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ELEMENTS OF
X-RAY DIFFRACTION
ADDISON-WESLEY METALLURGY SERIES
MORRIS
COHEN, Consulting EditorCidlity ELEMENTS OF
X-RAY
DIFFRACTIONGuy ELEMENTS
OF PHYSICALMETALLURGY
Norton ELEMENTS OFCERAMICSSchuhmann METALLURGICAL ENGINEERING VOL. I: ENGINEERING PRINCIPLES Wagner THERMODYNAMICS OFALLOYS
ELEMENTS OF
X-RAY DIFFRACTION
by
B. D. CULLITY
Associate Professor ofMetallurgy University ofNotre
Dame
ADDISON-WESLEY PUBLISHING COMPANY, INC.
READING, MASSACHUSETTS
ADD1SON-WESLEY PUBLISHING COMPANY,
Inc.Printedni the UnitedStatesofAmerica ALL RIGHTS RESERVED. THIS BOOK, OR PARTS THERE-
OF,
MAY NOT
BE REI'RODlCED INANY FORM WITHOUT
WRITTEN PERMISSION OFTHE
PUBLISHERSLibraryofCongressCatalog
No
56-10137PREFACE
X-ray
diffraction is a tool for the investigation of the fine structure of matter. This techniquehad
itsbeginningsinvon
Laue's discoveryin1912
that crystals diffract x-rays, themanner
of the diffraction revealing the structure of the crystal.At
first, x-ray diffractionwas used
only for the determination of crystal structure. Later on,however,
other useswere
developed,and today
themethod
is applied, not only to structure deter- mination,but
to such diverseproblems
as chemical analysisand
stressmeasurement,
to thestudy
ofphase
equilibriaand
themeasurement
of particle size, to the determination of the orientation ofone
crystal or theensemble
of orientations in a polycrystalline aggregate.The purpose
of thisbook
is to acquaint the readerwho
hasno
previousknowledge
of the subject with the theory of x-ray diffraction, the experi-mental methods
involved,and
themain
applications.Because
theauthor
is
a
metallurgist, the majorityof these applications aredescribedinterms
of metalsand
alloys.However,
little orno
modification of experimentalmethod
isrequiredfortheexaminatiorrof nonmetallicmaterials,inasmuch
as the physical principles involveddo
notdepend on
the materialinvesti- gated.This book
should thereforebe
useful to metallurgists, chemists, physicists, ceramists,mineralogists,etc.,namely,
toallwho
use x-raydiffrac- tionpurelyasa laboratorytool forthesortofproblems
alreadymentioned.
Members
ofthis group, unlike x-ray crystallographers, arenotnormally concerned with
thedetermination ofcomplex
crystal structures.For
this reason the rotating-crystalmethod and
space-group theory, thetwo
chief tools inthe solution ofsuch
structures, are described onlybriefly.This
isa book
of principlesand methods
intended for the student,and
not a referencebook
for theadvanced
research worker.Thus no
metal- lurgicaldata
are givenbeyond
those necessary to illustrate thediffractionmethods
involved.For
example, the theoryand
practice of determining preferred orientation are treated in detail,but
the reasons for preferred orientation, the conditions affecting itsdevelopment, and
actual orien- tationsfound
in specificmetalsand
alloys arenotdescribed,because these topics are adequately covered in existing books.In
short, x-ray diffrac- tionisstressedratherthan
metallurgy.The book
is divided intothreemain
parts: fundamentals, experimentalmethods, and
applications.The
subjectof crystal structure isapproached
through,and based
on, the concept of the point lattice (Bravais lattice), because the point lattice ofa
substance isso closely related to its diffrac-tion pattern.
The
entirebook
is written interms
of theBragg law and can be
read withoutany knowledge
of the reciprocal lattice.(However,
a brieftreatmentof reciprocal-latticetheoryisgiveninan appendix
forthosewho
wish to pursue the subject further.)The methods
of calculating the intensities of diffractedbeams
are introduced early in thebook and used
throughout. Since arigorous derivation ofmany
ofthe equations for dif- fracted intensity is too lengthyand complex a matter
for abook
of this kind,Ihave
preferreda semiquantitativeapproach
which, althoughitdoesnot
furnisha
rigorous proofofthefinal result, atleastmakes
it physically reasonable. This preference isbased on my
conviction that it is betterfor
a
studenttograsp the physicalrealitybehind
amathematical
equationthan
tobe
able to glibly reproducean
involvedmathematical
derivation ofwhose
physicalmeaning he
isonlydimly
aware.Chapters on
chemical analysisby
diffractionand
fluorescencehave been
included because of the present industrialimportance
of these analyticalmethods.
InChapter
7 the diffractometer, thenewest
instrument for dif- fraction experiments, isdescribed insome
detail; here the materialon
the various kinds of countersand
their associated circuits shouldbe
useful,not
only to thoseengaged
in diffraction work,but
also to thoseworking with
radioactive tracers or similarsubstanceswho
wish toknow how
theirmeasuring
instruments operate.Each
chapter includes a set of problems.Many
of thesehave been
chosen to amplify
and
extend particular topics discussed in the text,and
as
such
theyform an
integral part of the book.Chapter
18 containsan annotated
list ofbooks
suitableforfurtherstudy.The
reader shouldbecome
familiarwith at leastafew
of these, ashe
pro- gressesthrough
this book, in order thathe may know where
to turn foradditional information.
Like
any author
ofa technical book, Iam
greatly indebted to previous writerson
thisand
alliedsubjects. Imust
alsoacknowledge my
gratitude totwo
ofmy former
teachersattheMassachusetts
Institute ofTechnology, Professor B. E.Warren and
ProfessorJohn
T.Norton:
theywillfindmany
an echo
of theirown
lecturesin these pages. ProfessorWarren
has kindly allowedme
to usemany problems
of his devising,and
the adviceand encouragement
of ProfessorNorton
hasbeen
invaluable.My
colleague atNotre Dame,
Professor G. C.Kuczynski,
has read theentirebook
asitwas
written,
and
his constructive criticismshave been most
helpful. Iwould
alsolike to
thank
thefollowing, each ofwhom
has read one ormore
chap-ters
and
offered valuable suggestions:Paul
A. Beck,Herbert Friedman,
S. S.
Hsu, Lawrence
Lee,Walter
C. Miller,William
Parrish,Howard
Pickett,
and Bernard Waldman.
Iam
also indebted to C. G.Dunn
for the loan of illustrative materialand
tomany
graduate students,August
PREFACE
VllFreda
in particular,who have
helped with the preparation of diffraction patterns. Finallybut
notperfunctorily,Iwish
tothank Miss Rose Kunkle
for her patience
and
diligence inpreparing thetyped
manuscript.B.
D. CULLITY Notre Dame, Indiana
March,
1956CONTENTS
FUNDAMENTALS
CHAPTER
1 PROPERTIES OFX-RAYS
11-1 Introduction 1
1-2 Electromagnetic radiation 1
1-3
The
continuousspectrum . 41-4
The
characteristic spectrum 61-5 Absorption . 10
1-6 Filters 16
1-7 Productionof x-rays 17
1-8 Detectionof x-rays 23
1 9 Safety precautions . 25
CHAPTER
2THE GEOMETRY
OF CRYSTALS 29^2-1 Introduction . 29
J2-2 Lattices . 29
2-3 Crystalsystems 30
^2-4 Symmetry
342-5 Primitiveand nonprimitive cells 36
2-6 Lattice directionsand planes * . 37
2-7 Crystalstructure J 42
2-8
Atom
sizesand coordination 522-9 Crystalshape 54
2-10
Twinned
crystals . 552-11
The
stereographic projection . . 60CHAPTER
3 DIFFRACTION I:THE
DIRECTIONS OFDIFFRACTED BEAMS
783-1 Introduction .
.78
3-2 Diffraction f. 79
^3-3 The
Bragg law *'
. 84
3-4 X-rayspectroscopy 85
3-5 Diffraction directions- 88
3-6 Diffractionmethods . 89
3-7 Diffraction undernonideal conditions . 96
CHAPTER
4 DIFFRACTION II:THE
INTENSITIES OFDIFFRACTED BEAMS
. 1044-1 Introduction 104
4-2 Scattering
by
an electrons . . 1054-3 Scattering
by
anatom
>,./
1084-4 Scattering
by
a unitcell*/
. Ill4-5
Some
useful relations . 1184-6 Structure-factorcalculations
^
1184-7 Application to
powder method
' 1234-8 Multiplicityfactor 124
4-9 Lorentz factor 124
1-10 Absorptionfactor 129
4-11 Temperaturefactor 130
4-12 Intensities of
powder
patternlines 1324-13
Examples
ofintensitycalculations 1324-14
Measurement
ofx-rayintensity 136EXPERIMENTAL METHODS
LPTER 5
LAUE PHOTOGRAPHS
1385-1 Introduction 138
5-2
Cameras
. 1385-3 Specimenholders 143
5-4 Collimators .
.144
5-5
The
shapesofLaue
spots . 146kPTER 6
POWDER PHOTOGRAPHS
..149
6-1 Introduction . 149
6-2 Debye-Scherrer
method
. 1496-3 Specimen preparation
....
1536-4 Filmloading . . 154
6-5
Cameras
forhighand
low temperatures . 1566-6 Focusing cameras
...
. 1566-7 Seemann-Bohlin camera . 157
6-8 Back-reflection focusing cameras . .
.160
6-9 Pinholephotographs . 163
6-10 Choiceofradiation .
.165
6-11
Background
radiation . 1666-12 Crystal
monochromators
. 1686-13
Measurement
of line position 1736-14
Measurement
of lineintensity . 173VPTER 7
DlFFRACTOMETER MEASUREMENTS
1777-1 Introduction . . . 177
7-2 General features
....
1777-3
X-ray
optics . . . - 1847-4 Intensitycalculations .
...
1887-5 Proportional counters . . . . 190
7-6 Geiger counters . .
... .193
7-7 Scintillation counters . - 201
7-8 Sealers .
... .... .202
7-9 Ratemeters . - 206
7-10
Use
ofmonochromators
211CONTENTS
XIAPPLICATIONS
CHAPTER
8 ORIENTATION OF SINGLECRYSTALS
. . . 2158-1 Introduction . .
....
2158-2 Back-reflectionLaue
method
. ..215
8-3 TransmissionLaue
method ....
. 2298-4 Diffractometer
method
'
.
...
2378-5 Settinga crystalina requiredorientation . 240
8-6 Effect of plastic deformation . 242
8-7 Relativeorientation oftwinned crystals 250
8-8 Relativeorientation of precipitateand matrix . . . 256
CHAPTER
9THE STRUCTURE
OF POLYCRYSTALLINEAGGREGATES
. 2599-1 Introduction . 259
CRYSTAL SIZE
9-2 Grain size 259
9-3 Particle size . 261
CRYSTAL PERFECTION
9-4 Crystal perfection .
....
2639-5
Depth
ofx-ray penetration . . 269CRYSTAL ORIENTATION
9-6 General .
.272
9-7 Textureofwire androd (photographicmethod) . . . 276 9-8 Textureofsheet (photographicmethod) 280 9-9 Textureofsheet (diffractometermethod) . . 285
9-10
Summary
. . 295CHAPTER
10THE DETERMINATION
OFCRYSTAL STRUCTURE
. . . 29710-1 Introduction . . 297
10-2 Preliminary treatmentofdata . . . 299
10-3 Indexing patterns ofcubiccrystals 301
10-4 Indexing patternsofnoncubiccrystals (graphical methods) 304 10-5 Indexing patterns ofnoncubiccrystals (analyticalmethods) .
.311
10-6
The
effectofcelldistortion onthepowderpattern . . . 314 10-7 Determinationofthenumber
ofatomsina unitcell ..316
10-8 Determinationof
atom
positions . 31710-9
Example
ofstructuredetermination....
. 320CHAPTER
11 PRECISEPARAMETER MEASUREMENTS
....
32411-1 Introduction
....
32411-2 Debye-Scherrercameras
.... ....
32611-3 Back-reflectionfocusing cameras 333
11-4 Pinholecameras 333
11-5 Diffractometers 334
11-6
Method
of leastsquares.335
11-7 Cohen's
method ....
33811-8 Calibration
method
. . 342CHAPTER
12 PHASE-DIAGRAMDETERMINATION
. . . 34512-1 Introduction . 345
12-2 Generalprinciples . . 346
12-3 Solid solutions . 351
12-4 Determinationofsolvuscurves (disappearing-phasemethod) 354 12-5 Determinationofsolvuscurves (parametricmethod) 356
12-6 Ternarysystems 359
CHAPTER
13ORDER-DISORDER TRANSFORMATIONS
36313-1 Introduction . 363
13-2 Long-rangeorderin
AuCus
36313-3 Other examplesof long-range order 369
13-4 Detection of superlatticelines 372
13-5 Short-range orderand clustering 375
CHAPTER
14CHEMICAL
ANALYSIS BY DIFFRACTION 37814-1 Introduction 378
QUALITATIVE ANALYSIS
14-2 Basicprinciples 379
14-3
Hanawait method
37914-4 Examplesof qualitative analysis 383
14-5 Practicaldifficulties 386
14-6 Identification of surface deposits 387
QUANTITATIVE ANALYSIS (SINGLE PHASE)
14-7 Chemicalanalysis
by
parameter measurement 388 QUANTITATIVE ANALYSIS (MULTIPHASE)14-8 Basicprinciples . . . 388
14-9 Directcomparison
method
. . . 39114-10 Internalstandard
method
. . . 39614-11 Practicaldifficulties . . . 398
CHAPTER
15CHEMICAL
ANALYSISBY FLUORESCENCE
40215-1 Introduction .
...
40215-2 General principles . . 404
15-3 Spectrometers
...
. 40715-4 Intensityandresolution . . . 410
15-5 Counters
....
. 41415-6 Qualitative analysis
.... ...
41415-7 Quantitativeanalysis
...
. . 41515-8 Automaticspectrometers . . 417
15-9 Nondispersiveanalysis
...
. 41915-10
Measurement
ofcoating thickness 421CONTENTS
xiilCHAPTER
16CHEMICAL
ANALYSIS BYABSORPTION
. . . 42316-1 Introduction . . .
...
42316-2 Absorption-edge
method
. ....
42416-3 Direct-absorption
method
(monochromaticbeam)
. 427 16-4 Direct-absorptionmethod
(polychromaticbeam) 42916-5 Applications . . 429
CHAPTER
17 STRESSMEASUREMENT
....
43117-1 Introduction . 431
17-2 Appliedstressandresidualstress . . 431
17-3 Uniaxialstress . . 434
17-4 Biaxialstress . 436
17-5 Experimental technique (pinholecamera) 441
17-6 Experimentaltechnique (diffractometer) 444 17-7 Superimposedmacrostress andmicrostress 447
17-8 Calibration 449
17-9 Applications 451
CHAPTER
18 SUGGESTIONS FORFURTHER STUDY
. 45418-1 Introduction 454
18-2 Textbooks . 454
18-3 Referencebooks . 457
18-4 Periodicals 458
APPENDIXES
APPENDIX 1 LATTICE
GEOMETRY
. 459Al-1 Plane spacings 459
Al-2
Cellvolumes . . 460Al-3
Interplanarangles . . . 460APPENDIX 2
THE RHOMBOHEDRAL-HEXAGONAL TRANSFORMATION
462 APPENDIX 3WAVELENGTHS
(INANGSTROMS)
OFSOME
CHARACTERISTICEMISSION LINES
AND ABSORPTION EDGES
. . . 464 APPENDIX 4MASS
ABSORPTION COEFFICIENTSAND
DENSITIES . 466APPENDIX 5
VALUES
OF siN28 . 469APPENDIX 6
QUADRATIC FORMS
OFMILLER
INDICES . . . 471APPENDIX 7
VALUES
OF (SIN0)/X . . . 472APPENDIX 8
ATOMIC
SCATTERINGFACTORS
. 474APPENDIX 9 MULTIPLICITY
FACTORS FOR POWDER PHOTOGRAPHS
. *. 477APPENDIX 10 LORENTZ-POLARIZATION
FACTOR
478APPENDIX 11 PHYSICAL
CONSTANTS
. 480APPENDIX 12 INTERNATIONAL
ATOMIC
WEIGHTS, 1953 481APPENDIX 13
CRYSTAL STRUCTURE DATA
482APPENDIX 14
ELECTRON AND NEUTRON
DIFFRACTION 486A14-1 Introduction .
...
. . 486A14r-2 Electrondiffraction
...
. 486A14-3
Neutrondiffraction....
. 487APPENDIX 15
THE RECIPROCAL
LATTICE . . 490A15-1 Introduction .
.... .490
A15-2
Vectormultiplication ....
490A15-3 The
reciprocallattice . ....
491A15-4
Diffractionand thereciprocallattice . 496A15-5 The
rotating-crystalmethod
. 499A15-6 The
powdermethod
. 500A15-7 The Laue method
. . 502ANSWERS
TO SELECTEDPROBLEMS
. 506INDEX ...
509CHAPTER
1PROPERTIES OF X-RAYS
1-1 Introduction.
X-rays were
discovered in 1895by
theGerman
physicist
Roentgen and were
sonamed
becausetheirnaturewas unknown
at the time. Unlike ordinary light, these rays
were
invisible,but
they traveled in straight linesand
affected photographic film in thesame way
aslight.
On
the other hand,they were much more
penetratingthan
lightand
couldeasily passthrough
thehuman
body,wood,
quite thick pieces of metal,and
other"opaque"
objects.It is not
always
necessarytounderstand
a thing in order to use it,and
x-rayswere
almostimmediately put
to useby
physicians and,somewhat
later,
by
engineers,who wished
tostudy
the internal structure ofopaque
objects.By
placing a sourceofx-rayson one
side ofthe objectand
photo- graphic filmon
the other, ashadow
picture, orradiograph, could bemade,
the less dense portions of the object allowing a greater proportion of the x-radiation to passthrough than
themore
dense. In thisway
the point of fracture in abroken bone
or the position of a crack in a metal casting could belocated.Radiography was
thus initiatedwithout any
precise understanding of the radiation used, because itwas
not until 1912 that the exact nature of x-rayswas
established. In that year thephenomenon
of x-ray diffractionby
crystalswas
discovered,and
this discovery simultaneouslyproved
thewave
nature of x-raysand
provided anew method
for investigating the fine structure of matter.Although
radiography is avery important
tool in itselfand
has awide
field ofapplicability, it is ordinarily limited in the internal detail it can resolve, or disclose, to sizes of the orderof 10""1cm.
Diffraction,
on
the other hand,can
indirectly reveal details of internal structure ofthe order of 10~~8cm
in size,and
it is with thisphenomenon, and
itsapplications to metallurgical problems, thatthisbook
isconcerned.The
properties of x-raysand
the internal structure of crystals are here described in the firsttwo
chapters as necessary preliminaries to the dis- cussion ofthediffraction ofx-raysby
crystalswhich
follows.1-2 Electromagnetic radiation.
We know today
that x-rays are elec- tromagneticradiation ofexactly thesame
natureaslightbut
ofvery much
shorter wavelength.
The
unit ofmeasurement
in the x-ray region is theangstrom
(A), equal to10~
8cm, and
x-rays usedin diffractionhave wave-
lengthslyingapproximately
inthe range 0.5-2.5A,whereas
thewavelength
of visible light is of the order of6000A. X-rays
thereforeoccupy
the1
1megacycle
10_
1 kilocycle
IQl
FIG. i-i.
The
electromagneticspectrum.The
boundariesbetweenregions are arbitrary, sinceno sharpupperorlowerlimitscan beassigned. (F.W.
Sears,Optics,3rded.,Addison-Wesley Publishing
Company,
Inc., Cambridge,Mass., 1949)region
between gamma and
ultraviolet rays in thecomplete
electromag- neticspectrum
(Fig. 1-1).Other
unitssometimes
used tomeasure
x-raywavelength
are theX
unit(XU) and
the kiloX
unit(kX =
1000XU).*
The X
unit is only slightly largerthan
the angstrom, the exact relationbemg lkX=
1.00202A.It is
worth
while to review brieflysome
properties of electromagnetic waves.Suppose
amonochromatic beam
of x-rays, i.e., x-rays of a single wavelength, is travelingin the x direction (Fig. 1-2).Then
it has asso-ciatedwith it
an
electric fieldE
in, say, the ydirectionand, at rightangles to this, amagnetic
fieldH
in the z direction. If theelectric field is con-finedtothe xy-planeasthe
wave
travels along,thewave
issaid tobe plane- polarized. (In a completely unpolarized wave, the electric field vectorE
and hence
themagnetic
field vectorH
canassume
all directions in the*Fortheoriginoftheseunits,see Sec. 3-4.
1-2]
ELECTROMAGNETIC RADIATION
FIG. 1-2. Electric and magnetic
fields associated with a
wave
movingin the j-direction.
t/2-plane.)
The magnetic
field is ofno
concern to us hereand we need
not consider it further.In the plane-polarized
wave
con- sidered,E
is not constant with time but variesfrom a maximum
in the+y
directionthrough
zero toamaxi-
mum
in the y directionand back
again, atany
particular point in space, say x=
0.At any
instant of time, sayt=
0,E
variesin thesame
fashionwith distance alongthex-axis.
If both variations are
assumed
to be sinusoidal, theymay
be expressed in theone
equationE = Asin27r(-
- lA
(1-1)where A = amplitude
of thewave,
X=
wavelength,and
v=
frequency.The
variation ofE
is not necessarily sinusoidal, but the exactform
of thewave
matters little; theimportant
feature is its periodicity. Figure 1-3shows
thevariation ofE
graphically.The wavelength and
frequency are connectedby
the relation cX
-
-. (1-2)V
where
c=
velocity of light=
3.00X
1010 cm/sec.Electromagnetic radiation, such as a
beam
ofx-rays, carriesenergy,and
therateofflowofthisenergythrough
unit area perpendiculartothedirec- tion ofmotion
ofthewave
iscalled the intensity I.The
average value of the intensity is proportional to the square of theamplitude
of thewave,
i.e., proportional to
A
2. In absolute units, intensity ismeasured
inergs/cm
2
/sec,
but
thismeasurement
isa difficultone and
isseldom
carried out;most
x-ray intensitymeasurements
aremade on a
relative basis in+E
-E
+E
i
(a) (b)
FIG. 1-3.
The
variationofE, (a) withtat a fixedvalueofx and (b) withxat a fixed value oft.4
[CHAP.arbitrary units, such as the degree of blackening of a photographic film
exposed
to the x-raybeam.
An
accelerated electric charge radiates energy.The
accelerationmay,
of course,
be
either positive or negative,and
thus a charge continuously oscillatingabout some mean
position acts asan
excellentsource of electro-magnetic
radiation.Radio
waves, forexample, areproduced by
theoscil-lation of charge
back and
forth in the broadcasting antenna,and
visible lightby
oscillating electrons in theatoms
of the substance emitting the light. In each case, the frequency ofthe radiation is thesame
as the fre-quency
oftheoscillatorwhich
produces it.Up
tonow we have been
considering electromagnetic radiation as wavemotion
in accordance with classical theory.According
to thequantum
theory, however, electromagnetic radiation
can
also be considered as astream
of particles calledquanta
orphotons.Each photon
has associated withitan amount
ofenergyhv,where
hisPlanck's constant (6.62X 10~
27erg-sec).
A
link is thus providedbetween
thetwo
viewpoints, becausewe
can use the frequency of thewave motion
to calculate the energy of the photon. Radiation thus has a dual wave-particle character,and we
will use
sometimes one
concept,sometimes
the other, to explain variousphenomena,
givingpreferenceingeneraltotheclassicalwave
theorywhen-
ever it is applicable.1-3
The
continuous spectrum.X-rays
areproduced when any
electri- cally charged particle of sufficient kinetic energy is rapidly decelerated.Electrons are usually
used
for this purpose, the radiation beingproduced
inan
x-ray tubewhich
contains a source of electronsand two metal
elec- trodes.The
high voltagemaintained
across these electrodes,some
tens ofthousands
of volts, rapidlydraws
the electrons tothe anode, or target,which they
strikewith very
high velocity.X-rays
areproduced
at the pointofimpact and
radiateinalldirections. Ife isthe chargeon
theelec- tron (4.80X 10~
10esu)and
1) the voltage (inesu)*across the electrodes, then thekineticenergy (inergs) of *the electronson impact
isgivenby
the equationKE - eV =
\mv*, (1-3)
where m
is themass
of the electron (9.11X 10~
28gm) and
v its velocityjust before impact.
At
a tube voltage of 30,000 volts (practical units), this velocity isabout
one-third that of light.Most
of the kinetic energy oftheelectrons strikingthetargetisconvertedinto heat,lessthan
1percent being transformedinto x-rays.When
the rayscoming from
the target areanalyzed, they arefound
to consistofa mixtureof differentwavelengths,and
thevariation ofintensity* 1volt (practical units)
= ^fo
volt(esu).1-3]
THE CONTINUOUS SPECTRUM
1.0 2.0
WAVELENGTH
(angstroms)FIG. 1-4. X-ray spectrumof
molybdenum
asa functionofappliedvoltage(sche-matic). Line widths nottoscale.
with
wavelength
isfound
todepend on
thetube
voltage. Figure 1-4shows
the kind of curves obtained.The
intensity is zeroup
to a certain wavelength, called the short-wavelengthjimit (XSWL), increases rapidly toamaximum and
thendecreases, withno
sharp limiton
the longwavelength
side.*
When
the tube voltage is raised, the intensity of allwavelengths
increases,and both
the short-wavelengthlimitand
thepositionofthemax- imum
shift to shorter wavelengths.We
are concernednow
with thesmooth
curves in Fig. 1-4, those corresponding to applied voltages of20 kv
or less in the case of amolybdenum
target.The
radiation repre- sentedby
such curves is called heterochromatic, continuous, or white radia- tion, since it ismade
up, like white light, ofrays ofmany
wavelengths.The
continuousspectrum
isdue
tothe rapiddecelerationoftheelectrons hittingthe targetsince, asmentioned
above,any
deceleratedcharge emits energy.Not
everyelectronisdecelerated inthesame way, however; some
arestoppedin
one impact and
giveup
alltheirenergyat once, whileothersare deviated this
way and
thatby
theatoms
of the target, successively losing fractions of their total kinetic energy until it is all spent.Those
electronswhich
are stopped in oneimpact
will give rise tophotons
ofmaximum
energy, i.e., to x-raysofminimum
wavelength.Such
electrons transferall their energyeV
intophoton
energyand we may
write[CHAP.
c he
12,400
(1-4)
This equation gives the short-wavelength limit (in angstroms) as a func- tion of the applied voltage
V
(in practical units). Ifan
electron is not completely stopped in one encounterbut
undergoes a glancingimpact which
onlypartially decreasesitsvelocity,then
only afractionofitsenergyeV
is emitted as radiationand
thephoton produced
has energy lessthan
hpmax- Interms
ofwave
motion, the corresponding x-ray has a frequency lowerthan
vmaxand
awavelength
longerthan
XSWL-The
totalityofthese wavelengths, rangingupward from
ASWL, constitutes thecontinuous spec- trum.We now
seewhy
the curves of Fig. 1-4become
higherand
shift to theleft as the applied voltage is increased, since the
number
ofphotons
pro-duced
per secondand
the average energy perphoton
areboth
increasing.The
total x-ray energy emitted per second,which
is proportional to the areaunder one
of the curvesof Fig. 1-4, alsodepends
on theatomic num-
berZ
ofthetargetand on
thetube currenti, thelatterbeinga measure
of thenumber
of electrons per second striking the target. This total x-ray intensityisgivenby
/cent spectrum
= AlZV,
(1-5)where A
isa
proportionality constantand m
isa constant with a valueofabout
2.Where
largeamounts
of whiteradiation are desired, it isthere- forenecessarytouse aheavy metal
liketungsten (Z=
74) asatargetand
ashigh a voltageas possible.
Note
that the material oftthe targetaffects the intensity
but
not thg.wftV dfinfi^h distribution Of t.hp..p.ont.iniiniia spec- trum,1-4
The
characteristic spectrum.When
the voltageon an
x-ray tubeis raised
above
a certain critical value, characteristic of the target metal, sharp intensitymaxima appear
at certain wavelengths,superimposed on
the continuous spectrum. Sincethey
are sonarrow and
since theirwave-
lengths are characteristic ofthe targetmetal
used,they
are called charac- teristic lines.These
lines fall into several sets, referred to asK,
L,M,
etc., in the order of increasing wavelength, all the lines together
forming
the characteristic spectrum of themetal used
as thetarget.For
amolyb- denum
target theK
lineshave wavelengths
ofabout
0.7A, theL
linesabout
5A,and
theM
lines still higher wavelengths. Ordinarily only theK
lines are useful in x-ray diffraction, the longer-wavelength lines being too easily absorbed.There
are several lines in theK
set,but
only the1-4]
THE CHARACTERISTIC SPECTRUM
7 three strongest are observed innormal
diffraction work.These
are thectz,
and Kfa, and
formolybdenum
theirwavelengths
are:0.70926A,
Ka
2: 0.71354A, 0.63225A.The
iand
2components have wavelengths
so close together that they are notalways
resolved as separate lines; if resolved,they
are called theKa
doublet and, if not resolved, simply theKa
line* Similarly,K&\
is usually referred to as theK@
line,with
the subscript dropped.Ka\
isalways about
twice as strong asKa%,
while the intensity ratio ofKa\
toKfli
depends on atomic number but
averagesabout
5/1.These
characteristic linesmay
be seen in theuppermost
curve of Fig.1-4. Since the critical
K
excitation voltage, i.e., the voltage necessary to exciteK
characteristic radiation, is20.01kv
formolybdenum,
theK
linesdo
notappear
in the lower curves of Fig. 1-4.An
increase in voltageabove
the critical voltage increases the intensities of the characteristic lines relative to the continuousspectrum but
does not change their wave- lengths. Figure 1-5shows
thespectrum
ofmolybdenum
at35 kv on
acompressed
vertical scale relative tothat ofFig. 1-4;the increasedvoltage has shifted the continuousspectrum
to still shorterwavelengths and
in- creased the intensities of theK
lines relative to the continuousspectrum but
has notchanged
their wavelengths.The
intensity ofany
characteristic line,measured above
the continuous spectrum,depends both on
the tube current iand
theamount by which
the applied voltageV
exceeds the critical excitationvoltage for that line.For
aK
line, the intensity isgivenby
IK
line= Bi(V - V K
)
n
, (1-6)
where B
is a proportionality constant,VK
theK
excitation voltage,and n
a constant with a value ofabout
1.5.The
intensity of a characteristic linecan
be quite large:for example, in the radiationfrom
acopper
target operated at30 kv, theKa
linehas an
intensityabout 90
times that ofthewavelengths immediately
adjacent to it in the continuous spectrum.Be-
sides being very intense, characteristic lines are alsovery
narrow,most
ofthem
lessthan 0.001A wide measured
at half theirmaximum
intensity,as
shown
in Fig. 1-5.The
existence of this strongsharp Ka. line iswhat makes
a great deal of x-ray diffraction possible, sincemany
diffraction experiments require the use ofmonochromatic
orapproximately mono-
chromatic radiation.*
The
wavelength ofanunresolvedKa
doubletisusuallytakenasthe weighted average of the wavelengths of its components,Kai
being given twice the weight ofKa%, sinceit is twiceas strong.Thus
thewavelengthoftheunresolvedMo Ka
lineisJ(2
X
0.70926+
0.71354)=
0.71069A.[CHAP.
60
50
.5 40
1
3020
10
Ka
*-<0.001A
0.2 0.4 0.6 0.8 1.0
WAVELENGTH
(angstroms)FIG. 1-5. Spectrumof
Mo
at35kv
(schematic). Line widths notto scale.The
characteristic x-ray lineswere
discoveredby W. H. Bragg and
systematizedby H.
G. Moseley.The
latterfound
that thewavelength
ofany
particularlinedecreasedastheatomic number
ofthe emitterincreased.In particular, he
found
a linear relation (Moseley's law)between
the square root ofthe line frequency vand
theatomic number Z
:= C(Z -
er), (1-7)
where C and
<r are constants. This relation is plotted in Fig. 1-6 for theKai and Lai
lines,thelatterbeing thestrongestline intheL
series.These
curves show, incidentally, thatL
lines arenotalways
of longwavelength
: theLai
line of aheavy metal
like tungsten, for example, hasabout
thesame wavelength
as theKa\
line of copper,namely about
1.5A.The
1-4]
THE CHARACTERISTIC SPECTRUM
3.0 2.5 2.0
X (angstroms)
1.5 1.0 0.8 0.7
80
70
60
I
W
w
50
u s
4030
20-
10
T
I I IT
I
1.0 1.2 1.4 1.6 1.8 2.0 2.2
X
109FIG. 1-6. Moseley'srelationbetween
\/v
andZ
for twocharacteristic lines.wavelengths
of the characteristic x-ray lines of almost all theknown
ele-ments have
been precisely measured,mainly by M. Siegbahn and
his associates,and
a tabulation of thesewavelengths
for the strongest lines of theK and L
series will befound
in
Appendix
3.While
the cQntinuoi^s_srjex;truri^js causedbyjthe T^^^^dej^tignj)^
electrons
by
the target; the origin of^
M
shellatoms
j3i_tl^_taj^J)_jrnaterial itself.
To understand
thisphenomenon,
itis
enough
to consideran atom
ascon- sisting ofacentralnucleussurrounded by
electrons lying in various shells (Fig. 1-7). If one of the electronsbombarding
the target has sufficient kinetic energy, itcan knock an
elec- tron out of theK
shell, leaving theatom
inan
excited,high-energystate,FlG
^
Electronic transitions inan at0
m
(schematic). Emission proc- esses indicated byarrows.One
ofthe outerelectronsimmediately
fallsintothevacancy
intheK
shell,emitting energy in the process,
and
theatom
is once again in itsnormal energy
state.The
energy emitted is in theform
of radiation of a definitewavelength and
is, in fact, characteristicK
radiation.The
Jff-shellvacancy may
be filledby an
electronfrom any one
of the outer shells, thus giving rise to a series ofK
lines;Ka and K&
lines, for example, resultfrom
the filling of a K-shellvacancy by an
electronfrom
theLOT M
shells, respectively. Itispossible tofilla7-shellvacancy
eitherfrom
theL
orM
shell, sothatone atom
of thetargetmay
be emittingKa
radiation while its neighbor is emitting Kfi\ however, it is
more
probable that a jf-shellvacancy
will be filledby an L
electronthan by an M
elec-tron,
and
the result is that theKa
line is strongerthan
theK$
line. Italso follows that it is impossible to excite one
K
line without exciting allthe others.
L
characteristic lines originate in a similarway: an
electronis
knocked
out oftheL
shelland
thevacancy
is filledby an
electronfrom some
outer shell.We now
seewhy
there should be acritical excitationvoltage for charac- teristic radiation.K
radiation, for example,cannot
be excited unless the tube voltage issuch
that thebombarding
electronshave enough
energy toknock an
electron out of theK
shell of a targetatom.
IfWK
is thework
required toremove a K
electron, then the necessary kinetic energy ofthe electronsisgivenby
ynxr = WK-
(1~8)It requires less
energy
toremove an L
electronthan
aK
electron, since theformerisfartherfrom
the nucleus;ittherefore followsthat theL
excita- tionvoltage is lessthan
theK and
thatK
characteristic radiation cannotbe produced
without L,M,
etc., radiationaccompanying
it.1-6 Absorption.
Further
understanding of the electronic transitionswhich can
occur inatoms can be
gainedby
considering not only the inter- actionofelectronsand
atoms,but
alsotheinteraction ofx-raysand
atoms.When
x-rays encounterany form
of matter,they
are partly transmittedand
partly absorbed.Experiment shows
that the fractional decrease in the intensity 7 ofan
x-raybeam
as it passesthrough any homogeneous
substanceisproportionaltothe distancetraversed, x. Indifferentialform,-J-/.AC,
(1-9)where
theproportionality constant /u is called the linear absorption coeffi- cientand
isdependent on
the substance considered, its density,and
thewavelength
ofthex-rays. Integration ofEq.
(1-9) gives4-
-
/or**,
(1-10)where
/o=
intensity of incident x-raybeam and
Ix=
intensity of trans- mittedbeam
afterpassingthrough
a thicknessx.1-5]
ABSORPTION
11The
linearabsorptioncoefficient/zisproportionaltothe densityp,which means
thatthe quantity M/Pisa constantofthe materialand
independent of itsphysical state (solid, liquid, orgas). This latterquantity, called themass
absorption coefficient, is the one usually tabulated.Equation
(1-10)may
then berewritten inamore
usableform
:(1-11)
Values of the
mass
absorption coefficient /i/p are given inAppendix
4 for variouscharacteristic wavelengths used in diffraction.Itis occasionallynecessary to
know
themass
absorption coefficient ofa substance containingmore
than one element.Whether
the substanceisa mechanical mixture, a solution, or a chemicalcompound, and whether
itis in the solid, liquid, or gaseous state, its
mass
absorption coefficient is simply the weighted average of themass
absorption coefficients of its constituent elements. IfWi,w
2, etc., are the weight fractions ofelements1, 2, etc., in the substance
and
(M/P)I, (M/p)2j etc., theirmass
absorptioncoefficients, then the
mass
absorption coefficient of the substance is givenby
- =
Wl (-J + W2
(-J +
. ... (1-12)The way
inwhich
the absorption coefficient varies withwavelength
gives the clue to the interaction of x-raysand
atoms.The
lower curve of Fig. 1-8shows
this variationfora nickel absorber; it is typical of all materials.The
curveconsists oftwo
similarbranches separatedby
a sharp discontinuity calledan
absorption edge.Along
eachbranch
the absorp- tion coefficient varies withwave-
length approximately according to a relation oftheform
M P
where
k=
aconstant,with adifferent value for each
branch
of the curve,and Z =
atomicnumber
ofabsorber.Short-wavelength x-rays are there- fore highly penetrating
and
are0.5 1.0 1.5 2.0 2.
X (angstroms) FIG. 1-8. Variation with wave- lengthofthe energy per x-ray
quantum
andofthe massabsorption coefficient of nickel.12
termed
hard, while long-wavelength x-rays areeasilyabsorbed and
are said tobe
soft.Matter
absorbs x-rays intwo
distinct ways,by
scatteringand by
trueabsorption,
and
thesetwo
processestogethermake up
thetotal absorptionmeasured by
the quantityM/P-The
scattering ofx-raysby atoms
issimilarin
many ways
tothescattering ofvisible lightby
dustparticles in theair.Ittakes placeinall directions,
and
since the energy inthescatteredbeams
does notappear
in the transmittedbeam,
it is, so far as the transmittedbeam
is concerned, said tobe
absorbed.The phenomenon
of scattering will be discussed in greater detail inChap.
4; it isenough
to note here that, except for thevery
light elements, it is responsible for onlya
small fraction of the total absorption.True
absorption is causedby
electronictransitions within the
atom and
is best consideredfrom
the viewpoint of thequantum
theory of radiation. Just asan
electron of sufficient energycan knock
aK
electron, for example, out ofan atom and
thus cause the emissionofK
characteristic radiation, so alsocanan
incidentquantum
ofx-rays, provided it has the
same minimum amount
ofenergy WK-
In thelatter case, the ejected electron is called a photoelectron
and
the emitted characteristic radiation is called fluorescent radiation. It radiates in all directionsand
has exactly thesame wavelength
asthecharacteristic radia- tion causedby
electronbombardment
of a metal target. (In effect,an atom
with a #-shellvacancy always
emitsK
radiationno matter how
thevacancy was
originally created.) Thisphenomenon
is the x-ray counter- part of the photoelectric effect in the ultraviolet region of the spectrum;there, photoelectrons
can
be ejectedfrom
the outer shells ofametal atom by
the action ofultraviolet radiation, provided thelatterhas awavelength
less
than a
certain critical value.To
say that the energy of theincoming quanta must
exceed a certain valueWK
isequivalenttosaying that thewavelength must
be lessthan
a certain value X#, since theenergy
perquantum
is hvand wavelength
isinverselyproportional to frequency.
These
relationsmay be
writtenhe
where
VK and \K
are the frequencyand
wavelength, respectively, of theK
absorptionedge.Now
consider the absorption curveof Fig. 1-8in light of the above.Suppose
that x-rays ofwavelength 2.5A
are incidenton
a sheet of nickeland
that thiswavelength
is continuously decreased.At
firstthe absorptioncoefficientis
about
180cm
2/gm, but
as thewavelength
decreases, the frequency increasesand
so does theenergy
perquantum,
asshown by
theupper
curve, thus causing the absorption coefficient to decrease, since the greater theenergy
of aquantum
themore
easily itpasses