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CH
APT
ER
3X
-RA
Y D
IFFR
AC
TIO
N
IN C
RY
ST
AL
1
I.X-Ray
II.
Diffraction
III.
Diffraction of Waves by
Crystals
IV.
X-Ray Diffraction
V.
Bragg Equation
VI.
X-Ray Methods
VII.
Neutron & Electron Diffraction
Bertha Röntgen’s
Hand 8 Nov, 1895
X-R
AY
�X
-rays
we
redis
cove
red
in
189
5by
the
Germ
an
physic
ist
Wilh
elm
Co
nra
d
Rö
ntg
en
and
were
so
nam
ed
2
beca
use
the
irnatu
rew
as
unknow
natth
etim
e.
�H
ew
as
aw
ard
ed
the
Nobe
l
prize
for
physic
sin
1901.
Wilh
elm
Conra
d R
öntg
en
(1845-1
923)
X-R
AY
PR
OPER
TIE
S
�X
ray,
invis
ible
,h
igh
lyp
en
etr
atin
ge
lectr
om
ag
ne
tic
rad
iatio
no
fm
uch
sh
ort
er
wa
ve
len
gth
(hig
he
rfr
eq
ue
ncy)
tha
nvis
ible
ligh
t.T
he
wa
ve
len
gth
ran
ge
for
Xra
ys
isfr
om
ab
ou
t1
0-8
mto
ab
ou
t1
0-1
1m
,th
eco
rre
sp
on
din
gfr
eq
ue
ncy
ran
ge
isfr
om
ab
ou
t3
×1
01
6H
zto
ab
ou
t3
×1
01
9H
z.
3
X-R
AY
EN
ER
GY
�E
lectr
om
agne
tic
rad
iatio
nd
escri
be
das
ha
vin
gp
acke
tso
fe
nerg
y,
or
pho
tons.
Th
een
erg
yo
fth
ep
ho
ton
isre
late
dto
its
fre
qu
en
cy
by
the
follo
win
gfo
rmu
la:
νh
E=
νc
=hc
E=
4
x-r
ay
≈10
-10
≈1A
°E
~1
04 e
v
=W
avele
ngth
,
ע=
Fre
quency ,
c =
Velo
city o
f lig
ht
λν
c=
λE=
λhc
E=
λ
λ
PR
OD
UC
TIO
N O
F X
-RA
YS
�V
isib
lelig
ht
ph
oto
ns
an
dX
-ra
yp
ho
ton
sa
reb
oth
pro
du
ce
db
yth
em
ove
me
nt
of
ele
ctr
on
sin
ato
ms.
Ele
ctr
on
so
ccu
py
diffe
ren
te
ne
rgy
leve
ls,
or
orb
ita
ls,
aro
un
da
na
tom
'sn
ucle
us.
5aro
un
da
na
tom
'sn
ucle
us.
�W
he
na
ne
lectr
on
dro
ps
toa
low
er
orb
ita
l,it
ne
ed
sto
rele
ase
so
me
en
erg
y;
itre
lea
se
sth
ee
xtr
ae
ne
rgy
inth
e
form
of
ap
ho
ton
.T
he
en
erg
yle
ve
lo
fth
ep
ho
ton
de
pe
nd
s
on
ho
wfa
rth
ee
lectr
on
dro
pp
ed
be
twe
en
orb
ita
ls.
6
�X
rays
ca
nb
ep
rod
uce
din
ah
igh
lye
va
cu
ate
dg
lass
bu
lb,
ca
lled
an
X-r
ay
tub
e,
tha
tco
nta
ins
essen
tia
llytw
oe
lectr
od
es—
an
an
od
em
ad
eo
fp
latin
um
,tu
ng
ste
n,
or
an
oth
er
he
avy
me
tal
of
hig
hm
eltin
gp
oin
t,a
nd
aca
tho
de
.W
he
na
hig
hvo
lta
ge
isa
pp
lied
be
twe
en
the
ele
ctr
od
es,
str
ea
ms
of
ele
ctr
on
s(c
ath
od
era
ys)
are
acce
lera
ted
fro
mth
eca
tho
de
toth
ea
no
de
an
dp
rod
uce
Xra
ys
as
the
ystr
ike
the
an
od
e.
X-R
AY
TU
BE
7
Eva
cu
ate
d g
lass b
ulb
An
od
eC
ath
od
e
pro
du
ce
Xra
ys
as
the
ystr
ike
the
an
od
e.
Monoch
rom
atic
and B
road
Spect
rum
of X-r
ays
�X
-ra
ys
ca
nb
ecre
ate
db
yb
om
ba
rdin
ga
me
tal
targ
et
with
hig
he
ne
rgy
(>)
ele
ctr
on
s.
�S
om
eo
fth
ese
ele
ctr
on
se
xcite
ele
ctr
on
sfr
om
co
resta
tes
inth
em
eta
l,w
hic
hth
en
reco
mb
ine
,p
rod
ucin
gh
igh
lym
on
och
rom
atic
X-r
ays.
Th
ese
are
refe
rre
dto
as
410
8
mo
no
ch
rom
atic
X-r
ays.
Th
ese
are
refe
rre
dto
as
ch
ara
cte
ristic
X-r
ay
line
s.
�O
the
re
lectr
on
s,
wh
ich
are
de
ce
lera
ted
by
the
pe
rio
dic
po
ten
tia
lo
fth
em
eta
l,p
rod
uce
ab
roa
dsp
ectr
um
of
X-r
ay
fre
qu
en
cie
s.
�D
ep
en
din
go
nth
ed
iffr
actio
ne
xp
eri
me
nt,
eith
er
or
bo
tho
fth
ese
X-r
ay
sp
ectr
aca
nb
eu
se
d.
410
AB
SO
RPT
ION
OF X
-RA
YS
�T
he
ato
ms
tha
tm
ake
up
yo
ur
bo
dy
tissu
ea
bsorb
vis
ible
ligh
tp
ho
ton
sve
ryw
ell.
Th
ee
ne
rgy
leve
lo
fth
ep
ho
ton
fits
with
va
rio
us
en
erg
yd
iffe
ren
ce
sb
etw
ee
ne
lectr
on
po
sitio
ns.
...s
om
eth
ing
you w
on't
see v
ery
oft
en
(Vis
ible
Lig
ht)
9
po
sitio
ns.
�R
ad
iow
ave
sd
on
'th
ave
en
ou
gh
en
erg
yto
mo
ve
ele
ctr
on
sb
etw
ee
no
rbita
lsin
larg
er
ato
ms,
so
the
yp
ass
thro
ug
hm
ost
stu
ff.
X-r
ay
ph
oto
ns
als
op
ass
thro
ug
hm
ost
thin
gs,
bu
tfo
rth
eo
pp
osite
rea
so
n:
Th
ey
ha
ve
too
mu
ch
en
erg
y.
Lig
ht)
X-r
ay
Genera
tion o
f X
-rays
(K-S
hell
K
nock
out)
An
ele
ctr
on
ina
hig
he
ro
rbita
lim
me
dia
tely
falls
toth
elo
we
re
ne
rgy
leve
l,re
lea
sin
gits
extr
ae
ne
rgy
inth
efo
rmo
fa
ph
oto
n.
It's
ab
igd
rop
,so
the
ph
oto
nh
as
ah
igh
en
erg
yle
ve
l;it
isa
nX
-ra
yp
ho
ton
.
10
Th
e f
ree e
lectr
on
co
llid
es
wit
h t
he t
un
gste
n a
tom
,kn
ockin
g a
n e
lectr
on
ou
t o
f a
low
er
orb
ital.
A h
igh
er
orb
ital
ele
ctr
on
fil
ls t
he e
mp
typ
osit
ion
, rele
asin
g i
ts e
xcess
en
erg
y as a
ph
oto
n.
�A
larg
er
ato
mis
mo
relik
ely
toa
bso
rba
nX
-ra
yp
ho
ton
inth
isw
ay,
be
ca
use
larg
er
ato
ms
ha
ve
gre
ate
re
ne
rgy
diffe
ren
ce
sb
etw
ee
no
rbita
ls--
the
en
erg
yle
ve
lm
ore
clo
se
lym
atc
he
sth
ee
nerg
yo
fth
ep
ho
ton
.S
ma
ller
ato
ms,
wh
ere
the
ele
ctr
on
orb
ita
lsa
rese
pa
rate
db
yre
lative
lylo
w
Abso
rpti
on o
f X
-rays
11
wh
ere
the
ele
ctr
on
orb
ita
lsa
rese
pa
rate
db
yre
lative
lylo
wju
mp
sin
en
erg
y,
are
less
like
lyto
ab
so
rbX
-ra
yp
ho
ton
s.
�T
he
so
fttissu
ein
yo
ur
bo
dy
isco
mp
ose
do
fsm
alle
ra
tom
s,
an
dso
do
es
no
ta
bso
rbX
-ra
yp
ho
ton
sp
art
icu
larl
yw
ell.
Th
eca
lciu
ma
tom
sth
at
ma
ke
up
yo
ur
bo
ne
sa
rem
uch
larg
er,
so
the
ya
reb
ett
er
at
ab
so
rbin
gX
-ra
yp
ho
ton
s.
DIF
FR
AC
TIO
N
�D
iffr
actio
nis
aw
ave
ph
en
om
en
on
inw
hic
hth
ea
pp
are
nt
be
nd
ing
an
dsp
rea
din
go
fw
ave
sw
he
nth
ey
me
et
an
ob
str
uctio
n.
�D
iffr
actio
no
ccu
rsw
ith
ele
ctr
om
ag
ne
tic
12
�D
iffr
actio
no
ccu
rsw
ith
ele
ctr
om
ag
ne
tic
wa
ve
s,
su
ch
as
ligh
ta
nd
rad
iow
ave
s,
an
da
lso
inso
un
dw
ave
sa
nd
wa
ter
wa
ve
s.
�T
he
mo
st
co
nce
ptu
ally
sim
ple
exa
mp
leo
fd
iffr
actio
nis
do
ub
le-s
litd
iffr
actio
n,
tha
t’s
wh
yfirs
tly
we
rem
em
be
rlig
ht
diffr
actio
n.
Wid
th b
V
ariable
(500-1
500 n
m)
Wavele
ngth
Consta
nt
(600 n
m)
Dis
tance d
= C
onsta
nt
LIG
HT
DIF
FR
AC
TIO
N
�L
igh
td
iffr
actio
nis
ca
used
by
ligh
tb
en
din
ga
rou
nd
the
ed
ge
of
an
ob
ject.
Th
ein
terf
ere
nce
pa
tte
rno
fb
rig
ht
an
dd
ark
line
sfr
om
the
diffr
actio
ne
xp
eri
me
nt
ca
no
nly
be
exp
lain
ed
by
the
ad
ditiv
e
na
ture
of
wa
ve
s;
wa
ve
pe
aks
ca
na
dd
tog
eth
er
tom
ake
a
bri
gh
ter
ligh
t,o
ra
pe
ak
an
da
thro
ug
hw
illca
nce
le
ach
oth
er
ou
t
13
bri
gh
ter
ligh
t,o
ra
pe
ak
an
da
thro
ug
hw
illca
nce
le
ach
oth
er
ou
t
an
dre
su
ltin
da
rkn
ess.
Th
us
Yo
un
g’s
lig
ht
inte
rfe
ren
ce
ex
pe
rim
en
t p
rov
es
th
at
lig
ht
ha
s w
av
eli
ke
pro
pe
rtie
s.
LIG
HT
IN
TER
FER
EN
CE
14
Const
ruct
ive &
Dest
ruct
ive W
aves
�C
on
str
ucti
vein
terf
ere
nce
isth
ere
su
lto
fs
ync
hro
niz
ed
lig
ht
wa
ves
that
ad
dto
geth
er
toin
cre
ase
the
ligh
tin
ten
sit
y.
�D
estr
ucti
veĐn
terf
ere
nce
.re
su
lts
wh
en
two
ou
t-o
f-p
ha
se
lig
ht
wa
ves
ca
nc
el
ea
ch
oth
er
ou
t,re
su
ltin
gin
da
rkn
es
s.
15
inte
ns
ity.
Lig
ht In
terf
ere
nce
16
Dif
fract
ion fro
m a
part
icle
and s
oli
d
Sin
gle
pa
rtic
le�
To
un
de
rsta
nd
diffr
actio
nw
ea
lso
ha
ve
toco
nsid
er
wh
at
ha
pp
en
sw
he
na
wa
ve
inte
racts
with
asin
gle
pa
rtic
le.
Th
ep
art
icle
sca
tte
rsth
ein
cid
en
t
17
Th
ep
art
icle
sca
tte
rsth
ein
cid
en
tb
ea
mu
nifo
rmly
ina
lld
ire
ctio
ns
So
lid
ma
teri
al
�W
ha
th
ap
pe
ns
ifth
eb
ea
mis
incid
en
to
nso
lidm
ate
ria
l?If
we
co
nsid
er
acry
sta
llin
em
ate
ria
l,th
esca
tte
red
be
am
sm
ay
ad
dto
ge
the
rin
afe
wd
ire
ctio
ns
an
dre
info
rce
ea
ch
oth
er
tog
ive
dif
fra
cte
db
ea
ms
A c
rysta
l is
a p
eriodic
str
uctu
re
( unit c
ells
are
repeate
d r
egula
rly)
Solid
Sta
teP
hysic
sde
als
how
the
wa
ves
are
pro
pa
gate
dth
rough
su
ch
perio
dic
str
uctu
res.
Inth
ischapte
rw
estu
dy
the
Dif
fract
ion o
f W
aves
by C
ryst
als
18
thro
ugh
su
ch
perio
dic
str
uctu
res.
Inth
ischapte
rw
estu
dy
the
cry
sta
lstr
uctu
reth
rough
the
diffr
action
of
photo
ns
(X-r
ay),
nuetr
ons
and
ele
ctr
ons.
Diffr
action
X-r
ay
Neutr
on
Ele
ctr
on
The g
enera
l princib
les w
ill b
e t
he s
am
e f
or
each t
ype o
f w
aves.
Dif
fract
ion o
f W
aves
by C
ryst
als
�T
he
diffr
actio
nd
ep
en
ds
on
the
cry
sta
lstr
uctu
rea
nd
on
the
wa
ve
len
gth
.
�A
to
ptica
lw
ave
leng
ths
such
as
50
00
an
gstr
om
sth
e
su
pe
rpo
sitio
no
fth
ew
aves
sca
tte
red
ela
stica
llyby
the
19
ind
ivid
ua
la
tom
so
fa
cry
sta
lre
su
lts
ino
rdin
ary
op
tical
refr
actio
n.
�W
he
nth
ew
avele
ng
tho
fth
era
dia
tio
nis
com
pa
rab
le
with
or
sm
alle
rth
an
the
latt
ice
co
nsta
nt,
on
eca
nfin
d
diffr
acte
db
ea
ms
ind
ire
ctio
ns
qu
ite
diffe
ren
tfr
om
the
incid
en
tra
dia
tio
n.
Dif
fract
ion o
f W
aves
by C
ryst
als
�The
structure
ofa
crystal
can
be
dete
rmin
ed
by
stu
dyin
gth
ediffractionpattern
ofa
beam
of
radia
tio
nin
cid
ent
on
the
cry
sta
l.
20
�B
eam
diffr
action
take
spla
ceonly
incertain
specific
directions,m
uch
as
light
isdiffr
acte
dby
agra
ting.
�B
ym
easuri
ng
thedirectionsofthediffraction
and
the
corresponding
intensities,
one
obta
ins
info
rmatio
nconcern
ing
the
crystal
structure
responsib
lefo
rdiffr
action.
X-R
AY
CR
YST
ALLO
GR
APH
Y
�X
-ra
yc
rys
tall
og
rap
hy
isa
techn
iqu
ein
cry
sta
llog
rap
hy
inw
hic
hth
ep
att
ern
pro
du
ce
db
yth
ed
iffr
actio
no
fx-r
ays
thro
ug
hth
eclo
se
lysp
ace
dla
ttic
eo
fa
tom
sin
acry
sta
lis
reco
rde
da
nd
the
na
na
lyze
dto
reve
alth
en
atu
reo
fth
at
latt
ice
.
21
�X
-ra
yd
iffr
actio
n=
(XR
D)
X-R
ay C
ryst
all
ogra
phy
�T
he
wa
ve
leng
tho
fX
-ra
ys
is
typ
ica
lly1
A°,
co
mp
ara
ble
toth
e
inte
rato
mic
sp
acin
g(d
ista
nce
s
be
twe
en
ato
ms
or
ion
s)
inso
lids.
22
�W
en
ee
dX
-ra
ys:
eVx
mx
hc
hc
hE
ray
x
3
10
10
3.12
10
1=
==
==
−−
λυ
ωh
Cry
stal Str
uct
ure
Dete
rmin
ation
A c
rysta
l behaves a
s a
3-D
diffr
action g
rating f
or
x-r
ays
�In
adiffr
actio
nexp
eri
ment,
the
spacin
gof
lines
on
the
gra
ting
can
be
ded
uced
from
the
sep
ara
tion
of
the
diffr
action
maxim
a
Info
rma
tio
na
bo
ut
the
str
uctu
reo
fth
elin
es
on
the
23
Info
rma
tio
na
bo
ut
the
str
uctu
reo
fth
elin
es
on
the
gra
tin
gca
nb
eo
bta
ine
db
ym
easu
rin
gth
ere
lative
inte
nsitie
so
fd
iffe
ren
to
rde
rs
�S
imila
rly,
me
asu
rem
en
to
fth
ese
pa
ratio
no
fth
eX
-ra
y
diffr
actio
nm
axim
afr
om
acry
sta
la
llow
su
sto
de
term
ine
the
siz
eo
fth
eu
nit
ce
lla
nd
fro
mth
ein
ten
sitie
sin
ten
sitie
so
fo
f
diffr
acte
dd
iffr
acte
db
ea
ms
be
am
so
ne
one
ca
nca
no
bta
ino
bta
inin
form
atio
nin
form
atio
na
bo
ut
ab
ou
tth
eth
e
arr
an
ge
me
nt
arr
an
ge
me
nt
of
of
ato
ms
ato
ms
with
inw
ith
inth
eth
ece
llce
ll..
X-R
ay D
iffr
act
ion
W.
L.
Bra
gg
pre
sente
da
sim
ple
expla
nation
of
the
diffr
acte
dbeam
sfr
om
a
cry
sta
l.
24
Th
eB
ragg
derivation
issim
ple
but
is
convin
cin
gonly
sin
ce
itre
pro
duces
the
corr
ectre
sult.
X-R
ay D
iffr
act
ion &
Bra
gg E
quati
on
�E
ng
lish
physic
ists
Sir
W.H
.B
rag
ga
nd
his
so
nS
irW
.L.
Bra
gg
de
ve
lop
ed
are
latio
nsh
ipin
19
13
toe
xp
lain
wh
yth
ecle
ava
ge
face
so
fcry
sta
lsa
pp
ea
rto
refle
ct
X-r
ay
25
cry
sta
lsa
pp
ea
rto
refle
ct
X-r
ay
be
am
sa
tce
rta
ina
ng
les
of
incid
en
ce
(th
eta
,θ
).T
his
ob
se
rva
tio
nis
an
exa
mp
leo
fX
-ra
yw
av
ein
terf
ere
nc
e.
Sir W
illiam Henry
Bragg (1862-1942),
William LawrenceBragg (1890-1971)
o1
91
5,
the
fath
er
an
dso
nw
ere
aw
ard
ed
the
No
be
lp
rize
for
ph
ysic
s
"fortheirservicesintheanalysisofcrystalstructurebymeansof
Xrays".
Bra
gg E
quati
on
�B
rag
gla
wid
en
tifie
sth
ea
ng
les
of
the
incid
en
t
rad
iatio
nre
lative
toth
ela
ttic
ep
lan
es
for
wh
ich
diffr
actio
np
ea
ks
occu
rs.
�B
rag
gd
eri
ve
dth
eco
nd
itio
nfo
rco
nstr
uctive
26
�B
rag
gd
eri
ve
dth
eco
nd
itio
nfo
rco
nstr
uctive
inte
rfe
ren
ce
of
the
X-r
ays
sca
tte
red
fro
ma
se
to
f
pa
ralle
lla
ttic
ep
lan
es.
BR
AG
G E
QU
AT
ION
�W
.L.
Bra
gg
co
nsid
ere
dcry
sta
lsto
be
ma
de
up
of
pa
ralle
lp
lan
es
of
ato
ms.
Incid
en
tw
ave
sa
rere
fle
cte
dsp
ecu
larl
yfr
om
pa
ralle
lp
lan
es
of
ato
ms
inth
ecry
sta
l,w
ith
ea
ch
pla
ne
isre
fle
ctin
go
nly
ave
rysm
all
fra
ctio
no
fth
era
dia
tio
n,
like
alig
htly
silv
ere
dm
irro
r.
�In
mir
rorl
ike
refle
ctio
nth
ea
ng
leo
fin
cid
en
ce
ise
qu
al
toth
ea
ng
leo
fre
fle
ctio
n.
27
an
gle
of
refle
ctio
n.
өө
Dif
fract
ion C
ondit
ion
�T
he
diffr
acte
dbeam
sare
found
tooccur
when
the
reflections
from
pla
nes
of
ato
ms
inte
rfere
constr
uctively
.
28
�W
etr
eat
ela
stic
scatt
ering,
inw
hic
hth
e
energ
yof
X-r
ay
isnot
changed
on
reflection.
Bra
gg E
quati
on
�W
he
nth
eX
-ra
ys
str
ike
ala
ye
ro
fa
cry
sta
l,so
me
of
the
mw
ill
be
refle
cte
d.
We
are
inte
reste
din
X-r
ays
tha
ta
rein
-ph
ase
with
on
ea
no
the
r.X
-ra
ys
tha
ta
dd
tog
eth
er
co
nstr
uctive
lyin
x-
ray
diffr
actio
na
na
lysis
in-p
ha
se
be
fore
the
ya
rere
fle
cte
da
nd
aft
er
the
yre
fle
cte
d.
29
Incident angle
Reflected angle
Wavelength
of X-ray
Total Diffracted
Angle
θ
θ=
θ=
λ=
θ2θ
θ2
=
Bra
gg E
quati
on
�T
he
se
two
x-r
ay
be
am
str
ave
lslig
htly
diffe
ren
td
ista
nce
s.
The
diffe
ren
ce
inth
ed
ista
nce
str
ave
led
isre
late
dto
the
dis
tan
ce
be
twe
en
the
ad
jace
nt
laye
rs.
�C
on
ne
ctin
gth
etw
ob
ea
ms
with
pe
rpe
nd
icu
lar
line
ssh
ow
sth
e
diffe
ren
ce
be
twe
en
the
top
an
dth
eb
ott
om
be
am
s.
30
Th
e li
ne C
E i
s e
qu
ivale
nt
to t
he d
ista
nce b
etw
een
the t
wo
laye
rs (
d)
diffe
ren
ce
be
twe
en
the
top
an
dth
eb
ott
om
be
am
s.
sin
DE
dθ
=
Bra
gg L
aw
�T
he
len
gth
DE
isth
esa
me
as
EF
,so
the
tota
ld
ista
nce
tra
ve
led
by
the
bo
tto
mw
ave
ise
xp
resse
db
y:
sin
EF
dθ
=
sin
DE
dθ
=
31
�C
on
str
uctive
inte
rfe
ren
ce
of
the
rad
iatio
nfr
om
su
cce
ssiv
ep
lan
es
occu
rsw
he
nth
ep
ath
diffe
ren
ce
isa
nin
teg
ral
nu
mb
er
of
wa
ve
len
gh
ts.
Th
isis
the
Bra
gg
La
w.
2sin
DE
EF
dθ
+=
2sin
nd
λθ
=
Bra
gg E
quati
on
wh
ere
, d
is t
he
sp
acin
g o
f th
e p
lan
es a
nd
n is th
e o
rde
r o
f d
iffr
actio
n.
�B
rag
gre
fle
ctio
nca
no
nly
occu
rfo
rw
ave
len
gthλ
θn
d=
sin
2
32
�B
rag
gre
fle
ctio
nca
no
nly
occu
rfo
rw
ave
len
gth
�T
his
is w
hy w
e c
an
no
t u
se
vis
ible
lig
ht.
No
diffr
actio
n o
ccu
rs w
he
n
the
ab
ove
co
nd
itio
n is n
ot
sa
tisfie
d.
�T
he
diffr
acte
db
ea
ms
(re
fle
ctio
ns)
fro
ma
ny
se
to
fla
ttic
epla
ne
sca
no
nly
occu
ra
tp
art
icu
lar
an
gle
sp
rad
icte
db
yth
eB
rag
gla
w.
dn
2≤
λ
Sca
tteri
ng o
f X-r
ays
from
adja
cent
latt
ice p
oin
ts A
and B
X-r
ays
are
incid
ent
at
an
ang
leon
one
of
the
pla
nes
ofth
eset.
There
will
be
constr
uctive
inte
rfere
nce
of
the
wave
s
scatt
ere
dfr
om
the
two
successiv
ela
ttic
ep
oin
tsA
and
Bin
the
pla
ne
ifth
edis
tances
AC
and
DB
are
equal.
θ
33
the
pla
ne
ifth
edis
tances
AC
and
DB
are
equal.
θθ
AB
CD
2θ
Const
ruct
ive inte
rfere
nce
of w
aves
scatt
ere
d fro
m the s
am
e p
lane
If the s
catt
ere
d w
ave m
akes t
he s
am
e a
ngle
to
the p
lane a
s
the incid
ent
wave
The d
iffr
acte
d w
ave looks a
s if it h
as b
een r
eflecte
d f
rom
the
pla
ne
34
pla
ne
We
consid
er
the
scatt
ering
from
lattic
epo
ints
rath
er
than
ato
ms
because
itis
the
basis
of
ato
ms
associa
ted
with
each
latt
ice
poin
tth
at
isth
etr
ue
repeat
unit
of
the
cry
sta
l;T
he
latt
ice
po
int
isanalo
que
of
the
line
on
optica
ldiffr
action
gra
ting
and
the
basis
repre
sents
the
str
uctu
reofth
elin
e.
Dif
fract
ion m
axim
um
Co
he
ren
tsca
tte
rin
gfr
om
asin
gle
pla
ne
isn
ot
su
ffic
ien
tto
ob
tain
ad
iffr
actio
nm
axim
um
.It
isa
lso
ne
cessa
ryth
at
su
cce
ssiv
ep
lan
es
sh
ou
ldsca
tte
r
inp
ha
se
35
�T
his
will
be
th
e c
ase
if th
e p
ath
diffe
ren
ce
fo
r
sca
tte
rin
g o
ff t
wo
ad
jace
nt p
lan
es is a
n in
teg
ral
nu
mb
er
of
wa
ve
len
gth
s
λθn
d=
sin
2
Labellin
g the refl
ect
ion p
lanes
�T
o la
be
l th
e r
efle
ctio
ns, M
ille
r in
dic
es o
f th
e p
lan
es
ca
n b
e u
se
d.
�A
be
am
co
rre
spo
ndin
gto
ava
lue
of
n>
1co
uld
be
ide
ntifie
db
ya
sta
tem
en
tsuch
as
‘the
nth
-ord
er
36
ide
ntifie
db
ya
sta
tem
en
tsuch
as
‘the
nth
-ord
er
refle
ctio
ns
fro
mth
e(h
kl)
pla
ne
s’.
�(n
hn
kn
l)re
fle
ctio
n
Th
ird
-ord
er
refle
ctio
n fro
m (
11
1)
pla
ne
(33
3)
refle
ctio
n
n-t
h o
rder dif
fract
ion o
ff (hkl)
pla
nes
�R
ew
ritin
g th
e B
rag
g la
w
wh
ich
ma
kes
n-t
ho
rde
rd
iffr
actio
no
ff(h
kl)
pla
nes
of
λθ
=
sin
2nd
37
wh
ich
ma
kes
n-t
ho
rde
rd
iffr
actio
no
ff(h
kl)
pla
nes
of
sp
acin
g‘d
’lo
ok
like
firs
t-o
rde
rd
iffr
actio
no
ffp
lan
es
ofsp
acin
gd
/n.
�P
lan
es o
f th
is r
ed
uce
d s
pa
cin
g w
ou
ld h
ave
Mill
er
ind
ice
s (
nh
nk n
l).
X-r
ay s
truct
ure
analy
sis
of N
aC
l and K
Cl
The GENERAL PRINCIBLES of X-RAY STRUCTURE ANALYSIS to
DEDUCE the STRUCTURE of NaCl and KCl
Bra
gg u
sed a
n o
rdin
ary
spectr
om
ete
r and m
easure
d th
e inte
nsity o
f
specula
r re
flection fro
m a
cle
aved face o
f a c
rysta
l
38
found s
ix v
alu
es o
f
fo
r w
hic
h a
sharp
peak in inte
nsity o
ccurr
ed,
corr
espondin
g to t
hre
e c
hara
cte
ristics w
avele
ngth
s (
K,L
and M
x-
rays)
in f
irst and s
econd o
rder
(n=
1 a
nd n
=2 in B
ragg law
)
�B
yre
peating
the
experim
ent
with
adiffe
rent
cry
sta
lfa
ce
he
could
use
his
eqn.
tofind
for
exam
ple
the
ratio
of
(100)
and
(111)
pla
ne
spacin
gs,
info
rmation
that
confirm
ed
the
cubic
sym
metr
yof
the
ato
mic
arr
angem
ent.θ
Details of structure
Deta
ilsof
str
uctu
rew
ere
than
deduced
from
the
diffe
rences
betw
een
the
diffr
action
patt
ern
sfo
rN
aC
land
KC
l.
�Majordifference;
absence
of
(111)
reflection
inK
Clcom
pare
dto
a
weak
but
dete
cta
ble
(111)
reflection
inN
aC
l.
39
�T
his
arises
because
the
Kand
Cl
ions
both
have
the
arg
on
ele
ctr
on
shell
str
uctu
reand
hence
scatter
x-r
ays
alm
ost
equally
where
as
Na
and
Cl
ions
have
diffe
rent
scatt
ering
str
ength
s.
(111)
reflection
inN
aC
lcorr
esponds
toone
wavele
ngth
of
path
diffe
rence
betw
een
neig
hbouring
(111)
pla
nes.
Experimental arrangements
for x-ray diffraction
�S
ince
the
pio
neering
work
of
Bra
gg,
x-r
ay
diffr
action
has
becom
ein
toa
routine
techniq
ue
for
the
dete
rmin
ation
of
crs
yta
l
40
techniq
ue
for
the
dete
rmin
ation
of
crs
yta
l
str
uctu
re.
Bra
gg E
quati
on
Sin
ce
Bra
gg
'sL
aw
ap
plie
sto
all
se
tso
fcry
sta
lp
lan
es,
the
latt
ice
ca
nb
ed
ed
uce
dfr
om
the
diffr
actio
np
att
ern
,m
akin
gu
se
of
ge
ne
ral
exp
ressio
ns
for
the
sp
acin
go
fth
ep
lan
es
inte
rms
of
the
irM
ille
rin
dic
es.F
or
cu
bic
str
uctu
res
ad=
41
No
teth
at
the
sm
alle
rth
espa
cin
gth
eh
igh
er
the
an
gle
of
diffr
actio
n,i.e.
the
sp
acin
go
fp
ea
ks
inth
ed
iffr
actio
np
att
ern
isin
ve
rse
lyp
rop
ort
ion
alto
the
sp
acin
go
fth
ep
lan
es
inth
ela
ttic
e.
Th
ed
iffr
actio
np
att
ern
will
refle
ct
the
sym
me
try
pro
pe
rtie
so
fth
ela
ttic
e.
2sin
dn
θλ
=22
2
dh
kl
=+
+
Bra
gg E
quati
on
Asim
ple
exa
mp
leis
thedifferencebetween
the
series
of(n00)reflections
fora
sim
ple
cubic
andabodycentredcubic
lattice
.F
or
the
sim
ple
cub
icla
ttic
e,
all
va
lue
so
fn
will
giv
eB
ragg
pe
aks. How
ever,
for
the
bod
ycen
tred
cub
icla
ttic
eth
e(1
00)
pla
ne
sare
inte
rlea
ved
by
an
eq
uiv
ale
nt
42
the
(10
0)
pla
ne
sare
inte
rlea
ved
by
an
eq
uiv
ale
nt
se
ta
tth
eha
lfw
ay
positio
n.
At
the
ang
lew
here
Bra
gg
'sLaw
wou
ldg
ive
the
(100
)re
fle
ction
the
inte
rlea
ved
pla
ne
sw
illg
ive
are
flection
exa
ctly
out
of
pha
se
with
tha
tfr
om
the
prim
ary
pla
ne
s,
wh
ich
will
exa
ctly
can
ce
lth
esig
na
l.T
he
reis
no
sig
nal
from
(n00
)p
lan
es
with
odd
va
lues
ofn
.T
his
kin
do
fa
rgu
men
tle
ad
sto
rule
sfo
rid
en
tify
ing
the
lattic
esym
me
try
from
"mis
sin
g"
refle
ction
s,
wh
ich
are
ofte
nq
uite
sim
ple
.
Types
of X
-ray c
am
era
Th
ere
are
many
types
of
X-r
ay
cam
era
to
sort
out
reflections
fro
mdiffe
rent
cry
sta
l
pla
nes.
We
will
stu
dy
only
thre
ety
pes
of
X-r
ay
photo
gra
ph
that
are
wid
ely
used
for
the
sim
ple
43
photo
gra
ph
that
are
wid
ely
used
for
the
sim
ple
str
uctu
res.
1.Laue p
hoto
gra
ph
2.R
ota
ting c
rysta
l m
eth
od
3.P
ow
der
photo
gra
ph
X-R
AY
DIF
FR
AC
TIO
N M
ET
HO
DS
X-Ray Diffraction Method
44
Laue
Rotating Crystal
Powder
Ori
en
tati
on
Sin
gle
Cry
sta
lP
oly
ch
rom
ati
c B
ea
mF
ixe
d A
ng
le
Latt
ice c
on
sta
nt
Sin
gle
Cry
sta
lM
on
och
rom
ati
c B
eam
Vari
ab
le A
ng
le
Latt
ice P
ara
mete
rsP
oly
cry
sta
l (p
ow
dere
d)
Mo
no
ch
rom
ati
c B
eam
Vari
ab
le A
ng
le
LA
UE M
ET
HO
D
�T
he
La
ue
me
tho
dis
ma
inly
use
dto
de
term
ine
the
orie
nta
tio
no
fla
rge
sin
gle
cry
sta
lsw
hile
rad
iatio
nis
refle
cte
dfr
om
,o
rtr
an
sm
itte
dth
rou
gh
afixe
dcry
sta
l.
�T
he
diffr
acte
db
ea
ms
form
arr
ays
of
45
�T
he
diffr
acte
db
ea
ms
form
arr
ays
of
sp
ots
,th
at
lieo
ncu
rve
so
nth
efilm
.
�T
he
Bra
gg
an
gle
isfixe
dfo
re
ve
ryse
to
fp
lan
es
inth
ecry
sta
l.E
ach
se
to
fp
lan
es
pic
ks
ou
ta
nd
diffr
acts
the
pa
rtic
ula
rw
ave
len
gth
fro
mth
ew
hite
rad
iatio
nth
at
sa
tisfie
sth
eB
rag
gla
wfo
rth
eva
lue
so
fd
an
dθ
invo
lve
d.
Back
-refl
ect
ion L
aue
Meth
od
�In
th
e b
ack-r
efle
ctio
n m
eth
od
, th
e f
ilm is p
lace
d b
etw
ee
nth
e
x-r
ay s
ou
rce
an
d t
he
cry
sta
l.T
he
be
am
s w
hic
h a
re d
iffr
acte
d
in a
ba
ckw
ard
dir
ectio
n a
re r
eco
rde
d.
46
�O
ne
sid
eo
fth
eco
ne
of
La
ue
refle
ctio
ns
isd
efin
ed
by
the
tra
nsm
itte
db
ea
m.
Th
efilm
inte
rse
cts
the
co
ne
,w
ith
the
diffr
actio
nsp
ots
ge
ne
rally
lyin
go
na
nh
yp
erb
ola
.
X-R
ay
Fil
m
Sin
gle
Cry
sta
l
Tra
nsm
issi
on L
aue M
eth
od
�In
th
e t
ran
sm
issio
n L
au
e m
eth
od
, th
e f
ilm is p
lace
d b
eh
ind
the
cry
sta
l to
re
co
rd b
ea
ms
wh
ich
are
tra
nsm
itte
d t
hro
ug
h
the
cry
sta
l.
47
�O
ne
sid
eo
fth
eco
ne
of
La
ue
refle
ctio
ns
isd
efin
ed
by
the
tra
nsm
itte
db
ea
m.
Th
efilm
inte
rse
cts
the
co
ne
,w
ith
the
diffr
actio
nsp
ots
ge
ne
rally
lyin
go
na
ne
llip
se
.
X-R
ay
Fil
mS
ing
leC
rys
tal
Laue P
att
ern
The
symmetry
of
the
spot
pattern
reflects
the
symmetry
of
the
crystal
when
viewed
along
the
direction
of
the
incident
beam.
Laue
method
isoften
used
todeterm
ine
the
orientation
of
single
crystals
by
means
of
48
crystals
by
means
of
illuminating
the
crystal
withacontinuosspectrum
ofX-rays;
�Singlecrystal
�Continous
spectrum
ofx-
rays
�Symmetry
ofthe
crystal;
orientation
Cry
stal st
ruct
ure
dete
rmin
ation b
y L
aue m
eth
od
�T
here
fore
,th
eLaue
meth
od
ism
ain
lyused
todete
rmin
eth
ecry
sta
lorienta
tion.
�A
lthough
the
Laue
meth
od
can
als
obe
used
todete
rmin
eth
ecry
sta
lstr
uctu
re,
severa
l
49
dete
rmin
eth
ecry
sta
lstr
uctu
re,
severa
lw
avele
ngth
scan
reflect
indiffe
rent
ord
ers
from
the
sam
eset
of
pla
nes,
with
the
diffe
rent
ord
er
reflections
superim
posed
on
the
sam
espot
inth
efilm
.T
his
makes
cry
sta
lstr
uctu
redete
rmin
ation
by
spotin
tensity
diffu
cult.
�R
ota
ting
cry
sta
lm
eth
od
overc
om
es
this
pro
ble
m.
How
?
RO
TA
TIN
G C
RY
ST
AL M
ET
HO
D
�In
the
rota
ting
cry
sta
lm
eth
od,
asin
gle
cry
sta
lis
mou
nte
dw
ith
an
axis
norm
al
toa
monochro
matic
x-r
ay
beam
.A
cylin
dri
cal
film
ispla
ce
d
50
Acylin
dri
cal
film
ispla
ce
daro
un
dit
and
the
cry
sta
lis
rota
ted
about
the
chosen
axis
.
�A
sth
ecry
sta
lro
tate
s,
se
tso
fla
ttic
ep
lane
sw
illa
tsom
epo
int
ma
ke
the
co
rre
ct
Bra
gg
ang
lefo
rth
em
on
ochro
ma
tic
incid
en
tbea
m,
and
at
tha
tpo
int
ad
iffr
acte
db
ea
mw
illbe
form
ed
.
RO
TA
TIN
G C
RYSTA
L
METH
OD
Lattic
econsta
nt
of
the
cry
sta
lcan
be
dete
rmin
ed
by
means
of
this
meth
od;
for
a
giv
en
wavele
ngth
ifth
eangle
at
whic
ha
θ
51
reflection
occurs
isknow
n,
can
be
dete
rmin
ed.
hkl
dθ
22
2
ad
hk
l=
++
Rota
ting C
ryst
al M
eth
od
Th
ere
fle
cte
db
ea
ms
are
loca
ted
on
the
su
rfa
ce
of
ima
gin
ary
co
ne
s.
By
reco
rdin
gth
ed
iffr
actio
np
att
ern
s(b
oth
an
gle
sa
nd
inte
nsitie
s)
for
va
rio
us
cry
sta
lo
rie
nta
tio
ns,
on
eca
nd
ete
rmin
eth
esh
ap
ea
nd
siz
eo
fu
nit
ce
lla
sw
ell
as
arr
an
ge
me
nt
of
ato
ms
insid
eth
ece
ll.
52
arr
an
ge
me
nt
of
ato
ms
insid
eth
ece
ll.
Film
TH
E PO
WD
ER
MET
HO
D
Ifa
po
wd
ere
dsp
ecim
en
isu
se
d,
inste
ad
of
a
sin
gle
cry
sta
l ,th
en
the
reis
noneedto
rotate
the
sp
ecim
en
,b
eca
use
the
rew
illa
lways
be
som
ecry
sta
lsa
ta
no
rie
nta
tio
nfo
rw
hic
h
53
som
ecry
sta
lsa
ta
no
rie
nta
tio
nfo
rw
hic
h
diffr
actio
nis
pe
rmitte
d.
He
rea
mo
no
ch
rom
atic
X-r
ay
bea
mis
incid
en
to
na
po
wde
red
or
po
lycry
sta
llin
esa
mp
le.
Th
ism
eth
od
isu
se
ful
for
sa
mp
les
tha
ta
re
difficu
ltto
ob
tain
insin
gle
cry
sta
lfo
rm.
TH
E P
OW
DER
MET
HO
D
Th
ep
ow
der
me
tho
dis
use
dto
de
term
ine
the
va
lue
of
the
lattic
ep
ara
me
ters
accura
tely
.L
attic
ep
ara
me
ters
are
the
ma
gn
itu
de
so
fth
eu
nit
ve
cto
rsa
,b
an
dc
wh
ich
de
fin
eth
eu
nit
ce
llfo
rth
ecry
sta
l.
54
Fo
re
ve
ryse
to
fcry
sta
lp
lan
es,
by
ch
ance
,oneor
more
crystals
will
be
inth
ecorrectorientation
tog
ive
the
co
rrect
Bra
gg
an
gle
tosa
tisfy
Bra
gg
'se
qua
tio
n.
Eve
rycry
sta
lp
lan
eis
thu
sca
pa
ble
of
diffr
actio
n.
Ea
ch
diffr
actio
nlin
eis
ma
de
up
of
ala
rge
nu
mb
er
of
sm
all
sp
ots
,e
ach
fro
ma
se
pa
rate
cry
sta
l.E
ach
sp
ot
isso
sm
all
as
tog
ive
the
ap
pe
ara
nce
of
aco
ntin
uo
us
line.
The P
ow
der M
eth
od
�If
am
on
och
rom
atic
x-r
ay
be
am
isd
ire
cte
da
ta
sin
gle
cry
sta
l,
the
no
nly
on
eo
rtw
od
iffr
acte
d
�If
the
sa
mp
leco
nsis
tso
fso
me
ten
so
fra
nd
om
lyo
rie
nta
ted
sin
gle
cry
sta
ls,
the
diffr
acte
d
�A
sa
mp
leo
fso
me
hu
nd
red
so
f
cry
sta
ls(i
.e.
ap
ow
de
red
sa
mp
le)
sh
ow
tha
tth
ed
iffr
acte
d
55
the
no
nly
on
eo
rtw
od
iffr
acte
d
be
am
sm
ay
resu
lt.
sin
gle
cry
sta
ls,
the
diffr
acte
d
be
am
sa
rese
en
tolie
on
the
su
rfa
ce
of
se
ve
ral
co
ne
s.
Th
e
co
ne
sm
ay
em
erg
ein
all
dir
ectio
ns,
forw
ard
sa
nd
ba
ckw
ard
s.
sa
mp
le)
sh
ow
tha
tth
ed
iffr
acte
d
be
am
sfo
rmco
ntin
uo
us
co
ne
s.
Acir
cle
of
film
isu
se
dto
reco
rd
the
diffr
actio
np
att
ern
as
sh
ow
n.
Ea
ch
co
ne
inte
rse
cts
the
film
giv
ing
diffr
actio
nlin
es.
Th
elin
es
are
se
en
as
arc
so
nth
efilm
.
Debye S
cherr
er C
am
era
Avery
sm
all
am
ount
of
po
wd
ere
dm
ate
rial
isseale
din
toa
fine
cap
illary
tube
made
from
gla
ss
that
does
not
diffr
actx-r
ays.
The
specim
en
ispla
ce
d
56
The
specim
en
ispla
ce
d
inth
eD
eb
ye
Sch
err
er
cam
era
an
dis
accura
tely
alig
ne
dto
be
inth
ecentr
e
of
the
cam
era
.X
-rays
ente
r
the
cam
era
thro
ug
ha
colli
mato
r.
Debye S
cherr
er C
am
era
The
pow
der
diffr
acts
the
x-r
ays
inaccord
ance
with
Bra
ggs
law
topro
duce
cones
of
57
pro
duce
cones
of
diffr
acte
dbeam
s.
These
cones
inte
rsect
astr
ipof
photo
gra
ph
icfilm
locate
din
the
cylin
drica
lcam
era
topro
duce
achara
cte
ristic
setofarc
son
the
film
.
Pow
der dif
fract
ion fil
m
Wh
en
the
film
isre
mo
ve
dfr
om
the
cam
era
,
fla
tte
ne
da
nd
pro
ce
sse
d,
itsh
ow
sth
ed
iffr
actio
n
line
sa
nd
the
ho
les
for
the
incid
en
ta
nd
58
tra
nsm
itte
db
ea
ms.
Appli
cati
on o
f X
RD
1.
Diffe
ren
tia
tio
nb
etw
ee
ncry
sta
llin
ea
nd
am
orp
ho
us
ma
teri
als
;
2.
De
term
ina
tio
no
fth
estr
uctu
reo
fcry
sta
llin
em
ate
ria
ls;
XR
D is a
no
nd
estr
uctive
te
ch
niq
ue
. S
om
e o
f th
e u
se
s o
f
x-r
ay d
iffr
actio
n a
re;
59
2.
De
term
ina
tio
no
fth
estr
uctu
reo
fcry
sta
llin
em
ate
ria
ls;
3.
De
term
ina
tio
no
fe
lectr
on
dis
trib
utio
nw
ith
inth
ea
tom
s,
an
dth
rou
gh
ou
tth
eu
nit
ce
ll;
4.
De
term
ina
tio
no
fth
eo
rie
nta
tio
no
fsin
gle
cry
sta
ls;
5.
De
term
ina
tio
no
fth
ete
xtu
reo
fp
oly
gra
ine
dm
ate
ria
ls;
6.
Me
asu
rem
en
to
fstr
ain
an
dsm
all
gra
insiz
e…
..e
tc
Advanta
ges
and
dis
advanta
ges
of X-r
ays
Advantages;
�X
-ray is t
he c
heapest,
the m
ost convenie
nt
and
wid
ely
used m
eth
od.
�X
-rays a
re n
ot absorb
ed v
ery
much b
y a
ir, so
60
�X
-rays a
re n
ot absorb
ed v
ery
much b
y a
ir, so
the s
pecim
en n
eed n
ot be in a
n e
vacuate
d
cham
ber.
Disadvantage;
�T
hey d
o n
ot in
tera
ct
very
str
ongly
with lig
hte
r
ele
ments
.
Difraction Methods
Diffr
actio
n
X-r
ay
Ne
utr
on
Ele
ctr
on
61
Diffe
rent
radia
tion
sourc
eof
neutr
on
or
ele
ctr
on
can
als
obe
used
indiffr
actio
n
experim
ents
.
The
physic
al
basis
for
the
diffr
action
of
ele
ctr
on
and
neutr
on
beam
sis
the
sam
eas
that
for
the
diffr
action
of
Xra
ys,
the
only
diffe
rence
bein
gin
the
mechanis
mofscatt
ering.
Neutr
on D
iffr
act
ion
�N
eu
tro
ns
we
red
isco
ve
red
in1
93
2a
nd
the
irw
ave
pro
pe
rtie
sw
as
sh
ow
nin
19
36
.
E =
p2/2
m p
= h
/λE
=E
ne
rgy
λ=
Wa
vele
ng
th
p
=M
om
en
tum
62
�λ
~1
A°;
En
erg
yE
~0
.08
eV
.T
his
en
erg
yis
of
the
sa
me
ord
er
of
ma
gn
itu
de
as
the
the
rma
le
ne
rgy
kT
at
roo
mte
mp
era
ture
,0
.02
5e
V,
an
dfo
rth
isre
aso
nw
esp
ea
ko
fth
erm
aln
eu
tro
ns.
p=
Mo
me
ntu
m
mn=
Ma
ss
of
ne
utr
on
= 1
,67
.10
-27k
g
Neutr
on D
iffr
act
ion
�N
eu
tron
doe
sno
tin
tera
ct
with
ele
ctr
on
sin
the
cry
sta
l.T
hu
s,
un
like
the
x-r
ay,
wh
ich
issca
tte
red
en
tire
lyb
ye
lectr
ons,th
en
eu
tro
nis
sca
tte
red
en
tire
lyb
yn
ucle
i
�A
ltho
ugh
un
ch
arg
ed
,ne
utr
on
has
an
intr
insic
mag
ne
tic
63
�A
ltho
ugh
un
ch
arg
ed
,ne
utr
on
has
an
intr
insic
mag
ne
tic
mom
en
t,so
itw
illin
tera
ct
str
ong
lyw
ith
ato
ms
an
dio
ns
inth
ecry
sta
lw
hic
ha
lso
ha
ve
ma
gn
etic
mo
me
nts
.
�N
eu
tron
sare
mo
reu
se
ful
than
X-r
ays
for
dete
rmin
ing
the
cry
sta
lstr
uctu
res
of
so
lids
con
tain
ing
light
ele
me
nts
.
�N
eu
tron
sou
rce
sin
the
wo
rld
are
limite
dso
neu
tron
diffr
actio
nis
ave
rysp
ecia
lto
ol.
Neutr
on D
iffr
act
ion
Neutr
on
diffr
action
has
severa
la
dva
nta
ges
over
its
x-
ray
counte
rpart
;
�N
eutr
on
diffr
action
isan
imp
ort
ant
toolin
the
investig
atio
nofm
agnetic
ord
ering
thatoccur
insom
em
ate
rials
.
64
ofm
agnetic
ord
ering
thatoccur
insom
em
ate
rials
.
�Lig
ht
ato
ms
such
as
Hare
better
resolv
ed
ina
neutr
on
patt
ern
because,
havin
gon
lya
few
ele
ctr
ons
toscatter
the
Xra
ybeam
,th
ey
do
not
contr
ibute
sig
nific
antly
toth
eX
ray
diffr
acte
dpatt
ern
.
Ele
ctro
n D
iffr
act
ion
Ele
ctr
on
diffr
actio
nh
as
als
ob
ee
nu
se
din
the
an
aly
sis
of
cry
sta
lstr
uctu
re.
Th
ee
lectr
on
,lik
eth
en
eu
tro
n,
po
sse
sse
sw
ave
pro
pe
rtie
s;
02A
≈λ
eVmh
mkE
40
22
2
22
2
==
=λ
h 65
Ele
ctr
on
sa
rechargedparticles
an
din
tera
ct
str
on
gly
with
all
ato
ms.
So
ele
ctr
on
sw
ith
an
en
erg
yo
fa
few
eV
wo
uld
be
co
mp
lete
lyabsorbed
by
the
specim
en
.In
ord
er
tha
ta
ne
lectr
on
be
am
ca
np
en
etr
ate
into
asp
ecim
en
,it
ne
ce
ssitas
ab
ea
mo
fve
ryh
igh
en
erg
y(5
0ke
Vto
1M
eV
)a
sw
ell
as
the
sp
ecim
en
mu
st
be
thin
(10
0-1
00
0n
m)
eVm
mE
ee
40
22
2=
==
λ
Ele
ctro
n D
iffr
act
ion
Iflo
we
lectr
on
en
erg
ies
are
use
d,
the
pe
ne
tra
tio
nd
ep
th
will
be
ve
rysm
all
(on
lya
bo
ut
50
A°)
,a
nd
the
be
am
will
be
refle
cte
dfr
om
the
su
rfa
ce
.C
on
se
qu
en
tly,
ele
ctr
on
diffr
actio
nis
au
se
fulte
ch
niq
ue
for
su
rfa
ce
str
uctu
restu
die
s.
66
au
se
fulte
ch
niq
ue
for
su
rfa
ce
str
uctu
restu
die
s.
Ele
ctr
on
sa
resca
tte
red
str
on
gly
ina
ir,
so
diffr
actio
n
exp
eri
me
nt
mu
st
be
ca
rrie
do
ut
ina
hig
hva
cu
um
.T
his
bri
ng
s
co
mp
lica
tio
na
nd
itis
exp
en
siv
ea
sw
ell.
X-Ray
Neutron
Electron
Diffraction Methods
67
λ=
1A
°
E ~
10
4eV
inte
rac
t w
ith
ele
ctr
on
Pe
ne
tra
tin
g
λ=
1A
°
E ~
0.0
8 e
V
inte
rac
t w
ith
nu
cle
iH
igh
ly P
en
etr
ati
ng
λ=
2A
°
E ~
15
0 e
V
inte
rac
t w
ith
ele
ctr
on
Le
ss
Pe
ne
tra
tin
g
