beam is of lower intensity than one would expect for a specimen of no

(a) (h)

FIG. 4-18. AbsorptionⁱⁿDebye-Scherrer specimens:(a)general case, (b)highly absorbing specimen.

4-10

_Absorption factor. Still another factor affecting the intensities of the diffracted rays

must

be considered,

and

that is the absorption

which

takes _place in the

specimen

^itself.

The _specimen

in the

Debye-Scherrer method

has the

form

ofa verythin cylinder of

powder

placed

on

the

camera

axis,

and

_Fig. 4-1 8_(a)

shows

the cross section of such a specimen.

For

the low-angle ^reflection

shown,

absorption ^of a _particular ray ⁱⁿ the inci-dent

beam

occurs along a

path

^{such as}

AB]

^at

5

a small fraction of the incident energy ^{is diffracted}

by

^a

powder

particle,

and

_absorption of this diffracted

beam

occurs along the

path BC.

_Similarly, for a high-angle reflection, absorption ^of

both

the incident

and

diffracted

beams

occurs along a

path

^{such as}

(DE + EF). The

net result is that the diffracted

beam

is of lower _intensity

than one would

_expect for a

specimen

^of

no

absorption.

A

calculation ofthis effect

shows

that the relative absorption ^increases as 6 decreases, for

any

given cylindrical specimen.

That

this

must

be so can

be

seen

from

_Fig. 4-1 8_(b)

which

_{applies to} a

specimen

(for example, tungsten) ^of very high absorption.

The

incident

beam

is

very

rapidly absorbed,

and most

of the diffracted

beams

originate ⁱⁿ the thin surface layer

on

the left side of the

specimen

,-fbackward-reflected

beams then undergo very

^little absorption, but forward-reflected

beams have

to pass

through

^the

whole specimen and

are greatly absorbed.^ Actually, the forward-reflected

beams

in this case

come

almost_entirely

from

_{the top}

and bottom

_edges of the specimen.* This difference in absorption

between

*

The

_{powder patterns reproduced in Fig.} 3-13

show

this effect.

The

lowest-angle^lineineach pattern ^issplitin two, because the

beam

diffracted through the center of the specimen ^{is so}highly absorbed. ^{It is} important^to keep ^the possi-bility of this

phenomenon

ⁱⁿ

mind when

_examining Debye-Scherrer photographs, or_splitlow-angle^lines

may

^beincorrectly interpreted asseparatediffraction lines

from twodifferent setsof planes.

130

DIFFRACTION

^III

THE

INTENSITIES

OF DIFFRACTED BEAMS

[CHAP. 4 high-0

and

low-0 reflections decreases as the linear absorption coefficient of the

specimen

decreases,

but

the absorption ^is

always

greater ^for the low-0 reflections. (These

remarks apply

only ^to the cylindrical

specimen used

inthe

Debye-Scherrer method. The

_absorptionfactor

has an

_entirely

different

form

for the _flat-plate

specimen used

ⁱⁿ

a

diffractometer, as will

be shown

in Sec. 7-4.)

Exact

calculation of the absorption ^{factor for} a cylindrical

specimen

^is oftendifficult, so it is fortunate thatthiseffect can usually

be

_neglected in the calculation of diffracted intensities,

when

the

Debye-Scherrer method

isused. Justification of thisomissionwill be

found

ⁱⁿ the next section.

4-11 Temperature

^factor.

So

far

we have

considered a _crystal as

a

collection of

atoms

located at fixed points ⁱⁿ the lattice. Actually, the

atoms undergo

^{thermal vibration}

about

their

mean

_positions

even

at the absolute zerooftemperature,

and

the

amplitude

^ofthisvibration increases as the

temperature

^{increases. In}

aluminum

at

room

temperature, the average displacement ^of

an atom from

its

mean

_position ^is

about

0.17A,

which

is

by ^no means

negligible, being

about

_{6 percent} ^of the distance of closest

approach

^{of the}

mean atom

_positions in this crystal.

^Thermal

agitation decreases the intensity of a diffracted

beam

because

it has theeffect of

smearing

^{out the lattice} planes;*

atoms

can

be

_regarded as lying

no

_longer

on mathematical

_{planes but} rather in platelike regions of ill-definedthickness.

Thus

the reinforcement of

waves

scattered atthe

Bragg

angle

by

^variousparallel planes, the reinforcement

which

iscalled a diffracted

beam,

^{is not} as perfect as ^{it is for}

a

_crystal with fixed atoms.

This reinforcement _requires that the

path

difference,

which

is a function of the plane spacingd,

between waves

scattered

by

adjacent planes be

an

integral

number

of wavelengths.

Now

the thickness of the _platelike

"planes'

' in

which

the vibrating

atoms

lie _is,

on

the average, 2?/,

where u

isthe average displacement ^of

an atom from

its

mean

_position.

Under

these conditions reinforcement is

no

_{longer perfect,}

and

^it

becomes more

imperfect ^as the ratio

u/d

increases, i.e., as the

temperature

increases, since that increases _u, or as increases, since high-0 reflections involve planes ^of

low d

value.

TThus

the intensity of

a

diffracted

beam

decreases as the temperature ^is raised, and, ^for a constant temperature, thermal vibration causesa greater decrease ⁱⁿthereflected intensity athigh angles

than

at

low

_angles./

The

temperature ^effect

and

the previously discussed absorption ^effect in cylindrical specimens ^therefore

depend on

angle ⁱⁿ opposite

ways

and, to a first approximation, ^cancel each other. In

back

reflection, for

exam-ple, theintensity ofadiffracted

beam

isdecreased very^little

by

absorption

but very

greatly

by

^thermal agitation, while in the

forward

direction the reverse ^is true.

The two

effects

do

not exactly ^cancel

one

other at all

4-11]

angles; however, ^if the

comparison

of line intensities is restricted to lines not_differingtoo_greatly in 6values, the absorption

and temperature

^effects can be safely ignored. This is.a fortunate circumstance, ^since

both

of theseeffects are ratherdifficultto calculate exactly.

It should be

noted

here that thermal vibration ofthe

atoms

of a _crystal does not cause

any

broadening ^of the diffraction lines; they

remain

_sharp right

up

^to the melting point, but their

maximum

_{intensity gradually} de-creases. Itisalso

worth

noting that the

mean _amplitude

of

atomic

vibra-tion is not a function of the

temperature

^{alone but}

depends

^also

on

the elastic constantsofthe _crystal.

At any

given temperature, the less "stiff"

the _crystal, the _greater the vibration

amplitude

^u.

This means

that

u

is

much

_{greater at}

_any

one temperature ^fora_soft, low-melting-point metal like lead

than

^{it is} _for, say, tungsten. Substances with

low

melting points

have

quite large values of u

even

at

room temperature and

therefore yield rather poorback-reflection photographs.

The

thermal vibration of

atoms

has another effect

on

diffraction pat-terns. Besides decreasing the intensity ^of diffraction _lines, it causes

some

general coherent _scattering in all directions. This is called temperature-diffuse scattering; ^it contributes only ^to the _general

background

^{of the} pattern

and

its intensity gradually ^increases

with

26. Contrast

between

lines

and background

naturally ^{suffers, so} this effect is a very undesirable one, leading ⁱⁿ

extreme

cases to diffraction lines in the back-reflection region scarcely distinguishable

from

the background.

In the

phenomenon

^of temperature-diffuse scattering

we have

another example,

beyond

^{those alluded to in} Sec. 3-7, of scattering at

non-Bragg

angles.

Here

_again it is not _surprising that such _scattering should _occur, since the displacement ^of

atoms from

their

mean

positions constitutes a kind _{of crystal} imperfection

and

leads to a _partial

breakdown

of the con-ditions necessary ^for perfect destructive interference

between

_rays scat-tered at

non-Bragg

angles.

The

effect of thermal vibration also illustrates

what

has

been

called

"the

approximate law

^of conservation of diffracted energy.

"

This

law

states that the total energy ^diffracted

by

^a particular

specimen under

par-ticular experimental conditions ^is roughly ^constant. Therefore,

anything done

toalterthe_physicalconditionofthe

specimen

^{does notalterthetotal}

amount

of diffractedenergy

but

_onlyitsdistribution in space.

This "law"

isnotatall rigorous,

but

it does

prove

helpful ⁱⁿconsidering

many

diffrac-tion

phenomena. For

_example, at

low

temperatures there ^is very ^little

background

scattering

due

to thermal _agitation

and

the diffraction lines are relatively intense; ^ifthe

specimen

^is

now

heatedtoa high temperature, the lines will

become

_quite

weak and

the energy

which

^is lost

from

the lines will

appear

ⁱⁿ

a

_spread-out

form

as temperature-diffuse scat-tering.

132

DIFFRACTION

^II:

THE

INTENSITIES

OF DIFFRACTED BEAMS

[CHAP. 4

4-12

Intensities of

powder

pattern ^lines.

We

^are

^now

^{in a} position to gather together thefactors discussedinprecedingsections into

an

_equation fortherelative intensity of

powder

pattern ^lines:

Y

¹

⁺

^C0s22g>

)

, (4-12)

\sin²6cos6/

where

^I

=

_{relative integrated intensity} (arbitrary units),

F ⁼

structure factor,

p =

multiplicity factor,

and

6

= _Bragg

_angle. In arriving at ^this equation,

we have

omitted factors

which

are constant for all lines of the pattern.

For

_example, ^all that ^is retained ofthe

Thomson

_{equation (Eq.}

4-2) ^is the polarization factor (1

+

^cos226), with constant factors, such asthe intensity^of theincident

beam and

the charge

and mass

ofthe elec-tron, omitted.

The

intensity ^of a diffraction ^{line is} also directly propor-tional tothe irradiated

volume

ofthe

specimen and

^inversely proportional tothe

camera

radius,

but

these factorsare again constant^forall diffraction lines

and may ^be

neglected.

Omission

^of the temperature

and

_absorption

factors

means

that Eq. (4-12) ^isvalid only ^forthe

Debye-Scherrer method and

then only ^{for lines} fairly close together

on

the pattern; ^{this latter} restriction ^is not as serious as ^it

may

^sound.

Equation

(4-12) ^is also re-stricted to the

Debye-Scherrer method

because of the _particular

way

ⁱⁿ

which

the Lorentz ^factor

was

determined; other

methods,

^{such as those} involving focusing cameras, ^{will require} a modification ^of the Lorentz factor given ^here. In _addition, the individual crystals

making up

^the

powder specimen must have

completely

random

orientations^if

Eq.

(4-12)

isto apply. Finally, ^itshould be

remembered

that thisequation givesthe relative integrated intensity, ^i.e., the relative area

under

the curve of in-tensity^{vs. 20.}

It should

be noted

that "integrated intensity" ^is not _really intensity, since intensity ^is-expressed in

terms

of energy crossing unit area per ^unit oftime.

A beam

^diffracted

by

^a

powder specimen

^carries

a

certain

amount

of energy per unit time

and one

could _quite properly ^{refer to} the total power^ofthe diffracted

beam.

Ifthis

beam

^{isthenincident}

^on

^a

measuring

device, such as photographic ^{film, for} a certain length ^of

time and

if a curve of diffracted intensity ^vs. 26 is constructed

from

the

measurements,

then the area

under

thiscurve_givesthetotal energyⁱⁿthediffracted

beam.

This is the quantity

commonly

^{referred to as integrated intensity.}

A

more

descriptive

term would be

"total diffracted energy,"

but

the

term

"integrated intensity" has

been

too long entrenched ⁱⁿ the

vocabulary

^of x-raydiffraction to

be changed now.

4-13 Examples

of intensity calculations.

The

use of

Eq.

(4-12) ^will be illustrated

by

^the calculation ofthe _position

and

relative intensities of

4-13] 133

the diffraction lines

on

a

powder

pattern ^of copper,

made

with

Cu

Ka.

radiation.

The

calculations are

most

readily carried out in tabular form, as in

Table

4-2.

TABLE

4-2

Remarks:

Column

2:Since copper ^is face-centered cubic,

F

isequal ^to 4/Cu ^{for lines}of un-mixedindicesand zero for lines of mixed indices.

The

reflecting planeindices,^all

unmixed, are written

down

in this column in order of increasing values of _(h² -f-fc

2

+

^Z2), from Appendix ^6.

Column

4: Foracubic _crystal, valuesof sin²6 aregiven by Eq. (3-10)^:

sm"0 =

j-gC/r -h /r -h r;.

In _{this case,} X

=

_{1.542A (Cu}

_Ka)

and a

=

_{3.615A (latticeparameter} of copper).

Therefore, multiplication of the integersⁱⁿcolumn 3 by X²/4a²

=

_0.0455

givesthe values of sin² listed in column 4. In this and similar calculations, slide-rule accuracy^isample.

Column

6:

Needed

to determine the Lorentz-polarization^factorand _(sin0)/X.

Column

7: Obtained from Appendix^7.

Needed

todetermine /Cu

-Column

8:

Read

fromthe curveofFig. 4-6.

Column

9:Obtained fromthe relation

F

²

=

_16/Cu²

-Column

10:Obtained from Appendix^9.

134

DIFFRACTION

^II:

THE

INTENSITIES

OF DIFFRACTED BEAMS

_{[CHAP. 4}

Column

^11: Obtained from Appendix^10.

Column

^12:Thesevaluesarethe product^ofthe valuesⁱⁿ columns9, 10, and 11.

Column

^13:Valuesfrom column12recalculated to givethefirstlinean_arbitrary intensity of ^10.

Column

14: Theseentries give the observed intensities, visually estimated ac-cording ^to the following simple scale, from the pattern

shown

in Fig. 3-13(a) (vs

=

_verystrong,s

=

_strong,

m ⁼

medium,

w =

_weak).

The _agreement

obtained here

between

observed

and

calculatedintensities is satisfactory.

For

_example, lines 1

and

2 are observed to be of strong

and medium

intensity, their respective calculated intensities being ¹⁰

and

4.0. Similar

agreement

^{can be}

found by comparing

^the intensities of

any

pair ^of neighboring ^{lines in} the pattern. Note, however, that the

com-parison

must

be

made between

lines

which

arenot toofarapart: ^for

exam-ple, the calculated intensity^{of line} 2 isgreater

than

that of^line 4,

whereas

line 4 is observedto be _stronger thanline 2. Similarly, the strongest ^lines

on

the pattern ^{are lines 7}

and

8, while calculations

show

line 1 to be_strongest. Errorsof this kind arise

from

the omission^ofthe absorption

and temperature

^factors

from

the calculation.

A more

complicated ^structure

may ^now

^be considered,

namely

^{that of} the zinc-blende

form

of ZnS,

shown

in Fig. 2-19(b). This

form

^of

ZnS

^is

cubic

and

hasa lattice

parameter

^of5.41A.

We

^{will calculate the relative}

intensities ofthe first six lines

on a

_pattern

made

with

Cu Ka

radiation.

As

always, the ^first step ^is to

work

out the structure factor.

ZnS

has four zinc

and

four sulfur

atoms

_perunit _cell, located in the following posi-tions:

'Zn:

\

\ \

+

face-centering translations, S:

+

face-centering translations.

Since the structure is face-centered,

we know

that the structure factor will be zeroforplanes ^of

mixed

indices.

We

^also

know, from example

^(e)

of Sec. 4-6, that the

terms

in the structure-factor equation corresponding to the face-centering translations

can be

factored out

and

theequation^for

unmixed

indiceswritten do\vn ^at once:

|F|

2

is obtained

by

multiplication of the

above by

^its

complex

conjugate:

This equation reduces to the^followingform:

|F|

2

=

₁₆

I/!,

2

+

^/Zn2

+

2/s/Zncos^*-(h

+

^k

+

J

4-13] 135 Furthersimplification ^is possible forvarious special cases:

\F\

2

=

_16(/s²

+ /

^Zn2)

when

_(h

+

^k

+

^{I) is}odd; (4-13)

\F\

2

=

16(/^s

-

_/Z

n)

2

when

_(h

+

^k

+

1} is

an odd

_multiple of_2; (4-14)

|^|

2

=

_16(/s

+

/zn)

2

when

_(h

+

^k

+

I) is

an even

_multiple of2. (4-15)

The

intensity calculations are carriedoutin

Table

4-3, with

some columns

omitted for thesake of brevity.

TABLE

4-3

Remarks:

Columns 5 and 6: These values are read from scattering-factor curves _plotted from the data of Appendix ^8.

Column _{7: \F\~}is obtainedby ^theuseofEq. (4-13), (4-14), or (4-15), depending on theparticular values ^of hklinvolved. Thus,Eq. (4-13) ^is used forthe 111 re-flection and _{Eq. (4-15)} forthe 220reflection.

Columns 10 and 11:

The

_agreement obtained here between calculated and ob-served intensities is again satisfactory. In _{this case,} the agreement ^is good

when

any pair of^linesis compared,^{becauseof thelimited rangeof6valuesinvolved.}

One

further

remark on

intensity calculations^isnecessary. In the

powder

method, two

sets of planes with ^different Miller indices can reflect to the

same

_point

on

the film: for example, the planes (411)

and

₍₃₃₀₎ in the cubic system, ^since they

have

the

same

value of _(h²

+

^k2

+

^I2)

and hence

the

same

spacing, or the planes (501)

and

_{(431) of} the tetragonal system,

JLJO

DIFFRACTION

^II:

THE

INTENSITIES

OF DIFFRACTED BEAMS

_[CHAP. 4 since they

have

the

same

values of (h?

+

^fc2)

^and

^I2.

^{In such}

^{a case, the}

intensity of each reflection

must

be calculated separately, since ⁱⁿ general the

two

^will

have

different multiplicity

and

structure _factors,

and

then

added

to find thetotal intensity ofthe line.

4-14 Measurement

ofx-rayintensity.

In

the

examples

just given, the observed _intensity

was

estimated simply

by

^visual

comparison

^of

one

^line with another.

Although

^this simple procedure ^is satisfactory ⁱⁿ a sur-prisingly large

number

^of_cases, there are

problems

ⁱⁿ

which

a

more

precise

measurement

of diffracted intensity ^is necessary.

Two methods

are in general use

today

^for

making

^such

measurements,

^one

dependent on

^the photographic^effectof x-rays

and

the other

on

the_abilityofx-rays^{to ionize} gases

and

cause fluorescence of _light in _crystals.

These methods have

already

been mentioned

briefly inSec. 1-8

and

^will be described

more

fully in

Chaps.

⁶

and

_7, respectively.

PROBLEMS

4-1.

By

adding Eqs. (4-5) and (4-6) and simplifying the sum, show ^that

E

3,

the resultantofthesetwosinewaves, ^{is also}asine wave,^ofamplitude

A

3

=

_[Ai²

+ ^A

2

4-2. Obtainthe same result

by

solving the vectordiagram ^ofFig. 4-11 for the right-angle triangle ^ofwhich

A

3is the hypotenuse.

4^3. Derive simplified expressions for

F

² for diamond, including the ^rules gov-erning observedreflections. Thiscrystal^iscubicand contains 8 carbonatoms_per unit_cell, located inthefollowing positions:

000 HO $0i OH

Hi Hi Hi Hi

4-4.

A

certain tetragonal crystal has fouratoms of the

same

_{kind per} unit_cell, located at

H.

^{i i, \ f,}

H-(a) Derivesimplified expressions for

F

^2.

(b)

What

isthe Bravaislatticeof this crystal?

(c)

What

are the valuesof

F

²forthe 100, 002, 111,and Oil reflections?

4-6. Derive simplified expressions for

F

² for the wurtzite formof ZnS, includ-ing the rules governing observed reflections. This_crystal ishexagonal

and

con-tains2

ZnS

_perunit _cell, located inthe following positions:

Zn:000, Hi

S:OOf,Hi

Note that these positions involve a

common

translation, which

may

^{be factored}

outofthestructure-factor equation.

4-6. In Sec. 4-9, inthe part devotedto scattering

when

the incidentand scat-tered

beams make

_{unequal angles} witli the _{reflecting planes,} it is stated that

"rays^scattered

by

^allother planes^{are in}phase with the corresponding rays scat-tered

by

^thefirstplane." Provethis.

4-7. Calculate the_position (interms^of6) and the integrated intensity(in rela-tive units) of the firstfive lines on the

Debye

pattern^{of silver}

made

with

Cu Ka

radiation. Ignore the temperatureand _absorptionfactors.

4-^8.

A

Debye-Scherrer pattern ^of tungsten

(BCC)

^is

made

with

Cu Ka

radia-tion.

The

firstfour lines on this pattern were observed to have the following 8 values:

Line 6

1 20.3

2 29.2

3 36.7

4 43.6

Index these lines (i.e., determine the Miller indices of each reflection

by

^{the use}

ofEq. (3-10) and Appendix ⁶⁾ and calculate their relative integrated intensities.

4-9.

A

Debye-Scherrer pattern ^is

made

of gray tin, which has the

same

struc-tureasdiamond,^with

Cu Ka

^radiation.

What

aretheindicesofthefirsttwolines

onthe_pattern, and

what

istheratio oftheintegrated intensity ^ofthefirstto that ofthesecond?

4-10.

A

Debye-Scherrer pattern ^is

made

of the intermediate phase InSb with

Cu Ka

radiation. This phase has thezinc-blende structureandalatticeparameter of 6.46A.

What

are theindices of the first twolines on the _pattern, and whatis

theratio ofthe integrated intensity^ofthefirst tothesecond?

4-11. Calculate the relative integrated intensities of the first six lines of the Debye-Scherrer patternof zinc,

made

with

Cu Ka

^radiation.

The

indicesand ob-served6valuesof theselinesare:

Line hkl 6

(Line 5 ^is

made _up

of two unresolved lines from _planes of very nearly the

same

spacing.)

Compare

your ^{results with the} intensities observed ⁱⁿ the pattern

shown

in Fig.3-13(b).

CHAPTER

5

In document WITHIN THE (pagina 140-149)