations are possible. As is the case
in the homogeneous TGIC's, the c-axis correlations in the heterogeneous form
Span the gammut from infinite range to very short range order.

When the two intercalating species form separate layers within the same
gallery, the resultant compound is referred to as a trilayer TGIC. The most

highly ordered form is 3D superlattice structure shown in Fig. 1.50. Many

l3

derivative structures from that shown in Fig. 1.5a are possible and the c-
axis correlations will vary correspondingly. Since the detailed discussion
of each of these structures is not of import in this thesis, and the reader
is referred elsewhere for this information21

The last type of TGIC depicted in Fig. 1.5a is the localized TGIC. In
this type of TGIC , each intercalating species appears to be a single phase,
and they are phase separated structurally or chemically into two distinct
entities or islands. In principle, the two types of islands may have 'a 3D
superlattice form in analogy with the other TGIC's and will also have many
deviative structural forms and their associated c-axis correlations.

The TGICs no doubt broaden the field of GIC study as noted above. One
of the important features of the TGIC is the ability to vary the composition
at constant stage. The BGIC's, though simple in their structural and chemi-
cal properties and in the ease with which they can be synthesized impose
intrinsic limitations on the types of measurements and phenomena which can
be explored using them. For instance, the staging phenomenon which is so
fundamentally intriguing gives rise to a relatively narrow range of composi-
tions over which a given stage can exist in BGIC's. However, this intrisic
limitation encountered in 3610's can be eliminated if one considers the
TGIC's. Because TGIC's can contain two species in the same gallery, the
stoichiometry of one of them, relative to the carbon atoms, may be held con-
stant while the stoichiometry of the other may be varied over a large range
without altering the stage of the GIC. The composition "handle" makes
TG IC's an ideal arena for studying order-disorder and order-order structural
phase transitions in a 2D binary alloy in analogy to the famous studies of
3d Cu Au 23. The prototypical study of this problem can be addressed by

3
forming the dual-alkali TGIC's system "1-1;“.ny where M and M' are heavy

14

alkali metalszl4 where y = 8 for the stage-1 compound and 12h, n 2 2, for

higher stages, and 0<x<1. Of’further significance is the fact that the

structure of the mixed intercalated layer in at least one TGIC K(NH3)u 3C2“,
is that of'a simple 2D liquid25"27 instead of the complex domain structures
28

of the BGIC's. This system also provides a testing ground for 2D kinetic
studies.

The TGIC'S not only expand the range of structurally based properties
and phenomena which can be probed with graphite materials, but also provide
new opportunities for electromagnetic investigations. With TGIC's it is
possible to prepare mixed donor-acceptor compounds which exhibit intriguing

behavior. For example, K(NH )xCZA exhibits a constant stage composition de-

29

3

pendent metal-insulator transition Another TGIC which contains an

admixture of magnetic and nonmagnetic species displays very unusual magnetic

30

phenonmena

112 Neutron Scattering31'3u

The neutron scattering technique has been one of the most powerful
tools in condensed matter physics because of the basic properties of‘
neutron.

The energy E of a neutron with a wave vector k is

E = h2k2/2m (1.1)
where m is the neutron mass. Using m = 9.383x108ev, we have
(ha/2m) = 2.08 meV A2 (1.2)
From this equation we see that a neutron with a wave vector k a few A"1 pos-

sesses an energy of about a few meV which is the typical energy of a lattice

vibrwation and of spin excitations in magnetic insulators. If we rewrite the

15

energy in terms of the neutron de Broglie wavelength, we have the following
relation:

l = (h2/2mE)1/2 = 9.0ue‘1/2AA (1.3)
where E is in units of meV. From this formula we see that a neutron with
energy E = 25meV has a wavelength A = 1.81 A which is on the order of the
interatomic distance in condensed matter. In consequence, the diffraction
of neutrons from condensed matter can display pronounced interference
effects.

There are other energy units frequently used in neutron spectroscopy.

We list the conversion of them for convenience.

1sz = 1012 s'1 = n.1umev
1meV = 0.2u THz = 8.07 em“ = 11.61 K (1.u)
1 THz = 33.35 cm‘1 = u8.02 K.

The penetration depth of neutrons in matter is very long, because they
are uncharged, and are only scattered by nuclear forces. Therefore, neutron
scattering is well suited to the study of bulk properties. The nature of
the nuclear force results in a random variation of the scattering cross sec-
tion for various nucleii, in contrast to the smooth dependence of the
scattering cross-section an atomic number as in the case of x-ray
diffraction. The scattering length for different isotopes can be very dif-

ferent, for instant, hydrogen has a scattering length of -.37 cm'12, while

the deuteron has a length of 0.667 CID-12. This fact provides us an ad-
vantage for many neutron scattering studies.

In a neutron scattering event, there are two conditions that have to be
Satisfied, namely, energy conservation and momentum conservation. The con-

servation of energy requires that the energy transferred to the target be

equal to the energy loss of the scattered neutron, i.e.,

16

ha = 5-51 = (h2/2m)(k2 -k'2

) (1.5)
where E and E' are the initial and final energy of the neutron respeCtiver.
The conservation of momentum requires that

6 = k-k' = G+q (1 6)
where O is defined as the scattering vector, k and k' are the initial and
final neutron wavevectors, G is a reciprocal lattice vector and q is the
deviation of 0 from the reciprocal lattice vector. Note that if q = O and k
= k', we simply get Bragg's Law, written as

l = 2dsine (1.7)

where 9 is the half scattering angle and d is the interplanar spacing.

In a neutron experiment, one actually measures the so called double

2
differential cross-section SOSE’ this quantity can be generally expressed as

 

 

d—g-fi-g,=‘§'§wp,1<twlvlti>12 anthers”) (1.8)
where V is the scattering potential, 1k) and lk'> are the initial and final
states of the neutron, and IA) and ll') are the initial and final states of
the target system. The weight p)‘ for the state ll) satisfies a normaliza-

tion condition.
{p : l (1.9)
l l

The horizontal bar stands for any relevant average over and above those
included in the weights pk.

We can calculate the scattering intensity for any neutron scattering
process for a specific system from Eq. (1.8), however those calculations are
usually very lengthy. There are many texts where this is done in great

31

detailifi'fi Here, following the treatment of Lovesey , we reiterate a few

or the highlights.

17

STATIC APPROXIMATION

The static approximation amounts to neglecting the term El'El' in the
6-function in Eq. (1.8). This means that the energy transferred to and from
the scattering system is much smaller than the energy of the incident
neutron or equivalently the condition for the validity of the static ap-
proximation is that the time taken for the neutron to cross from one atom to

the next is small compared to the characteristic oscillation or relaxation

time of the scattering system. The cross-section then reduces to

E'giaTg" =2 pA |<k'l'|V|kA>|25(hm). (1.10)
w

Note that the factor (k'/k) in (1.8) is unity since the scattering is
elastic. Integration with respect to E', yields the final form

do_
do‘

Here we have used a notation that will be used throughout this chapter,

<~|k'|Vlk>12 >. (1.11)

 

namely,

<(....)> = {pA<ll(....)ll>,
A

and the condition

{mm = 1. (1.12)
i

It is worth noting that the static approximation is distinct from
purely elastic scattering. In the latter case, the final states are identi-
cal with the initial states, i.e., A' = l in (1.10). In contrast, the
static approximation includes all possible final states.

In chapter 2, we use this approximation to determine the structure of
K(NH3)C2u where the rotational energy of NH3 is small compared with the in-

cident thermal neutron energy, therefore it would be valid to sum over all

the rotational states.

18

COHERENT AND INCOHERENT SCATTERING

The scattering interaction between a neutron and a nucleus can be
described by the Fermi pseudopotential35. In a system with a rigid array of
N nuclei, the position vector of the 2th nucleus being denoted by R, and the

scattering length of ch nucleus being described as b2, then the interaction

potential is

 

‘ - 21m2 .
V(r) = m §b26(r-RQ). (1.13)
Insert (1.11) into (1.12), and after some manipulations, one has
do .» *
d0 - Qg'exp{ik (RR-82,)lb2,bQ. (1.1a)
For 2:2', we have
a a _ 2
b2,bQ - 82, 8g - lbl
. * 2 2
But if 2'=2, b2,b2 = leI =Ib| ,
so that in general
* - 2 2 - 2
bQ.bQ - lbl +62,2.(|b| ‘lbl ). (1.15)

The scattering which derives from the first term in (1.15) is called
coherent scattering, and that from second term refers to incoherent

scattering. Then from (1.15) and (1.1M) we have

92. - (92} ,[92]
d0 ' d0 coh d0 incoh’

where the coherent cross-section is
[fifilcoh = 1512|Xexp(i§oRQ) 2 (1.16)
2
and the incoherent cross-section

(Salincoh = N11812-16121 = Nlb-5l2 (1.17)

There are two physical origins of incoherent scattering. First, it

originates from different isotopes of the atom from which the neutron is

19

scattered. Since the neutron interacts with the nuclei, the scattering
power varies from isotope to isotope. Because these isotopes are randomly
distributed throughout the sample, the scattering length varies from site to
site. The second source of incoherent scattering is the coupling of the
neutron spin with the nuclear spins. If the nucleus has a spin of I, then
the neutron-nuclear system posesses two spin values I + g or I - g . These
two spin states produce different scattering lengths, and they are randomly
distributed in the sample resulting in spin incoherence. Table 1.3 lists
the scattering lengths, the coherent and total cross-sections for the ele-
ments involved in this thesis.

As can be seen from Eqs. (1.16) and (1.17) coherent scattering reflects
strong interference effects allowing the study of collective properties
whereas incoherent scattering only contains information about an individual
nucleus. The physical reasons for this are as follows: Because the scat-
tering length varies from one isotope to another and also depends on the
nuclear spin orientation (relative to the neutron), the neutron does not see
a crystal of uniform scattering potential but one in which the scattering
length varies from one point to the next. It is only the average scattering
potential that can give interference effects and thus coherent scattering
proportional to lblz. The deviations from the average potential are ran-

domly distributed and therefore reflect the individual nucleus; they

10

-211

20

Table 1.3 Scattering lengths, coherent and total cross-sections for

the isotopes and elements of relevance in this thesis.

 

 

b 00 0tot
0.58 9.23 H.2A
H -0.37“ 1.76 81.50
0 0.667 5.59 7.60
Mg' 0.52 3.A0 3.7
Fe 0.95 11.3” 11.8
Al 0.35 1.59 1.5
Ca 0.99 3.02 3.1
Si .0.N2 2.22 2.2
C 0.665 5.6 5.6
N 0.937 11.0 11.5
K 0.379 2.19 2.30

 

Scattering length 8 in units of 10"12

cm .

2

cm, cross-section in units of

21

therefore give incoherent scattering proportional to the mean-square devia-

tion, i.e. to lb—bl2

DOUBLE DIFFERENTIAL CROSS-SECTION

In the previous paragraph, we discussed the coherent and incoherent
scattering in the static approximation. This approximation does not hold in
general. We now write the double differential cross-section in terms of the
coherent and incoherent scattering cross-sections. Then

d20 01

8538' = kRG11 Scohm’ w) + __ usinc(6’w)] (1'18)
Here Sc0h(0,w) and Sinc(0,m) represent the dynamic structure factors for the
coherent and incoherent parts of the scattering including both elastic and
inelastic contributions. One finds that

(0, w) _2%EN Z i:<exp(-iO-FJ,(0))exp(i0-r

J(t))>e'i“’tdt (1
JJ'

.19)

3cab

and

S nc(0,111) = 211111N E l:<exp(iO-FJW))exp(i0-FJ(t))>e'imtdt (1.20)

 

For most complex problems it is convenient to deal with the intermediate
scattering function I(Q, t) defined as

1(o,c) - ——{ <exp(-io-F

J.(0))exp(10-Fj(t))> (1.21)
JJ'

Then

 

1 a .
Scoh(0,m) - 2n“ [mexp(-imt)I(Q,t)dt (1.22)
We can further express I(0,t) into the 0 space Fourier transform of‘ai func-
tion of real space and time, the pair correlation function,G(r,t), which
represents the ensemble averaged probability of finding a particle at (nosi-

tion 3 at time t if a particle was at the origin at time 0. Therefore,

J
(0, w) =

Scoh

             

22

- -———I[IG(r, c)exp(10.F)aF]e'1“taat (1.23)

Similar arguments apply to the incoherent scattering, in this case, sue have
the time dependent self-correlation function GS(F,t) which in turn is the
probability of finding a particle at E at time t if the same particle is at

the origin at t = 0. Then

             

 

S “C(é’m) = -thdt
= 2nh I[IGS (r, t)exp(i0 r)dr'e tdt (1.24)
where 13(0,t) defined as
13(0,t) = $Z<exp(-10°rj(0))exp(iO-rJ(t))> (1.25)

J

Now we are going to calculate the coherent and incoherent one-phonon
scattering function. In general we can write
0(F,c) = 0(F,m)+0'(3,t) (1.26)
So that 9
11m G'(r,t) : 0
where G(r,m) is known as the Patterson function giving rise to the elastic
scattering and is used as an aid to structure determination ixiir-ray scat-
tering and neutron diffraction by crystals.
Similarly, we define
03(F,t) = 03(F,m) + a'S(F,c) (1.26a)

Here again Ila G;(r,t) = 0

 

 

Therefore,
0 ' -. v «o 9
(37213211113331 = '11 11° 12-2—1113 1..“ e M1111» explidwmwmt) (1.27)
o .
d2 O inel N C k. 1 ~1wt + . -o . 9
Ira—devil“ ‘ 11 1. 1721-171“ 6 W expilé'rlcslrv“ “-28)

One-phonon process

23

Now we consider a Bravais lattice of N atoms with their equilibrium
positions given by the lattice vectors
3:11a1+22a2+23a3,
The displacement of the atom at the 2th site is denoted by 0(2). “Thus

the position vector of this atom is

 

AQ:I+E(2) (1.29)
It can be shown that the correlation function is
0(F,t) = 75%77fi X Idkexp(-ik-F)exp[ik°(§'-§)l<exp{ik°u(Q)}exp{ikou(2‘,t))>
(1.30)
and that
Gs(r,t) = (2;)’N é Idkexp(-ik-r)<exp{ikou(Q)}exp[ik-0(Q,t)]> (1.30a)

To calculate the double differential cross-section we need to evaluate the

relation

(epr-ikeu(2))eprik-u(2',t)}>

exp[$<-{ko0(2)}2-{k-0(2')}2+2k-0(Q)k-u(Q',t)>]

exp(-2v(k))exp{<to&(2)tofi(i',t)>1

exp{-2v(k)}-[1+<E-&(1)E-&(i',t)>+....1 (1.31)

II

where exp{-2W(k)} is the Debye-Waller factor, and the formula is expanded to
the one-phonon process, and we also need

-————TET]1/203(q)exp(iq-l){8 (a)+2*(-5)1 (1.32)

‘ h
u (2) = I l
a e ZNMm
1.11 J J J
which describes the displacements in terms of annihilation and creation

4..

J J

dex _1. In other words, we expand the displacements by normal modes.

Operators a (a), the polarization components 03(5) and a mode irr-

(a) and 3

Subtracting the elastic scattering from Eqs. (1.26) and (1.26a), we obtain

the one-phonon coherent and incoherent inelastic scattering cross-sections

24

O , 6, J 2
inel = ck (2n)’ mieXPI 2W(Q)}{l o (g)|

 

 

(d --—-.-2°col
deE Auk vo j,q wJ(q)
x[nJ(a)a[u+uJ(6)1510+a—1)+1nJ(q)+1laiw-wjlq)15(0-a-i)1 (1.33)
0. Je
inel _._i k; 1 |0°0 (g)|2
[deE'linc 8n k i—epr-2W(Q)}X, wJ(q)
x[nJ(q)8{w+wJ(q)}+Inj(q)+1}SIw-wJ(Q))I (1.3“)
where nJ(q) is the Bose-Einstein thermal occupation factor given by
n.(a) = _ g‘ (1.35)
J exp(hmj(q)/kT)-1

The cross-section in Eq. (1.33) is the sum of two terms. The first, which

contairus the expression 6[u+w (6)}5(0+q-i), represents a scattering process

J

in which one phonon is annihilated and the second term, containing

SIm-m (5)}5(0-q-i), represents a process in which one phonon is created.

J
The two 6-functions in each process represent conservation of energy
and momentum, i.e.

E.

E t hwj(9)

E' = E x 0-1. (1.36)

These conditions determine the phonon dispersion relations m (5)133 a func-

J

tion of a. In chapter 3, we will address to this equation again.

The one-phonon incoherent cross-section Eq. (1.3A) can be further

reduced to
(d3:§.)i:§1 = lg-fi : expi- 2w(0)1{10 51(51121fl-Jégalin111.11 (1.31a)

where the average {IO-53(qnzlav is taken over modes with frequency (10, 2(0))
18 the phonon density of states.

The generalization of these results to non-Bravias lattice is
Straightforward. It can be shown that the one-phonon inelastic coherent

cros s-sect ion becomes

25

 

(dng'):2:1 - '5. (5101Z Z, 2“) —_L_.(Q)|§6d exp{- ”d (6)+160d}[6. °J(Q)}M- 1/2 2
° T J 9 J
x[nJ(a)5{w+wJ(a)}5(6+Q'T)+{DJ(Q)+1}6[w-mJ(q)}5(6-q-T)] (1.33a)

where d is the dth atom in a unit cell, and we define the dynamic structure

factor for the unit cell,
H316) = {Bdexpi-wd(6)+16-EJ{6-63(5))M“/2 (1.37)
q d

Similarly, we can obtain the one-phonon inelastic incoherent cross-

section

2
(dgdg.]:::l = —X{|bd I 2-|bd I 2)exp( -2H d(6)}-; quldo oJ(q)|2

(Q)6{w+wj (q)}+ [n J(q)+1}5{m-w (Q)}] (1.3ub)

"WW 1
This formula is not as useful as that for a simple lattice, because the
weight |6o53(5)12 in general is unknown. However, for a hydrogenated
sample, the incoherent scattering will be dominated by the contribution from

the protons, so we get back to the Eq. (1.3“) approximately.

26

1.3 Neutron Scattering Instruments
The neutron scattering measurements in this thesis were performed with

a triple axis spectrometer (BT-u) at the National Bureau of Standards

Reactor. The total power of the reactor is 105

1O1‘5

N giving a total flux of
neutrons 8-1. A schematic diagram of a triple-axis neutron
spectrometer is depicted in Fig. 1.6. A beam of neutrons from the reactor
passes through a collimator and impinges on a monochromator which selects a
particular wavelength from the distribution in the reactor by use of a Bragg
reflection. The monochromatic beam then passes through a second collimator
before being diffracted by the sample, where the angle w in Fig. 1.6 defines
the sample orientation with respect to the incoming beam, and the angle 4)
defines the scattering angle. A third collimator was placed before an
analyzer crystal which measures the final energy of the scattered neutrons,
again via the Bragg law. The neutrons then pass through a final collimator

3

before entering a He detector. Several crystals can be-used as
monochromators and analyzers, the most common being pyrolytic graphite and
copper as is the case at BT-u.

The triple axis spectrometer is ideal for constant scattering vector
(5) measurements and for quasi-elastic scattering. Two modes of operation
are possible. The most desirable one is the fixed final energy mode so that
the analyzer crystal is fixed at a certain angle selecting a constant final
energy and the monochromator is scanned giving a varying initial energy.
The energy resolution for this mode is shown in Fig. 1.7. There are two
distinguishing features in this diagram. First, the energy resolution is
degraded as the energy transfer increases, and second, the resolution change

takes place before the'monitor detector placing just before the sample. The

first feature gives an advantage for phonon creation measurement, since the

27

MONOCHROMATOR
29M
:0 _ ANALYZER
71' ‘ -....
e I
TARGET \ 4, 29A
. \(yn .
\ ‘
DETECTOR

Fig. 1.6 A Sohematic representation of a triple-axis spectrometer.

28

 

6 I r T I

Monochromotor
5 _. PG Cu

. o 20/20
. o 40/40 .

hAw (meV)
w 4:.

I I

1 1

N
I
l

 

 

c) .4"‘ " | l . .1, l
O ' 20' 4O 60 80 IOO

Incident Energy (meV)

Fig. ‘l.7 The resolution of one-half of a triple—axis instrument as a func-
tion of the energy passed by that half for two different collimations and
two different cystals for BT-u. (By one-half of an instrument, we mean
collimator-monochromator-collimator/collimator-analyzer-collimator.) Tk><3b-

tain the total instrumental resolution, one has to find both half instrument

resolutions then add them in quadrature.

29

intensity of a phonon group decreases with increasing m as shown in Eq.
(1.35), and the reduced resolution at the larger energy transfer compensates
for the intensity loss giving an enhancement overall. The second feature
obviates the need to correct the data for the varying resolution function.
The changing reflectivity of the monochromator is corrected by the monitor
detector. However, it is necessary to make one correction in this mode,
which comes from the higher order harmonic contamination. A filter is used
to remove the harmonics from analyzer and must be placed after the sample.
Therefore, the harmonic contamination in the incident beam causes the
monitor detector to overcount. This effect can be simply measured by using
a powder sample without a filter and comparing the intensity of a given
Bragg peak with its M2 counterpart. This gives the first order correction
since we ignore the higher order contamination as well as the changing
reflectivity of the powder sample. Fig. 1.8 gives the correction factor as
a function of incident energy, determined using a Si powder sample ((111)
reflection) for BT-H at NBS.

The other mode of measuring phonon groups, which is less often used, is
fixed initial energy. In this method, the main advantages are using the
high flux part of the neutron distribution from the reactor, and the
avoidance of swinging the entire instrument around the monochromator since
the monochromator angle is fixed in this case. However, one has to pay the
price of making more corrections including corrections for the varying
resolution function, the varying reflectivity of the analyzer crystal, and
the fact that the efficiency of the detector may depend on the final energy.

Another type of neutron scattering instrument for inelastic and
quasieiastic work is the time-of—flight spectrometer (TOP). The schematic

outline of a time-of-flight spectrometer is shown in Fig. 1.9. A

3O

 

 

 

 

 

. 2.00—
2
0
L13.
.5.
73
g 1.50—
o
(J
.3.
‘3
g

LOO 5'

L, l L, L, 1 L, L
0 IO 20 3O 40
Eka‘fl19\/)

Fig. 1.8 The monitor correction factor in the fixed-final-energy mode as
function of the initial energy. This particular figure was obtained at BT-u,
a triple—axis instrument located at the N85 reactor. When data nave teen
taken 1J1 the fixed-final-energy mode, they must be mutipiied by this factor

to account for the 1:2 contamination in the monitor detector.

31

MONOCHROM ATOR [70%

& DETECTOR

  

é BANK
CHOPPER TARGET d
' E
Q“
% MONITOR

Fig 1.9 A schematic outline of a time-of-flight spectrometer.

32

monochromator selects an initial energy E20 by a Bragg reflection. Then the
monochromatic neutron beam is chopped in short bursts, the energy of the
neutrons is measured by the flight time of the neutrons from the chopper to
the detectors. An array of detectors is placed in the scattering plane al-
lowing one to simultaneously collect data for several wave vectors. This
type of instrument is well suited for studying the dynamics of liquids or
powder samples for all wave vectors can be measured at once. Single crystal
work is also possible, but the interpretation of the spectra is not straight
forward. In this case, the triple-axis spectrometer is more suitable.

For the study of the structure of GIC's by use of elastic neutron scat-
tering, three major scattering configurations, i.e. the [002], inplane, and
c. scan are employed. They are schematically shown in Fig. 1.10. The inci—
dent neutron wave vector is E. and the scattered neutron wave vector is E',
where for elastic neutron scattering k. = k' . The inplane scan can be
achieved simply by rotating the sample 90° about an axis perpendicular to
the scattering plane from the [002] scattering configuration. The [0021 and
inplane scans are also called 9-29 scans. The c.-scan is more complicated
and computer control is used to keep the scattering vector along the AB line
(Fig. 1.10.).

For inelastic neutron scattering, constant-6 mode measurement has been
used most often. From Eqs. (1.33) and (1.34), we know that the dynamical
structure factor is proportional to (6-5812, therefore, 6 selection can be
employed to measure different polarized excitations. Fig. 1.11 illustrates
three configurations for phonon dispersion measurements. With configuration
I the frequencies of the longitudinal modes propagating along the c direc-

tion can be measured unambiguously as a function of wave vector. Using

33

x

 

 

 

 

 

[001] scan

 

 

 

 

 

 

inplanar scan

1.303 A scnematic diagram for (a) [001} scan and (D) ;noian¢ scar.

002

   

001

   

HKO

 

Fig. 1.10b A schematic diagram of a c*-scan in reciprocal lattice. AB line

. 1. '
represents a (102) c -scan.

3S

 

 

 

 

 

Fig 1.11 Some configurations for phonon dispersion measurements of GIC.

36

configuration II, we can determine the frequencies of the transverse acous-
tic modes and the low energy optic modes propagating along the [002]
direction, i.e., TA(0,0,§) and TO(0,0,§) modes. In configuration III, the
shear modes can be measured. Note that all of the aforementioned configura-

tions can be used the for constant-energy method.

1.

10.

11.

12.

13.

14.
15.

16.

37

References

 

K.V. Klitzing, Rev. Mod. Phys. 38, 519 (1986)

"Synthetic Modulated Structures", edited by L.L. Chang and B.C.
Giessen, (Academic Press, Orlando, Florida, 1985); Wharifinmntional
systems, Heterostructures and Superlattices", edited by G. Bauer, F.
Kuchar and H. Heinrich,(Springer Verlag, Berlhn 198A),Eknid State
Sciences, 53.

M.S. Dresselhaus and G. Dresselhaus, Adv. Phys. 39, 139 (1981).

S.A. Solin, Adv. Chem. Physics 39, 955 (1982).

S.A. Safran, Phys. Rev. Lett. 55, 937 (1980).

S.A. Safran and D.R. Hamann, Phys. Rev. Lett. 5g, 1910 (1979).

M.E. Misenheimer and H. Zabel, Phys. Rev. Lett. 33, 2521 (1985).

G. Kirczenow, Phys. Rev. Lett. 33, A37 (1984).

H. Zabel, S.C. Moss, N. Caswell and S.A. Solin, Phys. Rev. Lett. {13,
2022 (1979).

A. Erbil, l\.R. Kortan, R.J. Birgeneau and M.S. Dressehaus, Phys._Rev.
ggg, 6329 (1983).

H. Zabel, S.E. Hardcastle, D.A. Neumann, M. Suzuki,and A. Magerl, Phys.
Rev. Lett. 31, 2091 (1986).

H. Zabel, A. Magerl, A.J. Dianoux and J.J. Rush, Phys. Rev. Lett. 29:.
2099 (1983).

A. Hagerl, H. Zabel, and 1.8. Anderson, Phys. Rev. Lett. 33, 222
(1985).

M. Suzuki, Phys. Rev. B33, 1387 (1986).

R. Clarke, J.N. Gray, H. Homma, and M.J. Winokur, Phys. Rev. Lett. 31”
1407 (1981).

M.J. Hinokur and R. Clarke, Phys. Rev. Lett. 33, 811 (1985).

17.
18.

19.
20.
21.

22.

23.

29.
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27.

28.

29.

30.
31.

32.

33.

38

M. Kobayyashi and I. Tsujikawa, J. Phys. Soc. Jpn. 9Q, 1995 (1979).
L.A. Pendrys, R.A. Nachnik, F.L. Vogel, and P. Lagrange, Synthetic
Metals 3, 277 (1983).

Y. Iye and S. Tanuma, Phys. Rev. 333, 9583 (1982).

A. Herold, C.R. Lebd, Seances Acad. Sci. 333, 838 (1951).

S.A. Solin and H. Zabel, Advances in Physics (in press).

H.E. Stanley, Introduction to Phase Transitions and Critical Phenomena,
Oxford, New York, (1971).

P. Hansen, Physics Metallurgy, Cambridge Univ. Press, Cambridge,
England, (1978).

P.F. Chow and H. Zabel, Syth. Met. 1, 293 (1983).

X.H. Qian, D.R. Stump, B.R. York, and S.A. Solin, Phys. Rev. Lett. fiflJ
1271 (1985). ‘

X.H. Qian, D.R. Stump and S.A. Solin, Phys. Rev. §33, 5756 (1986).

‘Y.B. Fan, S.A. Solin, D.A. Neumann, H. label and J.J. Rush, Phys. Rev.
132, 3386 (1987).

Y.Y. Huang, D.R. Stump, S.A. Solin and J. Heremans, Solid State Com.
g, 969 (1987).

Y.Y. Huang, Y.B. Fan, S.A. Solin, J.M. Zhang, P.C. Eklund, J. Heremans,
and 6.6. Tibbets, Solid State Comm. Q9, 993 (1987).

H. Suzuki, 1. Oguro and Y. Jinzaki, J. Phys. 911, L575 (1989).

S.H. Lovesey, Theory of Neutron Scatteringyfrom Condensed Matter,
(Clarendon Press, Oxford, 1989).

G.L. Sqires, Introduction to the Theory of Thermal Neutron Scattering,

 

Cambridge Univ. Press, Cambridge, England, (1978).

Neutron Scattering, edited by G. Kostorz, (treatise on materials sciece

 

and technology; vol.15) (Academic Press, New York, 1979).

39

39. G.E. Bacon, Neutron Diffraction, (Clarendon Press, Oxford, 1975).
35. 1;.D. Landau” and B.M. Lifshitz, Quantum Mechanics, vol. 3 of Course of

Theoretical Physics, Pergamon Press, Oxford, New York, Toronto, Sydney,

Paris, Frankfurt, (1977).

40

Chapter II.
Elastic Neutron Scattering Studies of the Structure of
Graphite Intercalated with Potassium and Ammonia
2.1 INTRODUCTION

The ternary graphite intercalation compounds (GIC's) K(NH with

3)xc29
compositions in the range 0 S x S 9.38 are a class of GIC's which exhibits

3

unusual structural,"2 optical, electrical“ and kinetic5 properties. Ftu~
instance, diffuse X-ray scattering studies"2 coupled with Monte Carlo com-
puter modeling indicate that the potassium-amonia layers in K(NH3),,.38C2,,
constitute the two-dimensional (2D) structural analogue of the extensively
studied bulk three-dimensional (3D) metal-ammonia solutions6 which are
notable for the metal-insulator (MI) transition which they exhibit.6 In
this recent study,"2 it was shown that a semiquantitative fit to the
"liquid" structure factor can be obtained from a structural model in which
all potassium ions are symmetrically 9-fold coordinated to ammonia
molecules. One very interesting aspect of this work was that, unlike all

7,8

binary GIC's studied to date, the scattering could be described without

including any modulation of the intercalant layer due to the graphite host.
Thus, the potassium-ammonia layers could be treated as a "simple" 2D liquid
at room temperature, presumably due to an enhanced intercalant-intercalant
interaction. The origin of this increased interaction may be the back-

transfer of electron charge from the carbon layers to the intercalant layers

3,9,10

when “29 is amoniated. This back-transfer, which amounts to about

0.2 e/K for x = 9.38, has been established by (002) X-ray diffraction

9 proton NMR measurements,10 and optical reflectivity measurements.3

0

This effect may increase the intralayer binding in the K-NH3 liquid while it

studies,

would decrease the graphite-intercalant interaction. The back-transfer of

41

charge could therefore partially explain the reduction of the C33 elastic
force constant in 1((ND3),,.38C2,,11 compared to the corresponding binary com-
pound Kczu.

In addition to exhibiting some structural properties which are
analogous to bulk metal~ammonia solutions, recent electricalu and optical
reflectivity measurements3 of K(NH3)XC2,, indicate related electrical
behavior. Thus the opportunity exists of employing potassium-ammonia-
graphite to study the 2D MI transition as well as the electronic crossover

from 2D to 3D behavior by varying the stage of the K-NH GIC as well as the

3
composition. (A stage n GIC is one in which nearest pairs of intercalant
layers are interspersed by n graphite layers in a structure that exhibits
long range c-axis stacking order.) Moreover, the K—NH3 graphite system has
been shown to be the first GIC in which the intercalation and staging

5

kinetics are dominated by simple 20 diffusion rather than by the complex

Lifshitz processes12 associated with the formation of multidegenerate 20

domains13

characteristic of binary GIC's.

Since each of the unusual properties of the I(--NH3 ternary GIC's is
intimately related to the structure of the intercalant K-NH3 layer, it is
imperative to determine that structure as completely as possible. But it is

19,15 that a

well known from many studies of A-B binary disordered systems
definitive structure is difficult to determine from X-ray data alone because
the structure factor is the convolution of three pair correlation functions
which result from A-A, A-B, and B-B structural correlations. Thus, in
principle, it is necessary to carry out three distinct scattering experi-
ments to sort out the individual contributions to the structure factor. To

date, however, our knowledge of the K-NH3 intercalant structure derives

solely from an in-plane (th) and c-axis (h : const., k = const., Q) diffuse

92

scattering X-ray diffraction study. Accordingly, the purpose of this paper
is to report elastic neutron scattering studies of K<ND3)xC29 and to use our
results to further test the X-ray derived structural model of the 2D
potassium-ammonia liquid intercalant layer. In this case, the use of
neutrons complements the X-ray results quite well since the neutron scatter-
ing mostly reflects the ammonia-ammonia correlations while the X-rays are
almost equally sensitive to K-K and K-ammonia correlations. Therefore, by
referring the results to a particular structural model, it should be pos-

sible to reasonably characterize the structure.

2.1 EXPERIMENTAL METHODS AND RESULTS

Pyrolytic graphite with a typical mosaic spread of about 5° was cut

3 which were intercalated

to form binary stage-2 xczu GIC's using the standard 2-bulb method.16 These

into thirty pieces of dimension 1.5 x 1.0 x 0.1 cm

pieces were then transferred in a glove box to an aluminum sample can and
assembled in a package in which the specimens were separated by thin
aluminum spacers and stacked with their c-axes well aligned. Care was taken
to provide a region for subsequent expansion of the sandwich height upon
ammoniat ion. The aluminum can was compression-sealed with a tapered stain—

7 to which was welded a stainless steel bellows valve for

less steel cap1
evacuation of the sample holder and admission of anmon ia vapor.

To reduce the incoherent neutron scattering cross section, but enhance
the coherent neutron scattering cross section, deuterated amonia was used
to synthesize the ternary GIC's studied here. The N03 which was purchased
with a stated purity of 99.95% was further purified by repeated exposure to

sodium metal using procedures that are described in detail elsewhere.6’9

93

The neutron diffraction studies reported here were performed with a
triple-axis spectrometer located at the National Bureau of Standards re-
search reactor. All of our measurements were carried out with incident
neutrons monochromated to an energy of 35.0 meV by a pyrolytic graphite
monochrometer [(002) reflection]. A 2 inch thick pyrolytic graphite filter
was used to remove higher orders from the neutron beam. No analyzer crystal
was used for the in-plane scans. The transmission of the sample was es-
timated to be in excess of 95% making multiple scattering corrections
unnecessary. The Placzek cOrrections,18 which arise from inelastically
scattered neutrons, were also estimated to be less than 21. The fast
neutron background was found to be negligible as well. All measurements
were performed at room temperature.

The ammoniation of KC29 was carried out by in-situ exposure of the
sample to ND3 at a vapor pressure of ~9.5 atm. which was provided by a
remote liquid amonia reservoir held at room temperature. Intercalation to
the 9.5 atm. saturation composition of K(ND3)u.3C2u required a post exposure
period of approximately three hours during which repeated (002) scans were
taken to monitor the intercalation process. These scans are shown in Fig.
2.1. Note first from Fig. 2.1a that the reflections observed prior to ex-
posure to ammonia can be indexed to a pure stage-2 structure and provide a
confirmation that the specimen was not degraded in the glove box transfer
process. During the first hour of exposure, the system evolved from a pure
stage-2 binary “29 phase through a coexistence region (Fig. 2.1b) to a
stage-1 ternary phase with composition K(ND3)3.2C2,,. The ammonia content
was determined by comparing the measured basal spacings of the stage-1 ter-

9

nary phase with previous X-ray measurements of the variation of basal

spacing with composition. The post exposure period from approximately one

44

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

.11
(c) 3
a a F
I: s
g i
> L 1L
5 -1
N
E (b) .. g a
E = = 3 11' 3
55 § 8 s TA
2 an a
w 1.1 E A ’2’
5) a as” a
II II II
1 31
g L [ l—I

 

0.5 1.0 1.5 2.0 2.5 3.0
C2/(3r1

Fig. 2.1 The time evolution of the (00!) elastic neutron scattering
diffraction patterns of K(ND3)XC24 over the range 0 < x < 9.3 corresponding
to an elapsed time of the exposure of KC2u to NH3 from 0 hrs (panel a) to >
3 hrs (panel c). The reflections are indexed according to the notation
kn(002) where k = B or T signifying respectively a binary or ternary GIC and
n is the stage number. The abscissa is normalized in units of Q , the wave

vector correspond. to the (001) reflection of KC29'

45

to three hours is characterized by increased ammonia uptake of the pure
stage-1 ternary phase until the saturation composition of 1((ND3),,.3C2,4 was
reached as indicated in Fig. 2.1c.

In addition to (002) scans, two other types of diffraction scans were
employed in our studies. To measure the mosaic spread of the ternary phase
and at the same time establish the c-axis stacking sequence, a (2.99 0 ll)
so-called c' scan was taken. The results of this scan are shown in Fig.
2.2. The central broad peak of that figure indicates a mosaic spread of
about 10° while the sharp carbon layer Bragg reflections at Q/ch = :1
result from an eclipsed ...A/A/A/... c-axis stacking sequence where A and /
represent, respectively, crystalline carbon layers and liquid K-ND3 interca-
late layers. The small shoulders at $0.89"1 in Fig. 2.2 are Bragg
reflections from the aluminum sample can and spacers.

We carried out an additional c“ scan along the (202) direction in order
to ascertain the degree of c-axis correlation between the disordered inter-
calate layers. The details of this scan which are shown in Fig. 2.3 will be
discussed in section III. but we note here that it was chosen because it
contains no carbon layer-derived Bragg reflections.

To conclude the presentation of our experimental results we show in
Fig. 2.9a (solid line) an in-plane (hkO) scan of the diffuse scattering from
the K-ND3 liquid layers. "For clarity, the graphite and aluminum Bragg
reflections have been removed from the diffraction pattern of Fig. 9a. For
comparison and reference we also show in Fig. 2.9b (solid line) the cor-

1’2 of the in-plane diffuse scattering from the

responding x-ray measurement
potassium-ammonia layers in graphite. As can be seen from Fig. 2.9, the X-
ray and neutron diffractiion patterns are quite similar, the major

distinction between them being the strength of the first peak at 0.8A'1.

COUNTS*10“

‘I
Fig. 2.2 A (2.93 0 I) c scan of K(

is the c-axis repeat distance.

N03)

46

n.3C2u°

Note:

Q

 

2nd, where d

47

21 24

sz
o 1215 1a

 

-1.25-1.o -.75-.5 -.25 o .25 .5 .75 1.0 1.25
02““)

Fig. 2.3 A (2.0 o i) c* scan (dots) of K(ND3)u 3C21° The solid line was
computed from the square of Eq. (2.6) of the text. Note: 0 = 21/d, where

d is the c-axis repeat distance.

48

 

INTENSITWARBITRARY UNITS)

 

Fig. 2.9 (a) A comparison of the calculated (dashed line) and measured

(solid line) in-plane diffuse neutron scattering intensity from K(ND3)u 3C2“

(b) The X-ray diffuse scattering result (solid line) and the calculated

fit (dashed line) from the 9-fold cluster model.

49

This peak results from K-K scattering contributions1’2 (see section V below)
which, given the similar numbers of electrons on potassium and ammonia, are
similar in strength to the K-NH3 and NH3-NH3 contributions that generate the
second peak. In contrast the neutron crossection for potassium is much
weaker than that of ND and the first peak in the hr1flane scattering is

3

correspondingly much weaker than the second.

50

2.3 CALCULATIONS AND ANALYSIS

Two methods of analysis were applied to the data of Fig. 2.9a. In one,
the real-space structural model of the atomic coordinates which had been
previously applied"2 to explain the in-plane diffuse X-ray scattering (Nata
of Fig. 2.9b was used to calculate the corresponding neutron diffraction
pattern of Fig. 2.9a. in the other, the planar radial distribution function
(RDF) of the K-ND3 liquid was determined from the Bessel—fluxndon back-
transform of the neutron data. Although this latter method yields an REW‘
which is the sum of three pair correlation functions as noted above, the
neutron scattering intensity is dominated by ammonia scattering and the

back-transform represents primarily the ammonia-ammonia pair correlations.

A. The Rggl-Space Model Method

 

In a static neutron scattering experiment, the basic quantity that is

measured is the structure factor given by18

[$6]:(1)h : (IX EJ exD (16°fiJH2) (2.1)
.1

J is the position of the 3th nucleus, bJ

denotes that b is averaged over random nuclear spin-oriented and random

J

isotope distributions and the angular brackets represent the average over

where O is the scattering vector, R

all possible configurations of the nuclei. Using an appropriate structural
model for the scattering medium (see section 3.9), we can calculate the
coherent elastic cross section from Eq. (2.1).

We index the ammonia molecules by the label Q. The position of the vth

nucleus relative to the center of mass of the 11th molecule is Fle)’ we

denote the position of a potassium atom by the label 5K. So

51
I - 9' + rv(2) — rv'(Q)
fi-fi : §-§K'+FU(Q)

for A—K
1K - 1K, for K-K

for A-A

where A and K represent, respectively, ammonia and potassium. The cross

section then becomes

el

[dfilcoh z (génexP[iQ.(l-§')lv(R) vv(§1)bv(2)bv'(2')exP{ié.(rV(Q)v_rV'(Q'))}>

+ < '3 exp [(iO-(I-I ,)} ‘8 exp(iO°F )>
2% K K V(§) v(Q) V(Q)

. < X B: exp[(iO-(IK-§k)}> (2.2)

2K210

Averaging over all 2D directions and separating the intramolecular and in-

termolecular scattering, we have

do el

'—2 -2 ,
[Halcoh = NKbK + bK 2 E2. JO(D'IIK - IKI)
K K

+‘E < ‘S exp11o-F ) X J (6-11 - I 1)>
K vgfl) v(2) v(Q) 2’2K 0 K

. 1,5,5 . 331,2) . ~A< { 3,3,. exp{iD'(Fv-Fv.)]>
v(Q):v‘(2)

52

. (z bvbv,exp{10'(FU-FV.)1 X JO{O°(I-§')}> (2.31
vv' Q'ii

In Eq. (2.3) the first term is the self-scattering from the potassium
nucleus, the second term is the internuclear scattering from potassium, the
third term is the scattering from potassium-ammonia correlations, the fourth
term is the self-scattering from the ammonia molecule, while the fifth term
is the scattering from the correlations of different nuclei in the same
molecule, and finally, the sixth term is the intermolecular scattering from
ammonia.

To further clarify the important fifth and sixth terms of Eq. (2.3) we

introduce the molecular form factor
51(6) = {Ev exp(iD-FV) (2.11)
1)

which describes the self interference within a single molecule and is due to

the fact that a molecule is not a point scatterer of neutrons. From NMR10

19 it is known that the ammonia

and quasielastic neutron scattering results,
nmlecules are rotating rapidLy about more than one molecular axis.
Therefore, for the fifth term in Eq. (2.3) it is possible to treat the am-
monia molecule as a uniform sphere of scattering length density. Then for

the intramolecular scattering term one obtains

__ 31 NW ) _ 31 (011' )
11,16 bubD(—éfi-A-—)2 . 6 b02(—'&§T-E—)2]2. (2.5)
A A ‘

where “A is the number of ammonia molecules, J1 is the spherical Bessel

function, and RA is the radius of the ammonia molecule which was used as a

fitting parameter.

53

To calculate the form factor in the sixth term it is also possible to
consider the ammonia molecule spherical in the intermolecular scattering

term with an average radius R if one assumes that there are no directional

A
correlations between the amonia molecules. Then the molecular form factor,
which contains no information on the orientation of the ammonia molecule

with respect to the graphite planes, is given by

_ _ 3J,( QRA) .
F161 =11:N + 31.1,) . TEA—— (2.6)
and the sixth term in Eq. (2.3) becomes
2
1<r'(61>1 . ,.§,Jolo . (I - 1111 (2.7)

, .
A and RA were varied

independently and that a best fit was obtained with the values 1.939 and

Note that in the fitting proceedure used, the radii R

1.55A, respectively. The slightly larger value of R2, is not unexpected

since it is the average radius derived from the distribution P(r) of g_D_

A is the fixed

radius of a 32 molecule. Moreover, the background scattering is sensitive

discs which are used to model the in-plane structure while B

to the value of RA

fit included only a constant background term small variations due to the

while the structure in the scattering is not. Since our

Debye-Haller factor, multiple scattering, Placzek effects etc. may influence
the derived value of RA. Finally, inspection of the intramolecular scatter-
ing fifth term shows that this term contains virtually no features at the

small Q's in question and yields only a sloping feature peaked at Q = O

 

54

which is characteristic of any form factor. Thus there is no physical sig-
nificance of a small difference in the ammonia molecular radius between the
fifth and sixth terms.

From Eqs. (2.3)-(2.6) we obtain the final form we need to calculate the

diffuse scattering pattern.

— b2 — 2
1% W) = + b X N(r .) J (Or .1
Qcoh KbK F KiK' KK o KK

. bK<F'(Q)> X N(rAK) Jo(QrAK)

A,K
3a (0 a') 31 ( QR')
- 2 -— 2 ‘--— 1 A 2 -— 2 1 A 2
2
+ |<F'(Q)>| I Mr .11 (Or .1. (2.8)

where Mr”) is the number of atoms (molecules) at a distance r from an

11
atom (molecule) of type i to an atom (molecule) of type J. We use this for-
mula in section V below to calculate the neutron scattering crossection from

the computer-generated liquid structure of Fig. 2.6.

B. The Back-Transform Method
Now we address the theoretical method of the Bessel-function back

transform. In Eq. (2.8), we change the sum into an integral and rewrite the

formula as
do el -— 2 '— 2 - 2
(d 0)coh NKbr ‘ NA(bN * 3bD )
= 2n Ig‘EkzPK K(r) JO (Qr)rdr + 2n IO bK<F' (Q)>PK A(r) J0(Qr)rdr

55

. 21 1; 1<51K101>129,,(r1 J0(Qr)rdr (2.9)

where P1 (r) is the pair correlation function for atoms of type i and J, and

J
has a similar physical meaning to N(riJ).

The last term in Eq. (2.9) was derived from the last two terms of Eq.

(2.8) as follows

3J (02') 31 (03')
'—- 1 A 2 — 2 1 A 2
N [6b b (-———7—-—) + (-———T———>1 + 1<F'<o)>1 X ~(r .1 J (0r 1
A N D QR, D QR, ,,,. AA 0 AA
2 2
= |<F'(Q)>| NO(Q,r)JO(O) + 1<F'(o)>1 X N1r,,.)Jo(or,,.) (2.101

AiA'

Since NO(Q,O) is a slowly varying function of Q, Eq. (2.10) can be ap-

proximated as

|<F'(Q)>|2[NO(O)JO(O) + X N(r,,.)Jo(Qr,,)l
Aifi'

= I<r'(o)>l2 X N(r,,.)Jo(Qr,,.)
A,A'

= 1<r'(o)>12 - 21 13 P,,(r)Jo(Qr)rdr. (2.11)

0202

We now introduce a convergence factor e- and a sharpening factor

1/32(Q)15 into Eq. (2.9), where g(Q) decreases with increasing 0 and a is an

adjustable parameter. We take g(Q) = I<F'(Q)>| and define i(Q) as

-'2
- (finch - NKbK

1<r'(o)>12

«— 1- 2
- N (b + 3b )
A N D (2.12)

Then, the Bessel-function back-transform is given by

56

m 2 2
P(r) = 1 Qi(Q)e‘° Q

J (rQ)dQ
0 o

-'2
b P (r) 2 2

2 o drdQ

= 2n $mQr
|<F'(Q)>|

a a 'E |<F'(Q)>|P
. 2n 1 1 Or k g“
0 O I <F'(Q)>|

(r) _02Q2
JO(Qr)JO(Qr)e drdQ

 

m m -a2Q2
+ 2“ IO I0 Qr P,,(r) Jo(Qr)Jo(Qr)e

drdQ

a 2n 9,,(r) a P,,(r) (2.13)

In the last step of Eq. (2.13). we used the fact that contributions from K-K
and K-A arernegligible. In our calculation, the scale factor is not impor-

(r)

tant, and the transform of a constant background only contributes to PM
at r = O.
In computing the back transform of the experimental data of Fig. 2.9a,

we normalized (%i%):cl>h to 1 at large Q and normalized F'(Q) to 1 at Q = 0, so

do

 

(-) - 1
i(Q) : d0 data 2 (2.1“)
|<F'(Q)>|
do . .
where (d0)data - 1 is equal to zero at large Q. To av01d the difficulty as-

sociated with zero values of the sharpening function (the denominator) in

’1

57

Eq. (2.19), we incorporate an appropriate cut-off value,(flw in the in-
tegrals of Eq. (2.13) when calculating the back-transform of the neutron

data.

2.9 THE STRUCTURAL MODEL

He now discuss briefly the computer-generated structural model which we
used to fit the in-plane X-ray scattering1’2 (Fig. 2.9b) from potassium-
ammonia liquid layers in K<NH3)9.38C29 and which we now futher test by
incorporation into the Real-Space Method to calculate the corresponding
neutron diffraction pattern (Fig. 2.9a). The salient features of that model
are illustrated in Figs. 2.5 and 2.6 and are enumerated below.

1. The 11* ions and the (N113) ND3 molecules are treated as hard in-
finitely thin discs oriented with their planes parallel to the
carbon planes and located at the mid-point of the gallery along the
graphite c-axis.

2. The radii of the K* discs are identical and fixed at a value of
1.96A which gave the best fit to the x-ray diffraction pattern con-
sistent with the charge exchange to the graphite layer.

3. The model assumes a distribution of intermolecular distances, r

KA

and r which is Justified by the liquid dynamics of the

AA’

10.)) for which S(Q) represents an ensemble average.

system
9. There are two types of ammonia molecule in the liquid, those which
are 9-fold coordinated to potassium ions (open circles in Fig. 2.6)
and those which are dynamically unbound (hatched circles in Fig.
2.6). The 9-fold coordinated K--ND3 clusters which result are as-
sembled in a' symmetric configuration so that the lines connecting

the center of the K+1disc to each of the ND3 discs in a cluster

make equal angles of n/2.

58

1

1*“ 6.63 A —-_"

 

 

Fig. 2.5 (a) A schematic view of the tilted ammonia molecule between the
graphitic planes. The arrows indicate rapid rotation of the molecule about
its 3-fold axis which rapidly precesses about the graphite c-axis. The tilt
angle between these two axes is a. (b) The maximum cross-sectional
projection of the ammonia molecule onto a plane parallel to the graphite
planes. This projection defines the "hard disc" radius r. (c) The
distribution of hard disc radii that results from the tilt angle

distribution.

59

 

Fig. 2.6 Schematic representation of the mechanism by which 9-fold
coordinated potassium-ammonia clusters are assembled to simulate a 2D liquid
structure. Open (hatched) circles represent bound (free) ammonia moleCules
while the solid circles represent potassium ions. (b) Labels for reference

.in the text. (c) A portion of the simulated liquid structure.

60

5. The computer—generated collection of K-ND3 clusters and free ND3
discs is assembled (Fig. 2.6) in such a way as to maximize the in-

plane density. The model contains 500 K+ discs and 2165 ND discs

3
satisfying the stoichiometry K(ND3),.33C2M where 2165/500 : 9.33.

6. The effect of the graphite substrate potential on the structure of

the K-ND3 liquid has been ignored.

He also generated O-fold and 3-fold cluster models of the liquid struc-
ture of the K--NH3 layers using the proceedures described above. The X-ray
diffraction patterns calculated from these models yielded significantly
inferior fits to the measured patterns as discussed in detail elsewhere.2
For the O-fold cluster model, the most notable deficiencies were a total
loss of the first diffraction peak at 0.811.1 and a shift in the second peak
from the correct position of 2.0A-1to 2.3A'1. Similar discrepancies oc-
curred with the 3-fold model but the loss of intensity in the first peak was
not total. It is also noteworthy that the 9-fold cluster model was the only

one tested which yielded an in-plane density which was in agreement with the

experimental ly measured value .

2.5 CALCULATING THE DIFFRACTION PATTERN

In order to apply Eq. (2.8) to the above described structural model, it
is necessary that there be no c-axis correlation between the individual
potassium ammonia layers that contribute to the scattering. To experimen-
tally confirm this point which has been initially established using x-ray
scattering2 we measured a (2.0 0 2) c” scan in the manner described i11.sec-
tion 2.2. This scan is shown in Fig. 2.3 and the lack of short range order
diffuse scattering is direct evidence that the potassium-amonia layers are

uncorrelated. The solid line in Fig. 2.3 represents the square of the

"1

61

molecular form factor calculated from Eq. (2.6) for the corresponding posi-
tions in reciprocal space and provides an excellent fit to the experimental
data.

The results we obtain by applying Eq. (2.8) to the 9-fold cluster model
of Fig. 2.60 are shown in Figs. 2.7 a, b, and c which correspond respec-
tively to the potassium-potassium, potassium-amonia and amonia-ammonia
contributions. Notice immediately that the K-K contribution is cnmly about
1/80 of that from A-A and that the sum of the three contributions (Fig. 2.7d
and Fig. 2.9, dashed line) is essentially indistinguishable from that due to
A-A alone. Moreover the weight of each contribution is only dependent on
the stoichiometry of the sample and the stoichiometry of our model was
chosen to be the same. Clearly then, ammonia contributions do indeed
dominate the neutron scattering. This gives us the advantage of treating
the system as monatomic and greatly simplifies the analysis of the back
transform (see section 2.3).

In Fig. 2.9 we compare our calculated diffraction pattern with the ex-
perimentally measured pattern. -Only four fitting parameters were used to
achieve the fit shown in that figure. These are the molecular radii R, =
1.93A and R, = 1.55A, a constant background, and a scale factor. Note that
the molecular radii are indeed quite similar and are both physically
reasonable. As can be seen, the fit is excellent in the range 0 2 0.811"1
but reveals serious discrepancies at lower values of Q. There are two
reasons why these discrepancies did not appear in the X-ray studies. First,
in X-ray diffraction the contribution from ammonia was much weaker, only
half of the contribution from potassium. Second, x-ray data was obtained
from the difference of two x-ray scattering intensity profiles using a

"sample in/sample out" technique which may yield a large error at low Q.

‘. -n

62

 

(a)

 

 

 

 

 

k9

 

 

 

5 C 7

3 4
(“A“)

Fig. 2.7 Calculated contributions to the inelastic neutron scattering
intensity from (a) potassium-potassium correlations, (b) potassium- ammonia
correlations and (c) ammonia-ammonia correlations together with (d) the sum

of (a), (b), and (c).

63

We can gain insight into the low Q discrepancy by considering the
Bessel-function back-transforms shown in Fig. 2.8. In that figure the solid
curve (Fig. 2.8b) is the ammonia radial density distribution deduced from
the back-transform of the experimental data of Fig. 2.9 while the dashed
curve (Fig. 2.8a) is the corresponding function determined from the 9-fold
cluster structural model. Clearly, the latter exhibits much more sharp
structure, the envelope of which is similar to the former. The sharp struc-
ture in Fig. 2.8a has physical meaning and corresponds to puwnninent
(distances in the model. For instance, the first peak in Fig. 2.8a cor-
responds to the nearest neighbor distance, a-b of Fig. 2.6b, the second peak
represents the next-nearest neighbor distance, a-d of Fig.. 2.6b, etc. But
the broad first peak of Fig. 2.8b indicates that in the real potassium-
ammonia 2D liquid, the first and second neighbor peaks are indistinguishable
as are the third and fourth neighbor peaks. Thus, in the real fluid, the
correlations are not as strong as indicated by the model.

The strong correlations in the model are a direct manifestation of the
9-fold symmetry imposed on the K-ND3 clusters. If that symmetry was to be
relaxed so that the angular position of the ammonia molecules around the
central potassium was random, interneighbor distances would be much more
dispersed and the first two peaks in Fig. 2.8a would merge as would the
third and fourth peaks. Note, however, that the second and third peaks
would not merge since there is a K+ ion between atoms a and e in Fig. 2.6b.
Also note that the high frequency component of Fig. 2.8a is a characteristic
of the symmetric cluster while the low frequency component (the envelope)
results from the 9-fold clustering and is independent of the angular sym-
metry in the cluster. .Therefore, from the Bessel-function back-transform,

it is clear that the high frequency components contribute to the

64

“

 

1‘1

zwflgul-go)

 

 

 

~‘-----

~

 

 

o 2 4 a 3 10121416
1112)

Fig. 2.8 The radial density distribution of K(ND3)u 3C2, deduced from (a)

the 9-fold coordinated cluster model (dashed line) and (b) the Bessel-

funCtion back-transform of the experimental data (solid line) of Fig. 9a.

65

low Q discrepancy. Thus, if those components were absent from the radial
density distribution as they are in Fig. 2.8b, the low Q discrepancy in the
calculated diffraction would be minimized if not altogether absent.
Finally, it is worth pointing out that the oscillations in Fig. 2.8b extend
to large r due to the damping in Q-space which is imposed by the ammonia
molecular form factor. Those oscillations are certainly not real,tmt

rather are an artifact of the Bessel function back transform.

66

2.6 SUMMARY AND CONCLUSIONS

While the 9-fold cluster model contains the essential features of the
structure of the 2D K-ND3 liquid in graphite, the real liquid contains
clusters with a much lower degree of rotational symmetry. This is, of
course, to be expected since we know from NMR and quasielastic neutron scat-
tering measurements that at room temperature the ammonia molecules are
simultaneously rotating rapidly about their 3-fold axis while it precesses
rapidly about the graphite c-axis. The molecules are.also diffusing
rapidly. An additional stimulus to reducing cluster symmetry is the pres-
sure from unbound molecules which nudge between clusters and angularly
displace their bound neighbors. Thus it would indeed be puzzling if the
clusters preserved a high degree of internal symmetry. In principle, we
could take the symmetry breaking into account analytically by parameterizing
a probability distribution for the angular displacement of the ammonia
molecules in a cluster. But we maintain, as in our X-ray work, that such an
approach would require enough additional parameters to mask the essential
physics and its structural consequences.

A few additional conments are warranted. In our 2D model for the dif-
fraction, we used a 3D molecular form factor to compute the scattering.
This approach is valid because the 20 model only relates to the positions of
the molecular centers of mass in the graphite galleries. We have also
tacitly assumed that the anmonia molecules are "rigid". Also our treatment
of' the molecules as spherical is very reasonable since they are in complex
rapid motion and the structure factor takes an ensemble average over all
orientations.

Finally, the calculated X-ray pattern‘"2 exhibits a peak of excess

amplitude at 9.8A-1 while no such peak is evident in the neutron

67

I
calculation. Though the X-ray data does show a small peak in the 9.811-
region the calculated peak may result from the use of an oversimplified X-

ray form factor which was computed from fNH = fN + 3f“. That form factor
3

decreases much more slowly with increasing Q than does the more accurate
spherical type of form factor. Therefore, a high Q peak could have been in-
adaquately quenched in the X-ray calculation. In addition, a convergence
factor was not used in the X—ray analysis. Thus the X-ray calculation ex-

hibited more noise than did the neutron calculation.

10.

11.

12.

13.

19.

15.
16.
17.
18.

68

References
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1271 (1985).
X.W. Qian, D.R. Stump and S.A. Solin, Phys. Rev. B33, 5756 (1986).
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69

19. D. Neumann, Y.B. Fan, H. Zabel, S.A. Solin and J.J. Rush, to be pub-

lished.

70

CHAPTER III.
Inelastic Neutron Scattering Studies of stage-1

and stage-2 K-Ammonia Intercalated Graphite

3.1 INTRODUCTION

In this chapter we address the time correlated properties of the
K-ammonia graphite intercalation compound, or in other words the information
contained in G'(r,t) and Gé(r,t). The use of coherent inelastic scattering
allows one to investigate the properties of G'(r,t) which gives dynamic in-
formation such as phonon dispersion and librational excitations. One can
measure the q-dependence of the excitation modes, which will yield informa-
tion on their propagating nature. The phonons always display some
dispersion which indicates that they are a collective excitation with a
specific phase relation between the vibrating particles. Librations may or
may not display such dispersion depending on whether or not they are collec-
tive excitations or are Einstein oscillators due to single-particle motions
with no coherent phase relations,

Layered solids and some anisotropic molecular crystals have been exten-

1-3

sively studied due to their variety of kinetic, dynamical phenomena

including orientational phase transitions, translation-rotation coupling,
etc. Solids of this type are characterized by tight binding between atoms
in a molecular unit which in turn is loosely bound to other molecules or to
the.surroundlng atoms and by the strong anisotropy between interlayer and

intralayer interactions. Some systems which have been heavily investigated

include organic compounds adsorbed on the surface of graphiteuq; ammonia

intercalated in TaSZB; H,2 and HNO3 9-11

intercalated in various forms of clay

intercalated in graphite
12-19

; and water

71

Neutron scattering has been a powerful tool to study elementary excita-
tions such as collective lattice vibrational modes, intramolecular modes,
librational modes ("vibrational" excitations of a molecule which oscillates
as a single unit in a hindered potential), molecular rotations, rotational
tunneling and translational diffusion in a molecular crystal. One of the
most important advantages of the neutron scattering technique is the ability
to study the details of the interaction between those excitations. A fruit-
ful system for such a study is the mixed alkali halide-cyanides. An
interesting result from this system is that, in the low cyanide concentra-
tion limit, the [110] transverse acoustic branch (polarization [110]) splits

due to phonon-libron hybridization15'16.

17

Another split phonon dispersion
curve has been reported in the BGIC RbC8 but the mechanism of this split-
ting has not been determined unambigiously. It is probably due to coupling
between the transverse optical mode of Rb and the [OOIIL phonon of graphite.
Recently, we reported a coherent inelastic neutron scattering study of
phonon-libron coupling in the layered TCIG, K(ND3)XC2,,18. In that paper,
several models have been proposed to explain the splitting. However, none
of these models could consistently account for all of the experimental
results. Here we present a more detailed analysis of the experimental data
and provide a new model which consistently explains all of the data accumu—
lated to date.

Despite the fact that the first K-anmonia TGIC was synthesized over 30

years ago‘g, systematic physical studies of this material have appeared only

very recently. X-ray and neutron diffraction studie320-22 have revealed
that the structure of the intercalant layer is a two-dimensional liquid at

room temperature and that the liquid structure factor can be reasonably well

accounted for by a model in which four ammonia molecules are attached to a

72

single K-ion forming an ammonia-K cluster and the rest of the ammonia
molecules are treated as free. The structure is depicted schematically in
Fig. 3.1. In addition, NMR spectra have indicated on the average, the
three-fold symmetry axis of the ammonia molecule lies parallel to the basal

23-25. A neutron quasielastic scattering study26 (see

plane of the graphite
Chapter-ILV) has shown that there are three different diffusion modes in the
liquid layer. First, the ammonia molecule diffuses translationallj1 in the
gallery; second, it spins about its C3-symmetry axis; and third, the C3 axis
reorients or precesses about the graphite c-axis. Resistivity27 and optical
reflectance measurements28 as a function of ammonia concentration have
recently been interpreted in terms of a two-dimensional metal-insulator
transition analogous to the three dimensional metal-insulator transition in

29. This transition was attributed to the back

bulk metal-ammonia solutions
transfer of electrons from the carbon plane to the intercalant-layer where
they are localized.at lower ammonia concentration and delocalized above
x:9.127. The intercalant-graphite force constant e are expected to be
reduced by back charge transfer and the associated lattice expansion which
results from ammoniation. In this chapter, we will present inelastic
neutron scattering measurements of the [001] longitudinal phonon dispersion
of K(NH3)xC29’ and discuss the effects of charge transfer and lattice expan-

sion on the force constant. We also will provide a detailed discussion of

the mechanism fer the libron-phonon coupling in K-ammonia TGICs.

3.2 EXPERIMENT AND RESULTS
Details of sample preparation have already been addressed in Chapter II
and the general methodology for neutron scattering has been considered in

(Huapter I. Therefore, we will only brieffly address here some additional

 

 

Fig. 3.1 A schematic Diagram of the inplane structure of K(NH The

3’u.3C21°

circles represent KT ions, the open triangles represent NH3 molecules which

are associated with a 13 ion, and the shaded triangle represents a "free"

ammoniua molecule.

 

 

 

__- .3,

 

74

aspects of the experimental procedure. After all measurements of K(ND3)XC2,
were performed at x = 9.3, the ND3 concentration was reduced to x = 3.1 by
slowly removing ammonia vapor until a pressure of 0.20 atm was reached in
the sample can. The ammonia fraction was determined using the results of

30

York and Solin who measured the composition dependence of the c-axis lat-
tice parameter. The lattice parameter shift upon cointercalation was
measured by repeated (002) scans and was found to be consistent with that
measured by York and Solin30. The stage-2 sample was prepared in the same
way as the stage-1 specimen but at a lower ND3 pressure. Neither the
K(ND3)3.,C2,, nor the stage-2 1((ND3)2.3C28 compounds were pure stage

samples31.

Although the stage of interest was the dominant phase in each
sample, care must be taken so that all phonons were measured in more than
one Brillouin zone to assure that they were indeed associated with the peri-
odicity of the phase of interest.

The neutron scattering results presented here were obtained using two
triple axis spectrometers (BT-9 for stage-1 and BT-9 for stage-2 ) located
at the National Bureau of Standards Reactor. Pyrolytic graphite was used as
monochromator and analyzer crystals in both stations. All coherent scatter-
ing data present in this section were obtained in the constant Q mode with a
fixed final energy of 13.7 mev and collimation of 90'-20'-90'-90' for the
acoustic branch and 90'-90'-90'-80' for the optic branch, respectively. A
one inch graphitefilter was used to remove harmonic contamination. Other
instrumental configurations were also used to eliminate the possibility of
spurious observations. All of the measurements were performed at room
temperature.

The measured [001] longitudinal phonon dispersion curves of stage-1

32

K(ND3), 3C2, are shown in Fig. 3.2. It is well known that for a stage-n

75

25 1 r

 

 

10-

Energy (meV)

 

 

 

 

 

 

0 0.1 0.2 0.3 0.4 0.5

Fig. 3.2 [001]L phonon dispersion of stage 1 K(NH3)u 3C2,. The solid

represents a fit using the dynamical matrix, A, described in the text.

 

line

 

76

compound there are 1) optical branches, and 1 acoustic one, of longitudinal
phonon modes. In Fig. 3.2, we see clearly three phonon branches. Careful
examination and comparison with the stage-1 binary counterpart indicates
that the lower two branches of K(ND3)9.3C29 actually belong to the same
acoustic branch which has been split into two parts due to the coupling of
the phonon to a libron with an energy of about 6.7 mev. This phoncnrelibron
coupling will be discussed in detail later. The [001] L phonon dispersion
of the stage-1 compound K(ND3)3.,C22 are shown in Fig. 3.3. As can be seen,
the phonon-libron coupling is also present in this compound but the libra-
tional energy has increased slightly to 7.1 mev and the splitting of the
acoustic branch has been slightly reduced in comparison with K(ND3)9.3C29°
The representive phonon-libron coupling effects are shown in Fig. 3.9
for three different values of wavevector q in units of 21/10. The solid
line in that figure represents a two Gaussian fit to the experimental data.
For q = 0.275 and 0.325, two peaks are evident. The less intense peak is
broadened compared to the resolution function of the instrument while the
more intense one is nearly resolution limited. At q = 0.30, the two peaks
are almost equally intense and their combined width is broad compared to the
resolution. Since the phonon lifetime is-so long. that all phonon peaks
should be resolution limited, we assume that the broadened weak peak is a
libron-like excitation. An attempt was made to observe the librational mode
directly. The result of this effort is shown in Fig. 3.5. We performed the
measurement at Q = (0,0,3) (q = 0) in order to avoid inelastic scattering
from nearby phonons. The solid line in Fig. 3.5 again represents a two
Gaussian fit to the data. The first fits the "background" from other in-

elastic scattering as well as elastic scattering, the second fits the

librational mode yielding an energy of 7.1 mev and a width much broader than

77

 

 

 

25 i I 1 1
20*- _
0
3 15- _
0D
.5
>~
E."
G)
,5 10-

 

 

 

 

1
O 0.1 0.2 0.3 0.4 0.5

Fig. 3.3 [OOIJL phonon dispersion of stage 1 K(NH3)3 1C2,. The soliti line

represents a fit using the dynamical matrix, A, described in the text.

 

 

Counts

 

 

 

 

 

Fig. 3.9 Representlve phonon groups attrit275, 0.300, 0.325 for tne com-
pound K(NH3)u 3C2,. The solid lines are fits of tne data to a dLMTLJL two
Gaussians. The splitting due to phonon-libroniununlng can be seen clearly,

and the positions from such fits were plotted in Fig. 3.2.

79

 

75 1

 

 

 

 

3.0

6.0 9.0
Energy (meV)

I I f 1
5
9.
3.5
g d
o ‘
'5(Lnl-L'.f“£. l L 1 ""0021.--“ .

72.0

Fig. 3.5 Inelastic scan at 5:0 (Q=(0,0,3)). The solid line is a fit to two
Gaussian functions, one was fixed at 5:0 with a resolution limited width.
The second peak is due to the librational excitation and has an energy of

7.1 mev and a width of 3.5 mev.

80

the instrumental resolution of 1.1 mev. The libron excitation energy of 7.1
mev is in good agreement with the value of libron-phonon coupling deduced
directly from the phonon dispersion curves. This indicates that the libron
is essentially dispersionless along the c-axis, implying that there is no
collective motion in adjacent galleries. ‘This is consistent with the ab-
sence of positional correlations observed in the elastic scattering22

The measured phonon dispersion of the stage-2 compound K(ND3)2.3C2u is
shown in Fig. 3.6. In spite of considerable effort, neither the upper opti-
cal branch nor the phonon-libron splitting was found. The failure to
observe the upper optical mode could be due to the higher excitation energy
and the lower concentration of N03 in the stage-2 compound. The absence of
the splitting is due to the fact that the energy of the librational excita-
tion, which exists at ‘ 10 mev according to our model (see next section),
falls within the Brillouin zone boundary gap of the stage-2 compound causing
the phonon-libron coupling to disappear. Therefore, in order to determine
the existence of this libron mode, one would have to observe the excitation
directly and unambiguously. However, several facts make this impossible.
First, the 1((ND3)2.3028 was not a pure phase, so that some stage-1 regions
were also present; since the libron is dispersionless, it is impossible to
distinguish whether the observed scattering was due to a libron from stage-2
or if it came from the stage-1 regions. Second, the lifetime of the libron
is short and .its intensity is very weak. Finally, the ammonia concentration
is low and this in turn causes the total scattering cross-section to be

reduced so that the observation becomes even more difficult.

81

3355 1 i 1 1

 

2K)'- '-

 

Energy (meV)

 

 

 

 

 

Fig. 3.6 [001]L phonon dispersion of stage 2 K(ND3) The 50111 line

2.3C21-

represents a simple two force constant Born-von Karman model discussed in

the text .

82

3.3 DISCUSSION

In order to interpret the observed phonon dispersion relations, a 3x3
dynamical matrix, A, was employed in Ref. [18] to account for the split
phonon dispersion curve. The dynamic matrix A contains a virtual crystal,
one-force constant Born-von-Karman model, an Einstein oscillator of energy 0

representing the libration, and a coupling term with strength coefficient Y.

—a— l(1+eiq1) 10(1-ei2q1)
A: 1-(1 e ‘qi ) :2— 0
Ya(1-e ian) 0 (12

Here 0 is the intercalate-graphite force constant per carbon atom, M is the
areal carbon mass density, m is the areal intercalate mass density per carb-
on atom, and the I is the c-axis lattice constant. The best fit to the
measured phonon dispersion relations using this dynamic matrix yielded the
parameters given in Table 3.1. From this table we see that 0. the force
constant per carbon atom, is essentially independent of the N03 concentra-
tion, x. This implies that the K-graphite interaction is the dominant
factor in determining 0 since the ratio of K to C is nearly constant in the
two stage-1 compounds, and also that the elastic interaction is probably
more important than the electrostatic interaction in determining 0. This is
in agreement with the fact that the K-ND3 interaction is strong and the N03-
C interaction is weak. It should be noticed that the increase of back
charge transfer and lattice parameter with ammoniation tend to decrease 4).
On the other hand, the increase of x must strengthen the ammonia-graphite

interaction. Therefore it is difficult to draw any unique conclusion from

the charge transfer effect, since it depends on the

83

Table 3.1. Best fit parameters for the [001]L modes of two stage 1 K-

ammonia intercalated graphite compounds by use of dynamic matrix A.

 

Compounds c (dyn cm) 1 (mev) Y (mev)
K(ND3)u.3C2, cU9U 0.7 0.70
K(ND3)3.,C2, 2120 7.1 0.99

 

84

competition between those interactions. The slight reduction of the libra-
tion energy with increased x can be interpreted by the fact that the c-axis
lattice parameter increases as x increases, reducing the curvature of the
potential and therefore the energy of the libration.

In Fig. 3.6, the solid line represents the fit with a two parameter
Born-von-Karman model for the stage-2 compound K(ND3)2.3C28. Because of the
lack of the upper branch of the optical mode in this sample, we were unable
to fit the data in an unambiguous way. A graphite-graphite nearest-neighbor
force constant of 2900 dyn/cm, consistent with the values obtained for the

stage-2 binary compounds33

, was assumed, and. the intercalate-graphite force
constant, 0:2100 dyn/cm, obtained from the stage-1 results and determined to
be independent of the intercalate concentration, gives the best fit.

The sound velocity v8 and elastic constant 033 , determined from the
initial slope of the acoustic branch is given for K(ND3)xC29 as, well as the
values for the stage-1 KC8 in Table 3.2. It clearly shows that there is es-
sentially no difference between the two compositions of the K-ammonia,
compounds studied here, but both v8 and C33 are substantially reduced com-
pared to the binary KC8' The possible reasons accounting for this reduction
were proposed. The first, and most important effect was the large expansion
of the c-axis of the ternary compound. The second possible reason may be
the back charge transfer to the intercalate layer in the K-ammonia compound.
From the previous discussion concerning 0, we know that the electronic ef-
fects are unimportant within the accuracy of the present data. The sound
velocity and C33 for the stage-2 ternary compound are also shown in Table
3.2 along with the results for the binary stage-2 compound KC29' The reduc-
tion of C for the ternary compound compared to the binary compound is

33

apparent for the stage-2 compounds, though it is not as dramatic

85

Table 3.2 Elastic constants and sound velocities for K-ammonia inter-

calated graphite along with the analogous binary compounds.

 

 

Compounds Stage 035(i10110yn cm) Vj(~105 cm/sec)
x02,(N03),.3 1 1.71 0.10 3.22 0.10
KC2u(ND3)3., 1 1.02 0.10 3.18 0.10
KC2u(ND3)2.3 2 2.99 0.10 3.90 0.10

kc8 1 9.85 0.1113 9.91 0.073
x02, 2 3 71 0 158 9.33 0 15a

 

aAfter H.Zable and A.Magerl, Phys. Rev. 825, 2963 (1982).

86

as it was for stage—1. The fact that stage-1 ternary compound is softer
than the stage-2 ternary compound is because stage-1 contains more "soft"
intercalate-graphite bonds than does the stage-2 ternary.

The large width of the librational peak indicates that the librational
potential is probably quite anharmonic so that the splitting between levels
is not at all constant.

Note that the coupling constant 1 scales approximately with the ammonia
concentration”. This seems to be evidence that the librons couple to the
phonons individually. However, further study shows that this scaling
phenomenon may be an artificial effect since this phenomenological descrip-
tion is not universal.

Consider a dynamical matrix B which is similar but not identical to A:

_g_ 7L<1+eiq1) 0
B: —L(1+e'q i ) ‘%"‘ Ya(1—e21q1)
I“ I
1 0 Yc(1-e12ql) 02 1

In this matrix all terms are the same as in matrix A, except the coupling
term is placed at a different position (compare with A). The quality of the
fit for both matrices is equivalent and the fitting parameters are given in
Table 3.3. As can be seen, the force constant 0 and the libration energy a
are the same to within experimental error as those from matrix A given in
Table 3.1. The only difference is in the coupling constant Y which for
matrix B is independent of ammonia concentration. This leads us to conclude
that the coupling interaction is due to K-graphite, which contradicts the
previous model. Thus, both dynamical matrices are deficient and any inter-

pretation for the libron-phonon coupling based on them is necessarily

87

Table 3.3 Best fit parameters for the [001]L modes of two stage 1 1(-

ammonia intercalated graphite compounds by use of dynamic matrix B.

 

 

Compounds 0 (dyn/cm) d (mev) 1 (mev)
K(ND3)9.3L29 1 2090 1 0.7 1 1.22
1 1 1
. . 3 ,.
1(N031,.,t2, 1 -1.0 1 7 1 1 1.23

 

88

ambiguous. In addition, the coupling term contains exp(2qI) which imply
that there are coupling interactions between second nearest layers. This is
physically unreasonable and in conflict with the fact that the intercalate
layers are uncorrelated and the intercalate-intercalate interaction is very
weak. Therefore, a new approach to the problem is warranted. In that ap-
proatmiide construct a dynamical matrix according to a vibrational model and
test the resultant fit to the phonon dispersion curves.

Several possible models which are independent of the description by the
dynamical matrix A wereldiscussed in reference [18]. We consider those
models first. As discussed in reference [18], it is unlikely that the
libration is due to the oscillation of an individual ammonia molecule dipole
in a potential provided by the K-ion since the energy estimated for this mo-
tion is much higher than the measured value of ” 7 mev. In contrast, the K-
ammonia bond-bending (umbrella) mode in which motion occurs principally out
of the basal plane was estimated to be roughly in the right energy range.
We approximate this bond-bending mechanism by the force constant model
depicted in Fig 3.7. The dynamical matrix for this model can be determined

rigorously and is given as C below.

89

 

Fig. 3.7 A schematic diagram for the umbrella model described in the text.
K represents K+ ion, A represents ammonia molecule, and C represents a carb-

on layer.

90

 

 

 

 

 

-(2¢K.¢,) 1,11+e1811 1,11.eiql1
I—Mc 711C111, 711,11K
1,(1+e"q11 -(2¢K+1'1 ,.

C‘ 711 11 11 711 M
c x x A x
.. ' I
1,(1+e 1“ 1 ,. -(21,+1')
JMCM, JMAMK MA

 

 

Here the carbon-ammonia interaction is represented by 0,, the K-ammonia
interaction (out-of—plane) by 0k and the carbon-potassium interactitni is by
41. Although, there are three fitting parameters in this matrix, it can not
fit the split phonon dispersion curves of Figs. 3.2 and 3.3. Even when ad-
ditional fitting parameters are employed, using the dynamical matrix C'

where

 

a, 1,11.e‘qI) 1211+eiql1
_ -in
C'- Y,(1+e ) 02 Y3
-in
Y2(1+e ) Y3 a3

 

a satisfactory fit cannot be obtained. Thus we conclude that the bond-
bending model of Fig. 3.7 is not applicable to the libration in K(ND3)xC29’

Now consider the "washing machine" mode in which the K-ammonia cluster
rotates about an axis which passes through the K-ion and is perpendicular to
the graphite basal plane. The fact that this excitation would only take
place in the basal plane does not eliminate it from consideration since the

observed splitting of the [001]L acoustic phonon branch in the stage—2 RbC8

91

compound was found to be attributable to a coupling between the in-plane mo—

3“ indicate that the

tion of the Rb and the out-of-plane phonon. NMR results
graphite layers provide an out-of-plane potential which prevents the am-
monia's C3 axis from lying out of the basal plane. This result as well as
the fact that the graphite carbon layer and the K-ammonia cluster have diffi-
ferent symmetry would provide a coupling to the [001]L phonon. The coupling
Hamiltonian can be determined easily
H : -Yo(xn-xn_,)v + $0202 (3.1)

where v is the third degree of freedom representing the angular displacement
of K-ammonia cluster. Y is the coupling constant. The term containing
x

x indicates that the narrower the gallery is the stronger is the hin-

n' n-1

derance to rotation. The dynamical matrix, D, for "washing machine" motion

can be derived by use of Eq. (3.1). We have

-ef' 7%;(1+eiql) Ya(1-e1q1)
- .2. -in .2.
D- JHIIH+e ) m 0
Ya(1-e-iq1) O 02

Here all symbols have the same physical meaning as in the dynamial matrix A.
Notice that matrices A and D are almost identical except for the coupling

qI i2qI

term, which now contains e1 instead of e The former derives from the

term xn-x in Eq. (3.1) and is apparently more physically meaningful. The

n-1
positive square roots of the eigenvalues of this matrix are shown as the
solid curves in Figs. 3.8 and 3.9 . As can be seen, the fit is as good as
the one obtained from matrix A (see Figs. 3.2 and 3.3, all fits have ap-
pnoximately same chi square). The best fit parameters corresponding to

these dispersion relations are given in Table 3.9. Comparing Table 3.9 with

Tables 3.2 and 3.3, we see that the force constant and the libration energy

92

 

ZOFIUVIIIIIIUTUFTTTTII[gr ‘

15—

 
 

p—a
O
l
l

 

Energy(mev)

on
I

 

 

 

LILllllLllillllLllllll

0
0 0.1 0.2 0.3 0.4 0.5
0120/1.)

 

Fig. 3.8 [001]L phonon dispersion curve of stage 1 K(ND3), 3C2, fitted by

dynamical matrix 0 (see text) for "washing machine" mode.

93

 

20 'IrIIIIIlrIIrIIrIrlrrr

 

H
01
l
L

 

p_a
O
|

 

 

Energy(mev)

01
I

 

 

 

lllllllll'lllllllLlllLl

0
0 0.1 0.2 0.3 0.4 0.5
«Zn/Io)

 

Fig. 3.9 [001]L phonon dispersion curve of stage 1 K(ND3)3 ,0}, fitted by

dynamical matrix D for "washing machine" mode.

94

Table 3.9 Best fit parameters For Ume[CNH1L modes of two stage 1 K-

ammonia intercalated graphite compounds by use of dynamic matrix D.

 

 

Compounds 0 (dyn cm) i (mev) Y (mev)
K(N051,.502, 3110 0.7 0.78
K(ND )- C . 2280 7.1 0.78

55.129

 

95

for all these matrices are the same to within experimental error. The force
constant and the sound velocity are determined by the slope of the lowest
energy branch and are the same as those determined from dynamical matrix A.
Discussion of these parameters has already been presented. We will now con-
sider the mechanism of the libron-phonon coupling.

Notice that the libration energy scales approximately with 1/7111 (Fig.
3.10), where m is the areal intercalate mass density per carbon atom, and
that the energy is independent of dynamical matrix used to describe the
libron-phonon coupling. This is evidence that the libration excitation is
due to the K-ammonia complex. The coupling strength can be represented by
Ya which also scales with 1//m. Nevertheless, it is consistent with the
fact that the c-axis lattice parameter increases as the ammonia concentra-
tion increases, reducing the coupling interaction. It is worthwhile to
point out that the fit depends crucially on the coupling term whether it
contains (1-exp(in)) or (1+exp(in)). The former come from the out-of-
plane to in-plane coupling (without displacement of the center of mass along
the c-axis) and the latter represents a coupling between two out-of-plane
excitations. A reasonable fit can be only achieved for the former case
which strongly suggests that the splitting is due to transverse mode cou-
pling to the longitudinal phonon. This conclusion is also consistent with
the fact that the librational mode can barely be observed in the [001]
direction without coupling.

We conclude that the actual physical motion of the observed librational
excitation is best characterized as a "washing machine" mode: The K-ammonia
clusters tend to partially reorient themselves about an axis through the K+
ion and parallel to the graphite c-axis in response to a force which is

provided by the reorientational activation energy. The clusters can not

96

10

Energy1mev)

 

00 0.03 0.06 0.09 0.12 0.15
1/ l m

Fig. 3.") The relation between librational energy a and the areal

intercalate mass.

97

rotate freely since sterically (see Fig. 3.1) other clusters block rotation.
The cluster thus bumps back and forth and this motion constitutes the libra-
tional excitation. The relaxation of these excitations forms the
reorientational motion of the complex. Quasielastic neutron scattering
results (see next chapter) show that the characteristic time of reorienta-
tion is of the same order as the libron lifetime. This is evidence that the
reorientation happens on a much slower time scale than the libron does and
is thus a mechanism for libron damping.

Let us consider a simple model of a K-ammonia complex rotationally os-
cillating in the potential of the reorientational activation energy. For a

35

classical oscillator , 2T = U and the oscillation frequency is

U 1/2
m=1?)
where I is the moment of inertia of K-ammonia complex about! the rotational
axis. Using a metal-ammonia distance of roughly 3 A and an average complex

9 mev-Az/cz.

K("DB)9.3’ we obtain 11175911101 With U = 80 mev3u (see next
chapter), this will give an energy for the libration of about 6.9 mev which
is in good agreement with the observed value of 6.7 mev for x = 9.3.
Similarly, for K(ND3)3., we have m = 7.5 mev which is also in good agreement
with the observed value of 7.1 mev. The quantitative values for this over-
simplified model are not very meaningful. The important thing is that such
a simple model gives the right order of magnitude for the librational ex-
citation energy. This simple calculation suggests that the model for
libration is reasonable.

It is worthwhile to point out that other mechanisms can also fit the
phonon dispersion data since the description is still phenomenological. For

instance, in the bond-bending model, if we introduce two transverse degrees

of motion for ammonia and K-ion, the dynamical matrix C will become SXS, and

98

the transverse longitudinal coupling term will contain (1—exp(in)).
Apparently, it will fit the dispersion data with two more extra transverse
branches of phonon. However, the energy for this model would be at higher
frequency since the stretching force between K and ammonia is strong. In
addition, one would expect that the energy will be independent of ammonia
concentration. The more realistic mechanism is the stretching mode between
K-ammonia complex. The Hamiltonian for this model will be same as Eq.
(3.1). The energy of excitation will scale with 1/1/m and the coupling con-
stant will be independent of ammonia concentration. All these predictions
agree with experimental observation and make the model very promising.
However, if we use a barrier of 80 mev, calculation shows that the mean
square displacement for this model will be about 2.3 A, which is an unaccep-
tably large value. The barrier is about two orders of magnitude higher. If
the barrier was 100 times lower, the thermal energy would be much higher
than the barrier and one would expect that the lifetime of the libron would
be so short as to render the definition of libron meaningless. In other
words, the stretching mode in the present barrier has a much higher energy

than 7 mev.

3.9 CONCLUSIONS

We have examined the [001]L phonon dispersion in three K—ammonia TGICs
and have shown that the previously proposed bond-bending mode model of the
libron motion is deficient. The libron-phonon coupling has to be associated
with a transverse mode. The coupling mechanism which is most consistent
with all experimental data acquired to date, is the "washing machine" mode

of the K-ammonia complex coupled to the [001]L acoustic phonon.

10.

11.

12.

13.

19.

15.

99

References
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W. Press; Single Particle Rotations in Molecular Crystals, Springer
Tracts in Modern Physics Vol. 92, (Springer-Verlag, Berlin, 1981).
P. Thorel, J.P. Coulomb and M. Bienfait, Surface Sci. 119, L93 (1982).
R.K. Thomas, Prog. Sol. State Chem. fl, 1 (1982).
B.R. Grier, L. Passell, J. Eckert, H. Patterson, D. Richter and R.J.
Rollefson, Phys. Rev. Lett. 33, 819 (1989).
R. Wang, H. Yaub, H.J. Lauter, J.P. Biberian and J. Suzanne, J. Chem.
Phys. 83, 3965 (1985).
C. Riekel, A. Heidmann, B.E. Fender and G.C. Sterling, J. Chem. Phys.
71, 530 (1979). '
P.C. Eklund, E.T. Arakawa, J.L. Zarestky, W.A. Kamitakahara and 0.0.
Mahan, Synth. Met. 1;, 97 (1985).
F. Batallan, 1. Rosenman, A. Magerl and H. Fuzellier, Phys. Rev. &,
9810 (1985).
J.P. Beaufils, 'I‘.L. Crowley, T. Rayment, R.K. Thomas and J.W. White,
Mol. Phys. 91, 1257 (1981).
S. Olejnik and J.W. White, Nat. Phys. Sci. gag, 15 (1972).
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(1979).
D.K. Ross and P.L Hall, Neutron Scattering methods of investigation
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16.

17.

18.

19.

20.

21.

22.

23.

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26.

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100

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331, 8929 (1988).

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X.W. Qian, D.R. Stumpp, B.R. York and S.A. Solin, Phys. Rev. Lett. 33”
1271 (1985).

X.W. Qian, D.R. Stumpp and S.A. Solin, Phys. Rev. 333, 5756 (1986).
‘Y.B. Fan, S.A. Solin, D.A. Neumann, H. label and J.J. Rush, Phys. Rev.
239.

T. Tsang, R.M. Franko, H.A. Resing, X.W. Qian and S.A. Solin, Sol.
State Comm. 33, 117 (1987).

T. Tsang, R.M. Franko, H.A. Resing, X.W. Qian and S.A. Solin, Sol.
State Comm. 33, 361 (1987).

R.M. Franko, T. Tsang, H.A. Resing, X.W. Qian and S.A. Solin, Phys Rev.
B, to be published.

D.A. Neumann, H. Zabel, Y.B. Fan, S.A. Solin and J.J. Rush, J. Phys. C,
‘33, L761 (1987)

Y.Y. Huang, Y.B. Fan, S.A. Solin, J.M. Zhang, P.C. Eklund, J. Hermans,
and 6.6. Tibbetts, Solid State Comm. 33, 993 (1987).

J.M. Zhang, P.C. Eklund, Y.B. Fan, and S.A. Solin, Phys. Rev. 331, 6226
(1987).

J.C. Thompson, Electrons in Liquid Ammonia, (Clarendon, Oxford, 1976).

 

B.R. York and S.A. Solin, Phys. Rev. ggg, 2963 (1982).

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32.

33.
39.

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101

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T. Tsang, R.M. Franko, H.A. Resing, X.H. Qian and S.A. Solin, Solid
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(1969).

102

Chapter IV.
Quasielastic Neutron Scattering Study of Rotations and Diffusion

in Stage-1 K-Ammonia Intercalated Graphite

9.1 Introduction

In Chapter I, we showed that many properties associated with the motion
of individual atoms can be determined by incoherent neutron scattering.
Dynamic properties, such as rotations and diffusion in molecular crystals
are typical subjects for this type of study. Usually, a large incoherent
neutron scattering cross-section is required. This makes the K-ammonia sys-
tem, which contains hydrogen, an ideal candidate for the study of rotational
and translational diffusive motions of molecules confined to a plane1'3.

The so called quasielastic neutron scattering (QENS) technique has
greatly enhanced our knowledge of the miscroscopic details of rotations and

diffusion in molecular crystalsu-7.

The actual quantity measured by QENS is
given in Eq. (1.28). The incoherent scattering function Sincm’w) is the
space and time Fourier transform of the self—correlation function 63(F,t),
now we return to Eq. (1.29)

3mm...) = Eli-IISQJR'Mt-‘dt (11.1)
where the intermediate scattering function 13(6,t) is the space Fourier
transform of 68(3,t) and can also be expressed by Eq. (1.25). In order to
understand the microscopic details of rotations and diffusion, we will
derive the incoherent scattering function for three simple models.
8

9.2 Translational Diffusive Motion

The diffusion motion is simply governed by Fick's law

éflgéfl. = DV2P(F,t) (14.2)

103

where P(F,t) is the particle density at position 1‘ at time t and D is the
diffusion constant. A solution of this equation is given by the self cor-

relation function
1
(9na(t)

where a(t,1) : D(t-I). For true diffusive motion, t>>1. We can then

 

cs(;,c) = exp(-r2/9a(t)), (9.3)

)3/2

neglect 1 and do a space Fourier transform of Eq. (9.3) yielding the ixiter-

mediate scattering function

 

2
13(6,t) = e'Q DC (a n)
which in turn yields a Lorentzian scattering function
2
1 DO
3. (Q,u) = - . (9.5)
inc uh (DQZ)2+m2

fflums, a difquive motion of particles is characterized by a Lorentzian line
shape with an energy width
r = 2hDQZ(9.6)
and zero energy scattering function
sinc(6,0) = 1/nhDQZ. (9.7)
These results have been experimentally verified for liquid argon at 85K by

Skold, et al.9

9.3 Translational Jump Diffusion1O

This is a hopping type motion of atoms on lattice sites in a crystal.
For the simplicity of mathematical treatment we assume that the Jump motion
is random, the Jumps are instantaneous, i.e., the residence time is much
longer than the flight time to the next lattice site, and the available lat-
tice sites form a Bravais lattice. Then the particle motion can be derived

from the rate equation

Q)
'0
'1‘
r?
v
H MD

[P(F+§i,t)-P(r,t)l, (9.8)

l.
nn. 1

1

Q.)
(T

109

where P(r,t) has the same meaning as in Eq. 9.2, T is the residence time and
the sum is taken over all 11 next nearest neighbor sites at distance Ei‘
Using the boundary condition P(r,0) = 6(3), the probability P(r,t) becomes
equivalent to Gs(r,t). Using the Fourier transform method to solve Eq.

(9.8), we obtain

exp( (9.9)

f(5)t
13(6’t) ' T )9

with

(1-e”16'§i). (9.10)
1

Fourier transformation with respect to time yields a Lorentzian line shaped

r16) =

11M:

1
n .
l

scattering function

 

S (6,w) = l—- f(6)/T , (9.11)
“m 1“ (um/112.13
with a FHHM
rinc = 2hr(6)/1, (9.12)

and a zero energy scattering function

1 1
Sinc(6'0) = 33 f?67' (9.13)

For Bravais lattices, each lattice site is an inversion center and we obtain
1 n/2
f(5) = -[n-2 i cosQ-I ]
“ i = 1 i

9 "/2 . 2 col
='- Sln -—-.
n . 2
l = 1

(9.19)

Therefore the line width is an oscillatory function with nodes at the
reciprocal lattice points 5 = 2uI/22 and the zero energy scattering function
has singularities at reciprocal lattice points where the line width vanishes

(see Fig. 9.1). Notice that at small Q's I‘dQZ/t, one can compare this with

105

Sinc (0.0)

 

U MC

277' 477'
01

 

Fig. 9.1 Incoherent scattering from random Jump diffusional motion.

(a)Full—width at half-maximum F plotted against Q! from relation (9.12). (b)

Zero-energy scattering function sinc(Q’0) from eq. (9.13)

106

the pure diffusive motion, F = 2hDQ2, obtaining an equivalent diffusion con-
stant for Jump diffusion, Dad/1H. This makes it possible to compare the
QENS results with those obtained using other methods and also to discern the
activation energy Eo via the Arrhenius relation

D : Doexp[ޤ%-] (9.15)
B

9.9 Rotational Jump Diffusion12

He now consider the case of a particle motion which is constrained to
instantaneous Jumps on an available lattice, forming a closed ring. For ex-
ample, benzene rotates by 60° around its c6-axis or ammonia rotates by 12N)°
around its 03~axis, and the rotation takes a time much shorter than the
residence time (see Fig. 9.2). For illustration, we only derive the scat-
tering function for the three site case (ammonia's Jump rotation), as
sketched in Fig. 9.2. A general discussion of more complex models can be
found in Refs. 12 and 13.

The self-correlation function can be written as

,n
Gs(r,t) =1 1.; 1numb-411) (9.16)

where R1 are the sites which form a ring and the function f1(t) is the prob-
ability that a particular atom is at site i at time t. Assuming that the
Jump rate is 1/1 and the initial condition f1(0) = 1 which means that the

particle 1 is at site 1 at t = O, we have the rate equation
df,(t)
dt

 

1
=2—T-igz [fi(t)-f1(t)]. (9.17)

and the conditions

f2(t) = F3(t), i i 1 fi(t) = 1

The solutions for Eq. (9.17) are

107

 

(o) (b)

Fig. 9.2 (a) Closed ring of lattice sites for rotational Jump diffusion,
(b) triangular arrangement of lattice sites for the model discussed in the

text.

108

f1(t) = g(1+2exD(-3t/21)

(9.18)
(t) = 3(1- -€Xp( '3t/2T).

f2 ,3
For small t, the probability to find particle 11at site 1 is greater than
that of other particles, and when t“), all fi(t) are identical and finite,

which give rise a 5(w) spike in the energy spectrum of Sinc(0,u). From Eqs.

(9.16) and (9.18), we can then obtain the intermediate scattering function
3
IS(Q,t) :1%(1+2exp(-3t/2I))r%(1-exp(-3t/21)) X exp(i0-Ri). (9.19)
i = 2
Fourier transform of Eq. (9.19) with respect to t, yields the scattering

function

1m((2- Z exp(16- ofii)) 3/2‘2
1 = 2 (3/21) +w

exp(10 8 ))5(m)}
i 2

The scattering function consists of two components, a broad Lorentzian with

3..l

 

S nc(6’w): 2

31(1+ (9.20)

3

IIP'Kp Iv

FHHM I‘~1/I due to the Jump diffusion within the triangle, and a narrow 5

component caused by the finite value of the correlation function at infinite

 

times.
Using IRZI = IR3I, and taking the powder average of’the scattering
function we obtain
1 3/21
s (6,6) =‘-— 111-1 (QR)}
nc uh 0 (3/21)2+w2
1 1
«glguomanamn _ (9.203)

The intensities of the two components are oscillatory functions of the scat-

tering vector, and the sum of two is a constant (see Fig. 9.3).

109

 

F~1/T.

Sinc (010)
3

 

Sinc (010)

 

 

 

Fig. 9.3 Scattering function for rotational Jump diffusion. (3) energy
spectrum of broad and narrow component, (b) dependence of Sinc(Q,0) on the

scattering vector for the broad component (solid line) and the narrow com-

ponent (dashed line).

110

9.5 General case

Apparently, the microscopic diffusion mechanism is not limited to the
above three simple examples. For instance, if the thermal energy is of the
same order as the static potential barrier separating the wells, the Jump
diffusion rate 1/1 becomes a distribution function of time, therefore the
Jump diffusion can be replaced by rotational difffusionu. In this case the

scattering function becomes1n

__I_1__
2 2
F191.)

where p is the radius of the rotating molecule, 9 is the angle between the

Sincm’w) = J3(stin8)6(m)+% X Jimpsine) (9.21)

l 1

axis of rotation and 0, and I‘ = l(l+1)DR with 0 representing the rota-

l R
tional diffusion constant. We see that the main features of the scattering
function in Eq. (9.21) are same as those of Eq. (9.20), however, the broad
peak now is composed of many different Lorentzians of varying widths, thus
making the total width of this component a function of 0.

Besides the molecular motions discussed in previous sections, there are
two other types of molecular motions, which have been extensively studied by
QENS, i.e. quantum mechanical free rotations and tunneling6’15. The quan-
tum mechanical free rotation can only occur when the molecule interacts so
weakly with its environment that it can be treated as a free rotator.
However, tunneling involves excitations between energy levels which are
split due to the overlap of the wavefunction between adJacent potential
wells. This provides a new method to measure the local potential. The
energy transfer for these two types of motion usually is of the order of
uev's. Therefore the measurement requires high resolution spectrometer such

as a backscattering spectrometer and has to be carried out at very low

temperature.

111

Usually, a system displays more than one type of diffusive motion. In
this case the total scattering function is the convolution of different
types of motion6 in which case the intermediate scattering function can be
written as

vib
13m) = Is (6.1;)1J113J(6,c) (9.22)

where IsVib(0,t) = exp(-02<u2)) is the Debye-waller factor and the product
is taken over different diffusive motions. The motions can be separated, if
they occur on different time scales, because they have different widths.
Even if they happen on the same time scale, some information can also be
deduced since they have different Q dependences of the zero-energy scatter-

ing functions.

9.6 Experiment

The samples of K(NH3)XC2u addressed in this chapter were prepared in
the same way as those described in Chapter II. The differences are the size
of the sample and the fact that hydrogenated amonia has been used here.
For quasielastic neutron scattering, the rule of thumb is that the multiple
scattering from the sample has to be less than 101, so that the sample
thickness was estimated to be less than 0.7 m. We used a 2X1.5x0.05 cm3
piece of pyrolytic graphite to form the stage-2 BGIC KC

2
technique described in Chapter I. The sample was then placed into an

9 using the 2-bulb

aluminum sample can and ammonia vapor was introduced to form the stage-1
TGIC K(NH3)9.3C29' The results presented here were again obtained using a
triple axis spectrometer (BT-9) located at the National Bureau of Standards
Reactor except for a fixed window scan [see Fig. 9.13 and associated dis-
cussion] which was obtained on the backseattering spectrometer IN10 at the

Institute Laue-Langevin, Grenoble, France. For the triple-axis

112

spectrometer, pyrolytic graphite ((002) reflection) was used for both the
monochromator and the analyzer. The fixed initial energy mode was used with
the incident neutron energy at either 9.9 mev (collimation 90'-20'-20'-90')
or 3.69 mev (collimation 90'-90-'90'-90') yielding instrumental resolution
about 120 uev respectively. A cooled Be filter was placed in the direct
beam to remove all harmonic contamination and also to eliminate the fast
neutron background. The data were corrected for the varying resolution
volume.

For reference, we also measured the elastic neutron scattering patterns
of K(ND3)9.3C29 at 5K (Fig. 9.9) and 200 K (Fig. 9.5). These patterns were
obtained using the deuterated sample discussed in Chapters II and III. He
already know from Chapter II that the inplane structure of the K-ammonia
layer is liquid-like at room temperature. As the temperature is decreased,
the inplane structure shows a disorder-order transition as evidenced by
Figs. 9.9 and 9.5. The ordered phase cannot be determined unambiguously
from these neutron scattering patterns alone. It is possibly a multiple
phase structure or a modulated 3x3 structure. In this chapter, we will dis-
cuss quasielastic results in both the low temperature ordered phase and the
high temperature liquid phase. We will further consider this ordered struc-

ture below.

9.7 Results and discussions

In Fig. 9.6, the quasielastic scans for several different Q's at room
temperature are shown. They were fitted by two component peaks (two
Lorentzians) broadened by the Gaussian instrumental resolution. The broader

Lorentzians have a width of 1.2 mev which is independent of 0 indicating the

113

 

 

Graphite

A1

 

 

 

CounIS
I

 

Al

1 L L l - l I 1
0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
(Mi-'1

 

 

 

 

 

 

 

 

 

Fig. 9.9 Inplane elastic scans of K0103)“ .1K29 at 5K, showing an ordered

phase, possibly mltiphase structure.

114

 

T=ZOOK

 

Graphite
J

Counts
I
Al

Al

 

 

 

 

 

 

2.0 2.5 3.0 .3.5 4.0
003'!)

o 0.5 1.0 1.5

Fig. 9.5 Inplane elastic scans of K(N03)u 1K2,4 at 200K, showing that the

intercalant is essentially a liquid at this temperature.

 

 

 

 

 

 

Intensity (arbitrary units)

 

 

 

 

 

 

 

 

[ - £41 l
. -3 -2 -1 0

Energy(mev)

1 2

Fig. 9.6 The quasielastic spectra of K(NH3)u 3C2Q at 300K for a scattering
vector at Q=0.6 A'1(a), 0:1.10 A"1 (6), 0:2.3 (1‘1 (c). Each spectrum was

fitted by two Lorentzian functions giving the widths plotted in Fig. 9.7.

116

existence of rotational diffusion of the NH3 molecule. The central peaks
are broader than the instrumental resolution which is evidence of transla-
tional difmsion. The widths vs. Q for ’I' = 300K are given in Fig. 9.7. As
expected,the width of the rotational component is independent of the scat-
tering vector, and the width of translational component is apparently
proportional to 02 for small 0. From the Q-dependence, we estimate the
translational diffusion constant to be 1x10'6cm2/sec at T=300K.
Quasielastic scans for different temperatures are given in Fig. 9.8.
At T = 10K, a good fit to the data was achieved by the use of a two com-
ponent peak (a S-function and a Lorentzian) broadened by instrumental
resolution. The width obtained from the fit is still 0.9 mev even at 10K.
As the temperature is increased, the width of this component increases. At
T = 150K, a width of 1.3 mev was obtained from the fit. As the temperature
is further increased to T = 200K, it is found that the first Lorentzian has
broadened into a flat background, but another Lorentzian with a width of 0.2
mev is clearly manifested in the scattering function. At room temperature,
this Lorentzian component has a width of 1.2 mev. The only reasonable ex-
planation for this is that the first Lorentzian at lower temperature is due
to the rotation of the NH3 molecule with lower activation energy and the
rotation which persists at room temperature has a higher activation energy.
Arrhenius plots indicate that the activation energy for these two rotations
are 2 mev and 80 mev respectively (Figs. 9.9 and 9.10).
.Fig. 9.11 shows the zero-energy transfer scattering function for the
second rotational component and the translational component of the
quasielastic line at room temperature. The fact that both components

quickly drop to zero intensity instead of oscillating indicates that the

117

 

O

------- ---------d--

1.0— —

T= 300K

0 Translation
0.5 - O'Rotoiion -

P(meV)

‘

 

 

Fig. 9.7 The width (obtained from the fits in Fig. 9.6) of the high tem-
perature rotational and translational components of the scattering plot ted
as a function of the scattering vector at 300K. From the Q dependece of the
broadening of the translational component, we obtain a diffusion constant of

1.019106 cmZ/sec.

 

T.10K. {-130.42 (“CV
.. o" 1 ago A" I"... I120 mev _,

  
 

-

 

 

 

 

)- -1
° . . m1
1' ugog 1‘; 11.25 am!
- o,,=2.oo A“ rm, - szpw -
é
& cll
§
§ . : r - i
Z 1' .200K ‘ ("2 10.21ch
§,—°".'.851 . qu'1ZOI‘CV «-
3

    

.M—“ 'r "finale-need.

 

 

 

T . m* r; ..J7MW
_°" . 1.10 r. 1“" .0226!“ .1
nuttchev

l - I
—o.. ~03 0.0 0.4 . 0.8
ENERGY (meV)

L

 

Fig.9.8 (a) A quasielastic spectrum of K(NH ) C at 10 K. The open
circles are the experimentally measured points, *h‘fifl' curve is a fit to a
5-mnction plus a Lorentzian, both convoluted with the Gaussian instrumental
resolution, and the full sqares represent the Lorentzian contribution to the
scattering. A scattering function of this type is indicative of a rota-
tional motion of the ammonia molecules. (b) A quasielastic spectrum at 150
K. This shows the broadening of the Lorentzian component of the scattering
function. The activation energy of this motion is estimated to be about 5
mev. (c) A quasielastic spectrum at 200 K (above the intercalant melting
transition). Here the previously observed Lorentzian has broadened into a
flat background. However the peak still includes both 8-function and a
Lorentzian component due to the appearance of a different reorientational
motion. ((1) A quasielastic spectrum at room temperature. The elastic com-

ponent has also been broadened due to translational motion of the ammonia
molecules.

119

 

 

tn([])(peV)
O)

    

£0 = 3.2 mev

 

 

 

4 L L L L
O 0.4 0.8 1.2
l/kT(meV")

Fig. 9.9 .Arrenius plot which shows the ln of the peak width as a function
of 1/kT for the low temperature rotation. The solid lines are drawn fknr ac-
tivation energies of 1.3 mev (best fit ) and 3.2 mev. From this we estimate

the activation energy to be 2 mev.

120

 

    

Eo= so mev'

 

 

 

' 1
0. . - 0.05 0.1
I / kT(me\/")

Fig. 9.10 Arrenius plot for the high temperature rotation. The solid line

represents an activation energy of 80 mev.

121

 

i i l
T=3>OOK

0 Translation
0 Rotation (x2)

Intensity

 

 

 

 

 

Fig. 9.11 The intensities of the high temperature rotational and the

translational compounds of the scattering plotted as a function of Q for

Tz300K.

122

motion is highly complex and cannot be easily separated into rotational and
translational components at these scattering vectors.

It is clear that the current results, obtained with a triple-axis
spectrometer located at a thermal neutron reactor, do not include quasielas-
tic spectra at enough values of the scattering vectors, to allow a
comparison of the intensities and widths of the various components of the
scattering function to a partimiuu~nmdel. Thus it is difficult to deter-
mine the Jump vectors of the rotational or translational motions, and to
assess if Jump vectors are a valid concept for any of these motions in any
of the temperature ranges studied. Nevertheless, it is possible to give a
reasonable assignment to each of the diffusive motions with the aid of other
experimental results.

The quasielastic scattering observed at low temperatures is almost cer-
tainly due to rotations of the hydrogens about the threefold molecular axis.
The rotation observed at high temperatures is assigned to the reorientation
of the K-NH3 complex and the translational motion is either the motion of
the complex as a unit, or of the individual ammonia molecule, or both. This
assignment is consistent with the fact that the scans performed with 0 per-
pendicular to the carbon planes show that the two motions appearing in the
liquid phase take place mostly parallel to the basal plane. This assignment

17 which indicate that at room tem-

is also in accord with recent NMR results
perature both spinning of the anmonia molecule about its threefold axis and
precession of this axis about the graphite c-axis are present and that botwi

5 Hz. when we include the

motions occur with frequencies greater than 10
librational motion discussed in Chapter III, the overall motion of ammonia

can be described in Fig. 9.12.

123

 

 

 

 

Fig. 9.12 A schematic diagram of the motions in K(NH3)u.3C2u. The molecule
spins about the three-fold molecular axis, reorients about a given K atom,
and translates either through the motion of the entire K-NH3 cluster or by
the ammonia moving between adJacent cluster. Note that only the spinning

motion occurs in the low-temperature ordered phase.

("7

13

('3

(3

C')

129

The ammonia molecule spins about its threefold axis at low temperature;
reorients about a given K-ion; and translates either through the motion of
the entire complex or by the ammonia moving between adJacent complexes; sun:
librates about the K atom (see Chapter III).

Fig. 9.13 shows a so called "fixed window" scan taken on the backseat-
tering spectrometer IN10 at the Institute Laue-Langevin with an energy
resoluth31cfl‘about 0.5 uev. The energy window for the scan is 15 uev.
Here we display the total intensipycn‘the elastic component of the
quasielastic line in this window as a function of the temperature for
Q = 1.68 904 directed both in and out of the basal plane. At high tempera-
ture above 185K, the inplanar structure is liquid-like so the elastic
component is weak, the scattering is mostly inelastic due to the Doppler
effect. When the temperature is decreased to 185K, a dramatic increase in
the elastic intensity occurs for Q parallel to the basal plane. This is a
clear signal of an inplane liquid-solid transition. Recalling that the
order-disorder transition we presented in the previous section occurs at
about the same temperature, leads us to conclude that this effect must due
to the same mechanism, i.e., the onset of translational diffusion and pos-
sibly also the reorientation of the NH3 molecule discussed earlier. As the
temperature decreased from 185K to about 30K, the intensity increases
linearly, this temperature dependence is consistent with the thermal thebye-
Waller factor in the high temperature limit. Below 30K the elastic
intensity increases for both Q in and out of the basal plane, indicating
that another low energy motion is frozen. The best candidate for this mo-
tion is certainly the rotation of the ammonia about its three-foLd symmetry

axis since it would display components in both of the directions

125

 

 

 

 

 

 

 

1
a —1
Q
X
75
E
.3 ' -
I I g
L l l 1
00 so 100 . 150 zoo

Temperature (K)

Fig.9.13 The elastic intensity as a fUnction of temperature for two direc-
tions of scattering vector Q both having a magnitude of 1.68 A'K.This
.intensity includes all of the counts within the resolution width of 1/2
uev.For T<30K, the increases quickly as temperature decreases, possibly in-
dicating the presence of rotational tunneling. For 30<T<180K shows a linear
decrease in the intensity with increasing temperature. Then For T=185K,
there is a sudden drop in the intensity of the elastic component for 0
parallel the graphite basal plane, which is associated with the inplane

transition.

126

and since the temperature for this motion to appear is at 30 K which cor-

responds to the 2 mev activation energy of this motion determined from Fig.

9.10.

In conclusion, the model of the spinning, reorientation and libration

qualitatively explains the quasielastic neutron scattering data. In order

to unambiguously determine the dynamics of this system more data are needed

for different temperatures and Q's.

10.

11.

12.

127

References

 

C. Riekel and A. Heidemann, B.E.F. Fender, and G.C. Stirling, J. Chenn
Phys. 11, 530 (1979).

1L Zabel,!L Magerl, A.J. Dianoux, and J.J. Rush, Phys. Rev. Lett.
.QQ.2099 (1983).

F. Batallan, I. Rosenman, A. Magerl, H. Fuzellier, Phys. Rev. 333, 9810
(1985).

111. Press, Single-Partical Rotations in Molecular Crystals, Springer
Tracts in Modern Physics vol. 92, (Springer-Verlag, Berlin, 1981).

J.H. White, in Dynamics of Solids and Liquids by Neutron Scattering,
edited by S.W. Lovesay and T. Springer, Springer Topics U1CMrrent
Physics vol. 3, (Springer-Verlag, Berlin, 1977), p. 197.

T. Springer, in Dynamics of Solids and Liquids by Neutron Scattering,
edited by S.M. Lovesay and T. Springer, Springer Topics in Current
Physics vol. 3,( Springer-Verlag, Berlin, 1977), p. 255.

H. Zabel, in Nontraditional Methods in Diffusion, edited by 6.8. Murch,
R.K. Birnbaum, and J.R. Cost, (The Metallurgical Society of AIME,
Harrendale, PA, 1989), p.1.

G.H. Vineyard, Phys. Rev., 119 999 (1958).

K. Shold, J.M. Rowe, G. Ostrowski, and P.D. Randolph, Phys. Rev. 33
1107 (1972).

C.T. Chudley and R.J. Elliott, Proc. Phys. Soc. 11, 353 (1960).

C.P. Flynn, Points Defects and Diffusion, (Clarendon Press, Oxford,
1972).

V.F.Sears, Can. J. Phys., 95 237 (1967).

13.

19.

15.

16.

128

T. Springer, Quasielastic Neutron scattering for the Investigation of
Diffusive Motions in Solids and Liquids, Springer Tracts hitKMern
Physics, vol. 69, Springer, Berlin, Heidelberg, New York, 1972.

J.A. Janik, J.M. Janik, K. 0tnes, and K. Rosciszewski, Physica 333,
259 (1976).

D.A. Neumann, H. Zabel, J.J. Rush, Y.B. Fan, and S.A. Solin, to be pub-
lished.

'T. Tsang, R.M. Franko, H.A. Resing, X.H.<Quuh and S.A. Solin, Solid

State Comm. 3g, 117 (1987).

 

129

Chapter V.
Neutron scattering studies of (CH3)3NH+-Vermiculite

5.1 INTRODUCTION

Recently, clay intercalation compounds (CIC's) have drawn the atten-
tions not only of chemists but also of solid state physicists due to the
novel chemical and physical properties”6 of these materials. Layered sili-
cates such as vermiculite as well as it's intercalation compounds, which are
the focus of this chapter, are unique among lamellar solids in their ability

to be pillared3

by robust intercalated guest ions which occupy specific lat-
tice sites in the interlayer galleries. These materials provide a testing
ground for the study of 2D layer rigidity and pillaring, the latter of which
is a special example of the more general phenomenon of intercalation.
Pillared clays also provide a unique system for the study of 2-dimensional
microporosity. The distinguishing feature of pillared clays is that the
gallery cations are robust 30 species which function in the water-free in-
terlayer space as molecular props. The pillaring cations can be uniform in
size and spacing within the galleries. Therefore, the enormous free volume
of accessible interior space that is derived from such an open structure has
significant practical applications in the field of catalysis and sieving.
In addition, the microporous structure provides a possible system for prob-
ing the 2D percolation process because one can accurately control (to one
part in 105) the number of open "bonds" in the planar network by controlling
the concentration ()1) in a mixed ion pillared clay such as A1-xBx-

Vermiculite. This control is especially important near the percolation

7

threshold where fractal8 behavior dominates the structural arrangement of

the intercalated species.

130

Mixed ion compounds of the type, A1_xBx-Vermiculite also provide a sys-
tem for the study of the dynamics of 2D molecular crystals formed by the
intercalant. Such a study can give information about the pillaring
mechanism. One expects that the molecular motion can be separated irnxa two
parts, the external modes related to displacements of and rotations about
the molecular center of mass, and the internal modes of much higher fre-
quency, related to vibrations with respect to the center of mass. Neutron
scattering is a powerful tool to elucidate these properties.

Vermiculite is built up from sheets of corner shared SiO,4 tetrahedra
and sheets of edge shared AlO6 octahedra to form a 2:1 layer structure such
as that shown in Fig. 5.1. Such a clay has a cation exchange capacity which
depends upon the substitution of lower valent atoms for aluminium in the oc-
tahedral sheet. The 2:1 clay layers are thus negatively charged because of
this and charge neutrality is provided by the cations which are present in
the galleries. Note that unlike graphite which is amphoteric, the clay
layers have a fixed negative charge. The intercalation process in these
compounds is equivalent to ion exchange but does not involve charge transfer
between the guest and host species. The basal oxygen layers are arranged in
a Kagome lattice whose hexagonal pockets form a triangular lattice of gal-
lery sites. In the interlayer galleries of vermiculite as well as other
clays chemical reactions can be selective, specific, and quite distinct com-
pared with the corresponding reaction in the free space.

9-11

'. Recently, attention has been focused on the structural and dynami-

1_xBx--vermiculite, 0 S X S 1, with A-

trimethylammonium = (CH3)3NH+ and B = tetramethylanmonium = (CH3)HN+. The

cal properties of the ternary CIC A

guests in this system area sufficiently robust to be considered as pillars

but structurally tractable enough to illucidate the physics of the pillaring

131

02
Si,Al

01

M9
01

 

 

Si,A|
02

   

Fig. 5.1 A schematic illustration of the tetrahedral and octahedral sites
in a 2:1 layered silicate (after ref [3]). Open circles are oxygen, closed
circles are cations in the tetrahedral (Si,Al) and octahedral (Al, Fe, Mg,
Li) positions. Hydroxyl groups are represented by doublcflrcles.(h1the
right side, the structure seen by [002] defraction is shown. The scattering;
length of each layer is treated as a fitting parameter Bi and the position

of each layer can be derived from the 0-0 distance do-o shown in table 5.1.

132

process. The concentration dependence of the normalized basal spacing dn(x)
of pillared vermiculite has been measured11 for the mixed layer system
[(CH3)uN+]x[(CH3)3NH+]1_x-Vermiculite compared with that of Cs_x RE -
Vermiculite. A phenomenological model which relates dn(x) to layer rigidity
and the binding energies of gallery and defect (non gallery) sites yields
excellent fits to the basal spacing data. However, the micromechanism which
determines the layer rigidity and the binding energies of the gallery and
defect sites remains to be clarified.

10,12

Theoretical and experimental lattice dynamical studies of interca-

lated layered silicates have been reported. Gupta, et al.12 investigated

the lattice dynamics of Cs xbe-Vermiculite using a force constant model

1-
which included both central and angular force and the virtual crystal
approximation. Experimentally, Raman scattering studies of the torsional
modes of the tetrahedral sheets of Cs1_xbe-Vermiculiteu and
[(CH3)3NH+]1_x[(CH3)uN]x-Vermiculite9 have been reported.

From the molecular crystal point of view, the multimethylammonium-
vermiculite clays are closely related to multimethylammonium halides. The
internal modes of multimethylammonium would, therefore, be expected to be
similar for these two type of materials, and their differences would reflect
the different environment of the multimethylammonium. Numerous studies have
been reported on the vibrational spectra of multimethylammonium halides
using IR, Raman and neutron scattering13-17.

In this chapter, we will present an elastic neutron scattering study of
the structure of trimethylammonium-Vermiculite intercalation compounds. We

also address the internal mode properties of the guest species using the in-

elastic neutron scattering and IR spectroscopic techniques.

133

5 . 2 EXPERIMENTS

The [(CH Min-Vermiculite used in this study was made from natural

3’3
Mg-Vermiculite from Llano, Texas, by using an ion exchange method. Selected
crystals of Mg-V were ground into powder in which the crystallites were a
(:11 microns in size. [(CH3)3NH+]-Vermiculite was prepared from this powder
by ion exchanging with (CH3)3NH+ in the presence of the EDTA anion and then
sedimenting the exchanged particles on a glass plate to form a self-
supporting oriented film with its c—axis perpendicular to the plate. Water
was removed by heating the sample in an oven over 12 hours at a temperature
of 100 C°. The large film was cut into many smaller pieces of size about
2x3 cm2 and these were mechanically stacked together with their c-axes well
aligned. A total weight of the sample was about 3gm. The deuterated sample

was prepared with same method using (CD ND+ instead of (C113 )3NH+. The

3’3
typical mosaic spread of the crystallite c-axes for both samples was about
30°.

All neutron scattering results presented here were obtained using a
triple axis spectrometer (BT9) located at the National Bureau of Standard
Reactor. The elastic scattering was acquired with a fixed energy of 13.7
mev and a collimation of 90'-20'-90'-90' . A one inch PG filter was used in
front of an analyzer to remove high order contamination. The diffraction
patterns of hydrogenated and deuterated trimethylaumonium vermiculite were
obtained at room temperature and are shown in Figs. 5.2 and 5.3
respectively.

The vibrational density of states was obtained from the inelastic

neutron scattering data using the incoherent approximation, that is , we can

replace the dynamic coherent scattering function with its incoherent

134

 

 

 

 

 

 

 

1 F f F 1 1 l 1 i
300- _
— C = 12.71 A _
2'3
‘1’ o
:3 200+- v _
.—e
X
(D A
..1 — <1-
1: o " W
:3 c3 “3
o A A v __ o
o 8‘ 8 8 ° ..
1001- o o 3 v- S __
8 .3 < 1 < a
0 JL L 1 L 1 1 L 1 1
O 2 4 6 8

Qz (in units of 21r/c)

Fig. 5.2 The elastic neutron scattering pattern in the [002] direction for
hydrogenated trimethylammonium-vermiculite. The peaks labeled by [(1NM1) are
associated with an impurity phase. The horizontal axis is normalized to the
first Brillouin zone which corresponds to a lattice period of 12.71A. Peaks

labeled Al are from the aluminum sampie holder.

135

 

 

 

 

 

 

 

 

 

 

 

1 1 1 FT 1 F 1 1 1
270? A _
A N
‘5 o
o O _
_ v 1 v A C — 12.71 A _
1 v
. o
0,1 o
0180- "' _
e-A
X
0)
‘5 " 23 ‘
A
:1 m o
o 8 3
0 90— e. _
.. a 3
A A 1" o
_ 1 s 3 E 5 c -
o a— _- v
1 3'1 2 “ “
o L 1 1 1 1 : ab— 1
O 2 4 6 8

Oz (in units of 217/ c)

Fig. 5.3 The elastic neutron scattering pattern in the [002] direction for
deuterated trimethylammonium-vermiculite. The peaks labeled by 1(00n) are
associated with an impurity phase. The horizontal axis is normalized to true
first Brillouin zone which corresponds to a lattice period of 12.71A. Peaks

labeled A1 are from the alumimum sample holder.

136

18,19

counterpart Within this formulism, the scattering functicni is given

by20
d’o k' n(m)+1

where Ii(m) is defined by

0.
l - 2 -2W.
fi:'\(6'fii) e 1>. (5.2)

Ii(w) =

Here k and k' are the magnitudes of the incident and final neutron wave vec-
tors, n(w) is the Bose occupation factor for a vibrational state of energy
1», and xi, a“) 01, and 61 are the atomic concentration, nuclear mass, total
scattering cross section and displacement vector of the ith atomic species,

respectively; H is the Debye-Haller factor, which can be assumed to be neg-

i
ligible at the sample temperatures in this study. The angle brackets
indicate an average over all sites of type i and over all modes of energy w.
Finally, g(ui) represents the density states for the ith type of atom, which
is defined by

g(mi ) = {5(w-w ), (5.3)

a i,a

where the sum is taken over all normal modes. Therefore, the scattering
does contain information about the vibrational density of states; however,
the contributions from different species are weighted by the value of 11(w).
From Eqs. (5.1) and (5.2), we see that the scattering cross section is

proportional to x o /m thus, for our system the scattering from hydrogen

1 i i’

will be dominant in the spectra.
The vibrational spectra were obtained using a Be-graphite-Be filter
analyzer assembly (the "trash" analyzer ). The analyzer was placed at a
scattering angle of 90° allowing Q to vary as a function of energy. For the

energy range from 20 to 110 mev, pyrolytic graphite [(002) reflection] was

used as a monochromator. The collimation was ll0'-20'-F.‘0'-110', yielding a

137

resolution of about 2-3 mev. At large energy transfer (35 mev 2 E 2 1110
mev), a Cu monochromator [(220) reflection] was used with a collimation of
60'-li0'-20'-140', yielding a energy resolution of about 2-5 mev. Since the
samples have a very large mosaic, out-of-piane vibrational mode measure-
ments, i.e., 6 parallel to the c-axis, were made with the sample fixed with
the c-axis parallel to the incident beam. For the in-planar modes, i.e.,
with Q perpendicular to the c-axis, it was rotated by 90°. The spectra for
both samples were obtained at liquid nitrogen temperature.

The contribution to the scattering from fast neutrons was measured and
subtracted. The scattering from the sample can was found to be negligible
and the multiphonon and multiple scattering were sufficiently featureless to
be ignored. A single vibrational density of state was obtained by sealing
the two different energy ranges according to the overlapping energy regions.
The "trash" analyzer spectrum of hydrogenated trimethylanmonium with (3 per-
pendicular to the c—axis of vermiculite is given in Fig. 5.“ and that with 5
parallel to the c-axis is given in Fig. 5.5. The patterns for deuterated
trimethylamonium-vermiculite are given in Figs. 5.6 and 5.7 for 6 perpen-
dicular to the c-axis and 6 parallel to the c-axis respectively.

The IR spectra were obtained on a IBM PC-AT based IR/lfll infrared
spectrometer. The peak positions are given by a built in data processor.
The IR spectra of the hydrogenated and deuterated samples are shown in Fig.

5.8. All IR data were collected at room temperature.

5.3 DISCUSSIONS
Elastic scattering
The elastic neutron scattering pattern along the [001] direction for

both the deuterated and hydrogenated are shown in Fig. 5.2 and Fig. 5.3,

138

 

 

 

 

 

I I I I l l l W
80 82
37 51 +3
31 + 3 ' l
28 + 58 .
.53 i i
g #44} $ + I” : ++ 103 Y++
8 i + + + # i a + +
1 11. + i
t ¢§ § fl: ++§ _ §§§
“ fl . F +”§§
l | I l L l I
20 5O 80 110 140

3 - Energy (meV)

Fig. 5.4 The spectrum of vibrational density of state of hydrogenated

trimethyl ammonium vermiculite with scattering vector 6 parallel to the

basal plane of the silicate sheet.

. ... 1..“

139

 

 

 

 

 

[ , F T F l V r
37
1°:
i
U, 28* i 51
a 1
5 f 1 82
o *1” 4‘
C.) fl ’
4‘ W1»
£1 1» NH, 3
+4 1 t. ‘1, 1 103
4 g" 5*” ¢ ¢¢¢
‘1 ¢”¢, ,H, 1‘
¢ o
0 .¢
1 I l l l L L
20 50 80 110 140

Energy (meV)

Fig. 5.5 The spectrum of vibrational density of state of hydrogenated
trimethyl ammonium vermiculite with scattering vector 6 perpendicular to the

basal plane of the silicate sheet.

140

 

6000

r rrrrfrfirTFI I I rFrr I I F[ r
30.,

5000

ft

4000

Counts

3000

 

zooo ' -..-

ril'U'IlI'IUIII'U
A
I ° l l L
All I‘ll ALLA [All 1141

 

1000‘lLLLLLgkLLLLLlLLLILPLPL

40 . 60 80 100 120
Energy(mev)

° .Fig. 5.6 The spectrum of vibrational density of state of deuterated
trimethyl ammonium vermiculite with scattering vector 0 parallel to the

basal plane of the silicate sheet.

 

.nfliZCC

141

 

 

 

 

550 1 r I 1 r 1 1

4.00L .—
3 27
5 - 51 32 103 -
o .. .

44 " 58 1
250 , ,   .. —
100 1 L 1 L 1 1 1
20 50 , 80 110 140

Energy (meV)

Fig. 5.7 The spectrum of vibrational density of state of deuterated
trimethyl ammonium vermiculite with scattering vector 0 perpendicular to the

basal plane of the silicate sheet.

 

IHANSMIIIANCE

IRANSMITTANCE

Fig. 5.8

ammonium

(”8“ cm-

142

 

 

 

1000 800 600 400

 

0.040: 105
0020-: 102

  

 

 

1000 ‘ 000 600 40°
wAvENUMBERS

The IR spectrum for deuterated (a) and hydrogenated (b) trimethyl
vermiculite .at room temperature. The adsorbtion band around 60 mev

1) is associated with the brucite layer optical modes, the band

around 80 mev (6M6 cm‘1) is due to the inplane modes of the silicate sheet.

143

respectively. The horizontal-axes of these diagrams are normalized to the
reciprocal lattice in the c-direction which corresponds to a repeat distance
of 12.71 A. The peaks at 5.“ and 6.30 are identified as the (111) and (200)
peaks from the sample can. Notice that there is another set of Bragg peaks
labeled as I(00n) which refers to an impurity phase from the origiiuil clay
mineral“. The peak at “.6 has not yet been identified, but is probably due
to an inplanar peak leaking inUJIIKH] direction since our samples have a
very large mosiac. All Bragg peaks have an instrumental resolution limited
width indicating that the layers are well ordered in the c-direction. In
order to obtain c-axis structural information, the structural form factor is
calculated according to the structure shown in Fig. 5.1.
The unit cell composition of trimethylamonium-vermiculite is

[(CH NH*]2-(Ai

3)3 0.30Feo.03"35.66)2(515.73A12.28)°20(°")u°
The form factor can be written as
P(Q) = 1X1 aniJexp(QRi)

where the sum is taken over all layers in a unit cell, n is the number of

ii
is the scattering length of the type J atom
th

atoms of type J in ith layer, bJ

and Ri is the distance of the 1

tion the Mg layer was chosen to be the origin.

layer from the origin. In this calcula-

Let B1 = ijnij which represents the total scattering length of layer i
J

in a unit cell. Then

F(Q) = {Biexp(QRi).
l

The Bi and R are listed in Table 5.1 for the ideal clay structure. The in-

i
tegrated intensities of the Bragg peaks are related to the form factor as

follows

 

J
0

 

(A

 

 

144

Table 5.1 A comparison between the calculated total scattering lengths Bi
of layer i in a unit cell of tne ideal structure shown in Fig. 5.1 and that
derived from a fit to the data of Figs 5.2 and 5.3 using a model described

in the text.

 

 

B. CaICulatlhn fit
1

BMg 6.15 5.90
1301 2.73 2.83 _____
BSi,Al 1.00 1.89
802 3.98 3.37
BI(H) -1.63 2.63
81(0) 19.19 8.50
RI 1.50 1.59
d 2.69 2.69
o-o

 

All 81's are in units of 10'12 cm, R and do-o are in units of A.

I

 

he.

this

fine

Sink

has
spb

11100

thj

 

 

11111

11

1115

2
lint(Q)°‘|F(Q)l . (5.2)

The integrated intensities were corrected for the Lorentz factor, which in

this case is approximately21

1 A

L °‘ _sin0 —Q (5.3)

where A is a factor independent of Q and can be treated as scaling factor.

Since the trimethylamonium is expected to be rotating in the gallery at

room temperature”, it is not a point scatterer and a molecular form factor

has to be used in the calculation. We treat the trimethylammonium as a
spherical molecule (many other models have been tried and the spherical

model is the best choice) with a radius of R (see Table 5.1). Therefore

I
the molecular form factor can be written as

31,100,)

F(Q)=B
THA QRI

(5.11)

where BI is the radius of trimethylamonium.
However, the calculation according to this ideal clay structure cannot

fit the observed experimental results, therefore we treat the B 's as fit-

i
ting parameters. Since the fitting parameters are correlated for deuterated
and hydrogenated samples, these two sets of data have been fitted
simultaneously21.

The best fit to the integrated intensities of the Bragg peaks are shown
in Figs. 5.9a and 5.9b for the deuterated and hydrogenated sample,
respectively. The fitting parameters are also given in Table 5.1 for com-
parison with the ideal structure. As can be seen from Figs. 5.9a and 5.9b,
the fits are reasonably good, and the parameters derived from these fits are
comparable to those for the ideal structure except for the intercalant

concentration. This deviation can be explained by the assumption that the

trimethylamonium has not fully substituted the ions which were originally

 

146

 

 

 

 

 

 

 

I 1 1 1 1 T l 1 r
' (a)
’ Deuterated

sample -
:0): _
'E
3
a d
E!
,é ..
‘9 L__m_fl_.l=_l:.__l—-
2: 2 4 6 8 I
E
2 r l 1 T I 1 ‘
.s (b)
B
E Hydrogenated
3 sample .
E.

 

 

 

 

 

2 4 5 ‘3
92 (in units of Zn/c)

Fig. 5.9 A comparison beteenw the integrated intensities obtained by
the neutron scattering measruements shown in Figs. 5.2 and 5.3 (solid bar)

and that from the best fit (open bar) using a model discussed in the text.

 

1117

in the gallery. The percentage x of trimethylammonium intercalated into the

clay gallery and the total scattering length of remaining ion Bto-rem in a

unit cell can be found by solving the following equations
Bto-rem+BH,calcx : BH,Fit
Bto-rem+BD,calcX BD,Flt

i.e. (5.5)

Bto_rem+2X(-O.815)x = 2.63

Bt0_rem+2x9.59x = 8.50

These equations give a solution B = 3.1 and x = 281. This result is

to-rem
consistent with the recent x-ray experiment and calculation that the per-

centage of nongallery sites in trimethylamonium is 70$ 11. Bto-rem : 3.1
corresponds to about one Mg plus one Fe ion remaining in the gallery, which
is very likely the case. Thus we conclude that the concentration of

trimethylamonium is about 30 1, and that the ion exchange is not complete.

Inelastic scattering

The spectra of the hydrogenated trimethylammonium-vermiculite for
two 0 directions are given in Figs. 5.11 and 5.5. In comparing them we see
that the peaks at 28 mev and 37 mev are isotropic whereas the peaks in the
rest of the spectrum are mostly associated with inplane modes. In fact,
some of the peaks for 0 parallel to the c-axis (Fig. 5.5) are simply the
leakage from the 0 perpendicular configuration into 0 parallel. This can be
shown simply as follows. The scattering intensity is proportional to (0-5)2
where a is the polarization direction of the mode. Assume that the mosiac
has a Gaussian distribution and that the mode measured is totally inplane.
We can calculate the ratio, R, of the peak intensities for the two

directions.

_ m ..A-

 

1118

 

 

[Q2] Isinzeexp(-0’/Bz)d9
R'dan‘ d (56)
- (do) Icoszeexp(-0’/02)de '
d0 1 d

For a mosiac of 35°, 0 = 0.7311 and we obtain R = 0.3. This value is about

0.

the same as the measured peak intensity ratio for all peaks above 50 mev.
So we conclude that the peaks above 50 mev in the 0 parallel spectrum are
indeed leakage peaks and the bands at 80 mev and 102 mev are from inplane
mode. Moreover, the bands around 51 mev are partially polarized.

By comparing the spectra of the hydrogenated and deuterated samples, we
see that the peaks at 28 mev, 37 mev, 51 mev and 58 mev are absent ifiwnn the
deuterated sample indicating that those peaks are associated with
trimethylammonium. Since the peaks above 80 mev are present in both samples
they are associated with the vermiculite host. The broad peak at 27 mev in
Fig. 5.7 is of the same origin as the peak at 37 mev in Fig. 5.5 but is
isotope down shifted due to the heavier deuteron mass. The peak at 1111 mev
in Fig. 5.7 is also isotope down shifted from the 58 mev peak in Fig. 5.11.
The peaks at 80 mev and 102 mev are also observed in the IR measurements for
both hydrogenated and deuterated samples. The fact that their positions are
independent of the isotope indicates that they are associate with the host.
Until now, we have identified the internal modes of the trimethlyammonium
guest in terms of their nonhost origin and their polarization.

Since the intramolecular interaction is much stronger than the inter-
molecular interaction, the intramolecular modes are effected very weakly by
the coupling between different molecule. Therefore, a comparison with other
solids which contain trimethylammonium will be helpful to identify the na-
ture of the internal modes of the ion in a clay host. The trimethylammonium
ion has C3v point group symmetry as do the methyl groups from which it is

composed. The internal modes of this ion have been heavily studied in the

1119

13

ionic salts which it forms The methyl groups are known to execute tor-
sional oscillations about their threefold axis with a characteristic energy
of typically 30 mev. In addition, the torsional mode coupling between these
methyl group rotors gives rise to a 5 mev splitting of the tortional mode
into components with a relative intensity ratio of 1/2 for the low energy
band relative to the high energy band. We have observed a similar effect in
the inelastic neutron scattering spectra of (CH3)3NH+-vermiculite. Notice
that the 28 and 37 mev bands of hydrogenated trimeltrylanlnonium vermiculite
(see Figs. 5.” and 5.5) have the requisite average enersy * 30 mev and a 1:2
intensity ratio for us to confidently assign them to the tortional split
mode. Specifically, we assign the 28 mev band as the v(A2) torsion mode
with A2 symetry and the 37 mev band as another torsion mode v(E) with 8
symmetry. Again by comparison with the salts of trimethylammnium, we can
also assign other (non torsional) modes. The 51 mev band is assigned as the
v(E,6NC3) bending mode of the N-C bonds with E symmetry and the peak at 58
mev is assigned as the v(A,,5NC,) bending mode. These trimethylammonium
modes have also been reported by several authors for other materials13-15.
A sumary of the above specified mode assignments is given in Table 5.2. As
can be seen, the splitting between the torsional modes increases con-
siderably in trimethylammonium (Tum-Vermiculite relative to that in the
trimethylamnium halides. The splitting comes from the coupling of top-top
interaction of the methyl groups and thus reflects the strength of this
interaction. The enhancement of the splitting in vermiculite could be
caused by the following reasons: first, the intercalated trimethylammonium
has changed its shape so that the top-top distance is effectively reduced

and this in turn increases the coupling; second, the indirect interaction

mediated by the clay layer has

 

150

Table 5.2 A comparison of the internal modes of trimethylammonium in
different materials. The splitting of the torsional modes in
trimethylammonium-vermiculite has increased relative to that in trimethylam-

monium salts.

 

Torsional
I torsional modes I bending modes I mode

splitting

 

 

 

 

 

 

 

I I
I 0(A2) I v(E) Iv(E,6NC3) I v(A1,5NC3) I v(E)-v(A2) I
I I I I I I

This work . 28.0 37.0 51 0 58 0 9.0
‘C”3)3"" ‘V I I I I I
I I I I I I I
I (CH3)3NHCl a I 30.9 I 36.1 I 39.3 I 57 7 I 5 5 I
I I I I I I I
I (CH3)3NHCI b I 32.6 I 37.7 I 50.1 I I 5.1 I
I I I I I I I

bfrom Ref. 16.

a from Ref. 13,

_.m

 

151

increased. Whatever its origin, it is surprising that these interactions
are stronger in the clay intercalation compounds than in the halide salts.

Since the lack of the detailed mode calculations on trimethylamon ium-
vermicnilite have not yet been carried out, we draw insight from mode
calculations on the alkali vermiculites‘z. According to these calculations,
the bands near 60 mev are due to the optical modes of brucite layer of the
host and the bands above 80 mev are assigned to inplane optical modes of the
silicate sheet. The large intensity of these peaks in our spectra is a con-
sequence of the fact that the oxygen layers contain 01-] groups which enhance
the scattering cross section.

The IR spectra for both the deuterated and hydrogenated samples IJT the
same energy range as the neutron measurements are shown in Fig. 5.8. The
absorption band near 65 mev correspond to the inelastic neutron scattering
peaks around 60-70 mev which are associated with the brucite layer modes.
Another strong absorption band from 80-90 mev also corresponds to peaks
around 80-90 mev in the neutron spectra. These bands are due to the inplane
optical modes of the silicate sheet. The peak at 102 mev in the neutron
spectra is also observed in the IR measurement. The two features at 102 and
105 mev in the IR spectra correspond to the broad peak at 103 mev in Figs.
5.5 and 5.6, but are not resolved due to the poor resolution of neutron

spectra in that energy range.

 

152

5.9 Conclusions

The elastic neutron scattering study shows that the trimethylanmmniium-
vermiculite clay has a c-axis repeat distance of 12.71 A. However, its
structure has not been determined unambiguously. There is evidence that
trimethylammonium has only partially substituted (301 of full exchange) for
the cations in the gallery of natural clay host. In order to determine the
structure unambiguously, additional Bragg peaks at higher 0 values are
required. The inelastic neutron scattering data shows that the internal
modes of trimethylammonium in vermiculite are consistent with the measure-
ments for trimethylammonium halides, but the top-top coupling interaction of
the methyl groups is stronger in vermiculite than in halide salts. In addi—
tion, the host modes determined from inelastic neutron scattering data are

in qualitative agreement with theoretical calculations.

10.

11.

12.

13.

14.

15.

153

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Functions f(x,5,8) and g(x,5,7) have same fitting parameters 5, and the
best fit of f(x,3,8) to data (fi’xi) (i:1,2,....,n) gives fitting
parameters 5,. Usually, g(x,0,,7) can not fit data (g1,xi) (i=1,...n)
no matter what 7 used. In this case, simultanously fit is needed
(varying one set of 5 to fit both functions instead of fitting them

sequentially.