part 1: surfaces physics - louisiana tech universityramu/msnt505/lec_notes/dobbins/surface... ·...
TRANSCRIPT
Par
t 1:
Sur
face
s P
hysi
csD
r. T
. Dob
bins
MS
E 5
05 S
urfa
ce a
nd S
urfa
ce A
naly
sis
Lect
ure
Ser
ies
Ref
eren
ce M
ater
ials
:1.
Kit
telC
., In
tro
du
ctio
n t
o S
olid
Sta
te P
hys
ics
Wile
y &
So
ns
(NY
) 19
96.
2.T
este
r J.
W.,
Th
erm
od
ynam
ics
and
Its
Ap
plic
atio
ns
Pre
ntic
e H
all (
NJ)
199
7.3.
Wes
t A
.R.,
So
lid S
tate
Ch
emis
try
and
Its
Ap
plic
atio
ns
Wile
y &
So
ns
(NY
) 19
84.
4.V
enab
les
J., I
ntr
od
uct
ion
to
Su
rfac
e an
d T
hin
Film
Pro
cess
esC
amb
rid
ge
Un
iver
sity
Pre
ss (
UK
) 20
00.
5.W
ebsi
te s
po
nso
red
by
the
UK
Su
rfac
e A
nal
ysis
Fo
rum
(U
SA
F)
htt
p:/
/ww
w.s
iu.e
du
/~ca
fs/s
urf
ace
(wri
tten
by
D.T
. Mar
x at
So
uth
ern
Illin
ois
U
niv
ersi
ty)
Ass
um
ed u
nd
erst
and
ing
of
Qu
antu
m M
ech
anic
s, C
ryst
allo
gra
ph
y, a
nd
Th
erm
od
ynam
ics.
Lec
ture
To
pic
s (P
art
1)--
-
Wh
at a
re s
om
e o
f th
e P
rop
erti
es o
f su
rfac
es?
The
sur
face
pro
pert
ies
we
will
con
side
r in
clud
e at
omic
den
sity
, sur
face
tens
ion,
an
d su
rfac
e en
ergy
.
Wh
at is
a S
urf
ace?
A
sur
face
is lo
cust
of p
oint
s w
hich
cla
ssify
the
boun
dary
bet
wee
n an
ob
ject
and
its
surr
ound
ings
.
Wh
y ar
e su
rfac
e at
om
s im
po
rtan
t?P
rope
rtie
s ar
e ty
pica
lly b
ased
upo
n bu
lk (
inte
rnal
) m
easu
rem
ents
. W
e ty
pica
lly a
re c
once
rned
with
sol
ids
havi
ng d
ensi
ties
of 1
023
atom
s/cm
3 . M
ost o
f tho
se a
tom
s ar
e w
ithin
the
solid
. H
owev
er,
surf
aces
bec
ome
impo
rtan
t whe
n w
e m
ove
into
nan
osci
ence
---
whe
re
man
y m
ore
of th
ose
atom
s ar
e su
rfac
e at
oms.
Or
whe
n ph
enom
ena
of in
tere
st o
nly
occu
rs a
t the
sur
face
.
Ho
w a
re S
urf
aces
Cla
ssif
ied
?
Sur
face
s ar
e cl
assi
fied
by th
e sp
acin
g be
twee
n su
rfac
e at
oms
and
the
# of
bon
ds e
ach
surf
ace
atom
form
s w
ith e
ither
oth
er s
urfa
ce a
tom
s or
ato
ms
in th
e bu
lk.
Wh
at a
re t
he
bro
ad c
ateg
ori
es o
f su
rfac
e re
acti
on
s w
e w
ill c
on
sid
er?
The
sur
face
rea
ctio
ns w
e w
ill c
onsi
der
are
subl
imat
ion
(rel
ease
of a
tom
s fr
om a
so
lid s
urfa
ce),
ads
orpt
ion
(upt
ake
of a
tom
s on
to a
sol
id s
urfa
ce),
epi
taxi
algr
owth
.
Lec
ture
To
pic
s (P
art
2)--
-W
hat
are
th
e cl
assi
fica
tio
ns
of
Su
rfac
e C
har
acte
riza
tio
n?
S
urfa
ces
may
be
char
acte
rized
with
res
pect
to th
eir
topo
grap
hy (
i.e. r
ough
ness
), c
hem
istr
y,
surf
ace
orie
ntat
ion,
and
thic
knes
s of
che
mic
ally
hom
ogen
eous
reg
ions
at t
he s
urfa
ce.
Wh
ich
Su
rfac
e C
har
acte
riza
tio
n T
ech
niq
ues
will
we
lear
n a
bo
ut
in t
his
lect
ure
?•
X-r
ay a
nd N
eutr
on R
efle
ctiv
ity•
X-r
ay P
hoto
elec
tron
Spe
ctro
scop
y•
Sec
onda
ry Io
n M
ass
Spe
ctro
met
ry•
Sca
nnin
g A
uger
Mic
rosc
opy
Wh
at a
re s
om
e o
ther
Su
rfac
e C
har
acte
riza
tio
n T
ech
niq
ues
of
pra
ctic
al im
po
rtan
ce in
res
earc
h?
•A
tom
ic F
orce
Spe
ctro
scop
y•
Sca
nnin
g T
unne
ling
Mic
rosc
opy
•N
ear-
IR S
pect
rosc
opy
Wh
at is
a S
urf
ace?
(110
) S
urfa
ce o
f GaA
s. S
urfa
ce a
tom
s (b
lue)
are
‘re
laxe
d’ (
i.e. n
ot c
onst
rain
ed in
3D
).
Sur
face
s ar
e de
fined
by
‘rela
xed’
ato
ms
(i.e.
not
con
stra
ined
in3D
as
thei
r in
tern
al c
ount
erpa
rts
are)
. D
angl
ing
bond
s fr
om th
ese
surf
ace
atom
s ar
e fr
ee to
rea
ct.
Rel
axat
ion
and
reco
nstr
uctio
n ar
e st
rong
ly in
fluen
ced
by th
e bo
ndin
g ty
pe in
the
bulk
mat
eria
l (i.e
. met
allic
, cov
alen
t, io
nic,
and
van
der
waa
ls)
Rel
axat
ion
of s
urfa
ce a
tom
s le
ads
to r
econ
stru
ctio
n (r
earr
ange
men
t of
atom
s ne
ar th
e su
rfac
e).
Wh
at is
a S
urf
ace?
---
Rev
iew
of
Mill
er In
dic
es in
Cry
stal
log
rap
hy
Mill
er In
dice
s ar
e us
ed to
iden
tify
the
surf
ace
term
inat
ing.
If it
is s
tate
d th
at th
e su
rfac
e is
a S
i(100
), th
at in
dica
tes
that
we
are
exam
inin
g a
surf
ace
whi
ch h
as 1
sur
face
ato
ms
spac
ed
at
.
Sili
con
Cry
stal
Str
uctu
re
•(1
00)
is t
he
set
of
pla
nes
(h
kl)
wh
ich
in
ters
ect
the
crys
tal a
t (1
/1, 1
/0, 1
/0)
or
(1,
,
)
∞∞
Ο
yx
z
a22
Th
is p
lan
e h
as
4(1/
8)+1
(1/2
) =
1 A
tom
Act
ivit
y--
-U
sin
g M
iller
Ind
ices
to
Def
ine
Ter
min
atin
g S
urf
ace
•W
hat i
s th
e nu
mbe
r of
ato
ms
and
atom
ic
spac
ing
for
a su
rfac
e te
rmin
atin
g at
the
(110
) pl
ane
of a
fcc
crys
tal?
HIN
T:
Rec
all (
100)
is t
he
set
of
pla
nes
(h
kl)
wh
ich
in
ters
ect
the
crys
tal a
t (1
/1, 1
/0, 1
/0)
or
(1,
,
)
∞∞
yx
z
FC
C C
ryst
al S
truc
ture
Ο∞
Act
ivit
y--
-U
sin
g M
iller
Ind
ices
to
Def
ine
Ter
min
atin
g S
urf
ace
•W
hat i
s th
e nu
mbe
r of
ato
ms
and
atom
ic
spac
ing
for
a su
rfac
e te
rmin
atin
g at
the
(110
) pl
ane
of a
fcc
crys
tal?
yx
z
FC
C C
ryst
al S
truc
ture
Ο
An
s:R
ecal
l (11
0) is
th
e se
t o
f p
lan
es (
hkl
) w
hic
h
inte
rsec
t th
e cr
ysta
l at
(1/1
, 1/1
, 1/0
) o
r (1
, 1 ,
)∞
Th
is p
lan
e h
as
4(1/
8)+2
(1/2
) =
1.5
Ato
ms
Ato
ms
are
spac
ed a
t:
a22
Wh
at is
a S
urf
ace?
Sur
face
ato
ms
(blu
e) u
nder
go r
elax
atio
n fo
llow
ed b
y re
cons
truc
tion.
(a)
rel
axat
ion
–lo
ss in
per
iodi
c or
der
in c
dire
ctio
n. (
b) r
econ
truc
tion
1 –
clas
sifie
d by
ch
ange
in a
tom
ic s
paci
ng in
a d
irect
ion.
(c)
re
cons
truc
tion
2 –
clas
sifie
d by
mis
sing
row
of a
tom
s.
Wh
at is
a S
urf
ace?
Sur
face
ato
ms
real
ize
a lo
ss in
cry
stal
line
orde
r. T
hese
ato
ms
take
on
a ‘p
seud
o’ r
ando
m c
onfig
urat
ion.
Suc
h a
nonc
ryst
allin
est
ruct
ure
is k
now
n as
‘am
orp
ho
us’
.
Vap
or
Ph
ase
Su
rfac
e A
tom
s
Cry
stal
line
So
lid
Ato
ms
Imag
es t
aken
fro
m w
ebsi
te –
Vis
ual
izat
ion
of
(210
) an
d (
310)
Gra
in B
ou
nd
arie
s:
htt
p:/
/ww
w.s
v.vt
.ed
u/c
lass
es/E
SM
4714
/Stu
den
t_P
roj/c
lass
95/m
uta
sa/m
uta
sa.h
tml
Wh
at is
a S
urf
ace?
---
Gra
in B
ou
nd
ary
(su
rfac
e b
etw
een
tw
o is
om
orp
ho
us
solid
s)
Sur
face
Ato
ms
typi
cally
hav
e a
stru
ctur
e in
term
edia
te b
etw
een
the
two
term
inal
ph
ases
with
the
exce
ptio
n of
not
ed s
olid
-sol
id s
urfa
ces
–ca
lled
grai
n bo
unda
ries
Cry
stal
lite
1
Cry
stal
lite
2
Gra
in
Bo
un
dar
yR
egio
n
Cry
stal
lite
2
Gra
in
Bo
un
dar
yR
egio
n
Cry
stal
lite
1
Wh
at is
a S
urf
ace?
Sur
face
s ex
ist b
etw
een
two
phas
es.
The
pha
ses
may
be
:•
two
solid
s•
a so
lid a
nd a
vap
or•
a liq
uid
and
a va
por
•a
solid
and
a li
quid
In a
ll ca
ses,
ther
e is
a fi
nite
leng
th o
f reg
ion
for
whi
ch th
e at
omic
pa
ckin
g/st
ruct
ure
unde
rgoe
s ch
ange
s. T
he a
tom
s in
this
reg
ion
are
the
‘sur
face
ato
ms’
.
Ofte
n th
e su
rfac
e at
oms
have
a s
truc
ture
inte
rmed
iate
bet
wee
n th
e tw
o te
rmin
al p
hase
s (w
ith o
ne n
oted
exc
eptio
n: s
olid
-sol
id s
urfa
ce/in
terf
ace)
Cry
stal
line
So
lidC
hara
cter
ized
by
Long
-ran
ge O
rder
Am
orp
ho
us
So
lidC
hara
cter
ized
by
Sho
rt-r
ange
Ord
er
Liq
uid
Cha
ract
eriz
ed b
y S
hort
-ran
ge O
rder
and
rigor
ous
atom
ic v
ibra
tion
Vap
or
Cha
ract
eriz
ed b
y N
o O
rder
Su
rfac
e T
her
mo
dyn
amic
s--
-T
reat
men
t o
f Q
uan
tita
tive
Su
rfac
e P
aram
eter
s
The
rmod
ynam
ics
is th
e fie
ld o
f sci
ence
whi
ch d
eals
with
the
mot
ion
of a
tom
s un
der
the
influ
ence
of t
herm
al d
rivin
g fo
rces
.
The
rmod
ynam
ic P
oten
tials
(i.e
. Int
erna
l Ene
rgy
(U),
Hel
mho
ltzF
ree
Ene
rgy
(F),
and
Gib
bs F
ree
Ene
rgy
(G))
hav
e ad
ditio
nal c
ontr
ibut
ion
due
to s
urfa
ce a
tom
s.
Con
trib
utio
n du
e to
bul
k at
oms:
dFbu
lk=
-S
dT–
PdV
+µd
N=
0 a
t con
stan
t T, V
, and
N.
Tot
al F
ree
Ene
rgy
cont
ains
add
ition
al c
ontr
ibut
ion
due
surf
ace
atom
s:dF
Tot
al=
dFbu
lk+
f sdA
dFT
otal
= f s
dAat
con
stan
t T, V
, and
N.
dFT
otal
= µ
dN+
f sdA
at c
onst
ant T
and
V.
f sis
the
surf
ace
exce
ss fr
ee e
nerg
y
Su
rfac
e T
her
mo
dyn
amic
s--
-S
urf
ace
Ten
sio
n a
nd
Su
rfac
e E
ner
gy
Su
rfac
e T
ensi
on
, γ,i
s th
e re
vers
ible
wor
k do
ne (
dW)
in c
reat
ing
a un
it ar
ea o
f new
sur
face
(dA
).
γ=
dW
/dA
= (
dFto
tal/d
A) T
,V
γdA
= µ
dN+
f sdA
whe
re µ
is th
e ch
emic
al p
oten
tial o
f the
ato
ms
and
N is
the
num
ber
of a
tom
s in
the
syst
em
Sin
ce
d
FT
otal
= µ
dN+
f sdA
at c
onst
ant T
and
V.
Rew
ritin
g γ
= -
µΓ+
f sat
con
stan
t T a
nd V
.w
here
Γ=
-dN
/dA
(and
dN
alw
ays
nega
tive
(-))
Co
ncl
usi
on
:A
dditi
on o
f ato
ms/
mol
ecul
es to
the
surf
ace
(incr
easi
ng N
---t
hat i
s, N
goe
s to
hi
gher
neg
ativ
e va
lue)
reg
ion
will
dec
reas
e th
e su
rfac
e te
nsio
n (γ
) vi
a in
crea
se in
Γ (Γ
is
prop
ortio
nal t
o –d
N).
Exa
mp
le 1
:A
soa
p fil
m lo
wer
s th
e su
rfac
e te
nsio
n of
wat
er b
ecau
se th
e so
apm
oele
cule
sfo
rm m
onol
ayer
sat
the
wat
er s
urfa
ce w
ith th
eir
‘hyd
roph
obic
’ end
s po
intin
g ou
tint
o th
e ga
seou
s re
gion
s.
Su
rfac
e T
her
mo
dyn
amic
s--
-S
urf
ace
Ten
sio
n a
nd
Su
rfac
e E
ner
gy
Exa
mp
le 2
:S
urf
acta
nt
(pol
ymer
mol
ecul
e w
ith h
ydro
ph
ob
icen
d gr
oup
and
hyd
rop
hili
cen
d gr
oup
is a
dded
to n
anop
artic
ulat
esu
spen
sion
s in
ord
er to
de
crea
se th
e dr
ivin
g fo
rce
for
part
icle
agg
lom
erat
ion.
γ=
µΓ+
f sw
here
Γ=
-dN
/dA
Hav
ing
Hig
h S
urfa
ce
Ene
rgy,
γ, n
anop
artic
les
will
Agg
rega
te to
red
uce
thei
r su
rfac
e ar
ea (
A)
Add
ition
of S
urf
acta
nt
to
nano
part
icul
ate
surf
aces
will
in
crea
se N
, thu
s de
crea
sing
sur
face
en
ergy
, γ.
No
need
for
aggr
egat
ion
to o
ccur
.
02
46
8
2468
0
45
90
135
180
225
270
315
Su
rfac
e T
her
mo
dyn
amic
s--
-W
ulf
fT
heo
rem
an
d S
urf
ace
En
erg
y
Wu
lff
Th
eore
mte
lls u
s th
at th
e eq
uilib
rium
cry
stal
lite
shap
e ha
s su
rfac
e pl
anes
of
min
imum
sur
face
ene
rgy,
γ,
Usi
ngW
ulf
fC
on
stru
ctio
n, w
e ca
n de
term
ine
the
equi
libriu
m s
hape
of c
ryst
allit
es
give
n on
ly γ
(hkl
) (i.
e. s
urfa
ce e
nerg
y fo
r gi
ven
(hkl
) m
iller
indi
ces.
Ste
ps:
1. P
lot p
olar
dia
gram
of γ
(θ).
2. T
ake
the
inne
r en
velo
pe o
f thi
s di
agra
m to
get
equ
ilibr
umsh
ape.
Wul
ffth
eore
m a
pplie
s to
InG
aAs
quan
tum
dot
str
uctu
res
---
whe
re w
e m
ay h
ave
pyra
mid
al s
hape
s gr
own
from
the
vapo
r ph
ase.
Exa
mpl
es
02
46
810
246810
0
45
90
135
180
225
270
315
02
46
810
246810
0
45
90
135
180
225
270
315
Iso
tro
pic
Su
rfac
e E
ner
gy
An
iso
tro
pic
Su
rfac
e E
ner
gy
Ter
race
•S
urfa
ce h
avin
g cr
ysta
lline
ord
er
Led
ge
•S
teps
form
ed a
t th
e bo
rder
of
terr
aces
Kin
k•
Def
ect f
orm
ed a
t th
e en
d of
ledg
es
Ad
ato
m•
Sin
gle
Ato
m
sitti
ng o
n a
terr
ace
or le
dge
surf
ace
Cla
ssif
icat
ion
of
Su
rfac
es b
y th
eir
Def
ects
(o
r Im
per
fect
ion
s)--
-Ter
race
, Led
ges
, Kin
ks a
nd
Ad
ato
ms
Su
rfac
e D
efec
ts (
or
Imp
erfe
ctio
ns)
---T
erra
ce, L
edg
es, K
inks
an
d A
dat
om
s
Ter
race
•T
erra
ce a
tom
has
5ne
ares
t nei
ghbo
rs
Led
ge
•Le
dge
atom
has
4ne
ares
t nei
ghbo
rs
Kin
k•
Kin
k at
om h
as 3
near
erst
neig
hbor
sA
dat
om
•Le
dge
adat
omha
s 2
near
est n
eigh
bor
•T
erra
ce a
dato
mha
s 1
near
est n
eigh
bor
Th
erm
od
ynam
ics
of
Su
rfac
e D
efec
ts
---T
erra
ce, L
edg
es, K
inks
an
d A
dat
om
s
Bin
ding
Ene
rgy
for
Ato
ms
at V
ario
us S
ites
Gib
bs F
ree
Ene
rgy
Equ
atio
nfo
r at
om tr
ansi
tion
from
terr
ace
to
ledg
e po
sitio
n.
∆G=W
ledg
e–
Wte
rrac
e
Wte
rrac
e–
Ene
rgy
requ
ired
to
brea
k 4
bond
s.
Wle
dge
–E
nerg
y re
quire
d to
form
5
bond
s.
Terrace
Site
Sta
bilit
y ha
s di
rect
pro
port
iona
lity
to
bind
ing
ener
gy.
The
hig
her
the
bind
ing
ener
gy, t
he h
ighe
r th
e si
te s
tabi
lity.
Arr
heni
usE
quat
ion
repr
esen
ts th
e T
empe
ratu
re d
epen
denc
e on
# o
f at
oms
unde
rgoi
ng tr
ansi
tion
(n).
n =
Nex
p(-∆
G/k
T)
whe
re N
is th
e #
of a
tom
s av
aila
ble
to p
artic
ipat
e in
the
tran
sitio
n
Sol
id-V
apor
Inte
rfac
e at
Equ
ilibr
ium
.In
terf
acia
l Are
a, A
Inte
rfac
ial T
hick
ness
, δ
Su
rfac
e an
d t
hei
r P
rop
erti
es--
-Pre
ssu
res
and
Fo
rces
Gra
die
nt
Ph
ysic
al P
rop
erti
es(d
ensi
ty, e
tc.)
bet
wee
n so
lid a
nd v
apor
pha
se.
For
ce o
n pl
ane
bδis
F=
Pbδ
-bγ
Whe
re γ
is s
urfa
ce te
nsio
n
surf
ace
ten
sio
n, γ
,is
the
reve
rsib
le w
ork
done
in
crea
ting
a un
it ar
ea o
f new
sur
face
.
Su
rfac
es a
nd
th
eir
Pro
per
ties
---S
urf
ace
Str
ess,
γS
V
Yo
un
g M
od
el (
dev
elo
ped
fo
r L
iqu
id
Su
rfac
es)
Ten
sion
Sur
face
(an
infin
itesi
mal
ly th
in e
last
ic
mem
bran
e) o
ccur
s at
the
inte
rfac
e. T
he s
um
of fo
rces
act
ing
on th
e le
ngth
of t
he in
terf
acia
l cu
rve
are
zero
. T
his
forc
e al
ong
a un
it le
ngth
(d
l) of
the
curv
atur
e su
rfac
e is
kno
wn
as th
e su
rfac
e st
ress
γS
Vre
port
ed in
uni
ts o
f [N
/m].
T
his
surf
ace
stre
ss m
ay b
e re
duce
d b
y in
crea
sing
the
leng
th b
etw
een
bond
s on
the
surf
ace.
γ SVdl
Ten
sion
Sur
face
A
BC
D
Su
rfac
es a
nd
th
eir
Pro
per
ties
---O
ther
Co
nce
pts
Usi
ng
Su
rfac
e S
tres
s, γ
SV
γ SL
γ SV
γ LV
θ
γ SL+
γ LVco
s(θ)
=γS
V
Oth
er C
on
cep
ts u
sin
g S
urf
ace
Str
ess,
γS
V•
Neu
man
n’s
Equ
atio
n-of
-Sta
te(J
. Col
loid
In
terf
ace
Sci
.148
(199
2) 1
90).
We
may
use
the
Neu
man
n’s
empi
rical
equ
atio
n to
det
erm
ine
γ SV.
Thi
s eq
uatio
n is
val
id fo
r S
urfa
ce S
tres
ses
smal
ler
than
72m
J/m
2(o
r 0.
072N
/m).
•F
orce
Bal
ance
Equ
atio
nm
ay b
e us
ed to
de
term
ine
the
inte
rfac
ial s
tres
s be
twee
n th
e dr
ople
t and
the
surf
ace,
γS
L.
We
can
calc
ulat
e th
is u
sing
the
mea
sure
d su
rfac
e st
ress
, γS
V,
surf
ace
tens
ion
betw
een
the
liqui
d an
d va
por,
γLV
, an
d th
e dr
ople
t con
tact
ang
le, θ
, par
amet
ers
in a
fo
rce
bala
nce
equa
tion.
()
[]1
exp
2co
s2
−−
−=
SVL
VL
VSVγ
γβ
γγθ
Su
rfac
e R
eact
ion
s--
-S
ub
limat
ion
S
ub
limat
ion
reac
tions
allo
w a
tom
s to
go
dire
ctly
from
the
solid
to g
as p
hase
.T
he c
hem
ical
pot
entia
ls o
f vap
or p
hase
and
sol
id p
hase
mus
t be
equa
l : C
ondi
tion
for
subl
imat
ion.
At l
ow v
apor
pre
ssur
e,
whe
re
DeB
rogl
ieW
avel
engt
hT
hus,
the
equi
libriu
m p
ress
ure,
pe
is g
iven
by
Now
, w
e ha
ve to
sel
ect a
µs.
Usi
ng a
mod
el w
hich
ass
umes
har
mon
ic v
ibra
tions
of f
requ
ency
,ν, a
nd a
mpl
itude
equ
al to
the
latti
ce
para
met
er o
f the
sol
id, w
e ha
ve th
e fr
ee e
nerg
y of
the
atom
as
We
find
that
at a
bsol
ute
zero
, Lo
is th
e su
blim
atio
n en
ergy
!
At h
igh
tem
pera
ture
s,
and
An
Arr
heni
us-t
ype
equa
tion
Gra
phic
al d
ata
of th
e fo
rmLo
g 10(p
e)=
A-B
/T
(s
ince
the
T-1
/2va
ries
rela
tivel
y sl
owly
, it c
an b
e ig
nore
d fo
r si
mpl
icity
)T
he c
onst
ant L
o is
foun
d w
ithin
B.
The
con
stan
t νis
foun
d w
ithin
the
cons
tant
A.
sv
µµ
=
)3
p/
kTln
(kT
vλ
µ−
=2/
1 )m
kT2
/(h
πλ
=
=kT
sex
p2/
5 )kT(
2/3 )
2h/
m2(
epµ
π
>
−
−<
+>
<+
==
kThex
p1
lnkT3
h23
oU
sN
/F
υυ
µ
><
+−
=υh
23o
UoL
=
−
−kTh
lnkTh
exp
1ln
υυ
−+
=kTh
lnkT3
oLs
υµ
−=
kToL
exp
2/1 )
kT(2/
3 )2
m2(
epυ
π
QM
is o
bey
ed b
y su
blim
ed a
tom
s!!!
−=
−kT
oLex
p2/
3 )3
k
2m
2(
2/1
T epυ
π
Su
rfac
e R
eact
ion
s--
-C
ryst
al G
row
th f
rom
th
e V
apo
r P
has
e
2. A
dsor
ptio
n ra
te, R
+, i
s gi
ven
by:
R+=
p/(
2πm
kT)1
/2
1.D
iffer
ence
bet
wee
n de
posi
tion
from
vap
or a
nd s
ublim
atio
n to
vap
or is
in
the
conc
ept o
f sup
ersa
tura
tion,
S.
S=
p/p e
. T
he c
hang
e in
che
mic
al
pote
ntia
l, ∆µ
, is
give
n by
: ∆µ=
kTln
S.
•P
osi
tive
∆µ
(p>p
e) le
ads
to c
on
den
sati
on
. •
Neg
ativ
e ∆µ
(p<p
e) le
ads
to s
ub
limat
ion
.
4.D
iffus
ion
of a
dato
mac
oss
surf
ace
is g
iven
by
diffu
sivi
ty, D
: D
=(ν d
a2/4
exp
(-E
d/k
T)
3.D
esor
ptio
nra
te, R
-, is
giv
en b
y: R
-=ν a
exp
(-E
a/kT
) ν a
is fr
eque
ncy
with
whi
ch a
tom
s le
ave
the
surf
ace
5. L
ifetim
e be
fore
des
orpt
ion,
τa,
is g
iven
by:
τa
=νa−1
exp
(Ea/
kT)
Aga
in, ν
ais
freq
uenc
y w
ith w
hich
ato
ms
leav
e th
e su
rfac
e
6. C
hara
cter
istic
dis
tanc
e, x
, the
ada
tom
may
diff
use
befo
re
leav
ing
the
surf
ace:
x=(
D τ
a)1
/2
Su
rfac
e R
eact
ion
s--
-C
ryst
al G
row
th f
rom
th
e V
apo
r P
has
e
Phy
sica
l Mea
ning
of x
: x=
(D τ
a)1/
2
Bin
ding
Ene
rgy
for
Ato
ms
at V
ario
us S
ites
Terrace
Sin
ce S
ite S
tabi
lity
is d
irect
ly
prop
ortio
nalit
y to
bin
ding
ene
rgy,
the
adat
omha
s lo
w s
ite s
tabi
lity
and
will
de
sorb
afte
r tim
e τ a
. G
iven
the
surf
ace
diffu
sivi
ty, D
, the
ato
m m
ust f
ind
a m
ore
stab
le s
ite w
ithin
a d
ista
nce,
x, i
n or
der
to
rem
ain
on th
e su
rfac
e an
d le
ad to
gro
wth
fr
om th
e va
por
phas
e.
Su
mm
ary/
Rev
iew
Wh
at a
re s
om
e o
f th
e P
rop
erti
es o
f su
rfac
es?
Th
erm
od
ynam
ic P
rop
erti
es
(Fre
e E
ner
gy,
Wu
lff
con
stru
ctio
n a
llow
s su
rfac
e en
erg
y to
pre
dic
t sh
ape
of
crys
tal)
, Den
sity
, Su
rfac
e T
ensi
on
(o
r S
urf
ace
En
erg
y)
Wh
at is
a S
urf
ace?
Ato
m S
tate
an
d S
tru
ctu
re
Wh
y ar
e su
rfac
e at
om
s im
po
rtan
t? R
eact
ion
s o
ccu
r at
su
rfac
e.
Ad
dit
ion
ally
, nan
ost
ruct
ure
dm
ater
ial h
as in
crea
sed
su
rfac
e to
b
ulk
ato
ms.
Ho
w a
re S
urf
aces
Cla
ssif
ied
? K
inks
, Ter
race
, Led
ges
, Ad
ato
ms
Wh
at a
re t
he
bro
ad c
ateg
ori
es o
f su
rfac
e re
acti
on
s co
nsi
der
ed?
Su
blim
atio
n a
nd
Cry
stal
Gro
wth
Co
nsi
der
th
e (0
01)
face
of
a fc
ccr
ysta
l.
Th
e s
ub
limat
ion
en
erg
y, L
o, i
s 3e
V a
nd
th
e E
inst
ein
fre
qu
ency
fac
tor,
ν,i
s 10
TH
z. U
se t
he
app
rop
riat
e fo
rmu
lati
on
s to
:
(a)E
xpre
ss t
he
loca
l eq
uili
bri
um
bet
wee
n a
dat
om
evap
ora
tio
n, R
-, a
nd
th
e ra
te o
f ar
riva
l, R
+, o
f at
om
s fr
om
th
e va
po
r to
th
e su
rfac
e to
fin
d t
he
con
cen
trat
ion
of
adat
om
mo
no
laye
r (M
L)
un
its.
Exe
rcis
e Q
ues
tio
n 1
(fr
om
Ven
able
ste
xt)
−+
=R
R
mkT
pR
π2=
+
−=
−kTE
Ra
aex
pυ
=m
kT
p π2
− kTE
aaex
pυ
=m
kT
p e π2
− kTE
aaex
pυ
=− m
k
Tp e
π2
2/1
− kTEa
aex
pυ
−=
−
kTL
kmT
po
eex
p)
2 (2/
33
22/
1υ
π
()
=
−
2/1
2/3
3
2 2
exp
)2 (
mk
kTL
kmo
π
υπ
− kTEa
aex
pυ
()
=
− kTL
kmo
exp
23
υπ
− kTEa
aex
pυ
()
=
− kTL
kmo
exp
23
υπ
− kTEa
aex
pυ
Exe
rcis
e Q
ues
tio
n 1
(fr
om
Ven
able
ste
xt)
(b)
Fin
d t
he
adat
om
con
cen
trat
ion
at
1000
K if
R =
1M
L/s
ec
()
=
− kTL
kmo
exp
23
υπ
− kTEa
aex
pυ
Lo
, is
3eV
an
d t
he
Ein
stei
n f
req
uen
cy f
acto
r, ν
, is
10T
Hz
− kTEa
aex
pυ
=1 M
L/s
ec
To
Co
mp
lete
th
e P
rob
lem
, So
lve
for
m (
mas
s o
f ad
ato
ms)
usi
ng
th
e ab
ove
par
amet
ers
in a
pp
rop
riat
e u
nit
s!!!
Exe
rcis
e Q
ues
tio
n 1
(fr
om
Ven
able
ste
xt)
Co
nsi
der
ho
w m
igh
t va
can
cies
(e.
g. e
mp
ty la
ttic
e si
tes)
-w
hic
h d
ecre
ase
the
Ein
stei
n v
ibra
tio
nal
freq
uen
cy o
f n
eig
hb
ori
ng
ato
ms
by
80%
--ef
fect
ad
sorb
ed M
L
con
cen
trat
ion
.
By
the
equ
atio
n,
Dec
reas
ing
th
e vi
bra
tio
nal
freq
uen
cy
will
dec
reas
e th
e su
blim
atio
n r
ate
(Rec
all R
HS
of
the
equ
atio
n is
R-)
, T
hu
s d
ecre
asin
g t
he
sub
limat
ion
rat
e w
ill in
crea
se t
he
adso
rpti
on
co
nce
ntr
atio
n.
()
=
− kTL
kmo
exp
23
υπ
− kTEa
aex
pυ