- Email: [email protected]
Theory for rotation and translation of islands on a substrate
Surface Science fi3 North-lIntland
86 (1979) 28-35 Publishing Company
THEORY FOR ROTATION
AND TRANSLATION
OF ISLANDS ON A
SUBSTRATE
Manuscript
received
in final form
1 November
1978
A theory is developed for rotation and translation of islands depasitcd on a substrate by assuming the harmonic potentials for both an island and the substrate themselves and a Gaussian type potential between them. The probability density that the island takes a set of values for the rotational and translational coordinates on the substrate surface is obtained by calculating the partition function in which integration is performed over all the other variables. The integration over the vibrational coordinates results in a set of effective parameters for the interaction potential. \L hich differ from the original ones by few percent in typical cascy. The integration over the rotational and translational coordinates other than those on the surface results in a prc-euponential factor, which affects the relative probability within one order of magnitude in typical cases.
1. Introduction
There has been experimental evidence [I] that crystallites deposited on substrates rotate and translate as a whole, but the results were not always mutually consistent [?I. MCtois et al. [3] recently reported that gold crystallites on (1 11) MgO surfaces rotate and translate at room temperature after deposition. Their results were obtained under ultra-high vacuum conditions and on freshly electron beam-cleaved surfaces. The electron beam effects were found to be minute on MgO surfaces [‘_I and the electron beam-cleaved ! 111) MgO surfaces are supposedly welldefined. Thus it is considered to be appropriate to carry out a theoretical analysis of the process on the basis of the interaction between the deposited island and the substrate. Reiss [4] performed an analysis on this basis with a schematic model and some of the results were qualitatively verified by experiments [5]. The present work intends to formulate a theory in a more rigorous manner by explicitly taking into account the motions of all the atoms, which enables one to calculate the activation entropy as well as the activation energy of the process.
29
2. Description
of rotation
and translation
for an island
Define the lattices separately for an island and for a substrate and assume that their interaction does not modify their respective lattices. To describe rotation and translation of an island a rotating and translating frame Z, which is fixed to the lattice {u& of the island, is introduced 161. Locate the origin of C at the center of the lattice and let it coincide with the center of mass of the island. We orient the axes X, y, z of C along the principal axes of the “moment of inertia” for the lattice {at}, and orient Z relative to the island such that
Z i
(aiyzj - aj,yi) = 0 ,
(1,
etc. ,
are satisfied, where zi denotes the z-component of the displacement ri from the lattice point ai of the ith atom, and etc. means the equations obtainable by permutation of the components, Apparently, we have
Zxi =0,
etc.
(2)
,
i
and the following six other equations:
22(iZj_vij-- @j_,jj)= 0 ,
etc.
(41
i
Eqs. (3) and (4) define the linear velocity k and the angular velocity CO,respectively, of X relative to the substrate lattice frame and they imply that there is no linear and approximately no angular momenta of the island relative to X. We define the angular orientation of an island by that of 2(x, ~1, z) relative to the substrate reference frame C&, n, {). To represent the orientation, let us employ Eulerian type angles 0, @and $Awhich make the momentum integral separable from the configurational integral. Supposing that the origin is common for the two frames C and X,, those angles may be chosen in the following three steps: (i) Rotate C about the z-axis by 0 so that the n-axis is in the yz-plane; -rr G 0 G 71. (ii) Rotate C about the x-axis by $ so that the y-axis coincides with the n-axis; -lr<#cp7r. (iii) Rotate IX about the _ri-axis by j/ so that the z- and x-axes coincide with the <- and g-axes, respectively; -rr < $ < 71. Take the positive sign for the angles when the required rotation is in a clo_ckwise sense about the direction of the axis of rotation. Define the rotation matrix R by
ii =
cos 0 cos I$ - sin 0 sin Q sin +
sin 0 cm I) + cm 0 sin Q, sin $I
--cos @ sin J,
-sin
cos 0 cos I$
sin 6
sin 0 sin I) - cos 0 sin Q cos j,
CDSCpcos \L i
B cos d,
cm 13sin + + sin 0 sin cp cos $
1.
(3
then a vector v(t, 77,f) from the origin before rotation becomes c\;r after a rotation given by (0, 4. $). where both r and /?r are represented with respect to &. The displacement LV due to a rotation is given by _Ir = (R ~~~ 7) T .
((,I
where / denotes the unit matrix. Translation of an island is defined by the displacement K of the origin of Z, which coincides with the center of mass of the island. relative to X,, and its components with respect to X:, are denoted as .Y, 1’ md %.
3. Partition
function
for the system of an island and the substrate
The kinetic energy T of an island is given by
where III denotes the mass of an atom, II the number of atoms in the island and I.\-, etc. the principal moments of inertia. The rotational--vibrational coupling term is neglected in eq. (7) and the products of inertia do not appear umler this appl-oximation owing to our choice of the axes. The prescrlt choice of the angles 0. $ and j/ results in the rela~i~~nsllip w, = 0 .
WI. = j, .
I lence the rotational /I i-c,!---ppr,y,y
w,-ti. Hmiltonian
(8) /_lr’jt of an island is given 1))
f p$/2ry,, + p;llf_-_- .
where po, 12~ and p$ are momenta conjugate lational Hamiltonian Hf’ of the island is
(‘)I to 0, q!~and $. respectively.
The trans-
where /7x, />y and pz denote the ~iol~l~llta conjugate to S. Y and %. Assutlii~lg 111~ harmonic potentials for both the island and the substrate themselves, the vibrational Hamiltonians HGh and HP” for the island and the substrate are given as usual in terms of the normal coordinates and their conjugate momenta. Consider next the interaction between the island and the substrate. To reduce mathematicai difficufties we employ a Gaussian type potential 171 lt(rij)
=
A cxp(-ar$)
-- R L’Xp( --$r$) ,
(Ii)
where rij denotes the distance between the ith atom of the island and the jth atom of the substrate. Denoting by ui and uSi the lattice vectors, measured from the origin, of the island when \J coincides with XS and of the substrate, respectively,
K. Nishioka
/ Theory for rotatim
and by ri and rsi the displacements rfj = (ai
U,j
+
gj)2
+
(ri
~
r,j)*
and tratnlatiort
31
of islatzds
from the lattice points, rfj is given by
+
2(Ui
where gi represents the displacement
-
Usj
+
gi)
(ri
-
rc_j)
(I’)
.
due to rotation and translation
and is given bq (13)
gi = R + @ -- 7) ai Since 4 and I) are considered
to be small, l? ~ ? may be approximated
Ccos 0 ~ Q$ sin 0 -1 R ~ ? = -sin 0
sin 0 + @J cos (1
-&
cos 0 m-1
Q 1
$ sin 0 - 0 cos 0
0
J, cos 1 + 4 sin 0
as
(14)
1
The {-axis is taken to be normal to the substrate surface. The interaction Uint between the island and the substrate is
potential
where the summation over i and j spans all the atoms in the island and the substrate atoms within a certain distance from the ith atom, respectively. The Hamiltonian H for the entire system including both the island and the substrate is given by 1, _ H” + H”‘t + flih
+ HVih s +u
i”t + 0”
,
(16)
where u” denotes the potential energy for the island and the substrate, excluding the interaction between them, with all the atoms at their lattice sites. When we neglect the normal coordinate dependence of the moments of inrrtia, the momentum integral‘ is separable from the configurational integral and the former may be obtained easily. To proceed further with the configurational integral, we employ the following approximation:
u inr
u
A ~~exp[-a(ui
I
+ A
i
ij
CCCxp[-*(ai- ~,i+gi)*] (-2N(ai
+ 2(u*[(Ui- a,j + gt) (ri - r,j)l
B
B ccCXP[ i j
- a,i +gi)‘]
j
j
P(ri - rsi)* +
20’ [(Ui -
Usi
+
Usj
+
gJ . (ri - rsj) -. cu(ri -
rs;)2
‘I
cC-pIkP(ai- as;+gi)‘] {---ZP(Uji
-
--P(ai - asj + gi121
gi)
(ri -
r,j)]
osj +
“}
gi)
(ri _
rsj)
(17)
This approximation may be justified by noting that CI(U~-a,i +gi)* = O( 1) and 0 < Q [7]. The first term in eq. (17) represents the interaction potential with all the
K. Nishioka / Thcor)’ for rotatiorl amI translatiorl of islands
32
atoms at the lattice sites and the second term the part which has to do with vibration of atoms. The magnitude of the second term is considered to be smaller than kT owing to the canonical weight factors with the harmonic potentials for vibration. Hence, we may employ the perturbation method [8] by regarding the second term as a perturbation. In proceeding with the first order perturbation calculation, let US approximate the vibrational Hamiltonians with those of the Einstein model. Then, the result is
(AU’“‘),
... s.I = _~~ ss
(AUint),,vi,
exp(-UF*/kT)
dX ... dii, (181
...
exp(-lJFt,“tlkT)
dX ,.. d$
where Upt and AU’“‘denote the first and second terms of eq. (17). In eq. (181, (AUFt)o,ti, represents the result of integration with respect to the vibrational coordinates and is given by (AU’“‘)e,vtb = A CCexp[-cy(ai i
X
~ uSi + gi)‘]
j
3Cu2kT(ai - asi t gi)
B
FFexp[-@ai~asi t gi)‘]
2f12kT(ai - asi + gi)’
(19) where K and K, denote the spring constants for the island and for the substrate, respectively. From the perturbation theory the configurational integral Z is related to the unperturbed function ZO and (AU’“’ jO by Z 21 ZO exp(-(AUi”‘)O/k7’)
(20)
,
which may be further approximated Z =ZO(l
as
- (AUint),JkT)
~k-3(ntN%$b
exp(mmUo/kT)J...
[exp(-U$/kT)
..
dX ... d$ ,
where N denotes the number of atoms of the substrate, Zg” the vibrational and lJ$ is given by
u$; =
uptt (AUi”t)O,vib = ~~I(,? , i
j
(21) integral
(32)
K. Nishioka / Theory for rotation and translation of islands
33
with uy
= A,rr exp [-+r&
- +,i + g#l
A,tr=~exp[-3ork~(~t~)],
aeff = cx- 2a=kT (;- f -:,)
- &ff exp[-Peff(ai- a,i +&I , ,
B,,,=Bexp[-3pkT(;++)]
3
Peff = P - @kT(;
+ ;)
.
(23) (24)
(25)
The result shows that the lattice vibration simply replaces the parameters in the interaction potential by their effective values. Note that the atoms may be regarded as fixed at the lattice sites and the effect of vibration is contained in the effective potential uhf. High temperature and small values of the spring constants result in weakening of the potential and widening of the distance to the minimum. Reduction of the spring constants in one order of magnitude results in about 10% reduction in the values of the parameters. A change in (Yand /I affects the degree of disregistry. To obtain the probability density that an island takes a set of values for X, Y and 19,we must carry out the configurational integral (21) over Z, @ and $. Since only small values in the magnitude of Z, $ and +!Jcontribute to the integral, let us expand Us: into a power series about Z = @= ri/ = 0 and neglect the terms higher than the second order. When the island possesses rotational symmetry, the following equations hold exactly for a set of values in (X, Y, .Q):
(au$;ja4), = (au.J~;/ati)O = (aYr&az a@), = (a”u$fa@a+jO= (aW$,‘lazaGjo= 0 Let us assume that eq. (26) approximately then it follows that U$
= Do + D,Z + D2Z2 + D,$’ f D,$’
(26)
holds for the other values in (X, Y, e), ,
(27)
where
DC,= Am
CCexpt-aeffw$)) - Beff CCexp(-fl,ffw$!)) i j i j
,
(33)
D, =:A,ff CCexp(-ru,ffw!??(-ol,ffw~~‘, r1 i j (29)
(30)
(31)
(32)
Het-e. w!/)ctc.
t ,v/,?z+ Employing
are defined by
t ,,~j,!?Z2 t ,$?)#
t ,t$“‘lJ/ 2 + ,,_
eq. (17) in the configurational
(33)
integral over Z, ci,and I$, it follows that
which is proportional to the probability density that au island takes a set of values for (X, Y. 6’). When we consider the translation and rotation described by (.I’. Y. 0 ), D, DT/4D2 plays the role of the effective potential. The pre-exponential fActor (D2D3D4)-1”2 affects the relative probability within one order of magnitude in typical cases. Note that D, etc. are all functions ofX. Y and 0 only, and the values of Z, o and $ are regarded as being zero in usin g the result. The activation entropy’ as well as the activation energy for the rotation and the translation can be calculated from the result of eq. (34). Application of the present theory to the kinetics in some systems will be published elsewhere [Oj.
References [ I] GA. Bassett, in: l’rtrc. I:uropean Keg. (‘onf. on I.‘lectron hliuoscopy, Vol. 1. I~ds. A.L. Flwvink and B.J. Spit (Vercniging v~,ur Hcctruncn Micr~~scopic. Dclt’t, 1961) p. 27tl; G. Honjo and K. Magi, J. Vacuum Sci. ‘l’e~l~n~~l.6 ( 1969) 576; A. Masson, J.J. RlGtois and K. Kern. Surt’acc Sci. 77 (1971) 463; H. Poppa, in: Epitasial Growth. Pt. A O\cadcmic Press, New York. 1975) p. 274; K. llcinemunn, J.J. MCtois and II. Poppa. in: hoc. 34th FMSA Meeting (Chitor. Baton Kou~e. 1976) p. 434; J.J. Mhois. K. Ilcincmmn and II. Poppa. Appl. Phyc. Letters 29 (1976) 134. 121 G. Hotlj~. K. Ta!iayan;igi. K. Kohayn~lli and K. Yapi. in: I’roc. 6th Intt~rn. V;icuu~n (‘~~npr.. I-ds. II. Kumqzlrcli and 7. Toya (I’uhl. Office ol.Jap:m.
J. .4ppl. I’hys.. Tokyo,
1974) p. 537.
K. Nishioka (31 [4] [5] [6] [7] [8] [9]
/ Theory for rotatiorl
arrd translation
35
of islands
J.J. Metois, K. Heinemann and H. Poppa, Phil. Mag. 35 (1977) 1413. H. Reiss, J. Appl. Phys. 39 (1968) 5045. L.T. Chadderton and M.G. Anderson, Thin Solid 1,ilms 1 (1968) 229. K. Nishioka and G.M. Pound, Acta Met. 22 (1974) 1015. T. Matsubara, Y. lwase and A. Momokita, Progr. Theor. Phys. 58 (1977) 1102. L.D. Landau and E.M. Lifshitz, Statistical Physics, 2nd ed. (Pergamon, New York, K. Nishioka and Y. Arimitsu, to be published.
1969).
86 (1979) 28-35 Publishing Company
THEORY FOR ROTATION
AND TRANSLATION
OF ISLANDS ON A
SUBSTRATE
Manuscript
received
in final form
1 November
1978
A theory is developed for rotation and translation of islands depasitcd on a substrate by assuming the harmonic potentials for both an island and the substrate themselves and a Gaussian type potential between them. The probability density that the island takes a set of values for the rotational and translational coordinates on the substrate surface is obtained by calculating the partition function in which integration is performed over all the other variables. The integration over the vibrational coordinates results in a set of effective parameters for the interaction potential. \L hich differ from the original ones by few percent in typical cascy. The integration over the rotational and translational coordinates other than those on the surface results in a prc-euponential factor, which affects the relative probability within one order of magnitude in typical cases.
1. Introduction
There has been experimental evidence [I] that crystallites deposited on substrates rotate and translate as a whole, but the results were not always mutually consistent [?I. MCtois et al. [3] recently reported that gold crystallites on (1 11) MgO surfaces rotate and translate at room temperature after deposition. Their results were obtained under ultra-high vacuum conditions and on freshly electron beam-cleaved surfaces. The electron beam effects were found to be minute on MgO surfaces [‘_I and the electron beam-cleaved ! 111) MgO surfaces are supposedly welldefined. Thus it is considered to be appropriate to carry out a theoretical analysis of the process on the basis of the interaction between the deposited island and the substrate. Reiss [4] performed an analysis on this basis with a schematic model and some of the results were qualitatively verified by experiments [5]. The present work intends to formulate a theory in a more rigorous manner by explicitly taking into account the motions of all the atoms, which enables one to calculate the activation entropy as well as the activation energy of the process.
29
2. Description
of rotation
and translation
for an island
Define the lattices separately for an island and for a substrate and assume that their interaction does not modify their respective lattices. To describe rotation and translation of an island a rotating and translating frame Z, which is fixed to the lattice {u& of the island, is introduced 161. Locate the origin of C at the center of the lattice and let it coincide with the center of mass of the island. We orient the axes X, y, z of C along the principal axes of the “moment of inertia” for the lattice {at}, and orient Z relative to the island such that
Z i
(aiyzj - aj,yi) = 0 ,
(1,
etc. ,
are satisfied, where zi denotes the z-component of the displacement ri from the lattice point ai of the ith atom, and etc. means the equations obtainable by permutation of the components, Apparently, we have
Zxi =0,
etc.
(2)
,
i
and the following six other equations:
22(iZj_vij-- @j_,jj)= 0 ,
etc.
(41
i
Eqs. (3) and (4) define the linear velocity k and the angular velocity CO,respectively, of X relative to the substrate lattice frame and they imply that there is no linear and approximately no angular momenta of the island relative to X. We define the angular orientation of an island by that of 2(x, ~1, z) relative to the substrate reference frame C&, n, {). To represent the orientation, let us employ Eulerian type angles 0, @and $Awhich make the momentum integral separable from the configurational integral. Supposing that the origin is common for the two frames C and X,, those angles may be chosen in the following three steps: (i) Rotate C about the z-axis by 0 so that the n-axis is in the yz-plane; -rr G 0 G 71. (ii) Rotate C about the x-axis by $ so that the y-axis coincides with the n-axis; -lr<#cp7r. (iii) Rotate IX about the _ri-axis by j/ so that the z- and x-axes coincide with the <- and g-axes, respectively; -rr < $ < 71. Take the positive sign for the angles when the required rotation is in a clo_ckwise sense about the direction of the axis of rotation. Define the rotation matrix R by
ii =
cos 0 cos I$ - sin 0 sin Q sin +
sin 0 cm I) + cm 0 sin Q, sin $I
--cos @ sin J,
-sin
cos 0 cos I$
sin 6
sin 0 sin I) - cos 0 sin Q cos j,
CDSCpcos \L i
B cos d,
cm 13sin + + sin 0 sin cp cos $
1.
(3
then a vector v(t, 77,f) from the origin before rotation becomes c\;r after a rotation given by (0, 4. $). where both r and /?r are represented with respect to &. The displacement LV due to a rotation is given by _Ir = (R ~~~ 7) T .
((,I
where / denotes the unit matrix. Translation of an island is defined by the displacement K of the origin of Z, which coincides with the center of mass of the island. relative to X,, and its components with respect to X:, are denoted as .Y, 1’ md %.
3. Partition
function
for the system of an island and the substrate
The kinetic energy T of an island is given by
where III denotes the mass of an atom, II the number of atoms in the island and I.\-, etc. the principal moments of inertia. The rotational--vibrational coupling term is neglected in eq. (7) and the products of inertia do not appear umler this appl-oximation owing to our choice of the axes. The prescrlt choice of the angles 0. $ and j/ results in the rela~i~~nsllip w, = 0 .
WI. = j, .
I lence the rotational /I i-c,!---ppr,y,y
w,-ti. Hmiltonian
(8) /_lr’jt of an island is given 1))
f p$/2ry,, + p;llf_-_- .
where po, 12~ and p$ are momenta conjugate lational Hamiltonian Hf’ of the island is
(‘)I to 0, q!~and $. respectively.
The trans-
where /7x, />y and pz denote the ~iol~l~llta conjugate to S. Y and %. Assutlii~lg 111~ harmonic potentials for both the island and the substrate themselves, the vibrational Hamiltonians HGh and HP” for the island and the substrate are given as usual in terms of the normal coordinates and their conjugate momenta. Consider next the interaction between the island and the substrate. To reduce mathematicai difficufties we employ a Gaussian type potential 171 lt(rij)
=
A cxp(-ar$)
-- R L’Xp( --$r$) ,
(Ii)
where rij denotes the distance between the ith atom of the island and the jth atom of the substrate. Denoting by ui and uSi the lattice vectors, measured from the origin, of the island when \J coincides with XS and of the substrate, respectively,
K. Nishioka
/ Theory for rotatim
and by ri and rsi the displacements rfj = (ai
U,j
+
gj)2
+
(ri
~
r,j)*
and tratnlatiort
31
of islatzds
from the lattice points, rfj is given by
+
2(Ui
where gi represents the displacement
-
Usj
+
gi)
(ri
-
rc_j)
(I’)
.
due to rotation and translation
and is given bq (13)
gi = R + @ -- 7) ai Since 4 and I) are considered
to be small, l? ~ ? may be approximated
Ccos 0 ~ Q$ sin 0 -1 R ~ ? = -sin 0
sin 0 + @J cos (1
-&
cos 0 m-1
Q 1
$ sin 0 - 0 cos 0
0
J, cos 1 + 4 sin 0
as
(14)
1
The {-axis is taken to be normal to the substrate surface. The interaction Uint between the island and the substrate is
potential
where the summation over i and j spans all the atoms in the island and the substrate atoms within a certain distance from the ith atom, respectively. The Hamiltonian H for the entire system including both the island and the substrate is given by 1, _ H” + H”‘t + flih
+ HVih s +u
i”t + 0”
,
(16)
where u” denotes the potential energy for the island and the substrate, excluding the interaction between them, with all the atoms at their lattice sites. When we neglect the normal coordinate dependence of the moments of inrrtia, the momentum integral‘ is separable from the configurational integral and the former may be obtained easily. To proceed further with the configurational integral, we employ the following approximation:
u inr
u
A ~~exp[-a(ui
I
+ A
i
ij
CCCxp[-*(ai- ~,i+gi)*] (-2N(ai
+ 2(u*[(Ui- a,j + gt) (ri - r,j)l
B
B ccCXP[ i j
- a,i +gi)‘]
j
j
P(ri - rsi)* +
20’ [(Ui -
Usi
+
Usj
+
gJ . (ri - rsj) -. cu(ri -
rs;)2
‘I
cC-pIkP(ai- as;+gi)‘] {---ZP(Uji
-
--P(ai - asj + gi121
gi)
(ri -
r,j)]
osj +
“}
gi)
(ri _
rsj)
(17)
This approximation may be justified by noting that CI(U~-a,i +gi)* = O( 1) and 0 < Q [7]. The first term in eq. (17) represents the interaction potential with all the
K. Nishioka / Thcor)’ for rotatiorl amI translatiorl of islands
32
atoms at the lattice sites and the second term the part which has to do with vibration of atoms. The magnitude of the second term is considered to be smaller than kT owing to the canonical weight factors with the harmonic potentials for vibration. Hence, we may employ the perturbation method [8] by regarding the second term as a perturbation. In proceeding with the first order perturbation calculation, let US approximate the vibrational Hamiltonians with those of the Einstein model. Then, the result is
(AU’“‘),
... s.I = _~~ ss
(AUint),,vi,
exp(-UF*/kT)
dX ... dii, (181
...
exp(-lJFt,“tlkT)
dX ,.. d$
where Upt and AU’“‘denote the first and second terms of eq. (17). In eq. (181, (AUFt)o,ti, represents the result of integration with respect to the vibrational coordinates and is given by (AU’“‘)e,vtb = A CCexp[-cy(ai i
X
~ uSi + gi)‘]
j
3Cu2kT(ai - asi t gi)
B
FFexp[-@ai~asi t gi)‘]
2f12kT(ai - asi + gi)’
(19) where K and K, denote the spring constants for the island and for the substrate, respectively. From the perturbation theory the configurational integral Z is related to the unperturbed function ZO and (AU’“’ jO by Z 21 ZO exp(-(AUi”‘)O/k7’)
(20)
,
which may be further approximated Z =ZO(l
as
- (AUint),JkT)
~k-3(ntN%$b
exp(mmUo/kT)J...
[exp(-U$/kT)
..
dX ... d$ ,
where N denotes the number of atoms of the substrate, Zg” the vibrational and lJ$ is given by
u$; =
uptt (AUi”t)O,vib = ~~I(,? , i
j
(21) integral
(32)
K. Nishioka / Theory for rotation and translation of islands
33
with uy
= A,rr exp [-+r&
- +,i + g#l
A,tr=~exp[-3ork~(~t~)],
aeff = cx- 2a=kT (;- f -:,)
- &ff exp[-Peff(ai- a,i +&I , ,
B,,,=Bexp[-3pkT(;++)]
3
Peff = P - @kT(;
+ ;)
.
(23) (24)
(25)
The result shows that the lattice vibration simply replaces the parameters in the interaction potential by their effective values. Note that the atoms may be regarded as fixed at the lattice sites and the effect of vibration is contained in the effective potential uhf. High temperature and small values of the spring constants result in weakening of the potential and widening of the distance to the minimum. Reduction of the spring constants in one order of magnitude results in about 10% reduction in the values of the parameters. A change in (Yand /I affects the degree of disregistry. To obtain the probability density that an island takes a set of values for X, Y and 19,we must carry out the configurational integral (21) over Z, @ and $. Since only small values in the magnitude of Z, $ and +!Jcontribute to the integral, let us expand Us: into a power series about Z = @= ri/ = 0 and neglect the terms higher than the second order. When the island possesses rotational symmetry, the following equations hold exactly for a set of values in (X, Y, .Q):
(au$;ja4), = (au.J~;/ati)O = (aYr&az a@), = (a”u$fa@a+jO= (aW$,‘lazaGjo= 0 Let us assume that eq. (26) approximately then it follows that U$
= Do + D,Z + D2Z2 + D,$’ f D,$’
(26)
holds for the other values in (X, Y, e), ,
(27)
where
DC,= Am
CCexpt-aeffw$)) - Beff CCexp(-fl,ffw$!)) i j i j
,
(33)
D, =:A,ff CCexp(-ru,ffw!??(-ol,ffw~~‘, r1 i j (29)
(30)
(31)
(32)
Het-e. w!/)ctc.
t ,v/,?z+ Employing
are defined by
t ,,~j,!?Z2 t ,$?)#
t ,t$“‘lJ/ 2 + ,,_
eq. (17) in the configurational
(33)
integral over Z, ci,and I$, it follows that
which is proportional to the probability density that au island takes a set of values for (X, Y. 6’). When we consider the translation and rotation described by (.I’. Y. 0 ), D, DT/4D2 plays the role of the effective potential. The pre-exponential fActor (D2D3D4)-1”2 affects the relative probability within one order of magnitude in typical cases. Note that D, etc. are all functions ofX. Y and 0 only, and the values of Z, o and $ are regarded as being zero in usin g the result. The activation entropy’ as well as the activation energy for the rotation and the translation can be calculated from the result of eq. (34). Application of the present theory to the kinetics in some systems will be published elsewhere [Oj.
References [ I] GA. Bassett, in: l’rtrc. I:uropean Keg. (‘onf. on I.‘lectron hliuoscopy, Vol. 1. I~ds. A.L. Flwvink and B.J. Spit (Vercniging v~,ur Hcctruncn Micr~~scopic. Dclt’t, 1961) p. 27tl; G. Honjo and K. Magi, J. Vacuum Sci. ‘l’e~l~n~~l.6 ( 1969) 576; A. Masson, J.J. RlGtois and K. Kern. Surt’acc Sci. 77 (1971) 463; H. Poppa, in: Epitasial Growth. Pt. A O\cadcmic Press, New York. 1975) p. 274; K. llcinemunn, J.J. MCtois and II. Poppa. in: hoc. 34th FMSA Meeting (Chitor. Baton Kou~e. 1976) p. 434; J.J. Mhois. K. Ilcincmmn and II. Poppa. Appl. Phyc. Letters 29 (1976) 134. 121 G. Hotlj~. K. Ta!iayan;igi. K. Kohayn~lli and K. Yapi. in: I’roc. 6th Intt~rn. V;icuu~n (‘~~npr.. I-ds. II. Kumqzlrcli and 7. Toya (I’uhl. Office ol.Jap:m.
J. .4ppl. I’hys.. Tokyo,
1974) p. 537.
K. Nishioka (31 [4] [5] [6] [7] [8] [9]
/ Theory for rotatiorl
arrd translation
35
of islands
J.J. Metois, K. Heinemann and H. Poppa, Phil. Mag. 35 (1977) 1413. H. Reiss, J. Appl. Phys. 39 (1968) 5045. L.T. Chadderton and M.G. Anderson, Thin Solid 1,ilms 1 (1968) 229. K. Nishioka and G.M. Pound, Acta Met. 22 (1974) 1015. T. Matsubara, Y. lwase and A. Momokita, Progr. Theor. Phys. 58 (1977) 1102. L.D. Landau and E.M. Lifshitz, Statistical Physics, 2nd ed. (Pergamon, New York, K. Nishioka and Y. Arimitsu, to be published.
1969).