Singular lines and singular points of ferromagnetic spin systems and of nematic liquid crystals

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Singular lines and singular points of ferromagnetic spin systems and of nematic liquid crystals

F.R.N. Nabarro

To cite this version:

F.R.N. Nabarro. Singular lines and singular points of ferromagnetic spin systems and of nematic

liquid crystals. Journal de Physique, 1972, 33 (11-12), pp.1089-1098. �10.1051/jphys:019720033011-

120108900�. �jpa-00207335�

1089

SINGULAR LINES

AND SINGULAR POINTS OF FERROMAGNETIC SPIN SYSTEMS

AND OF NEMATIC LIQUID ^CRYSTALS

F. R. N. NABARRO

(*)

Laboratoire de

Physique

des Solides

(**),

^Bâtiment

^510,

Université

Paris-Sud,

^Centre

d’Orsay, 91, Orsay,

^France

(Reçu

^{le 26}

juin 1972)

Résumé. ²⁰¹⁴ L’ensemble des

spins

^d’un

ferromagnétique

^{définit un}

champ

^{de vecteurs. Les}

lignes

de

singularité

^les

plus simples

^{de cet ensemble sont} donc des disinclinaisons de _rang 1, ^{et les}

points

de

singularité

^{sont du} type ^{de ceux}

analysés

par Poincaré. Un cristal

liquide nématique

^n’est pas

polaire

habituellement. En

conséquence,

^on peut transporter de manière continue un vecteur

aligné

avec 1’axe

nématique

^{à travers du « bon »}

cristal liquide,

^{et revenir au même}

point

^{avec le vecteur}

de

signe opposé.

^Les

lignes

^de

singularité

^les

plus simples

^{sont donc} des disinclinaisons de _rang 1/2.

Un circuit _parcouru deux fois ramène le vecteur selon son orientation initiale. Le

champ

^{des direc-}

teurs est donc semblable à la fonction d’onde d’un

spineur.

L’extension de cette idée au cas à trois dimensions permet ^une

description préliminaire

^des

singularités ponctuelles

^{dans les structures}

nématiques.

Abstract. ²⁰¹⁴ The

spins

^{in a}

ferromagnet

^{define a vector} field. Their

simplest

^line

singularities

^are

therefore disclinations of unit

strength,

^{and their}

point singularities

^{are of the} types

analysed by

Poincaré. A nematic

liquid crystal

^is

normally unpolarised.

^{As a}

result,

^{a vector}

placed along

^the

nematic axis at a

point

may be carried

continuously through

^«

good » liquid crystal

^{and return}

inverted. The

simplest

^line

singularities

^are therefore disclinations of half unit

strength.

^{A double}

circuit restores the vector to its

original

orientation. The director field thus has the character of the wave function of a

spinor.

The extension of this idea to three dimensions allows a

preliminary description

^{of the}

point singularities

of nematic structures.

LE JOURNAL DE PHYSIQUE TOME 33, NOVEMBRE-DÉCEMBRE 1972,

Classification

Physics ^Abstracts 02.00, 14.82, 16.40, ^{17 64}

1. The

singularities

^of

ferromagnetic spin systems. -

We

begin

^{with an} outline of Poincaré’s

analysis

^of

the

singularities

^{of vector} fields in two and three dimensions. We then use this

analysis

^{and the}

theory

of disclinations to consider the Bloch wall

separating

two

ferromagnetic

^domains

magnetised

ⁱⁿ

opposite directions,

^{the Néel} line which

separates

^two

portions

of the Bloch wall which have

opposite helicities,

^and the Bloch

point

^which

separates

^two Néel lines which have

opposite

disclination

strengths.

^{A similar}

analysis

^is

possible

^{for Néel}

walls,

^{Bloch lines and}

their

singular points.

1.1 POINCARÉ’S ANALYSIS IN TWO DIMENSIONS. ^-

We consider a vector field

(X, Y)

^{in the}

plane (x, y).

In Poincaré’s

analysis [1], (X, Y)

^{is the}

velocity

^{of a}

particle

^{situated at}

(x, y) ;

^{in our}

analysis

^{it is an}

unnormalised indicator of the direction of _magne-

tisation at

(x, y).

^We suppose

that,

^{in the}

neighbourhood

of the

origin,

where all the coefficients ^are real. Unless

Xo

^and

Yo

both

vanish,

the direction

(X, Y)

^is well determined at the

origin,

^{which is then an}

ordinary point.

^The

simplest singularities

^{are those for which}

Xo

⁼

Yo

⁼

0,

while _{ai ,}

bl, a2, b2

^{are not all zero.}

They

^{are in}

general

screw disclinations of

strength

^{± 1.} Disclinations of

larger integral strengths

may be

produced by

^the

confluence of these unit disclinations.

We look for the

regions

^{in which the vector field}

(X, Y)

^is

parallel

^or

antiparallel

^to the radius vector

(x, Y).

This

requires

which has non-zero solutions

only

^if

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019720033011-120108900

Since the coefficients are

real,

^{the roots of this}

quadratic

^{in A.}

present

^several

general

cases, which pass into ^one another

through special

^cases.

Case 1

(general).

^{The roots in  are}

^real,

^{and of}

opposite sign.

^{This is a}

^col,

^{and a screw} disclination of

strength

^{- 1.}

Along

^one real radius

(and

^its

opposite)

the field

(X, Y)

^is

parallel

^to the radius vector

(x, y).

Along

another real radius

(and

^its

opposite)

^the

vectors

(X, Y)

^and

(x, y)

^are

antiparallel.

Case 2

(special).

^{One root real and}

positive,

^the

other root zero. The field

everywhere

^{has one of the}

two directions

corresponding

^{to the non-zero}

root,

FIG. 1. ^- Three-dimensional Poincaré singular points : a) col, b) noeud, c) foyer, d) col-foyer.

1091

and there is one

straight

^line

through

^the

origin along

which

(X, Y)

vanishes. This allows a smooth transi- tion between cases 1 and 3. One

negative

^{root and}

one zero

gives

^{the same}

figure

with the direction of the field

(X, Y) reversed,

^{and we shall}

regard

^{such a}

simple

reversal of the field as trivial.

Case 3

(general).

^{Both roots}

real, positive,

^and

distinct. This is ^a

noeud,

^{and a screw} disclination of

strength

⁺ 1. All the lines of the field

(X, Y)

pass

through

^the

origin,

^{and there are two} distinct radii

(with

^their

opposites) along

^which

(X, Y)

^is

parallel

to

(x, y).

These four field lines are

straight ;

^{all other}

field lines are curved.

Case 3

(special).

^{The roots in are}

^real, positive,

and

equal.

All field lines are

straight

^lines

through

the

origin.

Case 4

(general).

^{The roots are}

complex conjugate.

The

pattern

^{is a}

foyer,

^{and a screw} disclination of

strength

^{+ 1. The}

trajectories

^{are distorted}

loga-

rithmic

spirals (with

^the

property

^that

they

^encircle

the

origin infinitely

^{often within an}

arbitrarily

^small

radius), having

^an

angle

between the vectors

(X, Y)

and

(x, y)

^which

depends only

^on

x/y.

^This

angle

is _very small when the

imaginary part

^{of À is}

small, giving

^a continuous transition from Case 3.

Case 5

(special).

^{Both roots are} pure

imaginary.

The

pattern

consists of closed

ellipses surrounding

the

origin,

^{is a} centre, and ^{a screw} disclination of

strength

^{+ 1.}

I.2 POINCARÉ’S ANALYSIS IN THREE DIMENSIONS. ^-

Poincaré’s

analysis [2]

for the three-dimensional case

is more

complicated,

^{and we shall} follow him in

considering only

^the

general

^{cases and one of the}

special

^{cases. The}

equation analogous

^to

(3)

^{is the}

cubic

This cubic

equation

with real coefficients

always

has one real root

À1, which,

^with

only

^a trivial loss of

generality,

^{we may take to be}

positive.

^{There are}

then four

general

^{cases and one}

important special

case.

Case 1

(general). À2

^and

Â3

^are

^real, negative

^and

distinct. This is a col.

Through

^the

origin

passes ^a

singular

surface. All the

trajectories

in this surface pass

through

^the

origin,

^so that this surface contains ^a two-dimensional noeud.

Through

^the

origin

^also

passes ^an axis

(not lying

^{in the}

surface),

^{which is a}

trajectory.

^{No other}

trajectories

pass

through

^the

origin.

^Sections

by

^surfaces

containing

^{the axis are} cols

(Fig. la).

^{With all arrows}

reversed,

^the

figure represents

^{the stream lines of two}

opposite jets

^of

water

impinging

^{on one another.}

Case 2

(general). À2

^and

À3

^are

real, positive,

^and

distinct from one another and from

Â,.

^{This is a}

noeud. All

trajectories

pass

through

^the

origin (Fig. lb).

The

figure represents

the lines of force of a

point charge.

Case 3

(general). Â2

^and

Â3

^are

complex conjugate.

Their real

parts

^are

positive.

^{This is a}

foyer.

^There

is a

singular

^{surface and an axis.}

Trajectories

^{lie in}

the

singular surface,

^{and form a} two-dimensional

foyer.

The axis is a

trajectory.

^{All other}

trajectories spiral

^{outwards around the axis} away from the

origin,

^{but cannot be traced back as far as the}

origin (Fig. 1 c).

Case 4

(special). Â2

^and

Â3

^are

purely imaginary.

This _may be a

foyer

^{or a}

col-foyer (Case 5),

^but may also be a

centre, again

^{with a}

singular

^{surface and}

an axis which is a

trajectory.

^The

trajectories

^{in the}

singular

^{surface are closed curves}

surrounding

^the

origin.

^Other

trajectories

^{lie on sleeves}

surrounding

the

axis,

^and

spiral

away from the

singular surface,

which

they

^{do not reach even if}

prolonged

^backwards

indefinitely.

Case 5

(general). Â2

^and

Â3

^are

complex conjugate.

Their real

parts

^are

negative.

^{This is a}

col-foyer.

The

trajectories

^{in the}

singular

^{surface form a}

foyer,

and the axis is a

trajectory.

The other

trajectories spiral

^{round the}

axis, contracting

^as

they

recede from the

singular surface,

^which

they

^{do not reach even}

if

prolonged

^backwards

indefinitely (Fig. Id).

I.3 BLOCH WALLS, NÉEL LINES, ^SINGULAR

(BLOCH)

POINTS ON NÉEL LINES. ^- We consider a

ferromagnetic crystal,

ⁱⁿ which the easy directions of the

magneti-

sation h are ± y. We suppose

(Fig. 2a, 3a, 4a, 5a)

that the

magnetisation

^lies

along

+ y

when x 0,

and

along - y when x > 0,

^{so that the}

plane x

^{= 0}

is the middle of a Bloch wall. In a Bloch

wall,

Fie. 2. - Bloch wall and uncharged region ^{of a} Néel line :

a) The Bloch wall in the plane x ^{= 0} separates ^two regions where the moments, shown by double-headed _arrows, lie along ::l: y. The moments in the middle of the wall lie along + ^z outside the closed Néel line, ^and along ^{- z} inside the Néel line.

The two crosses on the Néel line mark an uncharged region.

b) ^The arrangement ^of spins ^{in this} region. c) ^{The axes used}

in b). d) ^{The rotation of the} spins associated with the circuit r in b), ^and the direction of the magnetic ^{moment of this} region

of Néel line.

h rotates about the normal to the

wall, Ox,

^{in order to}

change from - y

^{to + y without}

introducing

^free

poles.

This rotation _may be in either _sense, so that the direction of h in the

plane x

^{= 0} may be either + ^z or - z. We suppose that a wall in which h in the mid-section lies

along

^{+ z contains a closed}

patch

with h in the mid-section

along -

^{z. The line}

separating

^{these two}

regions

^{is a Néel line.}

As in the case of a

crystal dislocation,

^{there are}

two

regions

^{on the Néel} line which have a

particu- larly simple

structure. The

first,

^{which we shall call}

an

uncharged region,

^{occurs where the} Néel line is

parallel (or antiparallel)

^{to the}

magnetic

field in the mid-section of the Bloch wall.

Figure

^{2b shows the}

spin

structure in ^a section

perpendicular

^to the Néel line at this

point.

The circuit r is taken around a line

parallel

^{to the z}

axis, and,

^{if we follow the}

spin

^vector

round a circuit far from the Néel

line,

^{it rotates once}

around the - x axis. The Néel line is thus an

edge

disclination of

strength

^{1. It is not a line of}

singu-

larities in the sense of

Poincaré,

because h is well defined

along

the disclination line. This

portion

^{of the}

line

clearly

^{carries a}

magnetic

^moment per unit

length along

^{+ x.} The second

region,

drawn in section

perpendicular

^to the Néel line at a

point

^{where the}

Néel line is

perpendicular

^{to h} in the mid-section of the Bloch

wall,

^{is shown in}

figure

3b. We shall call it a

region

of maximum

charge.

^{It is}

again

^an

edge

FIG. 3. ^- As for figure 2, ^{but at a} region of maximum charge

in the same Néel line.

disclination of

strength 1,

^{since a circuit r round} the z axis causes h to rotate once round the - x axis.

Again

^{it is not a Poincaré}

singularity,

^{and it}

has a moment per unit

length along

^{+ x. In}

addition,

it carries ^a

positive charge

per unit

length, since,

^on

the

axis, 8hx/8x

⁼

0, 8hy/8y = ^0,

^and

8hz/8z

> 0.

The conditions outside the core of the Néel line

(outside

^{the circuits r of}

Fig.

^{2 and}

3)

^can

equally

well be satisfied

by

structures in which the x _compo-

nents of all h vectors are the reverse of those shown in

figures

2 and 3. The disclination

strength,

^which

is determined

by

the directions of h vectors far from the _core, is unaltered. The

magnetic

^{moment per unit}

length

^is

reversed,

^{and the}

charge

per unit

length

ⁱⁿ

figure

3b is unaltered.

We now suppose that the Néel line has moment + x per unit

length

^{in some}

regions,

^{and moment - x}

per unit

length

^{in other}

regions.

^The

junction

^{of two}

such

regions

^of

opposite

^{moment is a Bloch}

point.

We shall consider the Bloch

points

ⁱⁿ

uncharged regions

^{and in}

regions

of maximum

charge

^{of a Néel}

line.

The Bloch

point

^{in an}

uncharged region

^{of a Néel}

line is shown in

figure ^4b,

ⁱⁿ which the

origin

^of

coordinates has been shifted to the centre of the

singularity. Here,

^the

largest

^bullets

represent

^the

configuration

^of

figure ^2b,

^{in a sheet close to the}

FIG. 4. ^- The Bloch point ^{where the} configuration ^of figure ²

meets an equivalent configuration ^of opposite ^{moment :} a) ^and c) ^{as in} figure ^2. b) ^The arrangement ^of spins. ^The configuration

in the sheet of spins nearest to the observer is the same as that in figure 2b, ^while the sheet of spins farthest from the observer has the x component ^{of each} spin ^reversed. Spins ^{drawn in} broken lines ^are interpolated ^between spins ^on lattice sites.

The double cross marks the centre. The single ^{crosses lie in}

the singular plane. d) ^The director of the axis of the col-foyer,

and the sense of circulation in the singular plane.

observer,

^while the smallest bullets

represent

^the

same

configuration

^with

hx reversed,

^{in a sheet far}

from the observer. The direction of h in the middle has to

interpolate simultaneously

^{between + x}

and - x,

^between + y ^{and -} y, and between + ^z and - ^z. The centre is thus ^a Poincaré

singular point.

By inspection,

^{we see}

that,

^{close to the} centre, ^the direction of h is

given by

The _eq.

(4)

thus takes the form

1093

or A 3

^{= -} 1. There are thus two

complex

^roots,

with real

parts

^of

opposite sign

^{to that of the real}

root. This is a

col-foyer.

^The

axis corresponding

to the real root À. = - 1 is

[111],

^{which we}

verify by noting

^that

Thus h close to the axis is

antiparallel

^{to the}

axis,

while h in the

singular plane spirals

around the centre. The

singular plane

^is

(111),

^{which contains}

the

complex eigenvectors (1,

^{ev, -}

(J)2)

^and

(1, (J)2,

-

co),

^{where ev and}

^(J)2

^{are the}

complex

^{cube roots}

of

unity.

^This

analysis

allows the intermediate sheets of

figure

^{4b to} be sketched.

For the Bloch

point

^{in a}

region

^{of maximum}

charge

on a Néel line

(Fig. 5b),

^{we have}

again

^{a Poincaré}

singular point,

^with

The determinant has roots +

1,

^{+ 1 and -}

1, giving

a col with three real

eigenvectors :

FIG. 5. ^- The Bloch point ^{where the} configuration ^of figure ³

meets an equivalent configuration ^of opposite ^{moment :} a) ^and c) ^{as in} figure ^3. b) ^{As in} figure 4b, ^{but for a} region ^{of maximum} charge. d) The direction of the axis of the col, ^{and the} divergence

of the spins ^{in the} singular plane.

The last

eigenvector

determines the axis

[110],

^while

the first two determine the

plane

^{of the}

noeud, (110).

Again

^the

knowledge

^{that the} h vectors near the

centre and close to the axis are

antiparallel

^{to the}

axis,

^{while h in the}

singular plane diverges radially

from the centre, allows the structure of the

singula- rity

^{to be sketched}

(Fig. 5b).

1.4 NÉEL WALLS, BLOCH LINES, SINGULAR

(NÉEL)

POINTS ON BLOCH LINES. ^- We

again

^{take a material}

with _easy directions of

magnetisation

^± y, and suppose ^at first that the

magnetisation

^is

along

+ y

when x «

^{0 and}

along - y ^{when x »}

^{0. The}

sample

is now taken to be a film thin in the z

direction,

^so

that, following

^Néel

[3],

^we may ^assume that the

demagnetisation

energy when

hz -:j::

^{0 is so}

large

^that

only

structures with

hz

^{= 0 occur.} Then h in the mid- section of the Néel wall is

parallel (or antiparallel)

to x. This

produces

^{sheets of}

magnetic charge

^{on the}

boundaries of the wall. To minimise the

exchange

energy, ^we make the lines of h curve inside the wall to fit

smoothly

^{with the lines} outside the wall

(Fig. 6).

FIG. 6. ^- A Néel wall at x ⁼ 0 in which the spins ^rotate from + y to - y while remaining ^{in the} x, y plane, ^{and Bloch}

lines of strengths =b ¹ lying along ^Oz.

Where a

region

^of moment + x

impinges

^{on a}

region

of moment - _x, we have a short line defect

parallel

to z which is

clearly

^{a screw} disclination of

strength

± ^{1. Since h is} indeterminate all

along

^the

line,

^{it is a}

Poincaré line of

singularities.

^{We call it a} Bloch line.

So

far,

^we have considered

only regions

^{of the Néel}

wall which are

parallel

^or

antiparallel

^{to h outside}

FIG. 7. ^- A Néel wall turns through ^two right angles. ^The chain-dotted line represents ^the region of maximum positive magnetic charge : a) ^{A bend free from} singularities, ^but unsym- metrical. b) ^A symmetrical ^{bend with a screw} disclination _{+ 1.}

c) ^A symmetrical ^{bend with a screw} disclination - 1.

1095

the wall.

Presumably

domains of reversed h _may be initiated far from the boundaries of the

specimen,

and will then be bounded

by cylindrical

^{Néel walls.}

The

portions

of wall which do not lie

parallel

^to

± ^h outside the wall

obviously

carry ^a

magnetic charge

per unit ^area. As is shown in

figure 7a,

^a Néel wall _{may turn}

through

^a

large angle

^without

introducing

any

singularity

^{into the h field. More}

symmetrical configurations

^{involve screw disclina-}

tions of

strength

^{± 1}

(Fig.

^{7b and}

c).

^{The chain-}

dotted lines in these

figures

^{are the}

regions

^{of maxi-}

mum

density

^of

positive magnetic charge.

The disclination lines in

figures 6,

7b and 7c ^are

regions

^of

high density

^of

exchange

energy. This energy is reduced if h is directed towards ± ^z

along

the

axis, curving

up towards the axis from distant

regions,

^{where h} lies in the

(x, y) plane.

The line is still a screw

disclination,

^{but it is no}

longer

^{a Poincaré}

singularity,

because h is well determined

along

^the

axis. We call it a modified Bloch line.

Suppose

^now that such a modified Bloch line has

hz

^{= + z in one}

portion

^and

hz

^{= - z in another}

portion.

^We will call the

meeting

^{of these}

portions

a Néel

point.

^Such

points

^{may be}

barely

^{stable in}

thin

films,

but may

perhaps

^{exist in}

magnetic

^materials

of orthorhombic

symmetry,

^with

± y

^an easy direction of

magnetisation,

^{± x}

intermediate,

^and ± ^z difficult.

Such a

point

^{is a Poincaré}

singularity,

^{of the kind}

which has been discussed

by

Feldtkeller

[4], [5]

^and

Dôring [6].

^{Here h is}

specified by

^its

polar angles 0.

^and qJm, which are

given

^{in terms of the}

polar angles 0,

ç of the direction of the vector from the centre to the field

point by

In cartesian

components,

The

eigenvalues

^are

given by

with roots

or

In the first case we have ^a three-dimensional

foyer

for -

n/2

^a

n/2,

^{and a}

col-foyer

^for

n/2 1 a

^n.

If a ⁼

n/2

^the

point

^{is a} centre. In the second case

the

point

^{is a} col. The form

(10),

^{with an}

appropriate

choice of _axes, is thus

capable

^of

representing

^the

singular points

^{of both} Néel and Bloch

lines,

^and

may

appropriately

be called a Feldtkeller microma-

gnetic point.

II. The

singularities

of nematic

liquid crystals.

^-

If the two-dimensional vector field

(X, Y)

^{has a}

singularity

^{at the}

origin,

^{we write}

We ^are concerned

only

^with the direction of

(X, Y)

at

(x, y),

^{and not with its}

magnitude,

^{and hence}

essentially

with the ratio

YjX. Changing

^to

polar

coordinates

(r, ç)

^{in the}

(x, y) plane,

^we may

replace (14) by

^the

equivalent equations

A circuit around the

origin keeps cp

^constant

[mod

²

n],

and ^so

(X, Y)

^is

single-valued.

Terms with

integral multiples

^of cp in

(15)

retain this

property,

^{and lead}

to

permissible

disclinations of

higher order,

^but

submultiples

^of cp lead to ratios

Y/X

^{which are not}

single-valued.

A nematic

liquid crystal

^can exhibit all the

singu-

larities of a vector array. It can also exhibit disclina- tions of

half-integral strength.

^{If we}

arbitrarily

^choose

one of the two axial directions at a

point

^{in a nematic}

as the director

vector,

the director need not necessa-

rily

fall back on itself after it has been

transported continuously

^{around a} circuit. A reversed director

corresponds

^{to the same}

physical

^{state of the}

system.

II. 1 THE SINGULARITIES OF A NEMATIC IN TWO DIMENSIONS. ^- The fact that the director may be reversed in a circuit around a

singularity (which

we will

place

^{at the}

origin),

^{and be restored after two}

circuits, suggests

^that

(15)

^{may be}

replaced

^{in a nematic}

by

We

again

look for the directions

(x, y) along

^which

(X, Y)

^is

parallel

^or

antiparallel

^to

(x, y).

This

requires

Eliminating À,

^and

putting t

⁼

tan 2

ç, ^we obtain

This cubic

equation

with real coefficients has either 1 or 3 real roots. There ^are thus 1 ^or 3 values of

tan 2

ç for which

(X, Y)

is radial. But

tan 2

9 determines

§ ç [mod n],

^{and so} ç is determined

[mod

²

n].

^There

are thus either 1 ^or 3 distinct directions in which

(X, Y)

^is

radial, corresponding

^{to the screw discli-}

nations of

order 2

^{and -}

2.

II.2 THE SINGULARITIES OF A NEMATIC IN THREE DIMENSIONS. ^- Before

discussing

^the

singularities

^of

a

nematic,

^{we note that it can be}

formally represented

by

^{a tensor which}

specifies

the direction of its double-

ended

director,

^{and the}

strength

^{of its}

anisotropy,

as

measured,

^for

example [7], by magnetic

suscep-

tibility.

^{This tensor must, in axes}

(xi,

x2,

X3),

^{be of}

the form

Qij,

^where

The

principal

^{axes of the tensor are}

and in these axes it has the form

The director lies

along x’, and,

^{if the direction}

cosines of

x3

^are ni, ^we have

We

represent

^the

position

^{of a}

point

^{in three dimen-}

sions

by polar

coordinates

(r, 0, qJ), with r > 0,

^and consider a

singularity

^{at r = 0.}

Corresponding

^{to the}

identity operations

ç - ç ± 2 n in two dimensions

we have the five

independent identity operations

The vector

(sin

^{0 cos} 9, sin 0 sin _ç, cos

0)

^{is an}

eigenfunction

of all five

operations,

^with

eigenvalue

+ 1 for each

operation,

ⁱⁿ

analogy

^{with the vector}

(cos

ç, sin

ç)

^{in two} dimensions. In two dimensions

we

analysed

^the

singularities

^{of a nematic}

by

^intro-

ducing

^{the vector}

(cos t

ç,

sin -1 p),

^{which is also}

an

eigenvector

^{of the}

symmetry operations ç - ç

^{± 2 n,}

but with

eigenvalue -

^{1 for each}

operation.

^The

corresponding eigenvector

^of the group

(23)

^is

(sin -10

^cos ç,

sin -L

^{0 sin} ç,

cos 1 0),

^{which has}

eigen-

values +

1,

⁺

1,

^-

1,

^-

1,

^{+ 1.}

By analogy

^with

(17),

^we may take the director ^to have

components

which are linear functions of the

components

^{of this} vector, and look for directions in which the director is

parallel

^or

antiparallel

^to the radius vector.

We therefore write

We eliminate À between

(24)

^and

(25),

^and

put tan 2 0 = t, obtaining

Similarly eliminating

^{À between}

(24)

^and

(26),

^we

find

If we substitute

(27)

ⁱⁿ

(28),

^{we obtain an}

equation

in sin 2 _ç and ^cos 2 _ç, which becomes an

algebraic equation

^{in T = tan} ç. This is

apparently

^{a sextic}

in

T,

but the coefficient of

T 6 vanishes,

^{so that T is}

determined as the root of a

quintic equation

^with

real coefficient. Since the number of

parameters

a,, ^..., c3

greatly

^{exceeds the}

degree

^{of the}

equation,

it is

unlikely

^that any identical relation exists among the coefficients of the

quintic.

^{It therefore has}

1,

³

or 5 real roots. Each root determines _{tan ç,} and hence

determines ç [mod n].

^Then

(27)

^determines

t ⁼

tan 2

⁰

uniquely,

^which

determines 2

⁰

[mod n],

or 0

[mod

²

n].

^The

pair

^{of roots} ç, ç + 7r lead from

(27)

^to

equal

^and

opposite

^values of t, and hence of 0.

But the

operation 0 --> - 0, o --> o

^{+ is an}

identity operation

^{which leads back to the same direction}

in space, and ^so each real T determines

just

^one

direction in space

along

which the director is

parallel

or

antiparallel

^to the radius vector. We note that the

opposite

direction in space will ^not in

general satisfy

this condition.

To see what

types

^of

singularity

^are

generated by

this _process, we note that the

polar angles 0,

9 ^are continuous functions of

position

^{on a}

sphere, except

when 0 ⁼ nn

(n

^{= ..., -}

1, 0, 1, ...). Here ç

^{is inde-}

terminate. The

components

^{of the vector}

(sin

^{0 cos} (p, sin 0 sin _ç, cos

0)

^which

generate

the Poincaré

singu-

larities of a vector field are

everywhere

^continuous

functions of

position,

because sin 0 ⁼ 0 at the

points

for which ₉ is indeterminate. This is no

longer

^true

for the field defined

by (24)-(26).

^When

0 = (2 n + 1) n,

the

components

^{of the vector}

(sin 2 8

^cos ç,

sin 2 B

^sin ç,

cos 2 0)

^are indeterminate. The

corresponding

^field

therefore has a line of

singularities along

^this

unique direction, though

^not

along

^the

opposite

^direction

0=2 nn. The

singular point

^{at the}

origin

^{is the end}

of ^a line of disclination.

We illustrate this

type

^of

singularity by

^the

simple example

in which the director is

given by

The direction 0 ⁼ 0 is a

unique non-singular principal

direction in which the director is

parallel (or

^anti-

parallel)

^to the radius vector ; the line ^{0 = n is a} line of disclinations

corresponding

^{to Case 3}

(special)

1097

of

paragraph

^I.

Any

^{section 9 = constant contains}

a nematic screw disclination of

strength + 2,

^and

circuits on the unit

sphere

^which pass

through

^the

singular point 0

^{= jr reverse} the director. Circuits which do not pass

through

^this

point

^{do not reverse}

the

director, and,

^{if we} agree ^to exclude such

circuits,

the field is one which is

possible

^{for vectors.}

A second illustration is

provided by

Again

^{0 = n is a}

disclination,

^{now of}

strength -

^1.

The section _ç ^-

0, n again

^{contains a} disclination of

strength + 2,

^{while the section} ç

= 2 ’TC, 1

^{n contains}

a disclination of

strength - 2

^{There are three non-}

singular principal directions, 0

⁼

0,

ç

indeterminate,

and 0

2 71

^with 9

= -1 7c or 3

^n. The field is

again possible

^{for a}

vector, provided

^circuits

passing through 0

^{= n are excluded.}

When B ⁼ 1 the field

has

non-singular principal

directions

along

⁰⁼⁰

and

along

all directions in the cone 0

= t

^{7T. For}

other

positive

^{values of 8 there are five}

principal directions,

^{0 =}

0,

({J

indeterminate,

^{and 0}

= î

^rc,

9 = 1 n7r.

In the field

given by

the line 0 ⁼ n is an

edge

disclination of unit

strength

which terminates at the

origin.

None of these fields is

essentially nematic,

^{nor is}

it an isolated

singular point,

^{and an} extension of an

argument

^{due to F. C. Frank}

[8]

^shows

why

^this

should be so. If the field is

essentially nematic,

^there

must be many circuits ^on the unit

sphere

^which

reverse the director. A small circuit on the unit

sphere

does not reverse the director. If there were no

singular points

^{on the unit}

sphere

^{at which the direction of the}

director was

indeterminate,

any ^«

reversing »

^circuit

could be shrunk

continuously

^{to a «}

non-reversing » circuit,

^{which is absurd. We have} introduced a

single singular point

^{at 0 = n, but a}

single singular point

on a

sphere

^{does not}

prevent

^an

arbitrary

^{circuit from}

being

^{shrunk into an} infinitesimal ^«

non-reversing

^»

circuit. The surface of ^a

sphere

^{must have at least}

two

singular points

if circuits are to be divided into two

classes,

those which when shrunk to infinitesimal size contain a

singular point,

^and those which do not.

This

argument

^indicates

that,

^{in order to}

generate

a

truly

^nematic

point singularity,

^{we must dissociate}

the

terminating

disclination line of

strength

± ¹ into two continuous disclination lines of

half-integral strengths.

If we take the

singularity

^of

(29),

^and dissociate the disclination of

strength

^{+ 1}

along [001]

^{into two}

disclinations each of

strength

⁺

2,

^which

separate

until

they

^lie

along

the directions

[100]

^and

[100],

the result is ^a uniform disclination line of

strength + 2 lying along

the directions

[100]

^and

[100],

^with

no

point singularity

^{at the}

origin.

The

singularity

^of

(30)

^{has a} disclination of

strength

- 1

along [001].

^{This can} dissociate into two dis- clinations of

strength - 2

^if their lines move towards the directions

[110]

^and

[110],

^or,

equivalently,

towards

[110]

^and

[110].

^The result in the former case

is a screw disclination of

strength - t lying along

the directions

[110]

^and

[110],

^and

passing through

the

origin.

^{There are}

non-singular principal

^directions

along [001 ],

^{and close to}

[021 ] and [021 ].

^{The directors}

on the unit

sphere surrounding

^the

origin,

^{as viewed}

along [001],

^{are shown in}

stereographic projection

in

figure

^{8a and b.}

FIG. 8. ^- The director field on a sphere surrounding ^{a nematic} point singularity, ⁱⁿ stereographic projection. ^{A screw discli-}

nation of strength - t lies along ^{the line}

[110]-[110].

a) Upper hemisphere, b) lower hemisphere.

Acknowledgments.

^{- I am}

grateful

^{to F. C.}

Frank,

J.

Friedel,

^{P.-G. de}

Gennes,

^{W. F.}

Harris,

^M.

Kléman,

M. J.

Stephen

^{and A. T.}

Quintanilha

^for

reading

^the

manuscript,

^{and for} discussions.

References

[1]

^POINCARÉ

(H.),

^{J. de Math.}

[3], 1881, 7, 375 ; 1882, 8,

251.

[2]

^POINCARÉ

(H.),

J. de Math.

[4], 1886, 2,

^151.

[3]

^NÉEL

(L.),

^J.

Physique Radium, 1956, 17,

^250.

[4]

FELDTKELLER

(E.),

^Z. angew.

Physik, 1964, 17,

^121.

[5] Ibid., 1965, 19,

^530.

[6]

^DÖRING

(W.),

^J.

Appl. Phys., 1968, 39,

^1006.

[7]

^{DE GENNES}

(P.-G.),

^Molec.

Cryst. Liq. Cryst., 1971, 12,

193.

[8]

^FRANK

(F. C.), 1971, private

communication.