114 DIFFRACTION II : THE INTENSITIES OF DIFFRACTED BEAMS [CHAP. 4

FIG. 4-9.

The

three-dimensional analogueof Fig. 4-8.

These two waves may

differ, not only ⁱⁿ phase,jbut^also ⁱⁿ

amplitude

^if

atom B and

the atonTstr-trre

ongih"^l^^d^fferent

^{kinds. In that case,}

v.ie amplitudes ^{of these}

waves

are given, relative to the

amplitude

^{of the}

wave

scattered

by

^a single electron,

by

the appropriate values ^of/, the

atomic

_scatteringfactor.

We now

seethat the

problem

of scattering

from

a unitcell resolvesitself into

one

of

adding waves

^{of different}

phase and _amplitude

in orderto find the resultant

wave. Waves

scattered

by

^{all the}

atoms

of the unit cell, includingtheoneatthe _origin,

must

be added.

The most

convenient

way

of carrying out ^this

summation

is

by

expressing each

wave

as &

complex

exponential ^function.

+E

FIG. 4-10.

The

additionof sinewavesof differentphaseand_amplitude.

4-4] 117

~-2

FIG. 4-11. Vectoraddition ofwaves. FIG. 4-12.

A

complex^plane.

wave

vector in the

The two waves shown

as full lines in Fig.

4-10

represent the variations in electric field intensity

E

with timetof

two

_rays

on any

given

wave

front in a diffracted x-ray

beam.

Their equations

may

^{be written}

EI

= _A\

sin (2irvt ^i), (4-5)

E

2

= A

2sin

(2wt -

_{$2). (}4_~^)

These waves

^are of the

same

_frequency v

and

therefore ofthe

same

wave-length A, but differ in amplitude

A and

in phase </>.

The

dotted curve

shows

their

sum E

3,

which

^is also a sine wave, but ^{of different}

amplitude and

_phase.

Waves

differing inamplitude

and

_phase

may

^alsobe

^{added by}

represent-ing

them

as vectors. In _{Fig. 4-11,} each

component wave

^is represented

by

^{a vector}

^whose

length ^is equal ^tothe amplitude ^ofthe

wave and which

is inclined to the:r-axis at

an

angle equal ^{to the}phase angle.

The

ampli-tude

and

phase ^ofthe resultant

wave

is then

found

simply

by adding

^the vectors

by

the parallelogram ^law.

This geometrical construction

may

^{be avoided}

^by

^use of the following analytical treatment, ⁱⁿ

which complex numbers

are usedto represent the vectors.

A _complex number

^{is the}

sum

of a ^real

and anjmaginary

num-ber, such as (a

+

^6z),

^where

^a

^and

^{6are real andjt}

^{= V-il}

^{is imaginary.}

Such numbers may

^beplottedⁱⁿ the

"complex

plane," ⁱⁿ

which

real

num-bers are plotted as abscissae

and

imaginary

numbers

as ordinates.

Any

point ^{in this} plane ^{or the}vector

drawn from

the _{origin to} ^this point then representsa _particular

complex number

_(a

+

bi).

To

find

an

analytical expression ^for a vector representing a wave,

we draw

the

wave

vector in the

complex

plane ^{as in Fig. 4-12.}

Here

_again the

amplitude and

phase^ofthe

wave

^isgiven

by

A, the length^ofthe_vector,

and

0, the angle

between

the vector

and

the axis of real

numbers. The

analytical expression forthe

wave

^is

now

the

complex number

(A ^cos<t>

+

iA

sin</>),since these

two

termsare thehorizontal

and

vertical

components

DIFFRACTION

II:

THE

INTENSITIES

OF DIFFRACTED BEAMS

_{[CHAP. 4}

md ON

of the vector.

Note

that multiplication of a vector

by

ⁱ

jtates it counterclockwise

by 90;

^thus multiplication

by

ⁱ converts the horizontal vector 2 into thevertical vector2i. Multiplication twice

by

i,

that _is,

by

ⁱ²

=

1, rotates a vector

through

¹⁸⁰ or reverses its sense;

thus multiplication twice

by

ⁱ converts the horizontal vector 2 into the horizontal vector 2 pointingⁱⁿ the opposite^direction.

If

we

write

down

the power-seriesexpansions ^{ofeix, cos}x_y

and

sin_x,

we

findthat

e^ix

=

_cos

_x +

^isinx (4-7)

or

Ae* = A

cos <t>

+ ^Ai

^sin4. (4-8)

Thus

the

wave

vector

may

be expressed analytically

by

^{either side of} Eq. (4-8).

The

_expression

on

the left is called a

complex

exponential function.

Since the intensity of a

wave

is proportional ^{to the} square ^{of its} ampli-tude,

we now need an

_expression for

A

², the square ^of the absolute value ofthe

wave

vector.

When

a

wave

^isexpressedⁱⁿ

complex

form, ^this

quan-tity ^is obtained

by

multiplying the

complex

expression ^for the

wave by

its

complex

conjugate,

which

is obtained simply

by

replacing ⁱ

by

^i.

Thus, ^the

complex

conjugate ^of

Ae

^l* is

Ae~

^l*.

We have

\Ae

l

*\

2

= _Ae

^l

_+Ae-* = A

², (4-9)

which

isthe quantity^desired. Or,using the other

form

_given

by

Eq. (4-8),

we have

A

_(cos

+

^isin

4)A(cos

^{< isin}<)

= A

²_(cos²<t>

+

^sin2</>)

==

A

^2.

We

return

now

to the

problem

^of

adding

^{the scattered}

waves from

each ofthe

atoms

inthe unit cell.

The _amplitude

ofeach

wave

^is_given

by

the appropriate value ^of

/

^{for the} scattering

atom

considered

and

the value of (sin0)/Xinvolvedinthereflection.

The

_phaseofeach

wave

^is_given

by

Eq. (4-4) ⁱⁿterms ofthehkl reflectionconsidered

and

the

uvw

coordinates ofthe atom.

Using

our previous relations,

we

can then express

any

scat-tered

wave

in the

complex

exponential

form

(4-10)

The

resultant

wave

scattered

by

alljbhe

atoms

ofthe unit cell is calledthe structure factor

and

isdesignated

by

^the

symBol

^{F. It"is"obtained}

by simply adding

together ^all the

waves

scattered

by

the individual

atoms>

If a unit cell contains

atoms

_{1, 2, 3,} . .. ,

N,

^{with fractional} coordinates Uivi!!,

u

₂v₂tt?

2, MS*>3MS, ^...

and atomic

scattering factors/i,/2,/a, ^{.. .,}

then the structure factor forthehklreflection isgiven

by

^

^{y e}2*i(hu2+kvi+lwti ⁱ /

g2iri(Au3-H;i>s-fIwi) ⁱ . . .

4-4]

SCATTERING BY A UNIT CELL

117

In document WITHIN THE (pagina 125-128)